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Why is the equivalence principle so important to general relativity?

Why is the equivalence principle so important to general relativity?


1 Answers

Saurabh Koranglekar
askIITians Faculty 10341 Points
one year ago
A derivation of Einstein's equation isn't why the Equivalence principle is central to GR. The reason that the equivalence principle is central to GR is in the fact that you can represent the gravitational field with a metric tensor at all--you can replace a force equation with a geodesic equation for a test mass precisely due to the fact that the geodesic that that test mass follows (or the "acceleration" felt by a Newtonian mass) is independent of the mass of that test11particle.

The equivalence principle, however, only selects out that one can represent gravity with a metric tensor. There are a great many other so-called "metric theories of gravity" that obey the equivalence principle, but are not general relativity--amongst other things, they will differ in the field equation for the metric tensor, or have extra fields in addition to the metric--the most famous of these is the Brans-Dicke theory, which treats Newton's constant as a scalar field coupled to the metric tensor. Most alternative metric theories have either been experimentally ruled out, or have had their additional fields constrained to the point where their values are consistent with zero (for instance, Brans-Dicke theory has a parameterωω, and tends to GR if1ω→01ω→0. Current data says thatω>4000ω>4000, or some similar number.).

11Note that this is generally only true if the mass of the test particle is "small" compared to the local curvature of the spacetime, and if it's motion is slow enough to not produce gravitational radiation comparable to its energy. Either of these effects will cause the test mass to perturb the background spacetime, and those effects will both be mass dependent and cause the test mass to not follow a geodesic of the background spacetime. Both of these approximations are true (to great precision, at least) of all of the planets, asteroids and comets orbiting the sun, amongst many other things.

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