# Length Contraction and SR

1. Oct 16, 2011

I'll try to be as specific as possible with my question.

From my reading of SR I understand that an object that is moving relative to me will be contracted in the direction of its motion. I have no issue with that and I accept it as a fact. Would I be right in thinking that there is also length contraction in GR in that if two people are at different "heights" (different potentials?) in a gravitational field then the one in the stronger field will have a shorter rule than the one that is in the weaker field?

If so, then I have a question (if not then please ignore the question):

Imagine I am static with respect to the horizon of a black hole (I guess I will have to idealise this to a non-rotating black hole as well to simplify the question). I find the mass of the black hole and then can calculate the distance to the horizon from my position. We will call this distance d.

I then start lowering a rope to the event horizon (we assume this is a special rope that is not stretched in any possible way for the purpose of this question but is still subject to the dilation effects) that is equal in length to d.

Will the rope reach the event horizon? Or will it be too short or too long?

2. Oct 16, 2011

### Drakkith

Staff Emeritus
I don't believe there is length contraction due to gravity. I've looked around and all I've seen producing length contraction is high velocity, not gravity.

3. Oct 16, 2011

### That fast guy

I'm in agreement with Drakkith, length contraction would depend on how fast you lowered the rope. The black hole would manipulate space-time, but I don't think it's the same thing...

4. Oct 16, 2011

Hmm, I'm surprised at that, but if there isn't then there isn't. Thanks guys.

5. Oct 16, 2011

### yoron

A event horizon is observer dependent. And yes, a gravity well will act the same way a (uniform) acceleration does as far as I understands it. In all gravity wells you can safely assume that the mass always will at the absolute 'center' just as Newton did in his shell theorem. So there will be a Lorentz contraction.

You have an observer A and an observer B

Then you get yourself a 'gravity well' like a neutron star. A strong one. B goes into it, A stays 'outside' the gravitation. When they meet again they compare clocks. B:s clock will say 10 m. A:s clock will say 11 m. The time dilation seen can also be transformed into a Lorentz contraction, and is..

To A all things inside the gravity well will contract as he looks, they will be smaller. But to B the opposite will be true. From inside that neutronstar space will expand as well as all objects in it, so to him A will become proportionally larger. This effect holds true for all things you can think of as I understands it. A have an electric charge which he sends in to B telling him before what he measured, he then measures it again when it arrived at B, and finds it reduced.. Now we let B measure it, and he will find it to be the correct charge for what A told him before sending it to him. What A says the charge to have now B will find incorrect. He then sends it back to A and measure it after it arrives. He now finds the charge to be stronger than than it was when at rest with him.

6. Oct 16, 2011

### That fast guy

but is that really a Lorentz contraction or is that just gravity messing with space time? or is it essentially the same? :P

7. Oct 16, 2011

### yoron

The Lorentz contraction is one of the trickiest thing I know of :)

If you think of it as a 'surface' like a transparent 'pool of contraction' shrinking everything, but in the direction you 'move', or in this case, the way 'gravity points'. But I think it's real, and if we use the 'muon example' the muon's don't see the same as the observer on Earths surface does.

And that do create two 'realities', one for the muon and one for me. You can refer to is as an effect between two 'frames of reference' as if it happens in between the SpaceTime I see and SpaceTime the muon sees, also assuming a same SpaceTime. Or you can assume it to be a effect of locality, which then makes it into something 'different', although translatable by radiation and gravity. The difference is subtle, but in the last what you see is what you get, in the first the Lorentz transformations etc will be the 'real reality' of what you see.

It's like Smolins 'phase space' where all definitions are uniquely true, with the difference that he defines them as unable to translate into each other, as I understood it. If we assume him to be right, then what he defines have to follow from what we see in the theory of relativity, which might make my idea of 'locality' a little more palatable for those finding it slightly too weird to contemplate :)

Because I agree, it's a weird one.

8. Oct 17, 2011

Yoron, thanks for your input. So does this apply to rules as well in that if I pass a 12 inch rule to my friend in the stronger gravitational field then from my perspective the rule will be shorter in the direction of the gravitational source (assuming enough difference in potential to be able to measure a real difference)? Thus the rope in my example won't reach the EH?

9. Oct 17, 2011

### Drakkith

Staff Emeritus
Again, I don't think so. According to my knowledge length contraction is not due to acceleration, but due to velocity. See here: http://en.wikipedia.org/wiki/Length_contraction

I'm not formally educated on the subject, so I could very well be incorrect.

10. Oct 17, 2011

### TrickyDicky

Think about the perihelion advance of Mercury's elliptical orbit due to the sun's gravitational influence distorting the space in Mercury's path. Even though this can't be technically called length contraction since this term only applies to the kinematic effect of SR, it is the closest equivalent effect in GR.

11. Oct 17, 2011

I know of the precession of the perihelion of Mercury but I'm not au fait with precisely why that happens. Is it because the distance that it travels has been reduced and thus it takes a tighter orbit than it would if there was no relativistic effect?

12. Oct 17, 2011

### TrickyDicky

Basically the space around the sun is distorted by the sun's gravity and given Mercury's elliptical orbit some parts of the ellipse are affected differently (for instance there is more distortion at perihelion than at aphelion) by this distortion, the result of this is that the orbit doesn't describe perfect closed ellipses like in Newtonian euclidean space but there is a progressive precession with every orbit (the ellipses don't close perfecly) a very small displacement but that is cumulative.

13. Oct 17, 2011

### yoron

If we assume that gravity can dilate a clock, then you also should be able to see it as a Lorentz contraction, as I think of it. They are each others counterparts, your frame witnessing a 'time dilation', your 'counterpart' witnessing a Lorentz contraction.

And why it is so has to do, in my eyes, with the fact that you nowhere ever will be able to define 'c' to another speed (locally). If time is durations, then we can split them, and the best 'splits' you ever will be able to make are defined by light speed in a vacuum. And where that 'propagation' ends, so will mainstream physics. So in my thoughts I see 'c' not as a sped, but more of a 'clock'. And 'c' is always 'c' locally, as I define it.

14. Oct 17, 2011

### yoron

But yes, I see what you mean, the symmetry (time dilation/Lorentz contraction) we see in a motion differs from the one we see with a gravity. What you might want to question there is if there ever is a space that isn't Lorentz contracted, even if just infinitesimally. If it is so then all assumptions about what a correct distance should be seen as will be arbitrarily. You can't use uniform motion for defining a same 'distance' even though all experiments done inside a black box scenario will come out the same. so what is the correct definition there? That the experiments tell you a truth, or that they lie?

If they tell you a truth then being 'at rest' is the same for all uniform motion, but 'distance' isn't locally, and neither is 'time', as defined from some other frame of reference.

But the contraction is there, although as defined from inside the gravity-well you might want to say that the 'space expands' and that the far observer 'speeds up' relative you. But they are the same, they must be, or Einstein got his GR equivalence wrong between Gravity/uniform constant acceleration. But it's tricky, and this is my view.

15. Oct 18, 2011

### harrylin

That's not a Lorentz contraction but an Einstein contraction.
You can read it (or hear it) here "from the horse's mouth" (p.196, 197):
http://www.alberteinstein.info/gallery/gtext3.html [Broken]

Harald

Last edited by a moderator: May 5, 2017
16. Oct 18, 2011

### Pengwuino

You will see length contraction in a curved spacetime. That's the point of the metric. The metric describes how an observer makes distance and time measurements in its frame of reference.

17. Oct 18, 2011

### ZealScience

I think curved space have a potential, which is approximated by Schwarzschild solution to be the time component of metric. Consequently, there is gravitational red-shift. But I think it doesn't mean that potential has no effect on space components. It's just a bit insignificant.

Since gravity is calculated using Ricci tensor which contains christoffel symbols and thus directly affected by different components of metric, there would be a length contraction to my perspective.

18. Oct 18, 2011

Is there a direction to this length contraction like in SR?

19. Oct 18, 2011

### harrylin

Yes you can read it in Einstein's paper* (to which I had given a wrong link, now corrected):
According to GR the contraction is along the radius (the height) and there is no tangential (horizontal) contraction.

*for convenience here once more: p.197 of
http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf [Broken]

Last edited by a moderator: May 5, 2017
20. Oct 18, 2011