What is the relationship between black holes and time dilation?

[johatfie]

Ok, I’m a layman with an interest in physics. I don’t know any of the math beyond high school physics and first semester college calculus (with trig) a long time ago. I’ve got pretty good handle on special relativity (for a layman) and an introduction to the concepts of general relativity.

Here’s my premise. From the frame of reference of a distant observer not experiencing any time dilation from a black hole, I have read/seen it explained several times that such an observer will never see a space ship, or whatever, cross the event horizon of a black hole since such a ship will be experiencing ever increasing time dilation from the effects of both special and general relativity as it accelerates toward the event horizon. From the frame of reference of said external observer, the ship will appear to freeze at or near the event horizon. Indeed, I have read/seen it said that we have never observed anything cross such an event horizon. Matter may be swirling around at speeds approaching C, but we have not observed any of it cross the event horizon.

1) How can a black hole be said to consume surrounding matter from the frame of reference of a distant observer? Wouldn’t such an observer see it continually accumulate near the event horizon, yet never go in? I guess what I’m getting at is how much time dilation would such a ship experience, relative to a distant observer, prior to crossing the event horizon? The impression I have received is that it would be an infinite amount, i.e. the ship would never cross the event horizon from the distant observer’s frame of reference.
2) Has any theory been put forth for the huge jets of matter sometimes seen being ejected from the center, perpendicular to the accretion disk at speeds approaching C?
3) Is the formation of an accretion disk related to “frame dragging”? Presumably from a rotating singularity.
4) Is "obital procession" related to or caused by "frame dragging"?thanks

 

[pervect]

For your first point, see any of the zillion threads on the same topic. While the observer 'at infinity' never sees the black hole form, the observer falling through the event horizon reaches the singularity in a finite proper time.

Once the analogy between the "black hole" event horizon and the Rindler horizon of an accelerated observer became known, people for the most part aware of the analogy abanonded the "frozen star" idea. Sea any of the aforementioned threads for more info on the Rindler horizon, associated with an accelerated observer, which provides a useful analogy.

For your second question, I'm afraid I don't know very much about the jets. I did find http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/990923a.html, but I'm not sure how helpful it will be. Maybe someone else will know more.

For your last question, orbital precession occurs even around a non-rotating black hole (Schwarzschild black hole). While frame dragging could cause addional precession, frame dragging is not needed to cause precession.

 

[Jheriko]

I think that the jets form because the material accumulates around the black hole faster than it can fall in, the resulting pressure forces infalling material to flow around the edges existing disk or to displace plasma already in the disk. The magnetic fields formed by the plasma of the accretion disc then confines the displaced plasma to near the rotational axis of the black hole. The energy gained through interaction with the disk and the magnetic fields will allow some of the matter to attain escape velocity, which creates the jets.

EDIT: Just read pervect's link, and it seems that although I am partly correct, astrophysicists are still looking for a good model that shows how the magnetic fields confine the jets and for how the material leaves the accretion disc. Interesting read.

 

[Chris Hillman]

Ok, I’m a layman with an interest in physics. I don’t know any of the math beyond high school physics and first semester college calculus (with trig) a long time ago. I’ve got pretty good handle on special relativity (for a layman) and an introduction to the concepts of general relativity.

Thanks, this kind of information is always helpful to those trying to frame a reply!

From the frame of reference of a distant observer not experiencing any time dilation from a black hole, I have read/seen it explained several times that such an observer will never see a space ship, or whatever, cross the event horizon of a black hole

Correct, except I'd replace "distant observer not experiencing any time dilation from a black hole" with "static observer far from an isolated black hole".

since such a ship will be experiencing ever increasing time dilation from the effects of both special and general relativity as it accelerates toward the event horizon.

No, in fact this "explanation" runs completely counter to the spirit as well as the letter of the law as laid down by gtr, if I might so put it.

The so-called "gravitational time dilation" is a straightforward curvature effect. In any curved manifold, initially parallel geodesics will in general converge (positive curvature) or diverge (negative curvature) as you run along one of them. Near the exterior of the event horizon of a black hole (in the simplest case, this situation is modeled by the Schwarzschild vacuum solution of the Einstein field equation of gtr, or EFE for short), two radially outgoing null geodesics corresponding to signals sent from an infalling observer will diverge. That means that when the signals are received by our distant static observer, the time between the two, as measured by an ideal clock carried by this static observer, will be larger than the time between the emission of the two signals, as measured by an ideal clock carried the infalling observer.

These two "ideal clocks" are assumed to be absolutely identical and in particular, by definition they always "run at the same rate" under any circumstances (a real clock, even an atomic clock, will be affected by acceleration and so on); the "relativity" in gtr can be taken to refer to the fact that when we compare identical ideal clocks located at different "places", we must expect discrepancies, depending upon the details of the ambient gravitational field, the relative motion of the observers, and the method by which the comparison is made (typically, lightlike signals, but these can in general take more than one path and there are other complications we probably don't want to get into here).

From the frame of reference of said external observer, the ship will appear to freeze at or near the event horizon.

Avoid "frame of reference" or "Lorentz frame" in gtr, since in str this term tacitly invokes Cartesian coordinates, which only exist in flat spacetime. The closest analogous concept valid in gtr is a frame field, a quartet of orthonormal vector fields (one timelike and three spacelike); a frame at one event is sometimes called a "local Lorentz frame" (a better term would be "infinitesimal Lorentz frame").

Also, the problem of describing optical effects in gtr is interesting and valid, but not the same as the problem of describing clock effects, so be careful here.

If you meant to ask what our distant observer, A, would literally -see- if our infalling observer, B, were say pointing a laser beam steadily in A's direction as he falls toward the hole, then A would see the spot of light redden and then very rapidly wink out as B nears the horizon. For a stellar mass black hole, in fact, this would happen in about 10^-5 seconds!

Indeed, I have read/seen it said that we have never observed anything cross such an event horizon.

Correct, and according to gtr (and similar theories which admit black holes), by definition, an exterior observer can never receive any signal from an observer behind an event horizon, although the inside observer can still receive signals from the outside (at least for a short time after falling past the horizon).

Matter may be swirling around at speeds approaching C, but we have not observed any of it cross the event horizon.

Right, and this is leads us to one of the most interesting observations of astrophysical black holes: astronomers have watched blobs of hot matter falling into supermassive black hole "candidates" (to be perfectly pedantic one can append that qualififier), and simply vanish. If the object in question had a surface, we'd expect to see a flash of light when the matter hits the surface and vaporizes, but this never seems to occur. This is of course just one of many lines of evidence which convinced mainstream astronomers, after decades of opposition, that black holes do exist in Nature.

Regarding "speeds approaching C", note that even in flat spacetime, there are in fact multiple distinct but operationally significant notions of distance valid in large regions, all of which agree in very small neighborhoods (in gtr, the latter fact can be understood as a consequence of the "strong equivalence principle").

1) How can a black hole be said to consume surrounding matter from the frame of reference of a distant observer? Wouldn’t such an observer see it continually accumulate near the event horizon, yet never go in?

That's basically the "frozen star" notion, which is based upon various misconceptions as indicated above. I was just about to say "the website http://casa.colorado.edu/~ajsh/schw.shtml (Andew Hamilton, JILA, University of Colorado) might help" when I noticed that you linked to this in the very next sentence! OK, I still think his pictures should help--- see the figures depicting the world line of an observer falling into the hole in the Eddington coordinate chart, Painleve chart, or Kruskal-Szekeres chart.

2) Has any theory been put forth for the huge jets of matter sometimes seen being ejected from the center, perpendicular to the accretion disk at speeds approaching C?

Yes, in fact more than one, in fact the exact mechanism which produces these jets remains a problem of great interest in astrophysics. The dominant model for some decades has been based upon the "advection dominated model" for hot ionized matter forming an accretion disk around a rotating hole, which is thought to lead to some material being ejected along the axis of rotation. This issue seems to involve relativistic physics, but to require electromagnetism, not just gravitation.

3) Is the formation of an accretion disk related to “frame dragging”? Presumably from a rotating singularity.

Gravitation but not neccessarily relativistic gravitation can lead to the formation of an accretion disk whenever you have stuff falling toward a massive object. This has much more to do with orbital angular momentum of the infalling material than with frame dragging.

According to gtr, curvature singularities should exist inside the horizon, but this is irrelvant since signals cannot escape from inside the horizon.

4) Is "obital procession" related to or caused by "frame dragging"?

To elaborate a bit on what pervect already told you:

Geodetic precession or de Sitter precession (see any gtr textbook for "the precession of the perihelia of Mercury") does not involve frame dragging; the classic formula provided by Einstein works the same for a rotating or nonrotating isolated massive object (not just a black hole). You can think of this effect as saying that a small object in a bound orbit around a massive object will exhibit quasi-Keplerian motion, but the long axis of the "almost elliptical orbit" will very slowly rotate over time at a steady rate. This effect has been confirmed in solar system observations (for Mercury, Venus, the Earth, and various asteriods) and also in binary star systems in which one or both "stars" are neutron stars or black holes, most notably the Hulse-Taylor binary.

According to gtr, a gyroscope orbiting a rotating object will experience an additional small precession called "Lense-Thirring precession". This effect does involve "gravitomagnetism" and the Standard "Gravity Probe B" satellite experiment has been testing it.

Chris Hillman

 

[johatfie]

Wow. Ok, I clearly know even less about gtr than I thought. And apparently I know nothing about black hole geometry except what I've heard in popular science.

Avoid "frame of reference" or "Lorentz frame" in gtr, since in str this term tacitly invokes Cartesian coordinates, which only exist in flat spacetime. The closest analogous concept valid in gtr is a frame field, a quartet of orthonormal vector fields (one timelike and three spacelike); a frame at one event is sometimes called a "local Lorentz frame" (a better term would be "infinitesimal Lorentz frame").

Ok, I think I'll have to go with "point of view" then. I had assumed Einstein would continue to use the frames of reference from str. I don't think I've ever seen much of an explanation for gtr. I had thought that I had been introduced to the most basic concepts, but now I'm not even sure that's true. I did put a book on my Christmas list that Pervect seems to recommend often, "General Relativity from A to B".

Right, and this is leads us to one of the most interesting observations of astrophysical black holes: astronomers have watched blobs of hot matter falling into supermassive black hole "candidates" (to be perfectly pedantic one can append that qualififier), and simply vanish. If the object in question had a surface, we'd expect to see a flash of light when the matter hits the surface and vaporizes, but this never seems to occur. This is of course just one of many lines of evidence which convinced mainstream astronomers, after decades of opposition, that black holes do exist in Nature.

Ok, so we have seen matter disappear though? That answers my question. I still don't understand how though. I'm kind of stuck on the "frozen star" model.

No, in fact this "explanation" runs completely counter to the spirit as well as the letter of the law as laid down by gtr, if I might so put it.

The so-called "gravitational time dilation" is a straightforward curvature effect. In any curved manifold, initially parallel geodesics will in general converge (positive curvature) or diverge (negative curvature) as you run along one of them. Near the exterior of the event horizon of a black hole (in the simplest case, this situation is modeled by the Schwarzschild vacuum solution of the Einstein field equation of gtr, or EFE for short), two radially outgoing null geodesics corresponding to signals sent from an infalling observer will diverge. That means that when the signals are received by our distant static observer, the time between the two, as measured by an ideal clock carried by this static observer, will be larger than the time between the emission of the two signals, as measured by an ideal clock carried the infalling observer.

Ok, I didn't understand anything between "gravitational time dilation" and "observer will diverge". I think I know what a "Schwarzschild black hole" is though. A stationary, non-rotating black hole that has already collapsed. Basically a black hole with the simplest possible math.

So, are you saying that there is only the appearance of time dilation under gtr? Ok, take two ideal clocks "at infinity". One travels to the black hole, gets near the event horizon, then safely returns to it's starting point next to the other ideal clock. Won't the traveling clock have experienced very large time dilation as compared to the stationary clock, even after we calculate out the effect from str?

I'm not sure how to rephrase my original question and set aside the observational distortions caused by gtr, but here goes. A spaceship travels to a black hole and crosses the event horizon. When it crosses the event horizon won't it be in the far, far, far distant future of the observer "at infinity"? The astronaut aboard the ship will experience time normally from his "point of view" at least until he is killed by the tidal forces, etc. But if he had a "crystal ball" and was able to observe the ideal clock of the observer "at infinity", wouldn't the clock at infinity be millions or billions of years ahead of his own clock?

That's basically the "frozen star" notion, which is based upon various misconceptions as indicated above. I was just about to say "the website http://casa.colorado.edu/~ajsh/schw.shtml (Andew Hamilton, JILA, University of Colorado) might help" when I noticed that you linked to this in the very next sentence! OK, I still think his pictures should help--- see the figures depicting the world line of an observer falling into the hole in the Eddington coordinate chart, Painleve chart, or Kruskal-Szekeres chart.

I think that was Pervect's link. I think the material there will require some study before I understand it.

Gravitation but not neccessarily relativistic gravitation can lead to the formation of an accretion disk whenever you have stuff falling toward a massive object. This has much more to do with orbital angular momentum of the infalling material than with frame dragging.

Chris Hillman

That's what I thought. So, there's no reason something could not go straight into a black hole without swirling around it assuming there's nothing else in the way and it's trajectory is directly toward the center of the black hole. Correct?


Thanks for taking the time to answer my questions. You obviously took quite a bit of time writing a response. I've been looking for some of the other posts that Pervect referred to and have not yet seen an explanation that I understand. I will have to study the material you both linked to, but the math and geometry involved is way beyond anything I have studied. It is fascinating though. I was once a physics major, so I love this stuff, but I just don't have the foundation necessary to understand a lot of it.

 

[Chris Hillman]

I enthusiastically second pervect's recommendation of the popular book by Geroch. In fact, I think it will prove the perfect book for you!

Enjoy!

 

[johatfie]

Yes, I hope it will, but in the mean time, could you help to clear up some of my misconceptions?

 

[johatfie]

anyone? anyone?

 

[MeJennifer]

Ok, so we have seen matter disappear though? That answers my question. I still don't understand how though. I'm kind of stuck on the "frozen star" model.

The matter does not go away, but we simply do not receive any light signals from it anymore. Since all the light emitted from this matter is heading for the singularity (and so is the matter itself of course).

Ok, I didn't understand anything between "gravitational time dilation" and "observer will diverge".

Ok let's look at Chris' quote:

The so-called "gravitational time dilation" is a straightforward curvature effect. In any curved manifold, initially parallel geodesics will in general converge (positive curvature) or diverge (negative curvature) as you run along one of them. Near the exterior of the event horizon of a black hole (in the simplest case, this situation is modeled by the Schwarzschild vacuum solution of the Einstein field equation of gtr, or EFE for short), two radially outgoing null geodesics corresponding to signals sent from an infalling observer will diverge.

What is means in the most simple terms (and therefore not exactly correct) I can think of is that the black hole is pulling the "rug" from under the the legs of the consecutive light pulses. So they appear to us, who are far away from the black hole, to come later and later. The closer to the singularity the fast the pulling so eventually the pulling of the "rug" is so fast that light has no time to make one shred of progress into our direction on the contrary it goes backwards while it still "thinks" it goes forwards.
Or think about moving walkaways, for a black hole you are standing on them and they go faster and faster the closer you get to the singularity. At one point there is no escape, since you obviously cannot walk faster on them than c.
Does that make a bit more sense?

So, are you saying that there is only the appearance of time dilation under gtr? Ok, take two ideal clocks "at infinity". One travels to the black hole, gets near the event horizon, then safely returns to it's starting point next to the other ideal clock. Won't the traveling clock have experienced very large time dilation as compared to the stationary clock, even after we calculate out the effect from str?

Yes of course, who says it is "appearance" it is as real as it gets.

I'm not sure how to rephrase my original question and set aside the observational distortions caused by gtr, but here goes. A spaceship travels to a black hole and crosses the event horizon. When it crosses the event horizon won't it be in the far, far, far distant future of the observer "at infinity"?

Well you really cannot say that since the spaceship has no way of coming back. They are forever disconnected. In other words there is no future event that they can share, so any comparison is useless.

Remember that in SR and GR there is no absolute time, no universal clock. Each mass object has its own sense of elapsed time. And this eapsed time depends on the path it traveled in space-time.

 

[pervect]

I thought we addressed at least some of them? What are you still puzzled by?

 

[dpyikes]

Doesn't time dilation keep singularities from ever forming?

I hope this question belongs in this thread. If someone on the surface of a star which just started collapsing into a black hole is receiving signals sent every second from an external point in "flat space", then the signals the surface receives will be very blue shifted -- As the density of the object increases, won't the time dilation with respect to the external signal source approach infinity before the surface can collapse completely? It seems like a singularity (a point of infinite density) could never form in the sense that the signals from the external observer would continue to strike the non-singular object that is in the process of collapsing throughout all future history.

 

[pervect]

I hope this question belongs in this thread. If someone on the surface of a star which just started collapsing into a black hole is receiving signals sent every second from an external point in "flat space", then the signals the surface receives will be very blue shifted -- As the density of the object increases, won't the time dilation with respect to the external signal source approach infinity before the surface can collapse completely? It seems like a singularity (a point of infinite density) could never form in the sense that the signals from the external observer would continue to strike the non-singular object that is in the process of collapsing throughout all future history.

As the observer continues to fall, he starts to move at faster and faster velocities towards the black hole. He may start out as a "stationary" observer, but if he falls into the BH, he does not remain stationary. This causes a redshift due to his motion - a very large redshift, because his infalling velocity approaches the speed of light (as measured relative to an observer "holding station")

The two effects work in opposite directions. If you imagine someone somehow "hovering" at a constant r coordinate, the blueshift becomes infinite at the event horizon. But this is not an infalling observer, it's a station-holding observer.

Calculating the red/blue shift depends on the trajectory. I'd have to look up the thread where this was discussed, but if you assume an observer free-falling from infinity (i.e. he has zero velocity at infinity) into a black hole, the total shift at the event horizon is a redshift which halves the frequency of light falling in "radially".

The redshift/blueshift also depends on direction in which he looks, the factor of 2 in the above example is for the observer who looks directly "up" at radially infalling light.

Besides the past thread, there's some discussion in http://casa.colorado.edu/~ajsh/singularity.html#redshift.map

(I think there's even more discussion elsewhere on this webpage about the issue).

 

[MeJennifer]

If someone on the surface of a star which just started collapsing into a black hole is receiving signals sent every second from an external point in "flat space", then the signals the surface receives will be very blue shifted

Correct.
The radius of the star will continue to shrink and eventually after it reaches a certain density the event horizon will be outside of what is left of the star. Then everything inside is trapped and cannot escape. The star will continue to collapse to a singularity in finite proper time.

-- As the density of the object increases, won't the time dilation with respect to the external signal source approach infinity before the surface can collapse completely?

From the perspective of an outside observer, the time dilation will be infinite.
But for everything inside the black hole that is of no matter, everything inside will collapse to a singularity in finite proper time.

It seems like a singularity (a point of infinite density) could never form in the sense that the signals from the external observer would continue to strike the non-singular object that is in the process of collapsing throughout all future history.

From the perspective of an outside observer that is true, but from the perspective of an inside observer that fact is of no influence whatsoever.

Note that there is no such thing as absolute time in GR so even if some event would take say a few billion years for one observer it could be ten minutes for another observer.

 

[dpyikes]

Even if there is no absolute time in GR, it seems to make sense to say that since any signal sent from flat space time into any black hole will (and always will) strike a collapsing, but not collapsed object, we can say that the universe contains no singularities. By the proper time of the surface of the collapsing objects, they will receive signals (and collisions with other objects such as black holes) from the outside (flat space-time) for all future history before they can complete their collapse, even if it only takes 10**-5 seconds by their own proper time to complete their collapse -- They "experience" those collisions all at once, just as they begin their collapse. Even by their own proper time, I'd say they never get through that 10**-5 seconds.

 

[pervect]

Even if there is no absolute time in GR, it seems to make sense to say that since any signal sent from flat space time into any black hole will (and always will) strike a collapsing, but not collapsed object,

Where did you get that idea?

Take a look at the Eddington-finklestein diagram for a pressureless dust collapsing into a black hole, for example.

For instance, http://casa.colorado.edu/~ajsh/collapse.html#finkelstein

The infalling light rays (yellow lines at 45 degree angle) will eventually strike, not the collapsing sphere of dust (the white curving line), but the singularity, a totally collapsed object (the vertical cyan line).

Thus a signal sent from flat space-time can and will strike a collapsed object (i.e. the singularity).

 

[dpyikes]

Pervect -- Thanks for the response and the link to the diagrams. I am still not seeing how it can be the case that the white line ever turns Cyan if those yellow lines are drawn as shown -- i.e., if you've got density that is approaching infinity, then the gravitational potential will approach infinity and the time dilation with respect to the source of those yellow lines ought to go to infinity too. I would think that either the yellow lines ought to be drawn never hitting the zero radius point or the white line ought to just go up the vertical axis and never hit the zero radius point.

 

[pervect]

Pervect -- Thanks for the response and the link to the diagrams. I am still not seeing how it can be the case that the white line ever turns Cyan if those yellow lines are drawn as shown -- i.e., if you've got density that is approaching infinity, then the gravitational potential will approach infinity and the time dilation with respect to the source of those yellow lines ought to go to infinity too. I would think that either the yellow lines ought to be drawn never hitting the zero radius point or the white line ought to just go up the vertical axis and never hit the zero radius point.

GR doesn't really have any concept of "gravitational potential". But it is true that the metric coefficients become singular inside the singularity. g_00, which represents time dilation outside the event horizon, becomes negatively infinite, while g_rr goes to zero.

Your notion of time dilation becoming infinite can be made to make some sense by observing that r and t "switch roles" inside the event horion - i.e., if you look at the Schwarascihld metric, inside the event horizon r becomes a time coordinate (because it has a negative metric coefficient) while outisde the event horizon r is a spatial coordinate.

Hence we look at g_rr instead of g_tt for time dilation inside the horizon, and we see that g_rr goes to zero - which is "infinite time dilation".

However, it simply doesn't follow that g_rr going to zero means that events "never happen" as viewed by the perspective of an infalling observer.

To really get into the detials, you'll probably need a textbook. The textbook "Gravitation", authors Misner, Thorne Wheeler (abbr. MTW) for instance, talks about the collapse of a pressureless dust on pg 859

Eddingtion Finklestein coordinates are talked about on pg 828

You'll probably be better off with a book like "Exploring Black Holes" (by some of the same authors as MTW) rather than MTW itself, though I can't guarantee that they'll go into the "dust collapse" model in detail.

I would suggest trying to understand the Eddingtion-Finklestein diagram of the Schwarzschild geometry first, then worry about the refinement of the collapse of the dust shell later.

Online, you can look at http://casa.colorado.edu/~ajsh/schwp.html for some of this information (but a textbook would still be a better bet).

If you look at the Schwarzschild metric:

ds^2 = -(1-2M/r) dt^2 + 1/(1-2M/r) dr^2 + r^2 (d theta^2 + sin^2(theta) dphi^2)

you can solve for the path of light by setting ds=0. (The lorentz interval of a ligthbeam is always zero)

For radially infalling light, dtheta=dphi=0

so you get

-(1-2M/r) dt^2 + 1/(1-2M/r) dr^2 = 0

This gives you some of the information you need as to how to assign coordinates by rescaling time (EF coordinates) so that light always appears to travel at 45 degree angles, which is the main point of an EF diagram.

 

[dpyikes]

Suppose signals are being sent from flat space at one a second to an object that is just about to collapse into a singularity, and the first signal has just arrived at the surface of the the object. Give the object some appropriate mass. Is there a calculation somewhere for how many signals will hit the surface before the object collapses to a singularity? We know the signals are very blue shifted. I claimed before it would be infinite.

 

[MeJennifer]

Your notion of time dilation becoming infinite can be made to make some sense by observing that r and t "switch roles" inside the event horion - i.e., if you look at the Schwarascihld metric, inside the event horizon r becomes a time coordinate (because it has a negative metric coefficient) while outisde the event horizon r is a spatial coordinate.

Pervect you mentioned this before in other topics.
Could you please support your assertion that r becomes a time coordinate?

 

[pervect]

I'm not sure if it will be online forever, but http://web.mit.edu/8.962/www/probset/pset11.pdf, a homework set, solves this.

The answer is that the dust cloud will collapse to a singularity in a proper time (as measured by a clock anywhere in the dust-cloud) of (pi/2)*R_0 /c (assuming a_max = 1 as advised in the problem set).

I believe that R_0 is given from the density of the cloud by the equation (8 pi / 3) R_0^2* rho =c^2/G, where rho is the density, but I could easily be screwing something up. Maybe we can get some other ambitious person to check this.

The problem is basically a time-reversed matter-dominated FRW cosmology - instead of working out the time from the "big bang" to the maximum radius, one works the problem in reverse, starting at the maximum radius, and running backwards to the big bang. This is because the interior metric is a FRW metric. (FRW = Friedmann Robertson Walker) - and because we are assuming the pressure is zero (this makes the cosmology matter dominated).

The homework problem does make it clear that the lengthscale R_0 is related to the density of the cloud, not its initial radius (the notation could be confusing). R_0 can be interpreted as the spatial curvature scale of the closed FRW cosmology.

Note that the problem set uses geometric units - I've added the G and c factors back in in this response (hopefully , correctly).

 

[Dr.Brain]

Ok, I’m a layman with an interest in physics. I don’t know any of the math beyond high school physics and first semester college calculus (with trig) a long time ago. I’ve got pretty good handle on special relativity (for a layman) and an introduction to the concepts of general relativity.

Here’s my premise. From the frame of reference of a distant observer not experiencing any time dilation from a black hole, I have read/seen it explained several times that such an observer will never see a space ship, or whatever, cross the event horizon of a black hole since such a ship will be experiencing ever increasing time dilation from the effects of both special and general relativity as it accelerates toward the event horizon. From the frame of reference of said external observer, the ship will appear to freeze at or near the event horizon. Indeed, I have read/seen it said that we have never observed anything cross such an event horizon. Matter may be swirling around at speeds approaching C, but we have not observed any of it cross the event horizon.

1) How can a black hole be said to consume surrounding matter from the frame of reference of a distant observer? Wouldn’t such an observer see it continually accumulate near the event horizon, yet never go in? I guess what I’m getting at is how much time dilation would such a ship experience, relative to a distant observer, prior to crossing the event horizon? The impression I have received is that it would be an infinite amount, i.e. the ship would never cross the event horizon from the distant observer’s frame of reference.
2) Has any theory been put forth for the huge jets of matter sometimes seen being ejected from the center, perpendicular to the accretion disk at speeds approaching C?
3) Is the formation of an accretion disk related to “frame dragging”? Presumably from a rotating singularity.
4) Is "obital procession" related to or caused by "frame dragging"?


thanks

For your first Question ..
Due to the tremendous speed the observed body possesses when approaching a black Hole , this would result in a SR's time dilation effect objectively known through Lorentz Trans. applications..So as per the observer , his time is slowed down to such an extent , that the events he is observing almost come to a hault , secondly , the images that you were receiving initially when the object was not very near to the BH , are now not being 'updated' , i.e because of the BH's property of arresting lights due to its dense gravity.

As far as the accretion disks are concerned , they are formed due to huge potential difference over small distances near a black hole , so energy is liberated in form of radiation ,when matter moves towards/away from the black hole resulting in Quasars too.

 

[pervect]

OK, a Newtonian analysis provides a much needed cross-check, and gets the same answer:

the time to collapse is independent of the inital radius of the cloud, and depends only on its density, rho.

This time is sqrt(3*pi / 8 rho G), so it's inversely proportional to the square root of the density.

The GR analysis has to work for very small clouds, so it must also work for clouds so small that they can be modeled by a Newtonian analysis.

In the Newtonian analysis, the position of a particle at any time t will be multiplied by some scale factor alpha(t), i.e.

r(t) = r0 alpha(t)

This is because the force from a constant density cloud is a Hooke's law force, i.e. the acceleration of a particle towards the center is proportional to the distance, so a particle twice as far away from the center initially has twice the acceleration towards the center.

The actual acceleration is G * (4 * pi * r^3) * rho / r^2
where the term inside the parenthesis is the total mass inside a radius r. You can see that this is equivalent to 4 * pi * G * rho * r, i.e. that acceleration is proportional to r

From inspecting the GR solution, we can confirm that

alpha(t) = (1/2) (1 + cos eta*t)

so that

r(t) = (r0/2)*(1 + cos eta*t)

Taking the second derivative, we find that r'' is negative and proportional to r, and we can solve for the value of eta

eta^2/2 = (4 * pi * G * rho)/3

When eta*t = pi, r(t) = 0, and the sphere has collapsed, i.e t = pi/eta is the time to collapse.

 

[DaveC426913]

Pervect you mentioned this before in other topics.
Could you please support your assertion that r becomes a time coordinate?

I am not sure if this is exactly the same thing but I have read that black holes in the right shape become time machines. If you approach them in the right way, one of your spatial axes is rotated with the time axis. When you make a right turn, you are now traveling along the T coord, and if you then leave the BH, you will have traveled in time. (This does NOT require a hypothetical lethal journey INTO the BH).

 

[pervect]

Pervect you mentioned this before in other topics.
Could you please support your assertion that r becomes a time coordinate?

Sure.

First, look at the Schwarzschild metric (in geometrized units, where c=1)

ds^2 = -(1-2m/r) dt^2 + dr^2/(1-2m/r) + r^2 (d theta^2 + sin^2(theta) dphi^2)

What is important is the signs of the metric coefficients. For r>2m we have

(-+++) for (dt,dr,dtheta,dphi)

The time coordinate gets a negative sign, just as it does in the Minkowski metric

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2

At r=2m, there is a coordinate singularity (i.e. one of the metric coefficients goes to zero, another becomes infinite)
at r<2m, we have

(+-+++) for (dt,dr,dtheta,dphi)

We can thus see that dt is spacelike (because it has a positive sign), and dr is timelike.

Check out also MTW, "Gravitation" pg 823 if you want a textbook reference

The most obvious pathology at r = 2M is the reversal there of the roles of t and r as timelike and spacelike coordinates.
...
What does it mean for r to change in character from a spacelike coordinate to a timelike one? The explorer in his jet powered spaceship prior to arrival at r=2M always has the option to turn on his jets and change his motion from decreasing r (infall) to increasing r (escape). Quite the contrary is the situation when he has once allowed himself to fall inside r=2M. Then the further decrease of r represents the passage of time. That unseen power of the world which drags everyone forward willy-nilly from age twenty to forty, and from forty to eighty also drags the rocket in from time coordinate r=2M to the later value of the time coordinate r=0. No human act of will, no engine, no rocket, no force (see exercise) can make time stand still. As surely as cells die, as surely as the traveler's watch ticks away "the unforgiving minutes,", with equal certainty, and with never one halt along the way, r drops from 2M to 0.

 

[MeJennifer]

Thanks Pervect, I appreciate all your help and sorry for that you sometimes have to deal with my slow catching up on things.

Yes, I understand everything you write here.
Sure the signs of the space and time coordinates change but is that enough evidence to consider that time and space coordinates actually swapped, or even stronger that the physical properties of space have become time and vice versa?
To me there is simply no mathematical validity to make that statement and as far as I know there is no rule in GR that disallowes any of those values to become negative.

To assume that imaginary time is equal to space and imaginary space is equal to time seems to me a bit of a leap.

Anyway, I suppose I just have to get used to this IMHO confusing terminology.

 

[pervect]

Note that you square the terms in the metric before adding them together. Because the square of a real number is always positive, a negative contribution to the metric would not correspond to negative space, but possibly "imaginary space".

However, this view doesn't turn out to be all that useful. (You'll occasionally see this POV, though, especially in SR, via the use of ict in the expression for the metric,i being of course the square root of negative one).

The difference between a timelike interval and a spacelike interval is well defined physically, and determined mathematically by the sign of the metric coefficients. Two points in space-time inside the event horizon that differ only by their 'r' coordinate are separated by a timelike interval, not a spacelike interval. As the long quote from MTW demonstrates, one really can view the 'r' coordinate inside the black hole as a time coordinate.

 

[Chris Hillman]

I am not sure if this is exactly the same thing but I have read that black holes in the right shape become time machines.

Are you sure that you are not thinking of certain hypothetical (and possibly dubious) scenarios which go by the name "time machines" in the literature? If so, the maximal analytic extension of the Schwarzschild vacuum solution (sometimes called the "Synge-Kruskal-Szekeres model" or "eternal black hole") is not a time machine in this sense.

But the more general Kerr vacuum (which models rotating black holes in gtr) does contain (deep inside the horizon) closed timelike curves, which if taken literally would be associated with a kind of time travel. However, it is believed that this feature is probably a mathematical artifact of the extreme symmetry of these exact vacuum solutions. Realistic models should closely resemble Kerr geometry in the exterior (as per the work of Price, Carter and others, on "no hair theorems"), but in the interior are generally expected to be qualitatively different from the Kerr vacuum. Finding a reliable qualitative description of a "generic" black hole model in gtr remains an open question; see http://arxiv.org/find/gr-qc/1/ti:+EXACT+black_hole_interior/0/1/0/all/0/1 (not all of these concern classical gtr models of black hole models) or http://arxiv.org/find/gr-qc/1/AND+OR+ti:+EXACT+black_hole+abs:+Poisson+abs:+Israel/0/1/0/all/0/1
(again, unfortunately, not all of these are terribly relevant).

If you approach them in the right way, one of your spatial axes is rotated with the time axis. When you make a right turn, you are now traveling along the T coord, and if you then leave the BH, you will have traveled in time. (This does NOT require a hypothetical lethal journey INTO the BH).

You can't "make a right turn" to turn a spacelike vector into a timelike one, even in flat spacetime. Perhaps better put: you cannot "swing" a spacelike vector into a timelike vector by using any Lorentz transformation. Can you recall exactly what you read and where? What you wrote cannot be quite right. "Make a right hand turn" sounds awfully suspicious for the reason I gave, and "if you then leave the black hole" obviously requires explanation!

People do often say, sloppily, that "r becomes timelike and t becomes spacelike inside the horizon". Or, a bit less sloppily, they might say that inside the horizon, the coordinate basis vectors \partial_r, \; \partial_t "become" timelike and spacelike respectively. Such statements are confusing and even misleading (as we see here) and therefore should be deprecated in popular books or in forums like this one.

In fact this claim refers to two nonoverlapping charts (in which the metric tensor happens to take the same functional form), the interior Schwarzschild chart (defined in a "triangle" in the usual block diagram) and the exterior Schwarzschild chart (defined in a "diamond" in the usual block diagram). In the exterior chart, \partial_r is spacelike and \partial_t is timelike, but in the interior chart, \partial_r is timelike and \partial_t is spacelike.

If you know about Killing vector fields (which describe self-isometries of a "symmetrical" spacetime), then the we can rephrase this in invariant (coordinate-free) language: the unique Killing vector which is timelike in the exterior region becomes null on the horizon and spacelike in the interior. In the exterior region, this Killing vector describes the symmetry "translation in time", i.e. it says that our spacetime is "stationary" in the exterior (in fact, "static", because this Killing vector is even irrotational, or equivalently, hypersurface orthogonal). In the interior region, not surprisingly, the spacetime is no longer stationary; the Killing vector still exists but it is spacelike. If you've seen the nifty diagrams in MTW depicting embeddings of spatial hyperslices, you know that in the interior region you can find cylindrical spatial hyperslices of form \bold{R} \times S^2. The translation symmetry of these hyperslices corresponds to the Killing vector \partial_t, which is spacelike inside the horizon.

In contrast, if you are studying some family of observers who fall through the horizon, such as the slowfall or Lemaitre observers, defined by some frame field and represented by some chart such as the infalling Eddington or Painleve charts which is well-defined at the horizon, the nonspinning frame fields carried by these observers do nothing odd as they pass through the horizon!

 

[pervect]

Personally, I don't think it's at all confusing to say that r is timelike (or that r is a time coordinate) inside the event horizon.

If we consider a set of points of varying r, and constant theta, phi, and t, the interval between any pair of points in this set will be timelike.

This is the same as the behavior of a set of points of constant x,y, and z and varying t in Minkowski space.

Just as we say that t is a time coordinate in Minkowski space, we can say that r is a time coordinate in the interior region (i.e. inside the event horizon).

 

[MeJennifer]

Just as we say that t is a time coordinate in Minkowski space, we can say that r is a time coordinate in the interior region (i.e. inside the event horizon).

Of course that is true, you could use any variable in a coordinate system but clearly using a variable called r to indicate time would be confusing!

Consider complexified space-time or spinors for a moment, would you for instance equate a space coordinate with a zero valued real part and a non zero valued imaginary part with time and vice versa?

With regards to Schwarzschild coordinates, inside a black hole it is not totally unreasonable, to me at least, to have imaginary space and time coordinates. In my opinion it is cleaner mathematically and less interpretative.
To give it a physicial interpretation as a switching of space and time is IMHO not warranted, but I do not believe that is what you are saying.

 

[quantum123]

Chris Hillman:
>>>>In the interior region, not surprisingly, the spacetime is no longer stationary;

Is this a contradiction to Birkoff's theorem? (it says spherical symmetric fields must be stationary)

 

[Chris Hillman]

Birkhoff's Theorem

I remarked "In the interior region, not surprisingly, the spacetime is no longer stationary", which prompted this query:

Is this a contradiction to Birkoff's theorem? (it says spherical symmetric fields must be stationary)

Not quite. Birkhoff's theorem is in essence a unicity result. One way to state the theorem would be along these lines: "any spherically symmetric local vacuum solution of the EFE must be locally isometric to one of the one-parameter family of Schwarzschild vacuum solutions". (I'd hate to be forced to define exactly what I mean by "spherically symmetric local solution", though.)

Stephani states the theorem like this: "every spherically symmetric vacuum solution is independent of t", but immediately adds that some caveats which directly address your question: in the interior region "t is not a timelike coordinate and r is not a spacelike coordinate...However the theorem still holds, although one would no longer describe the solution as static" in the interior region.

Other authors offer other statements and other proofs (in physicist's sense). One approach is to argue from spherical symmetry to the existence of a killing vector which behaves like our \partial_t, i.e. timelike in an asymptocally flat exterior region, spacelike in an interior region, and null on the locus (horizon) seperating these regions. In short, there is no contradiction with the interior region not being stationary.

 

[DaveC426913]

Are you sure that you are not thinking of certain hypothetical (and possibly dubious) scenarios which go by the name "time machines" in the literature?

By literature, do you include popular physics books such as Brian Greene's? (Not that this was in that one, but those are the kinds of book I read.)[/QUOTE]

You can't "make a right turn" to turn a spacelike vector into a timelike one, even in flat spacetime. Perhaps better put: you cannot "swing" a spacelike vector into a timelike vector by using any Lorentz transformation. Can you recall exactly what you read and where? What you wrote cannot be quite right. "Make a right hand turn" sounds awfully suspicious for the reason I gave, and "if you then leave the black hole" obviously requires explanation!

What I recall is this:

First, you need to build a black hole in the shape of a torus (granted, this is the fabulously advanced technology part).
If the BH is massive enough you can take advantage of the space-bending effect without crossing the event horizon.
In the right place, the time axis is bent 90 degrees (this happens regardless of whether your ship is there), thus, your ship is able to travel in a direction that, once it exits the BH, it will have traveled in time, rather than in space.

I grant this is horribly hypothetical. It's not like it's at all practically possible, but it suggests that the universe does not rule out time travel.

I also grant that my understanding is highly simplistic and popularized. I am open to enlightenment (though post-high school math is beyond me).

 

[pervect]

By literature, do you include popular physics books such as Brian Greene's? (Not that this was in that one, but those are the kinds of book I read.)

What I recall is this:

First, you need to build a black hole in the shape of a torus (granted, this is the fabulously advanced technology part).
If the BH is massive enough you can take advantage of the space-bending effect without crossing the event horizon.
In the right place, the time axis is bent 90 degrees (this happens regardless of whether your ship is there), thus, your ship is able to travel in a direction that, once it exits the BH, it will have traveled in time, rather than in space.

I grant this is horribly hypothetical. It's not like it's at all practically possible, but it suggests that the universe does not rule out time travel.

I also grant that my understanding is highly simplistic and popularized. I am open to enlightenment (though post-high school math is beyond me).

Offhand, I'd guess that your "black hole in the shape of a torus" is a spinning black hole, i.e. a Kerr black hole, which is often described in the popular literature and a rather bad Wikipeida article as a "ring singularity", though I don't think I've ever seen that description in a textbook.

There are other possibilities - an infinite rotating cylinder gives rise to a time machine, the Tippler time machine. At one time it was thought that a finite but long rotating cylinder would also form a time machine, but I believe that this is no longer felt to be the case.

You'll basically have to find your source before we can comment. For what it's worth, I have a vague recollection of reading something like you describe, a very long time ago before I knew much about GR, but I don't recall where.

 

[dpyikes]

I’m afraid I continue to be perplexed about the way time dilation works inside the event horizon, particularly for the case of a collapsing star. What I am interested in is a sequence of light rays which had originated from flat space, and are striking the surface of the star. Pervect, you write:
“http://web.mit.edu/8.962/www/probset/pset11.pdf, a homework set, solves this.
The answer is that the dust cloud will collapse to a singularity in a proper time (as measured by a clock anywhere in the dust-cloud) of (pi/2)*R_0 /c (assuming a_max = 1 as advised in the problem set).”

However, I am worried not about the proper time for the surface of the collapsing star (or anywhere in the star). I don’t doubt that that is finite. I am worried about the proper time for the source of signals in flat space “while” the collapse is happening in the sense that signals from flat space strike the star during the collapse.

Before the collapse, and before the creation of the event horizon, the rays are blue shifted. By the proper time of the source, they are leaving at one per second, while, just before the collapse of the star, they are arriving at the surface at many more than one per second. As the star begins to collapse, but before the event horizon breaks the surface of the star, the blue shifting will increase. Even more signals will strike the surface per second of proper time at the surface of the collapsing star. Naively, this rate appears to be going to infinity before the star collapses to zero radius. When the event horizon passes the surface of the star, perhaps something different will happen? As the light cone turns over, so that in the usual space time diagrams, the “right hand” part of the cone is now pointing to the star side of vertical. I take it this is the point where space and time “switch roles”. Does this have the effect of changing the behavior of the time dilation so that the increase of rate of signals striking the surface changes, and the time between strikes doesn’t approach zero? In the Eddington-Finklestein diagram at http://casa.colorado.edu/~ajsh/schwp.html, the ochre lines coming in at 45 degrees are a sequence of null surfaces originating from flat space which are striking the surface of the collapsing star at earlier times, and are striking the singularity at later times. If we draw those lines as one per second, I take it that as the star begins to collapse, the ochre lines get closer together as they go up the diagram. From the responses on this thread, I take it that the distance between the lines doesn’t go to zero as the radius of the collapsing star goes to zero. I would like to understand why the distances don’t go to zero before the star collapses to singularity. I will be getting a copy of MTW’s “Gravitation” soon, but it’s an open question whether I will be able to understand it.

 

[MeJennifer]

I am worried about the proper time for the source of signals in flat space “while” the collapse is happening in the sense that signals from flat space strike the star during the collapse.

Well, the proper time interval between the source and the surface of the collapsing star for "the source of signals" is exactly zero.

Perhaps the confusion is related to the term "shift". Nothing is shifted in the light between the source and the surface of the collapsing star.

Light, by itself, does not undergo any changes going from one gravitational potential to another. The observed "blue" shift is solely due to the difference in "clock" speeds between the emitter and absorber.

In other words, the absorbing atom observes a gravitational redshift if the emitting atom was "redder" and a gravitational blueshift if the emitting atom was "bluer".

Think of this analogy: if someone comes out of the freezing cold and takes a cold shower it does not feel cold at all, but for someone who sat in front of the fireplace it feels very cold.

 

[dpyikes]

MeJennifer –

You write:

“the absorbing atom observes gravitational redshift if emitting atom was "redder" and gravitational blueshift if the emitting atom was "bluer".”

In the relevant case, the absorbing atom is on the surface of the collapsing star, the emitting atom is at some point in flat space -- there is a gravitational blueshift at work in the sense both that if those light rays had been received in flat space, they would have been at a lower wavelength than those received at the surface, and that the number received per second by the proper time of a receiver in flat space would have been less than those received by the proper time of the surface of the collapsing star. When talking about the proper time for the surface of the star, I was not talking about the proper time of the light signal. I was talking about the proper time for the surface, say, between signal strikes. I am trying to understand why that proper time is not approaching zero as the radius of the star moves toward zero.

 

[MeJennifer]

I was talking about the proper time for the surface, say, between signal strikes. I am trying to understand why that proper time is not approaching zero as the radius of the star moves toward zero.

Let's see if I understand this properly:

An object, at a far distance from a black hole, sends every second a pulse in the direction of a collapsing star.
This pulse is received by an observer who is on the surface of this collapsing star. The star is soon to become a black hole.

Why does the proper time interval between consecutive pulses received by the observer on the surface not approach zero during the formation of the event horizon?

Did I formulate your question correctly?

 

[dpyikes]

Yes, MeJennifer -- It seems to me that the proper time between signals received ought to approach zero, unless there is something about time and space switching roles inside the event horizon that prevents this.

 

[George Jones]

However, I am worried not about the proper time for the surface of the collapsing star (or anywhere in the star). I don’t doubt that that is finite. I am worried about the proper time for the source of signals in flat space “while” the collapse is happening in the sense that signals from flat space strike the star during the collapse.

This is an interesting question, and I hope that sometime in the next few days I can find the time to do the calculation.

 

[pervect]

I’m afraid I continue to be perplexed about the way time dilation works inside the event horizon, particularly for the case of a collapsing star. What I am interested in is a sequence of light rays which had originated from flat space, and are striking the surface of the star. Pervect, you write:
“http://web.mit.edu/8.962/www/probset/pset11.pdf, a homework set, solves this.
The answer is that the dust cloud will collapse to a singularity in a proper time (as measured by a clock anywhere in the dust-cloud) of (pi/2)*R_0 /c (assuming a_max = 1 as advised in the problem set).”

However, I am worried not about the proper time for the surface of the collapsing star (or anywhere in the star). I don’t doubt that that is finite. I am worried about the proper time for the source of signals in flat space “while” the collapse is happening in the sense that signals from flat space strike the star during the collapse.

Before the collapse, and before the creation of the event horizon, the rays are blue shifted. By the proper time of the source, they are leaving at one per second, while, just before the collapse of the star, they are arriving at the surface at many more than one per second. As the star begins to collapse, but before the event horizon breaks the surface of the star, the blue shifting will increase. Even more signals will strike the surface per second of proper time at the surface of the collapsing star. Naively, this rate appears to be going to infinity before the star collapses to zero radius. When the event horizon passes the surface of the star, perhaps something different will happen? As the light cone turns over, so that in the usual space time diagrams, the “right hand” part of the cone is now pointing to the star side of vertical. I take it this is the point where space and time “switch roles”. Does this have the effect of changing the behavior of the time dilation so that the increase of rate of signals striking the surface changes, and the time between strikes doesn’t approach zero? In the Eddington-Finklestein diagram at http://casa.colorado.edu/~ajsh/schwp.html, the ochre lines coming in at 45 degrees are a sequence of null surfaces originating from flat space which are striking the surface of the collapsing star at earlier times, and are striking the singularity at later times. If we draw those lines as one per second, I take it that as the star begins to collapse, the ochre lines get closer together as they go up the diagram. From the responses on this thread, I take it that the distance between the lines doesn’t go to zero as the radius of the collapsing star goes to zero. I would like to understand why the distances don’t go to zero before the star collapses to singularity. I will be getting a copy of MTW’s “Gravitation” soon, but it’s an open question whether I will be able to understand it.

I believe I mentioned the relevant points for this part of the question earlier. This time around I'll ask some questions rather than give any more answers.

1) If you are standing still, and your friend is passing you by at a significant percent of light speed, do you both agree on the frequency of a light beam, or is there some relative red/blue shift due to the difference in your velocities when you both measure the frequency of the same light beam?

2) If a hovering observer "hovers" at a stationary r coordinate, what is the rerlative velocity between that hovering observer and an free-falling observer, in the limit as the hovering observer approaches the event horizon?

3) Does this relative velocity contribute to the observed red/blue shift, as per question 1?

 

[dpyikes]

Pervect -- I think I liked it better when you were answering the questions.

”1) If you are standing still, and your friend is passing you by at a significant percent of light speed, do you both agree on the frequency of a light beam, or is there some relative red/blue shift due to the difference in your velocities when you both measure the frequency of the same light beam?”

Assuming this takes place in flat space and each are sending signals to the other, there would be a red shifting of signals in both reference frames – The situation is symmetric: If both people sent and received a sequence of signals, both would receive the light at lower frequencies (and receive signals at lower rates) than would be received by people moving along with the two people respectively.

”2) If a hovering observer "hovers" at a stationary r coordinate, what is the relative velocity between that hovering observer and an free-falling observer, in the limit as the hovering observer approaches the event horizon?”

As I understand the question, the hoverer is hovering very close to the event horizon and the free faller is falling from infinity, and the question is what is his velocity relative to the hoverer just before the free faller passes the event horizon. I don’t know the answer – in fact, I wouldn’t have thought there is one single answer. The relative speed would be greatest if the free faller were falling radially. Anyway, I don’t know.

”3) Does this relative velocity contribute to the observed red/blue shift, as per question 1?”

Assuming the speed of the free faller is a significant percentage of the speed of light, there would be symmetric special relativistic red shifting of signals sent and received between the two observers, just as in case 1. If they are both just outside the event horizon, then there is no difference in gravitational potential between them, so I think there would be no gravitational time dilation between them.

Signals from a third point in flat space would have a blue shifted component to both of them because of gravitational time dilation. If the free faller is moving at some significant portion of the speed of light, this would red shift the signals from flat space, tending to cancel blue shifting to him.

I’m sorry I could not find the answer to my question in your previous posts. I take it the answer is that there is a finite number of signal strikes originating from flat space before the collapsing star reaches singularity?

 

[Jorrie]

johatfie wrote: "since such a ship will be experiencing ever increasing time dilation from the effects of both special and general relativity as it accelerates toward the event horizon."

Chris replied:
No, in fact this "explanation" runs completely counter to the spirit as well as the letter of the law as laid down by gtr, if I might so put it.

Chris, just to clear/educate my own mind, would it have been more correct if johatfie stated: "since such a ship will be experiencing ever increasing velocity time dilation and gravitational time dilation as it accelerates (or falls) toward the event horizon?"

I am asking this because in Schwarzschild coordinates, the ratio: proper-time to coordinate time can be expressed as the product of a gravitational time dilation factor and a velocity time dilation factor, e.g.,

dtau^2/dt^2 = -g_00 (1-v^2),

where g_00 is the time-time coefficient of the metric and v is the velocity in the local (infinitesimal) Lorentz frame.

Regards, Jorrie

 

[Chris Hillman]

Wild speculation, confusion due to misread popular books, etc.

By literature, do you include popular physics books such as Brian Greene's? (Not that this was in that one, but those are the kinds of book I read.)

No, I meant "research literature". I tend to assume that anyone who comes by here has some familiarity with the popular literature, and has been thoroughy confused by it.

Brian Greene's field of research uses gtr, but is not gtr. Among the popular books dealing with classical gravitation, I feel that the books by Geroch and by Wald are outstanding; see http://www.math.ucr.edu/home/baez/RelWWW/reading.html#pop

First, you need to build a black hole in the shape of a torus (granted, this is the fabulously advanced technology part).
If the BH is massive enough you can take advantage of the space-bending effect without crossing the event horizon.
In the right place, the time axis is bent 90 degrees (this happens regardless of whether your ship is there), thus, your ship is able to travel in a direction that, once it exits the BH, it will have traveled in time, rather than in space.

I grant this is horribly hypothetical. It's not like it's at all practically possible, but it suggests that the universe does not rule out time travel.

I also grant that my understanding is highly simplistic and popularized. I am open to enlightenment (though post-high school math is beyond me).

OK, Dave, I still don't know what you read or where (or are you saying you think you read the above in a book by Brian Greene?--- if so, I am confident that you have omitted essential context and also misread or misrecall whatever BG might have written), but "In the right place, the time axis is bent 90 degrees (this happens regardless of whether your ship is there), thus, your ship is able to travel in a direction that, once it exits the BH, it will have traveled in time, rather than in space" is wrong, if indeed it even rises to that level. (I recall the caustic phrase of Pauli, author of the first relativity textbook, that some claims are "not even wrong".)

I think it is best to say that there are some more or less wild speculations in the research literature about possible time travel (including speculation involving Lorentzian wormholes), but it is probably fair to say that most physicists currently think this is ruled out in Nature. Certainly the mainstream view is that it seems unlikely that humans can engage in time travel (into their absolute past).

 

[Chris Hillman]

Two quick comments

Hi, dpyikes and everyone else confused by this thread,

It seems to me that the proper time between signals received ought to approach zero, unless there is something about time and space switching roles inside the event horizon that prevents this.

Two quick comments:

1. "time dilation" is always relative: it concerns signals emitted by one observer and received by another; thus, this phrase has no meaning unless you specify which observers are involved; in addition, time dilation effects occur because of geodesic deviation of null geodesics; it is nonsensical to speak of "time slowing down" (no wonder you guys are confused!--- please, stop reading those awful popular books which use such terribly misleading language, and start reading the two popular books I recommended)

2. is is nonsensical to claim that "time and space switch roles" anywhere in any Lorentzian manifold (no wonder you guys are confused!--- please, stop reading those awful popular books which use such terribly misleading language, and start reading the two popular books I recommended).

In short: no wonder you guys are confused!--- please, stop reading those awful popular books which use such terribly misleading language, and start reading the two popular books I recommended in http://www.math.ucr.edu/home/baez/RelWWW/reading.html#pop .

 

[MeJennifer]

1. "time dilation" is always relative: it concerns signals emitted by one observer and received by another; thus, this phrase has no meaning unless you specify which observers are involved

Christ, I attempted to formulate dpyikes question to avoid such uncertainties. Please take a look at it in my prior posting so you can see exactly what he is asking.
Hope this helps!

 

[Chris Hillman]

Arghghgh!

Hi, Jorrie,

Chris, just to clear/educate my own mind, would it have been more correct if johatfie stated: "since such a ship will be experiencing ever increasing velocity time dilation and gravitational time dilation as it accelerates (or falls) toward the event horizon?"

To repeat: "time dilation" (and "red/blue shift") concerns signals emitted by one observer and received by another; it has no meaning unless you specify an ordered pair of observers.

I am asking this because in Schwarzschild coordinates, the ratio: proper-time to coordinate time can be expressed as the product of a gravitational time dilation factor and a velocity time dilation factor, e.g.,

dtau^2/dt^2 = -g_00 (1-v^2),

where g_00 is the time-time coefficient of the metric and v is the velocity in the local (infinitesimal) Lorentz frame.

"Proper time" has no meaning unless you specify which observer you are talking about.

I know which observers you are asking about here, but instead of answering your question, I'll let you try to reformulate it properly (no pun intended).

 

[Chris Hillman]

Tipler

There are other possibilities - an infinite rotating cylinder gives rise to a time machine, the Tippler time machine.

Tipler's (one p) model results from modeling the exterior field of a rotating cylindrical shell, or something like that. See also the Van Stockum dust and consider matching to a vacuum region (possible because the dust is a pressureless perfect fluid). Tipler has written some interesting papers (early in his career), e.g. on classification of curvature singularities, but has become quite eccentric in some of his popular writings, as you probably know...

Various regions in a pair of coaxial rotating cylindrical shells has been discussed in connection with supposed "Machian" (or "anti-Machian") phenomena in gtr (compare Lense-Thirring for a rotating spherical shell).

 

[Chris Hillman]

Computing a frequency shift

I’m afraid I continue to be perplexed about the way time dilation works inside the event horizon, particularly for the case of a collapsing star.

So-called "time dilation" concerns lightlike signals sent from one observer to another observer. This notion has no meaning unless you specify an ordered pair of observers. In the above, I guess you mean an observer on the surface of a collapsing spherical dust ball (in an Oppenheimer-Snyder model) sending time signals once per second by his ideal clock to a very distant static observer.

What I am interested in is a sequence of light rays which had originated from flat space, and are striking the surface of the star.

OK, now I guess that you mean a distant static observer sending signals once per second (by his clock) to an observer on the surface of a collapsing spherical dust ball in an OS model (not the same thing at all!).

Everyone, please, make the effort to try to specify which pair of observers you are talking about, or endless confusion will result! I'd also once again highly recommend the two popular books which I keep recommending (see link given in my immediately preceding posts in this thread above).

I am worried about the proper time for the source of signals in flat space “while” the collapse is happening in the sense that signals from flat space strike the star during the collapse.

There is no flat spacetime here. I doubt you were referring to the flat spatial hyperslices in the Painleve chart; rather, I guess that you were referring to
the fact that the exterior region is asymptotically flat, so that very far away from the collapsing dust ball, our OS spacetime is approximately Minkowskian.

I don't understand what you are worried about, but I can help you compute the frequency shift in this scenario (assuming I understand what you have in mind).

To construct the OS model, we match a region filled with collapsing dust (modeled using a portion of an FRW dust model with E^3 hyperslices orthogonal to the world lines of the dust) across a collapsing sphere to an asymptotically flat vacuum exterior region (modeled using a portion of the Schwarzschild vacuum solution). A key observation is that since a dust is a pressureless perfect fluid, there are no forces acting on the dust particles, so their world lines will be timelike geodesics. In particular, the world lines of observers on the surface will be timelike geodesics.

Now, as you would expect, our dust ball must be momentarily at rest at some r=r_0, where r is the Schwarzschild radial coordinate of the surface in the exterior region. To keep things as simple as possible, we should take the limit where r_0 \rightarrow \infty, and then observers riding on the surface of the collapsing dust ball will be "Lemaitre observers" (the ones involved in the Painleve chart, called "free-fall chart" by Andrew Hamilton; see the figure just above the one you mentioned). If we took a finite dust ball, we would use "Novikov observers".

Let me outline how Hamilton came up with the picture he labels "free-fall spacetime diagram".

In the Painleve chart (1921), the metric tensor can be written
ds^2 = -dT^2 + \left( dr + \sqrt{m/r} \, dT \right)^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)
= -(1-2m/r) \, dT^2 + 2 \, \sqrt{2m/r} \, dT \, dr + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty &lt; T &lt; \infty, \; 0 &lt; r &lt; \infty, \; 0 &lt; \theta &lt; \pi, \; -\pi &lt; \phi &lt; \pi
The world lines of the Lemaitre observers are the integral curves of
\vec{e}_0 = \partial_T - \sqrt{2m/r}\, \partial_r
That is:
T - T_0 = \sqrt{\frac{2 \, (r-r_0)^3}{9m}}
The world lines of radially ingoing light rays are the integral curves of
\vec{k} = \partial_T - \left( 1 + \sqrt{2m/r} \right) \, \partial_r
That is:
T - T_0 = -\left( r-r_0 \right) + \sqrt{8 m} \, \left( r-r_0 \right) - 4 m \, \log \frac{\sqrt{r}+\sqrt{2m}}{\sqrt{r_0}+\sqrt{2m}}
So now you can draw a bunch of light signals sent from a very distant static observer (once per second, by his ideal clock) to an observer standing on the surface of the collapsing dust ball. You should get the same picture as Hamilton (see "Freely fall spacetime diagram", where the bright yellow curves depict the world lines of some radially infalling null geodesics, and where the light green curves depict the world lines of some Lemaitre observers.)

We can augment the timelike unit vector field \vec{e}_0 with three spacelike unit vector fields, all mutually orthogonal, in order to obtain the frame field we would use to describe the physical experience of our Lemaitre observers:
\vec{e}_1 = \partial_r, \; \vec{e}_2 = \frac{1}{r} \, \partial_{\theta}, \; \vec{e}_3 = \frac{1}{r \, \sin(\theta)} \, \partial_{\phi}
Then \vec{e}_0, \; \vec{e}_1, \; \vec{e}_2, \; \vec{e}_3 is an example of an inertial nonspinning frame field, which is as close as we can come in a curved spacetime to a "local Lorentz frame". As I said, the timelike congruence defining the Lemaitre observers is obtained from \vec{e}_0; note that the null congruence of radially ingoing null geodesics is \vec{k} = \vec{e}_0 - \vec{e}_1. Do you see why we can see at a glance that this is a null vector field, indeed a "radial" null vector field? Can you guess what would be the null vector field defining the null congruence of radially outgoing null geodesics?

I stress that the Painleve chart is defined down to the curvature singularity at r=0 and has two delightful properties:

1. the hyperslices T=T_0 are locally isometric to E^3, with a polar spherical chart in which the radial coordinate is just the Schwarzschild radial coordinate,

2. the difference of Painleve time coordinate T_2-T_1 for two events on the world line of one of our Lemaitre observers directly gives the elapsed proper time as measured by ideal clocks carried by this observer.

So we want to compute the frequency of the signals upon reception by our surface riding Lemaitre observer, as measured by his own ideal clock. But now that I've gotten you started, maybe I should give you an opportunity to try to finish the computation...

Let me just go back to something I said above: you probably already appreciate that if we compute the frequency shift for signals sent from the surface-riding observer back to our very distant static observer, this turns out to be a red shift which diverges as the falling observer passes the horizon. Before carrying out the computation, from looking at the diagram in Andrew Hamilton's website, can you guess what should happen in the case of signals sent from our very distant static observer to our surface riding observer?

 

[pervect]

Pervect -- I think I liked it better when you were answering the questions.

I'm hoping that getting you to answer some questions will ease the communication problem and highlight some of the areas that need more explaining. And I think it has succeeded to some extent.

”1) If you are standing still, and your friend is passing you by at a significant percent of light speed, do you both agree on the frequency of a light beam, or is there some relative red/blue shift due to the difference in your velocities when you both measure the frequency of the same light beam?”

Assuming this takes place in flat space and each are sending signals to the other, there would be a red shifting of signals in both reference frames – The situation is symmetric: If both people sent and received a sequence of signals, both would receive the light at lower frequencies (and receive signals at lower rates) than would be received by people moving along with the two people respectively.

This isn't the answer I was looking for. This appears to be where the major confusion is.

Consider the case where you have a light beam coming "from infinity", the same as in your black hole problem, where you have a light beam falling into the black hole from r=infinity.

Assume that the light beam is pulsed once per second at its source, as in your black hole example.

The two observers, in different states of motion, measure the frequency of the light beam (or look at the color of the light beam).

I thought it was obvious that there will be either a red-shift or a blue-shfit of the measurement of the frequency of the light beam. This phenomenon is called "doppler shift".

There is a closely related point that may also be important here.

Suppose the first observer measures a light beam, comfing from infinity, to have a frequency of 800 terahertz (800e12 hz) , which should put it in the blue region of the spectrum.

The light beam is modulated so that it pulses at once per second according to this observer.

We can assume that this observer is stationary to the source, so that at the source the beam has a frequency of 800 terahertz as well, and is modulated so that it emits 1 second long pulses.

A second moving observer measures the same light beam, but because of the doppler shift he measures a frequency of only 400 terahertz (which should put the frequency of the light into the red region of the spectrum). The second observer will also observe a different timing of the pulses.

The first observer will observe that each 1-second pulse contains 800e12 wavelengths of the signal. Each wavelength has an observable "peak" in the electric field. The second observer will count the same number of wavelengths in the pulse - 800e12. This is because the number of peaks is independent of the observer.

Because the measured frequency of the radiation is only 400e12 hertz, according to the second moving obsserver, however, the second observer will also see that the pulses are not 1 second long, but 2 seconds long. A short way of saying this: any modulation on the light beam (in this case, amplitude modulation) gets red-shifted by exactly the same factor as the carrier does.

”2) If a hovering observer "hovers" at a stationary r coordinate, what is the relative velocity between that hovering observer and an free-falling observer, in the limit as the hovering observer approaches the event horizon?”

As I understand the question, the hoverer is hovering very close to the event horizon and the free faller is falling from infinity, and the question is what is his velocity relative to the hoverer just before the free faller passes the event horizon. I don’t know the answer – in fact, I wouldn’t have thought there is one single answer. The relative speed would be greatest if the free faller were falling radially. Anyway, I don’t know.

That's a fair answer. I should clarify the question a bit - I am assuming that the observer is free-falling radially. I can give you the answer, but you have to decide whether or not to believe it. In the limit, as the hovering observer gets closer and closer to the event horizon, the velocity at which the infalling observer passes the hovering observer approaches 'c'. This happens (for the radially falling observer, which is the only case I've worked out) regardless of the exact trajectory of the infalling observer or his initial (radial) velocity "at infinity" as he falls into the black hole.

Now, if we can get question #1 straightened out (which I thought was the easy part), you can hopefully see why this is relevant.

 

[dpyikes]

Pervect –

I think I have no problem with what you are saying about #1 – I was imagining A and B sending signals to each other rather than receiving signals from a third source. In your scenario, a source external to our two observers is sending light signals that are received once a second as measured by A while two signals a second are measured by B. Maybe A is moving toward the source, and B is moving away from it.

I believe I heard somewhere that an in-falling object from infinity passing the event horizon approaches c. I have no problem stipulating that.

As to the relevance of all this to the case of how many one second apart signals from approximately flat space will strike the surface of a star before it collapses to singularity, I am not so sure. Is the point that STR-Red shifting of the signals that strike the surface cancels the gravitational blue shifting of those signals? Maybe I heard somewhere that a collapsing star collapses at free fall, hence the surface of the collapsing star will be like the free falling observer. Since the surface will be receiving light STR-red-shifted wrt the almost flat source, the gravitational blue shifting of the in-falling light wrt the source will cancel with its STR-red shifting as it collapses. Maybe there will then be a finite number of strikes from (approximately) flat space time. Of course, the star is not collapsing from infinity, but rather starts with a zero velocity wrt the source. Maybe the surface gets to relativistic speeds wrt to the source pretty fast so this canceling takes place?