Markus Hanke

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 ☝️

You need to remember though that just because you don’t understand it, doesn’t mean it’s not useful or doesn’t work. It evidently does, because we are using those findings in practical applications. I myself do not understand in detail how a mobile phone is constructed, since electrical engineering is not my area of expertise. But it still works. The average person in any math or physics department at a university isn’t a genius, with very few exceptions - they’ve just decided to put in the work necessary to learn the concepts. In-depth mastery of any subject requires time and effort, that’s just how it is.

You know, that’s a pretty useful general guideline to have, so far as personal speculations in physics go 👍 The reality is that issues such as dark matter/energy etc have been deeply thought about for a long time by a large number of very brilliant minds. You can’t just dismiss and disregard that. It is therefore exceedingly unlikely that the next major breakthrough is going to happen on some social media forum. That being said, I think that most in the physics community agree that our current models are provisional, and that our understanding is evidently off somewhere. The problem is being looked at from all angles - not a day goes by where not a new paper appears on arXiv about proposed new particles, modifications of gravity, discretization of spacetime etc etc. It’s an area of very active research.

I rather suspect that the opposite might happen - he’ll become a bit of a legend in…well…let’s call it “certain circles”. Somewhat similar to what happened to Tesla.

No. There are time-dependent processes that do not involve motion, such as the decay of elementary particles for instance. Locally at any given location, all clocks always tick at exactly “one second per second” - so there is no meaningful way to say that it is different in different locations. The only thing that changes is the relationship of clocks in spacetime, but that’s not the same thing. Again, clocks don’t have different “speeds” - it’s only that clocks at different events are related in non-trivial ways. This may at first glance appear to say the same thing, but it doesn’t. I have personally done it twice - once in high school physics class with an apparatus basically consisting of an assembly of rotating mirrors, and once for fun using the classic setup involving marshmallows in a microwave. And there are many other ways to do it at home, it’s not really that difficult. Note though that the level of precision within such DIY tabletop experiments is naturally limited, so don’t expect too much in terms of accuracy of the final numerical value.

I absolutely agree with you, which is why, in my post, I added the caveat that it wasn’t entirely rigorous. I chose to use it anyway as I thought it might be the best fit to what I perceived (perhaps incorrectly?) to be the level of background knowledge the OP possesses. It’s not always an easy task to balance technical rigour with the needs of the audience. But +1 from me for the excellent explanation for what really happens 👍

Photons do not experience deceleration or acceleration. What happens in a medium other than vacuum is basically that they start interacting with electrons present there; you could perhaps say (not entirely rigorously) that they get absorbed and re-emitted, the process of which leads to a tiny but measurable delay. So the overall measured speed appears to be lower, even though the photons themselves always locally propagate at exactly c. But there is never any deceleration involved, since massless particles cannot travel at anything other than exactly c.