Incoming spacelike radial geodesic

[deadringer]

We need to show that using the Schwarzschild metric, an incoming radial spacelike geodesic satisfies r>= 1m/(1 + E^2)

I know that E = (1-2m/r) is constant, and I think that ds squared should be negative for a spacelike geodesic. I try substituting E into the metric and setting ds squared <=0 but this does not give the required expressions. I'm also unsure about the meaning of "incoming" - does this mean dr/ds or dr/dt < 0?

 

[Dick]

This is not making sense to me on so many levels. If E = (1-2m/r) is a constant then r is a constant. And that's hardly 'incoming'. What are the geodesic equations in the Schwarzschild metric with constant angular coordinates?

 

[deadringer]

Sorry I meant to say that E = (1-2m/r) dt/ds is constant (this is one geodesic equation).

 

[deadringer]

The other geodesic equation gives us d^2 s/dt^2 which I don't think is useful.

 

[Dick]

Ah, ok. Finally straightened out. And the '1' in the numerator of the radius inequality is supposed to be a '2'. I think your main problem is trying to set ds^2 to be negative in some vague way. Spacelike or timelike is determined by the sign of g*t*t - where g is the metric tensor and t is a tangent vector. Further, in the case of a nonnull geodesic you can set this quantity equal to plus or minus one (defining proper time). Which that is depends on whether the geodesic is timelike or spacelike (and your metric convention).

 

[deadringer]

Thanks very much Dick. It's all clear now.