Energy-Momentum Tensor: Validity in Relativity?

[davidge]

As you may know from some other thread, I was interested through the week in finding a general way of express the energy-momentum tensor that appears in one side of the Einstein's equation.
After much trials, I found that

$$T^{\sigma \nu} = g^{\sigma \nu} \frac{\partial \mathcal{L}}{\partial \partial_{\mu} \phi} \partial_{\nu} \phi - g^{\sigma \nu} \delta^{\mu}{}_{\nu} \mathcal{L}$$ where $\mathcal{L}$ is the Lagrangian density.

As I haven't found an expression like this one on web, I'm unsure about its validity in Relativity.

I would appreciate if someone could tell me whether this expression is valid or not.

I can post here links to the pdf's I have read and how I arrived at the above result.

 

[PeterDonis]

I was interested through the week in finding a general way of express the energy-momentum tensor

What you are doing is not general enough; you're assuming that the matter is a scalar field (that's what $\phi$ is).

The most general way of expressing the stress-energy tensor that appears on the RHS of the Einstein Field Equation is the Hilbert stress-energy tensor, given here:

https://en.wikipedia.org/wiki/Stress–energy_tensor#Hilbert_stress.E2.80.93energy_tensor

Note that this expression just involves $\mathcal{L}_{\text{matter}}$, i.e., the Lagrangian density for matter, without making any assumptions about the form that Lagrangian density takes.

I would appreciate if someone could tell me whether this expression is valid or not.

Even leaving out the above, your expression is not valid on its face because the indexes don't match up; you have $\sigma$ and $\nu$ as free indexes on the LHS but $\sigma$ and $\mu$ as free indexes on the RHS.

However, as above, the more general problem with your expression is that it assumes that the matter is a scalar field, which you shouldn't do if you want the most general expression possible.

Even for matter as a scalar field, though, your expression is not correct. You can use the general Hilbert form I linked to, plus the correct form of $\mathcal{L}_{\text{matter}}$ for a scalar field, which is not just $\phi$, to derive what the correct stress-energy tensor is for a scalar field.

 

[davidge]

Thanks Peter.

your expression is not valid on its face because the indexes don't match up; you have $\sigma$ and $\nu$ as free indexes on the LHS but $\sigma$ and $\mu$ as free indexes on the RHS.

I'm sorry, it was a typo. The correct would be inserting $\mu$ instead of $\nu$ on the left hand side.

you're assuming that the matter is a scalar field (that's what $\phi$ is).

I thought we could always represent matter as a scalar quantity, represented by a scalar field. Why it is not the case?

The most general way of expressing the stress-energy tensor that appears on the RHS of the Einstein Field Equation is the Hilbert stress-energy tensor

Unfortunately the Wiki article don't provide a derivation for it. But it's interesting to see its form. The Lagrangian density is dependent of what quantities?

 

[PeterDonis]

I thought we could always represent matter as a scalar quantity, represented by a scalar field.

I have no idea why you would believe that. In any case, it's false. All ordinary matter is fermions, which are spinor fields (spin 1/2), not scalar fields (spin 0). All gauge bosons in the Standard Model are spin-1 fields. The only spin-0 field in the Standard Model is the Higgs.

More generally, at the macroscopic level, the simplest model of matter is a perfect fluid, which is not the same as a scalar field, as is evident from looking at their respective stress-energy tensors.

The Lagrangian density is dependent of what quantities?

It depends on what kind of matter you are talking about. There is no general rule. That's why the Hilbert stress-energy tensor makes no assumptions about its form or what it depends on.

 

[PeterDonis]

the Wiki article don't provide a derivation for it.

Click on the "Einstein-Hilbert action" link.

 

[davidge]

I have no idea why you would believe that. In any case, it's false. All ordinary matter is fermions, which are spinor fields (spin 1/2), not scalar fields (spin 0). All gauge bosons in the Standard Model are spin-1 fields. The only spin-0 field in the Standard Model is the Higgs.

That's because I've seen many times phrases like "the field has a mass m"; "a massive field"; etc... (as when one talks about the Klein-Gordon Equation). So I related these things in my mind.

It depends on what kind of matter you are talking about. There is no general rule. That's why the Hilbert stress-energy tensor makes no assumptions about its form or what it depends on.

Ok

 

[PeterDonis]

I've seen many times phrases like "the field has a mass m"; "a massive field"; etc... (as when one talks about the Klein-Gordon Equation)

Sure, the scalar field can have a mass. But that doesn't mean the scalar field is the only field that can have a mass, which is what your belief that all matter can be described by scalar fields implies. You will see similar statements made about fermion fields and the Dirac equation in many texts. You will also see many texts talking about massive spin-1 fields in the context of showing what kinds of experiments might tell you whether photons have mass or not.

 

[haushofer]

By the way, if you want to understand the coupling of spin 1/2 to gravity, you need the vielbein-(or tetrad)formalism. See e.g. Carroll on GR or Samtleben on SUGRA.

 

[PeterDonis]

that doesn't mean the scalar field is the only field that can have a mass, which is what your belief that all matter can be described by scalar fields implies

Actually, even this isn't broad enough. Massless fields can also contribute to the stress-energy tensor. Typically the term "radiation" is used for massless fields (such as photons in the early universe), but as far as GR is concerned, stress-energy is stress-energy, and all types of stress-energy appear in the Einstein Field Equation. In other words, "matter" as the term is usually used is not general enough to describe all of the possible sources in the EFE.