# Lorentz contraction in circular particle accelerators

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1. Jun 15, 2014

### QuasiParticle

My question is essentially a variation of the Ehrenfest paradox in SR. But hopefully with some experimental data.

In the LHC, for example, a fixed number of particle bunches with some length are injected into the main ring. Now, as the velocity of the particles increases, the bunches would be expected to Lorentz contract to smaller length (for a stationary outside observer). However, at the same time the distance between successive bunches should also decrease. This seems paradoxical, since the circumference of the accelerator remains constant. What actually happens in the accelerator to the bunch lengths and distances between bunches? My understanding is that the bunches do Lorentz contract, but is there any explanation for the paradox? There has been some discussion of this in the forum before, but I was unable to find any satisfactory answer. Born rigidity, which is often evoked in the context of the Ehrenfest paradox, does not play a role here.

2. Jun 15, 2014

### Bill_K

The bunches do NOT contract. Nor do the spaces between them. They remain a certain fixed fraction of the circumference. As you point out, a bunch is not a rigid object, so the proper distance between particles in a bunch does not need to remain fixed.

3. Jun 15, 2014

### WannabeNewton

This is only true if one has Born-rigidity; in fact Born-rigidity is equivalent to the condition that the time-like congruence in question Lorentz contract locally by the appropriate amount in a local non-comoving frame. In your scenario the time-like congruence clearly will not be Born-rigid. Your claim that Born-rigidity plays no role is quite incorrect. Why would you assume it doesn't? Born-rigidity is a very general statement about the relative motions between neighboring particles of a time-like congruence hence not necessarily one about the rigidity of a single body.

The comments made above by Bill and myself carry over verbatim to the resolution of Ehrenfest's paradox.

4. Jun 15, 2014

### Staff: Mentor

In discussions on Lorentz contraction there is often some confusion. Some people talk about Lorentz contraction as though it were what happens to a rigid rod as it accelerates. However, length contraction is actually a disagreement between different inertial reference frames regarding the length of an object and has little to do with acceleration.

The "acceleration contraction" only applies to Born-rigid objects, and as you mention the bunch is not Born rigid. Bill_K and WannabeNewton are correct in their responses regarding that.

However, Lorentz contraction is happening. The separation of the particles in a bunch is greater in their momentarily co-moving inertial frame than in the lab frame. This is, IMO, one of the experimental proofs of length contraction. If it didn't happen then the particles would be more strongly interacting in their frame and would push the bunch apart beyond what the accelerator could handle.

5. Jun 15, 2014

Staff Emeritus
If we're talking about the real LHC, much of what is written is not relevant.

The space between bunches is determined by the RF structure of the beam. They are at fixed distances in the lab frame because we move the beam until they are at fixed distances in the lab frame.

The size of a bunch is Lorentz contracted, but since protons repel, the bunch would explode if not captured by the RF, and again you are in a situation where it's determined by the RF in the lab frame.

6. Jun 18, 2014

### QuasiParticle

Ah, ok, thank you all. So the Born rigidity does indeed play a role (due to its absence). And good point about the actual LHC.

The result that the bunches do not contract is somewhat counter-intuitive when viewed from SR point of view, though intuitive in the sense that it solves the mystery.

I'm not familiar with "time-like congruence". My understanding of GR (and related) is at the level of about 80 % of Carroll's book, though it's been a few years. But SR should be enough to describe the situation here, or is it impossible to do in flat spacetime.

I still don't understand why the bunches would not contract in the lab frame. Can someone provide a reference to something where this has been considered in detail.

7. Jun 18, 2014

### WannabeNewton

Time-like congruences are the natural language for describing Born-rigidity, be it in SR or GR, but there's no need to consider them for the purposes of your question.

To start with, you should start with an appreciably simpler scenario: a rod at rest in some inertial frame is accelerated along its length in such a way that each particle in the rod (idealized as a linear array) receives the exact same constant proper acceleration simultaneously. The ensuing motion of the particles will not be Born rigid because by construction the differential elements of the rod, or equivalently the distances between infinitesimally neighboring particles in the rod, remain constant as measured relative to the inertial frame. If the motion was Born-rigid then the distances between neighboring particles will Lorentz contract by the appropriate amount relative to the inertial frame. Note again that the rod fails to Lorentz contract in the inertial frame simply by construction of the problem: all points of the rod receive the same proper acceleration, simultaneously in this frame.

8. Mar 21, 2015

### QuasiParticle

WannabeNewton,

Sorry for the very late response, but your reply led me to the Bell's spaceship paradox etc. and everything is clear now Thanks!