Find SEM Tensor from Lagrangian of Temporal Variable

[Tertius]

TL;DR Summary

I want to obtain the SEM tensor a from a scalar field that has been integrated over all space.

It seems the field φ(t, xⁱ) could be integrated over all space to form a single temporal variable (which isn't a field anymore, but is just a function of time) as follows:

Φ(t) = ∫φ(t, xⁱ)dxⁱ​

Suppose we then assume a Lagrangian from this temporal variable to be:

L₁ = -1/2 Φ'(t)² + 1/2 b² Φ(t)²
​

From this we can turn to finding the SEM tensor. From Carroll pg. 164, the SEM tensor for a scalar field with a Lagrangian density of the form

L₂ = -1/2 g^μν(∇_μΦ)(∇_νΦ) - V(Φ)

is given by:

T_μν = ∇_μΦ∇_νΦ - 1/2 g_μν g^ρσ∇_ρΦ ∇_σΦ - g_μνV(Φ)
​

The only difference between the two Lagrangians is the sign and form of the "potential" term. So, substituting in the second term of the first Lagrangian in for the second term of the second Lagrangian, since neither depends on the variation of the inverse metric, we can obtain the following as the SEM tensor for L₁

T_μν = ∇_μΦ∇_νΦ - 1/2 g_μν g^ρσ∇_ρΦ ∇_σΦ + g_μνb² Φ(t)²
​

That is at least what we get if we change the sign of the last term. However, when I leave it a negative, i.e. (- g_μνb² Φ(t)²), I get an answer that makes more sense for the T₀₀ component (it looks like the hamiltonian total energy when I leave it negative, and has a subtracted term when positive). Am I making a mistake in doing this substitution?

 

[PeterDonis]

Summary:: I want to obtain the SEM tensor a from a scalar field that has been integrated over all space.

You can't. The stress-energy tensor is not an integral. There is a different such tensor for every event in spacetime; that's how tensors work. The stress-energy tensor of a scalar field at a given event depends only on the field and its derivatives at that event.

 

[Tertius]

Thanks for the response. I suppose I can write the question more clearly: I would like to obtain the SEM tensor from an object that is only a function of time. When computing the SEM tensor for cosmic dust, (w/o pressure terms) the density is only a function of time. I'm just wondering how to do this starting with a slightly more complicated temporal object like from the L₁ I listed above.

 

[PeterDonis]

I would like to obtain the SEM tensor from an object that is only a function of time.

You can't. Tensors are attached to points in spacetime, not times.

 

[PeterDonis]

When computing the SEM tensor for cosmic dust, (w/o pressure terms) the density is only a function of time.

No, the density is a function of points in spacetime. It just so happens that, in the particular spacetime you are talking about (the FRW spacetimes used in cosmology), the spacetime admits a choice of coordinates for which the density is only a function of the time coordinate. Another way of stating this property is that the spacetime is homogeneous and isotropic.

One can consider a scalar field in a homogeneous and isotropic spacetime that also shares that same property, of being homogeneous and isotropic. In that case, the scalar field will only be a function of the time coordinate in the standard FRW coordinates referred to above. But that does not mean you obtain the SET for the scalar field by integrating over space. You don't. You obtain it for a point of spacetime, just like any other SET; the field just happens to have the property that, for a set of points in spacetime that all have the same time coordinate in standard FRW coordinates, the SET for any point in the set is the same as the SET for any other point in the set.

 

[Tertius]

That makes sense. Yes, the FRW metric is built on co-moving coordinates, so yeah it would only represent a single point, but it just happens to be made into a universal point.

This was very insightful. much appreciated.

 

[Tertius]

the situation I am imagining is as follows:

1) there is no dark energy, so no spatial expansion
2) the universe is isotropic and homogeneous
3) it is filled with a single scalar field (which, because of condition 2 should appear only as a temporal variable in the stress energy tensor)

Is my approach I outlined in the original question a suitable method to find the stress energy tensor?

 

[PeterDonis]

1) there is no dark energy, so no spatial expansion

Wrong. There would still be expansion; it just wouldn't be accelerating. Although, with a scalar field, it actually would; a scalar field acts the same as dark energy (aka a cosmological constant) does as far as expansion is concerned. This is how inflation is modeled in cosmology; the inflaton field is a scalar field with a very high energy density, which therefore causes a rapidly accelerating expansion.

3) it is filled with a single scalar field (which, because of condition 2 should appear only as a temporal variable in the stress energy tensor)

This is still true if you leave out the "in the stress energy tensor" part. In a universe that is homogeneous and isotropic, when you are working in standard FRW coordinates, any variable can only be a function of the time coordinate.

Is my approach I outlined in the original question a suitable method to find the stress energy tensor?

You already know the stress-energy tensor as soon as you specify a scalar field. The SET of a scalar field is already known; you gave it yourself in the first post in this thread. If you want to simplify the expression, you need to use the FRW metric to find the components $g_{\mu \nu}$ and substitute them into the expression. No integration needed or wanted.