I Non-radial geodesics in Schwarzschild spacetime

[timmdeeg]

Consider a non-radial timelike geodesic outside the event horizon. Will it nevertheless cross the horizon radially or are non-radial geodesics also possible inside? I couldn't find any reference regarding a possible angle dependence in this respect.

 

[Orodruin]

The angular momentum will prevent it from crossing the event horizon "radially". Inside the horizon you have to realize that "radially" as in "in the direction of the r coordinate" is not a spatial direction, but a temporal one. However, to answer your question, there will naturally be geodesics where the angular coordinats $\theta$ and $\varphi$ are not fixed.

 

[timmdeeg]

The angular momentum will prevent it from crossing the event horizon "radially". Inside the horizon you have to realize that "radially" as in "in the direction of the r coordinate" is not a spatial direction, but a temporal one. However, to answer your question, there will naturally be geodesics where the angular coordinats $\theta$ and $\varphi$ are not fixed.

Thanks. Does it mean that the proper time to reach the singularity will exceed $\tau={\pi}GM/c^3$ (for the radial trajectory) depending on the angular coordinates? And if yes, is there an upper bound? It would be great if you could show the respective formula.

 

[Orodruin]

Thanks. Does it mean that the proper time to reach the singularity will exceed $\tau={\pi}GM/c^3$ (for the radial trajectory) depending on the angular coordinates? And if yes, is there an upper bound? It would be great if you could show the respective formula.

To be honest, I have not thought much about it, but I would not think so. My intuitive feeling is that it would actually decrease the proper time to reach the singularity.

 

[PAllen]

We had a recent thread touching on this. The longest possible proper time for any timelike worldline from horizon to singularity is attained by a radial geodesic corresponding to a drop from just outside the horizon. There is an exercise in MTW establishing this.

 

[PeterDonis]

The longest possible proper time for any timelike worldline from horizon to singularity is attained by a radial geodesic corresponding to a drop from just outside the horizon.

IIRC the discussion in that other thread, the paper linked to there, and the MTW exercise, only considered radial motion.

 

[PAllen]

IIRC the discussion in that other thread, the paper linked to there, and the MTW exercise, only considered radial motion.

No, MTW exercise 31.4 does not restrict itself to radial motion. That the maximizing geodesic is radial is something to conclude, not assume.

 

[PeterDonis]

MTW exercise 31.4 does not restrict itself to radial motion.

Ah, ok. I'll check it out when I have a chance to dig out my copy of MTW.

 

[Orodruin]

Ah, ok. I'll check it out when I have a chance to dig out my copy of MTW.

I thought all copies of MTW had collapsed to black holes by now ... if so you might have difficulties getting the information out ...

 

[PAllen]

I thought all copies of MTW had collapsed to black holes by now ... if so you might have difficulties getting the information out ...

I thought MTW disproved the principle of equivalence - it falls faster than everything else.

 

[PeterDonis]

I thought MTW disproved the principle of equivalence - it falls faster than everything else.

Hmm...maybe I'll dig up my copy of Wald as well as MTW and run the experiment. I'll need a vacuum chamber, though, to be sure I've eliminated possible confounding factors.

 

[timmdeeg]

No, MTW exercise 31.4 does not restrict itself to radial motion. That the maximizing geodesic is radial is something to conclude, not assume.

Thanks for mentioning that. In exercise 31.4 MTW note the hint: ... show that the geodesic of longest proper time lapse between $r=2M$ and $r=0$ is the radial geodesic, ...

Could one argue heuristically (having the light cone inside the event horizon in mind) that the proper time lapse of timelike non-radial geodesics decrease and approach $0$ as their trajectory approaches the null geodesics?