Markus Hanke

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”.