Markus Hanke

And that is a very valid concern! I appreciate you pointing it out. My main problem is, and always has been, that I’m just an amateur - I haven’t gone into the same amount of depth as someone who has formally studied those things, so there’s always the possibility that I’m missing something. That’s why it’s important to get reality checks from professionals like yourself Well, the proof is always in the pudding, meaning in whether the chosen approach fits observational data or not. Fact is that we have no evidence or indication of extra dimensions, microscopic or otherwise, much less of the extra required scalar field. Also, AFAIK some of the actual predictions that KK theory makes are different from those of QFT, and thus wrong - the Wiki article mentions the electron mass for example. So we can say that these two approaches to modelling EM are not equivalent, and that QFT fits the data much better. I guess this is why we don’t hear so often about KK theory - interesting attempt, but QED works better

It’s a unification of EM and GR, but it’s purely classical - EM appears when you add an extra spatial dimension to spacetime, which is curled up into a tiny circle, plus a scalar field and some more technical assumptions. As being classical, the model has no concept of field quanta. Can you specify what you mean by compatible with QFT? Kaluza-Klein is explicitly classical.

Yes, very valid point +1 I did not mean to make the thread above sound like the last words have been spoken on these matters...it's just that the connection between gravity and thermodynamics seems to be too strong to be a mere coincidence. And thermodynamics is "emergent" in the sense that it is a statistical macroscopic description of systems made up of microscopic constituents (or states). Lol, I do occasionally teach in real life (it's part of the "job" of being a monk), but of course that's not related to science. In general, I find it easier to explain things in writing than "live" in front of an audience.

What theories are you referring to, and what do you mean by “echo”?

That’s a very good point actually, I never looked at it from this angle before. It really is uncanny how few equations describe a very large set of seemingly different physical scenarios.

At the point where you leave the regime where the semi-classical approximation works well enough, ie at the point where you can no longer meaningfully average over things statistically, and the metric of spacetime itself ought to reflect the quantum properties of the source. This happens either at very low energies (single particles, or very small assemblies of particles), or at very high energies. Here we require full quantum gravity. But the problem is - if this entire research program with emergent gravity turns out to have some truth, then the concept of “quantizing GR” really doesn’t make much sense anymore (not that superpositions of metrics etc ever did make much sense to begin with). I am, at this point, really not clear about how meaningful the entire concept of “spacetime” even is in the quantum gravity regime, at least not in its familiar classical form. As I keep saying, I think new and deeper structures will ultimately be needed. PS. If gravity is a phenomenon of statistical emergence on larger scales, then there is also the possibility that the very notion of “quantum gravity” is simply meaningless. Consider an ocean with waves on its surface. The “water” on those scales is, microscopically, just an ensemble of molecules that obey the laws of fluid dynamics. But does it make sense to ask “where are the waves on a molecular level, and how do they function”? No. The notion is not meaningful, just as a single molecule isn’t “wet”. Likewise, perhaps the very notion of gravity isn’t meaningful at all on quantum scales, but only a statistical description of large scale systems. Just a thought.

Yes, very valid point +1 I did not mean to make the thread above sound like the last words have been spoken on these matters...it's just that the connection between gravity and thermodynamics seems to be too strong to be a mere coincidence. And thermodynamics is "emergent" in the sense that it is a statistical macroscopic description of systems made up of microscopic constituents (or states).

In the same sense that eg the Navier-Stokes equations are “emergent” from the statistics of particulate fluids and gases. The broader picture here is that, every time you start with QFT in general spacetimes with as-yet unconstrained metric, and add in the concepts of relative entropy plus some more technical tools (modular theory), the Einstein equations arise as consistency conditions. This is the same as, when you start with general particle ensembles and add in Newtonian forces and conservation principles, the emergent global “statistics” of these fluids are shown to obey Navier-Stokes. It that sense, NS is emergent. Same with gravity - the interplay of quantum fields and their relative entropies is consistent if and only if the background spacetime has a particular geometry that fits the configuration of quantum fields in question. Exactly. NB. The consistency conditions always turn out to be ordinary Einsteinian GR, not any of the many proposed alternatives or modifications.

The derivation obtains the semi-classical Einstein equations. The term "semi-classical" in this context means that matter fields (more generally: energy-momentum) are treated quantum mechanically, while the geometry of spacetime remains classical. Basically, they start with quantum fields, and show how classical spacetime curvature follows from them via the notion of relative entropy, plus some technical assumptions. They don't quantize spacetime geometry itself. The bridge between the two is given by the Bekenstein-Hawking entropy law, which is assumed in the derivation. That latter point is where I am somewhat confused though - the Bekenstein-Hawking law is itself a semi-classical result of QFT and GR, so pre-assuming it basically guarantees that you get from QFT to the Einstein equations. Thus, while an important technical result, the whole thing seems just a little bit like a tautology to me. But then again, maybe I am missing something, which is quite possible since QFT isn't my area of expertise. But my main point stands regardless - the very fact that one can associate thermodynamic entropy with certain types of horizons, while remaining fully consistent with both GR and QFT, indicates to me that gravity is quite possibly not a fundamental thing, but an emergent phenomenon, and this paper strengthens that position. Also, BH horizons having entropy at all seems to imply that the spacetime in the region enclosed by the horizon needs to have some sort of structure or microstates, and can't be smooth and trivial everywhere, or else the very notion of "entropy" associated with them wouldn't make sense. Just my two cents. PS. It is worth mentioning that the "relative entropy" referred to here is not standard classical thermodynamic entropy. It is something called Araki-Uhlmann relative entropy, and if I understand this concept correctly (I may not), it essentially measures how different an excited state of a quantum field is relative to the field's vacuum state.

It goes further than all that - quite recently, Dorau and Much managed to derive the full semi-classical Einstein equations from just relative entropy: https://doi.org/10.1103/lmq8-nsty If this result holds - and I’m fairly sure it will -, it will mean that gravity is not a fundamental interaction, but an emergent phenomenon.

What do you mean by this? Clearly, gravity across vacuum regions just as much as it does inside energy-momentum distributions.

Exactly.

Thanks I've a good bit of material to go through now, much of it pretty non-trivial, so I'll have to take it a step at a time. This is all new territory to me, as until quite recently I wasn't aware that such things as fractal and fractional calculus even existed.

Great, thanks :) Let us, for the time being, just say that I am curious as to what happens when you relax the notion of smoothness that underlies pretty much all our physical models. I’m also curious what would happen if dimensionality of space/time were allowed to vary with scale, even just minutely, and take on non-integer values in some regimes. I’ve also recently discovered the concept of the fractional (distinct from “fractal”) derivative, which naturally introduces a notion of non-locality into analysis, so I am curious as to that, too. I want to first learn what the literature says about these things, and, once I’m a little familiar with the tools of the trade, experiment a little myself, insofar as I am able to. I do have something particular in mind, and yes, it’s to do with spacetime, but I don’t know yet if that is viable even in principle, so I won’t go into it just yet. I’m sure I will have a lot of questions along the way!

Textbook recommendations, please I'm currently investigating an idea I've had, and in that context I need to familiarize myself with both local and non-local Fα-calculus ("fractal calculus") on fractal sets. I don't wish to go into the details of the project just yet, as right now it only exists in form of a very rough outline, and I need to to investigate first if it is in fact worthwhile pursuing at all (chances are it might not be). Suffice to say I need a mathematical toolset that generalizes ordinary multivariate calculus / differential geometry on smooth differentiable manifolds to fractal sets with non-integer Hausdorff dimensionality. So I'm looking for a text that introduces Fα calculus, including fractal derivatives and integrals, both of integer and fractional order (think Riemann-Liouville with fractal measure); a generalization of the usual differential operators (div, grad, curl,...) to fractal sets; differential equations on fractal sets; and ideally Dirichlet forms. I've got access to "Fractal Calculus and Its Applications" by Golmankhaneh, but I find it to be too technical for me as an interested amateur. I'm hoping perhaps someone here can recommend a text on the subject that is more accessible and builds intuition, rather than just listing definitions and lemmas? I've tried searching the Interwebz of course, but there appears to be surprisingly little literature on this particular subject - or perhaps I just didn't search for it right. Thanks in advance!

As measured by which clock?

I’m currently on a month-long long-distance hike in the Alps, and have been crossing the border between Germany and Austria multiple times along my route. I noticed that I can’t access the site in Austria - it gives the very problems described by others above -, but as soon as I’m on the German side and my phone connects to a German provider, all seems fine. Maybe just a coincidence, but it is strange.

All these things originate outside the event horizon. What they don’t mention is that adding torsion into our models of gravity has other consequences too - in particular, it modifies the Dirac equation, making it non-linear. We have not observed any of the associated effects that would arise from this.

The problem is mostly that there exist situations in nature where both gravity and quantum effects appear to be simultaneously non-negligible. Thus, it is reasonable to assume that there should exist some mathematical framework that can describe such situations in an internally self-consistent way. But you are right in that this framework taking the form of a single unified theory is largely an assumption based on what happened with the other fundamental interactions. Though I must say it is difficult to see what a possible alternative might look like.

I used to have this problem too, until I recently changed phones (the old one died after ~10 years), and thus upgraded to new versions of both OS and browser. Now the issue is gone completely. Looks like this is a local problem, not server-side.

Spacetime and its geometry are “there” not only in vacuum, but also in the interior of energy-momentum distributions. There is no situation where there is not spacetime, since there is nowhere one can not place rulers and clocks. I still don’t get what the “issue” here is…?

They are the current scientific consensus, and thus the best models we currently have. Take careful note of the word “currently”. Physics, like all sciences, is a process - as new data becomes available to us, the consensus may need to be updated, and occasionally radically reworked (“paradigm shift”, like from Newton to Einstein eg).

It depends what is meant by “precisely”. If you mean exactly, ie with no deviations at all, then I agree that this is probably not possible. In practice though it is often possible to minimize differences such that their effects on the evolution of the system are negligible, at least for some specified period of time.

How about the Vaidya class of black holes? These spacetimes are not asymptotically flat.

Nice way to visualise this +1