I've forgotten a lot of material and have been trying to go over it to get back up to speed. However, I can't figure out this one:(adsbygoogle = window.adsbygoogle || []).push({});

1, If [itex]P^\mu[/itex] is timelike and [itex]P^\mu S_\mu=0[/itex] then [itex]S^\mu[/itex] is spacelike.

I said that taking [itex]\eta_{\mu \nu} = (-1,+1,+1,+1),[/itex] we could deduce that [itex]P^2=P^0P^0-P^iP^i.[/itex] For this to be timelike, P^2 must be negative and so we can deduce that [itex]P^i>P^0.[/itex]

However, [itex]P^\mu S_\mu=P^0S^0-P^iS^i=0.[/itex]

Given that [itex]P^i>P^0[/itex], the only way for this to be true is if [itex]S^0>S^i \Rightarrow S^0S^0-S^iS^i>0[/itex] and so [itex]S^\mu[/itex] would be spacelike.

The one problem is that I took [itex]P^0P^0-P^iP^i<0[/itex] to imply [itex]P^0-P^i>0[/itex] which I don't think it does since [itex]P^\mu=(-4,1,1,1)[/itex] contradicts this and there is no requirement in the question that either of the vectors be future directed.

What am I doing wrong?

2, If [itex]P^\mu[/itex] and [itex]Q^\mu[/itex] are timelike and [itex]P^\mu Q_\mu<0[/itex] then either both are future directed or both are past directed. Again I proceeded in a similar fashion to above but seeing as I am unconvinced about that, I would like to seek some help for this also.

Thanks a lot.

P.S. How come no combination of LaTeX tags that I try ever works in this forum anymore?

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# Timelike, Spacelike and Null

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