Proving Energy-Momentum Tesor Relation

In summary, the energy-momentum tensor for any classical field theory can be proven to be equal to -2 times the functional derivative of the action with respect to the metric tensor. This can be shown by taking the functional derivative of the matter-gravity action and separating it into two parts, one corresponding to Einstein's equations and the other to the source of gravity. This is known as the covariant energy momentum tensor and is how textbooks define it in general relativity. However, the person asking the question is seeking a proof for this in the absence of gravity, specifically for a "free" matter field Lagrangian.
  • #1
samalkhaiat
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How can you go about and prove the following :
The energy-momentum tensor for any classical field theory = -2 X the functional derivative of the action with respect to the metric tensor.
 
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  • #2
Basically, if you have an action for matter interacting with gravity S = SG + SM and perform the functional derivative of S with respect to the metric tensor, you will get a first part corresponding to SG, which will be Einstein’s equations in vacuum, and a second part corresponding to SM, which will be the source of gravity. This is the covariant energy momentum tensor.
 
  • #3
You are right, this is how textbooks "define" the mater field E-M tensor in GR. This was not what I asked. I wanted a proof for the following;
given any "free" matter field Lagrangian, show that the symmetric E-M tensor is equal to -2 X the functional derivative of the "free" matter action with respect to the metric tensor.(I am asking for proof with no reference to gravity). thanks.
 

1. What is the Energy-Momentum Tensor Relation?

The Energy-Momentum Tensor Relation is a fundamental equation in physics that describes the relationship between energy, momentum, and the geometry of spacetime. It is a key component of Einstein's theory of general relativity and is used to calculate the gravitational effects of matter and energy on the curvature of spacetime.

2. Why is it important to prove the Energy-Momentum Tensor Relation?

Proving the Energy-Momentum Tensor Relation is important because it provides a rigorous mathematical foundation for understanding the relationship between energy, momentum, and gravity. It also allows us to make predictions about the behavior of matter and energy in the universe, such as the bending of light around massive objects like black holes.

3. How is the Energy-Momentum Tensor Relation derived?

The Energy-Momentum Tensor Relation is derived from the Einstein field equations, which describe the curvature of spacetime in the presence of matter and energy. These equations are based on the principle of general relativity, which states that the laws of physics should be the same for all observers, regardless of their relative motion.

4. Are there any experimental tests of the Energy-Momentum Tensor Relation?

Yes, there have been numerous experimental tests of the Energy-Momentum Tensor Relation. These include observations of the bending of starlight during a solar eclipse, the deflection of light from distant galaxies, and the precession of the orbit of Mercury. All of these experiments have confirmed the predictions of general relativity, providing strong evidence for the validity of the Energy-Momentum Tensor Relation.

5. Are there any alternative theories to the Energy-Momentum Tensor Relation?

There are currently no widely accepted alternatives to the Energy-Momentum Tensor Relation. However, there are ongoing efforts to develop theories that can better explain certain phenomena, such as the behavior of matter and energy at very small scales or the expansion of the universe. These theories may modify or extend the Energy-Momentum Tensor Relation, but they are still based on the fundamental principles of general relativity.

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