# Why is the equivalence principle so important to general relativity?

In its simplest form, equivalence principle states that the inertial mass and the gravitational mass should be the same. This is easy to understand.

But why is it so important to the formulation of General Relativity? To be more specific, I don't understand how the gravitational field equation:

can be derived from this principle.

-
It can't be derived from that principle alone. The equation is derived from several different observations as well as certain assumptions. So do you want to hear how to get to Eistein's equations, or why is equivalence principle important? These two questions are quite unrelated. –  Marek Jan 1 '11 at 15:31
@Marek, why and how equivalence principle is important in the formulation of Einstein's equations. I'm not really interested in knowing the step by step derivation to get Einstein's equation. –  Graviton Jan 1 '11 at 15:33
that is not a very valid question to ask so let me instead provide answer discussing these topics so that you see why it doesn't make sense. Hopefully it will be useful to you. –  Marek Jan 1 '11 at 15:35
I have posted the steps on how Einstein arrives to the field equations from the Equivalence Principle and other assumptions here –  Eduardo Guerras Valera Apr 22 '13 at 1:45

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 test$^{1}$ particle.

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 parameter $\omega$, and tends to GR if $\frac{1}{\omega}\rightarrow 0$. Current data says that $\omega > 4000$, or some similar number.).

$^{1}$Note 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.

-
In the Brans-Dicke model, if the test particle has a nonzero trace for its stress-energy tensor, and there's a gradient in the Brans-Dicke scalar, there will be violations from the equivalence principle. –  QGR Jan 9 '11 at 11:13

Equivalence principle states (very roughly) that movement of objects doesn't depend on their mass (so long as they are massive, of course). These important observation is what introduces (pseudo)Riemannian geometry into the theory of gravitation, because it essentially tells us that matter that is not acted on by other forces follows the geodesics of the given manifold.

In total, there are three key ingredients to General Relativity and only one of them relates directly to the Einstein's equations. Let me mention each one of them briefly:

1. General covariance This a requirement that every reference frame is good for doing physics and brings in the concept of manifold and diffeomorphism invariance.

2. Equivalence principle This is a requirement that the manifold be (pseudo)Riemannian and that test particles follow geodesics (a note on this at the end of the answer).

3. Interaction of matter and space-time in a certain manner and reduction to Newtonian gravity in a classical limit This is what gives you Einstein equations.

So it's not really well-posed question to ask what role does equivalence principle play in deriving Einstein's equations. Only correct answer would be: it lets you introduce the concept of (pseudo)Riemannian manifold so that you have some well-defined object to define the equations for in the first place. In particular the Einstein tensor $G_{\mu \nu}$ depends on a metric tensor, so you already have to know there is such a thing as a metric (which is what equivalence principle tells you) even before you start pondering whether there might be such a thing as Einstein's equation.

Note: One can have geodesics on other mathematical structures than just (pseudo)Riemannian manifolds. A manifold with connection is enough to have but there are other reasons people like to have metric tensor around (which induces a special kind of connection by itself).

-