# [SR] Constant four-velocity c

1. Mar 3, 2009

### NanakiXIII

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)

$$r_4 = (c t', x, y, z) = (\frac{c}{\gamma} t, x, y, z)$$

so that

$$v_4 = \.{r}_4 = (\frac{c}{\gamma}, \.{x}, \.{y}, \.{z})$$.

The length of this vector is c, as per

$$v_4 \cdot v_4 = c^2 (1 - \frac{v^2}{c^2}) + v^2 = c^2$$.

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.

2. Mar 3, 2009

### Fredrik

Staff Emeritus
Post #7 in this thread may be useful.

3. Mar 3, 2009

### JesseM

Brian Greene is the one who I've seem summarizing relativity in terms of the "everything travels at c through spacetime" idea, and I quoted the math he uses to justify this in post #3 here.