An exercise in eloquence: Basic GTR

[Crosson]

I have two questions that were on a (past due) take home test in my GTR class. There maybe many answers, but I am interested in the best answers.

1) Why is einstein's GTR a theory of gravity?

2) Why is einstein's GTR a geometric theory?

I welcome anyone who thinks they have an answer to post it, but I am particularly interested in the more sophisticated answers. (In other words, don't look at these as shallow novice questions). I will post my answers (for critique) after I see some of yours.

 

[jcsd]

The equiavlence princple is all you need, which staes grviataional and inertial forces are equiavelent.

The immediate result of this is that there is not always a globally inertial coordinate system (only in special case of a globally pseudo-Euclidean geometry) though locally there is always an inertial coordinate system. Compare to the axiomatixc definition of a manifold which can be thought of as locally Euclidean, but not always globally Euclidean and you should see that differential geometry could be used to model gravity.

 

[Crosson]

Here is my answer to #1: (sorry for the length)

GTR is a statement of the laws of physics that has the same form in all frames of reference. With respect to gravitation, Newton’s theory accomplishes this for all inertial observers:

$$\nabla^2 \Phi = 4 \pi G \rho$$

$$\vec{a} = -\nabla \Phi$$

For a simple reason, this expression of gravity cannot be generally covariant. With respect to a general transformation, the components of the acceleration vector will transform, so the gradient of the potential must transform, but the mass density in Poisson’s equation does not transform. (A classicist would say Poisson’s equation was correct, and that we had introduced pseudo forces through our choice of frame.)

Einstein firmly believed general covariance was possible, and that an updated law of gravitation which involved the mass energy equivalence of special relativity would be a necessary part of his General Theory. The natural covariant representation of mass-energy is called the stress-energy tensor. So, following this line of reasoning:

(Covariant Representation of Gravitational Field) = (Stress-Energy Tensor)