# A few paradoxes in STR

As you can guess from my username, I come from a biology background. I've recently had the dubious pleasure of discussing evolution with a guy who is also a major physics crank; now, my physics is good enough to dismiss most of his nonsense, but he gave me pause with the following problems (actually, I thought that I had an answer, but after reading about Bell's spaceship paradox, I'm confused). I'm assuming you guys can make short work of them :)

1. Two buildings are connected by a straight, rigid wall. The wall is strong enough to stand for centuries, but is made of crushable material. An alien vessel passes by the Earth at 99% speed of light.

From the reference frame of the buildings, absolutely nothing happens. From the reference point of the alien ship, two buildings and the wall suddenly got shorter. The wall either breaks apart since the distance between two buildings has been increased, or it is crushed from its own contraction (to great surprise of everyone on the ground).

My assumption was that the space in which the buildings and the wall are situated also contracts from the reference frame of the spaceship; therefore, everything stays "in proportion", so to speak. But upon reading the solutions to Bell's spaceship paradox (where the string does break, if I understand correctly), I find myself utterly confused.

2. Related to the previous. A train sits at the train station. It is composed of wagons that are touching each other, but are not physically connected otherwise. An alien vessel flies overhead at 0.99c.

Does the alien vessel "see" a bunch of shortened wagons with spaces between them, or does it "see" the whole train as being shortened (with wagons still touching)?

3. And again related to the previous. I have two rods, AB and CD, which are aligned lengthwise so they touch each other (point B of AB touches point C of CD). The rods are stationary relative to myself, and I measure their coordinates: length AB = x, length CD = x, length AD = 2x.

The rods suddenly accelerate to relativistic speeds, and then they stop accelerating, maintaining a constant velocity relative to my frame of reference.

I now measure length A'B' = x/2, and length C'D'= x/2. What is the length A'D' - i.e. did the rod remain in contact (in which case A'D' = x), or did they become separate?

Furthermore, let us say that during the period of acceleration, between my first and my second measurement, point A traversed a length of 10x. But if the rods remain in contact, point D traversed a length of 9x (10x-2*x+2*x/2). How is this possible?

*

I expect that the answer to these problems is humiliatingly simple. :)

Dale
Mentor
2020 Award

1. Two buildings are connected by a straight, rigid wall. The wall is strong enough to stand for centuries, but is made of crushable material. An alien vessel passes by the Earth at 99% speed of light.

From the reference frame of the buildings, absolutely nothing happens. From the reference point of the alien ship, two buildings and the wall suddenly got shorter. The wall either breaks apart since the distance between two buildings has been increased, or it is crushed from its own contraction (to great surprise of everyone on the ground).
Nothing happens in the ship's frame. Lorentz contraction is strain-free. The length of the object and the distance that it has to stretch contract by the same amount.

2. Related to the previous. A train sits at the train station. It is composed of wagons that are touching each other, but are not physically connected otherwise. An alien vessel flies overhead at 0.99c.

Does the alien vessel "see" a bunch of shortened wagons with spaces between them, or does it "see" the whole train as being shortened (with wagons still touching)?
The whole train is shortened, no spaces between the wagons.

3. And again related to the previous. I have two rods, AB and CD, which are aligned lengthwise so they touch each other (point B of AB touches point C of CD). The rods are stationary relative to myself, and I measure their coordinates: length AB = x, length CD = x, length AD = 2x.

The rods suddenly accelerate to relativistic speeds, and then they stop accelerating, maintaining a constant velocity relative to my frame of reference.

I now measure length A'B' = x/2, and length C'D'= x/2. What is the length A'D' - i.e. did the rod remain in contact (in which case A'D' = x), or did they become separate?
That depends on the details of the acceleration. You can get acceleration profiles that achieve either result. This is essentially the same answer as to the Bell's spaceship exercise.

That depends on the details of the acceleration. You can get acceleration profiles that achieve either result. This is essentially the same answer as to the Bell's spaceship exercise.

Thanks!

I'm still a bit confused at the third point (I don't quite "get" the answer to the Bell's exercise, so please bear with me).

Let's simplify the issue: one rod, AB, length x. It accelerates, then stops accelerating, traversing 10x during the acceleration period. Then, when there is no more acceleration, and the rod's velocity is constant, the rod is measured again from the same frame of reference.

Distance travelled by A is 10x. Distance traveled by B is 10x - z, where z is the length contraction. Therefore, from the frame of reference we are using, the two points have traveled at different velocities (different length covered over the same period of time)?

I'm guessing that simultaneity somehow affects this, but for the life of me I can't figure it out?

The length of spacetime the rod is embedded in changed, the rod didn't travel a certain distance while shrunk, then stretch back out without the back end crossing the distance.

The definition of distance used to measure the length of the rod was changed by it's motion, which results in it viewing everything else undergoing this same effect.

Just because a cosmic ray whipped past the Earth over my head a second ago doesn't mean I got flattened into a pancake in my frame of reference, only in the rays frame.

From the rays frame it was unchanged by its motion, while I observe that it was smushed flat by it's 99.999999~% of c velocity.

Dale
Mentor
2020 Award

I don't quite "get" the answer to the Bell's exercise, so please bear with me
Try this link for a detailed explanation:
http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.html [Broken]

Basically, if they accelerate at the same rate starting at the same time in the lab frame then the distance between the ships in the ship frame increases. If the distance between the ships in their frame stays the same then the acceleration is different.

Last edited by a moderator:

The length of spacetime the rod is embedded in changed, the rod didn't travel a certain distance while shrunk, then stretch back out without the back end crossing the distance.

I highly appreciate these answers, but I'm still managing to stay confused. Sorry. :(

Perhaps the problem is this: I'm not talking about the reality (I can grasp that there is one underlying reality, as described by spacetime intervals). I'm confused by what the observer would see.

In other words, I know there is no paradox, I'm unclear as to what the measurement would be and why.

The observer measures the stationary rod AB at time t0.

The rod then accelerates to some fraction of c away from the observer. It then stops accelerating, and just moves at that constant speed.

Then the observer (who remained where he was) measures the position of the rod again at time t1, marking the beginning and end of the rod at that instant as A'B'.

In his reference frame, does it appear to him that the distance AA' is longer then the distance BB'?

If so, why? If not so, why?

Ich

In his reference frame, does it appear to him that the distance AA' is longer then the distance BB'?
That depends on some details that are not mentioned in the question.
Let's assume we have a rocket drive that makes 0.866 c (gamma=2) in a certain proper time. No matter what it has to carry, it reaches end veloctiy in a certain distance x, as seen from the initial frame.
If you attach the drive to A, A will make distance x and B will make x-L/2. So B accelerated differently than A, with smaller acceleration but for a longer time. That has to do with simultaneity, it's like gravitational time dilatation - remember, acceleration ~ gravitation, the equivalence principle.
If you attach a second drive to D, D will make also distance x, AD remains constant. There will be a gap between B and C.
If you attach a drive to both B and C (at the same position), they will obviously stay at the same position and AD will shrink.
If you attach drives to A and B, AB will stay constant, as measured in the original frame, therefore be stretched by a factor of 2 in its new rest frame. Maybe the respective rod will break.

So, you see, actually accelerating objects is more complex than switching to a moving observer - which does nothing to the observed things. It always involves real forces, and it's not possible to acellerate an extended object with the same acceleration at each point without straining it. You have to define different accelerations for each point, in a certain way, to make it as stress-free as possible.

That depends on some details that are not mentioned in the question.

Thanks! :)

With this explanation and a bit more thought, I figured it out.