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Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it.

Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it.

===The right side of the equation: the stress-energy tensor <math>T_{\mu\nu}</math>===

===The right side of the equation: the stress-energy tensor===

In the [[Law of Universal Gravitation|Newtonian approximation]], the gravitational vector field is directly proportional to mass. In general relativity, mass is just one of several sources of spacetime curvature. The ''stress-energy tensor,'' <math>T_{\mu\nu}</math>, includes all of these sources. Put simply, the stress-energy tensor quantifies all the ''stuff'' that contributes to spacetime curvature, and thus to the gravitational field.

In the [[Law of Universal Gravitation|Newtonian approximation]], the gravitational vector field is directly proportional to mass. In general relativity, mass is just one of several sources of spacetime curvature. The ''stress-energy tensor,'' <math>T_{\mu\nu}</math>, includes all of these sources. Put simply, the stress-energy tensor quantifies all the ''stuff'' that contributes to spacetime curvature, and thus to the gravitational field.

In this way, we can see that the stress-energy tensor neatly quantifies ''all'' static and dynamic properties of a region of spacetime, from mass to momentum to electric charge to temperature to pressure to shear stress. Thus, the stress-energy tensor is all we need on the right-hand side of the equation in order to relate matter, energy and, well, ''stuff'' to curvature, and thus to the gravitational field.

In this way, we can see that the stress-energy tensor neatly quantifies ''all'' static and dynamic properties of a region of spacetime, from mass to momentum to electric charge to temperature to pressure to shear stress. Thus, the stress-energy tensor is all we need on the right-hand side of the equation in order to relate matter, energy and, well, ''stuff'' to curvature, and thus to the gravitational field.

===The left side of the equation: the Einstein curvature tensor===

===The left side of the equation: the Einstein curvature tensor===