What is "naturally used"? Does it hold only as the observer crosses the event horizon? How can they "use" them?
If this is all that it is, I don't understand why it needed the emphasis of special points for understanding of Kruskal diagrams.Ibix said:
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).Hill said:
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).Ibix said:
It depends on, for what the capital letters X, T are used. Maybe that are used for the standard Minkowski coordinates.Orodruin said:
He refers to this figure:Sagittarius A-Star said:
There are no Minkowski coordinates on Schwarzschild spacetime (apart from locally) but the coordinates discussed here are explicitly said to be Kruskal–Szekeres coordinates.Sagittarius A-Star said:
Yes, T is definitely not proper time.Hill said:
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).Orodruin said:
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 said:
I understand from him that they coincide in that local area.Sagittarius A-Star said:
But they aren't unless you apply the transformation I described in post #4 and clarified in post #5 based on @Ibix's post.Hill said:
But they don't unless you apply the transformation I described.Hill said:
Do you mean from his book? Or from some personal conversation with him?Hill said:
Yes. I think it means, one of (many) possible Minkowski coordinates there, which are related by rotation, translation, and boost.PeterDonis said:
From his book.PeterDonis said:
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.Hill said:
Unfortunately, I don't have the book to check this. I found the opposite statement in his related video at 46:00.Hill said:
Does this process involve a transformation on the KS coordinates that looks like what I described in post #4? It should.Hill said:
Yes, he rescales and offsets.PeterDonis said:
Ok, good. That makes a big difference, and from your earlier posts it was not at all clear that he was doing that.Hill said:
Sorry for that. Between the earlier posts and the last I re-read the derivation to find it out. Thank you.PeterDonis said:
Do you mean by translating along the curves of constant Schwarzschild coordinate time ##t## in the external region I ?Ibix said:
No, quite the opposite. Translating along the curves with only t changing. Everywhere.cianfa72 said:
Yes, sorry. They are the curves of constant Schwarzschild ##r,\theta,\phi## i.e. integral curves of timelike KVF in the exterior region.Orodruin said:
Yes, that is correct. Sorry, I interpreted wrongly Susskinds first two sentences after 46:00.Hill said:
Hill said:
Me too. That statement is still a mystery to me.Sagittarius A-Star said:
Hill said:
The plot thickens.Sagittarius A-Star said:
I think that this is just what he does.Sagittarius A-Star said:
The (X, T) coordinates in a Kruskal diagram are a set of coordinates used to represent the spacetime structure of a black hole in a way that avoids the singularities and coordinate artifacts present in the standard Schwarzschild coordinates. They allow for a more complete and accurate depiction of the black hole's event horizon and the regions inside and outside of it.
Kruskal coordinates are a transformation of the Schwarzschild coordinates (r, t) that removes the coordinate singularity at the event horizon (r = 2M). This transformation results in a new set of coordinates (X, T) that smoothly cover both the interior and exterior regions of a black hole, providing a clearer and more comprehensive view of the spacetime geometry.
Kruskal coordinates are important because they provide a more accurate and complete representation of the black hole spacetime. This allows physicists to better understand the nature of the event horizon, the behavior of objects near the black hole, and the structure of spacetime itself. They are particularly useful for visualizing and analyzing the causal structure of black holes.
In a Kruskal diagram, lines of constant Schwarzschild radius r appear as hyperbolas, while lines of constant Schwarzschild time t appear as straight lines radiating from the origin. The event horizon (r = 2M) is represented by the 45-degree lines (X = ±T). This depiction helps in understanding the causal relationships and the flow of time in the vicinity of a black hole.
The Kruskal diagram helps in understanding the event horizon by providing a coordinate system in which the horizon is not a singularity but a regular surface. This allows for a more intuitive grasp of how light cones behave near the horizon, how particles and information can cross the horizon, and how different regions of spacetime are connected. It shows that the event horizon is a boundary beyond which events cannot affect an outside observer.
