Exercise list
Lecture 7
Exercise 7.1
Use the various diagrams we drew to describe what a third person, say Charlie, following Alice sometime behind her would see of Alice and Bob at different times.
The diagram below describe the 3 observer (Alice, Bob and Charlie) which are originally on an uniformly accelerated path (describe by an hyperbola).
Bob remains on a uniformly accelerated trajectory, corresponding to hovering at a fixed distance outside the event horizon. From his perspective, neither Alice nor Charlie ever crosses the horizon.
The light emitted by them takes increasingly longer to reach him, and the frequency of that light is exponentially redshifted.
Their images appear to slow down, dim, and freeze near the horizon, asymptotically approaching its edge.
In coordinate time, it takes an infinite duration for Bob to see either of them disappear, even though both cross the horizon in finite proper time.
Alice, the first to fall freely toward the black hole, experiences nothing singular at the horizon. Her crossing of (r = 2M) occurs after a finite proper time and without any local sign of a boundary.
While approaching the horizon, she receives light from Bob that is increasingly blueshifted due to both the gravitational potential and her inward velocity. Bob’s clock appears to tick faster and faster as she falls.
Once she has crossed the horizon, she can still receive some of Bob’s later signals because light rays can still travel inward, but there exists a last message that can reach her before she hits the singularity at (r = 0).
Inside the horizon, all future-directed trajectories point toward smaller (r), so her future inevitably terminates at the singularity after a short finite proper time.
Charlie begins his fall later. While he remains outside, his experience is similar to Bob’s: Alice appears to approach the horizon more slowly, fading as she does.
When Charlie releases himself and follows a free-fall trajectory, his causal relation with Alice changes.
Because his motion nearly matches hers, light from Alice that had been highly redshifted for the stationary observers now reaches him with much less distortion. He catches up with her rapidly.
When Charlie crosses the horizon, both are already inside the same causal region, and they remain able to exchange light signals until the first of them reaches the singularity.
Communication between Alice and Charlie remains possible because the light rays connecting them, which correspond to the (+45^\circ) green lines in the diagram, are still within their mutual light cones even inside the horizon.
For each of them, the proper time from horizon crossing to the singularity is finite. Alice’s world line terminates first; after that, no further communication from her is possible.
Charlie can continue to receive light from her until her final moment and then proceeds to his own end at (r = 0).
From Bob’s perspective, both remain frozen near the horizon forever, while from the viewpoint of the falling observers, crossing the horizon and their mutual interactions proceed smoothly until the singularity is reached.
The red lines, representing the (-45^\circ) null directions from Bob, illustrate the one-way communication from the exterior to the interior that continues for a limited interval after horizon crossing, while the green lines depict the interior light rays that allow Alice and Charlie to see and signal each other until the end.