Interpretation of (X,T) coordinates in Kruskal diagram

[Hill]

TL;DR Summary

The points from "General Relativity: The Theoretical Minimum" by Susskind.

These are the points in the book:

3. The coordinates (X, T) are those naturally used by an observer who is near the horizon of the black hole, and is free-falling in the gravitational field of the black hole.
4. Indeed, even though space-time is not flat, the observer free-falling through the horizon perceives it as flat. Therefore, they might find it convenient to use the rectangular coordinates (X, T) to chart what is going on around them. T is by definition the proper time of the observer. It is sometimes called the coordinate time. Coordinate X of course measures distances with the stick of the observer.

What is "naturally used"? Does it hold only as the observer crosses the event horizon? How can they "use" them?

 

[Ibix]

If you are shut in a box you could establish a local Einstein frame in there. I would interpret "naturally used" to mean matching that locally - so orthonormal (or at least orthogonal) with the timelike vector parallel to your worldline.

However, this does not apply to Kruskal-Szekeres coordinates for a freely infalling observer. Their worldlines areonly instantaneously vertical on a Kruskal diagram and nit necessarily so near the horizon (although it can be arranged).

As I write that, I note that you can "boost" Kruskal-Szekeres coordinates by translating along the Killimg field that is timeike externally. I think that will let you select an arbitrary event on a worldline and select coordinates so the Kruskal-Szekeres $T$ basis vector is parallel to it there. So I think you could always choose Kruskal-Szekeres coordinates that are "naturally used" by a selected observer crossing the horizon. Perhaps that's what he means? Bit of a stretch though.

 

[Hill]

If you are shut in a box you could establish a local Einstein frame in there. I would interpret "naturally used" to mean matching that locally - so orthonormal (or at least orthogonal) with the timelike vector parallel to your worldline.

However, this does not apply to Kruskal-Szekeres coordinates for a freely infalling observer. Their worldlines areonly instantaneously vertical on a Kruskal diagram and nit necessarily so near the horizon (although it can be arranged).

As I write that, I note that you can "boost" Kruskal-Szekeres coordinates by translating along the Killimg field that is timeike externally. I think that will let you select an arbitrary event on a worldline and select coordinates so the Kruskal-Szekeres $T$ basis vector is parallel to it there. So I think you could always choose Kruskal-Szekeres coordinates that are "naturally used" by a selected observer crossing the horizon. Perhaps that's what he means? Bit of a stretch though.

If this is all that it is, I don't understand why it needed the emphasis of special points for understanding of Kruskal diagrams.

 

[PeterDonis]

What is "naturally used"? Does it hold only as the observer crosses the event horizon? How can they "use" them?

Point 3 at least needs some clarification. If you pick an event on the horizon and imagine free-falling worldlines that pass through that point, you can then scale and offset the Kruskal $X, T$ to match the coordinates of a local inertial frame centered on the chosen event, in which the free-falling worldlines, within the confines of that local inertial frame, will be straight lines. In other words, the standard $x, t$ Minkowski coordinates of the local inertial frame will be expressible as $k X - X_0, k T - T_0$, where $k$ is a constant factor that can be obtained from looking at the metric in Kruskal coordinates, and $X_0, T_0$ are the Kruskal coordinates of the chosen event. All this only works within the local inertial frame, and the frames will be different for different chosen events (since $X_0, T_0$ will be different).

Point 4 is just a somewhat clumsy way of saying that, within a single local inertial frame, spacetime looks flat.

 

[PeterDonis]

you can "boost" Kruskal-Szekeres coordinates by translating along the Killimg field that is timeike externally. I think that will let you select an arbitrary event on a worldline and select coordinates so the Kruskal-Szekeres $T$ basis vector is parallel to it there.

Yes, in the construction I described in my previous post just now, I did not mention that you might have to apply a Kruskal boost if you want a particular free-falling worldline to be at rest in the local inertial frame (i.e., to be the $t$ axis of the frame).

 

[Orodruin]

T is by definition the proper time of the observer.

Wait, what?
What does he mean by this?
There is a big chunky r-dependent prefactor in front of $dT^2$ in the metric.

 

[Sagittarius A-Star]

Wait, what?
What does he mean by this?
There is a big chunky r-dependent prefactor in front of $dT^2$ in the metric.

It depends on, for what the capital letters X, T are used. Maybe that are used for the standard Minkowski coordinates.

 

[Hill]

It depends on, for what the capital letters X, T are used. Maybe that are used for the standard Minkowski coordinates.

He refers to this figure:

 

[Orodruin]

It depends on, for what the capital letters X, T are used. Maybe that are used for the standard Minkowski coordinates.

There are no Minkowski coordinates on Schwarzschild spacetime (apart from locally) but the coordinates discussed here are explicitly said to be Kruskal–Szekeres coordinates.

He refers to this figure:
View attachment 338161

Yes, T is definitely not proper time.

 

[vanhees71]

Maybe it's the $T$ and $X$ as in the Wikipedia article?

https://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

I don't see in which sense $T$ is a "proper time".

 

[PeterDonis]

Yes, T is definitely not proper time.

Yes, that part of point 4 is simply wrong as it's stated. The rescaled $T$, according to the procedure I described earlier in this thread, will be the proper time of the free-falling observer who is at rest in the resulting local inertial frame centered on the chosen event at the horizon, since coordinate time in that frame is the same as proper time for a free-falling observer at rest in the frame. (As @Ibix noted, you might have to perform a Kruskal boost to get a particular chosen free-falling observer to be the one whose worldline is at rest in that frame.) But Susskind doesn't talk at all about the necessary rescaling (or the offset that has to be applied to center the frame on the chosen event on the horizon).

 

[Hill]

I think I understand what he means: the KS coordinates (X, T) are Minkowski coordinates for the locally flat spacetime near the horizon,

 

[Sagittarius A-Star]

I think I understand what he means: the KS coordinates (X, T) are Minkowski coordinates for the locally flat spacetime near the horizon,

It seems that Susskind uses the same capital letters X, T for both, the locally applied standard Minkowski diagram in your posting #12 and the Kruskal diagram in your posting #8, which includes the upper quadrant containing the singularity.

 

[Hill]

It seems that Susskind uses the same capital letters X, T for both, the locally applied standard Minkowski diagram in your posting #12 and the Kruskal diagram in your posting #8, which includes the upper quadrant containing the singularity.

I understand from him that they coincide in that local area.

 

[PeterDonis]

Moderator's note: Thread level changed to "I".

 

[PeterDonis]

I think I understand what he means: the KS coordinates (X, T) are Minkowski coordinates for the locally flat spacetime near the horizon

But they aren't unless you apply the transformation I described in post #4 and clarified in post #5 based on @Ibix's post.

I understand from him that they coincide in that local area.

But they don't unless you apply the transformation I described.

 

[PeterDonis]

I understand from him

Do you mean from his book? Or from some personal conversation with him?

 

[Hill]

But they aren't unless you apply the transformation I described in post #4 and clarified in post #5 based on @Ibix's post.But they don't unless you apply the transformation I described.

Yes. I think it means, one of (many) possible Minkowski coordinates there, which are related by rotation, translation, and boost.

Do you mean from his book? Or from some personal conversation with him?

From his book.

 

[PeterDonis]

I think it means, one of (many) possible Minkowski coordinates there, which are related by rotation, translation, and boost.

But Kruskal coordinates aren't any of the possible Minkowski coordinates unless you apply the kind of transformation I described. There is no local inertial frame at the horizon where you can just take the Kruskal coordinates and use them directly as the Minkowski coordinates.

 

[Sagittarius A-Star]

I understand from him that they coincide in that local area.

Unfortunately, I don't have the book to check this. I found the opposite statement in his related video at 46:00.

 

[Hill]

Unfortunately, I don't have the book to check this. I found the opposite statement in his related video at 46:00.

It does not seem opposite to me, but rather the same, although put more mildly:

(the last part said while gesturing to T and X)

 

[Hill]

The presentation in the book goes like this:

First, using Minkowski coordinates X and T, he develops flat metric in hyperbolic coordinates. Then, using KS coordinates X and T, he approximates Schwarzschild metric in the vicinity of horizon and shows that it is (approximately) the same as the flat metric in the hyperbolic coordinates developed earlier.

 

[PeterDonis]

using KS coordinates X and T, he approximates Schwarzschild metric in the vicinity of horizon and shows that it is (approximately) the same as the flat metric in the hyperbolic coordinates developed earlier.

Does this process involve a transformation on the KS coordinates that looks like what I described in post #4? It should.

The analogue of hyperbolic coordinates here are of course Schwarzschild coordinates.

 

[Hill]

Does this process involve a transformation on the KS coordinates that looks like what I described in post #4? It should.

The analogue of hyperbolic coordinates here are of course Schwarzschild coordinates.

Yes, he rescales and offsets.
Yes, they are.

 

[PeterDonis]

Yes, he rescales and offsets.

Ok, good. That makes a big difference, and from your earlier posts it was not at all clear that he was doing that.

 

[Hill]

Ok, good. That makes a big difference, and from your earlier posts it was not at all clear that he was doing that.

Sorry for that. Between the earlier posts and the last I re-read the derivation to find it out. Thank you.

 

[cianfa72]

As I write that, I note that you can "boost" Kruskal-Szekeres coordinates by translating along the Killing field that is timeike externally.

Do you mean by translating along the curves of constant Schwarzschild coordinate time $t$ in the external region I ?

 

[Orodruin]

Do you mean by translating along the curves of constant Schwarzschild coordinate time $t$ in the external region I ?

No, quite the opposite. Translating along the curves with only t changing. Everywhere.

The qualifier of ”that is timelike externally” just identifies the field. The field itself exists everywhere but is null on the horizon and spacelike inside the horizon.

 

[cianfa72]

No, quite the opposite. Translating along the curves with only t changing. Everywhere.

Yes, sorry. They are the curves of constant Schwarzschild $r,\theta,\phi$ i.e. integral curves of timelike KVF in the exterior region.

 

[Sagittarius A-Star]

It does not seem opposite to me, but rather the same, although put more mildly:

Yes, that is correct. Sorry, I interpreted wrongly Susskinds first two sentences after 46:00.

However, I can't find in the video the following of what you quoted in the OP from the book:

These are the points in the book:

4. ... T is by definition the proper time of the observer. ...

 

[Hill]

I can't find in the video the following of what you quoted in the OP from the book

Me too. That statement is still a mystery to me.

 

[Sagittarius A-Star]

Me too. That statement is still a mystery to me.

With a Google search, I found the cited point 4 in the preview of Google books (using Chrome browser):
https://books.google.de/books?id=Wx...ances with the stick of the observer.&f=false

It refers indeed to figure 5.

 

[Sagittarius A-Star]

I don't find the point 4 in the errata sheet:
https://www.lapasserelle.com/general_relativity/erratum.html

 

[Hill]

I don't find the point 4 in the errata sheet:
https://www.lapasserelle.com/general_relativity/erratum.html

The plot thickens.

 

[Hill]

It seems that Susskind uses the same capital letters X, T for both, the locally applied standard Minkowski diagram in your posting #12 and the Kruskal diagram in your posting #8, which includes the upper quadrant containing the singularity.

I think that this is just what he does.