How to survive in a black hole: Myth Debunked

[quantum123]

Published online: 18 May 2007; | doi:10.1038/news070514-21
How to survive in a black hole
There's no escape, but how can you maximize your remaining time?
Philip Ball

Aaaaaaaaaaargh: how to draw out your torturous fall into a black hole.

NASA

So there you are: you discover that your spaceship has inadvertently slipped across the event horizon of a black hole — the boundary beyond which nothing, not even light, can escape the hole's fearsome gravity. The only question is how you can maximize the time you have left. What do you do?

A common idea in physics is that you shouldn't try to blast your way out of there. Black holes, it's said, are like the popular view of quicksand: the harder you struggle, the worse things become.

But Geraint Lewis and Juliana Kwan of the University of Sydney in Australia say this is a myth. Their analysis of the problem, soon to be published in the Proceedings of the Astronomical Society of Australia1, shows that in general your best bet is indeed to turn on the rocket's engine. You'll never escape, but you'll live a little longer.

Falling into a black hole is a strange affair. Because the hole's gravity distorts space-time, a far-off observer watching an object crossing the event horizon sees time for that object appear to slow down — a clock falling into a black hole would appear, from the outside, to tick ever slower. At the horizon itself, time stops, and the object stays frozen there for the remaining lifetime of the Universe.

But this isn't how things seem to the in-falling object itself. Indeed, if the black hole is big enough, nothing noticeable happens when a spaceship crosses its event horizon — you could stray inside without realizing. Yet once inside, nothing can save you from being crushed by the hole's gravity sooner or later.

Live long and prosper

Clearly, an astronaut in that situation might prefer it to be later. For a supermassive black hole such as that thought to exist at the Galactic Centre, the survival time could be hours. To stretch it out for as long as possible, the astronaut might be tempted to turn on rocket thrusters and try to head outwards, away from the hole's fatal 'singularity' at the centre.

But best not to, according to some sources. An article on black holes on the cosmology website of the University of California, Berkeley, for example, says "the harder you fire your rockets, the sooner you hit the singularity. It's best just to sit back and enjoy the ride."

Lewis and Kwan say this is mistaken. They point out that the analysis is usually done by thinking about a person who falls into the black hole starting from a state of rest at the event horizon. In that case, it's true that accelerating away from the singularity by using the rocket thrusters will only speed your demise. The longest survival time possible in that instance is free fall. Because all paths lead inevitably to the singularity, trying to travel faster - in any direction - only takes you there quicker.

Long and winding road

But in general a person falling past the horizon won't have zero velocity to begin with. Then the situation is different — in fact it's worse. So firing the rocket for a short time can push the astronaut back on to the best-case scenario: the trajectory followed by free fall from rest.

"There is one longest road - the freefall road starting from rest - as well as many shorter roads," Lewis explains. "If you cross the event horizon on one of the shorter roads, you can fire your rocket to move you on to the longest road."

But this has to be done judiciously. "If you overdo it, you will overshoot the longest road and end up on a shorter road on the other side," says Lewis. So you only want to burn your rocket for a certain amount of time, and then turn it off. "Once you know how fast you have passed through the event horizon, it is reasonably straightforward to calculate how long you need to burn your rocket to get on to the best path," he says. "The more powerful the rocket, the quicker you get on to this path." Starship captains, take note.

There's nothing particularly surprising in this analysis, but black-hole experts say that debunking this common misconception could have an educational value. "It's a misconception I had when I first did relativity many years ago, and one which I have heard in discussions with others," says Lewis. "It has generated a substantial discussion on the Wikipedia entry about black holes."

He adds that "even Einstein had a very hard time attempting to fathom just what is going on as things fall into a black hole."




References
Lewis G. F. & Kwan J.et al. Publ. Astr. Soc. Australia, in press (2007).

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[Chris Hillman]

Relevant citations

From time to time, people ask if I've ever spotted any errors in MTW, which was published way back in 1973 and remains a standard resource for students of gtr, which good reason. Compare Exercise 31.4 in MTW with http://www.arxiv.org/abs/0705.1029, the eprint which the Nature article is referring to, which purports to correct an overly ambitious statement in the exercise. (The point made by Lewis and Kwan is that the argument suggested by MTW only applies assuming certain boundary conditions on the motion of the infalling observers.)

By the way, the observers who are momentarily (!) at rest at the event horizon are the Frolov observers whose motion I compared with the Lemaitre, slowfall, and static (external!) observers https://www.physicsforums.com/showthread.php?t=146912&page=5

"It has generated a substantial discussion on the Wikipedia entry about black holes."

That does ring a faint bell. Presumably, this "substantial discussion" would be somewhere in one of the archives of http://en.wikipedia.org/wiki/Talk:Black_hole but I can't find it. Anyone know what discussion Lewis was referring to?

 

[MeJennifer]

Maximizing proper time

A free falling (Lemaître) observer maximizes his proper time between infinity and the singularity.

As soon as an observer momentarily diverges from free falling he is going to accumulate less proper time compared to an observer who did not accelerate. In the case of an observer who is accelerating away from the black hole when he crosses the event horizon, the best that he can do is to make the accelerated time interval as short as possible by accelerating in the opposite direction until his prior acceleration is neutralized. But the "lost time" between the accelerations is unrecoverable.

With respect to these situations it is interesting to take a look at the Gullstrand-Painlevé coordinate chart of the Schwarzschild metric. It describes a flat background (not to be confused with the metric) and uses a so-called shift vector (as in the ADM formalism) representing the gravitational pull. With this chart it is much easier to separate the Lorentzian effects from the gravitational effects, and an extra plus is that the shift vector behaves in a Galilean as opposed to a Lorentzian fashion. Another advantage is that the chart maps both the external and internal regions of the black hole.

In Minkowski spacetime we can see the same kind of effects on proper time as in the case of the black hole with regards to acceleration, for instance in the twin thought experiment. Two twins A and B are free falling in spacetime, A accelerates away from B. From that moment on, A is going to accumulate less proper time than B. Close to his point of return A accelerates towards B until they again are at rest with each other. From that moment on, A and B are going to accumulate the same amount of proper time.

But as soon as A returns by accelerating towards B, A again is going to accumulate less proper time than B. And finally, when A is very close to B, A accelerates away from B until A and B are again at rest with each other. From that moment on, A and B are going to accumulate exactly the same amount of proper time. But A´s "lost time" between the accelerations is unrecoverable.

It is possible that in some curved spacetimes an observer can actually accumulate more proper time between two events by performing at least two carefully planned accelerations. This is the case if he is going to travel along a geodesic that is a local, but not a global maximum. If he manages to travel to, and then free fall along the global maximum geodesic then he could, in principle, record more proper time than if he would have ridden along the shorter geodesic. But this is definately not possible in case of a spacetime with a Minkowski or Schwarzschild metric.

 

[George Jones]

From time to time, people ask if I've ever spotted any errors in MTW, which was published way back in 1973 and remains a standard resource for students of gtr, which good reason. Compare Exercise 31.4 in MTW with http://www.arxiv.org/abs/0705.1029, the eprint which the Nature article is referring to, which purports to correct an overly ambitious statement in the exercise. (The point made by Lewis and Kwan is that the argument suggested by MTW only applies assuming certain boundary conditions on the motion of the infalling observers.)

Exercise 31.4 in MTW, including the boundary condition given in its hint, is completely consistent with the eprint.

I don't see any overly ambtitious statement in 31.4.

 

[Chris Hillman]

I don't see any overly ambtitious statement in 31.4.

From time to time, people ask me if I ever make mistakes. I reply: see post #2.

Alas, post #3 contains serious misstatements I can't keep correcting these, so good luck to lurkers.

 

[Cusp]

Some Wikipedia discussions on this issue

Black Holes

May I ask why you reverted my edit? The information I posted was perfectly correct. It is a well known fact of General Relativity that geodesics are paths of maximal proper time. --Jpowell 09:30, 1 March 2006 (UTC)

On my talk page you said:

"Let's say you and I are on two different rocket ships approaching a galactic black hole. After crossing the event horizon you accelerate away from the black hole, while I accelerate toward the black hole. Obviously my rocket ship is going to disintegrate first.

What you wrote may be correct under certain technical sense, but read contradictory to the general audience. -MegaHasher 17:45, 1 March 2006 (UTC)"

That is not what I said. I was saying that once you fall into a black hole it is better to turn your engine off and simply fall into the singularity if you want to survive for longer. Accelerating in any direction will hasten your demise. It is not wise to talk about what is obvious in a region of very highly curved spacetime, a lot of very weird effects come into play.

"General Relativity tells us that once past the event horizon an object will always move closer to the singularity. A consequence of this is that a pilot in a powerful rocket ship that had just crossed the event horizon who tried to accelerate away from the singularity would actually fall in faster. As the ship tried to move faster time dilation would mean the inevitable fall into the singularity would simply occur faster, or as an alternative way of viewing the situation, length contraction would bring the singularity closer to the ship. This is related to the fact that geodesics (or unaccelerated trajectories) are also paths of maximal proper time."

I would like to reinstate this, but don't want to get into an edit war with you, if you could suggest a rewording or a part that needs clarification that would be appreciated.

[edit] Black Hole Discussion mk2

"IMHO, only the first sentence of the paragraph is true; the last sentence may be true, but since the object is accelerating, may not be applicable. A possible revision may be "General Relativity tells us that once past the event horizon an object will always move closer to the singularity. A consequence of this is that a pilot in a powerful rocket ship that had just crossed the event horizon cannot avoid its eventual destruction by firing its rocket."

I'm currently studying General Relativity at university and am confident that all parts of the statement are true. It is the acceleration that causes the worldline of the pilot to diverge from a geodesic (geodesics are unaccelerated trajectories), hence a deviation from longest possible proper time.

Simply stating the point that you cannot avoid destruction is pointless as this is a very basic consequence of passing the event horizon. It is more physically interesting that acceleration speeds up your doom, think of the analogy of struggling against quicksand, you will simply be pulled in faster.

Anyway, statements like that already appear multiple times in the black hole topic. I have started an effort to trim the article toward 32k in size; if you can help me in that, it would be great. For example, look for similar statements that were raised multiple times, long winded technical mumbo jumbo, or multiple explanations for a single term. -MegaHasher 20:48, 1 March 2006 (UTC)

Sure I'll glady help. --Jpowell 21:02, 1 March 2006 (UTC)

http://en.wikipedia.org/wiki/User_talk:MegaHasher

 

[George Jones]

That is not what I said. I was saying that once you fall into a black hole it is better to turn your engine off and simply fall into the singularity if you want to survive for longer. Accelerating in any direction will hasten your demise.

This isn't correct. In fact, this is the mythconception (paper's term) debunked by the paper for which Chris has given a link in post #2.

For an astronaut that falls through the event horizon with a non-zero velocity relative to an observer "at rest" at (i.e., infinitesimally close to) the event horizon, the best course of action is to fire his rocket inside the event horizon.

 

[Dr.Calpol3]

You have to remember that time near to black holes slows down, so you would move a lot slower than you normally would.

 

[Dr.Calpol3]

Anyway, how do people know that they will die? They could be transported to another dimension, or even an alternative reality.

 

[George Jones]

You have to remember that time near to black holes slows down, so you would move a lot slower than you normally would.

This is what an observer outside sees, which isn't relevant here. This thread is about time as measured on the watch of someone who falls into the black hole.

Anyway, how do people know that they will die?

A person falls feet-first into a black experiences a tidal force that pulls his body apart. In terms of Newtonian gravity, the person's feet are closer to the object than his head, thus the feet experience more gravitational pull than the head. Ouch!

As an analogy, suppose that you are suspended with hands over your head by a (strong) rope tied about your wrists. A dangling rope is tied about your ankles, and more and more weight is added to this bottom rope. Eventually your body pulls apart.

 

[cristo]

Black Holes

May I ask why you reverted my edit? The information I posted was perfectly correct. It is a well known fact of General Relativity that geodesics are paths of maximal proper time. --Jpowell 09:30, 1 March 2006 (UTC) ...

Cusp, I don't really understand the point of your post. It appears that you have come here to ask why someone has changed what you have written on wikipedia, which seems a strange thing to do. Especially, since you are clearly talking directly to someone, but have not said to whom you are talking! At a guess, I'd say it was aimed at Chris, but only as I know he used to write on wikipedia.

Anyway, I suggest you take such a discussion back to wikipedia, as this is not the correct place for it.

 

[pervect]

I think what happened is that the discussion on the wikipedia was mentioned earlier in this thread, so it was simply quoted. A little bit of editing would have made it clearer that the discussion wasn't being reopened, just that the old discussion was being quoted because a question was asked about it specifically.

 

[pervect]

This isn't correct. In fact, this is the mythconception (paper's term) debunked by the paper for which Chris has given a link in post #2.

Of course, "Mythconceptions" was also a book, one of the sequels to "Another fine myth", but I suppose that's not really relevant to the thread. I'm afraid I couldn't resist the pun, though.

 

[Cusp]

My bad

As pointed out, I should have quoted the text from wikipedia - it's not mine, but is related to the discussion which prompted the paper. I did put the weblink at the end.

Also take a look at Sean Carroll's (excellent) no nonsense intro to GR at

http://preposterousuniverse.com/grnotes/grtinypdf.pdf

on pages 18-19 he talks about faling through the event horizon and makes the statement that

"What's worse, what we noted above that a geodesic (unaccelerated motion) maximized proper time - this means the more you struggle, the sooner you get there"

As pointed out in the paper, this is strictly true for the person starting from rest at the event horizon (Frolov observers) and for anyone else, a burst of acceleration can extend your lifetime.

I'm sure Sean knows this, but the wording in the text can lead to confusion.

 

[Chris Hillman]

Cusp: I agree with everything you said (including the self-critique!), but I am not sure you found the discussion referred to in the Nature item, since these two posters seem to be arguing over something different from the (oversimplified) remark in Carroll.

 

[Cusp]

"General Relativity tells us that once past the event horizon an object will always move closer to the singularity. A consequence of this is that a pilot in a powerful rocket ship that had just crossed the event horizon who tried to accelerate away from the singularity would actually fall in faster. As the ship tried to move faster time dilation would mean the inevitable fall into the singularity would simply occur faster, or as an alternative way of viewing the situation, length contraction would bring the singularity closer to the ship. This is related to the fact that geodesics (or unaccelerated trajectories) are also paths of maximal proper time."

I think this text, taken from the wikipedia discussion above, *is* essentially the essentially the "no struggle" view point. I guess it's just part of the discussion.

 

[Chris Hillman]

OK, it doesn't help that neither party wikisigned their comments, but it seems that MegaHasher removed a chunk of text from http://en.wikipedia.org/wiki/Black_hole, including this statement:

A consequence of this is that a pilot in a powerful rocket ship that had just crossed the event horizon who tried to accelerate away from the singularity would actually fall in faster.

J. Powell then queried why MegaHasher had removed this material (of which the above statement is only a part), and stated at User talk:MegaHasher that he wished to reinsert it. In his reply, MegaHasher did not mention the statement I quoted. IOW, as I stated, their discussion seems to have concerned something else.

Recall that the Nature on-line article stated:

"It's a misconception I had when I first did relativity many years ago, and one which I have heard in discussions with others," says Lewis. "It has generated a substantial discussion on the Wikipedia entry about black holes."

My question is: does anyone know where is this "substantial discussion"?

A quick check of archived talk pages at http://en.wikipedia.org/wiki/Talk:Black_hole failed to turn up anything which appears to answer to this description.

 

[Cusp]

I would guess what it meant is that disussions of this sort have taken place on the internet (including wikipedia), rather than only wikipedia, and that it was written on the run without expecting to be directly quoted.

Cusp == Geraint

 

[Chris Hillman]

I get it now

I would guess what it meant is that disussions of this sort have taken place on the internet (including wikipedia), rather than only wikipedia, and that it was written on the run without expecting to be directly quoted.

Cusp == Geraint

OK, check, thanks for clearing up both my spoken and unspoken questions! And welcome to PF, hope you stick around.

 

[rbj]

Anyway, how do people know that they will die? They could be transported to another dimension, or even an alternative reality.

i think the tidal forces ripping my head offa my torso as it is being ripped offa my bottom is going to at least hurt a little bit.

 

[blumfeld0]

i have a stupid question and i know i should know this but
when it is said that "geodesics (or unaccelerated trajectories) are also paths of maximal proper time" and hence the more you struggle, the sooner you get there (assuming you start from rest at the horizon).

1. what does maximal proper time mean- this is the time of the wristwatch of an observer falling into the hole. what does it mean to be maximal?
to me if you have maximal time, it means you have a lot of time but in this context it means just the opposite. is that correct?
i don't understand the terminolgy.

2. how does it follow that if you maximize proper time you get to r= 0 faster?

maybe 1 and 2 are the same questions.

thank you very much

 

[Chris Hillman]

1. In Riemannian geometry, if we consider a geodesic arc connecting two points, then make a "small" perturbation of this path while keeping the endpoints fixed, then the distance along the new path will have increased. In the same way, in Lorentzian geometry, if we consider a timelike geodesic path connecting two events in a spacetime model (a Lorentzian manifold), then make a "small" perturbation of this path while keeping the endpoints fixed, then the proper time along the new path (analogy of "distance") will have decreased. These two statements are analogous. The second one was (inadequately) summarized by a Wikipedian as "timelike geodesic arcs maximize proper time"--- which isn't even true as stated, since it overlooks a local versus global issue. To see why, consider the analogous Riemannian slogan: "geodesic arcs minimize distance". Now consider two non-antipodal points on a round sphere. They are connected by two great circle arcs, one shorter than the other--- so the long one certainly doesn't minimize the distance! The correct if somewhat imprecise statement is that the distance is minimized locally, i.e. for "sufficiently small" perturbations.

2. It doesn't follow; that was the point of the eprint we've all been discussing, http://www.arxiv.org/abs/0705.1029

 

[blumfeld0]

Thank you. Yes i figured it didn't follow but reading Sean's notes, it is made to sound like it follows.
"that a geodesic (unaccelerated motion) maximized the proper time – this means that the more you struggle, the sooner you will get there"

so in the end of section 2, Lewis and Kwan say
As will be shown later, this maximum time
applies to a faller who drops from rest at the event
horizon and anyone who starts falling from above the
event horizon and free falls into the hole will experience
less proper time on the journey from the event horizon
to the singularity.

less proper time, in this context, actually means they will survive longer?

(assuming they survive spahetttification, accretion disk radiation etc.)

 

[Chris Hillman]

Well, uhm, I should let Cusp answer any questions about this eprint! But we all seem to agree that Carroll's remark doesn't tell the whole story.

 

[blumfeld0]

Moreover, is there another way to explain maximal proper time using the schwarzschild metric or any other metric or plain math?
i guess i don't really get how you make a small perturbation in the path AND keep BOTH the endpoints fixed, hence increase the distance? any example please.

thanks

 

[Chris Hillman]

Re (1), suggest you consider paths in Minkowski spacetime first. Indeed, consider paths in the two-dimensional version with line element
ds^2 = dt^2 - dx^2, \; -\infty < t, \, x < \infty
Consider the timelike geodesic arc x = x_0, \; t_1 < t < t_2. Now sketch an arc with the same endpoints (you can even make it have the same tangents at the endpoints, if you like). What does the Pythagorean theorem tell you?

 

[blumfeld0]

hi.
ok after thinking about it i understand your qualitative explanation a lot better.
i think the pythagorean theorem tells me that if x = x0, then ds^2 will go down
and because propertime^2 = ds^2 then proper time will have decreased.
i could just be making that up. :-)
thank you

 

[Cusp]

so in the end of section 2, Lewis and Kwan say
As will be shown later, this maximum time
applies to a faller who drops from rest at the event
horizon and anyone who starts falling from above the
event horizon and free falls into the hole will experience
less proper time on the journey from the event horizon
to the singularity.

less proper time, in this context, actually means they will survive longer?

No - less proper time for the person who does not start from rest at the event horizon means that the time they tick off on their watch on their journey from the event horizon to the singularity will be less than that experienced by the person who does start at rest.

 

[MeJennifer]

Perhaps someone can make it crystal clear to me about some of the claims made in this topic and the referenced paper.

Consider observer A and B, far away from the black hole, in free fall, radially attracted by the black hole. After they pass the event horizon, observer B accelerates while A stays on the geodesic. It is possible for B to record a larger amount of proper time than A by carefully accelerating. So in other words while A remains on a geodesic, B accelerates but still manages to record a larger amount of proper time.

Is that what is claimed here or do I seriously misunderstand?

 

[Cusp]

Is that what is claimed here or do I seriously misunderstand?

Yes, that's what is claimed

 

[George Jones]

Cusp == Geraint

Sorry, I should have read post #6 more closely. I only scanned it, and I got confused by alternating passages in quotes and not in quotes. This shines a very ironic light on my comments in post #7 .

I hope your interesting paper becomes well known. I read it when v2 came out on the arxiv, and I, too, thought that there was a problem with one of the problems in Misner, Thorne, and Wheeler. After looking more closely at the problem in MTW, I decided there was no problem.

 

[MeJennifer]

Yes, that's what is claimed

So then it is presumed that different observers can reach the singularity at different coordinate times?

 

[Cusp]

So then it is presumed that different observers can reach the singularity at different coordinate times?

Yes - the figures in the paper show this, although it's not presumed, it's calculated.

 

[MeJennifer]

Yes - the figures in the paper show this, although it's not presumed, it's calculated.

Yes, I read the paper, observers reaching the singularity at different coordinate times.

So perhaps, in general, for the good understanding, which charts of the Schwartzschild metric show an observer reach the singularity in finite T?

 

[George Jones]

So then it is presumed that different observers can reach the singularity at different coordinate times?

Yes, different observers can reach the singularity at values of the coordinate t. See Figure 4. in the arXiv paper (link given in post #2).

I have bit of a beef with the terminology used by both you and the paper. As with Schwarzschild coordinates, t is a space coordinate inside the event horizon, since

g \left( \frac{ \partial}{\partial t} , \frac{ \partial}{\partial t} \right) = - \left( 1 - \frac{2m}{r} \right),

which is positive inside the horizon.

It is interesting to note that, unlike Schwarzschild coordinates, r is spacelike outside the event horizon, on the event horizon, and inside the event horizon, since

g \left( \frac{ \partial}{\partial r} , \frac{ \partial}{\partial r} \right) = \left( 1 + \frac{2m}{r} \right)

is always positive.

This is an example of what Penrose calls Woodhouse's Second (or is it first?) Fundamental Confusion of Calculus.

Let (t,r) be Schwarzschild coodinates. Define Eddington-Finkelstein coordinates by

T(t,r) = t + 2m ln \left| \frac{r}{2m} - 1 \right|
R(t,r) = r.

Even though R = r, \partial / \partial R is not the same as \partial / \partial r because lines of constant (T,\theta,\phi) are not the same as lines of constant (t,\theta,\phi).

 

[MeJennifer]

As with Schwarzschild coordinates, t is a space coordinate inside the event horizon, since

g \left( \frac{ \partial}{\partial t} , \frac{ \partial}{\partial t} \right) = - \left( 1 - \frac{2m}{r} \right),

which is positive inside the horizon.

A short OT reply here:
Looking at it from a complex Euclidean CE⁴ space with both a + - - - and a - + + + section, I am not at all convinced that we can equate a sign change in t with space.

 

[pervect]

A short OT reply here:
Looking at it from a complex Euclidean CE⁴ space with both a + - - - and a - + + + section, I am not at all convinced that we can equate a sign change in t with space.

I don't understand your difficulty. Which of the following are you having difficulty with?

1)A vector u can be either space-like or time-like, depending on the sign of
g_{ij} u^i u^u.

2) Given a coordinate system, (t, etc), a vector pointing in the direction of increasing t is \frac{\partial}{\partial t} . Similarly, a vector pointing in the direction of x is \frac{\partial}{\partial x}

3) Given 1 & 2, the only remaining issue is that George has made some sort of math error, but I don't think that's your claim.

 

[pervect]

This is an example of what Penrose calls Woodhouse's Second (or is it first?) Fundamental Confusion of Calculus.

It's "second fundamental confusion of calculus" in my edition of "The Road to Reality" - pg 190, fig 10.7.

I ran into some similar issues not very long ago.

 

[MeJennifer]

I don't understand your difficulty.

It is not related to some difficulty but a matter of interpretation of the reality condition.
I remember we had a discussion about this before where everybody claimed not to understand my "difficulty" so there does not seem to be a point to repeat this. It appears I can only have a serious discussion about this with myself.

1)A vector u can be either space-like or time-like, depending on the sign of
g_{ij} u^i u^u.

That´s like saying a number can be either positive or negative. There happen to be imaginary numbers as well.

Anyway, we really should get back on topic. So let´s just say that I seriously misunderstand things here and move on.

So the argument, if I understand the paper correctly, is that since free falling observers can arrive at the singularity at different coordinate times it must be possible for some of those observers to accelerate in such a way that the coordinate time of arrival changes, with a result that the total proper time is increased with respect to the case in which that particular observer would not have accelerated. Is that right?

 

[pervect]

Travelling a geodesic path means that the proper time between any two fixed points is maximized.

To apply the analysis, the points must be fixed. The problem is that "hitting the singularity" isn't such a fixed point - it can be and usually is a different event with different starting velociteis.

However, everyone seems to agree that the limiting case is an observer who falls in such a way that (dr/dtau) -> 0 at the event horizon, r=2M. This isn't just a local maximum, it's a global maximum. (At least that's what I gather, I haven't looked at this terribly closely). It's really is not too surprising that such a global maximum is also a local maximum (i.e. a geodesic).

Strictly speaking, however, this limiting-case observer isn't actually crossing the event horizon.

So for someone who actually crosses the event horizon, they have to accelerate, as quickly as possible, to achieve the same energy-at-infinity (zero) as the observer above, one who starts out with dr/dtau = 0 at 2M. After they have done this, their best bet is to "coast" the rest of the way in.

 

[MeJennifer]

The problem is that "hitting the singularity" isn't such a fixed point - it can be and usually is a different event with different starting velocities.

Oh?

Frankly, I think that this does not make a lot of sense (and it looks like I am the only one again), perhaps you could provide me with some specific references to what you say, so I can study this.

Hopefully the paper we are discussing is not debunking a myth by affirming another one, namely that "the" singularity is some local entity.

 

[pervect]

The best references (IMO) have already been give - MTW, $25.5 and pgs 824-832 and exercise 31.4 on pg 836 and the preprint.

For the preprint http://www.arxiv.org/abs/0705.1029 see figure 4

For the red line, the faller fires their rocket all the way to the singularity, while the dark blue, light blue and green turn off their rocket when e = 0.3, e = −0.3 and e = 0 respectively. An examination of the proper time in the left-hand panel reveals that it the path that settles on e = 0 that possesses the longest proper time.

also

In dropping from rest at the event horizon, the firing of a
rocket does not extend the time left, it only diminishes it

The only question is what is meant by "at rest at the event horizon" - I assume that this means that dr/dtau is zero at the horizon. The observer "at rest" can't be one with a constant r coordinate - such observers, usually called stationary observers, are timelike only outside the event horizon. The null geodesic of an outgoing light ray is stationary at the event horizon, but the path of an observer must be timelike, not lightlike or spacelike.

In any given coordinate system, two events are the same if and only if ALL coordinates for both events are the same. If any coordinate is different, the events are different.

As far as computing geodesics go, I'd strongly recommend reading a textbook or the preprint section on conserved quantities, aka "Killing vectors". For geodesic radial motion, the conserved energy parameter (which I call energy-at-infinity as that's what MTW calls it) gives dr/dtau as a function of r. (For general motion, you also need the angular momentum, but for radial motion, the angular momentum is zero and you only need the energy parameter).

This doesn't work for non-geodesics, but it should be enough to illustrate how energy influences geodesic path. Note that you'll have to pick the correct formula for the conserved quantity depending on what coordinates you use, but all coordinate systems will have some way of expressing the numerical value of this conserved quantity.

Exercise: pick two radial geodesic paths, which terminate on the same event. Trace them backwards in time. Do they ever cross?

 

[MeJennifer]

I was simply inquiring about this notion of observers hitting "the" singularity at different times.

If "the" singularity is supposedly something that is a collection of separate events then there is no point in talking about hitting it. Also if we time reverse the situation (something that is perfectly valid in relativity) and look at it as a white hole, I am led to believe that geodesics start from this singularity at different times? What is supposedly the meaning of time and space at the singularity, to me is seems it is uttter nonsense to claim that different observers hit "the" singularity at different times.

Everybody else seems to claim the contrary, hence my request for references.

 

[pervect]

I was simply inquiring about this notion of observers hitting "the" singularity at different times.

If "the" singularity is supposedly something that is a collection of separate events then there is no point in talking about hitting it.

Huh?

Look at a Penrose diagram, or a Kruskal-Szerkes diagram. I think you have MTW? If so look at figure 31.4.

If not, try

http://casa.colorado.edu/~ajsh/schwp.html#penrose
http://casa.colorado.edu/~ajsh/schwp.html#kruskal

Is the singularity a point on these diagrams (for the lurkers, NO!), or is it a line (for the lurkers, YES!).

For extra credit. Consider two nearby points on set of points (the line) that forms the singularity. These two points occur at the same Schwarzschild coordinate r, and at different Schwarzschild coordinate t.

Is the Lorentz interval between these points timelike, or spacelike?

Also if we time reverse the situation (something that is perfectly valid in relativity) and look at it as a white hole, I am led to believe that geodesics start from this singularity at different times? What is supposedly the meaning of time and space at the singularity, to me is seems it is uttter nonsense to claim that different observers hit "the" singularity at different times.

Everybody else seems to claim the contrary, hence my request for references.

I don't understand where you got your ideas, much less why you think "everyone else claims the contrary". If you think there are contrary claims, please cite them.

 

[MeJennifer]

Yes, I have MTW and most other published GR books.

Is the singularity a point on these diagrams (for the lurkers, NO!), or is it a line (for the lurkers, YES!).

I see, so are you saying the singularity is a line?

I don't understand where you got your ideas, much less why you think "everyone else claims the contrary". If you think there are contrary claims, please cite them.

You misunderstand me, I was asking for references that help me to show that I am supposedly wrong or misguided in thinking that the notion of hitting the singularity at different times does not make any physical sense.
So far none has been given.

For extra credit. Consider two nearby points on set of points (the line) that forms the singularity. These two points occur at the same Schwarzschild coordinate r, and at different Schwarzschild coordinate t.

Ok for extra credit my answer:
The notion above that the singularity is at charted points on the manifold is false. The singularity is not on the manifold.

 

[pervect]

The notion that geodesics hit the singularity at different "times" is indeed suspect. However, if you will go back over the thread, you should notice that I said "different events", and not "different times". The reason I phrased it this way is that while the topology of the set of events that form the singularity is that of a line, the separation between nearby points of that line is not timelike, but spacelike.

I was hoping to lead you to this conclusion on your own, but you are resisting. A large part of the problem seems to be your rejection of the classification of 4-vectors as space-like or time-like. I really don't understand what your problem is here, but I think I've done about all I can to explain, the rest is up to you.

I think I've given enough references - what in my opinion needs to happen next is that you have to start reading them. And if you do have a disagreement, it's time (past time) for you to start citing specific references.

I'm going to lock this thread to give you some more time to read, before you shoot your mouth off again. I intend to unlock in again in a few days when things have hopefully calmed down (and if they don't calm down, I'll relock it).