Physical Interpretation of the Einstein Tensor [Technical]

Son Goku

What would be its physical interpretation. Just wondering how people see it.

planck2

Son Goku wrote:

What would be its physical interpretation. Just wondering how people see it.

It relates the presence of energy and matter present in a local region of space-time ( described by the energy-momentum tensor T) to the curvature of the space time via the the Ricci curvature tensor, the metric and the Ricci scalar. Conservation of energy is related to the fact that the absolute covariant derivative of the metric vanishes.

Son Goku

I was more wondering do you see it as simply a mathematical combination of the Ricci curvature Tensor, the Ricci scalar and the metric or as an entity on its own.
As an entity on its own it is usually considered the moment of rotation of space, but I would tend to see it more as a combination of the other Tensors, rather than a "thing" on its own.

Basically do you see:

or


Sorry for being unclear.

planck2

It's not a thing on its own because it is defined as above. It may be interpreted as a moment, but it's not an interpretation that I have heard before, where do you find such an interpretation.

Son Goku

Gravitation, Wheeler Misner Thorne, 1973.
Must be unusual to interpret it on its own then.

You'll have to forgive me, Gravitation made a good case that it is a physical quantity, but equally it seems more to be a simple combination of the other Tensors and wasn't sure which was the more conventional way of viewing it.

planck2

Son Goku wrote:

Gravitation, Wheeler Misner Thorne, 1973.
Must be unusual to interpret it on its own then.

You'll have to forgive me, Gravitation made a good case that it is a physical quantity, but equally it seems more to be a simple combination of the other Tensors and wasn't sure which was the more conventional way of viewing it.

Well you see mathematically you can say that it is a combination of tensors which are curvature terms, but physically it is energy-momentum.I see what you mean: on its own G may be interpretted as a moment ( am I right, I haven't looked at MTW yet), but this wouldn't really tell us anything

Son Goku

MTW defines it as "the amount of vector valued moment of rotation [of space] in the element of 3-volume of arbitrary co-ordinates."

Although I agree that it doesn't really tell us much. I myself would prefer to think of it as a combination of the other Tensors.

planck2

I have just browsed MTW. I see that when talking about G as a moment they are using the language of differential forms and the Cartan calculus. What this formalism is really useful for is quick calculations of the curvarture, but it also a much handier way of showing that G is divergence free and hence the fact that T is also divergence free.