What is the Minkowski metric tensor's trace?

In summary: Spacetime and Gravitation discusses the use of the metric tensor in Minkowski space, including the trace equal to four. In summary, the conversation discusses raising and contracting indices using the metric tensor in Minkowski space. The use of the inverse metric and the consistency of notation is emphasized. The trace of the contracted tensor is equal to four. Carroll's textbook Spacetime and Gravitation is referenced as a source for further understanding.
  • #1
LCSphysicist
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I would appreciate if you explain to me how to get the Minkowski metric tensor's trace.
I am trying to follow the rule, that is, raising an index and the contract it.
Be ##g_{\mu v}## the metric tensor in Minkowski space.
Raising ##n^{v \mu}g_{\mu v}## and then, we need now to contract it.
Now, in this step i smell a rat (i learned this pun today, hope this mean what i think this means haha)
Can i simply say that ##\mu## is an index using Einstein notation? I am a little confused how to contract this and then reduced it to delta kronecker, which, in the end, will give us the trace equal four.
 
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  • #2
LCSphysicist said:
Be ##g_{\mu v}## the metric tensor in Minkowski space.
Raising ##n^{v \mu}g_{\mu v}## and then, we need now to contract it.
If ##g_{\mu\nu}## is the metric then the inverse metric should be ##g^{\mu\nu}##. If you are meaning the metric of flat space, typically that's denoted ##\eta_{\mu\nu}## and the inverse would be denoted ##\eta^{\mu\nu}##. There's nothing wrong with using ##g## instead of ##\eta##, but you need to use it consistently.

Apart from that, what you've written seems fine. If you want to think of it in several stages, first you would use ##g^{\rho\mu}## to raise an index, giving you ##g^{\rho\mu}g_{\mu\nu}##, which is indeed ##\delta^\rho_\nu##. Then you can contract over the upper and lower indices - i.e. you needed to set ##\rho=\nu##, which (give or take using ##g## or ##\eta##) is what you wrote. Writing the sums explicitly (so no summation convention implied) it's ##\sum_{\mu=0}^4\sum_{\nu=0}^4g^{\mu\nu}g_{\mu\nu}##.
 
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  • #3
Carroll's textbook Spacetime and Gravitation discusses it, I think it is in the first chapter.
 
  • #4
Carroll's textbook Spacetime and Gravitation discusses it, I think it is in the first chapter. And the solution is as you wrote the trace is four. Page 28 in Carroll's textbook
 

Related to What is the Minkowski metric tensor's trace?

1. What is the Minkowski metric tensor's trace?

The Minkowski metric tensor's trace is a mathematical quantity that measures the curvature of spacetime in the context of special relativity. It is a 4x4 matrix that describes the geometry of spacetime and is used to calculate distances and intervals in a flat, four-dimensional spacetime.

2. How is the Minkowski metric tensor's trace calculated?

The trace of the Minkowski metric tensor is calculated by taking the sum of the elements on the main diagonal of the 4x4 matrix. This is also known as the sum of the diagonal elements.

3. What is the significance of the Minkowski metric tensor's trace?

The trace of the Minkowski metric tensor is significant because it is a measure of the curvature of spacetime. A zero trace indicates a flat spacetime, while a non-zero trace indicates a curved spacetime. This is important in understanding the effects of gravity and the behavior of objects in space.

4. How does the Minkowski metric tensor's trace relate to special relativity?

The Minkowski metric tensor's trace is a fundamental concept in special relativity. It is used to calculate distances and intervals in a four-dimensional spacetime, which is necessary for understanding the effects of time dilation and length contraction in special relativity.

5. Can the Minkowski metric tensor's trace be applied to other theories of relativity?

Yes, the Minkowski metric tensor's trace can also be applied to other theories of relativity, such as general relativity. In this context, it is used to describe the curvature of spacetime in the presence of matter and energy. However, the calculation and interpretation of the trace may differ in these other theories.

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