On November 25 2015, Albert Einstein submitted a paper to the Prussian Academy of Sciences in Berlin presenting to the world, for the first time, the final form of his field equations relating the curvature of spacetime to the energy and momentum of matter, the final component of his general theory of relativity. (On a somewhat related note, this November also marks the 150th anniversary of the publication of Maxwell’s equations of electrodynamics.)

### The Einstein Field Equations:

where is the Ricci curvature tensor and is the stress-energy tensor.

A. Einstein, “Die Feldgleichung der Gravitation”, *Preussische Akademie der Wissenschaften, Sizungsberichte* (1915), 844-847.

(Yes, I realize that it looks like one equation, but the notation employed hides the fact that this represents a system of coupled equations.)

For the first time, a contender had arisen to topple Newton’s work as the dominant model for describing gravity. In fact, general relativity went one step further than Newton, describing not only the dynamics of gravity, but also a mechanism: specifically, the curvature of space-time as described by the field equations. This new mechanism could be summed up in a statement frequently attributed to John Archibald Wheeler, “*Matter tells spacetime how to bend* and spacetime returns the complement by telling matter how to move.”

Of course, general relativity did not spring into existence fully-formed overnight. It was the end result of a decade of work by Einstein (ever since he revealed his special theory of relativity in his *annus mirabilis* papers of 1905), culminating in a series of four papers (presented as a series of weekly lectures to the Prussian Academy) in November of 1915. The timing of those papers was fortuitous for Einstein, as David Hilbert was on the cusp of arriving at the same field equations from a different approach.

### The Road to General Relativity

- Einstein, Albert (1905-06-30). “Zur Elektrodynamik bewegter Körper” (“On the Electrodynamics of Moving Bodies”).
*Annalen der Physik***17**: 891–921.(The Special Theory of Relativity) - Einstein, Albert (1905). “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?” (“Does the Inertia of a Body Depend Upon Its Energy Content?”).
*Annalen der Physik***18**: 639–641 (Mass-energy equivalence in Special Relativity) - A. Einstein, “On the relativity principle and the conclusions drawn from it”,
*Jahrbuch der Radioaktivitaet und Elektronik*4 [English translation in*The Collected Papers of Albert Einstein*, vol. 2 (tr. by Anna Beck and consultant Peter Havas), Princeton University Press, 1989.] - A. Einstein, “Ueber den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes”,
*Annalen der Physik*35, 1911, 898-908. - A. Einstein, “Lichtgeschwindigkeit und Statik des Gravitationsfeldes”,
*Annalen der Physik (ser. 4)*,**38**, 1912, 355–369, link - A. Einstein, “Lichtgeschwindigkeit und Statik des Gravitationsfeldes”,
*Annalen der Physik (ser. 4)*,**38**, 1912, 355–369, link - A. Einstein & M. Grossmann, “Entwurf einer verallgemeinerten Relativitaetstheorie und einer Theorie der Gravitation”,
*Zeitschrift fuer Mathematik und Physik***62**, 1913, 225-261. - “Die formale Grundlage der allgemeinen Relativitaetstheorie”,
*Preussische Akademie der Wissenschaften, Sitzungsberichte*(1914), 1030-1085. - A. Einstein, “Zur allgemeinen Relativitaetstheorie”,
*Preussische Akademie der Wissenschaften, Sizungsberichte*(1915), 778-786. - A. Einstein, “Zur allgemeinen Relativitaetstheorie (Nachtrag)”,
*Preussische Akademie der Wissenschaften, Sizungsberichte*(1915), 799-801. - A. Einstein, “Erklaerung der Perihelbewegung des Merkur aus der allgemeinen Relativitaetstheorie”,
*Preussische Akademie der Wissenschaften, Sizungsberichte*(1915), 831-839. link - A. Einstein, “Die Feldgleichung der Gravitation”,
*Preussische Akademie der Wissenschaften, Sizungsberichte*(1915), 844-847. link - A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie” (“The Foundation of the General Theory of Relativity”),
*Annalen der Physik*,**Vol. 354**, Issue 7, pp. 769-822 (1916)

Einstein’s 1905 special theory of relativity was built upon two fundamental postulates: 1) The laws of physics are the same in all inertial (non-accelerating) reference frames, and 2) The speed of light in a vacuum is the same for all observers. From special relativity came the concept of the dilation of space and time, as well as the equivalence of mass and energy.

It did not take long for Einstein to come up with the idea that would allow him to extend his ideas of relativity to non-inertial (accelerating) reference frames, the equivalence principle, which basically says that the force due to gravity is indistinguishable from the pseudo-force experienced by an observer in an accelerating reference frame. The classic thought experiment for illustrating this is someone sealed into a windowless container being unable to distinguish being sitting on the surface of a planet or being towed through space by a rocket. Alternatively, the same observer would be unable to distinguish between falling freely toward the planet (or around it, in the case of orbit) and drifting through deep space far from any gravitation influence. In a 1907 paper, Einstein laid out his equivalence principle, pointing out its implication that inertial mass and gravitational mass are identical, and predicting the curving of light by gravitational sources.

Next, in a 1911 paper, Einstein further explored the idea of gravity deflecting light, as well as introducing the concept of time dilation due to gravity. Next, in a pair of 1912 papers, we see Einstein come to the conclusion that time cannot be warped while keeping space flat. This realization led him to conclude that he would have to turn to the non-Euclidean geometry of Riemann, Lobachevsky, and others to describe the distortions of spacetime being predicted by his nascent theory. Not being well-versed in the mathematics of such things (which had until then been purely within the bailiwick of mathematicians), Einstein enlisted the assistance of an old friend of his, Marcel Grossmann, which resulted in a joint paper in 1913. For that year and the next, Einstein grappled with what appeared to be fatal flaw in his theory, the “hole arguement” (described in a 1914 paper) in which it seemed to Einstein that the field equations he was trying to create for describing the behavior of spacetime might not be generally covariant. I’ll not belabor this point with a detour into describing what that means, but suffice it to say that it turned out that he was actually on the right track, and he was able eventually to construct a form of the equations which were, in fact, admissible as being generally covariant.

Finally we come to the magic month of November in 1914, with four papers delivered one paper per week, in which we see the final dash to the finish line as Einstein hammered his equations into their final form. As a bonus, in the third paper, he presented calculations in which his new theory precisely accounted for the anomalous procession of the orbit of the planet Mercury. This anomaly, first discovered in 1859 by the astronomer Urbain Le Verrier, had proven to be a major problem for Newtonian physics, which could account for only a portion of the precession, and which had driven astronomers on a wild goose chase looking for a postulated Planet Vulcan within the orbit of Mercury. But Einstein’s calculations solved the puzzle, and a week later, on November 25, 1915, he delivered the paper with the final form of his field equations. The following year, his theory of general relativity was published in its full form.

Of course, that was just the beginning. The next step was validating the theory experimentally, but we’ll save that story for another time.

# For more information:

- Arch and scaffold: How Einstein found his field equations |
*Physics Today* - On the Anniversary of Two Scientific Revolutions | Phil Plait | Bad Astronomy
- A Century Ago, Einstein’s Theory of Relativity Changed Everything | The New York Times
- 100th Anniversary of Einstein’s Theory of Relativity | NASA
- Relativity | MIT OpenCourseWare
- The Meaning of Einstein’s Equation | John C. Baez
- GENERAL RELATIVITY FROM A TO Z | Perimeter Institute
- 100 Years of General Relativity: Scientific American Special Issue
- My Love Affair With Relativity |
*Scientific American* - GENERAL RELATIVITY AT 100 |
*Nature* - History: Einstein was no lone genius |
*Nature* - Einstein Centenary | 2015: International Year of Light
- GR100 Events around the World | International Society On General Relativity & Gravitation
- Explaining Einstein’s Theory Of General Relativity On Its 100th Birthday | NPR
- The Equation That Blew Up The Cosmos | NPR
- Princeton celebrates 100 years of Einstein’s theory of general relativity | Princeton
- The Collected Papers of Albert Einstein | Princeton University Press
- Einstein Archives Online
- 100 years of general relativity | Max Planck Society
- The Renaissance of relativity | Max Planck Society

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