## Hermitian Metric - Calculating Christoffel Symbols

Hello,

I am trying to understand what the differences would be in replacing the symmetry equation:

g_mn = g_nm

with the Hermitian version:

g_mn = (g_nm)*

In essence, what would happen if we allowed the metric to contain complex elements but be hermitian? I am not talking about moving to a fully complex space. We still have the 4 real values of ct, x, y, and z. I have searched the interweb for help, but cannot find anything talking about this particular issue. I only know differential geometry (basically, calculus), so the articles using groups etc. a far over my head, and most articles posted that I found just move to the more complicated "complex space" which I am trying to avoid.

What I really want to know is how the Christoffel Symbols and the Ricci Tensor would differ for this kind of metric. Any help at all would be greatly appreciated!
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 Mentor Nothing would be any different. Because all real numbers are their own conjugate transpose.
 Yes I know, but that assumes the metric has only real values. I am not assuming that, so in essence the metric, Christoffel Sybols, and Ricci tensor would all allow complex values. Solving for the Christoffel symbols (from examples I've seen) involves writing out the equation for a zero covariant derivative of the metric, rotating indices, and then using the metric's symmetry to solve it out. Since the symmetry condition is nolonger the same, I can't use that method anymore and am stuck. g_mn || k = g_mn | k - g_jn G^j_mk - g_mj G^j_nk = 0 My inkling is that this should be replaced with: g_mn || k = g_mn | k - g*_jn G*^j_mk - g_mj G^j_nk = 0 ButI'm not entirely sure...

## Hermitian Metric - Calculating Christoffel Symbols

Here is the equation using the editor:

$\frac{\partial g_{\mu \nu}}{\partial x^{\tau}} - g_{\alpha \nu} \Gamma^{\alpha}_{\mu \tau} - g_{\mu \alpha} \Gamma^{\alpha}_{\tau \nu} = 0$

which I propose changing to:

$\frac{\partial g_{\mu \nu}}{\partial x^{\tau}} - g_{\alpha \nu} \Gamma^{\alpha}_{\mu \tau} - g^*_{\mu \alpha} \Gamma^{* \alpha}_{\tau \nu} = 0$

in the general Hermitian case. Is this correct?

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 Quote by thehangedman In essence, what would happen if we allowed the metric to contain complex elements but be hermitian?
You would end up with physical predictions that contain complex entities. We have no way to interpret this sort of information... What does it mean to have $3i$ apples?
 While tempting to debate this statement, it's really off topic. I'm more asking about how the other components of differential geometry change when we change the symmetry condition on the metric tensor.
 Recognitions: Gold Member Science Advisor Staff Emeritus Let's start with a simpler example. Suppose you have a Newtonian simple harmonic oscillator with equation of motion $d^2x/dt^2+bx=0$, where b is a constant. One way to solve that is to guess a solution of the fotm $x=e^{rt}$, where r is a constant. This give solutions $r=\pm i\sqrt{b}$. The most general solution can be found by taking a linear superposition of the two solutions, $x=c_1e^{rt}+c_2e^{-rt}$. When you pick the c's to match real-world initial conditions, you get a real-valued x. The Einstein field equations are different, because they're nonlinear. Therefore you can't find a family of complex-valued solutions for the metric and take linear superpositions of them to get a real-valued result. There are certain tricks that can sometimes be used to "realify" a complex-valued metric, but they aren't tricks that work in general. If you want to see an example of such a technique, I believe that's how Kerr originally found the Kerr spacetime.