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I heard this theory yesterday:

If something is not moving in space, then it is moving on the time axis at the speed of light.

I realize that in essence there is no object which can be considered as "not moving in space".

So my question is rather theoretical:

Given an object with zero velocity on all three axises, is it moving on the time axis at the speed of light?

Does the law of conservation of energy (or any other law in physics) imply that this condition must hold?

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  • $\begingroup$ in special relativity nothing is moving only in space or only in time, but in space-time :) $\endgroup$ – Nikos M. Oct 22 '14 at 14:29
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    $\begingroup$ Although this isn't an answer at all, I find it interesting that if you think about it, nothing is ever moving at all from light's perspective. Even if you are going 99.999% the speed of light WRT some other object (by the way, you are), you are still standing absolutely still from light's perspective. Also implies that there is just one "light" that is everywhere--the light you see now is the exact same light I see now. Lots of bizarre implications once you try to see things from light's POV. $\endgroup$ – Bill K Oct 22 '14 at 16:11
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Square the proper time $d \tau = dt / \gamma$, multiply by $c^2$, rearrange, and take the square root: $$ \left(\frac{d\tau}{dt}\right)^2 = \gamma^{-2} = 1 - \left(\frac{\bf{v}}{c}\right)^2 \Rightarrow \sqrt{\left(c \frac{d\tau}{dt}\right)^2 + {\bf{v}}^2} = c $$ The proper time $d\tau$ is the time elapsed in the frame moving with respect to the lab frame, in which the elapsed time is $dt$. So, $c \ d\tau / dt$ is the "speed through time," and

  • $\left|d{\bf r}/dt\right| = 0 \Leftrightarrow c \ d\tau / dt = c$. This is why people say "if something is not moving in space, then it is moving on the time axis at the speed of light."
  • $d\tau / dt = 0 \Leftrightarrow \left|d{\bf r}/dt\right| = c$. This is why people say light doesn't age, because it always travels at speed $c$ through space.
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    $\begingroup$ This is why people say "if something is not moving in space, then it is moving on the time axis at the speed of light." "People" is the popularizer Brian Greene. Relativists in general don't talk this way. $\endgroup$ – user4552 Oct 22 '14 at 15:12
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    $\begingroup$ Fair enough @Ben Crowell. I think I first heard this statement in Greene's The Elegant Universe. It's interesting that "light doesn't age" seems to be used more regularly than "objects at rest move through time at the speed of light," when in some sense they're on equal footing. Agree / disagree? $\endgroup$ – Eric Angle Oct 23 '14 at 13:29
  • $\begingroup$ "if something is not moving in space, then it is moving on the time axis at the speed of light." If this is the case, can we say that light moves through one axis only (Time or Space) ; If so which axis ? $\endgroup$ – Hammar Feb 12 '17 at 14:12
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    $\begingroup$ @Hammar Yes, light moves through 3 dimensional space only, at speed $c$. $\endgroup$ – Eric Angle Feb 12 '17 at 17:12
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Yes it is - well, sort of.

The coordinate invariant form of velocity is the four velocity, and the magnitude of any four velocity is always $c$ (or $1$). So even if you are stationary in space in your chosen coordinate system the magnitude of your four velocity is still $c$.

Whether moving at the speed of light on the time axis is a good way to state this is debatable. What it actually means is that $dt/d\tau = 1$.

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    $\begingroup$ You're probably aware that this is how Brian Greene talks about things - been watching his videos with my daughter - just to let you know where this is probably coming from if you're not. $\endgroup$ – Selene Routley Oct 22 '14 at 11:22
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I realize that in essence there is no object which can be considered as "not moving in space".

No object at all is moving in space if you are taking the point of view of its reference frame!

The law of conservation of energy is requiring that the energy of its mass ($e = mc^2$) is "transported through time", or in other words, that time is passing for the massive object.

The contrary is happening for massless particles such like photons: they are travelling at speed of light, but from their (hypothetical) point of view they have no proper time, and their energy is not transported through time.

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