Does General Relativity Suggest a Younger Age for a Free-Falling Twin?

[mysearch]

I am in the process of trying to learn about some of the details that support general relativity. Therefore, my question is a genuine inquiry and not intended as oblique speculation. At the moment, I only have a basic understanding of general relativity in terms of all the apparent complexity of tensors and differential geometry; therefore I am initially focusing on the Schwarzschild metric, as it seems to encompass most of the basic principles. However, it also seems to lead towards a number of anomalies that I have not been able to fully resolve, which I was hoping a member of this forum might help me clarify. As a cross-referenced, I have also raised another thread regarding `Effective Potential` that is unresolved, at least, in my mind.

Anyway, back to the question at hand, starting with the Schwarzschild metric. I prefer not to use geometric units so that I have keep track of the `real` units, e.g. $$Rs=2GM/c^2$$ which reduces to [2M] in geometric units. Therefore, the following form includes the speed of light [c].

$$c^2 d\tau = c^2\left(1-Rs/r\right)dt^2-\left(1-Rs/r\right)^{-1}dr^2 - r^2d\theta-r^2sin^2\theta d\phi^2$$

By only considering free-fall radial paths and circular equatorial orbits this expression can be simplified to:

$$c^2 d\tau = c^2\left(1-Rs/r\right)dt^2-\left(1-Rs/r\right)^{-1}dr^2-r^2 d\phi^2$$

Many standard texts then proceed to solve this equation by dividing through by $$[dt^2]$$ or $$[d\tau^2]$$ to resolve the implications of general relativity from the perspective of either a distant observer [dt] or an onboard observer $$[d\tau]$$. In the context of a free-falling observer, an anomaly appears to arise because the Schwarzschild metric suggests that the distant observer will `see` time [dt] stop for the free-falling observer at the event horizon [Rs]. However, the same equation suggests that the perspective for the free-falling observer is very different. At this point, some texts highlight the difference between a physical singularity and a coordinate singularity and make reference to specific variations of the Schwarzschild metric, e.g. Gullstrand-Painleve or Eddington-Finkelstein, which are said to avoid the coordinate singularity associated with the standard metric when [r=Rs]. So my expectation was that the Gullstrand-Painlevé solution would resolve many of the apparent anomalies arising from the Schwarzschild metric. However, on review, I am not sure that this variant simply avoids the coordinate singularity rather than resolving it. This is not a statement of fact, simply my understanding so far.

To establish some physical comparison between these two frames of reference, i.e. distant observer $$[dt=A]$$ and free-falling observer $$[d\tau=B]$$, we might consider that our two observers (A) and (B) are twins. Overall, there seems to be little argument that the twin free-falling into a black hole is usually on a one-way trip, i.e. at some point there will be no escaping the physical singularity hidden behind the event horizon. However, what if we assume that the our free-falling twin does not initially plunge into the black hole, but only goes as close as possible before returning to (A).

Will the free-falling twin be younger?

As far as I can see, theory and consensus seems to suggest that twin (B) will indeed be younger than twin (A). As such, the time dilation that results from the increased gravity and velocity is a real effect and not just an aberration of the mathematics. However, this line of thought appears to lead to the conclusion that time will freeze at the event horizon. In contrast, solutions of both the Schwarzschild metric and the Gullstrand-Painleve variant for $$dr/d\tau$$ suggest that:

$$dr/d\tau = -c\sqrt{\left(\frac{Rs}{r}\right)}$$

It appears this equation can even be solved by integration:

$$d\tau = -\left(\frac{1}{c}\right)\int\left( \frac{Rs}{r}\right)^{-1/2} dr$$

$$d\tau = -\left(\frac{1}{c}\right)\left[ \frac {2/3Rs} {\left(Rs/r\right)^{1.5}}\right]^{0}_{Rs} = \frac{2}{3}*\frac{Rs}{c}$$

If the integration is right, it suggests that it would take about 22us to travel from the event horizon to the central singularity of a small black hole with a mass [M] of about 4 solar masses. On the other hand, this would increase to ~6hours for the black hole said to exist at the centre of our galaxy. However, all this said, I am still left with the nagging question:

How old is twin (A) when twin (B) hit the black hole?

As a side issue, it would seem that relativity is restricted to describing a black hole purely in terms of mass and gravity without really considering the quantum implications of the underlying nature of matter. The following links extend this discussion beyond the guidelines of this forum; therefore they are only referenced for interest and not discussion:

http://www.physorg.com/news101560368.html
http://arstechnica.com/news.ars/pos...-the-black-hole-information-loss-paradox.html
http://www.sciencenews.org/articles/20040925/bob9.asp

However, I would be interested in any other references, which might expand on this aspect of modern research.

 

[Ich]

How old is twin (A) when twin (B) hit the black hole?

How would you define "when"?

As such, the time dilation that results from the increased gravity and velocity is a real effect and not just an aberration of the mathematics.

You're going too far. There is no local property of spacetime at the horizon that would make it special, or which would make time dilation more "real" than not. You demand that B is able to return to A, which is a global requirement, not a local one. In the case of B touching the event horizon, you demand an impossibility which makes the maths blow up.

 

[mysearch]

How old is my twin brother? Response to Ich

Let me try to respond in the spirit of genuine inquiry as indicated in my original post. You ask upfront, how would I define `when`:

Twins (A) and (B) are initially collocated in local spacetime. It was suggested that twin (B) might initially fall towards the black hole, but before reaching any point of no return, he does return to (A). It was my understanding, rather than a declaration of absolute fact, that twin (B) would be younger than (A). I pretty much thought this was a given within the confines of relativity, but let me be sure on this point:

• If yes, this led to my statement about the effects of time dilation being real.

• If no, I would be interested in knowing the details of your argument.

To be honest I not sure I followed all the points you are trying to make in the last paragraph of your post so I will break my response as follows:

“You're going too far.”
I am assuming this point is covered above.

“There is no local property of spacetime at the horizon that would make it special, or which would make time dilation more "real" than not. “ I thought there was something very special about spacetime at the event horizon in that the nature of space and time reverses, which then prevents anybody reversing their spatial direction, i.e. it is analogous to trying to travel back in time. Again, this is not a statement, simply a question based on my current understanding. Again, the issue with time dilation was in the context of twin (B) returning to (A) and being younger, as outlined above.

“You demand that B is able to return to A, which is a global requirement, not a local one.” I was unaware that my question would come across as a `demand`. This was not the intention. It is my assumption that (B) can return to (A) provided he does not get too close to the event horizon. I am also assuming that twin (B) will age at a slower rate due to the real effects of time dilation, at least, on his journey towards the event horizon. I am also assuming that this is a requirement of relativity, irrespective of the inference being global or local.

“In the case of B touching the event horizon, you demand an impossibility which makes the maths blow up.” Again, I was not demanding anything, simply asking questions. However, you do touch on a point that underpins the question I raised. Both the Schwarzschild metric and the Gullstrand-Painleve variant suggest that twin (B) passes through the event horizon and will end up at the central singularity in a finite time. However, as far as I could see, there is no correlation of how this time corresponds to that of twin (A). Now this may be the very point you are raising about the scope of local and global frames of reference, but at a some level I was assuming that they still have to be relative. So let me modify my question:

At the end of the finite local time it takes twin (B) to reach the singularity, is there any meaningful concept of how much relative time passes for twin (A)? If no, why?

 

[Ich]

If yes, this led to my statement about the effects of time dilation being real.

Yes. My comment only related to a concept of time dilation as a local property of objects or spacetime. Time dialtion is always a relation (under such and such circumstances clock B will read x seconds less than clock A, when compared following such and such rules), but the Schwarzschild metric sometimes seduces one to think that spacetime simply slows clocks down according to position.

I thought there was something very special about spacetime at the event horizon in that the nature of space and time reverses, which then prevents anybody reversing their spatial direction, i.e. it is analogous to trying to travel back in time.

Reverses in comarison to what? To the outside world, if suitable coordinates are used. Spacetime itself is perfectly ok there, no way to decide from local measurements whether you're beyond the horizon or not.

I was unaware that my question would come across as a `demand`.

I'm not a native speaker, so I might have chosen the wrong words. What I meant was that you set up your Gedankenexperiment such that B returns to A. You demand it from the maths, not from me. You're right that B's clock will read less time then A's when they reunite.

However, as far as I could see, there is no correlation of how this time corresponds to that of twin (A). Now this may be the very point you are raising about the scope of local and global frames of reference, but at a some level I was assuming that they still have to be relative. So let me modify my question:

At the end of the finite local time it takes twin (B) to reach the singularity, is there any meaningful concept of how much relative time passes for twin (A)? If no, why?

Well, relativity usually strictly adheres to the facts. That is, if you can come up with a method to really compare those times, relativity will tell you the outcome. If not, the question may simply be not a valid one, relating more to inherited concepts than physics.
to make it short: I don't know of any meaningful concept of comparing those clocks. There seems to be a time when light signals from th outside can lo longer reach B, and there is a http://www.mathpages.com/rr/s6-04/6-04.htm" that one would calculate from the metric, but still, how do you compare the clocks?
The interior of the black hole is (at least in one way) separated from the outside world. Relativity is very serious about that, it makes everything arbitrary that you can't observe.

 

[mysearch]

Response to Ich:

Thank you very much for clarifying your position on a number of points raised. However, in doing so, you raised some additional points, which I would like discuss further. Again, I will snip extracts from your last posting by way of reference:

“the Schwarzschild metric sometimes seduces one to think that spacetime simply slows clocks down according to position.”

Yes, I agree this could be a mistake. For clarification, my assumption on examining the implications of the Schwarzschild metric is that it suggests that relative time really does slow down with respect to a relative position within a gravitational field and/or a relative velocity. However, you seem to be suggesting that local time cannot be compared to another frame of reference, e.g.

“Reverses in comparison to what? To the outside world, if suitable coordinates are used. Spacetime itself is perfectly ok there, no way to decide from local measurements whether you're beyond the horizon or not.”

“I don't know of any meaningful concept of comparing those clocks. There seems to be a time when light signals from the outside can no longer reach B, and there is a time that one would calculate from the metric, but still, how do you compare the clocks?”

Let’s split the discussion into two clear distinctions:

Outside the event horizon:
Within the accepted `rules` of relativistic physics, twin (B) can approach the event horizon and return to (A). Ignoring the additional issue of relativistic velocity for now, the age difference between twins (A) & (B) can be calculated as a function of their respective position within the gravitational field and hopefully verified by experiment, at least, in principle. Of course, if we accept this position, twin (B) does not actually have to come back to (A) for us to be able to determine the relative time separation between the twins. Of course, as twin (B) gets closer and closer to the event horizon, redshift will effectively make twin (B) disappear, at least, from the visible spectrum. However, I can’t see that this should undermine the mathematical principles that allow us to calculate the effects of time dilation. If so, I would have thought we could map the relative separation of these two local time frames?

Inside the event horizon:
I have to be honest at this point and say that I do not yet share the confidence forwarded by many-respected source about what really happens when crossing the event horizon. However, putting these concerns to one side, if twin (B) is still alive, the open questions that come to mind are:

Can the local time of twin (B) be compared to the local time of twin (A)?
Does time inside the black hole have any meaning?

Now what I find difficult to resolve is a `NO` answer to the first question, but a `YES` to the second. At one level, relativity in the form of Schwarzschild metric implies that time stops for twin (A) and while the Gullstrand-Painleve variant bypasses this coordinate singularity, it seems to do so by making no statement about the external time frame of twin (A). I believe you have raised this very point:

“The interior of the black hole is (at least in one way) separated from the outside world. Relativity is very serious about that, it makes everything arbitrary that you can't observe.”

Of course, not being able to see something does not normally prevent physics describing what will happen, but the implication of the time separation of twin (A), outside the horizon, and twin (B), inside the horizon, seems to be far more profound, at least in terms of relatuve time. However, I find it difficult to comprehend events taking place within the horizon that cannot even be described in terms of another frame of reference outside the horizon.

 

[Ich]

For clarification, my assumption on examining the implications of the Schwarzschild metric is that it suggests that relative time really does slow down with respect to a relative position within a gravitational field and/or a relative velocity.

But what is "relative time"? It is time, measured at a certain position, suitably compared to the time a clock far away from the BH reads.

However, you seem to be suggesting that local time cannot be compared to another frame of reference, e.g.

I say that there is no concept of "local time" in GR other than that it flows with 1 s/s - per definition. What you call "local time" is not a result of local physical laws, it ist the result of a comparison with a clock far away. And if "local time" seems to stop, well, that's nothing to do with local spacetime, it is the result of the comparison with a clock far away. Which becomes impossible at or inside the event horizon, that's why the mathematics explodes.

Within the accepted `rules` of relativistic physics, twin (B) can approach the event horizon and return to (A). Ignoring the additional issue of relativistic velocity for now, the age difference between twins (A) & (B) can be calculated as a function of their respective position within the gravitational field and hopefully verified by experiment, at least, in principle.

Right.

Of course, if we accept this position, twin (B) does not actually have to come back to (A) for us to be able to determine the relative time separation between the twins.

No, he doesn't actually have to return. But keep in mind the the "relative time separation" shows what one will measure if B would return. If he can't, even in principle, this number loses its meaning and announces to do so by becoming infinite at the horizon.

I have to be honest at this point and say that I do not yet share the confidence forwarded by many-respected source about what really happens when crossing the event horizon.

Why don't you try, and tell me what happened.

Can the local time of twin (B) be compared to the local time of twin (A)?

No.

Does time inside the black hole have any meaning?

Yes.
Explanation: The failure of a comparison does not mean that there is no time for the infalling observer. As I'v stated before, this uncommom behaviour is not due to a physical "change in local time flow" or whatever, it is the result of a comparison that becomes physically impossible.

Of course, not being able to see something does not normally prevent physics describing what will happen, but the implication of the time separation of twin (A), outside the horizon, and twin (B), inside the horizon, seems to be far more profound, at least in terms of relatuve time.

You hae the same thing in Quantum Mechanics: there are physical limits to your ability to see things, and these limits do have profound physical implications. If you ignore them and say that e.g. a particle must have both certain momentum and a certain position, you run into trouble. Because that's a part of your prejudices, not of reality.

However, I find it difficult to comprehend events taking place within the horizon that cannot even be described in terms of another frame of reference outside the horizon.

Well, I find it difficult, too. But, after all, Quantum Mechanics is still worse.

 

[mysearch]

Response3 to Ich:

Again, many thanks for taking the time to work through the issues being raised. It has been really useful to debate some of the physical implications that arise from the theory of relativity. While coming to some understanding of the maths is important, some physical interpretation is also required to infer meaning. Of course, it is equally important to challenge the assumptions on which any interpretation is being made. Therefore, I will continue to think about the questions you pose, as highlighted in italics below.

“what is "relative time"? It is time, measured at a certain position, suitably compared to the time a clock far away from the BH reads.”

True, but it is also tangible in the sense that I can reconcile the tick of the clock in both frames of reference. This appears problematic when one frame of reference is inside the event horizon:

“I say that there is no concept of "local time" in GR other than that it flows with 1 s/s - per definition. What you call "local time" is not a result of local physical laws, it is the result of a comparison with a clock far away. And if "local time" seems to stop, well, that's nothing to do with local spacetime, it is the result of the comparison with a clock far away. Which becomes impossible at or inside the event horizon, that's why the mathematics explodes.”

Is `local time` not a concept that relativity supports in that its basic postulates define the physical laws that cause time to be localised to a given observer? I agree that time always ticks at 1 sec per sec for all observers, but it is something quite different to say you cannot even correlate time in one frame of reference to another. Of course, as far as I can see, the anomaly of not being able to correlate the relative time between any two frames of `local time` only occurs when they exist inside and outside an event horizon. If this is the case, are we not discussing the exception to the rule for which the physics is still speculative?

"But keep in mind the "relative time separation" shows what one will measure if B would return. If he can't, even in principle, this number loses its meaning and announces to do so by becoming infinite at the horizon."

I understand the point you are making, but in a way this is the central issue of debate. You also challenged me to answer the next question:

“What really happens when crossing the event horizon.”

While I am not really qualify to answer this question, I will summarise some of the assumptions, as I understand them so far:

1. Theory suggests that the tick of the clock slows within a gravitational field. This idea is apparently support by GPS. As such, the contention is that time for twin (B) slows relative to time for twin (A).

2. As a consequence, theory implies that time for twin (B), as determined by twin (A), slows to standstill at the event horizon. This would seem to imply that any relative motion with respect to (A) would be restricted by this time dilation.

3. However, solutions of both the Schwarzschild metric and the Gullstrand-Painleve with respect to proper time $$[d\tau]$$, i.e. twin (B), suggest a finite journey time from the event horizon to the central singularity for twin (B), i.e.

$$dr/d\tau = -c\sqrt{\left(\frac{Rs}{r}\right)}$$

$$d\tau = -\left(\frac{1}{c}\right)\int\left( \frac{Rs}{r}\right)^{-1/2} dr$$

$$d\tau = -\left(\frac{1}{c}\right)\left[ \frac {2/3Rs}{\left(Rs/r\right)^{1.5}}\right]^{0}_{Rs} = \frac{2}{3}*\frac{Rs}{c}$$

4. What could also be highlighted is that free-fall velocity $$dr/d\tau$$ is supported by classical physics, which might suggest that this velocity is more physically meaningful than that measured at the distant twin (A)

$$1/2mv^2 = GMm/r$$

$$v = \sqrt {2GM/r}$$

But noting $$Rs=2GM/c^2$$ we arrive back at:

$$v = c \sqrt {\frac{Rs}{r}$$

5. It is difficult to reconcile the relative velocity of the free-falling twin (B), as determined by twin (A), which falls to zero at event horizon according to $$dr/dt$$. However, this is possibly an unnecessary complication in this summary.

So in many ways, there seems to be assumptions, which both support and contradict both perspectives. However, the time dilation effect does appear to be real and supported by some level of physical verification, i.e. GPS. So returning to your last comment:

“The failure of a comparison does not mean that there is no time for the infalling observer. As I've stated before, this uncommon behaviour is not due to a physical "change in local time flow" or whatever, it is the result of a comparison that becomes physically impossible.”

While I agree that time will appear consistent for (A) and (B), we return again to the central issue, i.e. the inability to correlate relative time with respect to (A) and (B). The suggestion appears to be that we simply move time on 1 sec per sec for twin (B) even though the implication is that this can only occur if an ever-increasing amount of time passes for twin (A). To be honest, I find squaring the circle in this case more than difficult, while I struggle to find any meaningful interpretation in the idea that there can be no correlation between the tick of the clock in (A) and (B).