- #1

DrGreg

Science Advisor

Gold Member

- 2,253

- 734

## Main Question or Discussion Point

Coordinates are sometimes described as "null coordinates". An example in SR is the coordinate [itex]u = x - ct[/itex]. Another example is one of the coordinates in the Eddington-Finkelstein metric. But I've never seen an explicit rigorous definition of a null coordinate. The defining property seems to be that any hypersurface of constant coordinate should possess a null tangent vector at every event within it.

My first thought was to say that [itex]x^0[/itex] is a null coordinate if the tangent vector [itex](dx^0, 0, 0, 0)[/itex] is null. But it didn't take long, considering the above examples, to realise this was wrong. Should I, instead, be considering the cotangent covector [itex](dx_0, 0, 0, 0)[/itex]? If I haven't made a mistake in my maths, this seems to work for the examples above, but I can't convince myself it will always work.

And then what about "timelike" coords and "spacelike" coords? Can they be defined in a similar way?

If I have an explicit equation for a metric, e.g.

is there a quick way of inspecting this to classify each of the four coordinates as timelike, spacelike, or null?

My first thought was to say that [itex]x^0[/itex] is a null coordinate if the tangent vector [itex](dx^0, 0, 0, 0)[/itex] is null. But it didn't take long, considering the above examples, to realise this was wrong. Should I, instead, be considering the cotangent covector [itex](dx_0, 0, 0, 0)[/itex]? If I haven't made a mistake in my maths, this seems to work for the examples above, but I can't convince myself it will always work.

And then what about "timelike" coords and "spacelike" coords? Can they be defined in a similar way?

If I have an explicit equation for a metric, e.g.

[tex]ds^2 = -\left(1-\frac{2M}{r} \right) du^2 - 2 du dr + r^2 \left(d\theta^2 + \sin^2 \theta\, d\phi^2 \right) [/tex]

is there a quick way of inspecting this to classify each of the four coordinates as timelike, spacelike, or null?