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The problem I have is on finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry, which comes from the following paper : https://journals.aps.org/prd/pdf/10.1103/PhysRevD.90.064034

In particular I am trying to find the:

(a) Particular solution of the resulting equations, and

(b) The General solution of the resulting equations.

Currently I am trying to solve for $m$ , $\psi_{v}$ and $\psi_{r}$. in the following 2 equations.

\begin{equation} \frac{2\dot{m}(v,r)}{r^{2}}= \Psi^{2}_{v}+ \left(1-\frac{2m(v,r)}{r}\right)\Psi_{v}\Psi_{r}. \end{equation}

\begin{equation} \frac{m^{\prime\prime}(v,r)}{r}= \Psi_{v}\Psi_{r} - \frac{2m^{'}(v,r)}{r^{2}} \end{equation} Any help, or suggestions would be greatly appreciated.

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  • $\begingroup$ presumably $\dot{m}$ and $m'$ denote two different derivatives of $m$? Which is which? Also, are $\psi_v$ and $\psi_r$ related in any way? Are they constants? $\endgroup$
    – Willie Wong
    Commented Aug 24, 2017 at 12:56
  • $\begingroup$ m(dot) is the derivative wrt to {v} and m(prime) is the derivative wrt {r}. {ψv} partial derivative wrt {v} and ψr is partial derivative wrt {r} $\endgroup$
    – Dickson
    Commented Aug 25, 2017 at 16:40

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