# Time Dilation and Length Contraction near the Event Horizon of a BH

• Constantin

#### Constantin

A clock falling towards the event horizon of a black hole would appear slowed down to the point of being frozen in time (or almost).

But I'd like to understand properly what happens to the length contraction experienced by an observer falling together with that clock.
Would he experience length contraction ?
As length contraction reaches extreme levels, what would he be able to see? The whole Universe shrank to a miniscule size ?

I remember reading in another post that time dilation and length contraction always happen together. But I've only seen this explained for the case of relativistic speeds, and never for the case of a black hole.

Thanks

Are you asking what the free falling observer would experience or what a distant hovering observer would measure? If the free faller, is the black hole small or large? If the hoverer, how does he remotely measure the length of the free faller?

If the free faller, is the black hole small or large? If the hoverer, how does he remotely measure the length of the free faller?
I would be interested that too. I remember several times people have made statements to the effect that gravitation length contracts without showing even a thought experiment on how to even prove such statements.

I think all we can say is that the volumes between shells r1 and r2 in a Schwarzschild solution are larger than would have been expected if the spacetime was Euclidean. Stating that that is due to shrinking rulers is in my opinion sheer speculation.

[PLAIN]http://img850.imageshack.us/img850/6859/volumeratio.png [Broken]
(x in the horizontal axis is r)

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Are you asking what the free falling observer would experience or what a distant hovering observer would measure? If the free faller, is the black hole small or large? If the hoverer, how does he remotely measure the length of the free faller?

I'm interested in the length contraction experienced by the free falling observer.
The fact the black hole is large or small probably makes no difference, as I'm not interested in the tidal effects.

I'm interested in the length contraction experienced by the free falling observer.
The fact the black hole is large or small probably makes no difference, as I'm not interested in the tidal effects.
Then there is none. Time dilation and length contraction of an observer is never experienced by the observer. All that an observer would experience crossing an event horizon are whatever tidal forces might be there, so if you are not interested in those then there is nothing else.

I know the explanation for relativistic speed effect and there's both time dilation and length contraction involved.
An observer inside a ship with speed close to c can travel large distances in short amounts of time (time measured by him) due to the length contraction he experiences. So for example he can travel 5 light years in 1 year (his time).

I suspected gravity has similar effects.

Or do those effects only exist in the case of an observer with fixed position in the gravity well (for example an observer with fixed position near the event horizon), and not for a free falling observer ?
An easier to imagine example would be in the case of the Earth: an observer standing on the surface of the Earth, and a free falling observer heading towards the center of the Earth.

I know the explanation for relativistic speed effect and there's both time dilation and length contraction involved.
Not experienced by the inertially moving observer. Both time dilation and length contraction are things that are measured on other objects, not experienced by an observer. I.e. as I speed by I measure that your clock is dilated and your rods are contracted, but mine are fine according to me. Similarly, as I fall through an EH my clocks and rulers are undilated and uncontracted and behave completely normally according to me (neglecting tidal effects).

I understand he experiences length contraction only measured on other objects. So that's exactly what I'm interested in.

As length contraction reaches extreme levels, what would he be able to see? The whole Universe shrank to a miniscule size ? The BH beneath him shrank in a similar way ?

Are time dilation and length contraction experienced by both free falling observers and observers with fixed position in the gravity well ? Again I'm interested in the way they see the "rest of the Universe".

Length contraction is something that exists only to the stationary observer. To the falling observer, his time is normal time. It is difficult to say exactly what he would observe if he looked at the other observer, or out into space. It is possible, based on the fact that time would be passing much slower for him, that everything outside the event horizon would appear to be moving very fast, passing into the future at an alarming rate. It is also possible that looking forward into the black hole from the event horizon, the black hole would appear to be moving very slowly, perhaps at a complete standstill at its core. Without actual observation, it is difficult to calculate the small idiosyncrasies that may exist as an observer moves toward a black hole.

As length contraction reaches extreme levels, what would he be able to see? The whole Universe shrank to a miniscule size ? The BH beneath him shrank in a similar way ?
OK, so that kind of brings me back to a variation of my previous question. How does the free-faller measure lengths of distant objects? Unfortunately, there isn't a "standard" way to do that.

I have a proposal for an in principle experiment. Though having read various derivations of the idea of length contraction in a gravitational field from the point of view of distant observers, I've never actually seen this or any other proposed experiment analyzed.

Imagine two distant observers (to cover two cases) near each other, one exactly hovering in place, the other inertial (but having only at microscopic acceleration relative to the adjacent stationary observer due to the distance from the gravitating sphere; further posit zero initial relative motion).

They observe a series of locally 1 meter rulers placed end to end, proceeding radially away from the body but orthogonal to their line of sight to the body (i.e. extending off the side from the body). Will they see a change in angle subtended by the sequence of rulers? If so, that could certainly be considered radial length contraction from the point of view of distant observers.

That is a good experiment, but ideally you would want the same experiment to reproduce length contraction in flat spacetime. Unfortunately, the angle subtended will decrease even in flat spacetime, and the angle subtended will not necessarily reduce due to velocity.

That is a good experiment, but ideally you would want the same experiment to reproduce length contraction in flat spacetime. Unfortunately, the angle subtended will decrease even in flat spacetime, and the angle subtended will not necessarily reduce due to velocity.

Not quite sure what you mean by "the angle subtended will decrease even in flat spacetime". I like the idea of doing something equivalent for velocity based length contraction in flat spacetime. That is trickier, but here goes:

Imagine a very distant, super bright, light source flashing regularly. Imagine as series of locally 1 meter rulers going by, timed to arrive orthogonal to your line of sight to the light source at each flash. They go by at various speeds. I believe the complications of Terrell-Penrose rotations are inconsequential in this orientation (they are significant as an object approaches and recedes). Ok, now you can measure the angle subtended by rulers of various speeds. You would (in principle) readily measure length contraction this way.

Not quite sure what you mean by "the angle subtended will decrease even in flat spacetime".
The angle subtended by the n'th ruler in the series of rulers placed end to end is given by:
$$\theta=asin\left( \frac{n+1}{d} \right) - asin\left( \frac{n}{d} \right)$$
Where d is the distance between the observers (in units of whatever length the rulers are). This function is monotonically decreasing as a function of n.

I understand he experiences length contraction only measured on other objects. So that's exactly what I'm interested in.

As length contraction reaches extreme levels, what would he be able to see? The whole Universe shrank to a miniscule size ? The BH beneath him shrank in a similar way ?

Are time dilation and length contraction experienced by both free falling observers and observers with fixed position in the gravity well ? Again I'm interested in the way they see the "rest of the Universe".
For free falling observer you have number of effects. Let's say there is star directly behind BH and observer will see it as Einstein ring.
1. Increasing length contraction of universe along direction of motion will expand Einstein ring.
2. Increasing aberration due to acceleration will contract Einstein ring.
3. Increasing bending of light due to stronger gravity will expand Einstein ring.
So it's hard to say what will be summary effect. In flat spacetime acceleration creates apparent acceleration in backward direction if we can see only distant objects. Rather counterintuitive effect because of aberration.

If he is looking at some extended formation behind him then it should be like this:
1. Increasing length contraction of universe along direction of motion will expand formation.
2. Increasing aberration due to acceleration will expand formation.
3. Increasing bending of light due to stronger gravity will contract formation.
Again it's hard to say what will be summary effect but I guess it's apparent expansion.

The angle subtended by the n'th ruler in the series of rulers placed end to end is given by:
$$\theta=asin\left( \frac{n+1}{d} \right) - asin\left( \frac{n}{d} \right)$$
Where d is the distance between the observers (in units of whatever length the rulers are). This function is monotonically decreasing as a function of n.

Yes, but this goes in the opposite direction from presumed length contraction. Thus any deviation from this toward rulers closer the central body looking shorter than expected by this formula would be interpreted as length contraction.

zonde's answer is closest to what I wanted to hear.

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So far, the way I imagine the situation is:
The Universe, and any object in it, will look flattened like a pancake in the direction of gravitational attraction.

I know movement with relativistic speeds causes this apparent flattening in the direction of motion. I suspect gravitational fields cause a similar apparent flattening in the direction of attraction.

I still can't imagine properly the different effect on how they observe the Universe for a free falling observer compared to a stationary one.
A stationary observer would have to experience a force that counterbalances the gravitational attraction. As the escape velocity from nearby the event horizon is relativistic, that creates additional problems for imagining the situation.

Edit: As imagining the way these observers see the Universe might get complicated, we can just imagine the way they see a round object, like a star or planet. That star or planet would be a significant distance away from the BH, so that it's not affected by strong gravitational effects.

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1. Increasing length contraction of universe along direction of motion will expand Einstein ring. ...
I don't know how you come to this conclusion. What is the relationship between the size of the ring and length contraction? AFAIK the length contraction of the universe is not even defined.

zonde's answer is closest to what I wanted to hear.
It may be what you want to hear, but AFAIK it is speculative.

The Universe, and any object in it, will look flattened like a pancake in the direction of gravitational attraction.
Are you talking about a purely visual effect now? Perhaps similar to the way that PAllen was talking?

On the question of what you would see falling through and beyond the event horizon, here is description from an apparently reputable source:

On the question of what you would see falling through and beyond the event horizon, here is description from an apparently reputable source:

Hmm, at the event horizon is says this:

"Relative to an observer stationary in the Schwarzschild metric, our velocity has now reached the speed of light. Relative to an observer freely falling radially from rest at infinity, our velocity is (8/9)1/2 c = 0.94 c. "

I suppose I am not quite sure what Andrew Hamilton means here.

Hmm, at the event horizon is says this:

"Relative to an observer stationary in the Schwarzschild metric, our velocity has now reached the speed of light. Relative to an observer freely falling radially from rest at infinity, our velocity is (8/9)1/2 c = 0.94 c. "

I suppose I am not quite sure what Andrew Hamilton means here.

Me neither, actually. Seems like he needs to say more - falling from infinity to where? Seems implausible all free fallers from infinity (at whatever distance they are) see horizon crosser at the same speed less than c! (even though they see each other moving at various relative speeds). That would make this .94c behave like c!

I suspect either a simple oversight (more specification was intended) , or you were supposed to get to this page from some other page where a more specific definition of this observer was given. The 8/9 is intriguing. 9/8 SC radius is the limit from Buchdahl's theorem on smallest size you can possibly have a static spherically symmetric body. Maybe this is for an observer falling through the Buchdahl limit??

Hmm, at the event horizon is says this:

"Relative to an observer stationary in the Schwarzschild metric, our velocity has now reached the speed of light. Relative to an observer freely falling radially from rest at infinity, our velocity is (8/9)1/2 c = 0.94 c. "

I suppose I am not quite sure what Andrew Hamilton means here.

From

This is a plan view of our trajectory into the black hole. The free-fall trajectory is rather special, because it puts us (temporarily) into a circular orbit with no rocket power required (aside from the manoeuvering thrusters needed to take careful aim at the outset).

The circular orbit at 2 Schwarzschild radii is unstable, a type of circular orbit which exists in General Relativity, but not in Newtonian gravity. A tiny forward blast on the thrusters will send us back out to safety; a tiny reverse blast will send us into the black hole. The unstable orbit at 2 Schwarzschild radii is that orbit which corresponds to zero kinetic energy (zero velocity) at infinity.

I don't know how you come to this conclusion. What is the relationship between the size of the ring and length contraction?
If hovering observer right next to falling observer sees Einstein ring as having angular size $\alpha$ then for falling observer length contraction changes angular size of Einstein ring to $\alpha '$:
$$\alpha '=2 arctan(\gamma tan\frac{\alpha}{2})$$
where gamma is calculated using relative speed between hovering observer and falling observer.

AFAIK the length contraction of the universe is not even defined.
In SR you generally speak about length contraction not length expansion. But obviously we can say that moving observer undergoes length expansion in his own simultaneity "slice".
Like in this picture:

AB is contracted length of moving rod in simultaneity "slice" of "stationary" observer but AC is expanded length of moving rod in rod's simultaneity "slice".
Obviously length contraction of universe is just inverse of moving observer's length expansion.

It may be what you want to hear, but AFAIK it is speculative.
The only speculative part is that I use simultaneity of hovering observers as global simultaneity. But IMHO it's well motivated assumption as it is kind of null assumption i.e. it is assumption that spacetime (Eintein's aether) is stationary in respect to gravitating body in it's rest frame.

If hovering observer right next to falling observer sees Einstein ring as having angular size $\alpha$ then for falling observer length contraction changes angular size of Einstein ring to $\alpha '$:
$$\alpha '=2 arctan(\gamma tan\frac{\alpha}{2})$$
where gamma is calculated using relative speed between hovering observer and falling observer.
OK, that is at least clearly described (I assume this is for an observer falling from rest at infinity since other free-falling observers will have different velocities) and could be measured. However, I have never until this thread heard this quantity described as "length contraction of the universe". Do you have any mainstream source for that terminology?

It is fine if you want to use that measure, similar to what PAllen was suggesting, but you can not simply say "length contraction of the universe" and expect that people will understand that you are referring to this quantity, unless it is a mainstream term that I am simply ignorant about.

Well, as I may have said a few times before, the event horizon is a light-like (i.e. null) surface, so it's expected that the relative velocity of the event horizon relative to any timelike observer (this will include any infalling physical observer who has a rest mass) is 'c'.

The relative velocity between observer and horizon is most sensibly measured in the local Lorentz frame of the physical observer -- there isn't really any local Lorentz "frame" in which the event horizon is stationary, for the same reason there isn't any Lorentz "frame" in Special Relativity in which light is stationary.

One can consider the velocity relative to the horizon as a sort of limiting process, if you consider which observer is timelike and which is null, there's no real need for such a limiting process.

OK, that is at least clearly described (I assume this is for an observer falling from rest at infinity since other free-falling observers will have different velocities) and could be measured. However, I have never until this thread heard this quantity described as "length contraction of the universe". Do you have any mainstream source for that terminology?

It is fine if you want to use that measure, similar to what PAllen was suggesting, but you can not simply say "length contraction of the universe" and expect that people will understand that you are referring to this quantity, unless it is a mainstream term that I am simply ignorant about.
"Length contraction" is mainstream terminology.
Question in OP was about general observations of universe from perspective of falling observer as relativistic effects become extreme.

Anyways if we speak about relativistic optical effects then length contraction is not the dominant effect in most cases. Here is the link http://www.anu.edu.au/Physics/Searle/ that explains relativistic optical effects quite nicely.

Well, as I may have said a few times before, the event horizon is a light-like (i.e. null) surface, so it's expected that the relative velocity of the event horizon relative to any timelike observer (this will include any infalling physical observer who has a rest mass) is 'c'.

The relative velocity between observer and horizon is most sensibly measured in the local Lorentz frame of the physical observer -- there isn't really any local Lorentz "frame" in which the event horizon is stationary, for the same reason there isn't any Lorentz "frame" in Special Relativity in which light is stationary.

One can consider the velocity relative to the horizon as a sort of limiting process, if you consider which observer is timelike and which is null, there's no real need for such a limiting process.
If we take BH as an object viewed by outside observer then event horizon is stationary in respect to BH i.e. it is not expanding or contracting at 'c'.

I would say that you are making paradoxical statements that do not explain anything. They only add confusion.

"Length contraction" is mainstream terminology.
Obviously. But using "length contraction of the universe" to refer to the optical effect of the ring diameter is clearly not mainstream terminology.

Anyways if we speak about relativistic optical effects then length contraction is not the dominant effect in most cases.
So then why would you equate the two?

Obviously. But using "length contraction of the universe" to refer to the optical effect of the ring diameter is clearly not mainstream terminology.
Fine. I am open to your suggestions.
We have two observers side by side that are in relative motion in respect to each other. We have complete picture what is seen by one observer (let's exclude the part where observers see each other as it is not relevant in this case).
Now we want to know what sees the other observer.

What is mainstream terminology if one is describing such transformation?

So then why would you equate the two?
Hmm, where did you get this?

I said:
For free falling observer you have number of effects. Let's say there is star directly behind BH and observer will see it as Einstein ring.
1. Increasing length contraction of universe along direction of motion will expand Einstein ring.
2. Increasing aberration due to acceleration will contract Einstein ring.
3. Increasing bending of light due to stronger gravity will expand Einstein ring.
So it's hard to say what will be summary effect. In flat spacetime acceleration creates apparent acceleration in backward direction if we can see only distant objects. Rather counterintuitive effect because of aberration.

1. and 2. are local effects if we compare hovering observer and falling observer and these two effects completely determine change of geometry for observed "picture" between two observers (there is change in frequency and intensity of light as well).
3. is the same for both observers after factoring out 1. and 2.

Obviously I am not claiming that relativistic optical effects are limited to 1.

What is mainstream terminology if one is describing such transformation?
I don't think there is any. That is the problem. Gravitational time dilation is well defined for static spacetimes, but not gravitational length contraction. Both you and PAllen have suggested reasonable definitions, but since neither is standard it would be good for the OP to weigh in and describe what they are referring to in their question.

Hmm, where did you get this?
Sorry about misunderstanding. I was (overly) focusing on 1 simply because that was the one you mentioned in association with length contraction.

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I don't think there is any. That is the problem. Gravitational time dilation is well defined for static spacetimes, but not gravitational length contraction.
When two observers are next to each other we can speak about pure SR effects in order to compare what they see. SR holds locally and there is no need to speak about any GR effects.

When two observers are next to each other we can speak about pure SR effects in order to compare what they see. SR holds locally and there is no need to speak about any GR effects.
Right, but the OP is interested in GR effects.

Right, but the OP is interested in GR effects.
Well, right. But posts #6 and #8 seemed to say that it might be good to start with differences between what is seen by hovering observer and what is seen by falling observer.

At least we can assume that one of them is more "natural" observer and then we can find out what the other one is seeing. And yes, I was explaining it from perspective that more "natural" observer is hovering observer as we generally are that type of observers.