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I thought I'd expand on some of my earlier remarks, with a bit more sophistication this go-around, with respect to the original questions about interpretation.
As far as interpretation goes, it's convenient and widespread (though perhaps not universal) to regard tensor quantities as fundamental.
Tensors are defined by their transformation properties. Rank 0 tensors, or scalars, have the simplest transformation possible. They have the same value for all observers.
Pre-relativity, distance was a tensor - if you measure it in one reference frame, it has the same value in all. This is obviously the simplest possible transform law for a scalar - something that doesn't change. Post relativity, distance is no longer a tensor - it's dependent on the reference frame. Instead, the Lorentz interval becomes the candidate for the fundamental quantity of interest, because of its tensor nature.
Rank 1 and higher tensors are important to relativity as well, but I'm going to skip over all the mathematical details. I'd like to encourage people to find out more about tensors, but I'm not quite sure where to point them, alas.
How did the tensor nature of distance get lost? Well, the manner in which we transform between frames changed When you have a moving frame with coordiantes (t',x') and a stationary frame with coordinates (t,x), there's some mapping from (t,x) to (t', x').
Pre-relativity, the mapping was defined by the Gallilean transform, t'=t, x'=x-vt v being the relative velocity between frames. Post relativity, the mapping is the Lorentz transform, t' = \gamma(t - vx/c^2), x' = \gamma((x - vt).
Changing the the way in which we transform between frames, changed distance from a fundamental tensor quantity independent of the observer, to a less fundamental non-tensor quantity that is observer-dependent.
The reason for choosing the more complex and less intuitive Lorentz transform as the way to switch between frames boils down to agreement with experiment.
As far as interpretation goes, it's convenient and widespread (though perhaps not universal) to regard tensor quantities as fundamental.
Tensors are defined by their transformation properties. Rank 0 tensors, or scalars, have the simplest transformation possible. They have the same value for all observers.
Pre-relativity, distance was a tensor - if you measure it in one reference frame, it has the same value in all. This is obviously the simplest possible transform law for a scalar - something that doesn't change. Post relativity, distance is no longer a tensor - it's dependent on the reference frame. Instead, the Lorentz interval becomes the candidate for the fundamental quantity of interest, because of its tensor nature.
Rank 1 and higher tensors are important to relativity as well, but I'm going to skip over all the mathematical details. I'd like to encourage people to find out more about tensors, but I'm not quite sure where to point them, alas.
How did the tensor nature of distance get lost? Well, the manner in which we transform between frames changed When you have a moving frame with coordiantes (t',x') and a stationary frame with coordinates (t,x), there's some mapping from (t,x) to (t', x').
Pre-relativity, the mapping was defined by the Gallilean transform, t'=t, x'=x-vt v being the relative velocity between frames. Post relativity, the mapping is the Lorentz transform, t' = \gamma(t - vx/c^2), x' = \gamma((x - vt).
Changing the the way in which we transform between frames, changed distance from a fundamental tensor quantity independent of the observer, to a less fundamental non-tensor quantity that is observer-dependent.
The reason for choosing the more complex and less intuitive Lorentz transform as the way to switch between frames boils down to agreement with experiment.
