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Complex Metric Tensor

  1. May 29, 2012 #1
    I am working on the weak gravitational field by using linearized einstein field equation. What if the metric tensor, hαβ 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?
  2. jcsd
  3. May 29, 2012 #2


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    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.
  4. May 29, 2012 #3


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    Since you're working with the linearized field equations, on a Minkowski background presumably, yes you can just take the real part.
  5. May 29, 2012 #4
    What I mean here is that the component of the hmn 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 hmn. So if component of hmn 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?

  6. May 30, 2012 #5
    Are you using dx0=ct or dx0=ict? This could be part of the difficulty.

  7. May 30, 2012 #6


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    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.
  8. May 30, 2012 #7


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