I have a hard time trying to figure out Lorentz transformations at the moment; I don't believe in just accepting math equations on faith without understanding EXACTLY what is going on.
Also I should note that Rene Descartes' Cartesian geometry, i.e. his co-ordinate system, is a human invention. I would like to translate the Lorentz transformations into a more general form involving just simple jumps and matches of pattern.
But what happens: you go outside and look at the stars. Suddenly two bright stars seem to appear in the sky simultaneously. Turns out they are supernovas. But although they look as if they blew up at the same time; one of them could be much further away than the other, its light taking millions more years to reach you. So your experience of 'simultaneity' was relative.
Suppose the supernovas appeared to be extremely close together, like a double star. But that could be an illusion of perspective; one could be much further away beyond the other. So your experience of 'simultaneity' of space was relative.
(If the light from the supernovas travelled instantly to you, your experience of simultaneity in your-time would be accurate. If you could travel instantly from where you are to one then to the other, yet measure the distance between, you would have an accurate space-measure).
"Interval" is what they call the fixed relationship between two events mapped so as to account for space and time. If lots of people observed these two supernovas from many different distances and directions; pooling all their results would give an absolute space-time 'location'. An interval between two such space-time 'locations' could be very small.
True, it requires time to go from one place to another; so you can't go from one now-place to another now-place because by the time you get to the other now-place its become a future-place (from the perspective of your starting now-place. (Unless you travel instantly).
And it requires space to go from one time to another; sitting still you automatically go from now-here to next-here- you may think you are still in the same space but from a space-time perspective your old space is way back in time-space. Sounds a bit dodgy but I think that's how this space-time business must work.
On photons- Alex appears to be right. Thanks to R.W. Ditchburn's text "Light" I roughly follow what he's saying; it helps to know that 'mass' is a form of frequency (as is energy).
See pages 699-700 "Light" by R.W. Ditchburn (1976)
However, according to a philosophy text; the reductive fallacy is widespread and simple. People experienced gravity long before they knew the equation. The experience and the equation both exist.
I would say: Chess rules may describe a particular game of chess very mathematically well, but the game does not reduce to the math rules. The game is something other than that.