Four Velocity Sign of Time: \dot t>0?

In summary, four velocity is not defined for light-like paths and it is convenient to have the world-line parameter defined such that ##\dot{t}>0##. For massive particles, the proper time can be chosen as a natural world-line parameter, while for massless particles any affine parameter can be chosen. In both cases, it is natural to have ##\dot{t}>0##, representing a motion into the future.
  • #1
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Is it generally the case even with light like paths that ##\dot t>0##?
Is it generally the case even with light like paths that ##\dot t>0##?
 
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  • #2
A four velocity is not defined for a light like path.
 
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  • #3
Up to you, really. It is true that all future-pointing vectors will have the same sign in their time component, assuming your time coordinate is reasonably named and the spacetime has a global distinction between past and future. But there's nothing to stop you having your time coordinate increase towards the past, in which case all future-pointing four vectors would have negative time components.

As @Sagittarius A-Star points out, four velocity is not defined for null paths. However, you can define other four vectors tangent to null curves, such as the four momentum.
 
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  • #4
It's convenient to have the world-line parameter defined such that ##\dot{t}>0##. For massive particles you have time-like worldlines, and you can choose the proper time, ##\tau## as a natural world-line parameter. Then the four-velocity is "normalized": ##u_{\mu} u^{\mu}=c^2##.

For massless particles ("naive photons") of course you cannot choose proper time, because it's not defined but you can choose any affine parameter you like. Then you have ##\dot{x}^{\mu} \dot{x}_{\mu}=0##, i.e., light-like worldlines.

In both cases it is natural to choose ##\dot{t}>0##, where ##t## is the time-like coordinate since then with increasing world-line parameter you describe a motion into the future.
 
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