Markus Hanke

Thanks everyone, this is all valuable, in particular the fact that it is the boundary conditions that impose the precise form of the solution, rather than the equation itself. Which I of course knew before, but hadn’t thought about deeply enough. I’m currently investigating the differential forms formalism for all this, which I find very valuable too for building geometric intuition.

I’m noticing this exact pattern on the side of many users as well. They’re being told by their AI that they’ve stumbled across some amazing idea or another, and then the AI actively reinforces this to keep the user hooked - irrespective of whether there is actual value in the original idea or not. Because let’s face it, this is the only reason these tools even exist in the public domain - to get the user to spend as much time on them as possible, same as social media. It’s a business model, not some altruistic public service. AI isn’t designed to champion objective truth, it’s designed to increase screen time.

I’m not so sure about this actually. If you equip a properly designed robot with a suitably trained AI or AGI , I think this is precisely what will happen. It’s just a matter of time. In Japan they already have restaurants entirely staffed by robotic waiters.

The following may be an unpopular opinion, but I’ll say it anyway. When I went to school, we spent the first five years of math education doing pretty much nothing else but pen-on-paper arithmetic. Addition, subtraction, multiplication, division…over and over and over again, with increasingly large numbers and more decimal points. Since I left school 30+ years ago, never even once was I in a situation where I in fact had to do pen-and-paper arithmetic, at least to the best of my recollection. I’m unsure if now I even remember how to do it. The reality is that we live in the Information Age, and it’s a skill that’s basically never needed anymore. Of course one needs to have an understanding of what those operations mean, but being proficient in working out 6537.45/765.44 by hand on paper has kind of lost its relevance, IMHO. I think it’s enough to spent at most a year on this. On the other hand, had we gone further on the upper end, beyond single-variable calculus, perhaps into differential equations, calculus on manifolds etc, it would have been very helpful. Just my personal opinion. You can crucify me for it :)

I think the best case scenario would be that it strengthens one’s ability to ask the right questions. I for example don’t work in academia, but I have a strong interest in physics. I often play around with ideas in my head which require some mathematical investigation. Unfortunately I don’t have access to advanced CAS such as Maple, and in GR calculating stuff with pen and paper is generally cumbersome and error-prone, especially when it’s not something you do every day for a living. I can do it, because I’ve taught myself how to, but I often make silly mistakes. So nowadays I offload the cumbersome stuff to AI, and just focus on the overarching ideas (caveat - AI does get maths wrong, so one needs to check!!!). That requires me to consider carefully what questions to ask, and how the answers fit into an overall context. So I think AI might (!!!) ultimately help to focus better on the bigger picture, by automating the cumbersome details, just as calculators helped us focus on concepts rather than manual arithmetic. But again, one has to think about the answers one gets, because they are often flawed, meaningless, or straight out wrong. But of course, it really depends much on how people use it in practice. There are no straightforward answers. One thing is for sure though - AI is here to stay.

This is extremely helpful - I knew about the general analytical solution in terms of arbitrary functions, but hadn’t thought about what it actually means. So the above is an important piece of the puzzle. I will think about this some more. But already now, thanks so much to you all, you are very helpful, as always!

An important piece of the puzzle, thank you! Yes, that’s where I’m stuck - I’ve only ever seen the equation used as a wave equation, but I never got the geometric intuition as to why it is specifically waves, as opposed to something else. Interesting, thanks! I’ll have to think about this a little more, before I can comment. These operators are linear, but the dynamics of GR are not, so the field equations needed to be something a little more complex. Good point! But again - why waves in the first place? But thanks everyone for the inputs :) It still hasn’t quite “clicked” for me yet, so do keep it coming if you can!

I’m trying to develop a geometric intuition about what the d’Alembert operator actually signifies. Specifically, I’m looking for a geometric intuition as to why equations of the form \[\square =0\] have waves as solutions, as opposed to something else that “lives” on the light cone. I can see where the light cone comes in, and I also understand why analytically/algebraically the solutions to this PDE are waves; I’m just missing a geometric intuition as to where these “waves” come from, if that makes sense. I’m a very visual thinker, so having such an intuition is always really helpful to me. Any takers?

No it is not. Rs of the sun is not the same as Rs of earth, for example. It depends on the mass of the body in question, as well as the relative strength of gravity. Again, the gravitational potential depends on both the mass of the body as well as the relative strength of gravity. The potential function of the sun isn’t the same as that of earth. I don’t know what “measurement” has to do with the basic fact that not all bodies share the same potential. The orbit itself depends on M and G. So why do you observe those quantities, orbits etc to be different for different bodies? What is it about those bodies that makes them different? You asked me before what I think of all this. I’m sorry to say that the only fitting word that comes to mind regarding your reasoning here is “bizarre”, especially after this last reply of yours. For my part, I’m not interested in investing more time in this, but I wish you all the best.

Did you not just say that your solution depends on neither G nor M, yet the above substitution explicitly introduces both of those quantities? How do you find this scale quantity a? This, again, explicitly depends on both M and G. And it implicitly assumes you are in a spacetime that has a time-like Killing vector, and is asymptotically flat, or else no concept of gravitational potential exists. How do you find this quantity? And where does this come from?

How am I mistaken in that G and m don’t appear in the vacuum equations?

There are some important subtleties here to be aware of. These predictions you are referring to arise from a particular metric, the Schwarzschild metric, which is a solution to the Einstein vacuum equation \[R_❴\mu \nu❵=0\] Notice how neither G nor m appear anywhere in this equation - it is simply a geometric statement, a constraint on what form any possible metric can take in vacuum. So those physical quantities aren't part of the theory at all at this stage. It is only when you begin solving these differential equations that there naturally appear integration constants in the process - and to find the physical meaning of those constants, we look at a boundary condition which we impose, namely that in the weak field limit, GR should reduce to Newtonian gravity. It is only through this particular boundary condition that G and m arise - they are thus the result of boundary conditions, not GR itself.

Emphasis on “possibility”. I think the reason why particulate DM is currently the favoured model is because it explains the widest range of observations in the simplest way with the least amount of extra assumptions. We can infer from observations that DM clumps around gravitational sources; it fits well to data concerning both the early universe, and current large-scale structure; it naturally explains observations around collisions of galaxy clusters; and at least some of these models fit well into the Standard Model. Some of the other alternatives work better on specific subsets of the available data, but then fail on other subsets. But again, I’m personally a bit sceptical, and my gut feeling is telling me that we’re missing something important here. Thus I’m looking forward to more research in this area.

Personally, I’m not at all convinced that DM is necessarily particulate in nature. However, the possibility is still strong enough that it can’t be ruled out based on currently available data, so it’s good to explore what options there are in that regard. And not having to postulate any exotic new particles that are hard to fit into the Standard Model, is a big plus in my mind. Hence the reference to Witten’s idea.

Have you heard of the 2024 re-visit of Witten’s 1980s idea that DM could be composed of strangelets: https://arxiv.org/html/2404.12094v1 I think this is very interesting, and perhaps warrants further investigation, since it requires no new particles to be hypothesized. There’s also this quite recent paper: https://journals.aps.org/prd/abstract/10.1103/w1sd-v69d which basically finds that large-scale geodesics are modified substantially if gravity is quantized on small scales, irrespective of the details of said quantisation.

+1 You are completely right, this is actually an important distinction - thanks for correcting me on this 👍

It should be noted that a simple curve (1D manifold) has no intrinsic curvature - the Riemann tensor vanishes identically in 1D. But it can of course have extrinsic curvature when embedded in a higher dimensional space.

Great insight! Never thought about it from this particular angle, though in retrospect it seems obvious +1

That seems more reasonable to me - not that I’m an expert, this is quite a subtle question. My own guess - the field equations for torsion in ECT contain no derivatives, and at the same time torsion is completely determined by local matter fields. This implies that torsion vanishes in regions where T=0, and no wave-type equation exists for torsion to “radiate” through vacuum. So it can’t have any propagating degrees of freedom - it’s purely a local phenomenon subject to the local presence of matter.

To be honest, I’m not so sure about this. The energy-momentum that forms the source term in the Einstein equations comes from the Noether current associated with spacetime translations, whereas spin comes from Lorentz invariance. These are different things. It is not in fact possible, AFAIK, to define a unique 4-momentum vector for intrinsic spin, so I don’t see how it could - if taken in isolation - act as a source of gravity. Or am I missing something?

I was under the impression that a recent experiment has cast serious doubts on the viability of Bohmian mechanics: https://www.nature.com/articles/s41586-025-09099-4 This is essentially a direct conflict between what BM predicts in that situation, and the observed outcome. What it means is that, if I understand the implications correctly (and I’m not sure that I do), the concept of “particle” that BM is constructed on does not correspond to particles in the real universe.

I think it goes even deeper - it’s about the wish to reduce an increasingly complex world that inherently operates in shades of grey, to a simple and easy to understand world view that has just black and white. We are more comfortable with what we can emotionally understand. This is why all conspiracies, without exception, are based on “us the good guys” vs “THEM”.

I don’t think this is correct. k=0 just means that the universe is spatially flat, but that doesn’t imply that a(t) must necessarily be unity. You can have a spatially flat, metrically expanding/contracting universe. But in either case, the Riemann tensor does not vanish, irrespective of the value of k.

But metric expansion is a gravitational effect…? That’s purely spatial curvature though. The Riemann tensor as a whole does not vanish in FLRW spacetime, even for the case k=0.

Lol, nice one ☝️