NanakiXIII
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When looking at time dilation, I once came across a piece (I don't remember where) that said you could view time dilation as follows. Everything moves through four-dimensional spacetime at a constant velocity c. Something stationary is only moving in the time dimension. Something that has a velocity v in a spatial direction, however, cannot move at velocity c in the time dimension since its total velocity would change. Thus its velocity in the time dimension is decreased, hence time dilation.
I thought that was a nice thought, but I never came across anything that verified that it makes any sense. Today I was prompted to consider this statement a bit more quantitatively. It works if you take the vector (my reference frame being S and there being some reference frame S' with spatial velocity v with respect to S)
[tex]r_4 = (c t', x, y, z) = (\frac{c}{\gamma} t, x, y, z)[/tex]
so that
[tex]v_4 = \.{r}_4 = (\frac{c}{\gamma}, \.{x}, \.{y}, \.{z})[/tex].
The length of this vector is c, as per
[tex]v_4 \cdot v_4 = c^2 (1 - \frac{v^2}{c^2}) + v^2 = c^2[/tex].
So, defining this vector things might make sense. What I'm wondering, however, is how much sense it makes to define this vector. It takes the spatial co-ordinates from S and the time co-ordinate from S', which seems odd. Does differentiating ct' make any sense if you want the velocity in the time dimension?
I couldn't find anything about this and I have no idea what it would be called, which makes searching for it a bit difficult. It sounds somewhat like a layman's explanation and I'm fairly sure I came across this in a very much simplified treatment of Special Relativity, but I'd like to hear what anyone has to comment on it, whether it has any merit.
I thought that was a nice thought, but I never came across anything that verified that it makes any sense. Today I was prompted to consider this statement a bit more quantitatively. It works if you take the vector (my reference frame being S and there being some reference frame S' with spatial velocity v with respect to S)
[tex]r_4 = (c t', x, y, z) = (\frac{c}{\gamma} t, x, y, z)[/tex]
so that
[tex]v_4 = \.{r}_4 = (\frac{c}{\gamma}, \.{x}, \.{y}, \.{z})[/tex].
The length of this vector is c, as per
[tex]v_4 \cdot v_4 = c^2 (1 - \frac{v^2}{c^2}) + v^2 = c^2[/tex].
So, defining this vector things might make sense. What I'm wondering, however, is how much sense it makes to define this vector. It takes the spatial co-ordinates from S and the time co-ordinate from S', which seems odd. Does differentiating ct' make any sense if you want the velocity in the time dimension?
I couldn't find anything about this and I have no idea what it would be called, which makes searching for it a bit difficult. It sounds somewhat like a layman's explanation and I'm fairly sure I came across this in a very much simplified treatment of Special Relativity, but I'd like to hear what anyone has to comment on it, whether it has any merit.