- #1
kontejnjer
- 72
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Homework Statement
Show that if [itex]x^{\mu}[/itex] is timelike and [itex]x^{\mu}y_{\mu}=0[/itex], [itex]y^{\mu}\neq 0[/itex], then [itex]y^\mu[/itex] is spacelike.
Homework Equations
[itex]ds^2=\\>0\hspace{0.5cm}\text{timelike}\\<0\hspace{0.5cm}\text{spacelike}\\0\hspace{0.5cm}\text{lightlike}[/itex]
metric is [itex]diag (+---)[/itex]
The Attempt at a Solution
Don't know if this is the correct way, but here goes: assuming that [itex]x^\mu[/itex] is timelike, we can pick a reference frame in which [itex]x^\mu=(x^0,\vec{0})[/itex], so due to invariance [itex]x^{\mu}y_{\mu}=x^0 y_0=0\rightarrow y^0=0[/itex], but, since it is stated that [itex]y^{\mu}\neq 0[/itex], then [itex]\vec{y}\neq 0[/itex], and hence we have [itex]y^\mu y_\mu=y^i y_i=-(\vec y)^2<0[/itex], making [itex]y^\mu[/itex] indeed spacelike. Am I missing something here or is this the right procedure?