Do falling bodies ever slow down?

Question

I've had some discussions recently about black holes, and the issue of infalling bodies taking forever to reach the event horizon. That's essentially what Einstein said in his 1939 paper on a stationary system with spherical symmetry consisting of many gravitating masses. He said "it is easy to show that both light rays and material particles take an infinitely long time (measured in “coordinate time”) in order to reach the point r = μ/2 when originating from a point r > μ/2”.

Now, I'm a big fan of Einstein. But there seems to be two issues with this:

One is that Einstein concluded that black holes cannot form, but we have good evidence that there are black holes out there. The obvious example is Sagittarius A*. That's where's there’s something with a mass 4.28 million times the Sun, with a diameter of less than 44 million kilometres, and we can't see it. Surely it's just got to be a black hole.
The other issue is that falling bodies don't slow down. Imagine you drop a body at elevation A and it falls down to elevation B. The "force" of gravity relates to the first derivative of gravitational potential. Hence the bigger the difference in gravitational time dilation between elevations A and B, the faster the body falls past B. Then if you drop a body at elevation B it falls down to elevation C. Again the bigger the difference in gravitational time dilation between elevations B and C, the faster the body falls past C. In a typical gravitational field the force of gravity at B is greater than at A. Hence as the body descends, the acceleration increases as well as the falling speed.

Imagine a gedanken spaceship from which we've suspended a cable. We have clocks at different elevations, so we can measure the gravitational time dilation at each elevation. We can also release test bodies at each elevation and record the clock readings as they fall past other elevations:

At the end of the experiment we can reel in the cable and upload the recorded measurements to ascertain how our test bodies behaved. My understanding is that we will always find that time dilation always increases as we descend, that the falling body always accelerate downwards, and that both the acceleration and the falling speed always increases as the body descends. Is this correct? Or do falling bodies somehow stop accelerating? And do falling bodies ever slow down?

Since information cannot be retrieved once it has crossed the event horizon, I assume that the lowest data-collecting point (clock C) is outside the horizon. That being the case, and since the observer is local, is there a reason the motion of the test bodies would not be as expected (i.e. accelerating)? — Chappo Hasn't Forgotten, Feb 01 '19 at 06:17
@chappo : yes, the lowest collecting point is outside the event horizon. I don't think there's any reason why the motion of the test bodies wouldn't be as expected. — John Duffield, Feb 01 '19 at 11:17
@Rob Jeffries : I don't know. And as far as I do know, references frames have no objective existence, but all observers will agree that the lower the clock, the slower it's going. Following your comment here I'd be grateful if you'd answer this question. — John Duffield, Feb 01 '19 at 11:23
@Chappo if B and C are close enough to the black hole then the observer on the ship might see the falling object as passing C more slowly than it passes B. On the other hand if you ask the falling observer or compare the speeds measured by the observers at B and C, you would find that the observer is moving faster at C than B. There is no absolute notion of speed. — Steve Linton, Feb 01 '19 at 14:50
Would the downvoter and the "unclear what you're asking" closevoter care to elaborate on their perceived issue with this question? I think it's a cracker, one that gets right to the heart of a problem in contemporary physics. — John Duffield, Feb 01 '19 at 15:46
@JohnDuffield I'm afraid most people who have studied this are confident that relativity is pretty clear and consistent on these points and that there isn't a question, at least until you enter the parts where quantum mechanics starts to be significant, which this question doesn't. — Steve Linton, Feb 01 '19 at 17:37
@Steve Linton : Like Rob said, stationary observers at various elevations say the speed of the falling object is increasing. Meanwhile Bob riding the elephant crosses the event horizon in finite proper time, But distant observer Alice allegedly sees the elephant get closer and closer to the event horizon, slowing down due to GR time dilation. Even though the gradient in gravitational potential between A and B is why the object accelerates downwards, and the GR time dilation at some elevation represents the gravitational potential at that location. There is most definitely a problem. — John Duffield, Feb 01 '19 at 17:59
@SteveLinton while time dilation always slows the clock at a lower gravitational potential (it doesn't need to be a black hole for us to measure this), by definition the object must have accelerated from B to C. Since acceleration is an increase in velocity (distance over time), how much does the slower time affect the observed speed? Wouldn't the increased velocity (from acceleration) greatly exceed the effect of the time dilation? — Chappo Hasn't Forgotten, Feb 01 '19 at 23:33
@chappo most of the time yes, but very close to a black hole it turns out no — Steve Linton, Feb 01 '19 at 23:34

ProfRob · Answer 1 · 2019-02-01T13:59:26.223

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According to the standard interpretation of General Relativity (e.g. as presented in "Exploring black holes" by Taylor & Wheeler chapter 3, or "Black holes, white dwarfs & neutron stars" by Shapiro & Teukolsky, pp 343-345) then yes they do. But it depends on the frame of reference of the observer - there is no absolute answer.

According to an observer far from the black hole, the rate of change of radial coordinate with time (for an object that started falling radially inwards far from a non-rotating black hole) is given by $$\frac{dr}{dt} = -\left(1 - \frac{r_s}{r}\right)\left(\frac{r_s}{r}\right)^{1/2}$$ where $r_s$ is the Schwarzschild radius and $r$ and $t$ are Schwarzschild coordinates.

If we call this the infalling speed as measured by a distant observer then we can see by differentiation that it goes through a maximum at $r=3r_s$ and that $dr/dt \rightarrow 0$ as $r \rightarrow r_s$.

However, an observer accompanying the falling particle would totally disagree. To them, their velocity is given by $dr/d\tau$, the rate of change of $r$ with respect to the time on their clock. $$\frac{dr}{d\tau} = -\left( \frac{r_s}{r}\right)^{1/2}$$ which continues to increase up to and below the event horizon.

The latter appears to admit the possibility of faster than light travel, but no more so than me (correctly) saying that if you travel at close to the speed of light you can get to a star 10 light years away in much less than 10 years (as measured on your clock).

Finally we could have the point of view of stationary "shell" observers at fixed radial distances (outside the event horizon, because no stationary observer is possible below the event horizon). They would measure the speed of objects falling past them to be $$\frac{dr_{\rm shell}}{d\tau_{\rm shell}} = -\left(\frac{r_s}{r}\right)^{1/2}.$$ This means that the reports of stationary observers (which is the gist of your question I think) at increasingly lower heights, is indeed that the velocity of the falling object is increasing as it falls.

There is no paradox to these apparently contradictory points of view. Measurements of non-local events and phenomena are not bound to agree in General Relativity, where there isn't even agreement on what is meant by "now" or whose coordinate system in what frame of reference should be used in any particular circumstance.

edited Feb 01 '19 at 13:59

answered Feb 01 '19 at 12:43

ProfRob

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Thanks for the answer Rob, +1. Let's keep this simple by staying outside the event horizon. So, falling bodies increase their speed according to some observers, but slow down and stop after r=3r$_s$ according to others. According to Taylor & Wheeler. That sounds like a paradox to me. Surely something is badly wrong here? The infalling body does what it does, and what it does doesn't depend on some observer. I've got a great idea. Why don't you ask a question pointing this out, and then ask if anybody can resolve this paradox? – John Duffield Feb 01 '19 at 13:04
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Nothing's wrong, different observers always measure different speeds. Even Galileo understood that. All that is wrong is the now-dead concept that speed is a function of the object rather than a function of the coordinates used to analyze the object. – Ken G Feb 01 '19 at 13:56
Everyone would agree on actual events. For example if your observer C had a measuring rod and a clock, everyone might agree that the falling object passed the top end of the measuring rod when the clock said 0 and the bottom end when it said 1. But the spaceship. A, B and C would all have different ideas about how long it took the clock to go from 0 to 1 and consequently how fast the falling object was going. The falling observer might also disagree about the length of the measuring rod. – Steve Linton Feb 01 '19 at 14:05
If you hauled A, B and C up afterwards you would genuinely find that their clocks had recorded different amounts of total elapsed time, so this isn't just some kind of illusion. – Steve Linton Feb 01 '19 at 14:06
@Steve Linton : only they don't agree on actual events. See the elephant and the event horizon. – John Duffield Feb 01 '19 at 15:35
@Ken G : there's plenty wrong Ken. One observer says the infalling body slows down and stops, the other one says it falls through the event horizon. I hope Rob asks a question about the paradox, because I'm pretty sure the only way out of it is that both observers are wrong. PS: You do know I'm "the physics detective", don't you? Steve Linton, no problem re the hauled-up clocks. – John Duffield Feb 01 '19 at 15:40
@JohnDuffield I found a summary of the elephant and the event horizon -- the original is paywalled. Alice and Bob agree on events which take place outside the event horizon (except perhaps for quantum mechanical corrections which are a known unsolved problem) which seems to be what the question is about. If your clocks were dangling there, Alice & Bob would agree on what each of them said at the moment the elephant passed it. What happens at and within the event horizon Alice can never know, since no information about it will reach her (again excepting QM) so she and Bob can never meet to argu – Steve Linton Feb 01 '19 at 15:45
@Steve Linton: it's copyrighted, but here's a fair-use excerpt: “Let’s say Alice is watching a black hole from a safe distance, and she sees an elephant foolishly headed straight into gravity’s grip. As she continues to watch, she will see it get closer and closer to the event horizon, slowing down because of the time-stretching effects of gravity in general relativity… Little did Alice realise that her friend Bob was riding on the elephant’s back as it plunged toward the black hole. When Bob crosses the event horizon, though, he doesn’t even notice, thanks to relativity..." – John Duffield Feb 01 '19 at 15:56
@John Duffield No there's no paradox, any more than if I'm driving on the highway, and a car passes me, and I speed up to catch it, it will look like it slowed down and stopped. You are using absolute language in situations where only relative language is appropriate. It's nothing new to generate apparent paradoxes when you do that. You and I have encountered this same problem before. The definition of ineffective language is language that generates logical inconsistencies that don't exist. – Ken G Feb 02 '19 at 19:59
@Ken G : this is nothing like a car on a highway. This is falling bodies. They accelerate downwards wherever there's a gradient in gravitational potential. The latter is marked out by the time dilation on your string of clocks. Every observer agrees that the falling bodies are falling down, and that the time dilation increases as we descend. Alice knows it, Bob knows it, and Rob knows it. So it's rather more than ineffective language to claim that at some point, where gradient is steep, the descent slows down, then stops. Especially since it would appear that black holes do in fact exist. – John Duffield Feb 03 '19 at 10:18
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Not every observer agrees the time dilation increases, time dilation is a coordinate issue. The only, yes only, situation in which elapsed time is an invariant observable is when it is proper time, i.e., the time registered on a clock that is present at two events, and then the clock registers the proper time between those events along that path connecting them. That is the sole, and only, kind of time interval that is not a matter of coordinates. – Ken G Feb 04 '19 at 03:43
@ Ken G : the time dilation increases with depth. If it didn't, a gravitational field would not be present, and it is. Some observer might not notice it, but that doesn't make it go away. And in a gravitational field, observers fall down. They fall down faster and faster. They don't slow down and stop. Nor do they cross the event horizon at some blistering pace. Instead something else happens to them. Ask the question and I'll tell you what. – John Duffield Feb 04 '19 at 08:23

Do falling bodies ever slow down?

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