Complex Space Time: Interpret Metric Tensor gmn

[ngkamsengpeter]

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

Let say you have a system with stress energy tensor T_mn and consider g_mn=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.

 

[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.

 

[ngkamsengpeter]

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.

Let say I have a system with some real T_mn and I use g_mn=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?

 

[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.

 

[ngkamsengpeter]

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.

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

Thanks.

 

[Bill_K]

Let say I have a system with some real T_mn and I use g_mn=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?

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 ◻h_μν = T_μν, and if T_μν is real then so is h_μν. Do you have a simple example of what you're talking about?

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

 

[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).

 

[ngkamsengpeter]

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 ◻h_μν = T_μν, and if T_μν is real then so is h_μν. Do you have a simple example of what you're talking about?

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

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?

 

[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).

 

[ngkamsengpeter]

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).

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.

 

[dx]

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

g_ij = s_ij + ia_ij

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

Our problem is that of finding the field equations for the total field. The desired structure must be a generalization of the symmetric tensor. The group must not be any narrower than that of the continuous transformations of co-ordinates. If one introduces a richer structure, then the group will no longer determine the equations as strongly as in the case of the symmetrical tensor as structure. Therefore it would be most beautiful if one were to succeed in expanding the group once
more, analogous to the step which led from special relativity to general relativity. [...]

After many years of fruitless searching I consider the solution sketched in what follows as the logically most satisfactory.

 

[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).

 

[Bill_K]

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?

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!

Maple give me a solution of Legendre function with complex argument.

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

 

[WannabeNewton]

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

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

 

[dx]

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

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."

 

[ngkamsengpeter]

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 P_n(ix). For example, P₂(x) = (3x² -1)/2, so P₂(ix) = (-3x² -1)/2, which is real.

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

 

[WannabeNewton]

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

I found it, thanks!