# Running on a treadmill at close to light speeds

A debate has just cropped up.

A person running on a treadmill where the belt below him is travelling close to the speed of light.
(ignoring physical stress of course)

would time dilation effects occur?  I think they wouldn't to the person on the treadmill.

How about the belt?  Suppose it was made of rubber that perished after 5 years, would it hypothetically last for longer due to time dilation? Could you run on it for say 50 years and find it had only aged 1 year?
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Commented:
The runner would have partial dilation.  The feet are moving at near-c, to match the treadmill belt.  If not, the runner is going to be hurled off the belt.
The legs would be moving slower, hips slower and torso minor rotation/bounce and head relatively stationary.

I'm not sure about the belt, intuitively I'd say yes, it'd last longer, but I also feel I'm forgetting something
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Commented:
Since time dilation is a function of the relativevelocities of the objects, then you're looking at a difference of close to 2c between the feet and the belt (for 1/2 of the time).

This hasn't been studied much, but the oscillation of the velocity vector in the feet (i.e, ~c one direction, then the opposite) is surely going to have some very odd properties (not to mention the force due to the acceleration/decceleraion being enough to move the entire known universe).
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Commented:
One attacks this problem by carefully defining the frame of reference to use. There are several: the belt. the feet, the head. In some frames you have an acceleration with which to  contend. Accelerations are hard to handle in special relativity.
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Commented:
Since the belt is moving in different directions, any frame will see dilation.
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Commented:
The feet are exceeding the speed of the belt.  They must accelerate faster than the belt in order to slow down for the shock of landing.  The subsequent deceleration brings foot speed back down to the same speed of the belt.

This is the same on the kick.  The foot will accelerate as load is released, and will briefly move faster toward the buttocks.

So, the feet will age slower than both the belt and the butt.  Head bobbing up/down and fore/aft means the head will be younger than the butt also.
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Commented:
>>would time dilation effects occur?

Time dilation effects always occur.

>>A person running on a treadmill where the belt below him is travelling close to the speed of light.

The person will be running at a normal speed when he looks down at the belt. For him the belt is fixed and he is running on it. In fact for him the belt is moving in an oblated circle with the same angular velocity as he is running.

For the trainer standing in the gym near the belt the runner is running fractionly faster than the belt is moving. The belt is moving at near light speed. The trainer does NOT see the runner running faster than light but only a little bit faster than the belt and complains to the runner that he's not running fast enough! That would be a typical time dilation effect. For the trainer the runner on the belt is not "aging" enough (or as one might say normally) - which means that the effect over time is slower, ie: he's not running fast enough.

>>ignoring physical stress of course

Of course. The force required to turn a rubber belt around an angle of 360 degrees at near light speeds is almost unimaginable!

>>Suppose it was made of rubber that perished after 5 years

As far as I know there is no experimental evidence to support this sort of hypothetical question. Theorecitally the processes of decomposition are electromagnetic and that means that the processes will also suffer time dilation. Ostensibly one could imagine the one observer seeing the belt fall apart at the same "instant" that the other observer sees it still intact. I shall have to do the maths to sort that one out and take a rain check.
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Commented:
@BigRat: Whether the decomposition is electromagnetic in nature or not does not matter. The dilation applies also to e.g. proton decay (even though Einstein worked only on electromagnetism)

The runner will see a fast moving belt , hence a slowly decaying belt.
The belt will see a fast moving runner, hence a slowly aging runner.
A paradox *seems* to come up if we assume that both the runer and the belt perish after 5 years - will the runer collaps on an intact belt or the other way round?
The solution is of course synchronicity. If the very long belt deacys at one instant in its own frame of reference the runner will observe decay at different times for differnt parts of the belt.
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