- #1
spaghetti3451
- 1,344
- 34
Homework Statement
I need to prove the following:
a) If ##P^{a}## and ##Q^{a}## are time-like and ##P^{a}Q_{a}>0##, then either both are future-pointing or both are past-pointing.
b) If ##U^{a}##, ##V^{a}## and ##W^{a}## are time-like with ##U^{a}V_{a}>0## and ##U^{a}W_{a}>0##, then ##V^{a}W_{a}>0##.
Homework Equations
Using the 'mostly minus' convention, ##A^a## is time-like, null and space-like if ##A^{a}A_{a}## is ##>0,=0,<0## respectively.
A time-orientation is chosen by taking at will some time-like vector, say ##U^{a} = (1,0,0,0)##, and designating it to be future-pointing. Any other time-like or null vector ##V^{a}## such that ##g_{ab}U^{a}V^{b}>0## is also future-pointing, whereas if ##g_{ab}U^{a}V^{b}<0##, then ##V^{a}## is past-pointing.
The Attempt at a Solution
a) If ##P^{a}## is time-like future-pointing and ##P^{a}Q_{a}>0##, then (by definition) ##Q^{a}## is also future-pointing.
Simialarly, for ##P^{a}## past-pointing.
b) If ##U^{a}## is time-like future-pointing and ##U^{a}V_{a}>0##, then (by definition) ##V^{a}## is also future-pointing.
If ##U^{a}## is time-like future-pointing and ##U^{a}W_{a}>0##, then (by definition) ##W^{a}## is also future-pointing.
Since both ##V^{a}## and ##W^{a}## are future-pointing, ##V^{a}W_{a}>0##.
Similar argument for ##U^{a}## past-pointing.
Are my proofs sound?