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)