SR vs GR - Visual Difference between Length Contraction

[D.S.Beyer]

TL;DR Summary

Are there any visual differences between the length contraction of SR and the length contraction of GR?

The 2 Bowling Balls

Ball(a) & Ball(b)

(a) is in acceleration of 10m/s^2
(b) is in at fixed position in a gravitational field where g=10m/s^2

In both cases the observer is:
- perpendicular to the vector of acceleration
- distant enough to be in empty flat space

Question : In an instantaneous rest frame of the observer, what distinguishes (a) from (b)?

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I assume these are very different equations. I’ll take a stab at the first, but definitely stop and freak out once the math comes into play for the second.

(a)’s equation is a simple Lorentz contraction. Take the diameter of the ball, when v=0, measured at rest, and call that L. L = Li/√1-(v/c)^2 contracted in the direction of motion.

(b)’s length contraction is an visual effect of non-euclidian spacetime viewed from euclidian spacetime. So, I’d probably start with the Schwarzschild Metric for a non-spinning, non-charged black hole.

Take an embedding diagram for a 2D plane slice through the black hole, x,y being a euclidian coordinate system and z producing a parabolic curve such that distances measured on that curve correspond to proper length. (Thus z is a ‘unreal’ dimension used only for visual exposition of proper length). Use points r(1) & r(2) to define distance dr along the x-axis. r(1) & r(2), projected along the z-axis, meet the parabolic curve at points p(1) & p(2). Let the distance between p(1) & p(2) = dp = proper length

(please see slide 6 of ‘The Schwarzschild Metric : Relativity and Astrophysics, Lecture 34, Terry Herter. A2290-34.)

We see that dp > dr
dr would then be the visualized contracted length.

(...I think this is all I’ll need to make some semblance of an bizarre math question...)

If we replace dr with Li, and dp with L, such that dp = dr/√1-(v/c)^2,

what is the velocity?

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What that answer is or isn’t may not be significant, as I have elevated Lorentz Transformations to, for some reason, play at the same level as the Schwarzschild solution to Einstein’s Field Equations. Maybe it’s a way to create statements about illusions, like, “An SR length contraction of this much, is visually equal to observing, from flat space, an object at radius r from a black hole.”

For a brief moment I entertained the idea that the visual phenomena of idealized SR length contraction was actually a result of non-euclidian spacetime deformation. But upon a very minimal amount of searching, found that acceleration does not curve spacetime.

Thus, the phenomena of length contraction in SR and GR seem to stem from very different ‘places’, and can both be present, overlapping constructively or destructively, in a scenario. For example if ball(b) was also accelerating with respect to the observer, it would be both length contracted by curvature and by SR effects.

And so…

I come back to my initial question.
How can we tell the difference between the visual of SR and GR length contraction?
Is there anything about the visual, that defines either SR or GR?

If the planet on the left was somehow invisible, in an instantaneous rest frame of the observer, is there anyway to tell the difference between the two examples?

 

[Ibix]

The obvious difference is that one is accelerating, and the length depends on velocity not acceleration, so its length changes over time.

It's also not at all clear what you mean by "length contraction in GR". It's an SR inertial frame phenomenon. You could certainly measure the angular subtense of the rod, but to what would you compare it? If you move the rod to flat spacetime or just let it free-fall, it'll stretch (or de-compress) because it's elastic. And light paths between the observer and the ruler would be different in flat spacetime anyway. You can't compare it to a co-located ruler because the ruler would also be affected by whatever effect you hope to measure.

I don't think your question has an answer, in short. If you want to know the angular subtense in the two cases then you can determine that, as I say. But that isn't really "measuring length contraction".

 

[PeterDonis]

Are there any visual differences between the length contraction of SR and the length contraction of GR?

The GR phenomenon you are describing is not "length contraction".

Also, in both the SR and GR cases, there are other effects involved which complicate what is seen visually. In the SR case, you have Penrose-Terrell rotation. In the GR case, you have bending of light by the gravitating mass. So what you are imagining would be seen visually in both cases is wrong.

Furthermore, you appear to misunderstand how length contraction works in SR:

(a) is in acceleration of 10m/s^2

Length contraction in SR is determined by relative velocity, not acceleration. See further comment below.

(b) is in at fixed position in a gravitational field where g=10m/s^2

You appear to be relying on the equivalence principle here to make the two cases "the same", but, while the EP does say that locally, the two observers in question, (a) and (b), will make the same observations (they both will feel the same weight, they both will observe the same acceleration relative to themselves of a dropped object, etc.), it says nothing whatever about how they will appear to other observers who are far away from them.

Question : In an instantaneous rest frame of the observer, what distinguishes (a) from (b)?

Apart from the issues I mentioned above, which complicate what is seen visually, there is the obvious difference that, since length contraction in SR depends on relative velocity, not acceleration, the length contraction of (a) will appear to be increasing with time to the distant observer, whereas the apparent "length" of (b) to the distant observer (ignoring the fact, mentioned above, that what is going on in this case is not "length contraction") will not change with time.

Even if we fix that by making the velocity of the object in (a) constant, there is still the obvious difference that in (a), the object is moving across the visual field of the distant observer, whereas in (b), it isn't.

(a)’s equation is a simple Lorentz contraction.

A time-dependent one, since $v$ changes with time in your scenario.

(b)’s length contraction is an visual effect of non-euclidian spacetime viewed from euclidian spacetime.

What is non-Euclidean in the Schwarzschild case is space (more precisely, the geometry of the "static" spacelike slices--note that the "slice" you take later in your post is at a constant time, time is not involved at all), not spacetime. And, as already noted, the visual effect is more complicated than what you are imagining.

If we replace dr with Li, and dp with L, such that dp = dr/√1-(v/c)^2,

what is the velocity?

Meaningless. What you are doing here is nonsense.

 

[Dale]

Question : In an instantaneous rest frame of the observer, what distinguishes (a) from (b)?

The curvature of spacetime between the observer and the ball

 

[D.S.Beyer]

@PeterDonis
"The GR phenomenon you are describing is not "length contraction"."

Then, how do talk about this visual phenomena?
This came up in a previous post about Christmas Lights into a black hole. From the perspective of an outside observer, the light got 'squished' together near the horizon.

 

[PeterDonis]

how do talk about this visual phenomena?

As visual phenomena.

This came up in a previous post about Christmas Lights into a black hole. From the perspective of an outside observer, the light got 'squished' together near the horizon.

The lights appeared to be "squished together" near the horizon--but if you remember, there were other effects as well, for example, due to light bending, all of the lights appeared further away from the hole than they would have in flat spacetime, and the amount of the bending decreased with distance from the hole, so lights further from the hole appeared more "squished together" because the light from them was bent less.

In any case, none of these effects have anything to do with "length contraction" or with any analogy between such phenomena and the effects of relative motion in SR.

 

[PeterDonis]

As visual phenomena.

Which, btw, "length contraction" is not even in SR. If you look up Penrose-Terrell rotation, you will see that relative motion in SR causes an object to appear, visually, to be rotated, not contracted. The "length contraction" is what is calculated after accounting for the effects of light travel time on what is directly seen, visually.

 

[A.T.]

Which, btw, "length contraction" is not even in SR. If you look up Penrose-Terrell rotation, you will see that relative motion in SR causes an object to appear, visually, to be rotated, not contracted. The "length contraction" is what is calculated after accounting for the effects of light travel time on what is directly seen, visually.

@D.S.Beyer Here is a website that visualizes what @PeterDonis wrote about the difference between the relativisic effect (length contraction) and the visual effect (due to signal delay etc.):

https://www.spacetimetravel.org/fussball/fussball.html

 

[D.S.Beyer]

https://www.spacetimetravel.org/fussball/fussball.html

This is awesome! Thank you for this helpful link.

 

[A.T.]

This is awesome! Thank you for this helpful link.

You are welcome. Here is an interesting aspect of the resulting effect. You can sometimes see the back side of objects that approach you, but not the front side (hence the name "rotation"):

https://www.spacetimetravel.org/bewegung/bewegung5.html

 

[Grasshopper]

You are welcome. Here is an interesting aspect of the resulting effect. You can sometimes see the back side of objects that approach you, but not the front side (hence the name "rotation"):

https://www.spacetimetravel.org/bewegung/bewegung5.html

So THAT’s how Superman does it.

 

[Demystifier]

The obvious difference is that one is accelerating, and the length depends on velocity not acceleration, so its length changes over time.

For the case someone is interested, here are the details:
https://arxiv.org/abs/physics/9810017