I fear there's so much maths involved we'd need Latin, Greek, Cryillic, and Egyptian Hieroglyphs just to name all the variables...
But I think what Cox is getting at is that we move the same distance per unit time through
spacetime, but not necessarily through space (or, indeed, time). If we weren't moving through space, we'd be moving at
c through the time axis. If we were travelling at
c through space, we'd be stationary with respect to the time axis. If you imagine two axes, space at the vertical axis and time on the horizontal,
c is a straight line of a constant distance. This can move about, though, with its end point lying anywhere on a circle - as our spatial velocity increases from zero, this 'line' moves from the time axis up and around till it's right on the spatial axis - sort of like a speedometer?
This is a neat explanation of time dilation and length contraction. When stationary in space, we're zipping through time, nice as you like. As soon as we start accelerating, we move less quickly through time (time dilation) as this little arrow starts to move upwards. Its distance along the time axis shortens, and its distance along the spatial axis lengthens.
I don't know how clear that is, but in my head it makes perfect sense
Turns out my whole brain-fry was a result of something I overlooked in the text. Specifically, this:
