- #1

- 10

- 0

Show that a timelike vector cannot be orthogonal to a null vector.

Timelike:

X[tex]^{2}[/tex] = g(sub a b)X[tex]^{a}[/tex]X[tex]^{b}[/tex] > 0

Null:

X[tex]^{2}[/tex] = g(sub a b)X[tex]^{a}[/tex]X[tex]^{b}[/tex] = 0

In order for them to be orthogonal...

g(sub a b)X[tex]^{a}[/tex]Y[tex]^{a}[/tex] = 0

I know from the line element of Minkowski space-time you have

ds[tex]^{2}[/tex] = dt[tex]^{2}[/tex] - dx[tex]^{2}[/tex] - dy[tex]^{2}[/tex] - dz[tex]^{2}[/tex]

I have played around with substituting the line element into the equation that specifies if it is orthogonal, but to no success. Any help would be great!

Sorry for the bad writting, I couldn't get latex to do subscripts, it only did them in superscripts....

Thanks,

Chris