How much is the universe blueshifted at a black hole event horizon? [duplicate]

Question

If Alice falls into a black hole wearing a wrist-watch, an outside observer Bob sees Alice freeze at the horizon, her watch seems to have stopped. This also means that all light that originates from Alice is redshifted asymptotically to zero Herz.

Now my question is what Alice sees when she is just a few femtometers above the event horizon and looks back at Bob. Is Bob blueshifted and does his watch seem to spin to the end of time in a wink of her eye? Can Alice see the universe evolve to the end of time and see the black hole below her evaporate? In other words: if Alice is redshifted to zero to Bob, should not Bob be blueshifted to infinity to Alice?

I believe the actual amount of blueshift is not explained by the answers to question: What will the universe look like for anyone falling into a black hole? ?

Answer 1

You're right. Let's assume Alice is hoovering just above the horizon (let's not consider the fact that an enormous amount of energy is needed for this).
Her time is almost standing still wrt to the most parts of the Universe. She sees the hole evaporate in a flash (though when the hole gets smaller she'll have to maintain the infinitely small distance to the shrinking horizon.

When the hole has evaporated (how long this takes depends on the mass of the hole), she'll be in sync again with most parts a the Universe (where gravity is weak).

So, indeed, it works in both directions, but oppositely.

EDIT
After reading the comment below by @anneb, the situation gets more complicated.
Of course, Alice needs a huge energy supply (dependent on the BH's mass) of energy to hoover above the BH's horizon. In the part above I assumed to ignore this.
If she could fill her rocket with energy extracted from the BH and her rocket will use photons to propel the rocket and counteract the gravity of the hole. The photons will re-enter the hole though, so the mass of the BH will stay the same. So there will be no difference.

The question remains essentially the same if Alice is hoovering at a considerable distance. In that case, she still needs much fuel to counteract the BH's gravity. If this is done again by ejecting photons, the photons will be absorbed by the hole which thus gets heavier.
Alice makes sure that the distance to the horizon stays the same. While her tank (say positrons and electrons) gets emptier it will in time become easier to stay put where she is.
The BH's mass has increased, so the time to evaporate has increased also. In that case, it takes more time for Alice to get back in sync with the clocks in the major part of the Universe, where gravity is absent. The times on her clock and the times on the clocks in empty space will differ though.

It needs a calculation to find out if the rocket's mass, i.e., including the fuel (which we assume to be positrons) will be that high to form a BH. But since I just woke up, I'll do that later.