Divergence of Energy-momentum Tensor

In summary, to prove that the energy-momentum tensor is divergence-free, one can use the vacuum Maxwell equations or the fact that the divergence is equal to F_{ab}J^b (up to sign). This is because the energy-momentum tensor is related to the 4-Lorentz force per unit volume, which takes into account the work done by the electromagnetic field on charges.
  • #1
ClaraOxford
6
0
How do you prove that the energy-momentum tensor is divergence-free?

∂μTμν=0
 
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  • #2
I mean

∂[itex]_{\mu}[/itex]T[itex]^{\mu\nu}[/itex]=0

T[itex]^{\mu\nu}[/itex]=F[itex]^{\mu\alpha}[/itex]F[itex]^{\nu}[/itex][itex]_{\alpha}[/itex]-1/4F[itex]^{\alpha\beta}[/itex]F[itex]_{\alpha\beta}[/itex][itex]\eta[/itex][itex]^{\mu\nu}[/itex]


I don't know whether to use Lagrangian variables or the Einstein tensor or if there's a simpler way to just expand the tensor and work it out?
 
  • #3
use the fact that:
[tex]
\partial_\nu F^{\mu \nu} = J^\mu, \partial_{\mu} F^{\nu \rho} + \partial_{\nu} F^{\rho \mu} + \partial_{\rho} F^{\mu \nu} = 0, \; F^{\mu \nu} = -F^{\nu \mu}
[/tex]
 
  • #4
Dickfore said:
use the fact that:
[tex]
\partial_\nu F^{\mu \nu} = J^\mu, \partial_{\mu} F^{\nu \rho} + \partial_{\nu} F^{\rho \mu} + \partial_{\rho} F^{\mu \nu} = 0, \; F^{\mu \nu} = -F^{\nu \mu}
[/tex]

It won't be divergence-free if you use those equations. Instead use the vacuum Maxwell equations (above with J=0). Alternatively use the above to find the divergence to equal [tex]F_{ab}J^b[/tex] (up to sign).
 
  • #5
Sam Gralla said:
It won't be divergence-free if you use those equations. Instead use the vacuum Maxwell equations (above with J=0). Alternatively use the above to find the divergence to equal [tex]F_{ab}J^b[/tex] (up to sign).

Ah, of course. There is work done on charges by the electromagnetic field. The above energy gives the 4-Lorentz force per unit volume.
 

1. What is the energy-momentum tensor?

The energy-momentum tensor is a mathematical object that describes the distribution of energy and momentum in a physical system. It is used in Einstein's theory of general relativity to account for the curvature of spacetime caused by the presence of matter and energy.

2. Why is the divergence of the energy-momentum tensor important?

The divergence of the energy-momentum tensor represents the flow of energy and momentum in a physical system. By studying its behavior, scientists can understand how energy and momentum are conserved and how they affect the curvature of spacetime.

3. What does it mean if the divergence of the energy-momentum tensor is zero?

If the divergence of the energy-momentum tensor is zero, it means that energy and momentum are conserved in the system. This is a fundamental principle in physics, and it allows scientists to make predictions about the behavior of a system based on its initial conditions.

4. How is the divergence of the energy-momentum tensor calculated?

The divergence of the energy-momentum tensor is calculated using mathematical equations that involve the components of the tensor and the partial derivatives of those components. These equations can be solved using advanced mathematical techniques, such as tensor calculus, to determine the divergence of the tensor.

5. What are some real-world applications of studying the divergence of the energy-momentum tensor?

Studying the divergence of the energy-momentum tensor has many real-world applications, such as understanding the behavior of black holes and other astronomical objects, predicting the evolution of the universe, and developing new technologies that rely on the principles of general relativity, such as GPS systems and gravitational wave detectors.

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