Markus Hanke

Finnish is on my personal list of languages I’d like to learn (to some degree) before this old brain begins to fail me I’m fairly fluent in Norwegian, and can at least understand and read most Swedish too, plus some Danish. Problem here is the large number of local dialects, some of which are very different from the standard form of these languages, particularly in Norway. Is it the same in Finland? Just out of interest.

I’m afraid that makes little sense…spacetime (however many dimensions you give it) is either geodesically incomplete, or it isn’t. This is not an observer-dependent notion. Extra macroscopic dimensions would be easily noticeable in the effects their existence has on the laws of physics. For example, in a (4+1)D universe, radiation fields would fall off with distance according to an inverse cube law, rather than an inverse square law. This is evidently not what we see in the real world. This isn’t what happens in GR, so what you are attempting to formulate is a modified law of gravity. That is good and well, but the major problem is of course that such a modified theory must also correctly model all other gravitational scenarios, not just gravitational collapse. This is where all known modified gravity theories (and there are many at this point) ultimately fail - some do very well in specific scenarios, but fail miserably in other situations. Right now only GR provides the best fit for the largest set of available data. This is actually a pop-sci misconception - collapse processes don’t lead to “infinitely dense point”. But that aside, when it comes to avoiding singularities, this is really not so hard to do; for example, simply allowing torsion on your spacetime does the trick (the resulting theory is called Einstein-Cartan gravity). The trouble with all those modifications is that they have other consequences too; for example, the above model leads to extra terms in the Dirac equation, making it non-linear, and there’s currently no observational evidence of any deviations from the standard form of the equation. Thus - proposing models that fix specific issues is quite easy, but making these models also agree with all other available data, that’s hard.

How so? The singularity theorems of GR guarantee that the region below the horizon must be geodesically incomplete.

That’s true - but it also has many symmetries in GR, hence only 20 of those are functionally independent. The Riemann tensor can be decomposed into two parts - the Ricci tensor, and the Weyl tensor. Suppose you have a small ball of test particles freely falling in a spacetime. The Ricci tensor tells you how fast the volume of this ball changes with time, whereas the Weyl tensor measures how the shape of the ball gets distorted as it falls. In vacuum, shapes get distorted, but volumes are preserved during free fall (hence the Ricci tensor vanishes, but the Weyl tensor doesn’t). In the interior of sources, both shapes and volumes may change. Mass itself adds linearly, but the source term in the field equations is not mass, but the energy-momentum tensor.

What do you mean by this, exactly? Spacetime is not embedded in anything else, so when we are talking of geometry in this context, what we are referring to is intrinsic geometry on the manifold.

This is true, but one must bear in mind the local nature of the field equations - outside the system, in vacuum, the energy-momentum tensor is zero, so the equations you’re solving are actually just the vacuum equations \[R_{\mu \nu}=0\] There is no source term at all here, yet you’re still getting a curved spacetime, even far from the central source. This is precisely due to the non-linearity of the theory - the curvature inside the system can “bleed out” into the surrounding vacuum, because curvature at one point is itself a source of curvature for surrounding points. This is encoded in the non-linear structure of the equations themselves.

It means that the gravity of a macroscopic system is in general different from a simple sum of the fields of their individual constituent particles taken in isolation. It also means that just because a volume of spacetime is empty, that doesn’t necessarily mean that it is flat. One has to start with the correct initial and boundary conditions.

Gravity being self-coupling or non-linear (which means the same in this context) means - among other things - that you cannot simply add together gravitational fields of individual sources to obtain the field of a more complicated system. For example, the spacetime geometry around two bodies in close proximity is not just the sum of two Schwarzschild metrics, especially not if these bodies are in relative motion. This is why you get eg extra perihelion precession with Mercury, which you wouldn’t expect in a linear theory. Another example are gravitational waves - when they traverse an area of background curvature (like a massive body, or another wave front), they interact in ways that deviate from ordinary linear wave dynamics. In more technical terms, if you take two metrics, both of which are valid solutions to the Einstein equations, and add them together, then the result is in general not itself a valid solution to the equations. It also means that a gravitational field can exist in the absence of any “ordinary” sources; for example, exterior Schwarzschild spacetime is everywhere empty, yet nonetheless curved. This is because curved spacetime itself contains energy, which can act as a source for further gravity (caveat: this form of energy cannot be localised, unlike ordinary sources). This is in contrast to Newtonian gravity, which is completely linear.

Ok, this is basically what I had in mind. We now need to first make precise what we actually mean when we speak of observers and IFRs. I’m taking the following from Sachs/Wu, General Relativity for Mathematicians (1977), which is the standard formal definition. We start with defining spacetime to be a singly connected, time-oriented Lorentzian manifold (M,g), endowed with the Levi-Civita connection and a suitable metric. An observer on M is a future-pointing time-like curve with appropriate parametrisation. A reference frame on M is a vector field, the integral curves of which are observers as defined above. An inertial frame is one that is constant with respect to the covariant derivative. Thus, choosing an IFR means choosing a vector field on the manifold M that fulfils the above requirements. There may exist infinitely many such choices, or none at all, depending on the geometry and topology of M. This definition is clearly inconsistent with your idea, since it makes no sense to speak of connections, metrics, integral curves, and vector fields that span multiple manifolds that don’t map their points into each other 1-to-1. The best you could do here is consider a foliation in some higher-dimensional space, where each spacetime is a hypersurface of some constant parameter. But it doesn’t look like that’s what you’re doing. Since you say that each IFR is its own spacetime manifold, what does it formally mean to be stationary relative to spacetime? And what does it mean for a spacetime to be in motion relative to another spacetime? If each IFR is its own spacetime manifold, and events on these manifolds don’t map 1-to-1, then transmitting information between such manifolds would be a clear violation of local unitarity. This explicitly implies \[\nabla_{\mu} T^{\mu \nu}\neq 0\] on all manifolds involved. But you postulate that events are not the same in all IFRs, in the sense that there’s no 1-to-1 map between points on different manifolds. If different IFRs are different spacetimes, then accelerating means the test particle is leaving one spacetime and entering another. So it does cease to exist on the original spacetime, irrespective of any dependencies. How is this possible if there’s no 1-on-1 map between these manifolds? And what is the nature of these “dependencies”, since you also say that each IFR is subject only to its own causal relationships? There is a clear contradiction here. If the outcome of an experiment in an IFR is somehow dependent on non-local influences, we’d have spotted that by now. On the other hand, if there are no such influences, there is no mechanism to guarantee causal consistency between manifolds. But this is precisely not what you’re claiming…? Refer to your earlier examples with the moon, and photons/muons.

Fear of punishment - whether earthly, or by some god - is not the same as genuine morality.

To be honest, I’ve been silently reading along this thread, and it is still not clear to me exactly what this hypothesis you’re referring to actually is. You keep talking about events and causality, and how they change between IFRs, but those are all concepts that already presuppose the existence of spacetime endowed with a connection and a metric. At the same time you refer to something “more fundamental than spacetime”, which is meant to be implied by the hypothesis. I’m so far failing to make the connection. I don’t know - it was you who made this claim: If there’s no mapping (invertable or not) between those frames, there would be no transformation that relates them to one another; but you did say that such a transformation exists. So you have two entirely separate patches of spacetime, each with their own set of events, which can’t be mapped into one another by any 1-to-1 map. These patches would thus each have their own independent histories, which are not guaranteed to be mutually compatible (see your own example above). Yet the patches are still somehow IFRs in a Minkowski spacetime, and thus related via the usual SR transformations? Is that the idea?

If c wasn’t invariant, there’d be a plethora of unresolvable physical paradoxes, and the universe wouldn’t have evolved. An invariant c is a fundamental prerequisite for any internally self-consistent notion of spacetime and causality, among other things. To put it differently, in the abstract space of all conceivable sets of laws of physics, only those with an invariant c can give rise to a macroscopic, self-consistent spacetime.

I don’t see what such a transformation could possibly look like. For starters, massless particles have no rest frame (inertial or otherwise) associated with them, so it is not clear at all what it actually is that you’re mapping between. Furthermore, what would be the parameters of the map? It can’t involve v, since v=c for all IFRs, so the map would be 1-to-many. But if it’s not v, what else could it be, since that’s the only parameter whereby IFRs are related to one another? Also, there’s more than one massless particle in nature. What is it that determines that a particular set of IFRs map a muon into a photon, and not a gluon (or hypothetically a graviton)? And what about the fact that muons decay, and gluons are QCD-confined, whereas photons are free and stable?

I never heard of this before, and tbh I don’t see how this is even mathematically possible…? Naively it would seem to me that you can’t get the proper polarisation states in g-radiation fields with anything less than spin-2 quanta. But maybe I’m missing something.

I would even go so far as to say that FE is nothing but a conspiracy theory - the dynamic at the heart of this concerns a “them” hiding things from “us”; it is wholly about control and power in politics, and has little if anything at all to do with the science of planets. It’s about mistrust in authority.

None of this is what General Relativity actually says. What is the question or discussion point here?

This falls under the area of General Relativity. If you already have a background in maths, my recommendation would be the book Gravitation by Misner/Thorne/Wheeler.

I’d bet there’s something to it. In fact, I’d go so far as to say that complexity, chaos and emergence in general are seriously underrated and under-utilised in modern physics. Just my opinion though

I wouldn’t put it so strongly, it just means that the data places strong constraints on which models might be viable or not. On the other hand though we have good reason to believe that there is entropy associated with the horizon of BH’s - and since the concept of entropy only really makes sense for a system that exhibits discrete microstates in some form or anither, the interior region cannot be smooth and continuous empty space everywhere. So I’d still bet my money on some deeper structure that underlies classical spacetime, even if that turns out to not have anything to do with spin foams.

Shame, that…I had some hopes for LQG. But this is just how science works.

This is the trouble, really; all the known and proposed alternative models of gravity have some form of problem. Generally speaking, they tend to be able to model one particular (class of) phenomenon better than GR, but then fail spectacularly in other situations. Most of them are also mathematically complicated, and unwieldy to work with, and oftentimes they rely on additional assumptions (extra fields etc) for which we have no evidence.

Sabine Hossenfelder, for instance - though I wasn’t immediately able to find a reference (have to look some more later). The idea isn’t new, and isn’t mine either, but I think never really came to the forefront, since it’s essentially untestable given the current limitations in computing power.

I have experienced this switch from LaTeX to RTF numerous times also - for me this happens when I simply press “Edit” after submitting a post that contains LaTeX.

I can not, of course, be sure about such specifics either, nor even about whether or not anything special would happen at all in an n-body problem. It’s really just an hypothesis, based on emergent dynamics in non-linear systems. Basically I’m sceptical about both the particle as well as alternative-model options, so it’s good to have a third alternative. I agree with DanMP’s earlier comment that DM is a big part of our current model of the universe, so this is a very important issue. Well, I never have been trained. I simply base my thinking on what we do know - on situations with small n that can be exactly solved. For example the n=2 case; the spacetime of a binary body system is nothing like our naive idea of two Schwarzschild metrics superimposed. What happens for n= ~billions is anyone’s guess, because I don’t think those non-linearities smooth out. So maybe DM is really a chaos-theoretic problem.

Yes I get you, but to me this simply is extrapolating a model which we know works very well on solar system scales, to larger scales. After all, there is no immediate reason to assume that gravity works differently on galactic scales than on solar scales - while that could be so, we have no evidence that that’s actually the case. Therefore do I think it’s important to try and find ways to figure out what GR actually predicts for large n-body systems, rather than just simplified models with an unknown error factor. One might also say that possible alternatives such as MOND etc are “cheating”, because all of those models postulate things (extra fields, new universal constants etc) for which we have otherwise no evidence. At the very least, GR is the simplest possible metric model of gravity that is fully relativistic.