Why are Kruskal coordinates related to a freely falling observer?

[Haorong Wu]

TL;DR Summary

What is the reason that the Kruskal and tortoise coordinates are associated with a freely falling observer and a static observer?

I am reading a paper, A Pedagogical Review of Black Holes, Hawking Radiation and the Information Paradox.

On page 17, it reads that

Since Kruskal coordinates cover the entire black hole manifold, both interior and exterior, these coordinates are associated to observers freely falling into the black hole. Correspondingly, the vacuum that these observers will see, denoted by $\left | 0_K\right >$.

and

In contrast, the tortoise coordinates used in the mode expansion (3.26) for the scalar field operator are associated to observers that are at a constant distance outside the black hole, since the coordinates do not cover the interior. Correspondingly, the vacuum that these observers will see, denoted by $\left | 0_T\right >$.

I am not convinced that the two sets of coordinates are associated with different observers. I think the coordinate systems are independent of observers. The Kruskal coordinates can cover the maximally extended Schwarzschild spacetime, so why it is not associated with the static observer? Thanks.

 

[Demystifier]

Before understanding that, consider first Minkowski and Rindler coordinates in flat spacetime. Do you agree that Minkowski coordinates are related to inertial observers and Rindler coordinates to non-inertial ones?

 

[PeterDonis]

On page 17, it reads that

The quoted statement about Kruskal coordinates is quite misleading. It implies that any observer whose worldline is a straight timelike line in Kruskal coordinates is freely falling, which is not the case.

Also, the coordinates most naturally associated with a static observer are Schwarzschild coordinates in the exterior region, which do not use the tortoise coordinate.

I do not think this paper is a good source.

 

[PeterDonis]

Do you agree that Minkowski coordinates are related to inertial observers and Rindler coordinates to non-inertial ones?

This is of course true, but unfortunately it is a poor basis for the claims being made in the paper. The analogue to Minkowski coordinates is indeed Kruskal coordinates, but, as I noted in post #3, observers whose worldlines are straight lines in Kruskal coordinates are not freely falling. In other words, the Kruskal chart is not an "inertial" chart the way Minkowski coordinates in flat spacetime are.

Also, the analogue to Rindler coordinates is Schwarzschild coordinates, which, as I noted in post #3, do not use the tortoise coordinate.

 

[PeterDonis]

The Kruskal coordinates can cover the maximally extended Schwarzschild spacetime

Yes. However, this is irrelevant to the question of covering static observers, since those only exist in the exterior region, so you don't need a chart that covers the full maximal extension of the manifold to deal with them.

so why it is not associated with the static observer?

Because a static observer is not at rest in these coordinates.

 

[Sagittarius A-Star]

Before understanding that, consider first Minkowski and Rindler coordinates in flat spacetime. Do you agree that Minkowski coordinates are related to inertial observers and Rindler coordinates to non-inertial ones?

See W. Rindler: "Kruskal Space and the Uniformly Accelerated Frame":
https://pubs.aip.org/aapt/ajp/artic...kal-Space-and-the-Uniformly-Accelerated-Frame

He wrote about this topic also in his book "Essential Relativity: Special, General, and Cosmological", 2nd edition, on page 156, chapter "8.6 Kruskal Space and the Uniform Accelerated Field".

 

[Haorong Wu]

Thanks, @Demystifier, @PeterDonis, @Sagittarius A-Star. Below is my attempt to understand this question.

In the exponential, $e^{-i\omega x^0}$ of a mode function, the factor $\omega$ before the time coordinate $x^0$ had better be (or at least be related to) the frequency (energy) perceived by a specific observer. Therefore, $x^0$ should be related to the observer's proper time.

The Minkowski coordinates are natural to an inertial observer since it is flat as seen by that observer. If the coordinates are set in the observer's proper reference frame, then the coordinate time is the proper time.
The Rindler coordinates are natural to an accelerated observer since it has a metric $ds^2=e^{2a\xi}(-d\eta^2+d\xi^2)$ which is conformal to a flat metric and similarly, the accelerated observer's proper time is proportional to $\eta$ since $\xi$ is constant.
Also, in the Schwarzschild spacetime, the metric has the form as $ds^2=-(1-2M/r)(dt^2-dr^{*2})$ in the tortoise coordinates, which is also conformal to a flat metric; and for a stationary observer with a fixed $r^{*}$, the proper time is proportional to $t$.

However, I still have problems understanding why the Kruskal coordinates are chosen for a freely falling observer. In the Kruskal coordinates, the metric is $ds^2=32M^3e^{-r/2M}/r(-dT^2+dR^2)$, which is conformal to a flat metric, as well, but I could not verify that the Kruskal time $T$ is proportional to the observer's proper time.

Maybe I think the wrong way.

 

[Demystifier]

I could not verify that the Kruskal time T is proportional to the observer's proper time.

That's because it isn't. The Kruskal coordinates are related to the freely falling observer in a weaker sense. Such observer has access to the whole spacetime, and Kruskal coordinates cover the whole spacetime. That's all.

 

[PeterDonis]

The Minkowski coordinates are natural to an inertial observer since it is flat as seen by that observer.

Minkowski spacetime is flat, period. Flat vs. curved is not observer dependent.

Minkowski coordinates are natural to an inertial observer in Minkowski spacetime because in flat spacetime there are global inertial frames, and every inertial observer is at rest in one of them, and Minkowski coordinates are the coordinates of global inertial frames.

I still have problems understanding why the Kruskal coordinates are chosen for a freely falling observer.

They aren't. The most natural choices of coordinates for radial free-falling observers in Schwarzschild spacetime are Painleve coordinates (if they are falling in from rest at infinity) or Novikov coordinates (if they are falling from rest at a finite height). Again, the paper you reference in the OP is not a good source and its claims regarding coordinates are not valid.

 

[PeterDonis]

Such observer has access to the whole spacetime

Only if they are free-falling upwards out of the white hole region, coming to rest at a finite height, and then falling back into the black hole region. And even then a given observer can at most enter one of the two exterior regions, not both, so they only have access to three of the four regions.

All other free-falling observers are restricted to only two of the four regions in the maximal extension.

 

[Demystifier]

Only if they are free-falling upwards out of the white hole region, coming to rest at a finite height, and then falling back into the black hole region. And even then a given observer can at most enter one of the two exterior regions, not both, so they only have access to three of the four regions.

All other free-falling observers are restricted to only two of the four regions in the maximal extension.

What I had in mind was not the maximal extension, but only this:

 

[PeterDonis]

What I had in mind was not the maximal extension, but only this:

Ok. But the OP is asking about the maximal extension.