# Einstein vs. Hilbert: The Race for Relativity's Foundations

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The contest between Albert Einstein and David Hilbert to formulate the foundations of general relativity is a fascinating chapter in scientific history.

Hilbert delivered a significant talk titled *Die Grundlagen der Physik* (“On the Foundations of Physics”) on November 20, 1915, just five days prior to the release of Einstein’s pivotal paper, *Die Feldgleichungen der Gravitation* (“The Field Equations of General Relativity”). This talk introduced a theory of gravitation and detailed field equations that depicted how mass and energy curve spacetime.

By November 1915, Einstein had been refining his grand theory since his first special relativity paper in 1905. It took him a decade to connect spacetime curvature with the energy and momentum of matter and radiation. His field equations, a set of differential equations, completed this monumental task.

Hilbert, a renowned mathematician of his time, was present when Einstein discussed his theory at Göttingen University in early 1915. He had invited Einstein to present, and during the lecture, Einstein revealed his advancements and challenges in achieving general covariance in his equations.

By the fall of 1915, while Einstein struggled to make headway, Hilbert hinted at his own progress on a gravitation theory. Both men were fiercely competing to derive the correct equations, and by the end of November, each had succeeded.

The year 2019 marked the centennial of Eddington's expedition, which validated Einstein's general relativity theory during a solar eclipse. Since then, discussions continue regarding who should be credited with the priority of these foundational equations: Einstein or Hilbert.

## Timeline of Relativity Theory

The journey of relativity traces back to Newton and the discovery of gravitational force:

**1687**: Isaac Newton publishes*Philosophiæ Naturalis Principia Mathematica*, which includes his law of universal gravitation.**1706**: In*The Queries*, Newton queries the relationship between bodies and light.**1861–62**: James Clerk Maxwell introduces the first forms of Maxwell's Equations.**1883**: Ernst Mach critiques the concept of absolute space in*Die Mechanik in ihrer Entwickelung*.

At 18, Einstein was introduced to Mach's ideas by Michele Besso while studying at Zürich Polytechnic.

**1904**: Henri Poincaré advocates for modifications to Newton's gravity to align with later discoveries.**1905**: Einstein publishes his*Annus Mirabilis*papers, presenting special relativity, which unifies various concepts, excluding gravity.**1907**: While at the patent office in Bern, Einstein has a realization about free-falling observers and later publishes his findings, including the equivalence principle.

From 1908 to 1911, Einstein shifted his focus to quantum theory before returning to gravitation in June 1911.

**1912**: Hilbert begins axiomatizing the new theory of electrons.**1913**: Einstein, with Marcel Grossmann, publishes a paper outlining a generalized theory of relativity using the elevator analogy to explain the equivalence principle.**1914**: Einstein publishes a comprehensive paper, expressing dissatisfaction with his earlier scalar theory of gravitation.

In the summer of 1915, Einstein visited Göttingen University, where he delivered lectures that impressed Hilbert.

By November 1915, both men agreed on the necessity for a generally covariant theory.

### November 1915: The Key Developments

On November 4, Einstein submits a paper to the Prussian Academy of Sciences titled *Zur allgemeinen Relativitätstheorie* (“The Theory of General Relativity”), where he renounces his earlier scalar theory, proposing a new equation instead.

On November 7, he sends Hilbert the proofs from his November 4 paper, acknowledging previous shortcomings.

On November 11, Einstein communicates again, presenting a new, generally covariant equation, which aligns with his final formulation only under specific conditions.

On November 14, Hilbert responds enthusiastically, noting that his own solution differs from Einstein's, inviting him to Göttingen.

Later, on November 18, Einstein acknowledges Hilbert’s work, suggesting parallels in their findings and leading to further correspondence.

On November 25, Einstein submits what is recognized today as the correct form of his field equations, without referencing Hilbert's contributions.

## The Dispute: Who Came First?

Einstein's paper is dated November 25, 1915, while Hilbert's is dated November 20. It is known that Hilbert sent Einstein a copy of his work, likely before Einstein submitted his own.

For a long time, it was believed that both scientists independently discovered the field equations. However, this is now under debate.

### The Case for Hilbert's Priority

Some argue that Hilbert's correspondence influenced Einstein's work, particularly concerning the trace term in the field equations. Einstein’s references to prior work in his November paper do not include Hilbert, raising questions about his independence in reaching the correct form.

### The Case for Einstein's Priority

Conversely, many historians and physicists assert that both arrived at their conclusions independently, with priority rightfully attributed to Einstein due to his foundational contributions to the theory. Evidence, including drafts and communications, suggests that while Hilbert made significant contributions, Einstein’s work predated and influenced the development of relativity.

## What is Indisputable

It is universally acknowledged that Hilbert recognized Einstein's unique contributions to relativity. Despite some tension, both men maintained a professional respect for each other, culminating in Einstein's efforts to reconcile their relationship.

For further exploration of Einstein's theory, Leonard Susskind's lecture series on general relativity remains accessible, providing valuable insights into this pivotal scientific breakthrough.

This essay is part of a series on mathematical topics featured in Cantor's Paradise, a weekly Medium publication. Thank you for your attention!