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[1312.4068] Marcel Grossmann and his contribution to the final concept of relativity

[1312.4068] Marcel Grossmann and his contribution to the final concept of relativity

2024-03-13 11:05:29


Tilman Sauer


Institute for Theoretical Physics


Albert Einstein Heart for Elementary Physics


College of Bern, CH-3012 Bern, Switzerland


and


Einstein Papers Undertaking, California Institute of Expertise MC20-7


1200 E California Blvd, Pasadena, CA 91125, USA


E-mail: tilman@caltech.edu

1 Introduction

The historical past of normal relativity is a topic which has been written about extensively.1 Nonetheless, most historic accounts of the emergence of the idea give attention to Einstein’s function in it, or at the very least they inform the story from a viewpoint that’s largely outlined by our view of Einstein’s achievements. Indicative of this truth is the remedy of Marcel Grossmann’s contribution to the emergence of normal relativity within the secondary literature. Neither does a biographical account of his life and work exist nor has any try been made to research in some element his specific contribution to the genesis of the idea, even supposing a giant and profitable worldwide convention sequence is called after him. On this article, an try is made to fill these gaps to some extent. Whereas a full-fledged biography is past the scope of the current article, an try is made to sketch at the very least Grossmann’s biography and mental achievements by accounting for his revealed work. I additionally evaluation the Einstein-Grossmann concept, which has been analyzed repeatedly within the literature and in nice element,2, 3, 4, 5, 6, 7, 8, 9 however it is going to be accomplished right here particularly from Grossmann’s perspective. The account attracts primarily on revealed sources in addition to on paperwork within the Albert Einstein Archives.

2 Marcel Grossmann (1878–1936)

Marcel Grossmann was born on 9 April 1878 in Budapest, Hungary.10, 11, 2, 12
The son of a big machine-shop proprietor, he was the descendant of an outdated Swiss household, originating from Höngg, close to Zurich. He spent his childhood and attended major college in Budapest. On the age of 15 he returned
together with his mother and father to Switzerland and attended secondary college in Basel. Apparently, he was a mannequin pupil. A report card from April 1894 gave him the perfect grade (1) for effort in all topics, and the perfect grade for achievement in virtually all topics, aside from technical drawing (2), freehand drawing (2), and bodily train (3).

Refer to caption
Determine 1: Marcel Grossmann (1878–1936). ©ETH Bibliothek Zürich, Bildarchiv.

After graduating from center college, he entered the Swiss Polytechnic Faculty in Zurich, now the Swiss Federal Institute of Expertise (known as ETH within the following),
in its division VI, the Faculty for
Arithmetic and Science Academics, in October 1896. The division was divided into two sections. Grossmann enrolled in part VI A, which was headed by Adolf Hurwitz (1859–1919) and specialised in arithmetic, physics, and astronomy. The ETH counted a complete of 841 college students in 1896, however solely 11 college students enrolled in part VI A for the winter semester 1896/97. As is well-known, amongst Grossmann’s friends of part VI A have been Albert Einstein (1879–1955) and Mileva Marić (1875–1948), the one lady in that class.
Two different college students of Grossmann’s entry class, Jakob Ehrat (1876–1960) and Louis Kollros (1878–1959), remained in Grossmann’s cohort till the ultimate examinations in 1900. Not like in its 5 engineering college departments, in division VI there was no formal curriculum. Right here the course of lectures to be taken by the scholars was decided on a kind of particular person foundation every year by the pinnacle of the division. Nonetheless, comparability of the scholars’ transcripts reveals that required courses have been very a lot the identical for all college students in part VI A within the first two years. They included programs on calculus, analytic geometry, descriptive geometry, mechanics, projective geometry, and determinants within the first 12 months; within the second 12 months Grossmann and his fellow college students took courses on differential equations, infinitesimal geometry, projective geometry, quantity concept, geometry of numbers, mechanics, physics, concept of scientific reasoning, Kant’s critique of pure cause, geometric concept of invariants, advanced evaluation, potential concept, concept of particular integrals, and an introduction to celestial mechanics. As well as, the
college students have been free to take a selection of non-obligatory programs. The obligatory a part of their schedule comprised some twenty hours per week. Not surprisingly, spending the higher a part of the week with a handful of friends in the identical lecture room for 2 years, friendships have been certain to construct up. As is well-known, one such friendship quickly developed between Grossmann and Einstein. They might hang around after college in Zurich’s Café Metropol and speak about “all the pieces that may very well be of curiosity to younger people with open eyes” 13 147.

Within the third and fourth 12 months, the curriculum of the scholars in part VI A diversified considerably. Einstein and Marić attended lectures in physics and spent extra time within the laboratory, Grossmann, Ehrat, and Kollros continued their mathematical research. Grossmann was a conscientious and industrious pupil. He took notes throughout the lectures and labored them out in certain booklets, fastidiously and
meticulously, in a clear and neat handwriting. These notebooks, invaluable witnesses of the mathematical data and coaching of the day, have been later given to the ETH archives and are accessible to the general public. Einstein later recalled that Grossmann would lend him his Ausarbeitungen in order that he might use them to arrange for his examinations.

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Determine 2: A web page from Grossmann’s notes of Carl Friedrich Geiser’s lectures on infinitesimal geometry, given in the summertime time period 1898 on the Swiss Polytechnic. This web page reveals a pencil remark most likely made by Einstein. ETH archives, Hs 421:16, p. 105. Grossmann’s lecture notes can be found on-line on the ETH’s e-manuscripta platform.

On 27 July 1900, remaining oral examinations passed off for the part VI A category of 1896. The mathematicians Grossmann, Ehrat, and Kollros have been examined in advanced evaluation, geometry, arithmetic and algebra, theoretical physics, and in astronomy. The physicists Einstein and Marić have been examined in theoretical physics, experimental physics, advanced evaluation, and in astronomy. For the mathematicians, grades within the
mathematical topics have been doubled. As well as, the grade for a diploma thesis was quadrupled for analysis of the ultimate grade. Grossmann, who had written a diploma thesis on non-Euclidean geometry with Wilhelm Fiedler, scored a mean of 5.23, second in his class after Louis Kollros who scored 5.45. Ehrat scored a mean of 5.14, Einstein 4.91, solely Marić failed the examination with a rating of 4.0.

After acquiring his diploma, Grossmann obtained a place as Assistent to Otto Wilhelm Fiedler (1832–1912), full professor for descriptive geometry and projective geometry on the ETH since 1867. This was a typical profession step in an educational vita. It allowed Grossmann to acquire his Ph.D., supervised by Fiedler, already in 1902 with a thesis “On the Metric Properties of Collinear Buildings”15. The subject of the thesis was from the sphere of projective geometry, a subject of experience of Fiedler’s, and the intention of the thesis was to provide an in depth dialogue of the focal and metric properties of collinear planes and bundles with a particular emphasis on the facet of projective duality. It additionally prolonged the idea of the attribute of airplane centric collineations to arbitrary collineations and, given two collinear areas, to place this attribute into relation to the coordinates of the collinear planes and bundles. Outcomes from his thesis have been revealed in a short paper, which, nevertheless, appeared solely in 1905.16

On 31 August 1901, Grossmann was appointed to a place as an teacher at Thurgau Kantonsschule in Frauenfeld
12 316, observe 2. Throughout his tenure in Frauenfeld, he revealed an in depth account of what he referred to as Elementary Constructions of non-Euclidean Geometry17. An extract from that work appeared the identical 12 months additionally in Mathematische Annalen.18 With this work, which allegedly evoked reward by David Hilbert (1862–1943) 10 323, Grossmann established himself as an knowledgeable in non-Euclidean and projective geometry. The purpose of those investigations was to indicate that and the way one can carry out the elementary constructions of Euclidean geometry additionally for the case of non-Euclidean geometries, each hyperbolic and elliptic. Grossmann primarily based his constructions on the ideas of Cayley-Klein geometry. On this framework,19 the set of improper, infinitely distant factors of, say, two-dimensional hyperbolic geometry, is represented by an actual, non-degenerate conic part ΩΩOmegaroman_Ω within the (Euclidean) airplane, the so-called absolute conic part. Given such an absolute conic part, the hyperbolic airplane is then shaped by all factors inside ΩΩOmegaroman_Ω. If A𝐴Aitalic_A and B𝐵Bitalic_B are two such factors, and U1subscript𝑈1U_{1}italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and U2subscript𝑈2U_{2}italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT the factors of intersection between ΩΩOmegaroman_Ω and the straight line connecting A𝐴Aitalic_A and B𝐵Bitalic_B, then the gap r𝑟ritalic_r between A𝐴Aitalic_A and B𝐵Bitalic_B is given by the logarithm of the cross ratio

r=okayln(U1,U2,A,B),𝑟𝑘subscript𝑈1subscript𝑈2𝐴𝐵r=kln(U_{1},U_{2},A,B),italic_r = italic_k roman_ln ( italic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_A , italic_B ) , (1)

the place okay𝑘okayitalic_k is an actual fixed. This Cayley-Klein metric now permits the development of all elementary geometric objects by elementary geometric means, and Grossmann proceeds to indicate how the elemental building duties of Euclidean geometry can now be transferred to the non-Euclidean case, see Fig. 3 for an illustration of his technique.

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Determine 3: Grossmann was an knowledgeable in artificial constructions of non-Euclidean geometry within the framework of Cayley-Klein geometry. This determine (17 22, 18 504) illustrates the development of a triangle with given proper angle (at level C𝐶Citalic_C) and two arbitrary angles α𝛼alphaitalic_α and β𝛽betaitalic_β in a hyperbolic geometry represented by the conic part ΩΩOmegaroman_Ω. The within of the advanced ΩΩOmegaroman_Ω represents the hyperbolic airplane, its border the improper factors at infinity. Straight strains which have a given angle α𝛼alphaitalic_α with s1subscript𝑠1s_{1}italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or β𝛽betaitalic_β with s2subscript𝑠2s_{2}italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are tangents to the conic sections Okay(α)𝐾𝛼Okay(alpha)italic_K ( italic_α ) or Okay(β)𝐾𝛽Okay(beta)italic_K ( italic_β ), and the duty is to discover a widespread tangent to each Okay(α)𝐾𝛼Okay(alpha)italic_K ( italic_α ) and Okay(β)𝐾𝛽Okay(beta)italic_K ( italic_β ), which is proven to be an issue solely of second somewhat than of fourth diploma.

In 1905, Grossmann moved to Basel to show at his personal former college, the Oberrealschule. At the moment, on the idea of his revealed analysis,
he additionally grew to become Privatdozent on the College of Basel, an unpaid educational title with the privilege (and obligation) to lecture on the college that was wanted in an effort to be thought of for a
professorship at a German language college. In Basel, Grossmann revealed two geometry textbooks, one on
analytic geometry20 and one on descriptive geometry21. Generations of arithmetic and engineering college students, in Basel, on the ETH, and elsewhere, would study geometry from these textbooks and its numerous later editions.

When his educational trainer Fiedler requested for a depart on account of sick well being in 1906, Grossmann was requested to step in for him and to show the course on descriptive geometry on the ETH within the winter semester of 1906/07. Fiedler resigned for good
in June 1907 as of 1 October, his place was marketed, and Grossmann who was listed secundo loco was appointed his successor on 22 July 1907, after Martin Disteli (1862–1923) in Dresden had rejected a name. The preliminary appointment was for 3 years, but it surely was prolonged for one more ten years in 1910, as was traditional with such appointments.

If Grossmann’s profession from an formidable pupil to a professor on the ETH was somewhat direct and with out setbacks, his pal Einstein’s profession was much less so. As is well-known,12 Einstein’s makes an attempt to acquire an assistantship anyplace failed flatly regardless of numerous letters of software. To make issues extra
tough, Einstein and his fellow pupil Mileva Marić had fallen in love and, sadly, Marić had develop into pregnant within the spring of 1901, an unplanned circumstance that certainly contributed to her failing the ultimate examinations once more at her second try in summer time 1901.

Einstein had additionally utilized in July 1901 for the place on the Thurgau Kantonsschule that Grossmann would get hold of 12 Doc.122. However Grossmann’s father, Julius Grossmann (1843–1934), who was an outdated pal of Friedrich Haller (1844–1936), director of the Swiss Patent Workplace in Bern, advisable Einstein for a place at this workplace, and in June 1902 Einstein started to work there as a Technical Skilled. Since Einstein now had a tenured job, he was in a position to marry Mileva Marić in January 1903. Coincidentally, the identical 12 months, Grossmann acquired married to Anna Keller (1882–1967). When Einstein obtained his Ph.D. in 1905, he devoted his doctoral thesis to his pal Grossmann.23 A couple of years later, Einstein was contemplating his subsequent profession steps.
In 1908, he requested Grossmann for recommendation in making use of for a trainer’s place at a technical college in Winterthur 24 Doc. 71. Nothing got here of that, however shortly thereafter, he obtained his habilitation on the College of Bern, and, in 1909, he accepted an affiliate professorship on the College of Zurich. Though in shut proximity once more, little is understood about Grossmann’s and Einstein’s interactions throughout the latter’s tenure at Zurich College.

Grossmann, within the meantime, engaged himself in all points of arithmetic: analysis, instructing, and disciplinary group. In 1909, he confirmed how projective constructions of hyperbolic geometry will also be accomplished utilizing solely a ruler, if a set auxiliary circle is given.25 He lectured about projective geometry and projective constructions to Swiss lecturers.26 In 1910, he confirmed a conjecture within the subject of photogrammetry formulated by Sebastian Finsterwalder (1862–1951) in 1897. Photogrammetry, in some sense the inverse of projective geometry, is anxious with the issue of developing authentic three-dimensional objects, if solely two-dimensional pictures of these objects are given. In a report on the state of the sphere introduced to the Deutsche Mathematiker-Vereinigung, Finsterwalder had claimed that given 4 pictures of an object, one can at all times assemble the item uniquely as much as a conformal scale issue however the express building was virtually unfeasible 27 14–15. In a lecture to the Zurich Naturforschende Gesellschaft, Grossmann confirmed the conjecture in regards to the sensible impossibility of the particular building by displaying “in a purely geometric manner” that the related surfaces could be given as a bit between a developable floor of nineteenth diploma with one other floor of fifth diploma and that after subtracting improper options, there exist 56 of these surfaces.28
In the identical 12 months, he co-founded the Swiss Mathematical Society, whose presidency he held within the years 1916 and 1917. An essential early exercise of the Society was the inauguration of a long-term editorial undertaking publishing the writings of Leonhard Euler (1707–1783).29 In 1911, Grossmann revealed a
detailed fifty-page report on arithmetic instruction on the ETH, as a part of a broad report on arithmetic
schooling in Switzerland, requested by the Commmission internationale de l’Enseignement mathématique.30 Earlier, he had pushed for the institution of vacation programs for arithmetic lecturers at Swiss gymnasia and center colleges.31 Comparable programs had been established just a few years earlier in Berlin and Göttingen and have been supposed to complement
lecturers’ data within the mathematical sciences 32 146–147. When Einstein moved to Prague as a full professor in 1911, he was requested by Grossmann to lecture at a kind of vacation programs, a request that Einstein felt unable to show down 24 Doc. 266. In a lecture to the Zurich Naturforschende Gesellschaft on non-Euclidean geometry given of 29 January 1912, Grossmann additionally mentioned the query of the geometric nature of actual bodily house.33

With Grossmann now an essential participant in ETH’s college, it’s not shocking that he was additionally concerned in bringing Einstein again to Zurich. Within the winter time period 1911/1912, Grossmann took over the chair of the division for arithmetic and physics lecturers, and one of many first issues he should have accomplished on this capability is to put in writing to Einstein informally to sound him out about his willingness to just accept a suggestion to affix the ETH college. Ever since Hermann Minkowski (1864–1909) had moved to Göttingen in 1902 the professorship for greater arithmetic had been vacant, a state of affairs that was lamented by Grossmann and his colleagues. When Einstein, in a letter of 18 November 1911 24 Doc. 307, indicated that he would take into account coming again to his alma mater, the problem was mentioned two weeks later at a gathering of the Swiss Faculty Council and ETH’s president Robert Gnehm (1852–1926) was requested to enter into formal negotiations with Einstein. Even after Gnehm had initiated his negotiations and when Einstein was receiving competing gives from the Netherlands, Grossmann continued to help informally with recommendation as to the way to finest speed up the method 24 Docs. 319, 321. Grossmann should have been happy when, already on 30 January 1912, Einstein was appointed Professor of Theoretical Physics on the ETH efficient 1 October 1912, regardless of some inside opposition from his colleagues 24 Doc. 291. He had succeeded in securing a first-rate scholar for the science schooling of the ETH college students and he may very well be wanting ahead to having his good pal round as a colleague once more.

Throughout Einstein’s tenure on the ETH, he and Grossmann engaged in an intense and really profitable collaboration, during which Einstein’s physics coaching and Grossmann’s mathematical background got here collectively in a fruitful seek for a relativistic concept of gravitation. The collaboration culminated of their joint publication of an “Define (German: Entwurf) of a normal concept of relativity and a concept of gravitation.” This era and the collaboration will probably be mentioned in additional element under. Right here we are going to first proceed to provide an account of Grossmann’s life and work.

Grossmann’s mental biography after the Entwurf episode continues to be dominated by his analysis, instructing, and administrative duties as professor of arithmetic on the ETH. However throughout the battle and for a while after the battle years, Grossmann additionally engaged in patriotic actions. He wrote essays within the each day Neue Zürcher Zeitung, revealed pamphlets and gave lectures, during which he emphasised the necessity to strengthen the nationwide unity of the totally different components of Switzerland.34, 35, 36, 37, 38

Towards the tip of the battle and within the instant postwar interval, Grossmann engaged within the publication of a periodical. On 25 October 1918, an initiative committee issued an announcement, signed by Grossmann and two others, of the founding of a cooperative with the intention of publishing a brand new periodical referred to as the Neue Schweizer Zeitung and an invite to potential subscribers and donors. Members of the cooperative needed to be Swiss nationals. The primary difficulty of the Neue Schweizer Zeitung appeared on 20 December 1918, and it was then revealed twice weekly. The paper was supposed to supply a discussion board for open debate of all points regarding Swiss affairs, supporting the beliefs of Swiss democracy, federalism, and nationwide unity. As is obvious from an invite for subscribers, dated 27 December, Grossmann functioned as president of the board of administration and as such was additionally a member of the modifying committee. So was his brother Eugen Grossmann (1879–1963), a professor of economics on the College of Zurich. Each Marcel Grossmann and his brother frequently revealed contributions within the new paper. It continued to seem for 3 and a half years till 29 June 1922 when it stopped publication, and through this era Grossmann revealed some 40 contributions within the Neue Schweizer Zeitung. A observe, dated 20 June 1922, saying the tip of the paper on account of financial issues was nonetheless signed by Grossmann as president of the board of administration.

One other of Grossmann’s actions within the instant postwar years involved an effort to reform the nationwide Swiss laws and necessities for acquiring a secondary-school diploma that might qualify for college research, the so-called Maturitätszeugnis or, brief, Matura.39 The tutorial system in Switzerland has strictly been beneath authority of the cantons, the one affect that the federal authorities might tackle the necessities of secondary college diplomata was via laws for admission within the medical professions and thru entrance necessities to the Swiss Federal Polytechnic (ETH) college in Zurich. Since a wide range of totally different secondary colleges, gymnasia, center colleges, enterprise, skilled and vocational colleges existed within the numerous cantons with extensively totally different curricula, requirements, and examination guidelines, a committee was put in with the duty of formulating guidelines for requirements that might be acceptable nationwide. The talk and dialogue course of continued for a number of years and touched on a number of hotly disputed points. One level of competition was the problem whether or not data within the outdated languages Latin and Greek because it was taught within the conventional gymnasia ought to be required because it had been the case historically for college kids of the medical sciences. This difficulty collided with the want that so-called Realgymnasien, i.e., secondary colleges which focussed on the sciences somewhat than on the languages, ought to be allowed to arrange for college research on the idea of a science-centered curriculum. Additionally at stake have been numerous totally different education traditions within the numerous cantons.

Arithmetic was a core topic in all curricula and historically it was the ETH college themselves who selected the admission of their college students. 40
As a professor of arithmetic on the ETH, Grossmann was a member of the committee who have been put in control of formulating a proposal for reform, and he labored on this capability fairly critically and passionately. In a lot of public statements made at numerous events,41, 42, 43, 44, 45, 46 he argued for a reform that was guided by a spirit of each liberality and excessive scientific commonplace. He argued in opposition to overloading the required curricula by demanding an excessive amount of data of particulars in favor of furthering abilities of unbiased judgment and research. He additionally advocated a proposal in line with which the Realgymnasien ought to be given the privilege of awarding a federally acknowledged Maturitätszeugnis with out requiring the data of outdated languages. Fairly within the spirit of his patriotic actions throughout the battle, he additionally emphasised the necessity of teaching college students within the spirit of changing into unbiased, democratic, and patriotic residents. The talk typically degenerated into veritable polemics throughout which Grossmann at one level accused his personal former college, the Realgymnasium in Basel, the place he additionally had been on the school for 2 years, of an utter lack of scholarly commonplace.46 In a pointed polemic, Grossmann had mentioned publicly that Basel had an excellent Gymnasium however a really unhealthy Realschule. His level was that the Realschule wouldn’t sufficiently weed out unhealthy college students, its gradings and examinations being too lax and pleasant.The background for the polemic was that an alleged unhealthy status of Basel’s Realgymnasium undermined Grossmann’s argument for a science-based Maturitätszeugnis. The ultimate revision of the decree regulating a federal Maturitätsprüfung, which was handed on 20 January 1925 proved to be a defeat in some factors for
Grossmann’s place. It launched three kinds of Matura primarily based on outdated languages (A), new languages (B), and on sciences (C), however didn’t put the Matura of sort C on a par with the opposite two varieties,
as a result of it required as well as additional Latin examinations for college kids who wished to enter medical college.

Grossmann’s devotion as a trainer and pedagogue and his ardour for instructional coverage is mirrored in an essay 47 he wrote a 12 months after that “failed try” at a reform of the Swiss instructional system. In that essay, he mirrored on his experiences as a trainer and examinator, having examined “a number of hundreds” of his personal college students in addition to having participated in “a number of hundred” examinations by others all around the nation. In that essay, he once more expressed his conviction that not solely instructing usually but in addition

instruction in particular disciplines ought to […] primarily develop competencies, ought to create a state of mind that allows the younger particular person on the finish of his research to swim about and not using a swimming coach and with out swimming rings, even when the present would circulation in sudden instructions.

Fairly equally he expressed himself in a contribution to the Festschrift for his colleague August Stodola (1859–1912),48 which is one other passionate plea for the enlightening function of science and expertise in society and for the necessity of a sound schooling in these fields.

So far as his personal pedagogical efforts are involved—past his instructing on the ETH with numerous new editions of his textbook on descriptive geometry—he alerted lecturers to the importance of projective geometry and its idea of improper parts on the stage of secondary college schooling.49 It must also be talked about on this respect that he supervised 4 doctoral dissertations within the subject of non-Euclidean geometry.11

When it comes to his personal mathematical analysis, Grossmann went again to his subject of artificial geometry.
In 1922, he introduced a chat to the Swiss Mathematical Society on projective constructions of
elliptic geometries.50 In 1924, he mentioned full focal methods of airplane algebraic curves, extending a definition of focal factors for algebraic curves given first by Julius Plücker (1801–1868).51 A 12 months later, he gave an in depth geometric dialogue of the development of the horopter, i.e. the set of factors in house that, in geometrical imaginative and prescient, are imaged at corresponding factors of the attention’s retina, geometrically a third-order curve generated by two congruent line bundles similar to the fixation strains of the visible rays.52 In 1927,
Grossmann obtained a patent for “enhancements relating for the manufacturing of cams for looms.” The purpose of the patent was to outline rules for a machine that
would permit a exact and correct grinding of a particular a part of mechanical looms. As he defined in a
companion publication,54 his invention arose from geometric perception.
He realized that the relative movement of sure cams for looms with rollers mounted on shafts tailored for angular motion would represent an enveloping floor arising from a system of congruent surfaces, and identified that research of such enveloping surfaces has an extended custom in arithmetic since Gaspard Monge (1746–1818).
In 1930,
he revealed one other brief observe on constructions of circles and conic sections in projective and non-Euclidean geometry.55 In his final scientific publication, to be mentioned
under, Grossmann took difficulty with Einstein’s teleparallel method to a unified subject concept of gravity and
electromagnetism.56

Grossmann’s work within the twenties was severely hampered by signs of an uncurable sickness. We all know some particulars about Grossmann’s situation from a letter that he wrote on 12 March 1927 to Einstein’s pal Heinrich Zangger (1874–1954), the director of the institute for forensic medication on the College of Zurich (AEA 40-059). On this letter, Grossmann studies that he had at all times been wholesome till the 12 months 1915, when he first had a match of dizziness throughout a mountain hike and famous that he misplaced safety of grip in his proper hand throughout mountaineering. In the summertime of 1917, he started to pull his proper leg and had disturbances of speech. Since that point he had seen numerous medical doctors for signs of impediments of movement, primarily on his proper facet. In his letter, Grossmann requested Zangger for an opinion that might corroborate his personal conjecture that these signs have been attributable to intoxication on account of unhealthy circumstances within the lecture halls throughout the battle years and that subsequently he could be eligible for skilled incapacity compensation. Grossmann’s situation, in any case, had gotten worse over time. In the summertime semester
1924, he needed to take a depart from his instructing duties for well being causes. In the summertime of 1925, Einstein visited Grossmann in Zürich and wrote to Zangger that he had a “peculiar nervous ailment with palsy” however discovered him higher than he had anticipated. One other depart was granted for the winter time period 1925/26. On 10 March 1926 he was granted a partial dispensation of his instructing duties for the summer time time period 1926 however a month later, he needed to take a full depart for that semester, too; and he was additionally relieved from instructing duties within the winter time period 1926/27. When he requested Zangger for his opinion, his retirement as a professor was imminent. Zangger wrote to Einstein that he didn’t imagine within the causal nexus with the circumstances throughout the battle years however somewhat thought Grossmann was affected by a number of sclerosis. Zangger’s analysis seems to have been correct and, in any case, Grossmann requested for an early retirement, which was granted to him on 28 Might 1927, efficient 1 October 1927.

Marcel Grossmann handed on 7 September 1936, on the age of 58.

3 Grossmann’s Collaboration with Albert Einstein

Allow us to now return to the time of the scientific collaboration between Grossmann and Einstein.
Einstein left Prague on 25 July 1912 and registered his change of residence to Zurich on 10 August. With a household of 4, his sons being 8 and a couple of years of age, it should have taken just a few days to settle in. The Grossmann household might have been of assist to the Einstein household however Marcel himself was busy getting ready for a lecture on the fifth Worldwide Congress of Mathematicians which passed off in Cambridge from 22–28 August 1912.58 Only a few days later, from 8 to 11 September, the Schweizerische Naturforschende Gesellschaft held their annual assembly in Altdorf, and Grossmann introduced there a “projective proof of Lobatchevsky’s absolute parallel building.”59

We all know, nevertheless, that quickly after arriving in Zurich, Einstein and Grossmann began a collaboration that might virtually result in the invention of normal relativity and that might, in any case, end in Grossmann’s most well-known scholarly achievement. As Louis Kollros recalled in 1956, Einstein approached Grossmann for assist, saying:

Grossmann, it’s important to assist me, or else I’ll go loopy!

And, famously, on 29 October Einstein reported to Arnold Sommerfeld (1868–1951):

I’m now working solely on the gravitation downside and imagine that I can overcome all difficulties with the assistance of a mathematician pal of mine right here. However one factor is definite: by no means earlier than in my life have I troubled myself over something a lot, and I’ve gained monumental respect for arithmetic, whose extra refined components I thought of till now, in my ignorance, as pure luxurious!

With the intention to correctly assess Grossmann’s contribution to the genesis of normal relativity, it’s
essential to establish as concretely as potential the start line for his collaboration with Einstein. Sadly, we are able to solely speculate about how precisely their joint work took off. Due to this fact, it is going to be essential to recapitulate briefly Einstein’s efforts in generalizing particular relativity up till his transfer to Zurich 1 Vol. 1, 81–113.

Einstein had made a primary step of generalizing particular relativity by formulating the equivalence speculation in 1907.61 Within the following years, the issue had remained dormant with him till the summer time of 1911 when he got here again to the issue in Prague. At the moment, he realized that one of many penalties of the heuristic assumption of a strict equivalence between fixed linear acceleration and static homogeneous gravitation may truly be observable with the bending of star mild grazing the limb of the solar throughout a photo voltaic eclipse.62 The essential level was that the equivalence assumption implied that the speed of sunshine c𝑐citalic_c rely on the gravitational subject, i.e. the fixed c𝑐citalic_c grew to become a spatially variable operate c=c(x)𝑐𝑐𝑥c=c(x)italic_c = italic_c ( italic_x ). Particularly, Einstein deduced that the speed of sunshine would rely on the gravitational potential Φ(x)Φ𝑥Phi(x)roman_Φ ( italic_x ) as

c=c0(1+Φc2).𝑐subscript𝑐01Φsuperscript𝑐2c=c_{0}left(1+frac{Phi}{c^{2}}proper).italic_c = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 1 + divide start_ARG roman_Φ end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) . (2)

In early 1912, Einstein was shocked by a paper by Max Abraham (1875–1922)63 who claimed that this relation follows readily from postulating a generalization of Poisson’s equation of the shape

2Φx2+2Φy2+2Φz2+2Φu2=4πγνsuperscript2Φsuperscript𝑥2superscript2Φsuperscript𝑦2superscript2Φsuperscript𝑧2superscript2Φsuperscript𝑢24𝜋𝛾𝜈frac{partial^{2}Phi}{partial x^{2}}+frac{partial^{2}Phi}{partial y^{2}}+frac{partial^{2}Phi}{partial z^{2}}+frac{partial^{2}Phi}{partial u^{2}}=4pigammanudivide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG ∂ italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG ∂ italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG ∂ italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = 4 italic_π italic_γ italic_ν (3)

with an imaginary time u=ict𝑢𝑖𝑐𝑡u=ictitalic_u = italic_i italic_c italic_t, a gravitational fixed γ𝛾gammaitalic_γ and a mass density ν𝜈nuitalic_ν, along with
equations of movement

x¨=Φx,y¨=Φy,z¨=Φz,u¨=Φu,formulae-sequence¨𝑥Φ𝑥formulae-sequence¨𝑦Φ𝑦formulae-sequence¨𝑧Φ𝑧¨𝑢Φ𝑢ddot{x}=-frac{partialPhi}{partial x},ddot{y}=-frac{partialPhi}{partial y},ddot{z}=-frac{partialPhi}{partial z},ddot{u}=-frac{partialPhi}{partial u},over¨ start_ARG italic_x end_ARG = – divide start_ARG ∂ roman_Φ end_ARG start_ARG ∂ italic_x end_ARG , over¨ start_ARG italic_y end_ARG = – divide start_ARG ∂ roman_Φ end_ARG start_ARG ∂ italic_y end_ARG , over¨ start_ARG italic_z end_ARG = – divide start_ARG ∂ roman_Φ end_ARG start_ARG ∂ italic_z end_ARG , over¨ start_ARG italic_u end_ARG = – divide start_ARG ∂ roman_Φ end_ARG start_ARG ∂ italic_u end_ARG , (4)

the place the dots point out differentiation of the coordinates of a cloth “world level” with respect to its correct time. Additional reflection and correspondence, nevertheless, made it clear that Abraham’s argument was not as easy as it could have appeared since his utilization of four-dimensional vector calculus grew to become inconsistent with the idea of a variable c𝑐citalic_c. Abraham conceded to Einstein’s criticism by proscribing the light-cone relation to an infinitesimal line factor64

ds2=dx2+dy2+dz2c2dt2𝑑superscript𝑠2𝑑superscript𝑥2𝑑superscript𝑦2𝑑superscript𝑧2superscript𝑐2𝑑superscript𝑡2ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_d italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT – italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (5)

with variable c𝑐citalic_c.

Einstein, in any case, was pushed by this dialogue to additional consideration of the idea of static gravitation primarily based on the equivalence speculation. In late February 1912, he revealed a paper65 on the subject, during which he additionally represented the propagation of sunshine by the use of an infinitesimal line factor (5) and during which he prompt to generalize Poisson’s equation
with a differential equation of the static gravitational subject that he gave as

Δc=okaycρΔ𝑐𝑘𝑐𝜌Delta c=kcrhoroman_Δ italic_c = italic_k italic_c italic_ρ (6)

with gravitational fixed okay𝑘okayitalic_k and matter density ρ𝜌rhoitalic_ρ. Right here ΔΔDeltaroman_Δ denotes the spatial, three-dimensional Laplace operator.

He quickly discovered fault together with his differential equation (6). Only a few weeks later, in late March, he submitted a second paper66 with a modified differential equation, which he now gave as

Δc=okay{cρ+12okaygrad2cc}.Δ𝑐𝑘𝑐𝜌12𝑘superscriptgrad2𝑐𝑐Delta c=kbiggl{{}crho+frac{1}{2k}frac{operatorname{grad}^{2}c}{c}biggr{}}.roman_Δ italic_c = italic_k { italic_c italic_ρ + divide start_ARG 1 end_ARG start_ARG 2 italic_k end_ARG divide start_ARG roman_grad start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c end_ARG start_ARG italic_c end_ARG } . (7)

He interpreted the second time period in brackets as an vitality density of the gravitational subject.

Thus far, Einstein had made use of the heuristics of the equivalence speculation solely by contemplating fixed linear acceleration. However already these issues had proven him that he needed to work with infinitesimal
line parts. He additionally had realized that the issue would most likely contain non-linear differential equations, which have been wanted in an effort to correctly bear in mind the vitality density of the gravitational subject itself. Nonetheless, the idea was nonetheless a scalar concept for a single operate c𝑐citalic_c representing each the velocity of sunshine and the gravitational potential.

The following step was to take a look at stationary rotating coordinates. Though a lot much less explicitly documented,
it’s clear that remodeling the line-element (5) to rotating Cartesian coordinates will produce blended phrases with a coefficient that entails the rotation frequency ω𝜔omegaitalic_ω. Deciphering the rotation subject, represented by ω𝜔omegaitalic_ω, as a gravitational subject might have induced Einstein, sooner or later, to think about a normal
line factor

ds2=i,okay=14giokaydxidxokay𝑑superscript𝑠2superscriptsubscript𝑖𝑘14subscript𝑔𝑖𝑘𝑑subscript𝑥𝑖𝑑subscript𝑥𝑘ds^{2}=sum_{i,okay=1}^{4}g_{ik}dx_{i}dx_{okay}italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i , italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT italic_d italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_d italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT (8)

because the illustration of a normal gravitational subject, i.e., one during which the coefficients giokaysubscript𝑔𝑖𝑘g_{ik}italic_g start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT weren’t essentially produced by a coordinate transformation away from the Minkowski line factor however somewhat would signify a generic, independently given gravitational subject. Einstein, in any case, sooner or later
noticed the analogy between infinitesimal line parts occurring in his concept with the two-dimensional
line factor of a curved floor in Gaussian floor concept, about which he had realized in Geiser’s lectures on the ETH (see Fig. 2).

We don’t know whether or not Einstein made the transition from a scalar concept to a gravitation concept primarily based on the final line factor (8) earlier than he got here to Zurich and earlier than he started to speak with Grossmann about his downside. In later recollections, he reconstructed the start of their collaboration by posing to Grossmann a somewhat particular mathematical query. In 1955, he wrote:

The issue of gravitation was thus decreased to a purely mathematical one. Do differential
equations exist for the giokaysubscript𝑔𝑖𝑘g_{ik}italic_g start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT, that are invariant beneath non-linear coordinate
transformations? Differential equations of this type and solely of this type have been
to be thought of as subject equations of the gravitational subject. The legislation of movement of
materials factors was then given by the equation of the geodesic line.


With this downside in thoughts I visited my outdated pal Grossmann who within the meantime had develop into professor
of arithmetic on the Swiss polytechnic. He directly caught hearth, though as a mathematician he had a considerably skeptical stance in direction of physics.

This recollection might not have been completely correct or, at the very least, it was most likely too transient. It fully neglects one other essential step: the query of the right illustration of the gravitating mass-energy density. The transition from a scalar concept to a concept primarily based on a multi-component object additionally
implied a transition from a scalar mass-energy density to a multi-component mathematical advanced that concerned momentum circulation and stresses. We’re purposely avoiding the trendy time period “tensor” right here as a result of, as we are going to see, the introduction of the tensor idea, as we’re used to it now within the context of normal relativity, might have been the primary of Grossmann’s contributions when he started his discussions with Einstein.

There are two key paperwork, which give us perception into Grossmann’s function within the collaboration with Einstein.3, 67, 1 The primary doc is Einstein’s so-called Zurich Pocket book 5 Doc. 10, 1 Vols. 1–2. It is a certain pocket book of some 85 written pages, 57 of which include analysis notes, documenting the seek for a relativistic concept of gravitation within the interval between summer time 1912 and spring 1913. All entries are in Einstein’s hand however Grossmann’s title seems twice within the pocket book, at strategic locations, as we are going to see under. The analysis documented within the Zurich Pocket book leads immediately as much as the second doc of relevance, a two-part paper revealed within the Zeitschrift für Mathematik und Physik entitled “Define (Entwurf) of a Generalized Idea of Relativity and of a Idea of Gravitation.”68 The work was
accomplished by mid-Might 1913 and offprints (with unbiased pagination) have been obtainable earlier than 25 June 1913.
The paper was divided into two components, a bodily half, authored by Albert Einstein, and a
mathematical half, for which Grossmann signed accountable.

Taking clues from these two paperwork we are able to attempt to reconstruct Grossmann’s contribution to the emergence of normal relativity at this level of their collaboration. Giant components of the Zurich Pocket book
could also be reconstructed because the seek for a gravitational subject equation of the shape 68 p. 15, 1 Vol. 1-2, 113–312, 489–714

Γμν=κΘμν.subscriptΓ𝜇𝜈𝜅subscriptΘ𝜇𝜈Gamma_{munu}=kappacdotTheta_{munu}.roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_κ ⋅ roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT . (9)

Right here ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT stands for some operator appearing on the metric coefficients gμνsubscript𝑔𝜇𝜈g_{munu}italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT in a particular
manner. Simply what kind ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT would have is the issue that Einstein and Grossmann have been making an attempt to unravel. κ𝜅kappaitalic_κ denotes a relentless that will probably be proportional to the gravitational fixed, the
proportionality being decided on inspection of a limiting case during which the Poisson equation is being recovered. ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT denotes the (contravariant) stress-energy-momentum tensor of matter and fields. Two feedback are so as. First, because it stands, i.e., so long as ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT isn’t but specified, Eq. (9) isn’t a covariant equation, or somewhat, its covariance group is undetermined. Second,
though the equation was written with subscript indices, it was supposed as a contravariant equation. The contravariant character of Eq. (9) was expressed by the truth that Greek letters have been used for the portions ΓΓGammaroman_Γ and ΘΘThetaroman_Θ.

The notational peculiarities of the Entwurf elevate an essential level concerning Grossmann’s contribution to the emergence of the idea at this level. When Einstein approached Grossmann in regards to the arithmetic related to the metric gμνsubscript𝑔𝜇𝜈g_{munu}italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT it was in no way clear what standing the item gμνsubscript𝑔𝜇𝜈g_{munu}italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT truly had. In Gauss’s floor concept, the three unbiased metric parts g11subscript𝑔11g_{11}italic_g start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT, g12subscript𝑔12g_{12}italic_g start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT, g22subscript𝑔22g_{22}italic_g start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT of the two-dimensional line factor have been sometimes denoted by the letters E𝐸Eitalic_E, F𝐹Fitalic_F, G𝐺Gitalic_G. The truth that the metric parts are parts of a tensor is to be attributed to Grossmann. It was he who realized {that a} department of arithmetic had developed during which the Gaussian concept of surfaces featured solely as a particular instance. Famously, Grossmann alerted Einstein to the existence of the so-called absolute differential calculus, which had been introduced in a complete joint paper by Gregorio Ricci-Curbastro (1853–1925) and Tullio Levi-Civita (1873–1941) in 1901.71, 72, 73

However Grossmann did greater than merely discover out about and exploit absolutely the differential calculus for the aim at hand. He realized very clearly that “the vector evaluation of Euclidean house in arbitrary curvilinear coordinates is formally similar to the vector evaluation of an arbitrary manifold given by its line factor.”

Though in later recollections, Einstein credited Grossmann primarily for displaying him the related literature, we should assume that Grossmann truly helped make clear the very mathematical standing of the objects that
have been coming into the middle stage of their theoretical efforts.

In any case, Grossmann gave a brand new and self-contained exposition of what he referred to as “normal vector calculus” (“allgemeine Vektoranalysis”). In doing so, Grossmann explicitly discovered it pointless to seek advice from any geometric ideas:

In doing this I intentionally didn’t draw on geometrical instruments, as they contribute little to the illustration of the ideas of vector evaluation.

Within the Entwurf, Grossmann proceeds to provide an exposition of tensor calculus. He launched covariant, contravariant, and blended tensors for areas of arbitrary dimensions and of any rank. The usage of the phrase
“tensor” on this context is a novelty. Ricci and Levi-Civita had referred to as these objects systèmes covariants ou contrevariants, and so they had by no means thought of systèmes of blended transformation habits, i.e. with a mixture of covariant and contravariant indices. They’d used superscripts and subscripts to point contravariant and covariant transformation habits, aside from coordinate differentials, which at all times carried subscript indices. Grossmann launched a notation the place all indices have been written as subscripts and the transformation character was indicated by writing the item itself with a Latin, Greek, or Gothic character for covariant, contravariant, or blended tensors, respectively. Within the Entwurf, Grossmann outlined tensor operations of a sum, exterior product, interior product (contraction), of adjusting covariant to contravariant objects by contraction with the elemental tensor and vice versa (what we now name elevating and reducing indices), and he launched what we now name the hint of a (second-rank) tensor. With express reference to Elwin Bruno Christoffel, (1829–1900),76 he additionally launched covariant differentiation of a tensor, which he referred to as “enlargement” (“Erweiterung”). Subsequent, he launched a covariant idea of divergence by covariant differention of a tensor and contraction with the elemental tensor, and he outlined a generalized “Laplacian operation” as the mixture of an enlargement and a divergence.
He additionally paid some consideration to the particular case of antisymmetric tensors, a particular case of which is the absolutely antisymmetric Levi-Civita tensor.

Within the Zurich pocket book, there’s a web page (05R) on which Einstein deduced an equation that we now acknowledge because the covariant divergence of the energy-momentum tensor. Einstein there proceeds by wanting a the Euler-Lagrange equations for a Hamiltonian H=ds/dt𝐻𝑑𝑠𝑑𝑡H=ds/dtitalic_H = italic_d italic_s / italic_d italic_t and figuring out the change of energy-momentum and the ponderomotive power density for a “tensor of the movement of lots” given as

Θiokay=ρdxidsdxokayds,subscriptΘ𝑖𝑘𝜌𝑑subscript𝑥𝑖𝑑𝑠𝑑subscript𝑥𝑘𝑑𝑠Theta_{ik}=rhofrac{dx_{i}}{ds}frac{dx_{okay}}{ds},roman_Θ start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT = italic_ρ divide start_ARG italic_d italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_s end_ARG divide start_ARG italic_d italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_d italic_s end_ARG , (10)

from which he obtained the energy-momentum steadiness equation

νnxn(ggmνΘνn)12μνggμνxmΘμν=0,subscript𝜈𝑛subscript𝑥𝑛𝑔subscript𝑔𝑚𝜈subscriptΘ𝜈𝑛12subscript𝜇𝜈𝑔subscript𝑔𝜇𝜈subscript𝑥𝑚subscriptΘ𝜇𝜈0sum_{nu n}frac{partial}{partial x_{n}}biggl{(}sqrt{-g},g_{mnu}Theta_{nu n}biggr{)}-frac{1}{2}sum_{munu}sqrt{-g},frac{partial g_{munu}}{partial x_{m}}Theta_{munu}=0,∑ start_POSTSUBSCRIPT italic_ν italic_n end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG ( square-root start_ARG – italic_g end_ARG italic_g start_POSTSUBSCRIPT italic_m italic_ν end_POSTSUBSCRIPT roman_Θ start_POSTSUBSCRIPT italic_ν italic_n end_POSTSUBSCRIPT ) – divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = 0 , (11)

a relation that we readily establish, in fashionable notation, as (gTnm);n=0(sqrt{-g},{T^{n}}_{m})_{;n}=0( square-root start_ARG – italic_g end_ARG italic_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ; italic_n end_POSTSUBSCRIPT = 0. Within the Entwurf, Grossmann provides an express proof of the declare that the energy-momentum steadiness equation (11) is a typically covariant expression by displaying that it’s obtained as a covariant divergence of ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT.

Grossmann’s perception that the energy-momentum steadiness equation is a wonderfully legitimate, typically covariant relation shouldn’t be underestimated. It should have prompt that normal covariance could be a viable purpose if refined ideas of superior arithmetic have been made use of. It additionally confirmed that one half of the gravitation downside was already solved. Given a gμνsubscript𝑔𝜇𝜈g_{munu}italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT-field, the motion of matter is decided by a typically covariant equation of movement.

Allow us to return now to the issue of discovering a gravitational subject equation. The schema of a subject equation needed to appear like Eq. (9) above. The issue was to seek out candidates for the gravitation tensor ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT. There’s a web page within the Zurich Pocket book the place Grossmann’s title seems proper subsequent to
the Riemann-Christoffel tensor, see Fig. 4. Apparently, Grossmann had proven Einstein the related object that might
open a path in direction of absolutely covariant gravitational subject equations. Subsequent to a definition of the
Christoffel symbols of the primary form

[μνl]=12(gμlxν+glνxμgμνxl)matrix𝜇𝜈𝑙12subscript𝑔𝜇𝑙subscript𝑥𝜈subscript𝑔𝑙𝜈subscript𝑥𝜇subscript𝑔𝜇𝜈subscript𝑥𝑙start{bmatrix}munu lend{bmatrix}=frac{1}{2}biggl{(}frac{partial g_{mu l}}{partial x_{nu}}+frac{partial g_{lnu}}{partial x_{mu}}-frac{partial g_{munu}}{partial x_{l}}biggr{)}[ start_ARG start_ROW start_CELL italic_μ italic_ν end_CELL end_ROW start_ROW start_CELL italic_l end_CELL end_ROW end_ARG ] = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_l italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG – divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG ) (12)

we discover on that web page the Riemann-Christoffel tensor in absolutely covariant kind,

Riokaylm=(iokay,lm)subscript𝑅𝑖𝑘𝑙𝑚𝑖𝑘𝑙𝑚displaystyle R_{iklm}=(ik,lm)italic_R start_POSTSUBSCRIPT italic_i italic_k italic_l italic_m end_POSTSUBSCRIPT = ( italic_i italic_k , italic_l italic_m ) =(2gimxokayxl+2gokaylxixm2gilxokayxm2gokaymxlxi)absentsuperscript2subscript𝑔𝑖𝑚subscript𝑥𝑘subscript𝑥𝑙superscript2subscript𝑔𝑘𝑙subscript𝑥𝑖subscript𝑥𝑚superscript2subscript𝑔𝑖𝑙subscript𝑥𝑘subscript𝑥𝑚superscript2subscript𝑔𝑘𝑚subscript𝑥𝑙subscript𝑥𝑖displaystyle=displaystylebiggl{(}frac{partial^{2}g_{im}}{partial x_{okay}partial x_{l}}+frac{partial^{2}g_{kl}}{partial x_{i}partial x_{m}}-frac{partial^{2}g_{il}}{partial x_{okay}partial x_{m}}-frac{partial^{2}g_{km}}{partial x_{l}partial x_{i}}biggr{)}= ( divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG – divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG – divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_k italic_m end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) (21)
+ρσγρσ([imσ][klρ][ilσ][kmρ]),subscript𝜌𝜎subscript𝛾𝜌𝜎matrix𝑖𝑚𝜎matrix𝑘𝑙𝜌matrix𝑖𝑙𝜎matrix𝑘𝑚𝜌displaystyle+displaystylesum_{rhosigma}gamma_{rhosigma}biggl{(}start{bmatrix}im sigmaend{bmatrix}start{bmatrix}kl rhoend{bmatrix}-begin{bmatrix}il sigmaend{bmatrix}start{bmatrix}km rhoend{bmatrix}biggr{)},+ ∑ start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT ( [ start_ARG start_ROW start_CELL italic_i italic_m end_CELL end_ROW start_ROW start_CELL italic_σ end_CELL end_ROW end_ARG ] [ start_ARG start_ROW start_CELL italic_k italic_l end_CELL end_ROW start_ROW start_CELL italic_ρ end_CELL end_ROW end_ARG ] – [ start_ARG start_ROW start_CELL italic_i italic_l end_CELL end_ROW start_ROW start_CELL italic_σ end_CELL end_ROW end_ARG ] [ start_ARG start_ROW start_CELL italic_k italic_m end_CELL end_ROW start_ROW start_CELL italic_ρ end_CELL end_ROW end_ARG ] ) ,

subsequent to the phrases: “Grossmann tensor of fourth manifold.”

Refer to caption
Determine 4: Prime portion of web page 14L of Einstein’s “Zurich Pocket book” (AEA 3-006). This pocket book paperwork the seek for a generalized concept of relativity and a concept of gravitation throughout the interval of collaboration between Grossmann and Einstein. This web page reveals that Grossmann launched Einstein to the Riemann tensor as a mathematical useful resource for the final concept of relativity. ©The Hebrew College of Jerusalem, Albert Einstein Archives.

Since an object was wanted that had solely two free indices, Grossmann contracted the
Riemann tensor as soon as. This amounted to the next operation:

Gim=okaylγokayl(iokay,lm)=okay(iokay,okaym)subscript𝐺𝑖𝑚subscript𝑘𝑙subscript𝛾𝑘𝑙𝑖𝑘𝑙𝑚subscript𝑘𝑖𝑘𝑘𝑚G_{im}=sum_{kl}gamma_{kl}(ik,lm)=sum_{okay}(ik,km)italic_G start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ( italic_i italic_k , italic_l italic_m ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_i italic_k , italic_k italic_m ) (22)

and gave an expression that we now name the Ricci tensor.
The issue now was that should you kind the Ricci tensor like this and also you take a look at the
second by-product phrases, you discover the next phrases

Gim=okay(2gimxokay2+2gokayokayxixm2giokayxokayxm2gokaymxokayxi)+subscript𝐺𝑖𝑚subscript𝑘superscript2subscript𝑔𝑖𝑚superscriptsubscript𝑥𝑘2superscript2subscript𝑔𝑘𝑘subscript𝑥𝑖subscript𝑥𝑚superscript2subscript𝑔𝑖𝑘subscript𝑥𝑘subscript𝑥𝑚superscript2subscript𝑔𝑘𝑚subscript𝑥𝑘subscript𝑥𝑖G_{im}=sum_{okay}biggl{(}frac{partial^{2}g_{im}}{partial x_{okay}^{2}}+frac{partial^{2}g_{kk}}{partial x_{i}partial x_{m}}-frac{partial^{2}g_{ik}}{partial x_{okay}partial x_{m}}-frac{partial^{2}g_{km}}{partial x_{okay}partial x_{i}}biggr{)}+dotsitalic_G start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_k italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG – divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG – divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_k italic_m end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) + … (23)

Within the restrict that

gim=(100001000010000c2)+him+𝒪(hij2,(hij)2)subscript𝑔𝑖𝑚matrix100001000010000superscript𝑐2subscript𝑖𝑚𝒪superscriptsubscript𝑖𝑗2superscriptsubscript𝑖𝑗2g_{im}=start{pmatrix}1&0&0&0 0&1&0&0 0&0&1&0 0&0&0&-c^{2}finish{pmatrix}+h_{im}+mathcal{O}(h_{ij}^{2},(partial h_{ij})^{2})italic_g start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL – italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ) + italic_h start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT + caligraphic_O ( italic_h start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ( ∂ italic_h start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) (24)

the primary of the second by-product phrases in (23) reduces to the d’Alembertian (x4=ictsubscript𝑥4𝑖𝑐𝑡x_{4}=ictitalic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_i italic_c italic_t)

μ=142xμ2superscriptsubscript𝜇14superscript2superscriptsubscript𝑥𝜇2Boxequivsum_{mu=1}^{4}frac{partial^{2}}{partial x_{mu}^{2}}□ ≡ ∑ start_POSTSUBSCRIPT italic_μ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG (25)

however the different three second by-product phrases don’t vanish or tackle easy types which have
an affordable bodily interpretation. Einstein and Grossmann reasoned that these three phrases
“ought to vanish”.

Additionally they discovered a approach to make these phrases vanish. They imposed
a restrictive situation, allow us to name it the harmonic coordinate restriction, of the shape

okaylγokayl[kli]=okaylγokayl(2gilxokaygokaylxi)=0.subscript𝑘𝑙subscript𝛾𝑘𝑙matrix𝑘𝑙𝑖subscript𝑘𝑙subscript𝛾𝑘𝑙2subscript𝑔𝑖𝑙subscript𝑥𝑘subscript𝑔𝑘𝑙subscript𝑥𝑖0sum_{kl}gamma_{kl}start{bmatrix}kl iend{bmatrix}=sum_{kl}gamma_{kl}biggl{(}2frac{partial g_{il}}{partial x_{okay}}-frac{partial g_{kl}}{partial x_{i}}biggr{)}=0.∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT [ start_ARG start_ROW start_CELL italic_k italic_l end_CELL end_ROW start_ROW start_CELL italic_i end_CELL end_ROW end_ARG ] = ∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ( 2 divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG – divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) = 0 . (26)

On p. 19L of the Zurich Pocket book, it’s proven that with this restriction, the Ricci tensor
reduces to

2G~im=okayl(γokayl2gimxokayxldisplaystyle 2tilde{G}_{im}=sum_{kl}biggl{(}gamma_{kl}frac{partial^{2}g_{im}}{partial x_{okay}partial x_{l}}2 over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG 12γokaylxmgokaylxi+γokaylxmgilxokay+γokaylxigmokayxl)displaystyle-frac{1}{2}frac{partialgamma_{kl}}{partial x_{m}}frac{partial g_{kl}}{partial x_{i}}+frac{partialgamma_{kl}}{partial x_{m}}frac{partial g_{il}}{partial x_{okay}}+frac{partialgamma_{kl}}{partial x_{i}}frac{partial g_{mk}}{partial x_{l}}biggr{)}– divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_m italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG )
okaylρσ(γρσγokaylgiρxlgmσxokay+γρσγokaylgilxρgmσxokay).subscript𝑘𝑙𝜌𝜎subscript𝛾𝜌𝜎subscript𝛾𝑘𝑙subscript𝑔𝑖𝜌subscript𝑥𝑙subscript𝑔𝑚𝜎subscript𝑥𝑘subscript𝛾𝜌𝜎subscript𝛾𝑘𝑙subscript𝑔𝑖𝑙subscript𝑥𝜌subscript𝑔𝑚𝜎subscript𝑥𝑘displaystyle-sum_{klrhosigma}biggl{(}gamma_{rhosigma}gamma_{kl}frac{partial g_{irho}}{partial x_{l}}frac{partial g_{msigma}}{partial x_{okay}}+gamma_{rhosigma}gamma_{kl}frac{partial g_{il}}{partial x_{rho}}frac{partial g_{msigma}}{partial x_{okay}}biggr{)}.– ∑ start_POSTSUBSCRIPT italic_k italic_l italic_ρ italic_σ end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_m italic_σ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG + italic_γ start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_m italic_σ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ) . (27)

This expression is manifestly of the shape that within the weak subject restrict (24)
it reduces to the d’Alembertian as
anticipated. Within the Zurich Pocket book, Einstein noticed that this consequence was “safe; legitimate for
coordinates that fulfill the Eq. Δφ=0Δ𝜑0Deltavarphi=0roman_Δ italic_φ = 0.”

The difficulty with this reasoning was that Einstein and Grossmann at this level have been nonetheless
on the lookout for a illustration of ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT. Since Gimsubscript𝐺𝑖𝑚G_{im}italic_G start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT didn’t produce the right restrict
they have been now contemplating G~imsubscript~𝐺𝑖𝑚tilde{G}_{im}over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT as a candidate for ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT. However the covariance group
of G~imsubscript~𝐺𝑖𝑚tilde{G}_{im}over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT was restricted by the validity of the harmonic coordinate restriction.

Einstein subsequently needed to discover a physics justification for the harmonic coordinate restriction. This proved
to be a deadly stumbling block for the expression G~imsubscript~𝐺𝑖𝑚tilde{G}_{im}over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT as a candidate for ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT. He briefly conjectured that the harmonic restriction (26) would maybe cut up
into the 2 circumstances

okaylγokaylgilxokay=okaylgilγokaylxokay=0subscript𝑘𝑙subscript𝛾𝑘𝑙subscript𝑔𝑖𝑙subscript𝑥𝑘subscript𝑘𝑙subscript𝑔𝑖𝑙subscript𝛾𝑘𝑙subscript𝑥𝑘0sum_{kl}gamma_{kl}frac{partial g_{il}}{partial x_{okay}}=-sum_{kl}g_{il}frac{partialgamma_{kl}}{partial x_{okay}}=0∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG = – ∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG = 0 (28)

and

okaylγokaylgokaylxi=0.subscript𝑘𝑙subscript𝛾𝑘𝑙subscript𝑔𝑘𝑙subscript𝑥𝑖0sum_{kl}gamma_{kl}frac{partial g_{kl}}{partial x_{i}}=0.∑ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = 0 . (29)

Wanting on the weak subject restrict of those two circumstances they discovered that the second situation amounted to the concept that the hint of the weak subject metric be fixed, a situation clearly violated by the
static subject metric (5). In an advert hoc transfer to treatment these difficulties, Einstein added a hint time period to the weak subject restrict of the sphere equation, thus successfully writing down the weak subject type of the ultimate subject equations of normal relativity. However, alas, once more Einstein discovered an issue within the interpretation of those equations when he confronted them together with his additional heuristic necessities.78, 1

But, Grossmann confirmed Einstein a second approach to do away with the undesirable second by-product phrases of Gimsubscript𝐺𝑖𝑚G_{im}italic_G start_POSTSUBSCRIPT italic_i italic_m end_POSTSUBSCRIPT.

Refer to caption
Determine 5: Prime portion of web page 22R of the “Zurich Pocket book” (AEA 3-006). Grossmann, on the time of their collaboration, additionally confirmed Einstein a approach to extract a decreased amount from the Riemann tensor that transforms as a tensor beneath unimodular coordinate transformations. Though given up on the time, this specific tensor reappeared three years later as a candidate gravitation tensor within the first of Einstein’s well-known 4 memoirs of November 1915 which mark the breakthrough to the ultimate concept of normal relativity. ©The Hebrew College of Jerusalem, Albert Einstein Archives.

On p. 22R of the Zurich Pocket book, we discover Grossmann’s title once more subsequent to a candidate gravitation tensor 1 p. 451, see Fig. 5.
The technique was the identical. This time the restrictive situation to be imposed on the Ricci tensor was that the determinant of the metric
transforms as a scalar. As well as, a situation just like (28) was assumed to carry.
Grossmann informed Einstein to put in writing the covariant Ricci tensor within the following kind (see Fig. 5)

Gil=okay{iokayokay}xl¯{ilokay}xokay+okayλ{iokayλ}{λlokay}{ilλ}{λokayokay}¯,subscript𝐺𝑖𝑙subscript𝑘¯matrix𝑖𝑘𝑘subscript𝑥𝑙matrix𝑖𝑙𝑘subscript𝑥𝑘subscript𝑘𝜆matrix𝑖𝑘𝜆matrix𝜆𝑙𝑘¯matrix𝑖𝑙𝜆matrix𝜆𝑘𝑘G_{il}=sum_{okay}underline{frac{partialleft{start{matrix}ik kend{matrix}proper}}{partial x_{l}}}-frac{partialleft{start{matrix}il kend{matrix}proper}}{partial x_{okay}}+sum_{klambda}left{start{matrix}ik lambdaend{matrix}proper}left{start{matrix}lambda l kend{matrix}proper}-underline{left{start{matrix}il lambdaend{matrix}proper}left{start{matrix}lambda okay kend{matrix}proper}},italic_G start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT beneath¯ start_ARG divide start_ARG ∂ { start_ARG start_ROW start_CELL italic_i italic_k end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG } end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG end_ARG – divide start_ARG ∂ { start_ARG start_ROW start_CELL italic_i italic_l end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG } end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG + ∑ start_POSTSUBSCRIPT italic_k italic_λ end_POSTSUBSCRIPT { start_ARG start_ROW start_CELL italic_i italic_k end_CELL end_ROW start_ROW start_CELL italic_λ end_CELL end_ROW end_ARG } { start_ARG start_ROW start_CELL italic_λ italic_l end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG } – beneath¯ start_ARG { start_ARG start_ROW start_CELL italic_i italic_l end_CELL end_ROW start_ROW start_CELL italic_λ end_CELL end_ROW end_ARG } { start_ARG start_ROW start_CELL italic_λ italic_k end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG } end_ARG , (30)

the place

{iokayλ}=μγλμ[ikμ]matrix𝑖𝑘𝜆subscript𝜇subscript𝛾𝜆𝜇matrix𝑖𝑘𝜇left{start{matrix}ik lambdaend{matrix}proper}=sum_{mu}gamma_{lambdamu}start{bmatrix}ik muend{bmatrix}{ start_ARG start_ROW start_CELL italic_i italic_k end_CELL end_ROW start_ROW start_CELL italic_λ end_CELL end_ROW end_ARG } = ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_λ italic_μ end_POSTSUBSCRIPT [ start_ARG start_ROW start_CELL italic_i italic_k end_CELL end_ROW start_ROW start_CELL italic_μ end_CELL end_ROW end_ARG ] (31)

are the Christoffel symbols of the second form.
Since

Tiokay{iokayokay}=1glngxisubscript𝑇𝑖subscript𝑘matrix𝑖𝑘𝑘1𝑔𝑔subscript𝑥𝑖T_{i}equivsum_{okay}left{start{matrix}ik kend{matrix}proper}=frac{1}{sqrt{-g}}frac{partiallnsqrt{-g}}{partial x_{i}}italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≡ ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT { start_ARG start_ROW start_CELL italic_i italic_k end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG } = divide start_ARG 1 end_ARG start_ARG square-root start_ARG – italic_g end_ARG end_ARG divide start_ARG ∂ roman_ln square-root start_ARG – italic_g end_ARG end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG (32)

Grossmann argued that if g𝑔gitalic_g transforms as a scalar, then Tisubscript𝑇𝑖T_{i}italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT transforms as a vector (a “tensor of
1stst{}^{rm st}start_FLOATSUPERSCRIPT roman_st end_FLOATSUPERSCRIPTrank”), and therefore the underlined phrases in (30) signify the covariant by-product of a
vector and subsequently remodel as a second-rank tensor beneath unimodular transformations. This implies
that the remaining two phrases in (30)

Gilokayxokay{ilokay}+okayλ{iokayλ}{λlokay}superscriptsubscript𝐺𝑖𝑙subscript𝑘subscript𝑥𝑘matrix𝑖𝑙𝑘subscript𝑘𝜆matrix𝑖𝑘𝜆matrix𝜆𝑙𝑘G_{il}^{ast}equiv-sum_{okay}frac{partial}{partial x_{okay}}left{start{matrix}il kend{matrix}proper}+sum_{klambda}left{start{matrix}ik lambdaend{matrix}proper}left{start{matrix}lambda l kend{matrix}proper}italic_G start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≡ – ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG { start_ARG start_ROW start_CELL italic_i italic_l end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG } + ∑ start_POSTSUBSCRIPT italic_k italic_λ end_POSTSUBSCRIPT { start_ARG start_ROW start_CELL italic_i italic_k end_CELL end_ROW start_ROW start_CELL italic_λ end_CELL end_ROW end_ARG } { start_ARG start_ROW start_CELL italic_λ italic_l end_CELL end_ROW start_ROW start_CELL italic_k end_CELL end_ROW end_ARG } (33)

additionally remodel as a second-rank tensor beneath this restricted group of coordinate transformations.
Increasing the Christoffel symbols, we discover that

Gil=κα12xκ(γκα(giαxl+glαxigilxα))+κλ{iκλ}{λlκ}.subscriptsuperscript𝐺𝑖𝑙subscript𝜅𝛼12subscript𝑥𝜅subscript𝛾𝜅𝛼subscript𝑔𝑖𝛼subscript𝑥𝑙subscript𝑔𝑙𝛼subscript𝑥𝑖subscript𝑔𝑖𝑙subscript𝑥𝛼subscript𝜅𝜆matrix𝑖𝜅𝜆matrix𝜆𝑙𝜅G^{ast}_{il}=sum_{kappaalpha}frac{1}{2}frac{partial}{partial x_{kappa}}left(gamma_{kappaalpha}left(frac{partial g_{ialpha}}{partial x_{l}}+frac{partial g_{lalpha}}{partial x_{i}}-frac{partial g_{il}}{partial x_{alpha}}proper)proper)+sum_{kappalambda}left{start{matrix}ikappa lambdaend{matrix}proper}left{start{matrix}lambda l kappaend{matrix}proper}.italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_κ italic_α end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_κ end_POSTSUBSCRIPT end_ARG ( italic_γ start_POSTSUBSCRIPT italic_κ italic_α end_POSTSUBSCRIPT ( divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_α end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_l italic_α end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG – divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ) ) + ∑ start_POSTSUBSCRIPT italic_κ italic_λ end_POSTSUBSCRIPT { start_ARG start_ROW start_CELL italic_i italic_κ end_CELL end_ROW start_ROW start_CELL italic_λ end_CELL end_ROW end_ARG } { start_ARG start_ROW start_CELL italic_λ italic_l end_CELL end_ROW start_ROW start_CELL italic_κ end_CELL end_ROW end_ARG } . (34)

Assuming additional that the situation

κγκαxκ0subscript𝜅subscript𝛾𝜅𝛼subscript𝑥𝜅0sum_{kappa}frac{partialgamma_{kappaalpha}}{partial x_{kappa}}equiv 0∑ start_POSTSUBSCRIPT italic_κ end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_κ italic_α end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_κ end_POSTSUBSCRIPT end_ARG ≡ 0 (35)

holds, and utilizing

αγκαgiαxl=αgiαγκαxlsubscript𝛼subscript𝛾𝜅𝛼subscript𝑔𝑖𝛼subscript𝑥𝑙subscript𝛼subscript𝑔𝑖𝛼subscript𝛾𝜅𝛼subscript𝑥𝑙sum_{alpha}gamma_{kappaalpha}frac{partial g_{ialpha}}{partial x_{l}}=-sum_{alpha}g_{ialpha}frac{partialgamma_{kappaalpha}}{partial x_{l}}∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_κ italic_α end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_i italic_α end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG = – ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_α end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_κ italic_α end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG (36)

the contravariant metric might be pulled exterior the by-product and Gilsubscriptsuperscript𝐺𝑖𝑙G^{ast}_{il}italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT turns
into

G~il=12κγκα2gilxκxα+κλ{iκλ}{λlκ}.subscriptsuperscript~𝐺𝑖𝑙12subscript𝜅subscript𝛾𝜅𝛼superscript2subscript𝑔𝑖𝑙subscript𝑥𝜅subscript𝑥𝛼subscript𝜅𝜆matrix𝑖𝜅𝜆matrix𝜆𝑙𝜅tilde{G}^{ast}_{il}=frac{1}{2}sum_{kappa}gamma_{kappaalpha}frac{partial^{2}g_{il}}{partial x_{kappa}partial x_{alpha}}+sum_{kappalambda}left{start{matrix}ikappa lambdaend{matrix}proper}left{start{matrix}lambda l kappaend{matrix}proper}.over~ start_ARG italic_G end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_κ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_κ italic_α end_POSTSUBSCRIPT divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_κ end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG + ∑ start_POSTSUBSCRIPT italic_κ italic_λ end_POSTSUBSCRIPT { start_ARG start_ROW start_CELL italic_i italic_κ end_CELL end_ROW start_ROW start_CELL italic_λ end_CELL end_ROW end_ARG } { start_ARG start_ROW start_CELL italic_λ italic_l end_CELL end_ROW start_ROW start_CELL italic_κ end_CELL end_ROW end_ARG } . (37)

This once more was of the specified type of a single second by-product time period which reduces to the
d’Alembertian for weak fields and to the Laplacian for weak static fields plus phrases quadratic
within the derivatives of the metric which vanish within the weak subject restrict. However, once more, the
derivation of this decreased “gravitation tensor” got here at the price of stipulating two extra
restrictive circumstances, the unimodularity situation and situation (35). However, once more,
the bodily interpretation of those two restrictions proved inconceivable for Einstein and
Grossmann, and they also discarded this method as properly.

Within the Zurich Pocket book, the identical technique of deriving a gravitation tensor of the shape

Γil=αβγκα2gilxαxβ+𝒪(g)2,subscriptΓ𝑖𝑙subscript𝛼𝛽subscript𝛾𝜅𝛼superscript2subscript𝑔𝑖𝑙subscript𝑥𝛼subscript𝑥𝛽𝒪superscript𝑔2Gamma_{il}=sum_{alphabeta}gamma_{kappaalpha}frac{partial^{2}g_{il}}{partial x_{alpha}partial x_{beta}}+mathcal{O}(partial g)^{2},roman_Γ start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_κ italic_α end_POSTSUBSCRIPT divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG + caligraphic_O ( ∂ italic_g ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (38)

was explored some extra with but totally different restrictive circumstances however none of these makes an attempt
proved possible, and it’s unclear whether or not Grossmann had his hand in any of the extra outlandish
makes an attempt alongside these strains which can be recorded within the Zurich Pocket book.

When the “mathematical” technique of exploring the Riemann tensor as a useful resource for a derivation
of an acceptable gravitation tensor ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT was exhausted, Einstein and Grossmann,
in a transfer of reflection, altered their technique. All alongside, their first criterion in evaluating
the feasibility of candidate gravitation tensors was whether or not it was of the
kind (38), the place the
phrases quadratic within the first derivatives have been to be decided by suitably proscribing the
Riemann tensor for his or her wants.

One other heuristic requirement for the specified subject equations arose from the covariant divergence
equation for the stress-energy tensor (11). This relation implied a requirement on the sphere equations as a result of along with a subject equation of the shape (9) it implied

(gΓμν);ν=0,left(sqrt{-g},Gamma^{munu}proper)_{;nu}=0,( square-root start_ARG – italic_g end_ARG roman_Γ start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT ; italic_ν end_POSTSUBSCRIPT = 0 , (39)

as certainly it does within the remaining concept the place ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT is instantiated by the Einstein tensor.
Right this moment the contracted Bianchi identification (39) provides
a direct trace as to what the gravitation tensor ought to be. However keep in mind, the Ricci tensor Gilsubscript𝐺𝑖𝑙G_{il}italic_G start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT and its numerous reductions G~ilsubscript~𝐺𝑖𝑙tilde{G}_{il}over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT, Gilsubscriptsuperscript𝐺𝑖𝑙G^{ast}_{il}italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT, G~ilsubscriptsuperscript~𝐺𝑖𝑙tilde{G}^{ast}_{il}over~ start_ARG italic_G end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_l end_POSTSUBSCRIPT (cp. Eqs. (22), (27), (33), (37)), or
the Einstein tensor, for that matter, had already been excluded for causes of violating the anticipated
habits within the weak static restrict. The energy-momentum conservation (11)
implied a heuristic requirement on the gravitational subject equations as a result of its particular person phrases have been
interpreted realistically. Einstein argued that within the absence of any gravitational subject, the
relation reduces to the divergence relation

Tμν;ν=0.{T^{munu}}_{;nu}=0.italic_T start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ; italic_ν end_POSTSUBSCRIPT = 0 . (40)

Consequently, the time period

12ggμνxσΘμν12𝑔subscript𝑔𝜇𝜈subscript𝑥𝜎subscriptΘ𝜇𝜈frac{1}{2}sumsqrt{-g},frac{partial g_{munu}}{partial x_{sigma}},Theta_{munu}divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT (41)

was interpreted because the gravitational power density exerted on the ponderable matter ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT by the gravitational subject
gμν/xσsubscript𝑔𝜇𝜈subscript𝑥𝜎{partial g_{munu}}/{partial x_{sigma}}∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT / ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT .
This implied that on utilizing the sphere equation (9) the time period

12ggμνxσΓμν12𝑔subscript𝑔𝜇𝜈subscript𝑥𝜎subscriptΓ𝜇𝜈frac{1}{2}sumsqrt{-g},frac{partial g_{munu}}{partial x_{sigma}}Gamma_{munu}divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT (42)

needed to be expressed as a coordinate divergence in an effort to assure that vitality and momentum
conservation not be violated. The reasoning right here was to some extent by analogy.

In electrostatics, Einstein argued, the momentum transferred onto ponderable matter of cost density ρ𝜌rhoitalic_ρ by an electrical subject asscoiated with an electrostatic potential φ𝜑varphiitalic_φ is expressed by
φ/xν𝜑subscript𝑥𝜈-{partialvarphi}/{partial x_{nu}}– ∂ italic_φ / ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT. The Poisson equation,

Δφ=ρ,Δ𝜑𝜌Deltavarphi=rho,roman_Δ italic_φ = italic_ρ , (43)

ensures conservation of momentum by advantage of the identification

φxνμ2φxμ2=μxμ(φxνφxμ)xν(12μ(φxμ)2).𝜑subscript𝑥𝜈subscript𝜇superscript2𝜑superscriptsubscript𝑥𝜇2subscript𝜇subscript𝑥𝜇𝜑subscript𝑥𝜈𝜑subscript𝑥𝜇subscript𝑥𝜈12subscript𝜇superscript𝜑subscript𝑥𝜇2frac{partialvarphi}{partial x_{nu}}sum_{mu}frac{partial^{2}varphi}{partial x_{mu}^{2}}=sum_{mu}frac{partial}{partial x_{mu}}left(frac{partialvarphi}{partial x_{nu}}frac{partialvarphi}{partial x_{mu}}proper)-frac{partial}{partial x_{nu}}left(frac{1}{2}sum_{mu}left(frac{partialvarphi}{partial x_{mu}}proper)^{2}proper).divide start_ARG ∂ italic_φ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_φ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG ( divide start_ARG ∂ italic_φ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_φ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG ) – divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( divide start_ARG ∂ italic_φ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . (44)

By analogy, Einstein and Grossmann now tried to assemble the primary by-product phrases in such a manner
{that a} relation of the shape

Sum of differential quotients
=displaystyle== 12μνggμνxσ{αβxα(γαβγμνxβ)displaystylefrac{1}{2}sum_{munu}sqrt{-g},frac{partial g_{munu}}{partial x_{sigma}}biggl{{}sum_{alphabeta}frac{partial}{partial x_{alpha}}left(gamma_{alphabeta}frac{partialgamma_{munu}}{partial x_{beta}}proper)divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG { ∑ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ( italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG )
+additional phrases which vanish with the primary approximation}.displaystyle+textual content{additional phrases which vanish with the primary approximation}biggr{}}.+ additional phrases which vanish with the primary approximation } . (45)

Whereas the heuristics of this reasoning was primarily based on analogy, Grossmann offered Einstein with
a rigorous identification. He took the expression

U=αβμνgμνxσxα(gγαβγμνxβ)𝑈subscript𝛼𝛽𝜇𝜈subscript𝑔𝜇𝜈subscript𝑥𝜎subscript𝑥𝛼𝑔subscript𝛾𝛼𝛽subscript𝛾𝜇𝜈subscript𝑥𝛽U=sum_{alphabetamunu}frac{partial g_{munu}}{partial x_{sigma}}frac{partial}{partial x_{alpha}}left(sqrt{-g},gamma_{alphabeta}frac{partialgamma_{munu}}{partial x_{beta}}proper)italic_U = ∑ start_POSTSUBSCRIPT italic_α italic_β italic_μ italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ( square-root start_ARG – italic_g end_ARG italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG ) (46)

and reworked it, utilizing partial integration and the relations
g,σ=(1/2)ggiokaygiokay,σsqrt{-g}_{,sigma}=(1/2)sqrt{-g},g^{ik}g_{ik,sigma}square-root start_ARG – italic_g end_ARG start_POSTSUBSCRIPT , italic_σ end_POSTSUBSCRIPT = ( 1 / 2 ) square-root start_ARG – italic_g end_ARG italic_g start_POSTSUPERSCRIPT italic_i italic_k end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i italic_k , italic_σ end_POSTSUBSCRIPT
and
grs,l=grρgsσgρσ,l{g^{rs}}_{,l}=-g^{rrho}g^{ssigma}g_{rhosigma,l}italic_g start_POSTSUPERSCRIPT italic_r italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT , italic_l end_POSTSUBSCRIPT = – italic_g start_POSTSUPERSCRIPT italic_r italic_ρ end_POSTSUPERSCRIPT italic_g start_POSTSUPERSCRIPT italic_s italic_σ end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_ρ italic_σ , italic_l end_POSTSUBSCRIPT,
to indicate that the next identification holds

αβτρsubscript𝛼𝛽𝜏𝜌displaystylesum_{alphabetataurho}∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT xα(gγαβγτρxβgτρxσ)12αβτρxσ(gγαβγτρxαgτρxβ)subscript𝑥𝛼𝑔subscript𝛾𝛼𝛽subscript𝛾𝜏𝜌subscript𝑥𝛽subscript𝑔𝜏𝜌subscript𝑥𝜎12subscript𝛼𝛽𝜏𝜌subscript𝑥𝜎𝑔subscript𝛾𝛼𝛽subscript𝛾𝜏𝜌subscript𝑥𝛼subscript𝑔𝜏𝜌subscript𝑥𝛽displaystylefrac{partial}{partial x_{alpha}}biggl{(}sqrt{-g},gamma_{alphabeta}frac{partialgamma_{taurho}}{partial x_{beta}}frac{partial g_{taurho}}{partial x_{sigma}}biggr{)}-frac{1}{2}sum_{alphabetataurho}frac{partial}{partial x_{sigma}}biggl{(}sqrt{-g},gamma_{alphabeta}frac{partialgamma_{taurho}}{partial x_{alpha}}frac{partial g_{taurho}}{partial x_{beta}}biggr{)}divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ( square-root start_ARG – italic_g end_ARG italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG ) – divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG ( square-root start_ARG – italic_g end_ARG italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG )
=displaystyle== μνggμνxσ{αβ1gxα(γαβgγμνxβ)αβτργαβgτργμτxαγτρxβdisplaystylesum_{munu}sqrt{-g},frac{partial g_{munu}}{partial x_{sigma}}biggl{{}sum_{alphabeta}frac{1}{sqrt{-g}}frac{partial}{partial x_{alpha}}biggl{(}gamma_{alphabeta}sqrt{-g},frac{partialgamma_{munu}}{partial x_{beta}}biggr{)}-sum_{alphabetataurho}gamma_{alphabeta}g_{taurho}frac{partialgamma_{mutau}}{partial x_{alpha}}frac{partialgamma_{taurho}}{partial x_{beta}}∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG { ∑ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG square-root start_ARG – italic_g end_ARG end_ARG divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ( italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG ) – ∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_τ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG
+12αβτργαμγβνgτρxαγτρxβ14αβτργμνγαβgτρxαγτρxβ}.displaystyle+frac{1}{2}sum_{alphabetataurho}gamma_{alphamu}gamma_{betanu}frac{partial g_{taurho}}{partial x_{alpha}}frac{partialgamma_{taurho}}{partial x_{beta}}-frac{1}{4}sum_{alphabetataurho}gamma_{munu}gamma_{alphabeta}frac{partial g_{taurho}}{partial x_{alpha}}frac{partialgamma_{taurho}}{partial x_{beta}}biggr{}}.+ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_α italic_μ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_β italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG – divide start_ARG 1 end_ARG start_ARG 4 end_ARG ∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG } . (47)

With the intention to interpret this identification, Einstein and Grossmann launched a differential operator

Δμν(γ)=αβ1gxα(γαβgγμνxβ)αβτργαβgτργμτxαγνρxβsubscriptΔ𝜇𝜈𝛾subscript𝛼𝛽1𝑔subscript𝑥𝛼subscript𝛾𝛼𝛽𝑔subscript𝛾𝜇𝜈subscript𝑥𝛽subscript𝛼𝛽𝜏𝜌subscript𝛾𝛼𝛽subscript𝑔𝜏𝜌subscript𝛾𝜇𝜏subscript𝑥𝛼subscript𝛾𝜈𝜌subscript𝑥𝛽Delta_{munu}(gamma)=sum_{alphabeta}frac{1}{sqrt{-g}}cdotfrac{partial}{partial x_{alpha}}biggl{(}gamma_{alphabeta}sqrt{-g}cdotfrac{partialgamma_{munu}}{partial x_{beta}}biggr{)}-sum_{alphabetataurho}gamma_{alphabeta}g_{taurho}frac{partialgamma_{mutau}}{partial x_{alpha}}frac{partialgamma_{nurho}}{partial x_{beta}}roman_Δ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_γ ) = ∑ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG square-root start_ARG – italic_g end_ARG end_ARG ⋅ divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ( italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG ⋅ divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG ) – ∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_τ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_ν italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG (48)

and a gravitational stress-energy tensor

2κϑμν=αβτρ(γαμγβνgτρxαγτρxβ12γμνγαβgτρxαγτρxβ)2𝜅subscriptitalic-ϑ𝜇𝜈subscript𝛼𝛽𝜏𝜌subscript𝛾𝛼𝜇subscript𝛾𝛽𝜈subscript𝑔𝜏𝜌subscript𝑥𝛼subscript𝛾𝜏𝜌subscript𝑥𝛽12subscript𝛾𝜇𝜈subscript𝛾𝛼𝛽subscript𝑔𝜏𝜌subscript𝑥𝛼subscript𝛾𝜏𝜌subscript𝑥𝛽-2kappacdotvartheta_{munu}=sum_{alphabetataurho}biggl{(}gamma_{alphamu}gamma_{betanu}frac{partial g_{taurho}}{partial x_{alpha}}frac{partialgamma_{taurho}}{partial x_{beta}}-frac{1}{2}gamma_{munu}gamma_{alphabeta}frac{partial g_{taurho}}{partial x_{alpha}}frac{partialgamma_{taurho}}{partial x_{beta}}biggr{)}– 2 italic_κ ⋅ italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_α italic_μ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_β italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG – divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG ) (49)

and rewrote Grossmann’s identification (47) within the kind

μνxν(ggσμϑμν)12μνggμνxσϑμν=12κμνggμνxσΔμν(γ).subscript𝜇𝜈subscript𝑥𝜈𝑔subscript𝑔𝜎𝜇subscriptitalic-ϑ𝜇𝜈12subscript𝜇𝜈𝑔subscript𝑔𝜇𝜈subscript𝑥𝜎subscriptitalic-ϑ𝜇𝜈12𝜅subscript𝜇𝜈𝑔subscript𝑔𝜇𝜈subscript𝑥𝜎subscriptΔ𝜇𝜈𝛾sum_{munu}frac{partial}{partial x_{nu}}left(sqrt{-g},g_{sigmamu}vartheta_{munu}proper)-frac{1}{2}sum_{munu}sqrt{-g},frac{partial g_{munu}}{partial x_{sigma}}vartheta_{munu}=-frac{1}{2kappa}sum_{munu}sqrt{-g},frac{partial g_{munu}}{partial x_{sigma}}Delta_{munu}(gamma).∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG ( square-root start_ARG – italic_g end_ARG italic_g start_POSTSUBSCRIPT italic_σ italic_μ end_POSTSUBSCRIPT italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ) – divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = – divide start_ARG 1 end_ARG start_ARG 2 italic_κ end_ARG ∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG roman_Δ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_γ ) . (50)

By comparability with the conservation equation for matter (11),
they concluded that the amount
ϑμνsubscriptitalic-ϑ𝜇𝜈vartheta_{munu}italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT performed the function of gravitational stress-energy, and by comparability of (50)
with their heuristic equation (45),
they concluded that the gravitation tensor ΓμνsubscriptΓ𝜇𝜈Gamma_{munu}roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT
coming into the sphere equations (9)
reads

Γμν=Δμν(γ)κϑμν,subscriptΓ𝜇𝜈subscriptΔ𝜇𝜈𝛾𝜅subscriptitalic-ϑ𝜇𝜈Gamma_{munu}=Delta_{munu}(gamma)-kappacdotvartheta_{munu},roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = roman_Δ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_γ ) – italic_κ ⋅ italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT , (51)

which renders the gravitational subject equations within the kind

Δγμ(γ)=κ(Θμν+ϑμν).subscriptΔ𝛾𝜇𝛾𝜅subscriptΘ𝜇𝜈subscriptitalic-ϑ𝜇𝜈Delta_{gammamu}(gamma)=kappaleft(Theta_{munu}+vartheta_{munu}proper).roman_Δ start_POSTSUBSCRIPT italic_γ italic_μ end_POSTSUBSCRIPT ( italic_γ ) = italic_κ ( roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ) . (52)

Eqs. (48), (49), and (52) are the gravitational subject equations of the Einstein-Grossmann concept of their contravariant kind.

From (11) and (50) it additionally follows that

μνxν{ggσμ(Θμν+ϑμν)}=0,subscript𝜇𝜈subscript𝑥𝜈𝑔subscript𝑔𝜎𝜇subscriptΘ𝜇𝜈subscriptitalic-ϑ𝜇𝜈0sum_{munu}frac{partial}{partial x_{nu}}left{sqrt{-g},g_{sigmamu}left(Theta_{munu}+vartheta_{munu}proper)proper}=0,∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG { square-root start_ARG – italic_g end_ARG italic_g start_POSTSUBSCRIPT italic_σ italic_μ end_POSTSUBSCRIPT ( roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ) } = 0 , (53)

a relation expressing, in line with Einstein, the validity of conservation legal guidelines for the union of
matter and gravitational subject.

In just a few months of their collaboration, Einstein and Grossmann had succeeded in formulating a relativistic concept of gravitation, which employed an tailored model of Ricci’s and Levi-Civita’s absolute differential calculus and which was typically covariant in all its components, besides
for the gravitational subject equations. In the middle of their joint work, that they had been considering as candidate gravitation tensors the proper mathematical objects and had even thought of a linearized model of the ultimate subject equations of gravitation.

After their joint work of the Entwurf68 was obtainable in offprint kind (see Fig. 6), Einstein and Grossmann engaged in numerous
actions to promote and promote their outcomes.

Refer to caption
Determine 6: Title web page of the “Define of a Generalized Idea of Relativity and of a Idea of Gravitation” revealed by Einstein and Grossmann in early summer time 1913. This work, during which Einstein signed answerable for the ‘bodily half’ and Grossmann for the ‘mathematical half,’ already contained all parts of the ultimate concept of normal relativity, aside from the right, typically covariant gravitational subject equations. Particularly, it contained an account of tensor calculus geared to the needs of a relativistic concept of gravitation in Grossmann’s mathematical half. The Einstein-Grossmann concept of this “Define” was solely given up by Einstein within the fall of 1915, when he succeeded in his remaining breakthrough to the final concept of relativity.

On 9 September 1913, they introduced their concept to the 96th annual assembly
of the Schweizerische Naturforschende Gesellschaft, which passed off that 12 months in Frauenfeld.
Once more, they introduced the work
with the identical division of labor and accountability. Abstracts of their shows have been
revealed within the society’s Verhandlungen,79, 80
and longer variations have been revealed within the Vierteljahrsschrift of the Naturforschende
Gesellschaft in Zürich
81, 82.
French translations of the latter items appeared in
Archives des sciences physiques et naturelles.83, 84
Einstein alone introduced the work two weeks later, on 23 September 1913,
on the eighty fifth assembly of the Gesellschaft Deutscher Naturforscher und Ärzte in Vienna.
An account of his report was revealed beneath his title within the Physikalische Zeitschrift.85
Dialogue remarks following his presentation have been additionally revealed, however just a few
weeks later, Einstein addressed once more a remark by Hans Reißner (1874–1967), which he felt he had not satisfactorily answered.86 He additionally defended the Einstein-Grossmann concept in opposition to criticism by Gustav Mie (1868–1957).87

In its difficulty of 30 January 1914, the Entwurf was ultimately printed as an everyday article within the
Zeitschrift für Mathematik und Physik,88 a journal edited by the
utilized mathematicians Carl Runge (1856–1927) and Rudolf Mehmke (1857–1944). The journal print is essential for its addendum, signed solely by Einstein, which identified
two latest advances within the understanding of the unique Entwurf concept 5 Doc. 26. For one, Einstein right here formulated
the notorious “gap argument” (“Lochbetrachtung”) by the use of which he had satisfied himself
that typically covariant gravitational subject equations weren’t appropriate with primary assumptions
on causality and the concept that the metric parts are decided uniquely by the matter tensor
ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT.
He additionally gave a model of the fundamental Entwurf equations when it comes to blended tensor densities. Introducing
the portions

𝔗σνsubscript𝔗𝜎𝜈displaystylemathfrak{T}_{sigmanu}fraktur_T start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT =μggσμΘμν,absentsubscript𝜇𝑔subscript𝑔𝜎𝜇subscriptΘ𝜇𝜈displaystyle=sum_{mu}sqrt{-g},g_{sigmamu}Theta_{munu},= ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG italic_g start_POSTSUBSCRIPT italic_σ italic_μ end_POSTSUBSCRIPT roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT , (54)
𝔱σνsubscript𝔱𝜎𝜈displaystylemathfrak{t}_{sigmanu}fraktur_t start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT =μggσμϑμν,absentsubscript𝜇𝑔subscript𝑔𝜎𝜇subscriptitalic-ϑ𝜇𝜈displaystyle=sum_{mu}sqrt{-g},g_{sigmamu}vartheta_{munu},= ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG italic_g start_POSTSUBSCRIPT italic_σ italic_μ end_POSTSUBSCRIPT italic_ϑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT , (55)

the conservation legal guidelines (11) and (53) might be written within the (deceptively) easy kind as

ν𝔗σνxν=12μντgμνxσγμτ𝔗τνsubscript𝜈subscript𝔗𝜎𝜈subscript𝑥𝜈12subscript𝜇𝜈𝜏subscript𝑔𝜇𝜈subscript𝑥𝜎subscript𝛾𝜇𝜏subscript𝔗𝜏𝜈sum_{nu}frac{partialmathfrak{T}_{sigmanu}}{partial x_{nu}}=frac{1}{2}sum_{munutau}frac{partial g_{munu}}{partial x_{sigma}}gamma_{mutau}mathfrak{T}_{taunu}∑ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ fraktur_T start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_μ italic_ν italic_τ end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG italic_γ start_POSTSUBSCRIPT italic_μ italic_τ end_POSTSUBSCRIPT fraktur_T start_POSTSUBSCRIPT italic_τ italic_ν end_POSTSUBSCRIPT (56)

and

νxν(𝔗σν+𝔱σν)=0,subscript𝜈subscript𝑥𝜈subscript𝔗𝜎𝜈subscript𝔱𝜎𝜈0sum_{nu}frac{partial}{partial x_{nu}}left(mathfrak{T}_{sigmanu}+mathfrak{t}_{sigmanu}proper)=0,∑ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARG ( fraktur_T start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT + fraktur_t start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT ) = 0 , (57)

respectively, and the Entwurf subject equations flip into

αβμxα(gγαβgσμγμνxβ)=κ(𝔗σν+𝔱σν).subscript𝛼𝛽𝜇subscript𝑥𝛼𝑔subscript𝛾𝛼𝛽subscript𝑔𝜎𝜇subscript𝛾𝜇𝜈subscript𝑥𝛽𝜅subscript𝔗𝜎𝜈subscript𝔱𝜎𝜈sum_{alphabetamu}frac{partial}{partial x_{alpha}}left(sqrt{-g},gamma_{alphabeta}g_{sigmamu}frac{partialgamma_{munu}}{partial x_{beta}}proper)=kappaleft(mathfrak{T}_{sigmanu}+mathfrak{t}_{sigmanu}proper).∑ start_POSTSUBSCRIPT italic_α italic_β italic_μ end_POSTSUBSCRIPT divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ( square-root start_ARG – italic_g end_ARG italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_σ italic_μ end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG ) = italic_κ ( fraktur_T start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT + fraktur_t start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT ) . (58)

Einstein additionally continued work on the gravitation downside in a collaboration with the Dutch physicist
Adriaan Fokker (1887–1972).
In a joint paper, acquired by Annalen der Physik on 19 February 1914,
they reinterpreted a scalar gravitation
concept by the Finnish physicist Gunnar Nordström (1881–1923) utilizing the instruments of absolutely the differential calculus and in contrast the Nordström concept
to the Einstein-Grossmann concept.90
That comparability was then additionally topic in a presentation that
Einstein gave on 9 February 1914 to the Naturforschende Gesellschaft in Zurich.
Grossmann was current throughout Einstein’s presentation however was energetic solely as a participant within the dialogue.91

Einstein left Zurich on 21 March 1914 24 p. 636 and took up his new place as member
of the Prussian Academy in early April, after a brief go to with Paul Ehrenfest (1880–1943) in Leyden. Einstein’s transfer
to Berlin put an finish to his collaboration with Grossmann. However presumably within the remaining weeks
earlier than leaving Zurich, he and Grossmann penned one other joint publication,
which was revealed on 29 Might 1914 additionally in Zeitschrift für Mathematik und Physik.92

The start line of their second joint paper was the perception that Einstein had formulated within the
addendum to the journal print of the Entwurf. By means of introduction, they recapitulated the
achievement of the sooner Entwurf. That concept, they identified, contained two sorts of equations.
The primary form have been generalizations of equations within the particular concept of relativity, which govern the habits of matter or materials processes usually, for a given
gravitational subject. These equations had been proven to be typically covariant. They’d additionally established
a set of equations that decided the gravitational subject if the portions that decide the fabric processes are thought of as given. This equation was a generalization of Poisson’s equation and there
was no particular relativistic analog to it. They’d not been in a position to decide the covariance group
of these equations. It was solely identified that they might be covariant beneath linear transformations but it surely had
remained unclear which additional coordinate transformations would preserve the type of these subject
equations. The aim of their second observe then was to find out the utmost covariance group for the
Entwurf subject equations.

The primary paragraph gave the fundamental equations of the idea in blended tensor density kind, simply as
Einstein had accomplished within the addendum, see Eqs. (56), (57), and (58) above. The second paragraph reiterated the opening argument. Though first formulated by Einstein, it was right here introduced of their joint paper as a “proof that if an answer for the γμνsubscript𝛾𝜇𝜈gamma_{munu}italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT for given ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT is already identified, then the final covariance of the equations permits for the existence of additional options” 92 p. 218. This mathematical truth would indicate
that “a whole dedication of the elemental tensor γμνsubscript𝛾𝜇𝜈gamma_{munu}italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT of a gravitational subject with given ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT
by a generally-covariant system of equations is inconceivable” 92 p. 217. The outlet argument, simply as earlier Grossmann’s
identification (47), is a central tenet of the Einstein-Grossmann concept, and evidently Grossmann was involved
in regards to the invariant-theoretic penalties that it implied. The proof proceeds like this.
Think about a area (the “gap”) in four-dimensional space-time during which there are not any materials processes, i.e., Θμν=0subscriptΘ𝜇𝜈0Theta_{munu}=0roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = 0. Assume that the metric subject γμν(x)subscript𝛾𝜇𝜈𝑥gamma_{munu}(x)italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ) is uniquely decided by the given subject of Θμν(x)subscriptΘ𝜇𝜈𝑥Theta_{munu}(x)roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ), additionally within the matter-free area. Now introduce new coordinates xsuperscript𝑥x^{prime}italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT such that the brand new coordinates agree with the outdated ones exterior the matter free area and solely differ inside it.
Such a coordinate transformation would produce a reworked metric subject γμν(x)superscriptsubscript𝛾𝜇𝜈superscript𝑥gamma_{munu}^{prime}(x^{prime})italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) inside
the matter-free area, however depart ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT invariant throughout the complete space-time since
exterior the opening now we have x=xsuperscript𝑥𝑥x^{prime}=xitalic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_x and inside now we have Θμν=0=ΘμνsuperscriptsubscriptΘ𝜇𝜈0subscriptΘ𝜇𝜈Theta_{munu}^{prime}=0=Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 0 = roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT.
Typically covariant subject equations then would permit to have γμν(x)superscriptsubscript𝛾𝜇𝜈𝑥gamma_{munu}^{prime}(x)italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ) additionally as an answer and
therefore one obtains two totally different metric fields γμνsuperscriptsubscript𝛾𝜇𝜈gamma_{munu}^{prime}italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, γμνsubscript𝛾𝜇𝜈gamma_{munu}italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT appropriate with one and the identical matter subject ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT.

The model of the opening argument introduced of their joint paper basically reiterated the model that
Einstein had given earlier within the addendum. However the presentation of their joint paper instantly
proceeds to right an inaccurate argument of the addendum. Einstein had earlier believed that
Eq. (57) is essentially solely covariant beneath linear coordinate transformations. Einstein and Grossmann
now identified that this conclusion solely holds “if one assigns tensorial character to the portions 𝔱μν/gsubscript𝔱𝜇𝜈𝑔mathfrak{t}_{munu}/sqrt{-g}fraktur_t start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT / square-root start_ARG – italic_g end_ARG which, it turned out, isn’t justified” 92 p. 218. Plugging Eq. (57) into (58), they as an alternative now argued
that

Bσαβμν2xνxα(gγαβgσμγμνxβ)=0subscript𝐵𝜎subscript𝛼𝛽𝜇𝜈superscript2subscript𝑥𝜈subscript𝑥𝛼𝑔subscript𝛾𝛼𝛽subscript𝑔𝜎𝜇subscript𝛾𝜇𝜈subscript𝑥𝛽0B_{sigma}equivsum_{alphabetamunu}frac{partial^{2}}{partial x_{nu}partial x_{alpha}}biggl{(}sqrt{-g},gamma_{alphabeta}g_{sigmamu}frac{partialgamma_{munu}}{partial x_{beta}}biggr{)}=0italic_B start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ≡ ∑ start_POSTSUBSCRIPT italic_α italic_β italic_μ italic_ν end_POSTSUBSCRIPT divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ( square-root start_ARG – italic_g end_ARG italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_σ italic_μ end_POSTSUBSCRIPT divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG ) = 0 (59)

represents a “actual restriction on the selection of coordinate methods” 92 p. 219.

The majority of the paper was dedicated to giving a variational derivation of the Entwurf subject equations,
which might permit them to make some inferences in regards to the invariant-theoretic properties
of their concept. They confirmed that the sphere equation may very well be represented by the variational
precept

(δH2κμνgTμνδγμν)𝑑τ=0𝛿𝐻2𝜅subscript𝜇𝜈𝑔subscript𝑇𝜇𝜈𝛿subscript𝛾𝜇𝜈differential-d𝜏0intbiggl{(}delta H-2kappasum_{munu}sqrt{-g},T_{munu}deltagamma_{munu}biggr{)}dtau=0∫ ( italic_δ italic_H – 2 italic_κ ∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT square-root start_ARG – italic_g end_ARG italic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_δ italic_γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ) italic_d italic_τ = 0 (60)

with a gravitational Lagrangian

H=12gαβτργαβgτρxαγτρxβ.𝐻12𝑔subscript𝛼𝛽𝜏𝜌subscript𝛾𝛼𝛽subscript𝑔𝜏𝜌subscript𝑥𝛼subscript𝛾𝜏𝜌subscript𝑥𝛽H=frac{1}{2}sqrt{-g},sum_{alphabetataurho}gamma_{alphabeta}frac{partial g_{taurho}}{partial x_{alpha}}frac{partialgamma_{taurho}}{partial x_{beta}}.italic_H = divide start_ARG 1 end_ARG start_ARG 2 end_ARG square-root start_ARG – italic_g end_ARG ∑ start_POSTSUBSCRIPT italic_α italic_β italic_τ italic_ρ end_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT divide start_ARG ∂ italic_g start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_γ start_POSTSUBSCRIPT italic_τ italic_ρ end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT end_ARG . (61)

The core argument of their second paper consists of a proof of the invariance of the variational integral
H𝑑τ𝐻differential-d𝜏int Hdtau∫ italic_H italic_d italic_τ beneath normal coordinate transformations that solely respect the restrictive situation (59). They interpreted their consequence to the impact that the gravitational subject equations
possess the maximal covariance group that’s appropriate with the opening argument.

Their second joint paper solely appeared when Einstein had already moved from Zurich to Berlin to take up
his place as member of the Prussian Academy.
With Einstein’s absence from Zurich the
collaboration between the 2 mates successfully got here to an finish. There is just one thing more that
Grossmann contributed to the Entwurf concept. Already from Berlin, however earlier than 10 April 1914,
Einstein wrote to Paul Ehrenfest:

Grossmann wrote me that now he is also succeeding in deriving the gravitation equations from
the final concept of covariants. This is able to be a pleasant addition to our examination.

Grossmann’s letter, sadly, seems to have been misplaced, and we don’t have every other proof
of Grossmann’s consequence. Apparently, he had discovered a approach to get well the Entwurf subject equations (52) or (58) from the Riemann-Christoffel tensor (21)
utilizing the restrictive situation (59).

To recapitulate: Grossmann’s contribution to the Entwurf concept consisted within the following.

  • He clarified the mathematical basis of the idea primarily based on a normal line factor (8) and generalized the idea of a tensor
    to a construction of nlimit-from𝑛n-italic_n –th rank in m𝑚mitalic_m-dimensional house.

    See Also

  • He recognized absolutely the differential calculus by Ricci and Levi-Civita because the related
    mathematical toolbox for the issue of a relativistic concept of gravitation and reworked it
    right into a tensor calculus each with respect to notation and by generalizing it to blended tensor densities.

  • He proved that the conservation legislation for matter (11) was a typically
    covariant equation
    by displaying that it expresses the covariant divergence of ΘμνsubscriptΘ𝜇𝜈Theta_{munu}roman_Θ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT.

  • He recognized the Riemann tensor as a related and wealthy useful resource for the issue of
    developing typically covariant gravitational subject equations, and he confirmed Einstein a number of
    methods of extracting a second rank object from the Riemann tensor that might have the specified
    limiting kind within the case of weak static fields.

  • After the failure of the mathematical technique of developing a subject equation from the
    Riemann tensor, he proved the central identification (47) from which the gravitational subject equations
    of the Entwurf concept have been learn off.

  • In joint work with Einstein, he confirmed how the Einstein-Grossmann concept might be formulated in phrases
    of a variational precept and clarified its transformational properties in mild of the opening argument.

A couple of months after Einstein had moved to Berlin the battle broke out, a political course of occasions
that contributed to placing an finish to the energetic collaboration between Einstein and Grossmann, even
if their friendship was not affected by the political turmoil. In any case, it was Einstein who
continued to work on the gravitation downside. Within the fall of 1914, he wrote a primary complete
evaluation paper of the Einstein-Grossmann concept, during which he additionally gave a brand new exposition of the related
arithmetic.94 The evaluation begins with giving credit score to Grossmann’s contribution. Einstein wrote:

In recent times, I’ve labored, partially along with my pal Grossmann, on a generalization of the
concept of relativity.

Within the evaluation Einstein referred to the Einstein-Grossmann concept as a “normal concept of relativity” for the primary time within the title, somewhat than calling it a “generalized concept,” because it appeared within the title of the Entwurf. It was the Einstein-Grossmann concept as introduced on this evaluation that Einstein defended in opposition to criticism by the
mathematician Tullio Levi-Civita himself, and which he additionally defended in a course of lectures held in
the summer time of 1915 to the mathematicians and physicists in Göttingen.

In the summertime of 1915, plans have been additionally beneath approach to put together a brand new version of the gathering of papers on the “relativity precept” first edited in 1913 by Otto Blumenthal (1876–1944).96 Apparently, Arnold Sommerfeld who had initiated the primary version of the anthology and who additionally had contributed some annotation to Minkowski’s paper in it, had requested Einstein which of his later works on relativity ought to be included in an augmented second version. In his reply, Einstein talked about the 1914 evaluation paper but in addition prompt that he most popular to have not one of the latest papers included since not one of the expositions of the “normal concept of relativity” could be full and he supposed to put in writing a brand new self-contained presentation anyway. In that context, Einstein additionally commented on his collaboration with this pal:

Grossmann won’t ever lay declare to being co-discoverer. He solely helped in guiding me via the mathematical literature however contributed nothing of substance to the outcomes.

As is well-known, the breakthrough to normal covariance occurred only some months after Einstein’s go to to Göttingen, and was documented in a sequence of 4 memoirs97, 98, 99, 100 introduced to the Prussian Academy, during which
Einstein regained normal covariance,97, 98 succeeded within the computation of Mercury’s anomalous perihelion advance,99 and eventually accomplished his normal concept of relativity by publication of the Einstein equations.100 Within the introductory paragraph of the primary of these November papers, during which he proposed a concept of gravitation primarily based on what now we have referred to as the
“November tensor”1 Vol. 1, p.192, i.e., Eq. (33) above, covariant beneath normal unimodular transformations, Einstein once more talked about his collaboration with Grossmann. He wrote:

Thus I got here again to the concept of a extra normal covariance of the sphere equations, which I had given up three years in the past solely with a heavy coronary heart, once I labored along with my pal Grossmann. Certainly, we had come at the moment already very near the answer that will probably be given within the following.

Very equally, he expressed himself in a letter to David Hilbert, written on 18 November 1915, the day of his third memoir, during which he had succeeded in computing the right worth of Mercury’s perihelion advance on the idea of subject equations RμνTμνproportional-tosubscript𝑅𝜇𝜈subscript𝑇𝜇𝜈R_{munu}propto T_{munu}italic_R start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ∝ italic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT:

The issue was not to find typically covariant equations for the gμνsubscript𝑔𝜇𝜈g_{munu}italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT; for that is simply achieved with the help of Riemann’s tensor. Reasonably it was laborious to see that these equations are a generalization of Newton’s legislation. This perception I solely achieved within the final weeks […], whereas I had already thought of the one potential typically covariant equations, which now turned out to be the right ones, already three years in the past with my pal Grossmann. Solely with a heavy coronary heart did we give them up, because it had appeared to me that their bodily dialogue had proven their incompatibility with Newton’s legislation.

In a letter to his Swiss pal Heinrich Zangger, written on 9 December 1915, only a few days after the ultimate breakthrough, he wrote:

The attention-grabbing factor is that now the inital hypotheses I made with Grossmann are confirmed, and probably the most
radical of theoretical necessities materialized. On the time we lacked only some relations of a proper nature, with out which the hyperlink between the formulation and already identified legal guidelines can’t be attained.

A couple of months after the completion of the final concept of relativity by publication, Einstein revealed a complete exposition of the ultimate concept.101, 7 The paper begins with a page-long introductory paragraph, during which Einstein gave credit score to the mathematical traditions that he had constructed upon, singling out the contributions of Minkowski, in addition to of Gauss, Riemann, Christoffel, Ricci and Levi-Civita. The hommage to the mathematical custom ends with an expression of gratitude for Grossmann:

Lastly I wish to acknowledge gratefully my pal, the mathematician Grossmann, whose assist not solely saved me the trouble of learning the pertinent mathematical literature, however who additionally helped me in my seek for the sphere equations of gravitation.

4 Biographical Epilogue

Though their biographical and mental trajectories continued largely independently after their collaboration in Zurich, Einstein and Grossmann remained mates. Throughout a go to to the Grossmann household in Zurich in summer time 1919, the 2 mates apparently even talked in regards to the chance that Einstein would come again to Zurich69 Vol. 9, 72e, 74d. Grossmann tried to lure Einstein again however Einstein was discouraged by the thought of getting to face a full instructing load once more.

In early January 1920, on the peak of the general public curiosity in Einstein’s concept of normal relativity,
Marcel Grossmann revealed a two-piece article entitled “A New Worldview” within the Neue Schweizer Zeitung.103 Only a few weeks earlier, on 6 November 1919 at a joint session
of the Royal Society and the Royal Astronomical Society in London, it had been introduced that the outcomes of the British eclipse expedition had confirmed Einstein’s relativistic concept of gravitation. After Grossmann put Einstein’s achievement in a line with Galilei, Kopernikus, Kepler, and Newton, he wrote:

As a faculty day pal and fellow pupil of this nice physicist I is likely to be permitted to observe up on the suggestion of the editorial board and provides an understanding of the person and the work to a wider viewers,
and attempt to give an idea of the ingenuity and penalties of his concepts. For years already these concepts have engaged his colleagues however it is just in latest months that additionally in wider mental circles it’s identified that right here a revolution was begun and accomplished of all of our primary ideas in physics, astronomy, geometry, in addition to philosophical epistemology.

Grossmann continued to provide some biographical info on Einstein, identified that he obtained Swiss citizenship as a pupil on the ETH and had stored his Swiss nationality even after his transfer to Berlin. He emphasised Einstein’s pacifist stance throughout the battle, mentioning the counter manifesto 95 Doc. 8 with Georg Friedrich Nicolai (1874–1964) and Wilhelm Förster (1832–1921) in opposition to the notorious manifesto of the 93. He then gave a brief characterization of the particular concept of relativity. His dialogue of particular relativity ends with this remark:

One can clarify all these premises and penalties of the idea of relativity solely to these whose mathematical data and maturity of judgment go far sufficient. The mathematician possesses in his formal language a shorthand of pondering, which isn’t solely helpful however indispensable for extra sophisticated trains of thought. Simply because the ability of shorthand writing helps us following a lecture, it’s the mathematical language of formulation that allows us to create sophisticated trains of thought, which couldn’t be dropped at a conclusion with out it.

With the final concept of relativity, Grossmann wrote, Einstein even topped his personal prior achievement. He not solely generalized particular relativity but in addition succeeded in drawing experimentally verifiable penalties, which certainly had been confirmed. Einstein’s former collaborator, who helped with the arithmetic, then noticed in regards to the function of arithmetic within the genesis of normal relativity:

Laymen have a completely deceptive conception of the essence of mathematical and customarily scientific analysis. Additionally on this subject of human mind, one thing new is barely being created by instinct, by inventive creativeness. The nice mathematicians and physicists should not ‘good calculators,’ on this respect they’re outplayed by your common ready accountant; neither is somebody who performs the piano with virtuosity an excellent musician! Unique achievements in all fields of human data and functionality are creative achievements and observe their very own legal guidelines.


To an individual who witnessed Einstein’s first laborious makes an attempt within the years 1912 and 1913, because the composer of those strains did, they have to seem just like the ascent of an inaccessible mountain at nighttime of the evening, with out path or path, with out foothold or route. Expertise and deduction offered solely few and insecure handholds. All the upper now we have to worth this mental deed.

No claims to co-discovery by Grossmann certainly!

Additionally in 1920, Grossmann felt compelled to intervene on behalf of the Swiss physicists and mathematicians and to defend Einstein’s relativity concept in opposition to criticism
by a Bernese physicist. Eduard Guillaume (1881–1959), then a mathematician on the Swiss
Federal Insurance coverage Bureau, had been a colleague of Einstein’s on the Swiss
patent workplace. In 1909, the 2 had even accomplished some experimental work collectively 24 Doc. 143,
and, in 1913, Guillaume had translated into French Einstein’s brief model of the Entwurf introduced at
the Frauenfeld assembly.83
However starting in 1917, Guillaume had began to criticize the particular concept of relativity in a lot of
articles that appeared principally within the Archives des sciences physiques et naturelles.

On 5 February 1920, Grossmann forwarded one among Guillaume’s papers on the latter’s request 104 Doc. 300. Einstein’s response was brief and harsh: “Guillaume’s discover is silly
like all the pieces this man dashes off about relativity.” Presumably with reference
to this letter, Grossmann had referred to Einstein’s opinion about Guillaume in a bit that
he revealed on 15 June 1920 within the Neue Schweizer Zeitung.105 Guillaume complained about
Grossmann’s criticism in non-public correspondence with Einstein and tried to elucidate his essential views.
Einstein patiently continued their correspondence however admitted that he was unable to grasp
what Guillaume’s level actually was.69, 106

Refer to caption
Determine 7: Marcel Grossmann (1878–1936). ©ETH Bibliothek Zürich, Bildarchiv.

On 3 September 1920, Grossmann revealed a observe within the Neue Schweizer Zeitung accusing Einstein’s German colleagues of not supporting him in opposition to the anti-Einstein marketing campaign that had simply culminated with the notorious lectures by Paul Weyland (1888–1972) and Ernst Gehrcke (1878–1960) within the Berlin Philharmonic.107
A couple of days later, on 9 September 1920, Grossmann made one other try at getting Einstein again to Switzerland: “Are you continue to not ripe for Zurich but?” he requested in a letter, during which he additionally identified that “each our boys, who’re in the identical class on the Gymnasium, are already calculating with logarithms”69 Doc.142. Within the letter, he additionally requested for one more assertion on Guillaume, which he may then translate into French and ahead for publication in Geneva’s Archives des sciences physiques et naturelles. In Grossmann’s phrases, “a cult is forming round Guillaume that thinks it should right important factors of your ideas.” Einstein was happy “that our boys are classmates, like we have been,” and complied with the request by sending a harsh assertion on Guillaume’s work, stating that he was “unable to connect any sort of clear sense to Guillaume’s explications”69 Doc. 148. Einstein’s assertion didn’t get revealed. As a substitute, Grossmann himself wrote a short assertion within the Archives.108 In accordance with his observe, Guillaume had developed his interpretation of the Lorentz transformations on the worldwide convention of mathematicians in Strasbourg, and whereas it was inconceivable for Grossmann to evaluate the importance of the idea so far as the physics was involved, he might put the finger on the mathematical error that he dedicated.
Guillaume claimed that he had discovered a brand new invariant of the Lorentz transformations. However one knew that the Lorentz transformations are related to a line factor with fixed coefficients, which possesses no invariants in any respect. Solely differential quadratic types with variable coefficients possess an invariant operate, the curvature of house in line with Riemann and Christoffel, on which the gravitational subject equations are primarily based. Guillaume’s expression, however, was “an identification pure and easy. It doesn’t carry any bodily or mathematical curiosity.”

For the next years, now we have just a few objects within the Einstein Archives that doc Grossmann’s and Einstein’s ongoing friendship. In July 1922, Einstein jokingly contemplated to call a crusing boat that he had purchased used and that originally had sunk on account of a leakage both “letdown” (“Reinfall”) or “Grossmann”, presumably close to Grossmann’s earlier help in preserving him afloat, because it have been 109 Doc. 306. Later that 12 months, Einstein and his (second) spouse tried to see Grossmann in Zurich on their manner out for his or her Japan journey however missed him. A postcard that they despatched as an alternative additionally conveys congratulations to Grossmann’s son who had simply graduated along with Einstein’s son from the Realgymnasium of the Kantonsschule in Zürich. In August 1923, Grossmann congratulated Einstein on his choice to rejoin the committee of mental cooperation of the League of Nations. On the finish of the 12 months, Grossmann wrote once more. This time, he was congratulating Einstein to the nice efficiency of his son who had scored finest at examinations on the ETH that he, Grossmann, had taken. He additionally invited Einstein for a lecture, and he reiterated his supply that Einstein might come again to the ETH if he so wished. One other invitation to ship a lecture on the annual meeting of the Schweizerische Naturforschende Gesellschaft to happen in October 1924 in Lucerne was conveyed via correspondence by Grossmann. This time Einstein accepted. As talked about above, Einstein visited Grossmann once more in summer time 1925, as, certainly, he might need accomplished extra ceaselessly than is documented, at any time when he visited Zurich to see his sons.

Grossmann’s final scientific publication considerations a mathematical critique of the geometric basis of Einstein’s so-called teleparallel method to a unified subject concept of gravitation and electromagnetism.56 In summer time 1928, Einstein had revealed two brief notes within the Proceedings of the Prussian Academy, during which he launched his new method and its underlying geometric concept.111, 112, 113 The geometry of teleparallelism was, in actual fact, not new in any respect, however had been investigated by others earlier than, notably by Élie Cartan (1869–1951) within the early twenties. It was formulated then when it comes to tetrad fields, orthogonal vector fields haμsubscript𝑎𝜇h_{amu}italic_h start_POSTSUBSCRIPT italic_a italic_μ end_POSTSUBSCRIPT outlined at every level of a manifold such that

ahaμhaν=δμ