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Given a fluid spherically distributed with density $\rho(r)$ in 3-dimensions in flat-spacetime; the projected surface density $\sigma(R)$ (onto two dimensions) can be obtained by the well known formula:

\begin{equation} \sigma(R)=2\int_{R}^{+\infty}\dfrac{\rho(r)r}{\sqrt{r^2-R^2}}dr\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1) \end{equation}

Suppose now the fluid with density $\rho(r)$ is embedded in a static and spherically symmetric spacetime, with the metric: $$ds^2=-A(r)dt^2+A(r)^{-1}dr^2+r^2d\Omega^2.$$

How it gets modified equation (1) to obtain the equivalent $\sigma(R)$ in terms of $\rho(r)$ and now $A(r)$?

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  • $\begingroup$ The answer depends on what you are trying to achieve with the formula. It is possible to generalize the formula for (say) hyperbolic space (not spacetime) but you are also introducing coordinate dependent time component for the metric. $\endgroup$
    – A.V.S.
    Jan 16, 2020 at 4:09
  • $\begingroup$ I guess the question is quite clear. The integral (1) is in cylindrical coordinates, one can recognize as well the jacobian measure r dr. The part of the time dependence, I don't know what you mean; it is enough to set t=const. $\endgroup$ Jan 16, 2020 at 18:35
  • $\begingroup$ the question is quite clear No it is not. The formula is not an arbitrary expression in cylindrical coordinates but rather a consequence of properties of Laplace operator and cylindrical symmetry. How one can generalize the formula on spacetime depends on what one is going to do with it. $\endgroup$
    – A.V.S.
    Jan 16, 2020 at 19:24
  • $\begingroup$ so, the goal is that given a spherical mass distribution, to compress in one direction and obtain the corresponding surface density... like in a typical surface brightness problem.... Abel's deprojection formula allows to do this, as long as spacetime is flat i.e. euclidean.... if spacetime is non euclidean.... how is this surface density computed in terms of the 3d density and the metric coefficients $\endgroup$ Jan 16, 2020 at 21:40
  • $\begingroup$ You are aware that curved spacetime introduces gravitational lensing effects, so reducing the problem to a single function $\sigma(R)$ is not an option? Also I expect that the same mass distribution but at different blue/redshifts should be contributing differently, depending on the exact problem under consideration. So, what exactly are you trying to achieve? $\endgroup$
    – A.V.S.
    Jan 17, 2020 at 15:46

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