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## Main Question or Discussion Point

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

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