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I have a metric: $$g_{\mu\nu}=\begin{bmatrix}-c & 0 & 0 & 0\\0 & a(t) & 0 & 0\\0 & 0 & a(t)\space r & 0\\0 & 0 & 0 & a(t)\space r\space \sinθ\end{bmatrix}$$ Having the dimensions $(t, r, θ, \phi)$. It corresponds to this formula: $$ds^2=i^2c^2dt^2+a(t)^2(dr^2+r^2(dθ^2+ \sin^2θ\space d\phi^2))$$ Could someone show me a real simple example of how I can use the metric (not the formula) to find the distance? I understand reasonably well how to do the problem with just spatial dimensions, but I'm unable to make the leap to 4 dimensions.

I would like to, for example, find the radial distance, $r$, from $a(t)=1$ (present day) to $a(t)=\frac{1}{1091}$ (Epoch of Recombination). I understand how to do this analytically using the formula, but now I want to create a Mathematica program to validate the analytical solution using the metric.

Quark Soup
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  • Hello. What distance are you trying to find? Could you give a simple example please? – IntroductionToProbability May 12 '20 at 13:26
  • Have you tried 69423? – Vladimir May 12 '20 at 14:13
  • Hmmm. Do you have a known curve you want to find the distance of? Or do you want to find a geodesic between two known points? And just to be clear, do you want a numeric answer or a symbolic answer? – thorimur May 13 '20 at 05:40
  • @thorimur - The geodesic, $ds^2$, is going to be zero for any photon path. I want to find the radial distance, $r$, in the scenario I mentioned above. Stated more plainly, given an emission time of 380,000 years and an observation time of 13.8 billion years, what distance has a photon travelled. Again, I have the answer done analytically. Now I want to actually employ the matrix $g_{\mu\nu}$ to verify the answer. – Quark Soup May 13 '20 at 11:13
  • @thorimur - ... or, more accurately, I want to verify that the metric array accurately describes the topology. I'm reasonably confident that the formula does. – Quark Soup May 13 '20 at 12:40

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