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I would like to write down the arc-length parametrization of a curve in 3-dimensional Euclidean space, [tex]\gamma(\lambda)[/tex], then specify that in a certain spacetime this path is a null geodesic [tex]z^\mu[/tex] and solve for the metric of that spacetime.

My first question is, does this even make sense as something to attempt, or does the initial reliance on coordinates to write down the curve (and the assumption that it's done in Euclidean space) mean that this will fail? If it does make sense, then my second question is does there exist some sort of framework in which to work that you could point me to?

Also, when I convert the parametrized spatial curve into a null geodesic I think I will have something like [tex]z^\mu\stackrel{?}{=}(f(\lambda),\gamma(\lambda))\stackrel{?}{=}z_\mu[/tex]. Which = is correct? Should the curve gamma be the spatial component of the covariant or contravariant form of the geodesic?

My plan right now is to impose symmetry and invertibility on an otherwise arbitrary metric tensor [tex]g_{\mu\nu}[/tex] and solve for [tex]g^{\mu\nu}[/tex]. I'll then impose [tex]g_{\mu\nu}z^\mu z^\nu=0[/tex] and the geodesic equation to obtain DE's for the components of g.

I have attempted this for the simple case of trying to recover 2D Minkowski space [g=diag(1,-1)]. With [tex]z^\mu=(\lambda,\pm\lambda)[/tex] I found that [tex]g_{11}=g_{00}+f(x^0+x^1)[/tex] and [tex]g_{01}=g_{10}=\frac{1}{2}(g_{00}+g_{11})=g_{00}+\frac{1}{2}f(x^0+x^1)[/tex]. By choosing [tex]g_{00}\equiv 1[/tex] and [tex]f(x^0+x^1)\equiv -2[/tex] we see that the 2D Minkowski metric satisfies the equations, so that's reassuring. I used the plus or minus lambda in the geodesic because I wasn't sure whether the spatial curve [tex]\gamma(\lambda)=\lambda[/tex] should be covariant or contravariant (and I cheated a little by knowing what the resulting metric would do to it if it should be covariant) -- it turned out not to matter in this case, which didn't help me reach a conclusion :).

Thanks for any help!

Mike

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# Solving for spacetime

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