Insights The Schwarzschild Geometry: Part 3 - Comments

fresh_42

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Thanks, even though I might not have understood it all. I'm looking forward to the next part and perhaps some speculations of yours, what a future model would have to accomplish.

There's a little error in the first third "Here will will assume that the GR model holds for all ##r>0##."
 

PAllen

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In exploring the KS chart, I find it useful to look at lines of constant X. Lines of constant T either don't cover the whole chart, or must be treated as split before a certain T, and after another one. Each line of constant X simply goes from a minimum T to a maximum T, giving a smooth connected picture of the S2xR2 manifold. Each of these lines has a nice interpretation: the evolution of a 2-sphere from r > 0, to some maximum r, then down toward 0 again. The smallest maximum r is the horizon radius; for larger X, the maximum radius grows without bound.
 

fresh_42

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Why do you think this is an error?
I meant it should have been "we will". I guess I should have said mistake instead. I'm not very good at distinguishing between error, failure, mistake and fault. There's only one word for it in my language.
 
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I find it useful to look at lines of constant X
The only thing I would clarify here is that these lines are timelike, so the word "evolution" that you use means "evolution in time". We could make that more specific by postulating an observer who follows this line as his worldline. (An interesting exercise is to compute the proper acceleration of this observer; it is not zero.)

Lines of constant X either don't cover the whole chart, or must be treated as split before a certain T, and after another one.
I'm not sure I understand. "The whole chart" is not the entire range of ##T## and ##X## coordinates; it is only those pairs ##(T, X)## that satisfy ##T^2 - X^2 < 1##. Lines of constant ##X## have no breaks within this range, and cover it entirely.
 

PAllen

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The only thing I would clarify here is that these lines are timelike, so the word "evolution" that you use means "evolution in time". We could make that more specific by postulating an observer who follows this line as his worldline. (An interesting exercise is to compute the proper acceleration of this observer; it is not zero.)
I used the word evolution deliberately to capture the timelike nature. Taken together, we have defined a congruence of world lines filling the entire manifold. Yes, I am well aware these world lines are not geodesic. I am going to write a follow on post on some additional physical interpretation for this congruence.
I'm not sure I understand. "The whole chart" is not the entire range of ##T## and ##X## coordinates; it is only those pairs ##(T, X)## that satisfy ##T^2 - X^2 < 1##. Lines of constant ##X## have no breaks within this range, and cover it entirely.
Consider a line of constant T with T greater than the minimum T for the future singularity. Then, there are actually two such lines, not one and they are disconnected. This is fine, but the lines of constant X show you more directly how the whole manifold is connected.
 

PAllen

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In exploring the KS chart, I find it useful to look at lines of constant X. Lines of constant T either don't cover the whole chart, or must be treated as split before a certain T, and after another one. Each line of constant X simply goes from a minimum T to a maximum T, giving a smooth connected picture of the S2xR2 manifold. Each of these lines has a nice interpretation: the evolution of a 2-sphere from r > 0, to some maximum r, then down toward 0 again. The smallest maximum r is the horizon radius; for larger X, the maximum radius grows without bound.
Expanding on this, one may make an analogy between this family of sphere evolutions and the exterior of spherical body. Such exterior region can be covered by families of world lines, each such family being a sphere of observers leaving the surface, reaching a maximum, and then returning (no requirement that they they be inertial). They can be set up so their maxima are all at constant T in some chart, and they don't cross, thus forming a valid congruence filling the entire exterior spacetime of the body. An observation is that this shows that the exterior of a spherical body, taken as a manifold by itself, has topology S2xR2.

However, there are some key differences from the KS congruence I described. The ordinary body congruence would have spheres evolving from minimum to maximum arbitrarily close to the minimum. There would be no notion of a minimum maxima that is larger by a finite amount than the minimum. Nor would there be a duplicate exterior corresponding to negative X in the KS case. This exterior would be analogous to one KS exterior quadrant, with the SC radius standing in for the body surface.

What GR tells us is that this construction (for the exterior of a body) cannot be shrunken down smoothly to represent a point body. It has a minumum size below which we (if we preserve vacuum and spherical symmetry) we must change the geometry, in the way mandated by the KS interior region(s).
 
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The next thing we notice is that the two boundary hyperbolas are spacelike (because their slope is always less than 45 degrees). That means that they are best viewed, intuitively, as moments of time, not places in space.

Reference https://www.physicsforums.com/insights/schwarzschild-geometry-part-3/
should that be "are timelike (..."?
By two boundary hyperbolas I assume you mean the two thick lined hyperbolas at r=0.
 

martinbn

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should that be "are timelike (..."?
By two boundary hyperbolas I assume you mean the two thick lined hyperbolas at r=0.
Spacelike, not timelike.
 

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