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### IV.13 General Relativity and the Einstein Equations

*Mihalis Dafermos*

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Einstein’s formulation of general relativity represents one of the great triumphs of modern physics and provides the currently accepted classical theory that unifies gravitation, inertia, and geometry. The *Einstein equations* are the mathematical embodiment of this theory.

The definitive form of the equations,

was attained in November 1915; this was the final act of Einstein’s eight-year struggle to generalize his *principle of relativity* so as to encompass gravitation, which had been described in the earlier “Newtonian” theory by the *Poisson equation*

for the potential and mass density *µ*.

An obvious contrast between the Einstein equations (1) and the Poisson equation (2) is that the mysterious notation of the former makes it far less obvious what they even mean. This has given the subject of general relativity a reputation for difficulty and impenetrability. However, this reputation is to some extent unwarranted. Both (1) and (2) represent the culmination of revolutionary theories whose formulations presuppose a complicated conceptual framework. For better or for worse, however, the structure necessary to formulate Poisson’s equation has been incorporated into our traditional mathematical notation and school education. As a result, ^{3}, with its Cartesian coordinate system, and notions such as functions, partial derivatives, masses, forces, and so on, are familiar to people with a general mathematical background, while the conceptual structure of general relativity is much less so, both with respect to its basic physical notions and with respect to the mathematical objects that are needed to model them. However, once one comes to terms with these, the equations turn out to be more natural and, one might even dare say, simpler.