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:

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can be derived from this principle.

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    $\begingroup$ 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. $\endgroup$ – Marek Jan 1 '11 at 15:31
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    $\begingroup$ @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. $\endgroup$ – Graviton Jan 1 '11 at 15:33
  • $\begingroup$ 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. $\endgroup$ – Marek Jan 1 '11 at 15:35
  • $\begingroup$ I have posted the steps on how Einstein arrives to the field equations from the Equivalence Principle and other assumptions here $\endgroup$ – 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.

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  • $\begingroup$ 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. $\endgroup$ – 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).

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Here is a different answer from the above.

The Einstein Equivalence Principle (EPP) states that if you have a microscopic physics lab able to experiment with all the laws of physics, you cannot tell what reference frame you are in: inertial in the intergalactic void, or free falling into a neutron star.

In other words, the EPP states that any reference frame is good for doing physics, same as stated by General Covariance. Einstein just adopted the covariance terminology for some reason for the 1916 paper, probably to get away from assumptions and arguments over which version of the equivalence principle, as there were several.

So the bottom line is, the EPP is necessary for the derivation of GR. But of course there were other factors and choices that differentiate it from other metric theories in which the EPP holds, as one responder already noted.

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I don't think the equivalence principle is all that important in modern derivations of general relativity. Instead, a much more central role is given to diffeomorphism covariance. In fact, there are modifications of general relativity — like introducing dilaton couplings with dilaton gradients, or spinning particles moving in a spacetime with torsion — which still satisfy diffeomorphism covariance, but violate the equivalence principle.

Actually, the equivalence principle is more or less equivalent to the statement that freefalling particles only minimally couple to the metric tensor, and that there are no other geometrical fields, like a dilaton or a torsion tensor. So, if we make this assumption, and also assume the covariant dynamical equations for the metric tensor can only be at most second order in the derivatives, the Einstein field equations is just about the most general equation. And the source tensor that is being coupled to has to be symmetric and have a zero covariant divergence. According to the Coleman-Mandula theorem, just about the only field with these properties is the stress-energy tensor. So I guess you can kind of derive the Einstein field equations from the equivalence principle.

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  • $\begingroup$ how do spinning particles violate the equivalence principle? Can you elaborate on that? $\endgroup$ – user346 Jan 9 '11 at 9:00
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    $\begingroup$ @space_cadet: If we have nonzero torsion, freefalling spinning particles will follow different trajectories from spinless ones. Torsion is usually assumed to be zero in general relativity, but that's not absolutely necessary. $\endgroup$ – QGR Jan 9 '11 at 11:09
  • $\begingroup$ even with zero torsion one might still choose non-metric connection. I wonder whether such a theory (that would approach GR in some limit, presumably) has been proposed. $\endgroup$ – Marek Jan 9 '11 at 15:10
  • $\begingroup$ so how does that violate the EP? You can always have a connection with torsion which would encompass that difference between spinning and non-spinning particles, right? The reason I'm asking all these questions is not to criticize your answer but to draw out the best explanation on the difference between spinning and non-spinning particles and the importance of torsion for the EP. @Marek - Hehl and co-workers have ben working on such ideas for more than 40 years now. Also look up Kibble-Sciama-Einstein-Cartan theory. $\endgroup$ – user346 Jan 9 '11 at 19:19
  • $\begingroup$ Spinning particles don't move on geodesics even in normal GR. They satisfy the Papapetrou equations. $\endgroup$ – Stingray Mar 29 '11 at 18:38

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