# Problems with a complex stress-energy tensor

• Time Suspect
In summary, the conversation is about a stress-energy tensor, which is a tensor that describes the distribution of energy and momentum in a spacetime. The tensor is not symmetric, which is causing confusion for the speaker. They are trying to differentiate a complex scalar field, but are unsure if they are doing it correctly. The speaker then suggests using a different definition for the stress-energy tensor that is simpler and guarantees symmetry. With this new approach, the speaker was able to find a symmetric tensor.
Time Suspect
Hi , I am working with the following stress-energy tensor:

T$\mu\nu$=$\partial$$\mu$$\phi$$\partial$$\nu$$\phi$* - g$\mu\nu$($\nabla$$\mu$$\phi$$\nabla$$\mu$$\phi$* - m2$\phi\phi$*)

Where $\phi$ is a complex scalar of the form:
$\phi$(r,t) = $\psi$(r)eiwt

that obeys the Klein-gordon equation and $\phi$* is the complex conjugate, and g the metric tensor.

My problem is that i think this tensor is not symmetric as Ttr /= Trt by a minus sign on the term of the partial derivates.

I think my problem wasn't really clear, this tensor is supposed to be a stress energy tensor and therefore it should be symmetric but I'm finding that it isn't Ttr /= Trt. I'm not sure if I'm differentiating wrong the complex scalar field or I think originally instead of partial derivatives they were covariant derivatives but since it is a scalar quantity I thought it wouldn't matter.

Hope this gives a better idea of the problem at hand.

Thanks a lot.

So why don't you just symmetrize it.

Or better yet, forget the "canonical" stress-energy tensor and use the correct and much simpler definition, Tμν = 2δL/δgμν whose calculation typically involves no taking of derivatives and is guaranteed to come out symmetrical. All you have to remember is that the Lagrangian density contains a factor √(-g) which must also be varied, and δ√(-g)/δgμν = 1/2 √(-g) gμν.

Thanks Bill_K, with that and the Klein-Gordon Lagrangian I found a Tensor which is indeed symmetric.

Thanks a lot.

## 1. What is a complex stress-energy tensor?

The stress-energy tensor is a mathematical concept used in physics to describe the distribution of energy and momentum in a given space. A complex stress-energy tensor is a more advanced version that takes into account not only the magnitude, but also the direction of energy and momentum in a given system.

## 2. What are some common problems associated with a complex stress-energy tensor?

One common problem is the complexity of the calculations involved in using a complex stress-energy tensor. It requires a deep understanding of mathematical concepts and may be challenging for non-experts to use. Another issue is the potential for errors in the calculations, which can lead to inaccurate results.

## 3. How is a complex stress-energy tensor used in research?

A complex stress-energy tensor is used in various fields of physics, including general relativity, quantum field theory, and fluid dynamics. It is used to describe the behavior of energy and momentum in complex systems, such as black holes, fluid flows, and particle interactions.

## 4. Can a complex stress-energy tensor be simplified?

While a complex stress-energy tensor is necessary for accurately describing certain physical systems, it can often be simplified for easier use in calculations. This can be achieved through various techniques, such as using symmetry arguments or approximations.

## 5. How does a complex stress-energy tensor impact our understanding of the universe?

A complex stress-energy tensor allows scientists to better understand and model the behavior of energy and momentum in complex systems, such as the universe. It is a crucial tool in making predictions and testing theories about the fundamental laws of nature.

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