Alternatively, and thinking purely classically, if we do t -> -t, what happens to a perfect absorber, which takes in anything and reflects nothing impinging on it? It should spit out arbitrary things and reflect everything impinging on it. (Here I'm thinking of the Poynting vector for plane waves with their origin below the photon sphere and moving outwards vs just above the antihorizon and moving inwards -- the latter can't enter the WH).
(Maybe easier to think about if the BH absorption isn't total, like if we spin the black hole and have the plane waves (or their polarization) move against the rotation, and the WH is just time reversed -- the WH emits a bit of radiation which joins more outside the antihorizon, and the combination flies off to infinity, and not thinking too hard about the interior metric; although Wald deals with much of this in the text around eqn (12.4.18)).
> if we do t -> -t, what happens to a perfect absorber, which takes in anything and reflects nothing impinging on it?
It becomes a perfect emitter, which emits anything and reflects nothing impinging on it. In other words, a white hole.
To put it another way: the inside of a black hole is a region from which nothing can escape and into which everything can fall. The inside of a white hole is a region from which everything can escape and into which nothing can fall. Things coming out of a white hole are escaping from the inside. They aren't reflected.
Everything in Wald that you are referencing is describing what I just described. It is not describing what you described.
> It should spit out arbitrary things and reflect everything impinging on it.
The first is right, the second is wrong. The time reverse of "no reflection" is "no reflection". It is not "reflect everything". "No reflection" means complete transmission through the surface in question. Reversing time just reverses the direction of transmission, in this case from inward (black hole) to outward (white hole).
Throw (slowly, compared to c) a black hole (BH) through a Bonnor beam (BB). Throw a white hole (WH) through a Bonnor beam. Ignore technical difficulties in superimposing the exact solutions for the moment. The breaking of the axial/spherical symmetries is deliberate.
Qualitatively what happens, in your view, when the WH touches the beam? Is it just the time reversed picture of the BH intercepting the BB? There's a part of the first spacetime where some of the light is trapped in the BH as it flies off to infinity after its encounter with the beam. Is there a part of the second spacetime in which some of the light is trapped in the WH? If so, where is it if the initial conditions are a Bonnor beam at large spacelike separation from the white hole? Does the WH cut a gap into the beam?
Alternatively, take two widely separated BHs. Grind out the Raychaudhuri equation, bearing in mind the focusing theorem. Now take two widely separated WHs. What's different? How does a circular orbit of two WHs evolve? Outgoing GWs tend to circularize elliptical binary BH orbits. Can a circular binary WH orbit decircularize? How? ("Incoming GWs from initial conditions" is not very satisfying).
The problem I think is confusing the reversal of the entire BH spacetime with a WH stitched into an ordinary Minkowski in the asymptotic limit, or into a cosmology. (Bonnor is essesntially the latter). More broadly, a WH should be able to live with future-directed null geodesics.
Of course, I could only agree with a response like, "well such a WH is almost certainly unphysical", but maybe we could ignore that in order to think about what happens when a plane wave hits a WH.
> Is there a part of the second spacetime in which some of the light is trapped in the WH?
No. Nothing can get into a white hole from the outside.
> where is it if the initial conditions are a Bonnor beam at large spacelike separation from the white hole?
You are mistakenly thinking of a white hole as a "thing". It isn't. It's a region of spacetime that nothing can get inside. Or, to put it another way, it is a region of spacetime that is to the past of something else: either a black hole (if you are talking about the maximally extended Schwarzschild spacetime) or an object like a star that the white hole expanded into.
So if you "aim" anything at a white hole, you won't hit the white hole; you'll hit whatever thing the white hole is to the past of. Either a black hole, or an object that the white hole expanded into. That is what the Bonnor beam in your scenario would actually intersect.
Similar remarks would apply to your other scenarios; the actual objects you would end up dealing with would not be white holes, but whatever the white holes were to the past of.
Where are you getting this from?