Vaidya Metrics: Outgoing M(u) Conditions & Phys. Situations

[Tomas Vencl]

I would like to ask if the function M(u) in Vaidya metrics must fulfil any special conditions, or is it completely free ?
https://en.wikipedia.org/wiki/Vaidya_metric#Outgoing_Vaidya_with_pure_Emitting_fieldIn other words, when the outgoing Vaidya metrics describes the metrics of radiating body (for example Hawking radiation), probably the M(u) must have some special form, conditions etc. to be a physical situation description.
Does anyone know some details about this topic ?
Thank you.

 

[PeterDonis]

when the outgoing Vaidya metrics describes the metrics of radiating body (for example Hawking radiation), probably the M(u) must have some special form, conditions etc. to be a physical situation description

$M(u)_{, u}$ (the derivative of $M(u)$ with respect to $u$) needs to be negative for the outgoing Vaidya metric in order for the emitted radiation to have a positive energy density. As far as I know that is the only restriction (other than the obvious one that $M(u) \ge 0$).

 

[Tomas Vencl]

Thank you.
I try to understand deriving Vaidya metric, but in the wiki article they just changed the Schw. coordinates of the Schwarzschild metrics to Eddington-Finkelstein coordinates of Schw. metrics and then changed the constant M to M(u) with only remark, that this is still physically reasonable. I understand that this changed the metrics, but I do not see the proof for reasonability and mathematical correctness for this step. (I do not understand why during derivation to E-F coordinates They treat with M as a constant, not a function of u, but at the end They set M to be a function M(u) )
So can I just change the constant M in Schwarzschild coordinates to corresponding transformed function M(t ,r) and also will obtain reasonable solution (not static, non vacuum..) ?

 

[PeterDonis]

I try to understand deriving Vaidya metric

The Wikipedia page doesn't derive the metric, it just writes it down and shows its similarities with the Schwarzschild metric.

To derive the metric, you would assume spherical symmetry and that the only stress-energy present is ingoing or outgoing null dust.

So can I just change the constant M in Schwarzschild coordinates to corresponding transformed function M(t ,r) and also will obtain reasonable solution (not static, non vacuum..) ?

If you follow the assumptions I just described and work the problem in standard Schwarzschild coordinates, you will indeed find that you have a mass $M$ that depends on both $t$ and $r$. However, that solution will have a coordinate singularity, which is why the solution is normally done in ingoing or outgoing Eddington-Finkelstein coordinates, which do not have a coordinate singularity in the region of interest.