To illustrate what I believe is the answer, here are some examples making use of a spacecraft at three locations:
- outside the event horizon,
- at the event horizon, and
- inside the event horizon.
The black hole is non rotating, and contains a singularity at its center.
The spacecraft has a mirror which will reflect incoming photons back along the same
path from which they arrived. The incoming photons are created from a source at
a substantial distance from the black hole, and are aimed directly toward the
spacecraft. The source of photons, spacecraft, and black hole center are colinear.
The spacecraft has a powerful engine which can provide infinite acceleration.
Example 1: The spacecraft is maintaining position at a distance of 1.5
Schwarzschild Radius. For a 10 Solar Mass black hole, the spacecraft will
need to accelerate at 100 billion g (see https://www.spacetimetravel.org/expeditionsl/expeditionsl.html) to maintain a constant position at 1.5 Schwarzschild Radius.
When a photon arrives from the photon source it will be blue shifted to higher energy,
the photon will reflect off the mirror back to where it was created. On it's
journey back the photon will be red shifted back to it's original wavelength.
Example 2: The spacecraft is maintaining position at the Schwarzschild Radius. This is
not actually possible, because the spacecraft would have to be under an infinite
acceleration to maintain position at the event horizon. However, laying that aside,
when a photon arrives it would be blue shifted by an infinite amount, and would
therefore have infinite energy. The infinite energy photon, would reflect off the
mirror and on it's journey back to it's source, it would be red-shifted back to it's
original wavelength. So, a photon with infinite energy could escape from the black
hole's Schwarzschild Radius.
Example 3: The spacecraft is again maintaining position at the Schwarzschild Radius.
The spacecraft emits a normal (finite energy) photon in the outward direction. In
this case the photon would be red-shifted by a factor of infinity over 0 distance.
So this normal photon would not escape the black hole.
Example 4: The spacecraft is maintaining position inside the Schwarzschild Radius.
In this case, similar to Examples 1 and 2, it seems that the photon that was emitted
outside the black hole would enter the black hole, reflect off the mirror and
return back to it's source outside the black hole. Obviously this is wrong.
There are several problems with this scenario as described, which are:
- a) The photon can not leave the black hole.
- b) In order for the spacecraft to maintain position at 1.5 Rs required
an acceleration of 100 billion g, and to maintain position at Rs required
infinite g. Extrapolating it seems that more than infinite g would be
required to maintain position inside Rs; which doesn't make sense.
- c) Similar to (b) the energy of the photon arriving at the stationary spacecraft
inside the Schwarzschild Radius would have greater than infinite energy.
Also, doesn't make sense.
Fortunately black hole theory doesn't allow for stationary objects within the
Schwarzschild Radius. Any object inside the Schwarzschild Radius has an
inward motion which will bring the object to the singularity, where it will be absorbed.
No amount of thrust from our spacecraft's motor can counter this inward motion of
the spacecraft toward the singularity.
Now, suppose the photon enters the black hole and reflects off the mirror of the
spacecraft. Remember the spacecraft & mirror are moving inward toward the singularity.
The photon would reflect back on it's path. However, from the point of
view of a hypothetical stationary observer within the black hole, the reflected
photon would have lost energy due to Doppler shift. And the reflected photon, now having
less energy, would not have sufficient energy to leave the black hole.