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Consider a four-velocity vector $\mathrm{u}=(u^0,u^1,u^2,u^3)$. The zeroth component of position is $x^0=ct$.

How do I know if $\mathrm{u}$ is directed towards the future or towards the past? Is it correct to look at the zeroth component of $\mathrm{u}$ and

  • if $u^0>0$ then $\mathrm{u}$ is towards future
  • if $u^0<0$ then $\mathrm{u}$ is towards past

?

For example $\mathrm{u}=(3c,c,c,c)$ is towards future, while $\mathrm{u}=(-3c,c,c,c)$is towards past.


If this is not correct, what is the way to know if $\mathrm{u}$ is directed towards the future or towards the past?

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    $\begingroup$ What you have said is true assuming that your chosen coordinate $x^0$ increases towards the future. $\endgroup$ – Prahar Jun 6 at 15:17
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Yes, with a standard coordinate system the timelike basis vector is future directed so positive components indicate the future direction.

Note, the Lorentz transform preserves the future orientation of timelike vectors. In general, you can smoothly transform any spacelike vector into any other, but you cannot smoothly transform a future directed timelike vector to a past directed one.

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