# Complex space time?

1. Jun 17, 2013

### ngkamsengpeter

Is it possible that the metric tensor gmn consist of functions of complex variables?

Let say you have a system with stress energy tensor Tmn and consider gmn=V dt2+W dr2. Is it possible that the solution W or V turn out to be a complex function? And how do we interpret this complex metric tensor? Does the imaginary part have any physical meaning or should we just use take the real part of it?

Thanks.

2. Jun 17, 2013

### Bill_K

Not quite what you're asking, but one can use complex coordinates. A simple description of the Kerr metric places the source an imaginary distance ia along the z axis. Also, complex basis vectors, such as the Newman-Penrose tetrad formalism.

3. Jun 17, 2013

### ngkamsengpeter

Let say I have a system with some real Tmn and I use gmn=V dt^2+ W dr^2. Then I try to solve the Einstein Field equation using spherical coordinates. The solution V and W turn out to be complex. How do I interpret this complex V and W? Does the imaginary part have any physical meaning?

4. Jun 17, 2013

### WannabeNewton

Coordinates don't have any physical meaning. A choice of coordinates is just a choice of a convenient (or possibly inconvenient if you're in a bad mood) computational tool. In GR we work with real differentiable manifolds, meaning every point has a neighborhood that is homeomorphic to $\mathbb{R}^{n}$. On the other hand spinors are objects that live in complex vector spaces: http://en.wikipedia.org/wiki/Spinor and they are used in GR as well through spinorial tensors.

5. Jun 17, 2013

### ngkamsengpeter

Then should we just take real part of this complex metric tensor to calculate the geodesic equation and so on?

Thanks.

6. Jun 17, 2013

### Bill_K

I have to question whether this is possible. If it's possible in the full theory, it's possible in the linearized theory. In the linearized theory ◻hμν = Tμν, and if Tμν is real then so is hμν. Do you have a simple example of what you're talking about?

EDIT: I see you keep writing gmn in a Euclidean form. Is the issue as simple as t → it?

Last edited: Jun 17, 2013
7. Jun 17, 2013

### WannabeNewton

Also, to add on to Bill's post #2, you might want to take a look here: http://en.wikipedia.org/wiki/Newman–Penrose_formalism (the NP formalism shows up again when working with 2-spinor calculus because you can reduce it to a tetrad calculus using the NP formalism essentially).

8. Jun 17, 2013

### ngkamsengpeter

Basically my problem is the differential equation have a complex solutions. The solutions of this differential equation give the complex metric tensor. Should I just take real part of the complex solution?

9. Jun 17, 2013

### WannabeNewton

Well for example when considering the source free linearized Einstein equations in the Lorenz gauge $\partial^{c}\partial_{c}\bar{h}_{ab} = 0$, there can be complex solutions of the form $\bar{h}_{ab} = A_{ab}e^{ik_{\mu}x^{\mu}}$ (where $A_{ab}$ can have complex components as well) if and only if $k^{\mu}k_{\mu} = 0$ i.e. the wave 4-vector is null; these solutions describe plane gravitational waves in the linearized regime. When you actually want to compute down to earth observables you would want to take the real part (e.g. if we wanted to calculate the potentially detectable amplitude of the gravitational radiation field generated by two masses attached to opposite ends of an oscillator).

10. Jun 17, 2013

### ngkamsengpeter

Maple give me a solution of Legendre function with complex argument. So let say I want to find the geodesic equation, I just take the real part of it right?

Thanks.

11. Jun 17, 2013

### dx

Einstein considered a generalization of general relativity where the metric is replaced by

gij = sij + iaij

where 's' is symmetric and 'a' is anti-symmetric.

12. Jun 17, 2013

### WannabeNewton

Only when we are hunting for classical physical observables to be potentially measured do we have a need to take the real part of the above plane wave solution to the linearized Einstein equations in vacuum, in the Lorenz gauge (this is no different in spirit from plane wave solutions to Maxwell's equations in vacuum).

13. Jun 17, 2013

### Bill_K

Taking the real and imaginary parts works for linear equations, where the family of possible solutions are linear combinations of each other. Einstein's equations are nonlinear, and no, you can't just take the real part.

I'd say your complex solution is incorrect.

EDIT: Or maybe your solution is already real and you don't know it!

Look for an identity for Pn(ix). For example, P2(x) = (3x2 -1)/2, so P2(ix) = (-3x2 -1)/2, which is real.

Last edited: Jun 17, 2013
14. Jun 17, 2013

### WannabeNewton

I'm curious as to what the sketch is now. The quote stops right at the climax! Will there be a sequel :p

15. Jun 17, 2013

### dx

Read the last 2-3 pages of his essay "Autobiographical Notes" (I can send you the djvu if you can't find it)

If I remember correctly, he also talks about it in an appendix in his book "The Meaning of Relativity."

16. Jun 17, 2013

### ngkamsengpeter

Ok. I will have a look on the identity. Thanks.

17. Jun 17, 2013

### WannabeNewton

I found it, thanks!