Vaidya metric

The Vaidya metric is a generalization of the spherically symmetric s Schwarzschild metric . It applies to non-rotating and electrically neutral bodies, but - in contrast to the Schwarzschild metric - their mass increases over time due to emitted or absorbed massless radiation (" null dust ", e.g. photons or neutrinos , but no electromagnetic radiation ) - or decreases:

${\displaystyle {\begin{aligned}M(t)&\neq {\text{konst.}}\\\Leftrightarrow {\frac {dM}{dt}}&\neq 0\end{aligned}}}$

The Vaidya metric is therefore - again in contrast to the Schwarzschild metric - neither static nor stationary.

application

Both stars and black holes lose mass due to radiation transport or Hawking radiation or can increase in mass due to incident matter or radiation. Depending on whether the loss or increase in mass predominates, there is a monotonically decreasing or a monotonically increasing function. ${\displaystyle M(t)}$

With the Vaidya metric, however, such physical relationships can only be modeled with restrictions:

• On the one hand, it correctly takes into account massless radiation
• on the other hand, it is unphysical insofar as the change in mass is immediately effective in the whole space ( action at a distance ).

A metric as a consistent solution for vaporizing black holes has not yet been found. Therefore, combinations of the following metrics are used for analyzes and simulations:

• Vaidya metrics for the area in which the Hawking radiation is generated (near the event horizon ),
• for somewhat larger distances the Schwarzschild metric
• for large distances the Minkowski metric of flat spacetime .

Mathematical description

Vaidya metric

In the natural units and with , the line element of the Vaidya metric is in expiring Eddington-Finkelstein coordinates ${\displaystyle G=c=k_{\rm {B}}=1}$ ${\displaystyle \mathrm {d} \Omega ^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}\theta \ \mathrm {d} \phi ^{2}}$

${\displaystyle {\rm {d}}s^{2}=-\left(1-{\dfrac {2M(u)}{r}}\right){\rm {d}}u^{2}-2\ {\rm {d}}u\ {\rm {d}}r+r^{2}{\rm {d}}\Omega ^{2}}$

and in incoming Eddington-Finkelstein coordinates

${\displaystyle {\rm {d}}s^{2}=-\left(1-{\dfrac {2M(v)}{r}}\right){\rm {d}}v^{2}+2\ {\rm {d}}v\ {\rm {d}}r+r^{2}{\rm {d}}\Omega ^{2}.}$

Vaidya-Bonner metric

For electrically charged bodies, the Vaidya metric is expanded to include the Vaidya-Bonner metric

${\displaystyle {\rm {d}}s^{2}=-\left(1-{\dfrac {2M(u)}{r}}+{\dfrac {Q(u)}{r^{2}}}\right){\rm {d}}u^{2}-2\ {\rm {d}}u\ {\rm {d}}r+r^{2}{\rm {d}}\Omega ^{2}}$

and

${\displaystyle {\rm {d}}s^{2}=-\left(1-{\dfrac {2M(v)}{r}}+{\dfrac {Q(v)}{r^{2}}}\right){\rm {d}}v^{2}+2\ {\rm {d}}v\ {\rm {d}}r+r^{2}{\rm {d}}\Omega ^{2},}$

in which

• ${\displaystyle M}$ the mass equivalent and
• ${\displaystyle Q}$is the electrical charge of the central body.

The Vaidya-Bonner metric is reduced:

• with on the Vaidya metric ${\displaystyle Q={\rm {d}}Q=0}$
• with the Reissner-Nordström metric ${\displaystyle {\rm {d}}M={\rm {d}}Q=0}$
• with the Schwarzschild metric. ${\displaystyle {\rm {d}}M={\rm {d}}Q=Q=0}$

	M = const.	M ≠ const.
uncharged ( ) ${\displaystyle Q=0}$	Schwarzschild metric	Vaidya metric
loaded ( ) ${\displaystyle Q\neq 0}$	Reissner-Nordström metric	Vaidya-Bonner metric

Meaning of the coordinate time

The time coordinate of a field-free and sufficiently far from the mass remote stationary observer stands with the coordinates and relative ${\displaystyle t}$ ${\displaystyle M}$ ${\displaystyle u}$ ${\displaystyle v}$

${\displaystyle {\rm {d}}t={\rm {d}}u+{\frac {{\rm {d}}r}{1-2M(u)/r+Q(u)/r^{2}}},}$

${\displaystyle {\rm {d}}t={\rm {d}}v-{\frac {{\rm {d}}r}{1-2M(v)/r+Q(v)/r^{2}}}.}$

There is a constant ( ) with ${\displaystyle u}$ ${\displaystyle {\rm {d}}u=0}$

${\displaystyle {\frac {{\rm {d}}r}{{\rm {d}}t}}=+\left(1-{\frac {2M(t)}{r}}+{\frac {Q(t)}{r^{2}}}\right)}$

for radially outgoing radiation and

a constant ( ) with ${\displaystyle v}$ ${\displaystyle {\rm {d}}v=0}$

${\displaystyle {\frac {{\rm {d}}r}{{\rm {d}}t}}=-\left(1-{\frac {2M(t)}{r}}+{\frac {Q(t)}{r^{2}}}\right)}$

for radially incoming radiation.

Individual evidence

1. ↑ Kim, Choi & Yang: Black hole radiation in the Vaidya metric
2. ↑ Corvin Zahn: Visualization of the theory of relativity . Coordinate-free and interactive tools. Tübingen 2008 ( full text [PDF; accessed December 8, 2017] dissertation).
3. ^ Shaikh, Kundu & Sen: Curvature Properties Of Vaidya Metric
4. ↑ Thanu Padmanabhan: Gravitation: Foundation and Frontiers . Cambridge University Press, New York 2010, ISBN 978-0-521-88223-1 (English, full text in Google Book Search).