# Conservation of Electromagnetic Energy-Momentum Tensor

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• mjordan2nd
In summary, the goal is to show that \partial_\mu T^{\mu \nu}=0 for T^{\mu \nu}=F^{\mu \lambda}F^\nu_{\; \lambda} - \frac{1}{4} \eta^{\mu \nu} F^{\lambda \sigma}F_{\lambda \sigma} using the electromagnetic equations of motion \partial_\mu F^{\mu \nu}=0 and \partial_\mu F_{\nu \lambda}+\partial_\nu F_{\lambda \mu}+\partial_\lambda F_{\mu \nu}=0. The first step is to apply the partial derivative directly, which gives us three terms with three partial derivatives
mjordan2nd
I'm trying to show that $\partial_\mu T^{\mu \nu}=0$ for

$$T^{\mu \nu}=F^{\mu \lambda}F^\nu_{\; \lambda} - \frac{1}{4} \eta^{\mu \nu} F^{\lambda \sigma}F_{\lambda \sigma},$$

with the help of the electromagnetic equations of motion (no currents):

$$\partial_\mu F^{\mu \nu}=0,$$
$$\partial_\mu F_{\nu \lambda}+\partial_\nu F_{\lambda \mu}+\partial_\lambda F_{\mu \nu}=0.$$

Applying the partial derivative directly gives us

$$\partial_\mu T^{\mu \nu} = F^\nu_{\; \lambda} \partial_\mu F^{\mu \lambda}+F^{\mu \lambda} \partial_\mu F^\nu_{\; \lambda}-\frac{1}{4} \eta^{\mu \nu}F_{\lambda \sigma} \partial_\mu F^{\lambda \sigma}-\frac{1}{4} \eta^{\mu \nu}F^{\lambda \sigma} \partial_\mu F_{\lambda \sigma}.$$

The first term on the right-hand side drops out due to the first equation of motion. The last two terms can also be combined, but I'm not sure if it would be right to do this immediately as right now I have three terms with three partial derivatives. So my instinct was to try and put them in a form where I could use the second equation of motion. Unfortunately I couldn't see how to do that, especially with the $\frac{1}{4}$ factor on the last two terms. I then did what every physics student does and played with the expression in hopes of finding something simpler. I got a simpler expression, however I could not see how to proceed from my simpler expression to showing what I want to show. I will post my general thought process below, and would appreciate any help on proceeding beyond my final step.

The first thing I did was lower all indices on the tensors being differentiated, as that is in accordance with the form of my second equation of motion. So we got

$$\partial_\mu T^{\mu \nu} =\eta^{\nu \sigma} F^{\mu \lambda} \partial_\mu F_{\sigma \lambda}-\frac{1}{4} \eta^{\mu \nu} \eta^{\alpha \lambda} \eta^{\beta \sigma} F_{\lambda \sigma} \partial_\mu F_{\alpha \beta}-\frac{1}{4} \eta^{\mu \nu}F^{\lambda \sigma} \partial_\mu F_{\lambda \sigma}.$$

I next noticed that if I lowered all indices on the field tensors then each of my terms would have three metric factors, which looked more promising if I wanted to try and factor things out, so I got

$$\partial_\mu T^{\mu \nu} =\eta^{\nu \sigma} \eta^{\alpha \mu} \eta^{\delta \lambda} F_{\alpha \delta} \partial_\mu F_{\sigma \lambda}-\frac{1}{4} \eta^{\mu \nu} \eta^{\alpha \lambda} \eta^{\beta \sigma} F_{\lambda \sigma} \partial_\mu F_{\alpha \beta}-\frac{1}{4} \eta^{\mu \nu} \eta^{\alpha \lambda} \eta^{\beta \sigma} F_{\alpha \beta} \partial_\mu F_{\lambda \sigma}.$$

I initially thought that if I replace the dummy indices properly I might get something where I can factor out the metric factors, however since $\nu$ is not a dummy index I can't move it around freely, and it doesn't look like it's going to be very easy to factor out the metric factors. I therefore changed my plan to simplify the expression to see if I could get something that I could make sense of. Rather than focusing on making the metric part factorable, I tried to change the $F \partial F$ part into a form that might be factorable. In the first term I made the following replacements: $\sigma \leftrightarrow \alpha,$ $\lambda \leftrightarrow \beta.$ The second term was left alone. On the last terms I interchanged $F_{\alpha \beta} \leftrightarrow F_\lambda \sigma$ which I think is okay since both indices are contracted over. All of this gave me

$$\partial_\mu T^{\mu \nu} =\eta^{\nu \alpha} \eta^{\sigma \mu} \eta^{\delta \beta} F_{\sigma \delta} \partial_\mu F_{\alpha \beta}-\frac{1}{4} \eta^{\mu \nu} \eta^{\alpha \lambda} \eta^{\beta \sigma} F_{\lambda \sigma} \partial_\mu F_{\alpha \beta}-\frac{1}{4} \eta^{\mu \nu} \eta^{\alpha \lambda} \eta^{\beta \sigma} F_{\lambda \sigma} \partial_\mu F_{\alpha \beta}.$$

Combining the last two terms and factoring out what I can gave me

$$\partial_\mu T^{\mu \nu} = \eta^{\sigma \beta} F_{\lambda \sigma} \partial_\mu F_{\alpha \beta} \left( \eta^{\nu \alpha} \eta^{\lambda \mu} - \frac{1}{2} \eta^{\mu \nu} \eta^{\alpha \lambda} \right).$$

I can't see how to simplify this expression to zero, so I assume I made a wrong turn during my calculation somewhere. However, I can't see where. How do I use the equations of motion to show that the energy-momentum tensor is conserved here?

mjordan2nd said:
The last two terms can also be combined, but I'm not sure if it would be right to do this immediately as right now I have three terms with three partial derivatives.
You had the right idea but threw it away. Do combine these into one. Otherwise you have three terms with different prefactors which really does not help you. Your goal should instead be to rewrite the first term as two terms with the prefactor 1/2. Hint: use the symmetries of the field tensor.

I also suggest you work with ##\partial_\mu T^\mu_\nu## instead. It is equivalent and your ##\nu## being covariant will help a lot.

## 1. What is the Conservation of Electromagnetic Energy-Momentum Tensor?

The Conservation of Electromagnetic Energy-Momentum Tensor is a fundamental principle in physics that states that the total energy and momentum of an isolated system must remain constant over time. This principle is derived from the laws of physics, specifically from the law of conservation of energy and the law of conservation of momentum.

## 2. How does the Conservation of Electromagnetic Energy-Momentum Tensor apply to electromagnetism?

The Conservation of Electromagnetic Energy-Momentum Tensor applies to electromagnetism by stating that the total energy and momentum of an electromagnetic field must remain constant in any given space. This means that energy and momentum can be transferred between objects through electromagnetic interactions, but the total amount must remain the same.

## 3. What is the importance of the Conservation of Electromagnetic Energy-Momentum Tensor?

The Conservation of Electromagnetic Energy-Momentum Tensor is important because it is a fundamental principle that governs the behavior of energy and momentum in the universe. It allows us to accurately predict and understand the interactions between electromagnetic fields and matter.

## 4. Are there any exceptions to the Conservation of Electromagnetic Energy-Momentum Tensor?

There are no known exceptions to the Conservation of Electromagnetic Energy-Momentum Tensor. This principle has been extensively tested and has been found to hold true in all known cases. However, there may be situations where this principle is not applicable, such as in extreme conditions like black holes.

## 5. How does the Conservation of Electromagnetic Energy-Momentum Tensor relate to other conservation laws?

The Conservation of Electromagnetic Energy-Momentum Tensor is closely related to other conservation laws, such as the conservation of mass, energy, and momentum. In fact, these laws are interconnected and are all derived from fundamental principles of physics, such as the laws of thermodynamics and Newton's laws of motion.

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