# Gravitational redshift of temperature and electrostatic potential

Consider a charged black hole in four-dimensional Minkowski spacetime, with charge $Q$, mass $M>Q$:

$ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\Omega_2^2$, with

$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$.

When an observer at radial coordinate $r_1$ emits a photon, an observer at radial coordinate $r_2>r_1$ will perceive the photon with a redshifted wavelength. This is easy to interpret.

A similar thing happens for the temperature. If $\kappa$ is the surface gravity of the black hole, then the Hawking temperature is $T_H=\frac{\kappa}{2\pi}$. Due to gravitational redshift, the temperature measured by an observer at radial coordinate $r$ is

$T_{loc}(r)=\frac{1}{\sqrt{f(r)}}T_H$.

The redshift factor is the same as for a redshifted frequency, which I can understand by associating temperature with the inverse imaginary time period.

I interpret this redshift as a consequence of the particles which constitute Hawking radiation experiencing gravitational redshift. Is this correct?

Apparently, something similar also happens for the electrostatic potential. The electrostatic potential difference between the outer event horizon $r_+$ and infinity is given by $\Phi=\frac{Q}{r_+}$. However, the electrostatic potential between $r_+$ and some coordinate $r>r_+$, "blueshifted" from infinity to $r$, is given by

$\phi(r)=\left(\frac{Q}{r_+}-\frac{Q}{r}\right)\frac{1}{\sqrt{f(r)}}$.

(Source: Braden, Brown, Whiting and York, Charged black hole in a grand canonical ensemble, PRL Vol. 42 No. 10, 1990, equation 4.15.)

This expression seems to tell me that $\frac{Q}{r_+}-\frac{Q}{r}$ is the electrostatic potential difference between $r$ and $r_+$ as measured by someone at infinity, and the above expression is this same potential difference as measured by someone at $r$.

Is there a simple interpretation why the measured electrostatic potential should experience gravitational redshift as well, with the same redshift factor as a frequency? What is the wavelength that is being redshifted in this case, is it the one from the photons mediating the electromagnetic force? (These photons are virtual though, so can they actually be redshifted?)

## 1 Answer

I don't have the paper to hand, but by $$\phi$$ the authors could mean either $$A_0$$ or $$A^0$$, and these differ by a factor of $$f(r)$$.