Comment by raattgift
6 months ago
The problem is probably connecting an image of a tilted causal cone at p near but outside a regularized r_s in Schwarzschild spacetime with the idea that from p there is a limited number of null geodesics escaping to the asymptotically flat region, that the fraction decreases with decreasing regularized r coordinate, and that from p' at the trapping surface there are no such null geodesics, as all non-spacelike curves (accelerated or not) at any point on or interior to the trapping surface terminate at the singularity. The idea that the singularity inevitably lies in the future of any observer at p' is behind the "spacetime swap" notion.
Some of the problem is that Schwarzschild coordinates have surprises buried in them, and what \Delta r and \Delta t mean are not what most people tend to think.
Someone should do an ELI12 of Unruh's (ca. 2014) excellent (give or take varying the spelling of Martin Kruskal's surname) Schwarzschild BH global coordinates pedagogic review <http://theory.physics.ubc.ca/530-21/bh-coords2.pdf> and add in a bit on Fermi normal coordinates as a maybe-obvious not-a-chart follow-on to the commenary just above eqn (55). But on "maybe-obvious", Unruh has the choice line: "Since in a large number of cases, the single horizon coordinates were discovered long before Schild’s coordinates, this is an exercise in alternate reality – what could have so easily happened if only the generators of those coordinate systems had recognized what they had."
Hi. Sorry to bother you here. Could you point out what error am I making in my, I believe, very simple objection?
https://news.ycombinator.com/threads?id=scotty79&next=441240...
tl;dr The farrrr-from-the-horizon part of Schwarzschild spacetime is just not like our spacetime. Only near and outside the horizon (or better, in the absence of a horizon) does Schwarzschild become a decent physical approximation for anything in our universe.
Schwarzschild infinity is unphysical, while your notion of t(Earth) is physical because we can associate a worldline with the planet's centre of mass (COM), hold the COM at the spatial origin of a system of spacetime coordinates, and use whatever "timestamps" we like on the time axis. But we could decide that t(Earth)=infinity could be yesterday, or tomorrow, or a billion years ago, or a couple billion years from now; if we count of seconds before or after t(Earth)=infinity, we still have t'(Earth)=infinity, so it's not a very good choice of coordinate.
I think you have a misunderstanding that is probably beyond my ability to help you with, since we can't do interactive blackboard work in HN comments. The root of your problem seems to be mis-identifying the local time at Earth with the Schwarzschild time at infinity in the Schwarzschild solution. We aren't at infinity to any known black hole: between us and the most distant black holes we know of is expanding spacetime not found in Schwarzschild's solution; betwee us and the nearest black holes is substantially and lumpily curved spacetime and plenty of matter unlike Schwarzschild's unique pointlike mass surrounded by non-lumpy matterless vacuum; none of the astrophysical black holes are infinitely old today (whereas Schwarzchild black holes are infinitely old at every time, otherwise the spacetime would not be static); and in general exact solutions of the Einstein Field Equations -- even ones that are not eternal -- do not superpose cleanly with solutions for other black holes (and crucially there are no black hole mergers in Schwarzschild), ordinary stars, galaxies, clusters, and expanding spacetime. As an example: hover just above the apparent horizon of Sagittarius A*. Look at a stellar black hole in our galaxy. What do you make of infallers plunging towards the smaller black hole? What do you make of the evolution of mass of the stellar black hole, from your vantage point hugging an SMBH's horizon?
Short of taking a series of courses or finding an informal short-term tutor to walk you through particular things (you can find either at your local tertiary education school, like a community college or university), there are plenty of good textbooks on General Relativity. You seem to have found Wald's, which is probably the most rigorously and densely mathematical of several popular teaching choices, and it does not seem to have helped you. I'd guess you'd be better off with e.g. Carroll's Spacetime and Geometry or Wheeler's Gravity and Spacetime.
There is also the Israel-Darmois thin shell method, which is technically annoying but lets us cut the central part of an e.g. Schwarzschild solution and paste it into a cosmology populated with other such pasted-in subregions. We can then trace light rays from e.g. a quasar, across early expanding space to a SMBH or elliptical galaxy acting as a gravitational lens, and then across later expanding space to an approximation of our neighbourhood, adapting the rays at each shell boundary. Although there is very definitely a subregion of black hole solution in that kind of approach, the asymptotically flat part of Schwarzschild is cut away along with its distant infinities. One can compare this cutting and pasting to the Hill sphere of influence of Jupiter and those of its satellites, for example, if one were interested in a navigational plan like Juno's or JUICE's.
> Only near and outside the horizon (or better, in the absence of a horizon) does Schwarzschild become a decent physical approximation for anything in our universe.
I guess that's my point. Noting at or inside event horizon (of any kind, not just Schwarzschild solution) is physical. It's pure math, no matter how fun, has nothing to do with reality.
> But we could decide that t(Earth)=infinity could be yesterday, or tomorrow, or a billion years ago, or a couple billion years from now;
No, we cannot. Because as you stated t(Earth), by which I meant time as it passes on Earth, is physical... Now I think I should write t_Earth instead so it looks more like subscript not function application. So t_Earth = Infinity is the moment after all of the time already passed. After every finite moment already occurred. So t_Earth = Infinity doesn't really exist ever. It's purely mathematical concept. Abstract limit of the real thing.
> I think you have a misunderstanding that is probably beyond my ability to help you with, since we can't do interactive blackboard work in HN comments.
That's probably true. Even a blackboard wouldn't help because you seem to be interested mostly in minutia and specifics while my problem lies with general reality (pun intended) of all of this and which part is real and which is just extrapolation to times that don't exist and why physicists usually don't seem to care to differentiate between one part and the other.
> We aren't at infinity to any known black hole
I'm saying exactly opposite. The black hole (its event horizon to be precise) is at infinity (infinite time) to us. Any infalling object at Kruskal diagram crosses the line clearly labeled as t=infinity which in reality can't happen because there's simply isn't infinite amount of time in the universe.
> none of the astrophysical black holes are infinitely old today
To be honest that's another claim I just don't take on faith, especially in the light of the discovery of early developed galaxies and the fact that galaxies developing this early this fast would emit so much light that it would contribute to CMB (even up to 100% of it) which throws off all of our precise math theories of how everything started. I'm more inclined to believe that every astrophysical black hole existed at the the time when CMB was emitted (and before) and had exactly same size (of the event horizon) as it has today.
> and crucially there are no black hole mergers in Schwarzschild
Do you say that because they are mathematically impossible (which I would agree with) or just because Schwarzschild modeled just one black hole so there's nothing to merge with?
> hover just above the apparent horizon of Sagittarius A*. Look at a stellar black hole in our galaxy. What do you make of infallers plunging towards the smaller black hole?
Let's assume your trajectories are parallel to skip issues of special relativity. If he's closer to his event horizon than you are he's slowed down in time for you as he is for the rest of the universe outside (just slightly less). If he's farther away he lives at a pace accelerated relative to you, in the same manner that the outside world is accelerated for you. The specific math of how both of you tend to infinite time dilationas you approach your respective event horizons should show if relative time dilation between you tends to some ratio (or 1) or infinity. I don't know which is the case.
> There is also the Israel-Darmois thin shell method, which is technically annoying but lets us cut the central part of an e.g. Schwarzschild solution and paste it into a cosmology populated with other such pasted-in subregions.
I'm not that interested in pasting them statically far away. I'd really love to see what shape two Schwarzschild blackholes (and by that I mean their event horizons because, I don't believe anything beyond them is real to us) hitting each other could look like. Maybe (|)
Thanks for recommendations for further reading.
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