Gravity in general relativity

 

Gravity is a distortion in space-time, which is felt as a pull on things proportional to a combination of their energy and momentum. The source of the curvature is the stress-energy tensor and this makes no distinction between mass and energy. They are treated as related by the famous E=mc^2.

When you first learn about gravity in school, you learn Newton’s law: that the force of gravity between two objects, one of mass M1 and one of mass M2, has a strength proportional to the product M1M2. { F = (GM1M2)/r^2}

But that was true before Einstein. It turns out that Newton’s law needs to be revised: the Einsteinian statement of the law is (roughly) that for two objects that are slow-moving (i.e. their speed relative to one another is much less than c, the speed of light) and have energy E1 and E2, the gravitational force between them has a strength proportional to the product E1E2.

How are these two statements, the Newtonian and the Einsteinian, consistent? They are consistent because Einstein and his followers established that for any ordinary object, the relation between its energy E, momentum p and mass M [sometimes called “rest mass”, but just called `mass’ by particle physicists] is

 E^2 = (p c)^2 + (M c^2)^2

For a slow-moving object, p ≈ Mv (where v is the object’s velocity) and pc ≈ Mvc is much smaller than Mc^2. And therefore

E^2 ≈ (M c^2)^2 (i.e., E ≈ M c^2 for slow objects)

Since planets, moons, and artificial satellites all move with velocities well below 0.1% of c relative to each other and to the sun, the gravitational forces between them are proportional to

*E1E2 ≈ M1 M2 c^4

And since c is a constant, for such objects Einstein’s law of gravity and Newton’s law of gravity are completely consistent; the force law is proportional to the product of the energies and to the product of the masses, because the two are proportional to one another.

But for objects that have high speeds relative to one another, or for objects subject to extremely strong gravitational pulls (which will quickly develop high speeds if they don’t have them already), the Einsteinian law of gravity involves a complicated combination of momentum and energy, in which mass does not explicitly appear. This is why Einstein’s version of gravity even pulls on things like light, which is made from photons that have no mass at all.

In general relativity, the gravitational field is determined by solving the Einstein field equations.

G = [(8πG')/c^4]*T

Here T is the stress–energy tensor, G' is the Einstein tensor, and c is the speed of light.