What is the relationship between black holes and time dilation?

[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
$$ds^2 = -dT^2 + \left( dr + \sqrt{m/r} \, dT \right)^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)$$
$$= -(1-2m/r) \, dT^2 + 2 \, \sqrt{2m/r} \, dT \, dr + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$$
$$-\infty < T < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi$$
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?

 

[Chris Hillman]

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

No, no, no! Three emphatic negatives, because this is terribly wrong for at least three reasons.

First, you may be confusing coordinate speeds, which have no geometric or physical meaning (in general), with physical measurements of "distance in the large", and thus of "velocity in the large" wrt some observer and his ideal clock.

Second, there are many distinct operationally significant notions of "distance in the large" (even in flat spacetime, for accelerating observers), so this really this should be stated as "velocity in the large" wrt some observer and his ideal clock, and some method measurement.

Third, avoid suggesting that the world line of any infalling test particle becomes null (i.e. that the tangent vector to the curve becomes null) as it crosses the horizon. This is absolutely not what gtr says. True, some truly abysmal arXiv eprints make that claim--- which is one reason why those authors are mostly ignored on the grounds that someone who insists that 1+1=3 in integer arithmetic is obviously pretty darn confused. Learn from standard textbooks, that's my advice--- they are much less likely to mislead you than other sources.

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.

If the collapsing star is modeled as a pressure-free perfect fluid or dust, yes. I just mentioned that in my post immediately above, in fact.

If I am not helping, let me know and I will be quiet...

 

[Jorrie]

"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).

Hi Chris,

Thanks for the prompt reply. OK, I'll give it a go:

In Schwarzschild coordinates, the ratio: proper-time of a free-falling observer, moving at locally Lorentz velocity v relative to a Schwarzschild black hole of mass M, to the coordinate time (the time of a stationary observer far away from an isolated black hole) 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/c^2),

where g_00 = -1 + 2GM/(rc^2), r the Schwarzschild radial parameter and G and c have there usual meanings.

Regards, Jorrie

 

[pervect]

No, no, no! Three emphatic negatives, because this is terribly wrong for at least three reasons.

I'm afraid any confusion here can be mostly attributed to me. But I don't think there is any confusion here, at least I hope not. I think there are just some communication problems. I try to keep an open mind about the possibility that I may be confused on some important issues, in fact that's one of the reason I'm glad to see knowledgeable and outspoken people like Chris Hillman here.

First, you may be confusing coordinate speeds, which have no geometric or physical meaning (in general), with physical measurements of "distance in the large", and thus of "velocity in the large" wrt some observer and his ideal clock.

The notion of velocity that I am intending people to understand is not "velocity in the large", but the frame-field velocity. The accelerating, hovering-at-constant r observer has a well-defined frame-field. So does the infalling observer.

The velocity that the infalling observer measures when he passes the hovering observer at the same location in space-time using his local frame field is going to be the same as the velocity as the hovering observer measures when the free-falling observer passes him using HIS local frame-field.

[add]
Let me make this staement even more precise. Two observers at the same point in space-time share the same vector space for their tangent vectors. This is why we can compare the velocity of two observers in the general case if and only if they are at the same event - because this is the only way to guarantee that the tangent space is shared.I should add that I use this approach a lot. If it can be demonstrated that this approach has problems, it has to go, but I think it's OK.

The only thing that's slightly tricky here is that the acceleration of our hovering observer is approaching infinity. This means that the maximum size of his frame field is is going to be very small, approaching zero. I don't think this is a killer problem though. It just involves defining the neighborhood over which the velocity is measured as "small enough". This should always be possible for any observer located arbitrarily close to the event horizon, and that's all we need to do.

Second, there are many distinct operationally significant notions of "distance in the large"

Since we are doing "velocity in the small this objection shouldn't matter.

Third, avoid suggesting that the world line of any infalling test particle becomes null (i.e. that the tangent vector to the curve becomes null) as it crosses the horizon. This is absolutely not what gtr says.

Here I agree. But I was a bit more careful in how I worded my original claim, which is that the limit of the velocity of the infalling observer approaches 'c', relative to the velocity of the hovering observer, as the hovering observer gets closer and closer to the event horizon.

I should add that while the re-phrasing dpyikes has given my original statement is no longer rigorous, I don't think it's fair to expect that it should be. What I'm looking for is enough feedback to see if I've communicated my main points.

Technically speaking, what happens is that the gravitational blueshift approaches infinity as one approaches the event horizon, and the doppler redshift also approaches infinity. The result is that the problem of determining the total red-shfit isn't well defined in Schwarzschild coordinates. The mathematically rigorous solution to these difficulties is not use Schwarzschild coordinates, but to use coordinates that are well-behaved at the horizon. The non-rigorous solution that will turn mathematicians green is to blithely say "The infinite term on the numerator cancels out with the infinite term on the denominator" :-) - i.e. that the infinite blueshift is "cancelled out" by a corresponding infinite red-shift.

Statements like this can only be truly justified by working out the answer in well-behaved coordinates. But I don't think dpyikes has the background for this (as far as I know, I could be wrong), so I'm trying to keep things as simple as possible. I'm basically trying to provide some insight why the observed redshift from a stationary source at infinity to a free-falling (i.e. Painleve) observer is finite in very elementary terms.

And the simple (IMO) answer is that dpyikes intuition has come to the wrong conclusion because he has been ignoring the doppler shift - and the doppler shift is not ignorable.

 

[pervect]

Pervect –

Is the point that STR-Red shifting of the signals that strike the surface cancels the gravitational blue shifting of those signals?

Yes, that's basically the point I was trying to make in a nutshell. It looks like Chris Hillman has some issues with what I was trying to say, so stay tuned and see if we can get them sorted out.

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?

No matter how close you are to the event horzion when you drop something, the limit of the velocity as the object approaches the event horizon will be 'c'.

To do so over an arbitrarily short distance requires an infinite acceleration, but the acceleration of the station-holding observer IS infinite - so the relative acceleration of the free-fall observer and the station holding observer is also infinite.

The mathematical details were worked out in https://www.physicsforums.com/showpost.php?p=602558&postcount=29

where I use the "energy-at-infinity" E rather than the drop height as the appropriate parameter. Dropping an object into a black hole from very near the event horizon is equivalent to setting E in the expression in the above post to a very small number, approaching zero as a limit. The answer in non-geometric units is c*sqrt(E^2)/E. This is undefined when E=0, but for any other value of E, no matter how small, the answer is c.

[add]Actually, I oversimplified. For completeness, the exact expression in geometric units was

$$v = lim_{r \rightarrow 2M+} \frac{\sqrt{E^2 - (1 - \frac{2M}{r})}}{E}$$

here v is the velocity measured relative to a hovering observer at some r>2M

Note that another poster (George Jones) came up with a similar answer in post #31 in the same thread.

 

[Chris Hillman]

Hi again, Jorrie,

OK, I'll give it a go:

In Schwarzschild coordinates, the ratio: proper-time of a free-falling observer, moving at locally Lorentz velocity v relative to a Schwarzschild black hole of mass M, to the coordinate time (the time of a stationary observer far away from an isolated black hole) 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/c^2),

where g_00 = -1 + 2GM/(rc^2), r the Schwarzschild radial parameter and G and c have there usual meanings.

I was using the Painleve chart, which is well behaved on the horizon and in fact in the entire "future interior" region, not the (exterior) Schwarzschild chart, which is only defined in the exterior.

Hi, pervect,

It looks like Chris Hillman has some issues with what I was trying to say, so stay tuned and see if we can get them sorted out.

No matter how close you are to the event horzion when you drop something, the limit of the velocity as the object approaches the event horizon will be 'c'.

OK, you are talking about the velocity of the Painleve observer wrt to the frame of the static observers, who can only exist outside the horizon. Indeed, it makes little sense to try to carry out computations at the horizon using a coordinate chart which is badly behaved there! He did say above that he wants to study the physical experience of an observer standing on the surface of the collapsing star even inside the horizon.

That is why I am urging Jorrie to adopt the Painleve chart, which is one of the simplest charts which is well behaved on and inside the horizon. I don't know anyone would discourage him from becoming familiar with the Painleve chart when this chart is so simple and has so many virtues http://www.arxiv.org/abs/gr-qc/0001069. I guess that is our disagreement.

 

[Jorrie]

Hi again, Jorrie,

I was using the Painleve chart, which is well behaved on the horizon and in fact in the entire "future interior" region, not the (exterior) Schwarzschild chart, which is only defined in the exterior.

Hi Chris, thanks for the reference to the Painleve chart, I have studied this a bit before and will spend some more time on it.

I think you swapped references to dpyikes and myself a bit in your reply. I was not the guy who wanted to 'look inside' the event horizon, it was dpyikes. My previous (post #52) was strictly aimed at Schwarzschild coordinates, where I tried to correct my poor definitions of observers outside the horizon.

I agree with the problems that sloppy definitions of observers create, so
I will appreciate your comment for the sake of better communication.

Regards, Jorrie

 

[dpyikes]

Thanks

Hi,

I just wanted to thank MeJennifer, Pervect and Chris Hillman for their responses and patience. It is indeed the case that

"dpyikes intuition has come to the wrong conclusion because he has been ignoring the doppler shift - and the doppler shift is not ignorable"

I have learned a good deal from this and hope to continue doing so.

 

[Chris Hillman]

Frequency shifts observed by various infalling observers

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

Oh, I've done these computations and more, in various different ways. I was trying to nudge MeJennifer, Jorrie, and dpyikes towards discovering some nifty results of this kind on their own, but I can see that wasn't doing a very good job!

I think you swapped references to dpyikes and myself a bit in your reply. I was not the guy who wanted to 'look inside' the event horizon, it was dpyikes

I apologize to you both.

"dpyikes intuition has come to the wrong conclusion because he has been ignoring the doppler shift - and the doppler shift is not ignorable"

I have learned a good deal from this and hope to continue doing so.

Good, there's some cool stuff here. Did I understand that you wanted to follow the physical experience of an observer riding on the surface of a collapsing dust ball (in an OS model of gravitational collapse)? That in particular you wanted to compute the redshift of signals sent by this surface observer up to very distant static observers? And the frequency shift of signals sent by a distant static observer (or a distant star) to our surface riding observer? (The world lines of such signals would be respectively radially outgoing and radially ingoing null geodesics.)

If so, indeed, in the latter case, since the surface riding observer is moving radially away from the static observer, we can expect the gravitational blue shift (which would be observed by a static observer near the massive object) to counteract the Doppler shift. This suggests there might be a class of radially infalling observers who fall at just the right rate to ensure that they see no frequency shift at all of signals from distant static observers. And there is!

In the past, I posted some detailed computations concerning the physical experience of a half dozen classes of "interesting" observers in the Schwarzschild vacuum, including static observers, slowfall observers (accelerate radially outward with just the magnitude which would allow them to hover, according to Newtonian gravitation; in gtr gravity is a bit stronger so these observers fall slowly radially inwards), Lemaitre observers (fall radially in from rest "at infinity"), Novikov observers (fall radially in from rest at some finite radius), Frolov observers (whose spatial hyperslices are cylinders ${\mathbold R} \times S^2$, and whose world lines appear as radial rays in the interior Schwarzschild chart), and Hagihara observers (moving in stable circular orbits in the exterior region). It is very useful to write the corresponding frame fields in a number of charts, including the exterior and interior Schwarzschild charts, ingoing and outgoing Eddington charts, (ingoing) Painleve chart, Kruskal-Szekeres chart, and Penrose chart (at the very least--- the Lemaitre chart is another obvious choice).

See "Frame fields in general relativity" at http://en.wikipedia.org/wiki/User:Hillman/Archive to get started.

It is often advantageous to use a coordinate chart which simplifies computations as far as possible. It is a bit difficult to explain without sketches, but assuming that everyone recalled how I rediscovered the Painleve chart by pulling down the spatial hyperslices so that they become locally flat (in the sense of coordinate surfaces, and in the sense of intrinsic geometry!), I was trying to suggest that in this problem it is advantageous to pull down the ingoing or outgoing radial null geodesics to become straight (these constructions give two different charts, the ingoing and outgoing Eddington charts).

So I'd suggest starting with the slowfall observers (see the version of the Wikipedia article which was cited above) represented first in the ingoing Eddington chart, then the outgoing Eddington chart. Try to compute the ration of received frequency (measured by the slowly falling observer) to emitted frequency (measured by the distant static observer). Next, repeat for the Lemaitre observers. In the latter case, you should be able to reconcile your result (in the case of time signals sent from a distant static observer radially inward to a Lemaitre observer, studied using the ingoing Eddington chart) with the result you find using the method sketched by pervect. However, that method only gives results in the exterior region (and by taking a limit, at the horizon), while the method I am suggesting is simpler, more elementary (requires only derivatives, the Minkowskian "Pythagorean theorem", and similar triangles) and gives results in both the future interior and right exterior regions, i.e. the full domain covered by the infalling Eddington chart (which only covers half of the maximal analytic extension of the Schwarzschild vacuum solution, aka the "eternal black hole", but covers the entire exterior of a gravitational collapse model such as the OS model).

 

[Chris Hillman]

Just noticed this...

The notion of velocity that I am intending people to understand is not "velocity in the large", but the frame-field velocity. The accelerating, hovering-at-constant r observer has a well-defined frame-field. So does the infalling observer.

The velocity that the infalling observer measures when he passes the hovering observer at the same location in space-time using his local frame field is going to be the same as the velocity as the hovering observer measures when the free-falling observer passes him using HIS local frame-field.

Yes, exactly.

I should add that I use this approach a lot. If it can be demonstrated that this approach has problems, it has to go, but I think it's OK.

Now I am confused, because of course I was suggesting (in the Delphic manner) two methods, both using frame fields. So we must be talking at cross purposes. No doubt this happened because I have in mind a much wider selection of charts and of frame fields.

Be this as it may, after sufficient time has passed to allow dpyikes, Jorrie and maybe MeJennifer to try their hand at computing $\omega_{{\rm received}}/\omega_{{\rm emitted}}$ (for a distant static observer respectively sending and receiving signals from a Lemaitre or slowfall observer), I can give my trigonometric solution and compare with the approach suggested by pervect.

 

[Jorrie]

Hi Chris, you wrote:

... after sufficient time has passed to allow dpyikes, Jorrie and maybe MeJennifer to try their hand at computing $\omega_{{\rm received}}/\omega_{{\rm emitted}}$ (for a distant static observer respectively sending and receiving signals from a Lemaitre or slowfall observer), I can give my trigonometric solution and compare with the approach suggested by pervect.

I have tried my hand for a Lemaitre observer. Despite expecting hand slaps for not using Painleve coordinates, I worked in a good old Schwarzschild chart and hence had to stay outside of the event horizon. Here is how I understand it:

The velocity of the Lemaitre observer at Schwarzschild radial coordinate r relative to a locally stationary inertial observer equals the negative of the radial escape velocity, Ve = sqrt(2M/r) in geometric units. If we ignore spacetime curvature for the moment, both the Lemaitre observer and the distant static observer would have measured a wavelength redshift of the other's transmitter by a factor: sqrt((1+Ve)/(1-Ve)).

In curved spacetime, the distant static observer will receive the Lemaitre observer's transmitted wavelength as redshifted by a factor:
sqrt((1+Ve)/(1-Ve))/sqrt(1-2M/r) = 1/(1-Ve), for 2 < r/M < infty, basically dividing by the gravitational redshift factor. The observed redshift diverges at the event horizon, since r/M -> 2.

Likewise, The Lemaitre observer will receive the transmited wavelength of the distant static observer as redshifted by a factor:
sqrt((1+Ve)/(1-Ve))*sqrt(1-2M/r) = 1+Ve, approaching the value 2 as Ve -> 1.

These results are basically 'Newtonian' due to cancellation of the 'relativistic factors'. Provided that this effort is reasonably correct, I will try the Painleve chart next, also treating the inside of the hole.

Jorrie

 

[pervect]

Hi Chris, you wrote:

The velocity of the Lemaitre observer at Schwarzschild radial coordinate r relative to a locally stationary inertial observer equals the negative of the radial escape velocity, Ve = sqrt(2M/r) in geometric units.

Yes, I've gotten that result in the past. Note that it corresponds to setting E=1 in the more general expression for the velocity (relative to a stationary observer) of a radially infalling observer

$$\frac{\sqrt{E^2 - (1 - \frac{2M}{r})}}{E}$$

and that E=1 corresponds to a LeMaitre observer (someone who's velocity is zero at infinity).

Likewise, The Lemaitre observer will receive the transmited wavelength of the distant static observer as redshifted by a factor:
sqrt((1+Ve)/(1-Ve))*sqrt(1-2M/r) = 1+Ve, approaching the value 2 as Ve -> 1.

I've gotten this result before too, and two is the correct answer for the redshift of a radially infallling light beam as seen be a LeMaitre observer at the event horizon.

 

[Jorrie]

Yes, I've gotten that result in the past...

I've gotten this result before too, and two is the correct answer for the redshift of a radially infallling light beam as seen be a LeMaitre observer at the event horizon.

Thanks Pervect, it's comforting, but - I wonder what Chris would say...

Jorrie

 

[Chris Hillman]

Well done!

Hi again, Jorrie,

I have tried my hand for a Lemaitre observer. Despite expecting hand slaps for not using Painleve coordinates, I worked in a good old Schwarzschild
chart and hence had to stay outside of the event horizon.

Right, you used the method suggested by pervect, which combines the usual gravitational redshift (for a distant static observer and a static observer at some finite Schwarzschild radius) with the usual Doppler shift (valid for two local frames at event E), which is perfectly valid in the exterior region. And good, you used the correct velocity $v = \sqrt{2m/r}$ (taking the positive sign since the Lemaitre observer is falling away from the distant static observer), and then as you say
$$\frac{ \lambda_{{\rm received}} }{ \lambda_{{\rm emitted}} } = \sqrt{1-2m/r} \, \frac{ \sqrt{1+\sqrt{2m/r}} }{ \sqrt{1-\sqrt{2m/r}} } = 1+\sqrt{2m/r}$$
and as you observed this approaches a wavelength ratio of two as the Lemaitre observer approaches the horizon. So if our Lemaitre observer has a rear pointing spectrograph, he knows he is near the event horizon when he sees a redshift in his rear view mirror of about two. But if he only performs experiments inside the cabin of his rocket ship, then he won't notice anything in particular as he passes the horizon!

Next, because the interior region belongs to an real analytic extension of the result you found, by a computation which is valid only the exterior region, and since the formula you found continues to make sense on $0< r < 2m$, we might expect that it is also valid in the interior. To confirm that I suggested using a second method to directly compute the frequency ratio, which is valid on both regions. (More on that below.)

Also, it is worthwhile finding the radius as a function of the proper time of the Lemaitre observer remaining until impact at $r=0$, and then plugging into the expression you found above, you should find a certain power law expression for the observed redshift as a function of the proper time remaining until impact. So if our Lemaitre observer keeps an eye on his rearpointing spectrograph, he can estimate his remaining lifetime (assuming he doesn't turn on his rocket engine).

Likewise, going the other way, you found the correct redshift for signals sent radially from the Lemaitre observer back up to the distant static observer. It is interesting to express this in the proper time of the latter observer. His proper time agrees with Schwarzschild coordinate time (by definition of the exterior Schwarzschild chart!), so you need only compute $r(t)$ for the Lemaitre observer. The coordinate speed is $dt/dr = -1/(1-2m/r)/\sqrt{2m/r}$, so integrating we find
$$t-t_0 = -\sqrt{\frac{2r^3}{9 m}} - \sqrt{8 m r} + 2 m \, \log \left( \frac{1+\sqrt{2m/r}}{1-\sqrt{2m/r}} \right)$$
This is hard to invert! But notice that the first term dominates when $r \gg 2m$, while the third term dominates when $r \approx 2m$. So you should be able to find two approximate expressions from which you can see that the redshift obeys a power law for most of the infall, but very near the horizon obeys an exponential law.

Yes, I've gotten that result in the past. Note that it corresponds to setting E=1 in the more general expression for the velocity (relative to a stationary observer) of a radially infalling observer
$$\frac{\sqrt{E^2 - (1 - \frac{2M}{r})}}{E}$$
and that E=1 corresponds to a LeMaitre observer (someone who's velocity is zero at infinity).

Exactly. This is a third method (valid only in the exterior region, where the standard analysis via effective potentials of the null geodesics is valid).

Before we move on to the second method, note that the first method can be applied to other radially infalling observers, such as the slowfall observers. In the exterior Schwarzschild chart, the slowfall observer has tangent vector
$$\frac{1-m/r}{1-2m/r} \, \partial_t - \frac{m}{r} \, \partial_r$$
That is, in the exterior he has a world line of form
$$t-t_0 = -\frac{r^2}{2 m} - r - 2m \log ( r-2m)$$
Using the second method, can you find the wavelength ratio for signals sent from this observer back up to the distant static observer? And vice versa!

Here, recall that the slowfall observers, by definition, accelerate radially outward with just the right magnitude of acceleration (as a function of position) which would suffice to maintain their position according to Newton's theory of gravitation, namely $m/r^2$. But in gtr, the energy of the gravitational field itself gravitates, so the gravitational attraction is a bit stronger (roughly speaking), so these observers slowly fall radially inwards. How does this agree with the result you found for signals sent from the distant static observer to a slowfall observer? Can you find the wavelength ratio measured by the slowfall observer as a function of his proper time, and (suitable approximationate expressions for) the wavelength ratio measured by the distant static observer is his proper time?

Earlier I remarked that for signals sent radially from a distant static observer down to a radially falling observer, we should expect the usual gravitational blue shift to oppose the Doppler red shift. We could have guessed which would win out in the case of Lemaitre observers from your answer to the question about slowfall observers. This should help to explain why we found a redshift for both ingoing and outgoing signals in the case of a Lemaitre observer and a distant static observer.

OK, on to method two:

The ingoing Eddington chart is defined so that the ingoing radial null geodesics appears as horizontal coordinate lines. We can ensure this by setting $du = dt + dr/(1-2m/r)$, which integrates to $u = t + r + 2m \, \log(r-2m)$. The line element in the new chart is
$$ds^2 = -(1-2m/r) \, du^2 + 2 \, du \, dr + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$$
$$-\infty < u < \infty, \; \; 0 < r < \infty, \; \; 0 < \theta < \pi, \; \; -\pi < \phi < \pi$$
This chart is well defined on the right exterior and future interior regions, the same places where congruences of infalling observers are defined. It is perfectly suited for analyzing wavelength ratios for pairs of observers in which a more distant observer is sending signals radially downward to a closer observer. Can you figure out how to draw the light cones in this chart? The frame fields for static, Lemaitre, and slowfall observers? (Hint: if you know how to transform a vector field into a new chart, that's all you need since the frame fields are made up of unit vector fields.) Can you draw a diagram from which, using similar triangles, you can confirm our guess above about the general expressions for signals sent from the distant static observer down to the Lemaitre or slowfall observers? (The article "Frame fields in general relativity" in the version listed at http://en.wikipedia.org/wiki/User:Hillman/Archive should help if any of this seems confusing.)

The outgoing Eddington chart is defined so that the outgoing null geodesics appear as straight lines, by an expression very similar to the above. This chart is perfectly suited for analyzing wavelength ratios for pairs of observers in which a closer observer is sending signals radially outward to a more distant observer. Can you use the second method with this chart to confirm the expressions we found earlier? Note: the Lemaitre and slowfall frame fields are defined on the right exterior region and future interior region, whereas the outgoing Eddington chart is defined on the past interior and right exterior region (referring to the usual block diagram exhibiting the global causal structure of the maximal analytic extension of the exterior Schwarzschild vacuum). So, our results here only make sense for the exterior region. Of course, that is just what we expect since infalling observers cannot send signals outside the horizon if they have fallen into the future interior region!

You may be familiar with the (past) interior Schwarzschild chart in which the line element becomes
$$ds^2 = \frac{-1}{2m/t-1} \, dt^2 + \left( 2m/t-1 \right) \, dz^2 + t^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$$
$$0 < t < 2m, \; \; -\infty < z < \infty, \; \; 0 < \theta < \pi, \; \; -\pi < \phi < \pi$$
Observers who maintain constant spatial coordinates in this chart are called Frolov observers and they are geodesic observers who never emerge into either exterior region, although the Frolov congruence is also defined in the future interior region (it has a "caustic" at "the center of the X" in the usual block diagram, i.e. some pairs of world lines of Frolov observers intersect at that two-sphere). If you have seen the fine embedding diagrams of various spatial hyperslices inside the Schwarzschild vacuum as discussed in MTW, Gravitation, then you might recognize the spatial hyperslices above as the expanding (shrinking) cylinders ${\mathbold R} \times S^2$; in the Kruskal-Szekeres chart they appear inside the past (future) interior regions as hyperbolic arcs nested between the horizon and the past (future) curvature singularities.

Can you find the Frolov frame (in the future interior region) in terms of the ingoing Eddington chart? Can you then extend your computations above to obtain the wavelength ratio observed by a Frolov observer in the future interior region, for signals sent radially inward by a distant static observer? Can you re-express your answer as a function of the proper time of our Frolov observer? You should a blue shift evolving into a red shift! (Where does the transition occur?) Can you find the world lines of the Frolov observers in terms of the ingoing Eddington and Painleve charts? If you plot the world lines Lemaitre, slowfall and Frolov observers in the ingoing Eddington chart, can you "see" why we obtain redshift or blueshift?

 

[pervect]

I'd like to make a few comments about conserved quantities.

An easy way to solve for the equations of radial motion is to take advantage of the existence of conserved quantities.

This general approach is not restricted to the exterior region - one simply has to use well behaved coordinates such as the Painleve coordinates currently being discussed. While the effective potential was limited to the exterior region, this is a consequence of using the Schwarzschild coordinates to derive the conserved quantities.

I've talked about this before, briefly, BTW
https://www.physicsforums.com/showthread.php?t=126307

Consider the Painleve metric

$$-{d{{T}}}^{2}+ \left( d{{r}}+\sqrt {{\frac {2M}{r}}}d{{T}} \right) ^{2}+{r}^{2} \left( {d{{\theta}}}^{2}+ \sin ^2 \theta d {{\phi}}}^{2} \right)$$

Question: there appears to be a discrepancy between this and CH's version in
https://www.physicsforums.com/showpost.php?p=1205700&postcount=48?

Continuing:

This metric is not a function of T. Therefore it has "time-translation symmetry" over T. Therefore $\xi^a$ = (1,0,0,0) is a time-like Killing vector field.

It is an important theorem that if $\xi^a$ is a Killing vector field, and that $\gamma$ is a geodesic with a tangent vector (4-velocity) $u^a$, that $\xi_a u^a$ is a constant everywhere on the curve $\gamma$. See for instance Wald, pg 442.

Considering the above expression, we find that $\xi_a = g_{ab} \xi^b$, and because the only non-zero component of $\xi^b$ is $\xi^0$ which has a value of 1, we can say that

$\xi_a = g_{a0}$

Thus
$$\xi_0 = -1 + \frac{2M}{r}$$
$$\xi_1 = \sqrt{\frac{2M}{r}}$$

and the other components are zero

This in turn yields the expression

$$\left(-1 + \frac{2M}{r} \right) \frac{dT}{d\tau} + \sqrt{\frac{2M}{r}} \frac{dr}{d\tau} = \mathrm{constant}$$

For a radially infalling observer, the existence of this conserved quantity, allows us to solve a number of problems, such as finding the 4-velocity of a radially infalling observer. Note that we get a second equation from the fact that the magnitude of a 4-velocity is always equal to one (or minus one, depending on the sign convention being used). Thus the magnitude of the conserved quantity above (which can be considered to be a conserved energy since it is due to a time translation symmetry) is all we need to find both components $dT/d\tau$ and $dr/d\tau$ of a radially infalling observer.

 

[Chris Hillman]

I don't see any disagreement!

Hi, pervect,

This general approach is not restricted to the exterior region - one simply has to use well behaved coordinates such as the Painleve coordinates currently being discussed. While the effective potential was limited to the exterior region, this is a consequence of using the Schwarzschild coordinates to derive the conserved quantities.

Agreed, if you can define conserved quantities in a chart valid on a larger region, you can use them on that larger region.

$$-{d{{T}}}^{2}+ \left( d{{r}}+\sqrt {{\frac {2M}{r}}}d{{T}} \right) ^{2}+{r}^{2} \left( {d{{\theta}}}^{2}+ \sin ^2 \theta d {{\phi}}}^{2} \right)$$
Question: there appears to be a discrepancy between this and CH's version in
https://www.physicsforums.com/showpost.php?p=1205700&postcount=48?

Well, the Lemaitre coframe field can be written in the Painleve chart
$$\sigma^0 = -dT, \; \; \sigma^1 = dr + \sqrt{2m/r} \, dT, \; \; \sigma^2 = r \, d\theta, \; \; \sigma^3 = r \, \sin(\theta) \, d\phi$$
Then
$$ds^2 = -\sigma^0 \otimes \sigma^0 + \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3$$
which gives
$$ds^2 = -(1-2m/r) \, dT^2 + 2 \, \sqrt{2m/r} \, dT \, dr + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)$$
Hmm... looks I inadvertently ommitted a two in one of the equations of one of my posts in the thread you cited... is that the "discrepancy" you meant?

If anyone is worried that we might be writing the metric incorrectly in the Painleve chart, see Frolov and Novikov, Black Hole Physics, or http://www.arxiv.org/abs/gr-qc/0001069, where the Painleve chart is discussed extensively.

This metric is not a function of T. Therforre it has "time-translation symmetry" over T. Therfore $\xi^a$ = (1,0,0,0) is a time-like Killing vector field.

That's really awful notation (unfortunately very common) which is used only in physics; the entire rest of the mathematical world would write this (timelike) Killing vector as $\vec{\xi} = \partial_T$. See for example Stephani et al., Exact Solutions of Einstein's Field Equations, Cambridge University Press, 2nd edition, 2001, which has some fine introductory chapters surveying basic mathematical techniques useful for gtr.

For a radially infalling observer, the existence of this conserved quantity, allows us to solve a number of problems, such as finding the 4-velocity of a radially infalling obsever.

With respect to... what? My point was the static frame is only defined in the exterior.

Thus the magnitude of the conserved quantity above (which can be considered to be a conserved energy since it is due to a time translation symmetry) is all we need to find both components $dT/d\tau$ and $dr/d\tau$ of a radially infalling observer.

Just to be clear (I hope) to all: here, pervect is referring to a quantity which is invariant along (in particular) the proper time parametrized world line of a Lemaitre observer in the Painleve chart. As a parameterized curve, this contains more information than giving $T(r)$, say. Also, this is not the same thing at all as computing the physical velocity of this observer at some event wrt to another observer whose world line passes through that event. The latter amounts to expressing one frame field (anholonomic basis for the tangent space at each event on some world line) in terms of another.

 

[pervect]

Hi, pervect,

Hmm... looks I inadvertently ommitted a two in one of the equations of one of my posts in the thread you cited... is that the "discrepancy" you meant?

Yes -exactly.

If anyone is worried that we might be writing the metric incorrectly in the Painleve chart, see Frolov and Novikov, Black Hole Physics, or http://www.arxiv.org/abs/gr-qc/0001069, where the Painleve chart is discussed extensively.

I was worried about that very thing - when I had worked on this (quite a while ago), I had scrounged my metric from a homework problem someone had posted. When I saw that you had a different expression for the metric, I wasn't sure whether you had made a simple typo, or I'd somehow scrounged the wrong metric.

With respect to... what? My point was the static frame is only defined in the exterior.

Not really with respect to anyone in particular - as you point out, there isn't any static observer in the interior. However, the 4-velocity still exists as a mathematical entity.

The main use is that knowing the 4-velocity, i.e. dT/dtau and dr/dtau allows one to integrate to find T(tau) and r(tau), i.e. to solve for the geodesic curve corresponding to some particular energy. The LeMaitre observer will be a particular case having a specific value for the energy parameter. I think I did more on this topic in the other thread I mentioned.

 

[Chris Hillman]

The perils of transcribing from notes in a different format

Yes -exactly.

OK--- I apologize to all for my unfortunate transcription error (from my notes on computations related to the Painleve chart), which caused the confusion. Alas, it is all too easy to make an goof of this kind...

 

[Chris Hillman]

A small but important correction

Unfortunately, I can't edit the original post, but let me note a correct here. Thanks to pervect for noticing a typo in the following:

In the Painleve chart (1921), the metric tensor can be written
$$ds^2 = -dT^2 + \left( dr + \sqrt{m/r} \, dT \right)^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)$$
$$= -(1-2m/r) \, dT^2 + 2 \, \sqrt{2m/r} \, dT \, dr + dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),$$
$$-\infty < T < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi$$

If you multiply out the first equation, it doesn't match the second, because I accidently omitted a factor of two, writing $\sqrt{m/r}$ when I meant to write $\sqrt{2m/r}$. Sorry for any confusion!

Incidently, I just noticed a new paper http://www.arxiv.org/abs/gr-qc/0701061 coauthored by J. B. Griffiths, author of an important monograph on CPW solutions, which mentions the Ferrari-Ibanez CPW solution. I can't resist mentioning that in one of my questions to Jorrie above (where does the transition occur?), I was actually looking ahead to the time when, perhaps, we can discuss this important exact solution!

I suppose I really should add that this solution models two linearly polarized gravitational plane waves, having aligned polarizations and with specially chosen "amplitude profiles", which collide head on; this CPW model has the remarkable property that the "nonlinear interaction zone" which models the "aftermath" of the collision is locally isometric to one of the regions in the future interior of the Schwarzschild solution! Isn't that nifty?

The C-vacuum is related to one of those Weyl vacuums I just mentioned in another thread as a counterexample to the idea that gtr might not be quite as tricky as I sometimes claim!

 

[Jorrie]

Thanks, but Wow!

Hi again Chris.

Right, you used the method suggested by pervect, which combines the usual gravitational redshift (for a distant static observer and a static observer at some finite Schwarzschild radius) with the usual Doppler shift (valid for two local frames at event E), which is perfectly valid in the exterior region. ...

Thanks, I'm glad to be on the right track and appreciate your very detailed reply. But wow, it is a BIG mouthful for me.

As engineer, I was very chaffed when I got to grips with the Schwarzschild metric and chart. I more or less stopped there and only lately started to look a bit deeper (to the 'inside', pun intended). The guidance from yourself and Pervect is invaluable - I will surely work on the problems that you posted!

Regards, Jorrie

 

[George Jones]

Yes, I've gotten that result in the past. Note that it corresponds to setting E=1 in the more general expression for the velocity (relative to a stationary observer) of a radially infalling observer

$$\frac{\sqrt{E^2 - (1 - \frac{2M}{r})}}{E}$$

and that E=1 corresponds to a LeMaitre observer (someone who's velocity is zero at infinity).

Here's another way to look at it.

Suppose the radially infalling observer starts from rest (with respect to a static observer at $r = r_0 > 2M$) at $r = r_0$. Then, because the above speed is zero at $r_0$,

$$E = \sqrt{1- \frac{2M}{r_0}},$$

so the above speed becomes

$$\sqrt{1 - \frac{1- \frac{2M}{r}}{1- \frac{2M}{r_0}}}.$$