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Marcel Grossmann
Marcel Grossmann
Marcel Grossmann
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Marcel Grossmann (Hungarian: Grossmann Marcell; April 9, 1878 – September 7, 1936)[2] was a Swiss mathematician who was a friend and classmate of Albert Einstein. Grossmann came from an old Swiss family in Zürich. His father managed a textile factory. He became a Professor of Mathematics at the Federal Polytechnic School in Zürich, today the ETH Zurich, specializing in descriptive geometry.

Key Information

Grossmann was born to a Jewish family in Budapest.

Career

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In 1900 Grossmann graduated from the Federal Polytechnic School (ETH) and became an assistant to the geometer Wilhelm Fiedler.[3] He continued to do research on non-Euclidean geometry and taught in high schools for the next seven years. In 1902, he earned his doctorate from the University of Zurich with the thesis Ueber die metrischen Eigenschaften kollinearer Gebilde (translated On the Metrical Properties of Collinear Structures) with Fiedler as advisor. In 1907, he was appointed full professor of descriptive geometry at the Federal Polytechnic School.[4]

As a professor of geometry, Grossmann organized summer courses for high school teachers. In 1910, he became one of the founders of the Swiss Mathematical Society. He was an Invited Speaker of the ICM in 1912 at Cambridge[5] and in 1920 at Strasbourg.

Collaborations with Albert Einstein

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Albert Einstein's friendship with Grossmann began with their school days in Zürich. Grossmann's careful and complete lecture notes at the Federal Polytechnic School proved to be a salvation for Einstein, who missed many lectures.[6] Grossmann's father helped Einstein get his job at the Swiss Patent Office in Bern,[7] and it was Grossmann who helped to conduct the negotiations to bring Einstein back from Prague as a professor of physics at the Zurich Polytechnic. Grossmann was an expert in differential geometry and tensor calculus, the mathematical tools which would provide a proper mathematical framework for Einstein's work on gravity. Thus, it was natural that Einstein would enter into a scientific collaboration with Grossmann.[8]

It was Grossmann who emphasized the importance of a non-Euclidean geometry called Riemannian geometry (also elliptic geometry) to Einstein, which was a necessary step in the development of Einstein's general theory of relativity. Abraham Pais's book[9] on Einstein suggests that Grossmann mentored Einstein in tensor theory as well. Grossmann introduced Einstein to the absolute differential calculus, started by Elwin Bruno Christoffel[10] and fully developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita.[11] Grossmann facilitated Einstein's unique synthesis of mathematical and theoretical physics in what is still today considered the most elegant and powerful theory of gravity: the general theory of relativity. The collaboration of Einstein and Grossmann led to a ground-breaking paper, "Outline of a Generalized Theory of Relativity and of a Theory of Gravitation", which was published in 1913 and was one of the two fundamental papers which established Einstein's theory of gravity.[12]

Death

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Grossmann died of multiple sclerosis in 1936.[13] The community of relativists celebrates Grossmann's contributions to physics by organizing Marcel Grossmann meetings every three years.

Legacy

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The International Center for Relativistic Astrophysics presents the Marcel Grossmann Awards. Each recipient receives a silver casting of the T. E. S. T. sculpture [14] by the artist A. Pierelli. Each year, an institution is selected and between two and six individual scientists are selected. Past institutional winners include the Planck Scientific Collaboration (ESA), AlbaNova University Center, Institut des Hautes Etudes Scientifique (IHES) and others. Past individual winners include Shing-Tung Yau, Tsung-Dao Lee, Christine Jones Forman and Stephen Hawking.

See also

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Notes

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References

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Marcel Grossmann

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Marcel Grossmann (9 April 1878 – 7 September 1936) was a Swiss mathematician renowned for his expertise in differential and non-Euclidean geometry and for his crucial collaboration with Albert Einstein in formulating the mathematical foundations of general relativity. Born in Budapest, Hungary, to a Jewish merchant family, Grossmann relocated with his parents to Basel, Switzerland, in 1883, where he completed his secondary education. He enrolled at the Eidgenössische Technische Hochschule (ETH) in Zurich in 1896, studying mathematics alongside future physicist Albert Einstein, with whom he formed a close friendship. Grossmann graduated in 1900 with a diploma thesis on non-Euclidean geometry and earned his Ph.D. from the University of Zurich in 1902, with a dissertation on the metric properties of collineations. After brief stints as an assistant at and as a high school instructor in , Grossmann advanced to at the in 1905 before returning to ETH as an associate professor in 1907 and full professor of in 1908, a position he held until his early retirement in 1927 due to deteriorating health from . Throughout his career, he contributed to educational reforms in Swiss teaching and published works on and . Grossmann's most notable achievement came from his 1912–1914 partnership with Einstein, whom he had previously assisted in securing a position at the Swiss Patent Office in . Recognizing Einstein's need for advanced mathematical tools to extend to gravity, Grossmann introduced him to , absolute differential calculus (later known as tensor calculus), and the works of mathematicians like and . Their joint efforts culminated in the 1913 "Entwurf" paper, a precursor to that outlined field equations for gravitation using tensors, though it contained limitations later resolved by Einstein in 1915. Grossmann also co-authored a 1914 follow-up paper refining these ideas. In his later years, Grossmann focused on teaching and administrative roles at , while occasionally critiquing developments in unified field theories, as in his 1931 paper on . He died in at the age of 58 from complications of . Einstein, who regarded him as an irreplaceable friend and collaborator, expressed his condolences. Grossmann's legacy endures through the international Marcel Grossmann Meetings on , initiated in 1975 to honor his foundational contributions to the field.

Early Life and Education

Family Background and Childhood

Marcel Grossmann was born on April 9, 1878, in , , to Swiss parents Jules Grossmann and Katharina Henriette Lichtenhahn. His family traced its origins to an established Swiss lineage in Höngg, a district near , reflecting a heritage rooted in the region's industrial and mercantile traditions. Jules Grossmann, originally from but identifying as Swiss, managed a successful factory in that he co-founded in , specializing in the production of ; this enterprise provided the family with a stable middle-class lifestyle amid Hungary's growing industrialization. Grossmann had one , a brother named Eugen, and the family's environment emphasized education and intellectual pursuits, fostering a supportive atmosphere for the children's development. In 1893, at the age of 15, the Grossmann family relocated to , settling in , primarily due to Jules's professional commitments and the pull of their ancestral homeland. This move immersed Grossmann in Switzerland's rigorous educational culture, where he began to nurture his early fascination with , influenced by the family's valuing of scholarly achievement and the structured learning opportunities available in his new surroundings.

Education in Switzerland

Following the family's relocation from to in 1893, Grossmann attended the Oberrealschule there from 1893 to 1896, where he demonstrated strong academic aptitude, earning top grades in most subjects as noted in his 1894 preserved in the ETH Archives. This period of prepared him for advanced studies in , building on the familial support that facilitated the move. In 1896, Grossmann enrolled at the Eidgenössische Technische Hochschule (ETH) in , entering the Department VI A for training as a and under professors such as and Wilhelm Fiedler. He focused intensely on during his studies, excelling in courses on analytical, projective, and , where his meticulous note-taking and analytical skills stood out. Grossmann graduated in 1900 with a in , submitting a thesis on supervised by Fiedler and achieving a score of 5.23, placing second in his class of 11 students. Pursuing further specialization, Grossmann then sought a at the , as the at the time did not yet confer mathematical PhDs. Under the supervision of Otto Wilhelm Fiedler, he completed his PhD in 1902 with a titled die metrischen Eigenschaften kollinearer Gebilde (On the Metric Properties of Collinear Structures), which explored foundational aspects of , including metric properties in hyperbolic and elliptic contexts. This work highlighted his early proficiency in descriptive and , areas that would define his later contributions.

Professional Career

Academic Positions

Following his diploma from in 1900 and PhD in 1902, Grossmann began his academic career as an assistant to the geometer Otto Wilhelm Fiedler at the institution, where he contributed to teaching and research in . He then taught at the Cantonal School in from 1901 to 1905. In 1905, he briefly held positions as a teacher at the Oberrealschule in and as a privatdocent at the , focusing on mathematical education. In 1907, Grossmann was appointed associate professor of descriptive at and promoted to full professor in 1908, positions he held until his retirement in 1927 due to deteriorating health, during which he lectured extensively on geometric topics essential to and . As a faculty member, he took on administrative responsibilities, including serving on committees that proposed reforms to the at ETH to better integrate theoretical and applied aspects. Grossmann also played a key role in the broader mathematical community by co-founding the Swiss Mathematical Society in 1910 alongside Rudolf Fueter and Henri Fehr, an organization aimed at promoting mathematical research and collaboration in . He later served as its president from 1916 to 1917, guiding the society's early activities and international outreach.

Independent Research

Marcel Grossmann specialized in several branches of geometry throughout his career, with a particular emphasis on , , and descriptive geometry. His doctoral thesis, Über die metrischen Eigenschaften kollinearer Gebilde (1902), explored the metric properties of collinear structures within , laying foundational work in that examined how geometric figures maintain metric relations under projective transformations. In , Grossmann developed synthetic approaches using the Cayley-Klein framework, demonstrating projective constructions of solely with a , as detailed in his 1909 paper Projektive Konstruktionen in der hyperbolischen Geometrie. His expertise in descriptive geometry, which bridges theoretical and practical applications, informed his prolific output of textbooks, including Darstellende Geometrie (first edition 1906, with seven editions through 1932), designed to provide rigorous visual and spatial reasoning tools for students. A cornerstone of Grossmann's independent publications was Die fundamentalen Konstruktionen der nichteuklidischen Geometrie (1904), which outlined fundamental synthetic constructions in non-Euclidean spaces and received praise from for its clarity and innovation in handling absolute geometries without coordinates. This work built on his earlier explorations, such as Metrische Eigenschaften reziproker Bündel (1905), which delved into the metric properties of reciprocal bundles in , contributing to a deeper understanding of and invariance in geometric structures independent of later physical applications. Grossmann's research on absolute differential calculus and tensor methods emerged from his pre-1912 studies in , where he engaged with the foundational ideas of and ; by 1910, he had incorporated tensor notations in his lectures on infinitesimal geometry at , predating broader applications and emphasizing their role in invariant formulations of geometric problems. Grossmann actively contributed to mathematical societies, sharing his geometric insights through presentations that advanced synthetic methods. Following the founding of the Swiss Mathematical Society in 1910, which he co-established, he delivered papers on projective and non-Euclidean topics, including a 1922 talk on that revisited Cayley-Klein constructions for educational purposes. These engagements, often post-1910, highlighted his commitment to disseminating rigorous geometric principles within the Swiss academic community. At the Eidgenössische Technische Hochschule () in Zurich, where Grossmann held the professorship in descriptive from 1907, his research profoundly shaped the curriculum by integrating advanced geometric foundations into . His 1911 report, Der mathematische Unterricht an der Eidgenössischen Technischen Hochschule, advocated for a balanced emphasis on synthetic and analytic methods, influencing subsequent reforms and supervising four doctoral theses in between 1910 and 1920. Through textbooks like Analytische Geometrie (1906) and his descriptive series, Grossmann ensured that ETH students received training in precise spatial modeling, which became a model for technical institutions across .

Collaboration with Albert Einstein

Early Friendship at ETH

Marcel Grossmann and Albert Einstein first met as classmates at the Eidgenössische Technische Hochschule (ETH) in Zurich, where they both enrolled in the physics and mathematics section in 1896 and graduated in 1900. Their shared passion for mathematics and physics quickly fostered a close intellectual camaraderie, with Grossmann often engaging Einstein in discussions on advanced topics that extended beyond the classroom curriculum. Despite Einstein's occasional rebellious attitude toward formal lectures, which led to absences, Grossmann proved instrumental by sharing his meticulously prepared notes, helping Einstein stay on track with coursework in subjects like differential geometry and theoretical physics. This supportive dynamic continued after their graduation, as both faced challenges entering academia. In 1901, Grossmann, leveraging his family's connections in , played a key role in securing Einstein's position as a technical expert third class at the Swiss Patent Office in , providing Einstein with financial stability during a period of uncertainty. Their early correspondence reflected mutual encouragement, with Grossmann offering advice on Einstein's initial forays into independent research, such as his work on the , while Einstein valued Grossmann's insights into mathematical rigor. The friendship that developed at ETH evolved into a lifelong bond rooted in profound and unwavering personal loyalty, sustaining them through professional ups and downs without immediate collaborative projects. Grossmann's reliability as a and academic ally during their student years laid the foundation for this enduring relationship, marked by a rare blend of scholarly enthusiasm and genuine empathy.

Mathematical Assistance in General Relativity

In 1912, upon his return to , sought the mathematical expertise of his former classmate Marcel Grossmann to formulate a gravitational theory based on the . Grossmann, then a at the Eidgenössische Technische Hochschule (ETH), provided crucial guidance by introducing Einstein to advanced non-Euclidean geometries and tools essential for describing curvature. This assistance addressed Einstein's longstanding struggles with the mathematical framework needed to extend to accelerated frames and gravity. Grossmann specifically acquainted Einstein with , which models as a curved manifold using a gμνg_{\mu\nu} to define distances and angles in a coordinate-independent way. The , a symmetric second-rank tensor, determines the via the ds2=gμν dxμ dxνds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu, allowing gravitational effects to be encoded as variations in structure rather than forces. Building on this, Grossmann explained tensor calculus, including the (also known as absolute differential calculus), developed by and , which enables the manipulation of quantities invariant under coordinate transformations. These tools were pivotal for handling the complexities of , where physical laws remain form-invariant across all coordinate systems. Their collaboration culminated in the co-authored paper "Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation" (Outline of a Generalized Theory of Relativity and a Theory of Gravitation), published in 1913. In this work, Einstein contributed the physical interpretation while Grossmann authored the mathematical section, presenting the Entwurf theory with field equations of limited covariance. Grossmann guided Einstein in employing Christoffel symbols Γμνλ=12gλσ(∂μgνσ+∂νgμσ−∂σgμν)\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\sigma} (\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}), which quantify connections on the manifold and facilitate parallel transport of vectors. He also introduced covariant derivatives, denoted ∇μVν=∂μVν+ΓμλνVλ\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda} V^\lambda, which extend partial derivatives to preserve tensor character under curvature, essential for formulating laws like energy-momentum conservation in curved space. A cornerstone of Grossmann's input was the Riemann curvature tensor Rσμνρ=∂μΓνσρ−∂νΓμσρ+ΓμλρΓνσλ−ΓνλρΓμσλR^\rho_{\sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}, which measures spacetime deviation from flatness and serves as the basis for gravitational dynamics. Contracting this tensor yields the Ricci tensor Rμν=RμλνλR_{\mu\nu} = R^\lambda_{\mu\lambda\nu}, which Grossmann recommended for the Entwurf field equations Rμν=−κTμνR_{\mu\nu} = -\kappa T_{\mu\nu} (where TμνT_{\mu\nu} is the stress-energy tensor and κ=8πG/c4\kappa = 8\pi G/c^4), though restricted by coordinate conditions to avoid interpretational issues like the "hole argument." Despite these limitations, Grossmann's framework resolved Einstein's difficulties with non-Euclidean mathematics, enabling the breakthrough to fully generally covariant equations in November 1915: Rμν−12Rgμν=κTμνR_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \kappa T_{\mu\nu}. Einstein later acknowledged Grossmann's role, stating that his friend's help not only spared him from independently studying the literature but also directly aided the search for these equations.

Personal Life and Death

Marriage and Family

Marcel Grossmann married Anna Keller, the daughter of Eduard Keller, in 1903. The couple's first child, son Marcel Hans Grossmann Jr., was born on January 30, 1904, followed by their daughter Elsbeth Grossmann in 1909. The Grossmann family settled in , where Anna played a central role in maintaining household operations and managing family finances, allowing Marcel to focus on his academic pursuits amid his growing professional responsibilities. This domestic arrangement contributed to Grossmann's personal stability, providing a reliable support system during key periods of his career advancement. Insights into the family's dynamics and perspectives are detailed in the 2015 biography Marcel Grossmann: Aus Liebe zur Mathematik by Grossmann's granddaughter, Claudia E. Graf-Grossmann, which draws on personal family records to illuminate his home life.

Illness and Death

In the mid-1920s, Marcel Grossmann began experiencing significant health issues, with symptoms including dizziness, loss of grip, leg dragging, and speech difficulties first noted as early as 1915 but intensifying thereafter. By 1927, his friend and colleague Heinrich Zangger diagnosed the condition as , confirming earlier suspicions and leading to progressive mobility limitations primarily affecting the right side of his body. These challenges prompted repeated leaves of absence from his duties at the , including in the summers of 1924 and winters of 1925–1926 and 1926–1927, ultimately resulting in his early retirement on May 28, 1927, effective October 1 of that year. By the 1930s, Grossmann's health had deteriorated further, severely restricting his ability to teach and conduct ; his last appeared on January 16, 1931, marking a significant reduction in his scientific output. Despite these struggles, he received recognition for his contributions when elected as an honorary member of the Swiss Mathematical Society in 1935, a fitting capstone to his career shortly before his passing. Grossmann died on September 7, 1936, in at the age of 58, succumbing to the effects of after years of progressive decline. In response, wrote to Grossmann's widow, Anna, expressing profound sorrow and reflecting on their enduring bond: "One thing however is beautiful: We were and remained friends throughout life."

Legacy

Recognition in Mathematics

Marcel Grossmann, a Swiss mathematician, played a pivotal role in establishing the Swiss Mathematical Society (SMS) in 1910 alongside Rudolf Fueter and Henri Fehr, serving as its third president from 1916 to 1917 during a period of international tension caused by . His leadership helped maintain the society's unity and focus on advancing mathematical research in . In recognition of his longstanding contributions, Grossmann was awarded honorary membership in the SMS in 1935, alongside fellow founders Fehr and Fueter. Grossmann's expertise in geometry earned him significant esteem within Swiss academic circles, particularly for his work on non-Euclidean and descriptive geometry. Appointed as at the in 1905 based on his published research, he advanced to full professor of descriptive geometry at the Eidgenössische ( in 1907, a position he held until his retirement in 1927, succeeding Wilhelm Fiedler and supervising four doctoral dissertations in non-Euclidean geometry. His multivolume textbook on descriptive geometry, first published in 1915 and revised through 1932, became a standard reference, underscoring his influence in mathematical education and applied geometry. Additionally, his early engagement with tensor calculus, including expositions in collaborative works, was acknowledged in pre- and post-World War I publications as foundational for modern geometric frameworks. As a marker of institutional esteem, Grossmann's deep ties to —where he studied, earned his doctorate in , and later taught—reflected his status as a leading figure in Swiss mathematics. He was also elected a member of the Naturforschende Gesellschaft in , further affirming his standing among the country's scientific elite. These affiliations highlighted the respect afforded to his rigorous approach to and tensor methods throughout his career.

Enduring Influence

Marcel Grossmann's mathematical contributions were instrumental in establishing the rigorous framework for Albert Einstein's general theory of relativity, providing the tensor calculus tools that enabled the geometrization of gravity and profoundly shaped modern gravitational physics and cosmology. By introducing Einstein to Riemannian geometry and absolute differential calculus in 1912, Grossmann helped formulate the field equations that describe spacetime curvature, laying the groundwork for predictions like black holes, gravitational waves, and the expanding universe. This collaboration not only resolved Einstein's longstanding struggles with the mathematical structure of gravity but also influenced subsequent developments, such as the application of general relativity to cosmological models and astrophysical phenomena, including the detection of gravitational waves in 2015. Grossmann's advocacy and exposition of tensor calculus extended its applications far beyond relativity, embedding it as a cornerstone of and modern . In his 1914 paper with Einstein, he adapted concepts from and , popularizing the term "tensor" in physics and enabling covariant formulations essential for describing physical laws in curved spaces. This work facilitated broader uses in fields like and , where tensors model symmetries and interactions in higher-dimensional geometries, underscoring Grossmann's enduring role in bridging pure mathematics with physical applications. To honor Grossmann's legacy, the Marcel Grossmann Meetings were established in 1975 by Remo Ruffini and at the in , serving as a premier international forum for advances in , gravitation, and relativistic field theories. Held approximately every three years, these gatherings—now numbering over sixteen—foster global collaboration, with proceedings documenting key breakthroughs in theoretical and experimental relativity. Complementing the meetings, the Marcel Grossmann Awards, initiated in 1975, recognize exceptional contributions to gravitational physics, with recipients including luminaries like and , thereby perpetuating Grossmann's impact on the field. Biographical efforts have further preserved Grossmann's story, notably through his granddaughter Claudia Graf-Grossmann's 2015 book Marcel Grossmann: Für die Liebe zur Mathematik, translated into English in 2018 as Marcel Grossmann: For the Love of . This work, enriched by Tilman Sauer's analysis, details his personal and scientific life, emphasizing his , influences, and pivotal role in relativity while making his contributions accessible to broader audiences. Grossmann's commitment to mathematics education at ETH Zurich left a lasting imprint on Swiss academic traditions, shaping generations through his professorship and pedagogical innovations. As a professor of descriptive geometry from 1907, he authored influential textbooks like Analytische Geometrie (1906) and Darstellende Geometrie (1906), which became staples for engineering and mathematics students, while his 1911 report on ETH's mathematics instruction advocated for reformed curricula and teacher training. These efforts, including his involvement in national education reforms like the 1925 Matura standards, reinforced ETH's emphasis on rigorous, applied mathematics, influencing Swiss technical education long after his tenure.
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