The Speed of Light: Is a Second is a Second is a Second?

[Grimble]

I have had two things pointed out to me repeatedly: the first as in the label for this thread and that measurements in one frame are not necessarily equal to similar measurements in another frame; i.e. that the length of a metre measured in one frame is not the same as the length in another frame.
Yet, if the speed of light is the same in all frames then the length of a metre, which is defined according to the speed of light and a second is the same in every frame, then surely the metre, and all other measurements must be identical too?
That is not to say that a proper length as measured in a frame at rest, is the same as a coordinate metre in a moving frame, but that the proper metre, measured at the origin of any frame at rest must be equal to a proper metre measured in the same way in any other frame.

Or am I misunderstanding the terms here?

 

[Orodruin]

Yet, if the speed of light is the same in all frames then the length of a metre, which is defined according to the speed of light and a second is the same in every frame, then surely the metre, and all other measurements must be identical too?

No, you have a fundamental misunderstanding here regarding what a second is. There are two notions of time in special relativity. One of the notions is coordinate time, the t or t' that you use to describe when events occur. The second is proper time, which is the notion of time elapsed by a clock following a given world line.

The proper time for a clock is dependent only on the path taken in spacetime between two events and will be the same regardless of the frame you use to describe it. The coordinate time is what appears in Lorentz transformations and does differ between frames. It is the coordinate time you use when determining velocities together with coordinate length, which is also frame dependent.

 

[russ_watters]

A couple of things

First, distance equals speed times time. So if everyone measures the same speed of light, but different distances, they must also measure a different elapsed time. That's why time dilation and length contraction are two sides of the same coin.

Second, it was said above and I want to emphasize that these are measurements of the same object/event from different frames. If you measure the length of a meter stick sitting in front of you, with a laser and a clock, and your friend in a spaceship does the same with his meter stick and laser and clock, you'll get the same results. But if you measure HIS meter stick with YOUR laser and clock, you may not.