Does Hawking radiation depend on the coordinate system used?

Question

First things first, this question is different from the one I posted here, the coefficients I've found in this link are wrong, but the right coefficients suffer from the same strange relation:

\begin{align} |\alpha_{p,k;r_s,\infty}| &\propto |\omega_k+\omega_p|e^{\frac{1}{2}\pi\omega_p r_s} \\ |\beta_{p,k;r_s,\infty}| &\propto |\omega_k-\omega_p|e^{-\frac{1}{2}\pi\omega_p r_s} \end{align} With the same proportionality factor. So in the coordinate system I use, we don't have $|\alpha_{\omega,\omega'}|=e^{\pi\omega r_s}|\beta_{\omega,\omega'}|$

From this article (An Introduction to Black Hole Evaporation, Jennie Traschen) and from the original article of Hawking on his radiations, we can see that the tortoise coordinates and null coordinates/null Kruskal coordinates are used and then give the wright relation between the Bogoliubov coefficients.

In fact, it seems to me that the very structure of Bogoliubov coefficients can't give the relation $|\alpha_{\omega,\omega'}|=e^{\pi\omega r_s}|\beta_{\omega,\omega'}|$ in spherical coordinates, since in these we necessarily have $\Phi=e^{i\omega t}\phi(r)Y_l^m(\theta,\varphi)$.

So my question is: Is the Hawking radiation a coordinate dependent phenomenon?

Answer 1

The short answer is no, the Hawking radiation is not a coordinate dependent phenomenon. These (Kruskal-Szekeres and Tortoise) coordinates simplify the problem significantly when you are dealing with QFT in curved spacetime background (here Schwarzschild black hole background). Physically, any static observer at infinity would detect the same Hawking radiation rate. This also can be verified by use of the explicit form of Hawking radiation in terms of surface gravity ($\kappa$), i.e.,

$${T_{\rm{H}}} = \frac{\kappa }{{2\pi }}\,\,{\rm{where}}\,\,\kappa = \sqrt { - \frac{1}{2}({\nabla _\mu }{\xi _\nu })({\nabla ^\mu }{\xi ^\nu })}, $$ where $\xi _\nu$ is the Killing vector field satisfying the Killing equation, ${\nabla _\mu }{\xi _\nu } + {\nabla _\nu }{\xi _\mu } = 0$. Killing fields are the infinitesimal generators of isometries and each of them corresponds to a quantity which is conserved along geodesics. Obviously, this definition does not depend on the coordinates one is dealing with.

To make it more clear:

Kruskal-Szekeres coordinates have the advantage that they are well-behaved everywhere outside the physical singularity. In addition, they cover the entire spacetime manifold of the maximally extended Schwarzschild solution. For these reasons, KS coordinates which describes the entire spacetime are suitable for any observer freely falling through the horizon.

Tortoise coordinates are just suitable for a remote observer/frame at fixed location in black hole background since when an observer approaches a black hole event horizon, its Schwarzschild time coordinate grows infinite. The outgoing null rays in this coordinate system have an infinite change in $t$ on travelling out from the horizon. The tortoise coordinate is intended to grow infinite at the appropriate rate such as to cancel out this singular behavior in coordinate systems constructed from it.

Generally, in order to derive the Hawking temperature, you need to consider quantum fields from the viewpoints of different observers since there is no unique mode decomposition of the quantum fields. This is the main idea. The problem is to find the physical, well-behaved coordinates free from pathological properties. If you can construct such coordinates, you will find the same relation for the Hawking radiation.