Time Dilation, Length Contraction & Massive Bodies

[Bracken]

TL;DR Summary

Time dilation and length contraction both occur at relativistic speeds. Because time dilation occurs near massive bodies, does length contraction do too?

For instance, if a stream of muons were released from a box near a massive object and traveled on a straight path slowly (at a non-relativistic velocity) away from that object, a faraway observer would notice that the particles would take longer to decay than a muon typically would in empty space. This is because the presence of the massive object causes time dilation, which in turn, allows the muons to travel a farther distance before decaying, as seen from the observer's perspective. From the muons' perspective though, time is proceeding as usual, and they decay within the normal amount of time. But this causes a paradox. From the muons' perspective, there is no way to reach as far as the final position noted by the observer. In order for an agreement to exist between the observer and the muons as to how far away the particles make it before decaying, their path of travel away from the massive object must be contracted right?

So, taking this question a step further, if an individual were about to fall into an event horizon of a non-rotating black hole with their back turned to the singularity, would they observe a flattening of all objects in the universe onto the skin of the event horizon with them? As they approached the Schwarzschild radius, would the thickness of the universe from their perspective approach zero?

 

[PeterDonis]

Summary: Time dilation and length contraction both occur at relativistic speeds. Because time dilation occurs near massive bodies, does length contraction do too?

The question is based on a false premise: that the "time dilation" that occurs near massive bodies is the same "time dilation" that occurs at relativistic speeds. It isn't, even though the same term is sometimes used for both. They are two different things and there is no useful analogy between them.

 

[PeroK]

Summary: Time dilation and length contraction both occur at relativistic speeds. Because time dilation occurs near massive bodies, does length contraction do too?

For instance, if a stream of muons were released from a box near a massive object and traveled on a straight path slowly (at a non-relativistic velocity) away from that object, a faraway observer would notice that the particles would take longer to decay than a muon typically would in empty space. This is because the presence of the massive object causes time dilation, which in turn, allows the muons to travel a farther distance before decaying, as seen from the observer's perspective. From the muons' perspective though, time is proceeding as usual, and they decay within the normal amount of time. But this causes a paradox. From the muons' perspective, there is no way to reach as far as the final position noted by the observer. In order for an agreement to exist between the observer and the muons as to how far away the particles make it before decaying, their path of travel away from the massive object must be contracted right?

So, taking this question a step further, if an individual were about to fall into an event horizon of a non-rotating black hole with their back turned to the singularity, would they observe a flattening of all objects in the universe onto the skin of the event horizon with them? As they approached the Schwarzschild radius, would the thickness of the universe from their perspective approach zero?

On the first point, remote measurements of length are problematic in curved spacetime. Length contraction is measured in SR using a global inertial reference frame. No such frame exists in curved spacetime. In SR you measure the simultaneous position of each end of an object, using your global inertial reference frame and your global definition of simultaneity. In curved spacetime there is no unambiguous way to decide simultaneity for two spatially separated events. You still have the concept of local length contraction in curved spacetime, using a local inertial reference frame.

On the second point. Length contraction is what you measure. What you directly observe by light signals is something else. In any case, what you observe falling into a black hole would not be consistent with universal length contraction.

 

[Bracken]

Hello PeroK, thank you so much for your response. I had heard how muons are able to penetrate Earth's atmosphere by means of length contraction given their great velocity, and was hoping a similar analogy could be used to conceptualize the existence of length contraction near black holes. Whelp, so much for that concept. From a purely visual explanation though, if an experiment were set up with a box firing slow muons directly away from a massive body, what should a faraway observer observe as opposed to the following configuration without a massive body? Does the presence of the massive body change theta at all? (Under the assumption that the muon creates a flash of light when it decays).

 

[PeterDonis]

I had heard how muons are able to penetrate Earth's atmosphere by means of length contraction given their great velocity

It's length contraction from the point of view of the muon (the distance from the upper atmosphere to Earth's surface is much shorter in the muon's frame than in the Earth frame), but it's time dilation from the point of view of the Earth (the muons' half-life is extended due to their speed).

was hoping a similar analogy could be used to conceptualize the existence of length contraction near black holes

It can't. The analogy you are looking for doesn't exist.

Does the presence of the massive body change theta at all?

In principle, yes, but not because of any "length contraction" or "time dilation". The massive body changes the muons' trajectories--they decelerate as they move upward. So they won't get as far before they decay as they would in free space with no massive body nearby.

 

[PeroK]

@Bracken there are a number of conceptual problems with your experiment.

First, when you say "slow" muons, how are they escaping the large body? They would simply be pulled back by its its gravity.

Second, the path of light will be affected by gravity, hence you may not be able to interpret your angle $\theta$ the way you want.

Third, any measure of distance in this frame of reference from a point near the body to a point far from the body is not dependent on the muons. If you have such a measurement, then what length contraction are we talking about?

 

[PeterDonis]

the path of light will be affected by gravity, hence you may not be able to interpret your angle $\theta$ the way you want.

This is a valid point in principle, but in practice there will be a wide range of scenarios where the deceleration of the muons under gravity will be observable while the bending of light won't (it will be too small to matter). I think it's a good idea to get clear about the simpler scenario in which light bending is negligible first.

 

[PeterDonis]

when you say "slow" muons, how are they escaping the large body? They would simply be pulled back by its its gravity

They don't necessarily need to escape; they just need to be able to fly upward long enough to decay before they reach maximum altitude.

 

[PeroK]

They don't necessarily need to escape; they just need to be able to fly upward long enough to decay before they reach maximum altitude.

Okay. Let's say a muon lives for $1s$ proper time and it moves locally at $1m/s$. Locally it travels for $1m$. That's measured along a local metre stick.

How is the distant observer defining the distance between the two ends of the meter stick?

As you know, unlike in SR he/she can't import his/her own metre stick into the experiment.

Where does length contraction arise?

I'm not sure how you'd calculate the perceived angle $\theta$ here. But, in any case, its independent of any muon motion. It's just what a metre stick looks like in whatever spacetime we have here.

Finally, if the observer uses the local metre stick and his/her own clocks the he/she just gets a meaningless coordinate speed for the muon.

All we are left with here is what a metre stick looks like from a large distance and that isn't what length contraction is about.

With gravitational lensing you have length dilation , one could say.

 

[PeterDonis]

How is the distant observer defining the distance between the two ends of the meter stick?

This is a different issue from the one I was talking about in what you quoted. The issue I was talking about is that this...

Let's say a muon lives for $1s$ proper time and it moves locally at $1m/s$. Locally it travels for $1m$.

...is only true if there is no gravitational deceleration. The difference between the "no massive body" case and the "massive body present" case is observable using those local measurements alone. Given that, there will be a difference in the angle $\theta$ observed by the distant observer for those two cases, regardless of how the distant observer wants to try to relate that angle to a "distance" measured in whatever kind of coordinates he wants to use.

I'm not sure how you'd calculate the perceived angle $\theta$ here. But, in any case, its independent of any muon motion.

I'm not sure what you mean by "independent". It's certainly not independent of whether or not a massive body is present and affects the muon motion by gravitational deceleration. See above.

 

[PeroK]

...is only true if there is no gravitational deceleration. The difference between the "no massive body" case and the "massive body present" case is observable using those local measurements alone. Given that, there will be a difference in the angle $\theta$ observed by the distant observer for those two cases, regardless of how the distant observer wants to try to relate that angle to a "distance" measured in whatever kind of coordinates he wants to use.
I'm not sure what you mean by "independent". It's certainly not independent of whether or not a massive body is present and affects the muon motion by gravitational deceleration. See above.

We may be talking at cross purposes here.

1) there is no experiment under discussion here which does not involve the massive body. The massive body is always present.

2) the muon will live locally for a time ($\tau$) and travel a locally measured distance $d$. I wanted to assume these were $1s$ and $1m$ respectively.

3) Assume the length $d$ equates to the length of some fixed object, positioned locally.

4) the length of that fixed object is independent of any muons.

5) if there is length contraction between the local frame and a distant frame, then the length of that fixed object must be less than $d$ when measured remotely.

6) the OP, or you, needs to specify a measurement process that would produce this contracted length.

7) it's an interesting question what a metre stick near a massive body would look like " from the side". although what it looks like wouldn't normally be considered length contraction.

PS As the muon is traveling at non relativistic speed I'm taking the local frame and the muon frame to be effectively the same.

PPS that's why I said that any length contraction between the two frames is independent of the muons. Either there is length contraction or there is not and firing a few low speed muons about isn't going to affect that.

 

[PeterDonis]

there is no experiment under discussion here which does not involve the massive body

Yes, there is:

as opposed to the following configuration without a massive body?

The OP is asking to compare the "massive body present" case with the "no massive body" case.