Complex Metric Tensor: Meaning, Weak Gravitational Fields & Einstein Eqns

[ngkamsengpeter]

I am working on the weak gravitational field by using linearized einstein field equation. What if the metric tensor, h_αβ turn out to be a complex numbers? What is the physical meaning of the complex metric tensor? Can I just take it's real part?

Or there is no such thing as complex metric tensor and my system is not physically make sense?

 

[HallsofIvy]

I really don't know what you mean by a metric tensor turning out to be a "complex number". A tensor is NOT number to begin with. And if your coordinate system uses real numbers to as coordinates then the components of the tensor must be real numbers. Of course, in a given coordinate system, we can think of a tensor as represented by a matrix and so find its eigenvalues. It is possible for the eigenvalues of a general matrix to be complex but the metric tensor should always be represented by a symmetric matrix which must have real eigenvalues.

 

[Bill_K]

Since you're working with the linearized field equations, on a Minkowski background presumably, yes you can just take the real part.

 

[ngkamsengpeter]

I really don't know what you mean by a metric tensor turning out to be a "complex number". A tensor is NOT number to begin with. And if your coordinate system uses real numbers to as coordinates then the components of the tensor must be real numbers. Of course, in a given coordinate system, we can think of a tensor as represented by a matrix and so find its eigenvalues. It is possible for the eigenvalues of a general matrix to be complex but the metric tensor should always be represented by a symmetric matrix which must have real eigenvalues.

What I mean here is that the component of the h_mn turn out to be a complex numbers. That is h₁₁ is a complex numbers. In linearized einstein field equation, metric tensor is just sum of flat spacetime metric plus the small pertubation h_mn. So if component of h_mn is a complex number, so I would think that the metric tensor is also complex. Or should I just take the real part as what Bill_K said?

Thanks.

 

[Chestermiller]

Are you using dx⁰=ct or dx⁰=ict? This could be part of the difficulty.

Chet

 

[Bill_K]

Solving the linearized field equations amounts to solving the wave equation, so if your solution turns out to be complex there must be a fairly simple reason for it. Like you used exp(ikx-iwt) instead of sin and cos.

 

[julian]

You have to give more details on how you arrived at a comples metric.