- #1
Dixanadu
- 254
- 2
Hi guys,
So I have a vector which I need to show is timelike. The vector is
[itex] v^{\mu}=t^{\mu}-\frac{t^{\mu}\cdot X^{\mu\prime}}{X^{\mu\prime}\cdot X^{\mu\prime}}X^{\mu\prime} [/itex],
where [itex]t^{\mu}[/itex] is a timelike vector and [itex]X^{\mu\prime}[/itex] is spacelike, however these two vectors are not perpendicular so their dot product does not vanish.
I understand that in order to show that [itex]v^{\mu}[/itex] is timelike, I need to find [itex]v_{\mu}v^{\mu}[/itex] and show that this is greater than 0. So:
[itex]v_{\mu}v^{\mu}=t^{2}-2\frac{(t\cdot X')^{2}}{X^{\prime}\cdot X^{\prime}}+(t\cdot X^{\prime})^{2}[/itex]
So first of all I don't know if this is correct (yes I don't know how to take a dot product obviously lol) and secondly, even if it is, how do I show that this is greater than 0?
Thanks guys!
So I have a vector which I need to show is timelike. The vector is
[itex] v^{\mu}=t^{\mu}-\frac{t^{\mu}\cdot X^{\mu\prime}}{X^{\mu\prime}\cdot X^{\mu\prime}}X^{\mu\prime} [/itex],
where [itex]t^{\mu}[/itex] is a timelike vector and [itex]X^{\mu\prime}[/itex] is spacelike, however these two vectors are not perpendicular so their dot product does not vanish.
I understand that in order to show that [itex]v^{\mu}[/itex] is timelike, I need to find [itex]v_{\mu}v^{\mu}[/itex] and show that this is greater than 0. So:
[itex]v_{\mu}v^{\mu}=t^{2}-2\frac{(t\cdot X')^{2}}{X^{\prime}\cdot X^{\prime}}+(t\cdot X^{\prime})^{2}[/itex]
So first of all I don't know if this is correct (yes I don't know how to take a dot product obviously lol) and secondly, even if it is, how do I show that this is greater than 0?
Thanks guys!