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spaghetti3451
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Homework Statement
Prove the following:
a) If ##P^a## is time-like and ##P^{a}S_{a}=0##, then ##S^{a}## is space-like.
b) If ##P^a## is null and ##P^{a}S^{a}=0##, then ##S_{a}## is space-like or ##S^{a} \propto P^{a}##.
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.
The Attempt at a Solution
a) ##S^{a}S_{a} = S^{a}S_{a} + P^{a}S_{a} = (S^{a} + P^{a})S_{a} = (S^{a} + P^{a})(S_{a} + P_{a}) = S^{a}S_{a} + P^{a}S_{a} + P_{a}S^{a} + P^{a}P_{a} = S^{a}S_{a} + P^{a}P_{a} = ?##