Time Dilation Near a Black Hole: Accelerating Objects Approaching the Event Horizon

[Ivo Draschkow]

TL;DR Summary

If the object, at some point, were to start accelerating in the opposite direction and in this way reach the point where it is at rest relative to the black hole, what happens from the perspective of the distant observer as soon as it comes off the accelerator pedal?

First of all, I wish everyone a Happy New Year.

I am interested in your expertise on a special constellation, which I will first briefly describe.

If you observe an object that is approaching the event horizon of a black hole, it is said that at some point the distant observer will have the impression that it is slowing down - even though it is actually speeding up. The reason for this is the increasing time dilation - everything sounds plausible so far.

However, since the whole thing seems like a braking process, the following question arises regarding an idealized, non-rotating black hole:

If the object, at some point, were to start accelerating in the opposite direction and in this way reach the point where it is at rest relative to the black hole, what happens from the perspective of the distant observer as soon as it comes off the accelerator pedal? It can't get any slower, so how does the time dilation for the observed movement process manifest itself now? The object would accelerate again towards the event horizon and the external observer would also have to see this acceleration. At the same time, that would be paradoxical, because everything else that moves towards the event horizon is perceived in a decelerating form.

Or would the object, from the observer's perspective, simply remain at the appropriate distance from the event horizon at which it canceled out the acceleration effect of the black hole based on its own acceleration? That would also be paradoxical, since from its own perspective it would accelerate towards the event horizon again... The external observer would no longer receive this information - which would actually be akin to the event horizon having shifted to the point where the object came to a standstill.

I am very curious about your answers.

 

[Ibix]

In this context, it makes most sense to think of the object falling into the hole in finite time by its clock, and the light that comes from it being increasingly delayed in reaching the distant observer.

In that way of thinking it's fairly obvious what will be seen from a distance. The radial coordinate (in the natural-for-a-distant-observer Schwarzschild coordinate system) is always decreasing. The observed rate is initially zero when the object is hovering and tends to zero again as it approaches the horizon, so the observed rate must rise and then fall. It's possible to figure out numerical details if you want to do the work, although you probably need a computer because the equations will have to be solved numerically.

 

[Ivo Draschkow]

"so the observed rate must rise and then fall"

Thank you - but that is a very mathematical description. What does this mean for the observer, what is he going to see? Is he going to observe the object approaching the event horizon, or not?

 

[Ibix]

"so the observed rate must rise and then fall"

Thank you - but that is a very mathematical description. What does this mean for the observer, what is he going to see? Is he going to observe the object approaching the event horizon, or not?

Yes, it'll be seen to approach the event horizon, initially increasing speed but then slowing down again. Note that this is just distant observation - by its own clocks the infalling object drops into the hole in a short time.

Also note that, along with light taking longer and longer to get to the distant observer, it will also become dimmer and dimmer. In practice the object would simply fade into darkness very quickly.

 

[Ivo Draschkow]

Thank you and yes, i mean the POV of a distant observer.
So how is it possible that an observable acceleration initially occurs? This is difficult to imagine as it should be tending to zero the moment it occurs - shouldn't there be only infinitesimal progress after the hovering?

 

[Ibix]

So how is it possible that an observable acceleration initially occurs?

Because the slowing effect is due to the change in altitude. It's not radically different from the Doppler effect you get from a car accelerating away from you - each successive cycle of sound wave (light wave in the black hole case) takes longer to reach you because it has further to go. In the black hole case you have the addition of curved spacetime making "further" increase more rapidly than in flat spacetime, but that's all. The slowing effect wins at the event horizon and you never see the object cross, but you can see it increase or decrease observed fall rate depending on what it's doing.

 

[Ivo Draschkow]

To express myself more clearly:
The hover should not be at a large distance from the BH, but rather where falling objects are already observed to slow down.

So the object will still accelerate after the hovering, reach a certain speed in the eyes of the observer, and then slow down again?

 

[PeroK]

To express myself more clearly:
The hover should not be at a large distance from the BH, but rather where falling objects are already observed to slow down.

So the object will still accelerate after the hovering, reach a certain speed in the eyes of the observer, and then slow down again?

Physics isn't done by considering what an observer "sees". A black hole is described by a model of spacetime. We can consider an experiment where something falls close to the black hole event horizon and then escapes. The real physics is in that mathematical model/description. We may then consider what observations a distant observer would make of such an event. This is only one aspect of the physical scenario. What the distant observer sees is not the whole picture of the phenomenon.

Take two examples. The universe after the Big Bang and before the last scattering surface did not result in light waves that survived. There is no visual record of that time. The first events we can see are on the last scattering surface in the form of the CMBR. That does not mean that events prior to the CMBR emission did not take place. In fact, evidence of those events may be found in gravitational waves that have survived since that earlier time.

Likewise black hole mergers cannot be seen from Earth, but gravitational waves can be detected and used to corroborate the mathematical model of black hole mergers.

By focusing on what a distant observer sees, you are focusing unduly on only one aspect of the phenomena that take place in our universe.

 

[Ivo Draschkow]

Thanks for the insight and I am aware of what you say. You imply in your answer that I want to draw some important conclusions from the distant observer's visual data. That's not the case - this data is just one piece of the puzzle that interests me.

 

[Ibix]

but rather where falling objects are already observed to slow down.

That is not an absolute distance. An object dropped from rest at any radius will appear to a distant observer to initially accelerate then slow again, even if an object that was dropped from above it is already apparently slowing down. They are not on the same trajectory.

The radius at which the peak apparent velocity occurs depends on the launch height.

 

[PAllen]

In that way of thinking it's fairly obvious what will be seen from a distance. The radial coordinate (in the natural-for-a-distant-observer Schwarzschild coordinate system) is always decreasing. The observed rate is initially zero when the object is hovering and tends to zero again as it approaches the horizon, so the observed rate must rise and then fall. It's possible to figure out numerical details if you want to do the work, although you probably need a computer because the equations will have to be solved numerically.

Just fyi, the equations answering the following precise question:

What is redshift factor as a function of time for distant observer (observer at infinity), received from a test body dropped from a specified starting height (areal radius) from a BH?

can be computed in closed form. I have notes on this from long ago threads here that I could dig up if I were motivated. To me, this evolution in observed redshift is the only reasonable meaning to speed observed from a distance.

 

[Ibix]

Just fyi, the equations answering the following precise question:

What is redshift factor as a function of time for distant observer (observer at infinity), received from a test body dropped from a specified starting height (areal radius) from a BH?

can be computed in closed form.

D'oh - you're correct. You can get $t(r)$ for a radially in- or out-going timelike or null object in closed form, and that's all we need here. It's $r(t)$ that requires numerical work, but we don't care about that in this context.

 

[PAllen]

D'oh - you're correct. You can get $t(r)$ for a radially in- or out-going timelike or null object in closed form, and that's all we need here. It's $r(t)$ that requires numerical work, but we don't care about that in this context.

Also, I think the best definition of 'speed of infaller' as seen from a distance would be the speed corresponding to the ratio of redshift observed for the infaller to redshift for a colocated hovering body. We presume the latter is 'not moving', and the redshift ratio observed from afar is determined solely by the local relative speed between the infaller and the hoverer. This is the way relative colinear speed would be determined in any ordinary context.

The kicker is that while both redshifts factors go to infinity (or zero, depending on how you look at it), the ratio I believe does not. In fact, I think the speed inferred from the standard SR doppler formula applied to the observed ratio goes to c as the infaller approaches the horizon! The rate of increase in speed per this definition approaches zero, and the observability of the infaller lasts forever, but despite that, the relative redshift would show speed ever closer to c as time goes on for the distant observer.

 

[Orodruin]

Also, I think the best definition of 'speed of infaller' as seen from a distance would be the speed corresponding to the ratio of redshift observed for the infaller to redshift for a colocated hovering body. We presume the latter is 'not moving', and the redshift ratio observed from afar is determined solely by the local relative speed between the infaller and the hoverer. This is the way relative colinear speed would be determined in any ordinary context.

That definition has its own problems. For example, it will work well only for radial infall obseved from the same radial direction. Any deviation from that will spoil the idea.

The kicker is that while both redshifts factors go to infinity (or zero, depending on how you look at it), the ratio I believe does not. In fact, I think the speed inferred from the standard SR doppler formula applied to the observed ratio goes to c as the infaller approaches the horizon! The rate of increase in speed per this definition approaches zero, and the observability of the infaller lasts forever, but despite that, the relative redshift would show speed ever closer to c as time goes on for the distant observer.

If this speed tends to c, then the corresponding SR Doppler factor must tend to infinity as it is given by
$$
\sqrt{\frac{c+v}{c-v}}
$$

 

[PAllen]

That definition has its own problems. For example, it will work well only for radial infall obseved from the same radial direction. Any deviation from that will spoil the idea.

That’s why I said colinear.

If this speed tends to c, then the corresponding SR Doppler factor must tend to infinity as it is given by
$$
\sqrt{\frac{c+v}{c-v}}
$$

Yes, the redshift ratio would grow extremely slowly without bound

 

[Ivo Draschkow]

Thanks for all of your detailed answers. It seems to be all about the amount of redshift, which I misjudged completely.

If an observer with a telescope looks at the situation - will they actually see a slowing down of falling objects, regardless of the redshift?

 

[Ibix]

If an observer with a telescope looks at the situation - will they actually see a slowing down of falling objects, regardless of the redshift?

The redshift is the slowing down. If you ask a series of hovering observers how fast the falling object was going when it passed them each one will report that it was going faster than the previous one. But the time dilation or redshift (which ever way you want to look at it) increases as well, so your distant observer has to wait longer and longer to see it pass each one. At some point the "faster and faster" loses out to the "longer and longer" and you will see it slow down (but probably actually fade into blackness first).

Note that there are some complexities about what "faster" and "slower" even mean for a distant observer. I'm skating around them by talking about passing a sequence of evenly spaced hovering observers, but even that hides some complexity. That's kind of why there's a technical digression on what can be measured.

 

[PeroK]

Thanks for all of your detailed answers. It seems to be all about the amount of redshift, which I misjudged completely.

If an observer with a telescope looks at the situation - will they actually see a slowing down of falling objects, regardless of the redshift?

What does "slowing down" look like?

 

[PAllen]

One reason I bring up about observed redshifts is to address two different interpretations encountered (even among physicists) about near horizon behavior.

One is that the slowing you observe in some ideal imaging system is an optical effect in the same category as gravitational lensing. It is equivalent to observing a very slow motion picture of something occurring very fast (per information in the image). For example, if you see a motion picture of a bullet appearing to move 1 millimeter per second through a plywood board, and you see effects that would be expected for a bullet hitting the board at high speed, you conclude you are seeing a slow motion image. Not that the bullet is 'moving slow for you'. The comparison of redshifts between the infaller and colocated hovering observers tells you that the infaller is movng at near lightspeed relative to the hovering observers. So the reasonable interpretation is extreme 'optical slow motion'.

The alternative interpretation, which I would call just wrong, is that 'for external observers' the infaller is frozen near the horizon, essentially not moving at all. By looking at the compete information available, this view is not tenable. Instead, you are just getting extremely delayed slowed down images of near the horizon.

 

[PeterDonis]

I think the speed inferred from the standard SR doppler formula applied to the observed ratio goes to c as the infaller approaches the horizon!

That's correct. This fact is often misstated in pop science sources as the infaller "reaching the speed of light" as they reach the horizon, which of course they don't; their worldline remains timelike, and the fact that the horizon moves at $c$ relative to the infaller as the infaller crosses it is due to the horizon being lightlike.

 

[Ivo Draschkow]

What does "slowing down" look like?

I understand the idea behind your question. But if you get a random person to look through a telescope without all the knowledge you have, they will still describe something. A random person is likely to ignore the visual meaning of the redshift and focus only on the observed motion that they know from their non-relativistic environment

 

[Ivo Draschkow]

In order to understand your answers better myself, I would like to clarify my original question by presenting it as a process:

1. A fairly uneducated and long-lived observer who is at rest with respect to a distant Black Hole (=hovering), sees an object that seems to be accelerating (due to gravity) in the BHs direction. He uses a telescope which provides visual feedback on a fairly large screen.2. At some point, by purely measuring time and distance progress on his screen, he observes how the acceleration begins to decrease.

3. At some point he observes how the speed begins to decrease.

4. At some point he observes that the object no longer moves in the direction of the BH. Reason: it actively accelerates in the opposite direction and balances the BHs gravity. Of course, this information reaches the observer with a significant time delay, but that doesn't matter.

5. He sees a second object, which happens to be almost next to the first, which is also falling towards the BH and seems to be getting slower and slower.

6. What is he going to tell us about what he sees - again with a big time delay of course - after the first object takes off the gas? Let's assume that this happens exactly when the second object is at the same distance from the center of gravity as object 1 and both objects are very close to each other. Is he going to tell us, that the first speeds up and the second slows down?

 

[Ibix]

I'm afraid you can't get around the technical issues being raised here that easily. How do you measure distance progress on a screen? If you mean just to count pixels, the problem is that light paths are curved by gravity, so distance measured that way requires a degree of technical skill to unpick.

The question about two falling objects has a simple answer, though. The hovering one will start to fall, but the free-falling one is going faster and will always be (and appear to be) closer to the black hole. Remember that the "slowing" is in many senses like watching an increasingly slow-motion video - a faster object passes a slower one even if your video of the pass is slowed down.

Taking a step back, the fundamental problem is that curved spacetime takes a lot of our "obvious intuitive concepts" and throws them out on their ear. There is no clear definition of "the speed someone far from the hole measures for something close to the hole". That's not an issue with your question; it's just that such apparently simple concepts have multiple possible meanings when spacetime is strongly curved. That's why I've been careful to talk about passing hovering markers or rate of change of radial coordinate, since that probably corresponds to the concept used by whoever told you "things slow down as they approach the black hole". @PAllen is taking a different (and perfectly reasonable approach) and saying that nothing is slowing down at all.

So the problem is not really one you can fix by revising the question. You can certainly tighten up the question to the point that there is a unique answer, but in doing so you will have to make several more or less arbitrary choices about things like exactly what you mean by "measuring distance". (Radar or angular sizes? Measured from behind the infallers or off to the side?) And you could make different choices and get different answers.

So you will always see the second infalling object pass the first. You will aways see the first start to pick up speed (however you measure it) but it will never catch the second. Whether this is because the first begins to slow down too or because neither slows down will depend on how you choose to measure speed.

 

[PeroK]

I understand the idea behind your question. But if you get a random person to look through a telescope without all the knowledge you have, they will still describe something. A random person is likely to ignore the visual meaning of the redshift and focus only on the observed motion that they know from their non-relativistic environment

How do you observe the distance travelled? There are no signs in space.

If something is moving away from you, it's difficult to calculate how fast it's moving. Black hole or no black hole.

 

[PAllen]

I understand the idea behind your question. But if you get a random person to look through a telescope without all the knowledge you have, they will still describe something. A random person is likely to ignore the visual meaning of the redshift and focus only on the observed motion that they know from their non-relativistic environment

What about seeing the free faller hit a hovering (stationary with respect to distant stars) board. The behavior of the interaction seen in the image is characteristic, e.g. of baseball hitting a board at .9c. If you see this happening in slow motion, do you say the laws of physics changed or that you are simply seeing a slow motion image of a high relative speed collision?

 

[Ivo Draschkow]

...You will aways see the first start to pick up speed (however you measure it) but it will never catch the second....

This is exactly where my imagination fails. Given two objects that both slow down, it is clear that the slower one will always lag behind the faster one under the circumstances described. But why an object whose observed speed tends to zero will visually continue to slow down, whereas an object that was at rest would visually accelerate, still eludes me.

How should the curved space-time suddenly produce a visual feedback (no matter how complex that should be) to the distant observer that, albeit slowed down, corresponds to what the object itself is experiencing?
Maybe I just don't get it.

I understand that there are many observer perspectives, all of which may be different. I understand the analogy with the slow motion images. Nevertheless, the situation presented is not an illusion, because the passage of time actually changes.

 

[PeroK]

This is exactly where my imagination fails. Given two objects that both slow down, it is clear that the slower one will always lag behind the faster one under the circumstances described. But why an object whose observed speed tends to zero will visually continue to slow down, whereas an object that was at rest would accelerate, still eludes me. Maybe I just don't get it.

I understand that there are many observer perspectives, all of which may be different. I understand the analogy with the slow motion images. Nevertheless, the situation presented is not an illusion, because the passage of time actually changes.

Even to define speed in flat spacetime you need mathematics. It's impossible to communicate what is in your mind regarding objects moving and accelerating in curved spacetime without mathematics.

 

[martinbn]

This is exactly where my imagination fails. Given two objects that both slow down, it is clear that the slower one will always lag behind the faster one under the circumstances described. But why an object whose observed speed tends to zero will visually continue to slow down, whereas an object that was at rest would visually accelerate, still eludes me.

What exactly eludes you? The object is at rest and then it starts moving. How can it do that without accelerating first!

 

[Ivo Draschkow]

That's exactly what's paradoxical about it for me. It must be as you say - but I don't understand the mechanism of how, on the one hand, almost everything is curved in such a way that it appears to be slowing down, while only one special case is shown to be accelerating through the same curvature mechanism.

 

[martinbn]

That's exactly what's paradoxical about it for me. It must be as you say - but I don't understand the mechanism of how, on the one hand, almost everything is curved in such a way that it appears to be slowing down, while only one special case is shown to be accelerating through the same curvature mechanism.

If you rely only on intuition from Euclidean geometry and classical physics, you are not going to get far on this. You need to put in a lot of effort and learn som geometry and general relativity.

 

[Ivo Draschkow]

"You need to put in a lot of effort and learn som geometry and general relativity"

You're right, that's why. When I asked the question, I was hoping that maybe there was an easy way to think of it. Now I know that's not the case.

 

[martinbn]

"You need to put in a lot of effort and learn som geometry and general relativity"

You're right, that's why. When I asked the question, I was hoping that maybe there was an easy way to think of it. Now I know that's not the case.

This is the easy way.

 

[Ibix]

almost everything is curved in such a way that it appears to be slowing down,

That's the bit that's wrong.

Things deep in a gravity well will appear to move slower than the same processes next to you. But they only get slower if the thing you're watching goes deeper into the well. Both the objects in your last scenario are going deeper into the gravity well, one much faster than the other. So the visual slowing effect is more pronounced in that one - but it is only more pronounced because it is deeper in the gravity well. The very reason it appears to be slowed more is because it is ahead of the other object. And the fact that it's only depth in the well that matters is why you must see the object start to fall in order for the increased sliwing to happen.

Remember that the slowing effect can be directly attributed to the way light travels in curved spacetime from the objects to your observer. Seen locally, your scenario is (qualitatively) the same as if you crouch holding a ball while I stand over you and drop a ball and you release yours when mine passes your hand. My ball is always ahead of yours. It's just that it takes longer and longer for the light to reach us. But we could never see your ball catch mine because it's the position that defines the amount of extra delay. If we saw the slower catch the faster then they'd have the same degree of slowing and it'd genuinely be paradoxical for the slower one to have caught up.

 

[Ivo Draschkow]

Thanks, I think based on your last explanation I'm starting to see the error in my thinking. I was too caught up in the infinitesimal. It also occurred to me that the object could, for example, not fully release the gas and would still ultimately accelerate towards the BH.

 

[PAllen]

A simple way to picture this is to imagine the scene filmed from just above the action, noting that (if near a supermassive BH horizon), this situation is just like being above the surface of a large planet (except proper acceleration to hover is enormously larger). Then, what you see far away is nothing more than what that film would show slowed down e.g. a million times, and red shifted a million times as well (so you visible light would be radio waves). Everything in the same vicinity is slowed and red shifted by the same proportion.