Universal gravitation can be written , but with the gravitational potential it can also be written as a field equation.
The left side, , describes how the potential dips; the right side is the density of matter there. Where there is matter, the potential dips.
The upper part of the figure is the density and the lower part the potential; the potential well is deepest where the density peaks. The Einstein equation has the same form, but what dips is spacetime itself rather than a potential.
On the left, , the Einstein tensor, is built from the curvature of spacetime. On the right, , the energy-momentum tensor, describes how energy and momentum are distributed and how they flow.
The source of gravity is not mass alone; energy, momentum and pressure all enter. The energy of light curves spacetime too.
The figure varies the amount of matter on the right, and the more matter there is, the deeper the curvature of the grid on the left.
The indices and take the values for time and , , for space. Both and have components.
Since and , the independent components are the on the diagonal and the on one side of it, in all, and there are equations.
The left side contains second derivatives of the metric , which sets how distances in spacetime are measured, and the equations are coupled to one another. This is why they are hard to solve in general.
Setting gives . This does not mean that spacetime is not curved.
Outside a star there is no matter, yet spacetime is still curved. The solution Schwarzschild found in describes exactly this, and both the shift of Mercury's orbit and the bending of light passing the Sun can be calculated from it.
Gravitational waves are also solutions of this kind: curvature of spacetime travelling through space with no matter.
Write the equation as with an unknown coefficient . For weak gravity and slow motion, the time components become and .
Putting these in gives . To agree with Newton's , must be .
In SI units is about . That even the mass of the Sun bends a passing ray of light by only arcseconds comes from how small this coefficient is.