How to Prove the Energy-Momentum Tensor Identity?

In summary, the conversation discusses how to show that the double time derivative of the integral of a given equation is equal to the integral of another equation. The conversation delves into the use of the conservation of energy-momentum and the application of Stoke's Theorem. Ultimately, the solution is found through integration by parts.
  • #1
WannabeNewton
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Homework Statement


Show that [tex]\frac{1}{2}\frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\int_{V}\rho x^{j}x^{k}dV = \int_{V}T^{jk}dV [/tex].

Homework Equations


The Attempt at a Solution


[itex]\partial _{t}T^{t\nu } = -\partial _{i}T^{i\nu }[/itex] from conservation of energy - momentum. [itex]\partial_{t}\partial_{t}(T^{tt}x^{j}x^{k}) =(\partial_{t}\partial_{t}T^{tt})x^{j}x^{k} [/itex] since the x's are fixed coordinates of their respective volume element inside the source. So using the equality from conservation of energy - momentum I get [itex]\partial_{t}\partial_{t}(T^{tt}x^{j}x^{k}) =(\partial _{i}\partial _{m}T^{im})x^{j}x^{k}[/itex] and by using the product rule on [itex]\partial _{i}\partial _{m}(T^{im}x^{j}x^{k})[/itex] to solve for the right hand side of the previous equation I get [tex]\partial_{t} \partial_{t}(T^{tt}x^{j}x^{k}) = \partial _{i}\partial _{m}(T^{im}x^{j}x^{k}) - 2\partial _{i}(T^{ij}x^{k} + T^{ik}x^{j}) + 2T^{jk}[/tex] and this is where I am stuck. I don't know if what I am doing after this is exactly correct. For instance, [tex]-2\int_{V}\partial _{i}(T^{ij}x^{k})dV = -2\int_{\partial V}(T^{ij}x^{k})dS_{i} = 0[/tex] as per Stoke's Theorem and because [itex]T^{ij}[/itex] has to vanish at the boundary of the source so that the pressure differs smoothly from the source to the outside but I don't think I applied Stoke's Theorem correctly here. I did the same with the [itex]T^{ik}[/itex] also in the parentheses and for the first expression I did [tex]\int_{V}\partial_{m} \partial _{i}(T^{im}x^{j}x^{k})dV = \frac{\mathrm{d} }{\mathrm{d} x^{m}}\int_{V}\partial _{i}(T^{im}x^{j}x^{k})dV = \frac{\mathrm{d} }{\mathrm{d} x^{m}}\int_{\partial V}(T^{im}x^{j}x^{k})dS_{i} = 0[/tex] for the same reason as before so that [itex]\int_{V}\partial _{t}\partial _{t}(T^{tt}x^{j}x^{k})dV = \frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\int_{V}\rho x^{j}x^{k} = 2\int_{V}T^{jk}dV[/itex]. Could anyone tell me where and how I used Stoke's Theorem wrongly here and how I am supposed to correctly use it in the context of this problem? Thanks in advance.
 
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  • #2
try using the fact that

[tex] \int d^3 x T^{ij} = \int d^3 x [ \partial_k (T^{ik} x^j ) - ( \partial_k T^{ik} ) x^j ] [/tex]
 
  • #3
I tried that but I couldn't find any to use it to get to the answer? Could you show me how you would use it for one of the expressions or something along the lines? Thanks for the reply.
 
  • #4
I have attached the section of my cambridge notes covering this formula
 

Attachments

  • lecturenotes0112.pdf
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  • #5
Oh thank you very much. I didn't even think of integration by parts lol. Talk about a much simpler way of getting rid of surface terms.
 

Related to How to Prove the Energy-Momentum Tensor Identity?

1. What is the Energy-Momentum Tensor Identity?

The Energy-Momentum Tensor Identity is a fundamental law in physics that relates the conservation of energy and momentum to the curvature of spacetime in Einstein's theory of general relativity. It states that the divergence of the energy-momentum tensor, which describes the distribution of energy and momentum in space, is equal to the curvature of spacetime caused by the presence of matter and energy.

2. Why is the Energy-Momentum Tensor Identity important?

The Energy-Momentum Tensor Identity is important because it allows us to understand how energy and momentum are conserved in the presence of gravity. It also plays a crucial role in the formulation of Einstein's field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy.

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

The Energy-Momentum Tensor Identity is derived from the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy. By applying the principle of local energy-momentum conservation, which states that the total energy and momentum within a small volume of spacetime is conserved, we can derive the Energy-Momentum Tensor Identity.

4. Can you give an example of the Energy-Momentum Tensor Identity in action?

One example of the Energy-Momentum Tensor Identity in action is in the study of black holes. The energy and momentum of matter falling into a black hole contribute to the curvature of spacetime, and the Energy-Momentum Tensor Identity helps us understand how this curvature affects the behavior of particles and light near the event horizon.

5. How does the Energy-Momentum Tensor Identity relate to other principles in physics?

The Energy-Momentum Tensor Identity is closely related to other fundamental principles in physics, such as the conservation of energy and momentum, the equivalence principle, and the principle of general covariance. It also plays a crucial role in connecting the theories of general relativity and quantum mechanics, as it helps us understand the behavior of matter and energy on a microscopic level in the presence of gravity.

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