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The short answer is no, the Hawking radiation is not a coordinate dependent phenomenon. These (Kruskal-Szekeres and Tortoise) coordinates simplify the problem significantly. 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.

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.

The short is no, the Hawking radiation is not a coordinate dependent phenomenon. These (Kruskal-Szekeres and Tortoise) coordinates simplify the problem significantly. 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.

The short answer is no, the Hawking radiation is not a coordinate dependent phenomenon. These (Kruskal-Szekeres and Tortoise) coordinates simplify the problem significantly. 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.

The short is no, the Hawking radiation is not a coordinate dependent phenomenon. These (Kruskal-Szekeres and Tortoise) coordinates simplify the problem significantly. Physically, any static observer at infinity would detect the same Hawking radiation rate.

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. To do so, we need the following instructions:

i) Constructing a static frame with respect to the black hole for a remote observer (i.e., an observer located at the rest far away from the black hole).

ii) Determining an appropriate metric for a freely falling frame near/crossing the event horizon.

iii) Comparing quantum fields in both frames.

The static observer in black hole background observes Hawking radiation and the freely falling observer in black hole background would observe nothing. Having these two observers, you can derive the Hawking temperature in any black hole background, not only for Schwarzschild solution. However, working with a static, remote observer (at infinity) is more easier (so, conventionally, Hawking temperature, $T_{\rm{H}}$, is measured by an observer at infinity, but, an observer at $r$ would detect $T(r) = \frac{{{T_{\rm{H}}}}}{{\sqrt {\left| {{g_{00}}(r)} \right|} }}$). 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.

The short is no, the Hawking radiation is not a coordinate dependent phenomenon. These (Kruskal-Szekeres and Tortoise) coordinates simplify the problem significantly. 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.