What is the coordinate free stress-energy-momentum tensor

[brombo]

Without regard to a coordinate system (I only wish to consider special relativity) the stress-energy-momentum tensor defines a linear transformation from a 4-vector to a 4-vector. Let T be the linear transformation then b = T(a), a and b are 4-vectors. What is the physical meaning of a and b or for a and b arbitrary vectors what is the physical meaning of a⋅T(b).

 

[haushofer]

It depends on your choice of four-vector, I'd say. Take for example the energy-momentum tensor of a dust and contract it with the four-velocity. What do you get?

 

[Brage]

I am not sure what you mean but if you look for example at the Einstein-Hilbert Wikipedia page you can get a definition for the stress-energy tensor in terms of the variation of the lagrangian matter density w.r.t your metric.

 

[brombo]

For a Lagrangian, $\mathcal{L}$, in which the $\phi_{i}$ are vector fields the stress-energy-momentum tensor is give by the following coordinate free geometric algebra expression -
$$T(n) = \dot{\nabla}\left < \dot{\phi}_{i}\partial_{\nabla\phi_{i}}\mathcal{L}n\right >-n\mathcal{L}$$
where $\nabla$ is the 4-vector gradient, $n$ a 4-vector, $\left < A \right >$ the scalar part of the multivector $A$, and the overdot indicating that the partial derivatives of $\nabla$ only operate on $\phi_{i}$. The question is what is the physical meaning of $T(n)$?

 

[pervect]

Without regard to a coordinate system (I only wish to consider special relativity) the stress-energy-momentum tensor defines a linear transformation from a 4-vector to a 4-vector. Let T be the linear transformation then b = T(a), a and b are 4-vectors. What is the physical meaning of a and b or for a and b arbitrary vectors what is the physical meaning of a⋅T(b).

If you have MTW's "Gravitation", look at page 131.

If you let u be the 4-velocity of some observer, then T(u), where T is the stress-energy tensor regarded as a linear map in the manner you suggest, can be regarded as the density of energy-momentum seen by that observer - i.e the amount of energy/momentum contained in a unit volume.

MTW suggests other useful physical interpretations for the stress energy tensor. The other definitions don't use the notion of the tensor as a linear map from a vector to a vector, $T^a{}_b$ as you ask, but rather regard it as a linear map from two vectors to a scalar $T_{ab}$. It's a bit of a digression to give them all (as well as being more work), so I won't give these interpretations here, unless you are curious and ask.

 

[haushofer]

For a Lagrangian, $\mathcal{L}$, in which the $\phi_{i}$ are vector fields the stress-energy-momentum tensor is give by the following coordinate free geometric algebra expression -
$$T(n) = \dot{\nabla}\left < \dot{\phi}_{i}\partial_{\nabla\phi_{i}}\mathcal{L}n\right >-n\mathcal{L}$$
where $\nabla$ is the 4-vector gradient, $n$ a 4-vector, $\left < A \right >$ the scalar part of the multivector $A$, and the overdot indicating that the partial derivatives of $\nabla$ only operate on $\phi_{i}$. The question is what is the physical meaning of $T(n)$?

Wow, talking about disgusting notation :D

Instead of giving fancy-pancy coordinate free definition, let's start with the stresstensor of a dust. It is given by

$$T_{ab} = \rho u_a u_b$$

where rho is the energy density and u is the 4-velocity of the dust-particles. If you contract this with the four velocities twice, you get

$$T_{ab} u^a u^b = \rho$$

That's a start of an interpretation, right?

 

[brombo]

Thank you pervect. Your answer referencing MTW was exactly what I was looking for.