Dear Hector Zenil,
Thank you for trying to warn me about possible minor glitches, and for taking time to read my essay. I am genuinely interested in any problem that there may be with my approach, because I would not want to come up with a wrong theory. If there are some glitches in it, I would vey much like to know precisely what they are, to fix them and get the correct theory. But I'm afraid I don't understand where you see them. You said:
"I think there are possibly some minor logical glitches in constructing your case against some pre assumptions that you think are wrong in modern physics, in particular I think of assumptions 1 and 5. Assumption 1 says that Singularity theorems predict the breakdown of General Relativity (GR). As you know, I think one of the greatest, if perhaps not the greatest open problems in modern physics, because it could lead to a conciliation between GR and quantum mechanics (QM), is the issue of what happens inside a black hole and specially in the singularity point. I think most physicists would agree that if the equations of GR break that doesn't imply that reality breaks (this is connected also to your assumption 5)."
People really do claim that singularity theorems prove that GR breaks down. There are dozens of papers and books which start with this claim, in general to sell other "more radical" theories. And what I said is that A. singularity theorems are correct, and they lead to singularities and B. singularity theorems don't prove that GR fails. They only prove that singularities appear in some very general conditions. I agree that in GR singularities appear, but I show that they are not a problem, because the equations can be put in a form which works there too.
"I think most physicists would agree that if the equations of GR break that doesn't imply that reality breaks"
Yes, and I agree too. I don't reject this. I only reject the assumption that from the existence of singularities follows that GR breaks down. As I explained, the standard equations are those who break down. But I put them in a form which works without infinities at singularities. Otherwise, at non-singular points, they are equivalent. If we divide by the volume element to get Einstein's equation in the standard form, at singularities we divide by zero. If we don't make this "simplification", and keep the equations in the densitized form, then they are valid even at singularities.
IMO, it is a good thing to extend GR beyond the limits usually assumed. My approach makes it work at singularities, both for big bang and for black holes. I think I am correct, and I think this is a progress, because it fixes important problems in GR without extra assumptions, and without modifications which then should be proven to lead to the same predictions as GR. Fixing GR should not be viewed as an enemy of other theories.
This doesn't mean that I am against the "conciliation between GR and quantum mechanics (QM)". I am for it. In fact, my singularities seem to provide a way to make gravity renormalizable, by leading in a natural way to a dimensional reduction (section 7).
"The agreement I think is that we just don't understand and have no tools to explain what happens in such a limit situation. As you suggest, topology may be a source for better understanding, and I think that is an interesting idea discussed in good detail in your very fine essay."
Well, I hope these tools I developed help, and thank you for the appreciative comment. Even if GR should be modified and replaced, and even if by quantization it will become different, they may help, as many other tools developed in GR may be inherited in other theories. But it is possible that in the real world my solution doesn't work. We don't know the final theory, maybe it will incorporate GR (hopefully with my "bug fix"), maybe not.
"I am still bugged by the contradiction between your zero distance idea and the fact that QM seems to suggest there is a minimum length (and I also see your argument that there is no minimum mass, isn't it because photons are conventionally massless?)."
I'll try to explain this. Let me state from the beginning that I agree that the Plank length may be special, although I don't know yet how. But I think there is no experimental evidence or mathematical proof that there is a minimum length. You say "QM seems to suggest there is a minimum length". I am not aware of such suggestion from QM. In QM, the discrete spectra are obtained from equations which assume continuous space and time. There's no need to assume minimal length to get the discrete spectra of electrons in the atom for example. In fact, I don't know of a way which explains some quantum spectra of observables from the assumption of a minimal length. There is though the argument that to probe the Plank distance you need Plank energies, which would create tiny black holes. If it's correct that this prevents us from seeing what happens under the Plank scale, this doesn't mean that this distance is minimal. There may be a minimal distance, or there may be not. This argument can't distinguish between the two. Now, I don't say there is no special length. This may be true. But it doesn't mean it is minimal. I think that the existence of a minimal length is an open problem (although at this time I consider that it has little chances to be true).
Thank you for taking your time to read properly my essay and the papers on which it is based, and for warning me about possible dangers. I appreciate your comments, and if you feel that I did not answer properly to your possible objections, I hope you will find time to detail them.
Best regards,
Cristi
