# What would actually be observed during Lorentz contraction?

HJ Farnsworth
Greetings everyone,

I understand the derivation of the Lorentz transformations, and have not had trouble applying the concepts and the math to most elementary SR problems. However, something occurred to me recently which I have been unable to resolve.

Let’s say there are two tennis balls, initially at rest and 1m apart in the x-direction in our IRF. We find a way to suddenly accelerate each of them, simultaneously (in our frame) and in the same manner, to a given relativistic speed in the +x-direction.

HERE’S MY QUESTION: As I understand the Lorentz contraction, we should observe the distance between the two tennis balls decrease. But how does this happen – ie., do we see one tennis ball catching up to the other? This seems impossible, since both tennis balls were accelerated at the same time in the same manner. In fact, as far as I can tell, a spacetime plot of the situation should show their worldlines to be everywhere parallel and everywhere 1m apart, indicating that we would measure the distance between the two balls to be 1m everywhere, in apparent contradiction to the Lorentz contraction.

A slightly more complete thought experiment is below, for those of you who prefer, but my basic question is stated above.

We set up a tube, with cross-sectional area greater than that of a tennis ball, parallel to the +x-axis. Within the tube, separated by 1m as measured in our IRF, are two apparati that can detect an object going past them within the tube (each made of a laser and a photodetector or something). The tube is hooked up to a stop watch and a computer. The first apparatus (less far in the +x-direction) is programmed to send a time reading, t1, to the computer after having detected two objects. The second apparatus sends a time reading, t2, to the computer after having detected one object. The computer calculates delta t = t2 – t1.

Somewhere further in the negative-x direction, we have two things capable of hurling a tennis ball at relativistic speeds (Superman and the Hulk or whatever), which are each given a tennis ball. They are instructed to hurl the ball as fast as they can into the tube as soon as they see a light bulb turn on. A light bulb is placed so that it is equidistant to each of them.

The tube’s stopwatch is started, then the light bulb is turned on, and we eventually get a value for delta t.

My question: is delta t greater than 0 (indicating that the distance we measure between the two tennis balls has decreased), equal to 0 (distance has stayed the same) or less than 0 (distance has increased, for some extremely peculiar reason)? If the answer is “greater than”, in what manner, exactly, did this happen? If the answer is “equal to”, how is this reconcilable with the Lorentz transformation?

Thanks for any enlightenment you can provide.

-HJ Farnsworth

Staff Emeritus
Gold Member
Greetings everyone,

I understand the derivation of the Lorentz transformations, and have not had trouble applying the concepts and the math to most elementary SR problems. However, something occurred to me recently which I have been unable to resolve.

Let’s say there are two tennis balls, initially at rest and 1m apart in the x-direction in our IRF. We find a way to suddenly accelerate each of them, simultaneously (in our frame) and in the same manner, to a given relativistic speed in the +x-direction.

HERE’S MY QUESTION: As I understand the Lorentz contraction, we should observe the distance between the two tennis balls decrease. But how does this happen – ie., do we see one tennis ball catching up to the other? This seems impossible, since both tennis balls were accelerated at the same time in the same manner. In fact, as far as I can tell, a spacetime plot of the situation should show their worldlines to be everywhere parallel and everywhere 1m apart, indicating that we would measure the distance between the two balls to be 1m everywhere, in apparent contradiction to the Lorentz contraction.

-HJ Farnsworth

This is basically similar to the "Bell spaceship paradox". In it, two spaceships are linked together with a string and then accelerated equally. The paradox is that since the ships are accelerated equally, their distance apart will not change, but the string will undergo length contraction and break. However, in the frame of the Ships, the string doesn't contract, and shouldn't break.

The resolution is that you have to specify in which frame the acceleration is equal. In your example, you choose "our" frame. In this frame, the distance between the balls does not change. However, in the frame of either ball, it does. In this frame, the balls do not accelerate equally and the distance between them grows. Since the Lorentz contraction relates the proper length as measured in that frame to the length we measure in our frame, and the proper distance between the balls increases, the "length contracted" distance come out to be a constant value in our frame.

If instead, we decided that the accleration was equal in the frame of the balls, then they would measure the distance between each other as a constant, and we would see the distance as contracting.

This boils down to the fact that simultaneity is not absolute between the frames. Events that are simultaneous in one, (the balls attaining a specific speed for example) are not simultaneous in the other.

GrayGhost
Let’s say there are two tennis balls, initially at rest and 1m apart in the x-direction in our IRF. We find a way to suddenly accelerate each of them, simultaneously (in our frame) and in the same manner, to a given relativistic speed in the +x-direction.

Because you chose to accelerate them identically per your own POV, they must always remain 1m apart per you. What you will see is this ... the tennis balls length contract while their separation between center-of-mass remains unchanged. If they carried clocks that were synchronized prior, you would see that they remain always-in-sync as they go. On the other hand, from the POV of miniature passangers inside the tennis balls, they would disagree that they are accelerating identically, they would record their separation to become larger and larger with increased acceleration, and they'd record their own clocks to drop further and further out of sync as they accelerate.

The other option is this ...

The tennis balls accelerate always keeping the same distance between themselves as they go, per themselves. You would record their separation to contract more and more as they continue to accelerate. You would also record their clocks to drop out of sync. IOWs, you would not record their acceleration as identical.

GrayGhost

EDIT: Looks like Janus beat me to it. I'll leave the post up anyways.

Gold Member
Greetings everyone,

I understand the derivation of the Lorentz transformations, and have not had trouble applying the concepts and the math to most elementary SR problems. However, something occurred to me recently which I have been unable to resolve.

Let’s say there are two tennis balls, initially at rest and 1m apart in the x-direction in our IRF. We find a way to suddenly accelerate each of them, simultaneously (in our frame) and in the same manner, to a given relativistic speed in the +x-direction.

HERE’S MY QUESTION: As I understand the Lorentz contraction, we should observe the distance between the two tennis balls decrease. But how does this happen – ie., do we see one tennis ball catching up to the other? This seems impossible, since both tennis balls were accelerated at the same time in the same manner. In fact, as far as I can tell, a spacetime plot of the situation should show their worldlines to be everywhere parallel and everywhere 1m apart, indicating that we would measure the distance between the two balls to be 1m everywhere, in apparent contradiction to the Lorentz contraction.
In the original starting rest frame, the two balls will remain one meter apart. In a frame in which the two balls finally come to rest with respect to each other, they will be farther than one meter apart.

If instead, we decided that the accleration was equal in the frame of the balls, then they would measure the distance between each other as a constant, and we would see the distance as contracting.
I think this is not quite right--in order to keep the distance between the balls constant in their own rest frame, you'd have to do Born rigid acceleration, and this would actually result in the two balls experiencing different proper acceleration (i.e. different instantaneous coordinate accelerations in the inertial frame where they are both instantaneously at rest at a given moment). Essentially you'd want the balls to be at rest in Rindler coordinates, and as mentioned on this page:
We can imagine a flotilla of spaceships, each remaining at a fixed value of s by accelerating at 1/s. In principle, these ships could be physically connected together by ladders, allowing passengers to move between them. Although each ship would have a different proper acceleration, the spacing between them would remain constant as far as each of them was concerned.

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HJ Farnsworth
Thank you for those explanations, I think I understand now:

I was thinking that the Lorentz contraction automatically implied that the distance between the balls in our frame must decreases. However, the Lorentz contraction is actually a relation between the length in two given frames. The two tennis balls accelerate simultaneously in our frame, but not in either of theirs – as a result, they measure an increased distance between each other. The distance I measure is shorter than theirs, which is consistent with the Lorentz contraction.

So, applying the situation to Bell’s spaceship paradox – if the two ships are accelerated equally as observed by me, then from their point of view the rope will break since the distance between them increases. From my point of view, the distance between them does not increase, but the rope still breaks because the rope Lorentz contracts.

So all events occur in all frames, the Lorentz contraction is self-consistent, and the universe is a happy place.

Does this sound correct?

Again, many thanks.

-HJ Farnsworth

Mentor
2021 Award
I was thinking that the Lorentz contraction automatically implied that the distance between the balls in our frame must decreases. However, the Lorentz contraction is actually a relation between the length in two given frames. The two tennis balls accelerate simultaneously in our frame, but not in either of theirs – as a result, they measure an increased distance between each other. The distance I measure is shorter than theirs, which is consistent with the Lorentz contraction.

So, applying the situation to Bell’s spaceship paradox – if the two ships are accelerated equally as observed by me, then from their point of view the rope will break since the distance between them increases. From my point of view, the distance between them does not increase, but the rope still breaks because the rope Lorentz contracts.

So all events occur in all frames, the Lorentz contraction is self-consistent, and the universe is a happy place.

Does this sound correct?
Sounds right to me.

Gold Member
Yes.

phyti
Thank you for those explanations, I think I understand now:

I was thinking that the Lorentz contraction automatically implied that the distance between the balls in our frame must decreases. However, the Lorentz contraction is actually a relation between the length in two given frames. The two tennis balls accelerate simultaneously in our frame, but not in either of theirs – as a result, they measure an increased distance between each other. The distance I measure is shorter than theirs, which is consistent with the Lorentz contraction.

So, applying the situation to Bell’s spaceship paradox – if the two ships are accelerated equally as observed by me, then from their point of view the rope will break since the distance between them increases. From my point of view, the distance between them does not increase, but the rope still breaks because the rope Lorentz contracts.

So all events occur in all frames, the Lorentz contraction is self-consistent, and the universe is a happy place.

Does this sound correct?

Again, many thanks.

-HJ Farnsworth

Are you sure?

If the balls were connected by some rope, and accelerated so as to maintain a constant separation as measured by you, why would you see the rope break?

As for the Bell 'paradox', using the SR clock synch method, a moving observer A will calculate an inflated speed for a 2nd observer B moving relative to him. (Why is the velocity composition formula used with a 3rd observer in the rest frame?). The inflated speed is a result of the inflated distance because Einstein defines light speed to be c under all circumstances. The end result, the two ships aren't farther apart, and the rope doesn't break, just as in your ball example.

What you literally see is what was described except the ships or balls have to be offset a distance sideways to view them, therefore you see the length continuously change at a decreasing rate as it nears the horizon.
For the skeptics, work this example. If a rod moves in a horizontal straight path at constant speed from one horizon to the other, does it appear as a constant length?

Are you sure?

If the balls were connected by some rope, and accelerated so as to maintain a constant separation as measured by you, why would you see the rope break?

As for the Bell 'paradox', using the SR clock synch method, a moving observer A will calculate an inflated speed for a 2nd observer B moving relative to him. (Why is the velocity composition formula used with a 3rd observer in the rest frame?). The inflated speed is a result of the inflated distance because Einstein defines light speed to be c under all circumstances. The end result, the two ships aren't farther apart, and the rope doesn't break, just as in your ball example.
The rope does break because it's the rope's rest frame that determines the physical stress in the rope and the distance between the ships has increased in the ship's rest frame from what it was before they accelerated. This is an accepted result in relativity, did you read the Bell spaceship paradox article?
phyti said:
What you literally see is what was described except the ships or balls have to be offset a distance sideways to view them, therefore you see the length continuously change at a decreasing rate as it nears the horizon.
It's true that the apparent visual length changes depending on the direction of travel relative to the observer, but this is purely an optical effect and has no bearing on questions involving physical stresses like whether the rope breaks in the Bell spaceship paradox.

phyti
The rope does break because it's the rope's rest frame that determines the physical stress in the rope and the distance between the ships has increased in the ship's rest frame from what it was before they accelerated. This is an accepted result in relativity, did you read the Bell spaceship paradox article?

It's true that the apparent visual length changes depending on the direction of travel relative to the observer, but this is purely an optical effect and has no bearing on questions involving physical stresses like whether the rope breaks in the Bell spaceship paradox.

Yes, I've read the Bell paradox and its variations, and discovered there is not total agreement on its resolution.
I see the problem originating with the simultaneity/clock synch convention. There despite using relative light speeds for the transformation equations, he substitutes the propagation speed c. His pupose is obvious and he effectively issues a disclaimer in the 1961 book on SR and GR. It can be demonstrated by using space-time drawings WITHOUT clock synchronization and with relative speeds. No surprise there, because it is...relativity.

The visual effect was a separate response to someones question.

Yes, I've read the Bell paradox and its variations, and discovered there is not total agreement on its resolution.
Do you mean that any mainstream physicists disagree with the idea that the rope will break? (I don't think there are any non-crackpots who disagree with this, but if you think otherwise, please cite some specific sources) Or do you just mean that they have different arguments for why it will break? Of course this does not mean they are disagreeing with each other, there are often multiple arguments that can be used to reach the same conclusion, as with the various resolutions to the twin paradox.
phyti said:
I see the problem originating with the simultaneity/clock synch convention. There despite using relative light speeds for the transformation equations, he substitutes the propagation speed c.
Where does he use "relative light speeds for the transformation equations"? The speed of light is always c relative to any inertial frame, that's a basic principle of relativity.
phyti said:
His pupose is obvious and he effectively issues a disclaimer in the 1961 book on SR and GR.
phyti said:
It can be demonstrated by using space-time drawings WITHOUT clock synchronization and with relative speeds.
Space-time diagrams are based on inertial frames, so the different simultaneity conventions of different frames is built into them. In any case you aren't being very clear in any of your comments--what "can be demonstrated", and what do your vague criticisms of relativistic simultaneity have to do with the Bell spaceship paradox?

phyti
Do you mean that any mainstream physicists disagree with the idea that the rope will break? (I don't think there are any non-crackpots who disagree with this, but if you think otherwise, please cite some specific sources) Or do you just mean that they have different arguments for why it will break? Of course this does not mean they are disagreeing with each other, there are often multiple arguments that can be used to reach the same conclusion, as with the various resolutions to the twin paradox.

Where does he use "relative light speeds for the transformation equations"? The speed of light is always c relative to any inertial frame, that's a basic principle of relativity.