Einstein's 1915 Paper: The Field Equations of General Relativity

Einstein’s 1915 paper on general relativity, “Die Feldgleichungen der Gravitation” (The Field Equations of Gravitation), was presented in Berlin on 25 November 1915. In four pages Albert Einstein gave the final equations of general relativity, which describe gravity as the curvature of spacetime produced by matter and energy.

What the paper found

• The final field equations. Einstein wrote equations that hold in any coordinate system and link the curvature of spacetime to the energy and momentum of matter. They are still the equations physicists use today.
• Newton as a limit. For weak fields and slow motions the equations reduce to Newton’s law of gravity, so the old theory survives as an approximation.
• A theory that fits Mercury. In his paper of the previous week (18 November 1915) Einstein had shown that the theory explains the unexplained 43 arcseconds per century in the slow turning of Mercury’s orbit. This paper confirmed that the final equations keep that result.

How Einstein got there

The paper was the last of four short communications Einstein read to the Prussian Academy of Sciences in November 1915, after years of work and several false starts. Earlier versions of his theory were not valid in every coordinate system and could not account for Mercury. In the first three November papers he corrected the equations step by step; on 25 November he added the final term, involving the trace of the energy-momentum tensor, that made the theory consistent. The paper was issued on 2 December 1915, in the middle of the First World War.

The mathematician David Hilbert submitted a paper with related equations on 20 November 1915. In 1997 historians Leo Corry, Jürgen Renn and John Stachel reported that Hilbert’s surviving page proofs do not contain the explicit final equations, but part of the proofs is missing and the details are still debated.

Why it mattered

General relativity made predictions Newton’s theory could not. One was that starlight passing the Sun should bend by about 1.7 arcseconds, twice the value Einstein had derived in 1911. The 1919 solar eclipse expeditions led by Frank Dyson, Arthur Eddington and Charles Davidson reported a deflection consistent with Einstein’s value, and the result made him famous worldwide.

Within weeks Karl Schwarzschild found the first exact solution of the equations, the seed of black hole physics. In 1917 Einstein applied them to the whole universe in his cosmological paper, and Alexander Friedmann showed in 1922 that they allow an expanding cosmos. Modern cosmology, the study of black holes and gravitational waves all start from these equations.

What we know now

More than a century later the theory has passed every test. On 14 September 2015, almost exactly 100 years after this paper, LIGO detected gravitational waves from merging black holes, ripples in spacetime the theory predicts. In 2019 the Event Horizon Telescope imaged the shadow of the black hole in M87, matching general relativity’s prediction. The LIGO-Virgo-KAGRA network now uses each merger as a fresh test. The open problem is not observational but theoretical: general relativity has not been combined with quantum mechanics.

Who wrote it

Albert Einstein was 36 and a member of the Prussian Academy in Berlin. He wrote the paper alone. His 1921 Nobel Prize in Physics was awarded “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect”, not for relativity.

Read the original

Einstein, A. (1915). “Die Feldgleichungen der Gravitation.” Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (Berlin), pages 844-847. The German original is free to read as a scan on NASA ADS, and an English translation is on Wikisource. The paper has no DOI.

Questions people ask

What did Einstein publish in 1915?

In November 1915 Einstein presented four short papers to the Prussian Academy of Sciences. The last, on 25 November, gave the final field equations of general relativity. The previous one, on 18 November, used the theory to explain the anomalous motion of Mercury’s orbit.

What is general relativity in simple terms?

General relativity says gravity is not a force pulling across space but the curvature of spacetime caused by mass and energy. Objects move along the straightest possible paths in that curved spacetime, which is why planets orbit and why light bends near massive objects.

Did Hilbert discover the field equations first?

David Hilbert submitted a related paper five days before Einstein’s, on 20 November 1915. His surviving proofs do not show the explicit final equations, but part of them is missing. Most accounts credit Einstein with the theory; the exact sequence remains a subject of historical research.