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Homework Help: Construction of metric from tensor products of vectors

  1. Dec 29, 2017 #1
    1.
    The metric ##g_{\mu \nu}## of spacetime shall be constructed from tensor products of vectors (relevant are the unit vectors in the respective directions). One such vector shall be called ##A##.

    2. Relevant equations
    ##g_{\mu \nu} = \lambda \frac{A_\mu A_\nu}{g^{\alpha \beta} A_\alpha A_\beta} + .......##, where neither ##\lambda## nor the further terms shall involve ##A##. Now ##g## shall be differentiated w.r.t. ##A##, this is ##\frac{\partial g_{\mu\nu}}{\partial A_\tau}## .


    3. The attempt at a solution
    Everything works fine except one term originating from differentiating the denominator. It contains a factor ##A_\alpha A_\beta \frac{\partial g^{\alpha\beta}}{\partial A_\tau}##. I have no idea what to do with this expression. Is it zero? Or is it equal to to ##A^\alpha A^\beta \frac{\partial g_{\alpha\beta}}{\partial A_\tau}##? Or what else?

    Many thanks for any advice.
     
  2. jcsd
  3. Dec 29, 2017 #2

    George Jones

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    To find an expression for this factor, contact ##g_{\mu \nu} = \lambda \frac{A_\mu A_\nu}{g^{\alpha \beta} A_\alpha A_\beta} + .......## with ##A^\mu A ^\nu##, and take the derivative with respect to ##A_\tau##.
     
  4. Dec 29, 2017 #3

    George Jones

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    Actually, it might be better to look at
    $$0 = \frac{\partial}{\partial A_\tau} \left( \delta^\alpha_\beta \right) .$$
     
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