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
Onyx
120
3
TL;DR Summary
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##?
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.
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.