# Black hole time dilation

Would the universe end if you entered a black hole? What I mean by this is that due to time dilation would time elapse so fast for the universe outside the black hole relative to you inside it that all the stars would burn out and all that wouild be left would be other black holes?

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phinds
Gold Member
2019 Award
Would the universe end if you entered a black hole? What I mean by this is that due to time dilation would time elapse so fast for the universe outside the black hole relative to you inside it that all the stars would burn out and all that wouild be left would be other black holes?
I do understand you question, and am not sure of the answer, but if you entered a BH, the universe would end for YOU, that's for sure

I do understand you question, and am not sure of the answer, but if you entered a BH, the universe would end for YOU, that's for sure
Haha ok so assuming I can survive entering the black hole, is it possible that time would elapse so slowly there relative to the universe outside the black hole (apart from other similarly massive black holes) that the universe would essentially "run its course" during the time you spend in the black hole.

Edit: Oops misread your post I thought you said you didnt understand the question

phinds
Gold Member
2019 Award
I remember seeing other threads covering this so you might try a forum search.

The problems is not surviving since for a supermassive BH, there is nothing to survive out near the EH. Spaghetification doesn't occur w/ a SMBH until much closer to the singularity. You would not even notice crossing the EH. The problem is that you can't hover once inside, you HAVE to proceed to the singularity whether you like it or not. This does not, however, really answer your question, so keep looking.

jimmycricket
Chalnoth
Haha ok so assuming I can survive entering the black hole, is it possible that time would elapse so slowly there relative to the universe outside the black hole (apart from other similarly massive black holes) that the universe would essentially "run its course" during the time you spend in the black hole.

Edit: Oops misread your post I thought you said you didnt understand the question
Unfortunately, once you've entered the black hole, you are drawn towards the center in a short, finite time (from your perspective). Even if you were (somehow) able to survive the trip towards the center, that trip wouldn't take enough time for you to observe much that goes on outside the black hole.

jimmycricket
timmdeeg
Gold Member
Would the universe end if you entered a black hole? What I mean by this is that due to time dilation would time elapse so fast for the universe outside the black hole relative to you inside it that all the stars would burn out and all that wouild be left would be other black holes?
You would see the things far away extremely blue shifted due to gravitational time dilation (stars burning out) in case you would be hovering outside the black hole close to its event horizon. But it makes a big difference if you enter the black hole in free fall. Then (regardless if you are still outside or already inside the black hole), falling away from the stars outside causes redshift which is more or less canceled by the gravitational blueshift. I'm not sure about the net effect however.

Chronos
Gold Member
The external universe viewed by an infalling observer inside the event horizon is redshifted by a factor approaching 2. It is not blueshifted because you are falling away from the external universe, and you can only view photons approaching you from behind. There is no trajectory that permits photons from other directions to reach your eyes.

OK, so our indestructable observer has just fallen through what is the horizon as far as external observers are concerned.
He can observe only red shifted light coming from behind him, and that,s all.
If he could make sense of what he see behind him what does he see? - everything vastly sped up, though still looking very red.?
Galaxies and solar systems forming then dissipating in seconds?
Is there any idea of how much time passes from his own point of view until arrival at the dreaded singularity?

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Orodruin
Staff Emeritus
Homework Helper
Gold Member
OK, so our indestructable observer has just fallen through what is the horizon as far as external observers are concerned.
He can observe red shifted light behind him, and that's all.
Is there any idea of how much time passes from his own point of view until arrival at the dreaded singularity?
This would depend on the initial conditions. There is a continuum of different coordinate velocities you can pass the horizon with. And there are of course also different black hole sizes.

Nugatory
Mentor
Is there any idea of how much time passes from his own point of view until arrival at the dreaded singularity?
The proper time he experiences from horizon to singularity, assuming free fall, is something on the order of 15 microseconds for a sun-sized blackhole.

It scales linearly with the mass of the black hole.

PAllen
2019 Award
To answer the unanswered question, red shift correlates with observed clock rates. This must always be true, because a light emission process is effectively an oscillator clock. So, for the initial condition Chronos described, you would visually see the external universe going at half speed as you approached the singularity.

timmdeeg
Gold Member
The external universe viewed by an infalling observer inside the event horizon is redshifted by a factor approaching 2. It is not blueshifted because you are falling away from the external universe, and you can only view photons approaching you from behind.
If just falling away was to be considered one should expect that the infalling observer would see the stars outside infinitely red shifted while crossing the horizon at the speed of light. On the other side the light of the stars gains energy entering the gravity well of the black hole. But I'm not sure if this reasoning is correct. Anyhow, how do you calculate the factor 2?

Chronos
Gold Member
You need to chose the proper coordinate system to calculate the shift in frequency between emitted vs received photons. In Schwarzschild coordinates the infalling observer views the external universe at z=2 at the event horizon. Unfortunately, you cannot validly extend Schwarzschild coordinates beyond the event horizon. You need something like Panleve coordinates to do that - re: http://arxiv.org/abs/gr-qc/0001069. I believe this is what Hamilton used in his presentation at http://casa.colorado.edu/~ajsh/schw.shtml

timmdeeg
George Jones
Staff Emeritus
Gold Member
An earlier post of mine:
Actually, if the astronaut started falling from a great distance, the astronaut will see light red-shifted, i.e,, the distant outside universe will appear to run slow for the astronaut.

Suppose that observer A hovers at a great distance from a black hole, and that observer B hovers very close to the event horizon. The light that B receives from A is tremendously blueshifted. Now suppose that observer C falls freely from a great distance. C whizzes by B with great speed, and, just past B, light sent from B to C is tremendously Doppler redshifted. What about light from A to C? The gravitation blueshift from A to B is less that the Doppler redshift from B to C. As C crosses the event horizon, C sees light from distant stars redshifted, not blueshifted.

If observer A, who hovers at great distance from the black hole, radially emits light of wavelength $\lambda$, then observer C, who falls from rest freely and radially from A, receives light that has wavelength

$$\lambda' = \lambda \left( 1+\sqrt{\frac{2M}{R}}\right).$$
The event horizon is at $R = 2M$, and the formula is valid for all $R$, i.e., for $0 < R < \infty$. In particular, it is valid outside, at, and inside the event horizon.

See posts 5 and 7 in

I have since done the calculations using Painleve-Gullstrand coordinates that are vaild even on the event horizon.

timmdeeg
timmdeeg
Gold Member
An earlier post of mine:
... The gravitation blueshift from A to B is less that the Doppler redshift from B to C. As C crosses the event horizon, C sees light from distant stars redshifted, not blueshifted.

If observer A, who hovers at great distance from the black hole, radially emits light of wavelength λ \lambda , then observer C, who falls from rest freely and radially from A, receives light that has wavelength

λ ′ =λ(1+2MR − − − − √ ).
\lambda' = \lambda \left( 1+\sqrt{\frac{2M}{R}}\right).
Great, that's what I have been looking for, thanks George Jones and also Chronos.
So, the factor 2 is clarified.

Unfortunately my intuition is still a bit behind.
A sees B (hovering) extremely redshifted and C (in free fall) even more redshifted, because in the latter case the redshift consists of a gravitational and a kinematic component additional.
Passing by, C sees B extremely redshifted (purely Doppler) but sees A at z = 2 according to the formula you have shown. Would it be correct to say that also this redshift (z = 2) consists of a kinematic (not purely Doppler, because Doppler depends on a well defined relative velocity which in curved space-time is possible only locally) and a gravitational shift. So, it seems that the kinematic effect prevails over the gravitational effect. But why?
Kindly correct my reasoning.

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PAllen
2019 Award
Timmdeeg, I'm not sure there is a satisfying 'why', especially for what is observed by a freefaller past the horizon. Note that gravitational time dilation is only well defined for stationary spacetimes, and SR Doppler is well defined only over a spacetime region that may be considered flat (either due to small size or distance from any massive bodies). You are asking to extend these concepts beyond their applicability, especially to the interior case (the spacetime is not static or stationary, and curvature becomes unbounded as the singularity is approached).

It won't help your intuition necessarily, but there a couple of universal approaches to red shift that don't involve gravitational redshift at all (which is strictly a non-essential, derived concept in GR, that is useful only for the special case of stationary spacetimes).

1) Establish the 4-momentum of the emitted light pulse (it will be a null vector, whose time component will be the energy if the vector is expressed in a local frame of the emitter). Parallel transport this 4-momentum along the null geodesic path (must specify path, because, of course, parallel transport is path dependent) to the receiver. Compute the dot product with the receiver world line's tangent vector (4-velocity) at the event of reception. The ratio of the energy specified in emitter local frame and this dot product will be the red/blue shift.

2) Equivalently (not obvious at all), you can parallel transport the emitter 4 velocity along the null geodesic to the reception event, express the transported 4-velocity in the local frame of the receiver world line at that event, and apply the pure SR Doppler formua (as if the emitter were local, with transported 4-velocity).

Both of these methods are universally applicable in any GR situation, no matter how complex. They explain why there is really only one fundamental phenomenon, not two. So called pure gravitational redshift for static observers in a static spacetime falls right out of either of these computational methods. They also cover any combination of non-static observers, in any spacetime, as noted.

bcrowell
Staff Emeritus
Gold Member
The external universe viewed by an infalling observer inside the event horizon is redshifted by a factor approaching 2. It is not blueshifted because you are falling away from the external universe, and you can only view photons approaching you from behind. There is no trajectory that permits photons from other directions to reach your eyes.
This doesn't sound right to me. There can't be a fixed, finite Doppler shift for a free-falling observer observing an external signal, because two different free-falling observers could have different states of motion.

bcrowell
Staff Emeritus
Gold Member
Would the universe end if you entered a black hole? What I mean by this is that due to time dilation would time elapse so fast for the universe outside the black hole relative to you inside it that all the stars would burn out and all that wouild be left would be other black holes?
I think this is one of those questions that is very easy if you understand how to use Penrose diagrams, very hard otherwise. The question here is basically whether or not an event inside the event horizon has timelike infinity i+ in its past light cone. The answer is no if you look at the Penrose diagram.

By the way, it's a common misconception that the fate of all matter in the universe is that it will ultimately end up in black holes. Not true. In the distant future, due to the acceleration of cosmological expansion, there will be many cases where you have a single subatomic particle that is all alone inside its own cosmological horizon.

PAllen
2019 Award
This doesn't sound right to me. There can't be a fixed, finite Doppler shift for a free-falling observer observing an external signal, because two different free-falling observers could have different states of motion.
I assumed what was meant was the classic free faller from infinity. This is unique - the limit of free falling at zero speed from a static observer, as the static observer approaches infinite distance from the BH.

timmdeeg
Gold Member
1) Establish the 4-momentum of the emitted light pulse (it will be a null vector, whose time component will be the energy if the vector is expressed in a local frame of the emitter). Parallel transport this 4-momentum along the null geodesic path (must specify path, because, of course, parallel transport is path dependent) to the receiver. Compute the dot product with the receiver world line's tangent vector (4-velocity) at the event of reception. The ratio of the energy specified in emitter local frame and this dot product will be the red/blue shift.

2) Equivalently (not obvious at all), you can parallel transport the emitter 4 velocity along the null geodesic to the reception event, express the transported 4-velocity in the local frame of the receiver world line at that event, and apply the pure SR Doppler formua (as if the emitter were local, with transported 4-velocity).

Both of these methods are universally applicable in any GR situation, no matter how complex. They explain why there is really only one fundamental phenomenon, not two. So called pure gravitational redshift for static observers in a static spacetime falls right out of either of these computational methods. They also cover any combination of non-static observers, in any spacetime, as noted.
Thanks, PAllen, for your explanation. Yes, I think it is important to know that there is one phenomenon only, the observed frequency-shift, which is invariant.

Frankly speaking, I was inspired by authors like Peacok and Chodorowski,

http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.4573v1.pdf
http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.3536v3.pdf

who think of the cosmological redshift as being combined of kinematic and gravitational shifts. In the expanding FRW-model co-moving galaxies fall away from each other. Since the same is true in the Schwarzschild case regarding objects in radial free fall, I started reasoning why not thinking of said decomposition as well. Just to get a better feeling or intuition of the situation that the observer at R = 2M sees the far away exterior redshifted at z = 2 only, though moving locally with c.
But, of course, it is questionable whether the decomposition of the redshift makes any sense, because there is no invariant definition of a gravitational and a kinematic shift, which in combination yield the measured redshift.

timmdeeg
Gold Member
Oh yes, very interesting papers, thanks.

PAllen
2019 Award
Two papers that may be of interest relative to this subject: http://arxiv.org/abs/1201.4250, Exchange of signals around the event horizon in Schwarzschild space-time, http://link.springer.com/article/10.1007/s10714-012-1495-4/fulltext.html, On the nature of cosmological redshift and spectral shift in Schwarzschild-like and other spacetimes.
For the second of these papers, the author tries to distinguish a classic result of Synge from a newer equivalent derivation by Narlikar, and seems to argue that Narlikar's result is not universally true. I find this to be a mis-representation.

The problem starts from the summary of Synge's result (I have his book; it is a favorite of mine). The author describes Synge's result as if there were two independent results of the parallel transport, a relative speed and recession speed. The authors do concede that both are derived locally (at receiver) from parallel transport of the emitter 4-velocity, so only one transported vector is involved. However, the key point is that after accounting for Synge's quite unusual notations, his resulting formula is nothing but the standard SR Doppler formula for arbitrary relative motion, whence you have gamma(total relative speed) expressing the time dilation contribution, and relative speed along the light path expressing the classical Doppler aspect. The key, that I just worked through myself, is that after adjusting for Synge's notations, his formulas are truly exactly the same as e.g. http://en.wikipedia.org/wiki/Relativistic_Doppler_effect#Motion_in_an_arbitrary_direction .

Thus, Synge's result, which the author admits is general, already establishes the universality of redshift in GR as 'pure Doppler' given acceptance of using source 4-velocity transported along the connecting null geodesic.

[Edit: To flesh this out a bit, Synge's equation (45) in Ch. III, section 7 is identical to equation (1) in the above wikipedia link once it is understood that in Synge's unusual notation:

1 + v2 = γ2

and

vR = γ (v cos θ sub o), with v on the RHS having the normal meaning as wikipedia.

The key is that Synge's spatial speeds have scaling by γ built into them by the particular geometric definitions he is using.]

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I'm not sure if anyone has answered the question yet as I dont have the background to even begin to understand some of the concepts presented here. I see a lot of people talking about what you would observe while inside the black hole. What I was trying to get at is not what one would observe but what actually happens to the universe outside of the black hole during the time that would elapse between crossing the event horizon and reaching the singularity.

PAllen
2019 Award
I'm not sure if anyone has answered the question yet as I dont have the background to even begin to understand some of the concepts presented here. I see a lot of people talking about what you would observe while inside the black hole. What I was trying to get at is not what one would observe but what actually happens to the universe outside of the black hole during the time that would elapse between crossing the event horizon and reaching the singularity.
Ah, but starting from SR (let alone GR) any such question has no unique meaning. It all depends on the simultaneity convention adopted (equivalently, the foliation into time (per infaller) X space, chosen by the infaller). Only what you observe is devoid of convention - GR makes specific predictions for what you observe, and predicted observations don't depend on coordinate choices. But what is happening 'now' at a distance, is a choice of convention in all cases in GR and SR.

For a given moment on the infaller's world line after horizon crossing but before reaching the singularity, the only constraint on what distant event is considered 'now' is that it can't be in either the past or the future light cone of that event. Thus, you can choose, for the now time of (e.g. earth) anything after the last observed time on earth, up until infinite future (since none of the exterior is in the future light cone of an interior event). Thus, if at some moment you see 3PM today timestamped on signal from earth, you can say 'now' there is anything from 3PM + epsilon, up until infinite future. Pick whatever you want, all choices in this range are valid.