Yes, we can be certain. White holes can be considered in two ways: (1) a region of spacetime into which nothing can ever cross, with a surface that is a perfect reflector; or somewhat less commonly (2) a time-reversed black hole.
At scales of gigaparsecs, the universe appears to be both spatially homogeneous and isotropic, with matter denser at higher redshift. There is no way to generate this configuration of matter from a white hole. White holes are very much spatially anisotropic, that is you can point to where they are: left of you but not right of you, in front of you but not behind you, etc. The trajectories of the centres of momentum of galaxy clusters in every direction and at every redshift are diverging but not following an inertial radial. The really compelling accessible evidence against a white hole big bang is in the https://en.wikipedia.org/wiki/Lyman-alpha_forest from hydrogen clouds at different redshifts being backlit by bright background (and thus earlier in the universe) sources. This "stacking" is so hard to conceive for a white hole big bang that as far as I know, nobody has actually written down a reasonable explanation for it.
More technically, (general relativity) metrics at those "cosmologically large" scales are so different from the metrics of white holes that we would already have noticed.
Additionally, a white hole big bang has to account for the chemistry of the earliest visible stars, whose spectral lines indicate essentially hydrogen with a tiny amount of helium and lithium. The standard cosmology's big bang nucleosynthesis's chemical abundances would have to be duplicated by a white hole big bang, and as far as I know nobody knows how to make that work. How does whatever is spit out of a white hole interior, or reflected off a white hole boundary, eventually turn into atomic hydrogen (with a smattering of heavier atoms up to 7 atomic mass units)? How does it account for the cosmic microwave background, which implies that in some region of the white hole big bang universe atoms practically everywhere were so hot as to be completely ionized?
Likewise, the interior metrics of black holes are very different from metrics which reasonably describe observations at gigaparsec scales. So if one takes a "wormhole" maximal extension of a black hole spacetime and throws enormous amounts matter into the black hole end, what emerges at the white hole end is nothing like a cosmology. Nor is what's inside. So a reasonable prediction of our galaxy's future is not exiting from a white hole (with us presently inside), and a reasonable prediction of our past is not that our galaxy was spit out of a white hole (with us presently outside).
As far as I know, nobody has done a convincing treatment of a non-eternal white hole in General Relativity. We've had theoretical models for forming black holes (i.e., there is a past of the black hole where there is no black hole yet, so it's non-eternal) since the 1930s (<https://en.wikipedia.org/wiki/Oppenheimer%E2%80%93Snyder_mod...>), and models for evaporating black holes since Hawking 1970, so it's quite striking there's nothing similar for white holes. There is ample astrophysical evidence supporting a range of models for non-eternal black holes and their horizons. There isn't anything seen by our telescopes that looks like it might be a white hole horizon.
The accelerated expansion of the universe, discovered in 1998, kills off a bunch of vaguely related ideas, like we are in a tiny tiny fraction of a thin shell (thin as in small compared to the shell's diameter) of matter that is expanding inertially from an initial impulse (an "explosion", figuratively) acting on a dense phase of matter. By "tiny tiny" here we're talking about our visible universe occupying less than 10^-23 of the shell's thickness. This is from a family of ideas tossed around a bit before Alan Guth published his 1997 book <https://en.wikipedia.org/wiki/The_Inflationary_Universe>. One could imagine a reflector onto which the shell fell, and that reflector could resemble a white hole; you can get a cyclic cosmology by having the entire shell eventually reconverge on the central reflector. However, we now see the expansion of our universe is speeding up rather than slowing down, so this doesn't work.
Finally, we're learning a lot more about the gravitational wave background of our universe. It is reasonable to bet that studies of the polarization of the cosmic microwave background (e.g. <https://lweb.cfa.harvard.edu/~cbischoff/bicepkeck/>) will find evidence of gravitational waves at scales large enough and uniform enough to rule out any sort of hole-like boundary (i.e., a one-way-crossing-only surface which has a spatial direction as well as being in the past) during the inflationary epoch. And we are probably already at the ruling-out point with baryon acoustic oscillation galaxy filament https://svs.gsfc.nasa.gov/13768 evidence.
When I thought about white holes, I guess I thought there was a big source of energy pushing spacetime out from itself and the further you got the faster you went, basically the literal opposite of gravity. I guess I was imagining something other than what you described which is not very helpful or testable.
You're not all that far from describing dark energy.
Gravitational attraction tends to cause the trajectories of freely-falling ("floating in empty space") masses to converge. Dark energy tends to cause the trajectories of freely-falling masses to diverge. We see this in the sky: distant galaxies are freely-falling but their trajectories are away from each other (and from us).
As far as we can tell though, the "source" of dark energy is the cosmological constant. Physical interpretations usually start by slicing spacetime into spaces aligned along the cosmological time (the scale factor). In successive spatial slices ordinary matter, radiation, dark matter, and so forth all get sparser towards the future. We treat these as a sort of fluid or gas that dilutes away with the expansion of the universe, therefore the density of each fluid at a typical point differs from spatial slice to spatial slice (it's lower in the future). The cosmological constant is constant in every spatial slice. So if we were to turn it into one of these fluids, it would not dilute or thin out over time: its density at a typical point is always the same. Representing a geometrical feature (the cosmological constant) as a space-filling fluid with certain properties makes it easier to study what happens if the cosmological constant isn't constant . (Does it fluctuate? Does it get weaker or stronger over time or in the presence of overdensities of matter?)
Cosmic inflation introduces another of these fluids that like dark energy serves to separate rather than draw together the others. It's very very strong in the early part of the universe (the inflationary epoch), but disappears very early too. In some flavours of inflation, the "inflaton" field decays into standard-model matter; in others it just decays into weaker and weaker versions of itself and by the time there are electrons, positrons and photons it's basically so weak that it's undetectable.
The inflationary epoch somewhat matches your intuition that a big source of energy pushed space apart. The most important difference (at this "explain like I'm..." level) as far as I can tell is that there was no point in space that someone in the early universe could point to as the centre or source of inflation. A gravitating object has a centre of mass, and you can point to and away from that object, i.e. there is an up and down with respect to it, because it doesn't surround you like a shell. The inflaton field was strong everywhere, there was nowhere in space "outside" it.
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.
At scales of gigaparsecs, the universe appears to be both spatially homogeneous and isotropic, with matter denser at higher redshift. There is no way to generate this configuration of matter from a white hole. White holes are very much spatially anisotropic, that is you can point to where they are: left of you but not right of you, in front of you but not behind you, etc. The trajectories of the centres of momentum of galaxy clusters in every direction and at every redshift are diverging but not following an inertial radial. The really compelling accessible evidence against a white hole big bang is in the https://en.wikipedia.org/wiki/Lyman-alpha_forest from hydrogen clouds at different redshifts being backlit by bright background (and thus earlier in the universe) sources. This "stacking" is so hard to conceive for a white hole big bang that as far as I know, nobody has actually written down a reasonable explanation for it. More technically, (general relativity) metrics at those "cosmologically large" scales are so different from the metrics of white holes that we would already have noticed.
Additionally, a white hole big bang has to account for the chemistry of the earliest visible stars, whose spectral lines indicate essentially hydrogen with a tiny amount of helium and lithium. The standard cosmology's big bang nucleosynthesis's chemical abundances would have to be duplicated by a white hole big bang, and as far as I know nobody knows how to make that work. How does whatever is spit out of a white hole interior, or reflected off a white hole boundary, eventually turn into atomic hydrogen (with a smattering of heavier atoms up to 7 atomic mass units)? How does it account for the cosmic microwave background, which implies that in some region of the white hole big bang universe atoms practically everywhere were so hot as to be completely ionized?
Likewise, the interior metrics of black holes are very different from metrics which reasonably describe observations at gigaparsec scales. So if one takes a "wormhole" maximal extension of a black hole spacetime and throws enormous amounts matter into the black hole end, what emerges at the white hole end is nothing like a cosmology. Nor is what's inside. So a reasonable prediction of our galaxy's future is not exiting from a white hole (with us presently inside), and a reasonable prediction of our past is not that our galaxy was spit out of a white hole (with us presently outside).
As far as I know, nobody has done a convincing treatment of a non-eternal white hole in General Relativity. We've had theoretical models for forming black holes (i.e., there is a past of the black hole where there is no black hole yet, so it's non-eternal) since the 1930s (<https://en.wikipedia.org/wiki/Oppenheimer%E2%80%93Snyder_mod...>), and models for evaporating black holes since Hawking 1970, so it's quite striking there's nothing similar for white holes. There is ample astrophysical evidence supporting a range of models for non-eternal black holes and their horizons. There isn't anything seen by our telescopes that looks like it might be a white hole horizon.
The accelerated expansion of the universe, discovered in 1998, kills off a bunch of vaguely related ideas, like we are in a tiny tiny fraction of a thin shell (thin as in small compared to the shell's diameter) of matter that is expanding inertially from an initial impulse (an "explosion", figuratively) acting on a dense phase of matter. By "tiny tiny" here we're talking about our visible universe occupying less than 10^-23 of the shell's thickness. This is from a family of ideas tossed around a bit before Alan Guth published his 1997 book <https://en.wikipedia.org/wiki/The_Inflationary_Universe>. One could imagine a reflector onto which the shell fell, and that reflector could resemble a white hole; you can get a cyclic cosmology by having the entire shell eventually reconverge on the central reflector. However, we now see the expansion of our universe is speeding up rather than slowing down, so this doesn't work.
Finally, we're learning a lot more about the gravitational wave background of our universe. It is reasonable to bet that studies of the polarization of the cosmic microwave background (e.g. <https://lweb.cfa.harvard.edu/~cbischoff/bicepkeck/>) will find evidence of gravitational waves at scales large enough and uniform enough to rule out any sort of hole-like boundary (i.e., a one-way-crossing-only surface which has a spatial direction as well as being in the past) during the inflationary epoch. And we are probably already at the ruling-out point with baryon acoustic oscillation galaxy filament https://svs.gsfc.nasa.gov/13768 evidence.