**Required math: algebra**

**Required physics: special relativity**

Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 4; Problems 4.8, 4.9.

The electromagnetic field tensor* * is

If we contract with the metric tensor in flat space, we get

In the first line, we used ; in the second, and in the last line, we swapped the dummy indexes and . Thus the original quantity is equal to its negative, so

Now consider . First, since and for :

This is because the only sign change occurs when , which is when , so all elements change sign.

Then

This time, the sign change occurs when , so elements change sign. Thus lowering both indices in flat spacetime in rectangular coordinates changes the sign of all the electric field entries, but leaves the magnetic field unchanged.

Combining them, we get

To work this out, note that if we first formed the matrix product , then is the sum of the diagonal elements of this matrix product. That is

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Chris KranenbergGlenn,

Your solution to Problem 4.9 using the lowering of indices, an economical method, is not addressed in Chapter 4. A direct calculation without index manipulation may be more instructive for those who are seeing GR index notation for the first time. Also, Problem 4.11 asks to use index notation; the posted solution uses the vector notation.

Thank you.

Erms PereiraHello.

Do you know what is the metric obtained from Einstein’s equation related to a constant magnetic field?

http://physics.stackexchange.com/q/189787/71959

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