I think we are both confused on what “increasing the size of the Universe” means. Consider first a flat spacetime; there is no spatial limit—space coordinates may take any value. If you know the distance of the photon from the black hole (and the other masses influencing it), you know its energy, and vice-versa. Consequently the distance is not an independent variable. Knowing the initial energy of the system tells you how many states are available; all you can do is redistribute the energy between kinetic and potential. In this universe “increasing the size” is meaningless; you can already travel to infinity.
Now consider a closed spacetime (and your “only a mathematician” seems un-necessarily modest to me; this is an area of physics where I wish to tread carefully and consult with a mathematician whenever possible). Here the distance between photon and black hole is limited, because the universe “wraps around”; travel far enough and you come back to your starting point. It follows that some of the high-distance, low-energy states available in the flat case are not available here, and you can indeed increase the information by decreasing the curvature.
Now, a closed spacetime will collapse, the time to collapse depending on the curvature, so every time Omega makes you an offer, he’s giving you information about the shape of the Universe: It becomes flatter. This increases the number of states available at a given energy. But it cannot increase above the bound imposed by a completely flat spacetime! (I’m not sure what happens in an open Universe, but since it’ll rip apart in finite time I do not think we need to care.) So, yes, whenever Omega gives you a new offer he increases your estimate of the total information in the Universe (at fixed energy), but he cannot increase it without bound—your estimate should go asymptotically towards the flat-Universe limit.
With that said, I suppose Omega could offer, instead or additionally, to increase the information available by pumping in energy from outside the Universe, on some similarly increasing scale—in effect this tells me that the energy in the Universe, which I needed fixed to bound my utility function, is not in fact fixed. In that case I don’t know what to do. :-) But on the plus side, at least now Omega is breaking conservation of energy rather than merely giving me new information within known physics, so perhaps I’m entitled to consider the offers a bit less plausible?
I think we’re talking on slightly different terms. I was thinking of the Hubble radius, which in the limit equates to Open/Flat/Closed iff there is no cosmological constant (Dark energy). This does not seem to be the case. With a cosmological constant, the Hubble radius is relevant because of results on black hole entropy, which would limit the entropy content of a patch of the universe which had a finitely bounded Hubble radius. I was referring to the regression of the boundary as the “expansion of the universe”. The two work roughly similarly in cases where there is a cosmological constant.
I have no formal training in cosmology. In a flat spacetime as you suggest, the number of potential states seems infinite; you have an infinite maximum distance and can have any multiple of the plank distance as a separation. In a flat universe, your causal boundary recedes at a constant c, and thus peak entropy in the patch containing your past light cone goes as t^2. It is not clear that there is a finite bound on the whole of a flat spacetime. I agree entirely on your closed/open comments.
Omega could alternatively assert that the majority of the universe is open with a negative cosmological constant, which would be both stable and have the energy in your cosmological horizon unbounded by any constant.
In a flat spacetime as you suggest, the number of potential states seems infinite; you have an infinite maximum distance and can have any multiple of the plank distance as a separation.
No; the energy is quantized and finite, which disallows some distance-basis states.
But in any case, it does seem that the physical constraint on maximum fun does not apply to Omega, so I must concede that this doesn’t repair the paradox.
I think we are both confused on what “increasing the size of the Universe” means. Consider first a flat spacetime; there is no spatial limit—space coordinates may take any value. If you know the distance of the photon from the black hole (and the other masses influencing it), you know its energy, and vice-versa. Consequently the distance is not an independent variable. Knowing the initial energy of the system tells you how many states are available; all you can do is redistribute the energy between kinetic and potential. In this universe “increasing the size” is meaningless; you can already travel to infinity.
Now consider a closed spacetime (and your “only a mathematician” seems un-necessarily modest to me; this is an area of physics where I wish to tread carefully and consult with a mathematician whenever possible). Here the distance between photon and black hole is limited, because the universe “wraps around”; travel far enough and you come back to your starting point. It follows that some of the high-distance, low-energy states available in the flat case are not available here, and you can indeed increase the information by decreasing the curvature.
Now, a closed spacetime will collapse, the time to collapse depending on the curvature, so every time Omega makes you an offer, he’s giving you information about the shape of the Universe: It becomes flatter. This increases the number of states available at a given energy. But it cannot increase above the bound imposed by a completely flat spacetime! (I’m not sure what happens in an open Universe, but since it’ll rip apart in finite time I do not think we need to care.) So, yes, whenever Omega gives you a new offer he increases your estimate of the total information in the Universe (at fixed energy), but he cannot increase it without bound—your estimate should go asymptotically towards the flat-Universe limit.
With that said, I suppose Omega could offer, instead or additionally, to increase the information available by pumping in energy from outside the Universe, on some similarly increasing scale—in effect this tells me that the energy in the Universe, which I needed fixed to bound my utility function, is not in fact fixed. In that case I don’t know what to do. :-) But on the plus side, at least now Omega is breaking conservation of energy rather than merely giving me new information within known physics, so perhaps I’m entitled to consider the offers a bit less plausible?
I think we’re talking on slightly different terms. I was thinking of the Hubble radius, which in the limit equates to Open/Flat/Closed iff there is no cosmological constant (Dark energy). This does not seem to be the case. With a cosmological constant, the Hubble radius is relevant because of results on black hole entropy, which would limit the entropy content of a patch of the universe which had a finitely bounded Hubble radius. I was referring to the regression of the boundary as the “expansion of the universe”. The two work roughly similarly in cases where there is a cosmological constant.
I have no formal training in cosmology. In a flat spacetime as you suggest, the number of potential states seems infinite; you have an infinite maximum distance and can have any multiple of the plank distance as a separation. In a flat universe, your causal boundary recedes at a constant c, and thus peak entropy in the patch containing your past light cone goes as t^2. It is not clear that there is a finite bound on the whole of a flat spacetime. I agree entirely on your closed/open comments.
Omega could alternatively assert that the majority of the universe is open with a negative cosmological constant, which would be both stable and have the energy in your cosmological horizon unbounded by any constant.
As to attacking the premises; I entirely agree.
No; the energy is quantized and finite, which disallows some distance-basis states.
But in any case, it does seem that the physical constraint on maximum fun does not apply to Omega, so I must concede that this doesn’t repair the paradox.