Greg Egan on Nostr: I thought I’d say a bit about how these apparent distances were computed. Some of ...
I thought I’d say a bit about how these apparent distances were computed. Some of the intermediate calculations are quite messy, but it turns out it’s not too hard to find a simple formula for the apparent distance between two astronauts who fell from the same height at the moment when they cross the horizon.
d/M at horizon = β (1 – exp(–τ/β))
M is the mass of the black hole
β = 4 √[1 – 2M/r₀]
r₀ is the Schwarzschild r coordinate the two astronauts fell from
τ is the proper time that the second astronaut delayed their fall
We are using geometrical units, so masses and distances are measured in the same units.
In general, to compute the apparent distance between two astronauts who are both falling radially from the same Schwarzschild r coordinate, r₀, we need to know the location of the one who fell first, r₁, and the location of the one who fell after them, r₂, at the time they receive light from r₁. If the second astronaut delayed their fall for a proper time τ, we can find r₂ as a function of r₁ by matching two lapses of time, using any time coordinate that suits us:
Time for first astronaut to fall from r₀ to r₁
+ time for light to travel from r₁ to r₂
=
Time that passed while second astronaut waited at r₀
+ time for second astronaut to fall from r₀ to r₂
This equation can’t be solved explicitly for r₂, but given r₀, r₁ and τ, we can find the value of r₂ numerically.
Next, we look at what happens to a photon that travels from r₁ to r₂ with a small departure from radial motion. The parameter used to describe this is the “impact parameter”, b, of the photon, which basically says how far off-centre its line of motion will be from the black hole once it is far away. Radial motion is b=0.
d/M at horizon = β (1 – exp(–τ/β))
M is the mass of the black hole
β = 4 √[1 – 2M/r₀]
r₀ is the Schwarzschild r coordinate the two astronauts fell from
τ is the proper time that the second astronaut delayed their fall
We are using geometrical units, so masses and distances are measured in the same units.
In general, to compute the apparent distance between two astronauts who are both falling radially from the same Schwarzschild r coordinate, r₀, we need to know the location of the one who fell first, r₁, and the location of the one who fell after them, r₂, at the time they receive light from r₁. If the second astronaut delayed their fall for a proper time τ, we can find r₂ as a function of r₁ by matching two lapses of time, using any time coordinate that suits us:
Time for first astronaut to fall from r₀ to r₁
+ time for light to travel from r₁ to r₂
=
Time that passed while second astronaut waited at r₀
+ time for second astronaut to fall from r₀ to r₂
This equation can’t be solved explicitly for r₂, but given r₀, r₁ and τ, we can find the value of r₂ numerically.
Next, we look at what happens to a photon that travels from r₁ to r₂ with a small departure from radial motion. The parameter used to describe this is the “impact parameter”, b, of the photon, which basically says how far off-centre its line of motion will be from the black hole once it is far away. Radial motion is b=0.