Obtaining the matter Lagrangian from the stress energy tensor

[Bishal Banjara]

TL;DR

Generally, the Stress energy tensor is obtained from the Lagrangian. But is it possible to obtain matter Lagrangian (Lm) from the Stress energy tensor?

Basically, the stress energy tensor is given by $$T_{uv}=-2\frac{\partial (L\sqrt{-g})}{\partial g^{uv}}\frac{1}{\sqrt{-g}}.$$ It makes easy to calculate stress energy tensor if the variation of Lagrangian with the metric tensor is known. But it is possible to retrieve matter Lagrangian if the stress energy tensor is known? Is one of the possible way to solve is taking the integration of the above equation?

[Moderator's note: Some off topic content has been deleted.]

 

[dsaun777]

I think you are supposed to take the variation of the lagrangian with respect to the metric. But I think you can derive the Lagrangian density from the Einstein Hilbert action which is a functional of the metric tensor and Ricci scalar. The Ricci scalar can be calculated directly from the stress-energy tensor if you know the metric.

 

[PeterDonis]

I think you are supposed to take the variation of the lagrangian with respect to the metric.

Yes, that's correct. That's what the equation in the OP describes.

I think you can derive the Lagrangian density from the Einstein Hilbert action

The "Einstein-Hilbert action" is the Lagrangian density (technically the "action" is the integral over spacetime of the Lagrangian density, but that just means you read off the Lagrangian density from the integrand; there's no "derive" needed).

which is a functional of the metric tensor and Ricci scalar.

Yes.

The Ricci scalar can be calculated directly from the stress-energy tensor if you know the metric.

You don't even need the stress-energy tensor; the Ricci scalar is a function of the metric and its derivatives.

 

[PeterDonis]

taking the integration of the above equation

As @dsaun777 points out, that equation is for the variation of the Lagrangian density with respect to the metric. That is not the same thing as a derivative and so the equation cannot be integrated the way you are thinking.

 

[Bishal Banjara]

I think you are supposed to take the variation of the lagrangian with respect to the metric.

If this is solved, rest is simple.

 

[PeterDonis]

If this is solved, rest is simple.

As far as obtaining the stress-energy tensor from the Lagrangian, yes.

But in the OP you are asking about obtaining the Lagrangian from the stress-energy tensor. See post #4.

 

[Bishal Banjara]

The Ricci scalar can be calculated directly from the stress-energy tensor if you know the metric.

I know the metric, then what is the mathematical relation between the Ricci scalar and stress energy tensor?

 

[Bishal Banjara]

As far as obtaining the stress-energy tensor from the Lagrangian, yes.

But in the OP you are asking about obtaining the Lagrangian from the stress-energy tensor. See post #4.

Yes, I am asking to retrieve the case. But if there is way to explore the variation of matter Lagrangian density with metric tensor resolving the variation, then don't this makes sense to solve the problem?

 

[Bishal Banjara]

That is not the same thing as a derivative and so the equation cannot be integrated the way you are thinking

That was my mistake that I apparently saw partial differentiation in my own post. It is even mistake at this time also, please make it as variation. I have no edit option.

 

[PeterDonis]

I know the metric, then what is the mathematical relation between the Ricci scalar and stress energy tensor?

The Einstein Field Equation relates the Ricci tensor, Ricci scalar, and stress-energy tensor.

if there is way to explore the variation of matter Lagrangian density with metric tensor resolving the variation, then don't this makes sense to solve the problem?

What you are describing here is, again, deriving the stress-energy tensor from the Lagrangian using the variational method. If you use the full Lagrangian (including the Einstein-Hilbert term as well as the matter Lagrangian), the variational method just gives you the Einstein Field Equation. That has been known since 1915, when Hilbert published his derivation of the EFE by this method.

However, the "problem" that you say you are trying to solve is going in reverse--start with the stress-energy tensor and figure out what Lagrangian it came from by "integrating" the variational equation. But the variational equation is not a differential equation and can't be integrated that way.

 

[Bishal Banjara]

I followed the reverse back derivation of $$T_{\mu\nu}$$ in the equation $$T_{\mu\nu}=\frac{2\delta(\sqrt{-g}\mathcal{L}_m)}{\sqrt{-g}\delta{g^{\mu\nu}}}$$ multiplying by $$\sqrt{-g}/2$$ and reintroducing the intergand. Further, we get variation of matter action as $$\delta{S_M}=\dfrac12\int{T_{\mu\nu}\sqrt{-g}d^4x\delta{g^{\mu\nu}}}$$.This would lead to the expression of matter Lagrangian density as $$L_m=\dfrac12\int{T\sqrt{-g}d^3x}$$ where $T$ is trace stress energy tensor. This follows the lagrangian density $$\mathcal{L}_m =\dfrac T2$$ where, $T$ is obtained by contraction with $$g^{\mu\nu}$$. In terms of $$T_{\mu\nu}$$, we could extend this equation as $$\mathcal{L}_m=\dfrac12g^{\mu\nu}T_{\mu\nu}$$.
Is it correct?

 

[Bishal Banjara]

This would lead to the expression of matter Lagrangian density as

Correction: This would lead to the expression of matter Lagrangian (not density) as