How do you calculate the light cone for the following line element?

[Raziel2701]

Homework Statement

Consider the two-dimensional spacetime spanned by coordinates (v,x) with the line element

$$ds^2=-xdv^2 +2dvdx$$

Calculate the light cone at a point (vx)

The Attempt at a Solution

I don't even know how the light cone for flat spacetime is calculated. So if that one's easier to explain or understand I'd like to start there. In that one for instance, I don't know how it was calculated that 45 degree lines are reserved for things moving at lightspeed.

In the case of the line element of the problem, I don't know what it would look like compared to the flat spacetime element.

Ultimately I just don't know squat about manipulating these and extracting information from them.

 

[fzero]

The light cone is composed of all multiples of the null vectors at the point. A null vector $$n^\mu$$ satisfies $$g_{\mu\nu} n^\mu n^\nu =0$$.

For the flat metric in the usual form:

$$ds^2 = -dt^2 + dx^2,$$

this condition is just $$-(n^0)^2 + (n^1)^2=0$$. The solutions are $$n^0 = \pm a$$, $$n^1 = \pm a$$, where $$a$$ is any real number. These give the 4 lines that make $$45^\circ$$ angles with respect to the $$t,x$$ axes.

For your metric the calculation will be similar, but the solutions are very different.

 

[Raziel2701]

That makes a lot of sense, thank you very much.