Question

How close to the black hole must a space station be such that a signal
it sends to Earth using shortwave radio (roughly 0.1 meter wavelength)
gets redshifted to ultra long-wave radio (roughly 1000 meter wavelength)?
Assume the Earth is very far away from the black hole. Express your
answer as a multiple of the black hole’s Schwarzschild radius.

Answer 1

The equation for gravitational Doppler shift at infinite distance away is

Here

is the observed wavelength = 1000 m

is the emitted wavelength = 0.1 m

rs = Schwarzschild radius

and re = position where light is emitted

Therefore solving the above equation gives:

or

Therefore the signal has to be emitted at the black hole’s Schwarzschild radius,