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If an object moves close to the speed of light time will slow down in its reference frame as seen by a stationary observer. If my understanding is correct, this means that all movement - such as the ticking of a clock - would appear to be slowed down in the moving frame.

So, if the above is correct, then the movement of particles that compose the object will also appear to be slower which means they will have less kinetic energy and therefore the object itself will look colder for someone who measures it while being stationary.

Also, if this is true, then the object would also appear to give off less thermal radiation according to the stationary observer. But this doesn't seem to be right, since the properties electromagnetic waves should be the same in both reference frames.

What am I misunderstanding?

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But this doesn't seem to be right, since the properties electromagnetic waves should be the same in both reference frames.

You are right about everything. Except for that quote. So that must be the thing that you are misunderstanding.

So maybe give some more info about that part, unless this answer is perfect :)

As an example of properties electromagnetic waves not being same in different frames: energy and color of light are different in different frames.

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Not necessarily. If an object is moving away from you, it will appear red-shifted (ie cooler), but if it is moving towards you it will appear blue-shifted (ie warmer).

There are two separate factors that you need to distinguish, one being time dilation and the other being the Doppler effect.

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