Lightcones in advanced Eddington-Finkelstein coordinates

[peter46464]

I'm trying to understand the basics of GTR.

My textbook shows the worldline of an object freely falling into a Schwarzschild black hole in advanced Eddington-Finkelstein coordinates as neatly passing through a series of lightcones as it curves up right to left on the page toward the singularity. What I don't understand is why the worldline, after going through the origin of each lightcone, then travels neatly along each cone's axis. Is that deliberate or what? There's no explanation in the text.

And, is it sensible to similarly ask how a freely falling object would pass through a lightcone in Schwarzschild coordinates (in the region where r is greater than the Schwarzschild radius)?

Thank you

 

[Mentz114]

Do you mean traveling through the vertex ?

The faller is not falling through the vertices, rather, the worldline comes first and light-cones added. Any observer is always at the vertex of their current light-cone and any point on a worldline is a possible position of the observer on the world-line.

 

[peter46464]

I understand that. But why does the worldline follow the axis of the lightcone rather than any other path from the tip/vertex?

 

[Mentz114]

I understand that. But why does the worldline follow the axis of the lightcone rather than any other path from the tip/vertex?

Do you mean the axis of symmetery ?

This is because in the local frames defined by the tangent space of the worldline at each point, the light cone is always symmetrical because of the local isotropy of the speed of light. Probably.

 

[peter46464]

Yes, the axis of symmetry. I understand that the lightcones are symmetrical. But why does the textbook illustration show freely falling particles move along the axis of symmetry? I mean, there's any number of ways they could pass through the vertex and exit the cone. Why along the axis of symmetry?

Thank you

 

[Mentz114]

You will remember from special relativity that observers who are boosted wrt the elected rest frame pass asymmetrically through the light-cone ( which is global). The freely-falling observer is by definintion at rest in an inertial frame, so the light-cone is symmetrical around the world-line. That is, the axis to the worldline is in the direction of $\partial_0$ the timelike local basis vector.

You could say the light-cones are tilted by the coordinate transformation, which depends on M through the curvature.

 

[peter46464]

Thanks. That's sort of clearer, though I don't really understand "timelike local basis vector".

 

[PeterDonis]

But why does the textbook illustration show freely falling particles move along the axis of symmetry? I mean, there's any number of ways they could pass through the vertex and exit the cone. Why along the axis of symmetry?

I don't know which textbook you are using, but I suspect that by "freely falling particles" they really mean "particles freely falling from rest at infinity". In other words, they are not just any freely falling particles, but freely falling particles with a very specific initial condition. The other ways among the "any number of ways" that a freely falling particle could pass through the vertex of the light cone correspond to other initial conditions.

 

[peter46464]

I was looking at Lambourne (Relativity, Gravitation and Cosmology) p190 and Misner et al (Gravitation) p829 - which is more difficult for me to understand. However, you can clearly see in the Misner diagram that the worldline goes straight along the cone's axis of symmetry.