If we send a signal to an object closely orbiting a BH would the reflected signal be the redshifted original? Only that I am sure is that there would be time dilation of the signal...
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If the signal is literally reflected and we ignore any complications introduced by the orbital motion and Doppler effect, then the reflected signal will be received at the same frequency with which it was transmitted.
There is a gravitational blueshift for the light travelling inwards and a symmetric redshift as it travels outwards.
There is also a time dilation effect in the sense that the signal takes longer to be received than would be anticipated for a Euclidean estimate of the distance travelled.
