Conservation of Energy Momentum Tensor

In summary, the stress-energy tensor in Minkowski space with Minkowski coordinates satisfies the equation ##\partial_\alpha T^{\alpha \beta} = 0## only when there is no stress-energy present, in which case spacetime is flat. If there is a non-zero stress-energy tensor, then the correct equation to use is ##\nabla_\alpha T^{\alpha \beta} = 0##, which is a statement of the Bianchi identity.
  • #1
kent davidge
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Unfortunetly, I found across the web only the case where there's no source, in which case ##\partial_\alpha T^{\alpha \beta} = 0##. I'm considering Minkowski space with Minkowski coordinates here.

When there's source, is it true that ##\partial_\alpha (T^{\alpha \beta}) = 0## or is it ##\int \partial_\alpha (T^{\alpha \beta}) = 0##? Where now this latter ##T^{\alpha \beta}## is the result of the variation of the complete action (source included).
 
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  • #2
##\nabla_\alpha T^{\alpha\beta}=0## (edit: and not ##\partial_\alpha T^{\alpha\beta}=0##, as I incorrectly typed originally). Since the stress-energy tensor is the same as the Einstein tensor give or take a constant factor, this turns out to be simply a statement of the Bianchi identity.
 
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  • #3
kent davidge said:
When there's source, is it true that ##\partial_\alpha (T^{\alpha \beta}) = 0##

No, because if there's stress-energy present, spacetime is not flat, so you have to use the correct curved spacetime equation, which is

$$
\nabla_\alpha T^{\alpha \beta} = 0
$$

Ibix said:
##\partial_\alpha T^{\alpha\beta}=0##. Since the stress-energy tensor is the same as the Einstein tensor give or take a constant factor, this turns out to be simply a statement of the Bianchi identity.

Careful! If there is a non-zero stress-energy tensor, spacetime isn't flat. If spacetime is flat, then it is true that ##\partial_\alpha T^{\alpha \beta} = 0##, but only in the vacuous sense that ##T^{\alpha \beta} = 0##.
 
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  • #4
PeterDonis said:
Careful! If there is a non-zero stress-energy tensor, spacetime isn't flat. If spacetime is flat, then it is true that ##\partial_\alpha T^{\alpha \beta} = 0##, but only in the vacuous sense that ##T^{\alpha \beta} = 0##.
I knew it looked wrong as I was typing it. I should listen to my instincts more, apparently.
 

1. What is the Conservation of Energy Momentum Tensor?

The Conservation of Energy Momentum Tensor is a fundamental principle in physics that states that the total energy and momentum in a closed system remains constant over time. This means that energy and momentum can be transferred or transformed, but their total amount cannot be created or destroyed.

2. How is the Conservation of Energy Momentum Tensor related to the laws of conservation of energy and momentum?

The Conservation of Energy Momentum Tensor is a more general form of the laws of conservation of energy and momentum. It takes into account the fact that energy and momentum can be transferred between different forms, such as kinetic energy, potential energy, and thermal energy, and that they can also be exchanged between different objects within a system.

3. What are the practical applications of the Conservation of Energy Momentum Tensor?

The Conservation of Energy Momentum Tensor is a fundamental principle that is used in many areas of physics, including mechanics, thermodynamics, and electromagnetism. It is essential in understanding and predicting the behavior of complex systems, such as fluids, gases, and particles, and is also used in the development of technologies, such as engines, turbines, and energy storage devices.

4. How is the Conservation of Energy Momentum Tensor expressed mathematically?

The Conservation of Energy Momentum Tensor is expressed mathematically through the equations of motion, which describe the change in energy and momentum over time. These equations are based on the principles of conservation of energy and momentum, as well as other laws and principles, such as Newton's laws of motion and the laws of thermodynamics.

5. Are there any exceptions to the Conservation of Energy Momentum Tensor?

The Conservation of Energy Momentum Tensor is a fundamental principle that has been extensively tested and verified through experiments and observations. However, in certain extreme conditions, such as in the presence of black holes or in the early stages of the universe, it may not hold true. In these cases, other theories, such as general relativity, are needed to fully describe the behavior of energy and momentum.

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