Robert McNees
Black holes!
The plot below shows two observers in the Schwarzschild spacetime. One falls inward along a timelike radial geodesic (purple), another (green) remains at a fixed value of the radial coordinate r.
The infalling observer emits radial signals (red) at equal proper time intervals. These signals move outward along null geodesics, and the stationary observer receives them with decreasing frequency.
Robert McNees
Outside the black hole (r > 2M) there are two sorts of radial null geodesics: light rays can follow a path that moves outward (towards larger r) or one that moves inward (towards smaller r).
Inside (r < 2M) the two sorts of null geodesics both head inwards, towards smaller r.
Notice that at each point along the infalling observer's worldline, they find themself inside a "light cone" defined by the two sort of radial null geodesics emitted at their previous location, the instant before.
Robert McNees
When r > 2M, the observer has the option of heading in a direction of larger r, away from the black hole. This is accessible within their light cone at that instant. They just fire their thrusters, or whatever, to get off that infalling geodesic and fly away.
But once r < 2M, they cross into a region where this is no longer an option.
Robert McNees
In the region r < 2M, all the light cones have tilted over so that they lead inexorably inward.
Any observer in this region, geodesic or not, must move towards the singularity at r = 0 (the thick black line). There simply are no escape routes to larger (and safer) r.
What a remarkable thing! It's not like you could escape if you just had a more powerful rocket. The thing that traps one inside a black hole is *causality*.
Old One Eye
@mcnees Question coming from someone who doesn't understand a lick of what you've shared, as much as I've tried in the past few minutes......... how is this all impacted by space travel in the Star Wars universe, where typical interstellar cruising speed is 11Kx the speed of light?
Joseph Shoer
@mcnees This is why the Russian term for a black hole is "frozen star." Light from the dying star's surface appears to halt as it collapses in towards that limit, and photons reach observers less and less often. (I believe I remember this factoid from Thorne's "Black Holes and Time Warps.")
Robert McNees
As long as I'm posting diagrams, here is the Kruskal diagram for the Schwarzschild spacetime. Hyperbolas (blue) are constant r, and straight lines (red) are constant t. The 45 degree lines (purple) crossing the origin are the horizon, and the bold hyperbola (black) is the singularity.
Schwarzschild's coordinates cover the gray patch, Eddington-Finkelstein coordinates cover the orange patch, and Kruskal-Szekeres coordinates the entire (orange + green) shaded region.
Robert McNees
If you view the entire shaded region (orange and green) as its own spacetime, the interpretation is a wormhole connecting two distinct regions that are both "outside the black hole."
However, this sort of wormhole isn't traversable. An observer has to move along a path that, at each instant, remains inside their light cone from the instant before.
Light rays make 45 degree angles on this diagram, so an observer starting on the orange side can't cross the 45 degree line into the green region.
Robert McNees
(I am writing up notes and exercises for a group of students doing an independent / directed study course on general relativity. It would be easy to lose myself in this, and spend the whole day / week / month on it.)
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