# Apparent "centrifugal" like force on a uniformly accelerating relativistic object

So this is more of a special relativity question as it comes from someone who's only savvy in special relativity.

What I mean by this is if the answer is only describable in terms of general relativity, I'd simply prefer that you say so rather than try to explain it with concepts that are currently above my level.

Now I know I'm probably not even supposed to touch accelerating objects in relativity without first knowing the general theory, but I hope you'll humor me anyway and perhaps by the end specify the errors of my way.

Say that $$c=1$$ and we have some object made up of smaller evenly spaced point-like constituents (in this case spaced by 0.2 units initially). If we consider first that these points are detached, that is there is no binding force between them, if each point undergoes a constant acceleration they will each trace out similar hyperbolas as their world lines.

Now after doing some math (for one of said world lines), I found that from its perspective it would remain at the same distance from the other points (equilibrium if they were connected) if their world lines looked like this (first image).

Which are hyperbolas with oblique asymptotes of slope $$c$$.

Now this is interesting because the world lines of uniformly accelerated disconnected points look like this (second image).

Doing some math on these word lines, at least from the perspective of the bottom-most one, the other points would seem to be getting further away.

Now if we imagine that the points are connected, that is there exists some spring like force between them, by my reasoning the points will naturally "want" to follow the evenly spaced world lines of the second image, but would then observe the others getting farther away and thus pulled out of equilibrium.

This would mean that objects undergoing uniform acceleration would feel as though something was trying to stretch them apart, but I've never heard of something like this.

The types of answers I'd expect to this post would be.

a. Even in terms of special relativity your reasoning is wrong.

b. You need general relativity to explain why this is wrong.

or some third option that I'm not aware of.

Any answers would be appreciated, thank you.

• I am not sure what you mean by it seems to get stretched, but it is a standard SR acceleration textbook example that perfectly rigid bodies are forbidden by SR the moment you have any little bit of acceleration. The distances between the constituent point particles of any such body must change. I do not remember the details, but IIRC it was squashed not stretched. Commented Feb 29 at 5:08
• Are you perhaps talking about spatial contraction? If you are that's not exactly a physical squashing, in that none of the particles within the object measure a change in distance, only an outside observer. The effect I'm talking about could be measured internally within the object, if it indeed existed. Commented Feb 29 at 5:14
• The accelerating object can measure whatever change in distance the acceleration does. Acceleration is not a symmetry of SR. I just don't remember if it is squashed or stretched. Commented Feb 29 at 5:23
• – Sten
Commented Feb 29 at 17:31
• Also, I want to point out that special relativity has no problem at all describing accelerated motion or even accelerated reference frames (although you are only dealing with the former). That is just a common misconception.
– Sten
Commented Feb 29 at 17:39

This would mean that objects undergoing uniform acceleration would feel as though something was trying to stretch them apart, but I've never heard of something like this.

This is basically correct. More precisely, if an object is to remain rigid as it accelerates, then different parts of the object require different proper accelerations. The rear of the object has to accelerate faster than the front.

There are several ways to understand this. One is that in the frame of an inertial observer, the object length-contracts as it accelerates. To accomplish this, the rear clearly has to accelerate faster than the front.

We can also consider the perspective of observers on the accelerating object. Due to how surfaces of constant time rotate as you accelerate, these observers would say that the front of the object has been accelerating for a longer time than the back has. You can frame this more clearly in terms of gravitational time dilation: the rear is lower in the "gravitational potential", so its clock runs slower. Since the rear of the object has had less time to accelerate, it needs a greater acceleration to keep up.

Firstly, it is a mistake to imagine that problems involving acceleration are beyond the scope of special relativity; but they do tend to be more complicated than problems concerning inertial motion.

A particular complexity arises when you consider the acceleration of objects that are extended in their direction of motion, rather than point-like objects. That is because you need to specify the extent to which the different parts of the object accelerate in, or out of, synch with each other.

Bell's 'spaceship paradox' is an archetypal example of the problem. Suppose you have two spaceships initially at rest some distance apart, and they accelerate in the same direction along the line that separates them. If they accelerate in synch in the original frame, so that, for example, the front rocket is traveling with some speed v at the same instant as the rear rocket is travelling with the same speed v, then in their instantaneous rest frame, the front rocket must be travelling faster than the rear, owing to the relativity of simultaneity, which means that they are further apart in that frame. On the other hand, if the two rockets accelerated in such a way that they always remained the same distance apart in their instantaneous rest frame, they would seem to be accelerating at different rates in the initial frame, and the distance between them in that frame would shrink (owing to length contraction).

So, whether an extended object undergoes some kind of extension depends on the details of how the different parts of the object accelerate relative to each other.