GTR Postulates: What to Know About the Equivalence Principle

[A AM ARYA]

I know STR is based on two postulates viz.
1.All laws of physics are equally valid in all inertial frames of reference i.e There is no such experiment one can perform to know whether he is moving or not if he is in an inertial frame of reference.
2.The speed of light is constant(3*10^8 m/s approx.) in vacuum.
But what about the postulates of GTR?Is it only the equivalence principle or something else?

 

[dextercioby]

The 2nd one for SR is usually formulated:
The speed of light in vacuum has the same value (299792458 m/s) irrespective of the inertial system of reference it's being measured in.

 

[bcrowell]

A given theory can have more than one axiomatization. What you're referring to in #1 is Einstein's original 1905 axiomatization, which from the modern point of view is goofy. There are other axiomatizations, based on symmetry principles, that make more sense from the modern point of view. (This is the approach used by, e.g., Morin, Takeuchi, and my own SR book, http://lightandmatter.com/sr/ .)

Likewise there is no particular reason to expect there to be a unique axiomatization of GR. However, one possibility might be as follows. (1) Spacetime is a Hausdorff differentiable manifold with a Lorentzian metric. (2) The Einstein field equation holds.

In nontechnical language, statement #1 basically says that spacetime doesn't have to be flat or have the trivial topology, while #2 says "Spacetime tells matter how to move; matter tells spacetime how to curve" (Wheeler).

More broadly, GR can be described as the unique generalization of SR that has the equivalence principle and general covariance. This description is in some ways nicer because it allows for other formulations of the theory that are more or less equivalent to the standard one, such as ADM or Ashtekar, or the description of gravity as a spin-2 field on a flat background (Deser).

 

[samalkhaiat]

I know STR is based on two postulates viz.
1.All laws of physics are equally valid in all inertial frames of reference i.e There is no such experiment one can perform to know whether he is moving or not if he is in an inertial frame of reference.
2.The speed of light is constant(3*10^8 m/s approx.) in vacuum.
But what about the postulates of GTR?Is it only the equivalence principle or something else?

1) Pseudo Riemannian manifold $M^{4}$.
2) The metric tensor $g_{\mu\nu}$ on $M^{4}$ is a dynamical field satisfying second order partial differential equations.
3) The principle of equivalence.
4) The principle of general covariance.
1-4 lead uniquely to the Hilbert-Einstein Lagrangian. This in turn leads to the Einstein’s field equations of GR via an
5) Action principle.

 

[bcrowell]

1) Pseudo Riemannian manifold $M^{4}$.
2) The metric tensor $g_{\mu\nu}$ on $M^{4}$ is a dynamical field satisfying second order partial differential equations.
3) The principle of equivalence.
4) The principle of general covariance.
1-4 lead uniquely to the Hilbert-Einstein Lagrangian. This in turn leads to the Einstein’s field equations of GR via an
5) Action principle.

These have a lot of redundancy. 1, 2, and 5 would suffice. There also doesn't seem to me to be a clean logical separation between 2 and 5 in your formulation.

3 and 4 don't really work as postulates, because nobody has succeeded in formulating them in a way that's both rigorous and nontrivial. Re general covariance, see http://arxiv.org/abs/gr-qc/0603087 . Re the e.p., see http://arxiv.org/abs/0707.2748 . In any case, they're not needed as postulates. See also http://arxiv.org/abs/gr-qc/0309074v1 .

If you think the 2 postulates I listed in #3 are insufficient, and a much longer list like this is necessary, I would be interested to see a counterexample, i.e., a model of my postulates that is not a model of your postulates.

This is more of a technical quibble, but most people agree that you need to explicitly say that the spacetime is Hausdorff.

 

[samalkhaiat]

By $M^{4}$ it is meant a smooth oriented 4-manifold with unbounded, connected, paracompact and Hausdroff included in the term “manifold”.
1-4 allow you to derive the H-E Lagrangian.

 

[bcrowell]

1-4 allow you to derive the H-E Lagrangian.

It's impossible to say whether this claim is true unless you define in more detail what you mean by your postulates 3 and 4. As described in detail in the references I gave in my #5, these two postulates have no rigorous meanings as stated.

You also haven't addressed my question about demonstrating the necessity of your 3 and 4 by exhibiting a model -- but this would also depend on your being able to define more rigorously what you mean by 3 and 4.

 

[bcrowell]

1) Pseudo Riemannian manifold $M^{4}$.

BTW, a couple of side issues on this point. As written, this would allow 2+2 dimensions but exclude 2+1 or 5+1. I don't think anyone considers a theory of 2+2-dimensional spacetime to be GR, which is why I think "Lorentzian" works better than "pseudo-Riemannian." As a matter of taste, I also don't think it's a good idea to exclude n+1 dimensions with $n\ne3$. Cases other than n=3 are frequently of interest, even as descriptions of our own universe, and are usually included as part of GR.

 

[samalkhaiat]

I am aware of all the garbage talk about the meaning of general covariance and equivalence principle, and I really have no time to read more of that garbage. I get things done. See the PDF in this thread.
https://www.physicsforums.com/threads/einstein-hilbert-action-origin.838487/#post-5264554

 

[dextercioby]

Proving that a set of axioms A for a theory is equivalent to another set B (or that some axioms in the set A are actually theorems of the remaining axioms in the set A) is a nice logical and mathematical exercise. To my modest knowledge, there are only 2 books I know of that put axioms for GR: an elementary one (Ray d'Inverno's "Introducing Einstein's Relativity") and an advanced one (Sachs and Wu's "General Relativity for Mathematicians").

 

[bcrowell]

To my modest knowledge, there are only 2 books I know of that put axioms for GR: an elementary one (Ray d'Inverno's "Introducing Einstein's Relativity") and an advanced one (Sachs and Wu's "General Relativity for Mathematicians").

Cool. Those systems are likely to have been more carefully constructed than what we've come up with here. Can you describe them for us?