Energy Momentum Tensor Prerequisites: What Do I Need to Know?

[kent davidge]

I have a feeling that topics related to the Energy Momentum tensor are the most difficult part when learning Relativity. At least to me, it seems that the textbooks I'm reading assume that readers have a previous knowledge on some other area, maybe it's classical mechanics of fluids or something like that.
Note that I'm not talking about the Energy Momentum tensor itself, as in principle it's just a tensor and all you have to know about is Differential Geometry.

So does learning about Energy Momentum require some previous knowledge? If so, what?

 

[Orodruin]

It certainly helps if you took electrodynamics and solid mechanics. In essence they contain everything you need.

In electrodynamics, you already have the stress-energy tensor, but it is split into three parts, the energy density, the Poynting vector, and the electromagnetic stress tensor - just in the same way as 4-momentum is split into energy and momentum parts.

 

[kent davidge]

It certainly helps if you took electrodynamics and solid mechanics. In essence they contain everything you need.

those talks about perfect fluids and later, the energy momentum tensor for some stars.. are they contained in solid mechanics? because certainly they are not in electrodynamics

 

[Orodruin]

those talks about perfect fluids and later, the energy momentum tensor for some stars.. are they contained in solid mechanics? because certainly they are not in electrodynamics

Doesn’t matter. The idea is the same regardless of the continuum you are describing.

 

[pervect]

I have a feeling that topics related to the Energy Momentum tensor are the most difficult part when learning Relativity. At least to me, it seems that the textbooks I'm reading assume that readers have a previous knowledge on some other area, maybe it's classical mechanics of fluids or something like that.
Note that I'm not talking about the Energy Momentum tensor itself, as in principle it's just a tensor and all you have to know about is Differential Geometry.

So does learning about Energy Momentum require some previous knowledge? If so, what?

There's a couple of ways of describing the stress-energy tensor that I like, in addition to the one explained by Orodruin already.

One of the simplest is to find the stress-energy tensor of a swarm of particles. The ultra-short version of that is to consider that every particle has a number-flux four vector, a generalization of the charg-current four vector, and an energy-momentum four-vector.

Then the stress energy tensor is the tensor product of the number-flux four vector and the energy-momentum 4-vector. You know it's a tensor, because the tensor product of two tensors is another tensor. You sum the tensors from each particle in the swarm to get the tensor representing the swarm, they're linear.

t's possible, though not often done in textbooks, to envision fields as swarms of particles, rather than to use the Lagrangian formulation. I've seen this done in a few FAQ's, though. Particularly interesting is how you model a rope under tension with this approach. One winds up with particles of positive mass (forming the rope) exchanging particles of negative mass (possibly regarded as virtual particles) to create the tension forces in the rope.

A more abstract approach than the "swarm of particles" approach is to consider the stress energy tensor as the functional derivative of the Lagrangian density with respect to the metric.

Learning about the Lagrangian density formulation of fields from, say, Goldstein's "Classical Mechanics" , is recommended for this approach.

This still doesn't quite really motivate why one takes the functional derivative with respect to the metric to get the stress energy tensor, at least not to me. But it serves as a formal defnition, at least.

 

[Orodruin]

t's possible, though not often done in textbooks, to envision fields as swarms of particles, rather than to use the Lagrangian formulation. I've seen this done in a few FAQ's, though. Particularly interesting is how you model a rope under tension with this approach. One winds up with particles of positive mass (forming the rope) exchanging particles of negative mass (possibly regarded as virtual particles) to create the tension forces in the rope.

I use a the distribution function argument in my SR lecture notes to arrive at the equations of state for a gas of massive/massless particles (or with T >> m). It is rather effective and you never really need to know the actual distribution.