Markus Hanke

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

You’re absolutely right, and it was meant to be that, I once again forgot the conversion. This is what happens when you don’t do this stuff every day. Thanks for picking up on it 👍 I’m not entirely sure what “to second post-Newtonian order” actually means, but I presume this is an approximation of some kind? The full integral looks elliptic, so there shouldn’t be a closed-analytic form for the exact result.

Oh my, you are absolutely correct! My apologies. I took the expressions for E and L from my personal notes, without realising that they were in natural units, whereas the integral was in SI units. Silly amateur mistake on my side. Let’s try again - we have, this time in SI units, \[E=\gamma c^{2},\ L=\gamma v_{\infty}b\] with \[\gamma =\frac{1}{\sqrt{1-\frac{v_{\infty}^{2}}{c^{2}}}}\] Popping this into the original E-L integral, I get, in slightly different form \[\varphi =\int_{r_{\min}}^{\infty}\frac{\gamma v_{\infty} b dr}{r^2 \sqrt{\gamma^2 c^4 - \left(1 - \dfrac{2GM}{rc^2}\right) \left( c^2 + \dfrac{\gamma^2 v^{2}_{\infty}b^2}{r^2} \right)}}-\pi\] The units should be correct now, with the result being in radians - but perhaps it’s wise if you double check, since I’m doing all this pen-on-paper.

My pleasure. As a little exercise, I’ve reworked the integral to something more explicit (I personally never really liked the notation with E and L), and if I’m not mistaken this is what we get: \[\Delta \varphi = 2 \int_{r_{\min}}^{\infty} \frac{dr}{r^2 \sqrt{\dfrac{1}{b^2 v_\infty^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1 - v_\infty^2/c^2}{b^2 v_\infty^2} + \dfrac{1}{r^2} \right)}} - \pi\] So the deflection angle depends only on initial speed far away, impact parameter, and mass of the central object - as one would expect.

It is: \[\varphi =\int_{r_{\min}}^{\infty}\frac{dr}{r^2 \sqrt{\dfrac{E^2}{L^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1}{L^2} + \dfrac{1}{r^2} \right)}}\] For non-relativistic speeds and weak fields, this reduces to the Newtonian scattering formula. For v=c and massless test particles, you get the Schwarzschild light deflection formula. For strong fields and massive particles, the integral can be evaluated numerically.

This is certainly true for some of us, but one has to remember that autism manifests along a spectrum - some autistics have very profound difficulties with communication, whereas some others might be at a near-neurotypical level in that particular area, but might be really struggling with other things. It’s difficult, if not impossible, to generalise what the “typical” autistic person might be like.

I understand what you are trying to say here, but I’d like to highlight that it is only particular patterns / manifestations that one can improve on, given the right tools and strategies. Autism itself is a physiological difference in the human brain, you cannot snap out of it any more than you can snap out of being pregnant or having cancer. But you can find skilful ways to manage it.

I guess the difference is in the level of intensity - for autistic people the fixation on their hyperfocus can be very powerful, to the point that it is at the forefront of their inner lived experience much of their waking hours, and can often almost look like an obsession of sorts. Eg someone with a hyperfocus on Spongebob Squarepants might own all the relevant media, have SBSP bedlinen und brush their teeth with SBSP-branded toothpaste, while simultaneously knowing everything there is ever to know about SBSP. This can then also "bleed over" to other areas, for example when in a conversation they might inadvertently start to blabber about their hyperfocus ("infodumping") even though the initial interaction was about something entirely unrelated. This is not to say that neurotypical people don't have special interests or expertise in particular subjects, but the difference is in the degree / intensity of how this is experienced. Note also that this in isolation is not a defining indicator for someone being autistic, but it does form a part of a larger list of diagnostic criteria. Indeed.

It is very common for autistic people (at least the high functioning ones) to have areas of special interest, called a hyperfocus, which they get deeply fascinated by and perhaps over time come to know a lot about. Sometimes this can be the stereotypical maths, but it can just as well be LEGO, Marvel superheroes, or SpongeBob SquarePants. So no, not all autistics are maths geniuses - I know a lot of people in the autistics community, and not one of them fits that bill. But many of them are very knowledgeable at something.

You mean gravitational waves ;)