Possibility of Black Hole research

[timmdeeg]

Just think light cones.

Ah, that clarifies the issue, thanks.

 

[pervect]

Could you please elaborate a bit on the "other direction". By means of an Eddington-Finkelstein diagram I understand that the stern receives light from the bow but I fail to see the other way round. From the diagram it seems that the bow is in the past of the stern.

As I recall, the ingoing Eddington-Finkelstein diagram is different than the outgoing one. I'd recommend Kruskal-Szerkeres coordinates and/or a Penrose diagram, where light always travels at a 45 degree angle.

To quote Wiki

Kruskal–Szekeres coordinates have a number of useful features which make them helpful for building intuitions about the Schwarzschild spacetime. Chief among these is the fact that all radial light-like geodesics (the world lines of light rays moving in a radial direction) look like straight lines at a 45-degree angle when drawn in a Kruskal–Szekeres diagram (this can be derived from the metric equation given above. ... All timelike world lines of slower-than-light objects will at every point have a slope closer to the vertical time axis (the T coordinate) than 45 degrees. So, a light cone drawn in a Kruskal–Szekeres diagram will look just the same as a light cone in a Minkowski diagram in special relativity.

It shouldn't be too hard to see this that for a short enough space-ship, light from the stern can thus reach the bow, as well as the reverse, because locally the light is always traveling "faster" than any material object (the slope of the infalling object is more vertical).

This is only true in the appropriate limit of a "short enough" ship or "big enough" Schwarzschild radius.

Unfortunately, I didn't find any good diagrams of an infalling object in KS coordinates to illustrate this graphically.

The intuitive argument here is that if light from the bow couldn't reach the stern, or vica versa, an observer would obviously know that they crossed the event horizon when they lost sight of the other end of the ship. This just doesn't happen - in the limit of a short ship (which is the same as the limit of a large blac hole), one can't tell from inside the ship that one has crossed the horizon.

As one approaches the singularity , the space-time curvature keeps going, and it gets harder and harder to meet the requirements of a "short enough" ship.

 

[Ibix]

Unfortunately, I didn't find any good diagrams of an infalling object in KS coordinates to illustrate this graphically.

Your wish is my command:

This is the top right quadrant of a Kruskal diagram like the one (by @DrGreg, I believe) at Wikipedia. The fine grey line is the event horizon and the dark grey curve is the singularity (the shaded area doesn't correspond to anywhere in spacetime). The red lines show the final moments of three objects free-falling from rest at infinity. The two trailing objects emit blue light pulses as they cross the horizon - you can see that they catch up to the object in front, which has time to reply and receive a second reply (well, object one doesn't quite get to receive the second reply).

I actually set this up with a solar mass black hole ($R_S=3km$) and at about $1.5R_S$ the objects had Schwarzschild $r$ coordinates 100m apart.

 

[PeterDonis]

the ingoing Eddington-Finkelstein diagram is different than the outgoing one.

That's correct. However, the ingoing one is enough to analyze the experiment in question; it just isn't as easy to see the result as it is in a Kruskal diagram since only ingoing light rays go on 45 degree lines in the ingoing Eddington-Finkelstein diagram.

 

[pervect]

I did want to say a bit more about the "short space ship" condition. Material objects must follow a timelike worldline, and because of this no such object can ever go as fast as a light beam. So if a light beam and a material object have a race, if they both start at the same time and place, the lightbeam will always "be ahead of" the material object in a race.

However, if the accelerating object has a head start, even though it's slower than the lightbeam, it's possible (though a bit surprising) that the ligihtbeam will never be able to catch up. This is a feature of hyperbolic motion, for instance.

In the "short ship", limit, though, the space-time diagram will be linear, because terms of the second order will be negligible compared to terms of the first order. And for a linear diagram, the faster object will catch the slower one.

 

[pervect]

Your wish is my command:

Very nice! How did you calculalte and make the diagram?

 

[Ibix]

Very nice! How did you calculalte and make the diagram?

I took the geodesic equations in Schwarzschild coordinates and substituted $u$ and $v$ for $r$ and $t$. This works even at the horizon because the geodesic equation remains valid at and below the horizon - it's just its expression in Schwarzschild coordinates that goes wrong. I fed the result to one of scipy's numerical integrators and plotted results in pyplot. Confession: the light pulses were added by hand. My code can handle null geodesics, but I couldn't be bothered to do all the initialisation and intercept calculations in order to generate 45° straight lines.

I've had code lying around to do this for a while - I didn't do it all on Thursday night. It's not perfect, since you can't eliminate $r$ completely and you need to solve for it numerically. That generates numerical issues for long-lasting orbits since $u$ and $v$ grow while $u^2-v^2$ gets small, and $r$ depends on the latter (writing $(u+v)(u-v)$ postpones the problem, but ultimately doesn't solve it). And sometimes the integrator goes wrong near the singularity and objects skip merrily across its surface. But, those issues aside, the results are consistent with some analytical checking and a couple of things I found in the literature.

 

[timmdeeg]

Sorry for being a bit late and thanks for your response.

It shouldn't be too hard to see this that for a short enough space-ship, light from the stern can thus reach the bow, as well as the reverse, because locally the light is always traveling "faster" than any material object (the slope of the infalling object is more vertical).

Indeed, this is very convincing.

In the meantime I've found some Eddington-Finkelstein diagrams in Robert Geroch's nice book "General Relativity from A to B", which is written for laymen. One comes close to the space-ship scenario. Observer A falls into the black hole and after a while B follows him. Their wordlines are parallel and its easy to see that B receives light from A for some time beyond the horizon. In all the diagrams light goes up as time increases, so light emitted by B can't reach A. I'm not sure how to resolve this. In contrast in KS coordinates which @Ibix shows in #38 the situation is crystal clear.

 

[PeterDonis]

In all the diagrams light goes up as time increases, so light emitted by B can't reach A. I'm not sure how to resolve this.

If B falls in long enough after A, then light signals he emits inward won't reach A before A hits the singularity. To have A see any signals from B, you have to make sure the time between them falling in isn't too large.

 

[timmdeeg]

If B falls in long enough after A, then light signals he emits inward won't reach A before A hits the singularity. To have A see any signals from B, you have to make sure the time between them falling in isn't too large.

Heureka, I can see it now. https://www.researchgate.net/figure/Space-time-diagram-in-Eddington-Finkelstein-coordinates-showing-the-light-cones-close-to_fig4_260835665 is Fig. 93 in Geroch's book I've mentioned. It shows C freely falling and outgoing light, ingoing light according to the light cones. If I paint the worldline of B who jumped first together with the past light cone very close to that of C then it turns out that indeed B can receive light of C. It's a bit tricky though and as mentioned by others it's far better to be seen in a KS diagram.

 

[PAllen]

Heureka, I can see it now. https://www.researchgate.net/figure/Space-time-diagram-in-Eddington-Finkelstein-coordinates-showing-the-light-cones-close-to_fig4_260835665 is Fig. 93 in Geroch's book I've mentioned. It shows C freely falling and outgoing light, ingoing light according to the light cones. If I paint the worldline of B who jumped first together with the past light cone very close to that of C then it turns out that indeed B can receive light of C. It's a bit tricky though and as mentioned by others it's far better to be seen in a KS diagram.

There is something wrong with that diagram. It purports to show a timelike world line that does not remain inside its own future light cone. That is impossible. Your confusion is caused by a quantitatively inaccurate diagram. Consider especially the future light cone emanating from W, and the proposed world line beyond W. This cannot be right. It would make the free fall spacelike.

 

[timmdeeg]

Consider especially the future light cone emanating from W, and the proposed world line beyond W. This cannot be right. It would make the free fall spacelike.

Yes. The diagram which I mentioned in #43 doesn't show light cones. I was drawing the two parallel timelike geodesics close to each other and inside the respective light cones, with the ingoing null geodesics having an angle of 45° with the space axis.

 

[martinbn]

There is something wrong with that diagram. It purports to show a timelike world line that does not remain inside its own future light cone. That is impossible. Your confusion is caused by a quantitatively inaccurate diagram. Consider especially the future light cone emanating from W, and the proposed world line beyond W. This cannot be right. It would make the free fall spacelike.

It only looks like this. But in this diagram the light cones tilt as you go along the green/red world line. The closer to the singularity you are the more horizontal (open to the right) they are.

 

[PAllen]

It only looks like this. But in this diagram the light cones tilt as you go along the green/red world line. The closer to the singularity you are the more horizontal (open to the right) they are.

I know, but don’t think that affects my critique. The black dashed lines are paths of radially outgoing light, thus they form the left boundary of light cones. Along one of these dashes lines, the width of a light cone is shown constant. Following these rules of the diagram, the light cone at event W does not enclose the purported free fall path.

[edit: I should note that the issue is that this diagram is not remotely an accurate representation of Eddinton-Finkelstein coordinates. ]

 

[martinbn]

I know, but don’t think that affects my critique. The black dashed lines are paths of radially outgoing light, thus they form the left boundary of light cones. Along one of these dashes lines, the width of a light cone is shown constant. Following these rules of the diagram, the light cone at event W does not enclose the purported free fall path.

[edit: I should note that the issue is that this diagram is not remotely an accurate representation of Eddinton-Finkelstein coordinates. ]

I see, but I thought the diagram is only schematic.

 

[timmdeeg]

I see, but I thought the diagram is only schematic.

Altogether the 17 diagrams in this Book are very instructive. The first one shows just light cones with decreasing r getting increasingly tilted while their width decreases. There are various scenarios with one and two observers, with a rope going in, etc. The only scenario missing is the one discussed here.

 

[PAllen]

I see, but I thought the diagram is only schematic.

At least in this case, had the diagram been a little more precise with angles, and perhaps shown light cones along the infaller, it would have been less misleading.

I also just noticed some bad wording in the captions:

"The global time direction is in the direction of the axis of the cylinder". Taken literally, this is problematic, as the vertical direction is spacelike inside the horizon. But better wordings might sound pedantic, e.g. "the upper light cones are future pointing".