Markus Hanke

This is true. However, I would question whether the initial/boundary conditions required to arrive at these solutions are actually physically realisable - the standard Alcubierre metric would require exotic matter, and the Kerr metric - while being a very useful textbook case - requires asymptotic flatness amongst other things, and is unstable under perturbations, so arguably it wouldn’t arise in the real world, except as an approximation in its exterior region. I would argue that perhaps ensuring that initial/boundary conditions being as physical as possible would avoid most, if not all, such artefacts. That’s a very good point, I never looked at it that way before.

I’m trying to show you that your formalism requires some of the very things you reject. Orthogonality requires some notion of inner product. Gradients require derivatives and a metric, as do lengths. And so on. These are things you can easily research yourself, I don’t think it is always necessary to typeset LaTeX for stuff I would consider basic and easily found with a simple search, after one has been made aware of them. If you presume to be simplifying GR, you are operating at a level where you can be reasonably presumed to be able to do this. The problem is that you are so sure that your idea must be right, that you are no longer receptive to feedback. BTW, there already exists an almost purely algebraic formulation of GR - check out the papers by Geroch and Heller on Einstein algebra. Note though this still requires the concept of smoothness as an additional ingredient.

It seems to me that this is precisely what GR does - it tells you mathematically how the lightcones at each event in these circumstances are related to one another. That’s causal structure. The philosophical question is rather why not all such situations seem to be realised in nature, even though they are valid solutions to the relevant equations.

The whole thing starts with two fundamental observations about the world we live in: Events in the real world are separated - they can be spatially separated (things don’t all happen at the same place), temporally separated (not everything happens simultaneously), or both. Not all events are capable of influencing all other events - only events separated in particular ways are causally connected. In other words, the world is endowed with a causal structure (at least classically). The job of physics is to make models of aspects of the world we live in. Einstein, using pre-existing work by Minkowski, Riemann and others, realised that, if one uses a semi-Riemannian manifold endowed with a metric and a connection, and allows the metric to vary while holding the connection fixed (the Levi-Civita connection), one obtains a mathematical model wherein the points on the manifold are related in the same way as events in the real world are observed to be causally related. In small enough local patches this gives you SR with its light cones, whereas globally it gives you GR and hence gravity. In either case it boils down to causal structure. So the assumption that GR makes is that one can use a semi-Riemannian manifold endowed with the Levi-Civita connection and a varying metric as a good model for the causal structure of the world we live in. I highlighted “can” because no claim is made that this is the only possible way to model these dynamics. Many other models of gravity are known nowadays, not all of which are metric models either; but GR seems to be the simplest, and the one that works best AFAWCT. BTW, the Einstein equations don’t a priori assume any specific dimensionality. One advantage of tensor equations other than general covariance is that their form remains the same in any number of dimensions. The real world looks 4D, but GR works just fine in other situations too. Seems to me that it’s rather the other way around - despite multiple posters having patiently attempted to explain to you why, in your proposal, you are implicitly using some of the concepts you initially rejected, you’re still not getting it. I strongly suspect that’s because you don’t want to get it. To be honest, I think we’re done here.

Exactly 👍 That’s what I tried to point out before.

That’s interesting, I didn’t know that. Question is, how do you do this without already knowing the metric…? As you pointed out yourself. But when I asked you that question about the physical scenario earlier, you used the gradient in your answer, which is a differential operator.

Well, the standard formula is derived from the Schwarzschild metric, so it is an approximation to the extent that the SS metric itself is an approximation to the actual physical situation in the solar system. Actually quantifying this error would be not so easy - has anyone ever produced a numerical GR simulation of the solar system, taking into account the other sources of gravity, including the angular momentum of the sun itself? I wasn’t immediately able to find such a solution. It should be noted though that the standard formula matches the observed precession value to ~0.1%, so the approximation error is quite small when compared to actual observation.

You’re still not getting this - the point I was making was that the structures you are using already presuppose and require some combination of the things you have initially rejected as unnecessary, and I have highlighted some specific examples. Others here have attempted to point out the same. What you do with these criticisms we offered is up to yourself, it’s your hypothesis after all.

Exactly, and herein lies the issue - you explicitly said you weren’t going to use any coordinate systems, yet you talk about areal radius measured in meters. To meaningfully define the notion of “distance” between points, you need a metric space structure; the same is true for angles, and thus orthogonality. My question about gradients you haven’t answered at all; in short, you need a notion of derivatives to define the gradient, and, if the basis is not orthonormal, you explicitly need a metric as well. The point of all this isn’t numbers, but to show that you are tacitly using the very notions you are rejecting; that is why I asked about that scenario. You are literally going in circles, pun intended - and I note explicitly that the very notion of “circle” is meaningful only if you already have a metric space. Just throwing around the word “relational” doesn’t change these facts.

What is an “areal radius”? Since it is in units of meters, it sounds suspiciously like a distance (length) to me. What is an “algebraic output of a field measurement”? And how do you mathematically define the gradient of a scalar field without recourse to any derivatives? How do you even have a scalar field if there’s no manifold for it to live on? A field on what, exactly? What is “intrinsic orthogonality”? How do you tell if two directions (vectors) are orthogonal, in the absence of an inner product? What does “source of geometry” mean?

Thanks, but unfortunately that quote thingy never appears for me at all, not even for a single dead-Center word selection :/

I’m sorry, but this is utter nonsense. You said you aren’t going to use coordinates or metrics, but then you talk about radial and tangential directions, lengths and orthogonality, gradients, areas, components, conserved quantities, Killing vectors…none of which can meaningfully exist without a spacetime manifold endowed with a connection and a metric. Some of these things are also observer-dependent, which you’re not taking into consideration at all. Furthermore, nowhere do you actually take into consideration the nature and strength of the energy-momentum distribution; there’s only talk of some nebulous scalar, defined via a Schwarzschild radius, though this is most certainly not a Schwarzschild situation. if this were an EM field, where do you encode the specifics of the field? In reality the EM field cannot be represented by a single scalar. Also, real-world gravity isn’t linear, but nowhere do you account for that non-linearity - the overall gravity of this situation is not simply the sum of all gravitating sources. If I were to model this situation in standard GR, the result for the trajectory of the test particle is very different - not surprisingly, because it accounts for all relevant relativistic effects. It seems obvious to me that all you’re doing is to pull random stuff out of some LLM-AI. So I agree with the other posters on this thread that there’s not much of value here. My honest opinion.

I know, and I’m not at all blaming the mods here - I think you’re all doing a fantastic job :) It’s just a sign of the times I guess. Sure I will, I’ve no intention of disappearing, it’s just that I find it frustrating that everyone is suddenly the new Einstein, just because whatever LLM they’re using is designed to not outright tell them that their ideas are nonsensical. Science cranks have always been around, but since LLMs came along the issue has taken on a different quality - it often looks like they know what they are talking about, while in fact they don’t understand much of the subject matter, because they’ve never taken the time to study it in-depth. It’s just harder and more frustrating to debunk, because these LLMs are good at talking people into having the perception that they’re onto something, with superficially plausible-looking maths and all, even if they’re not. At least in the old days, the maths weren’t quite as easy to come by, you needed to actually make a real effort to develop it - nowadays, Lagrangians and tensor equations are a dime a dozen, even if they’re meaningless.

Unfortunately neither of these suggestions work for me - I’ve tried on Safari, Opera, Chrome and Firefox, scrolled and reloaded, but the partial quote button never appears. I haven’t actively posted here in a while (the flood of AI-generated posts/answers we’ve been seeing have spoiled the fun for me), but last time I did, this still worked fine :/

Does anyone have problems with the quote function? I can quote an entire post using the button underneath, but when I select just a snippet out of the post, I no longer get the option of just quoting that snippet…?

Suppose there’s a region filled with some form of energy-momentum distribution, perhaps a radiation field, not necessarily assumed to be uniform, plus some extended body with given mass located at a given point within that region. Suppose further a test particle enters this region, at some given point and instant, with some given initial velocity vector. Assume further that the test particle moves under the influence of gravity only. Can you show us exactly how you calculate the trajectory of this test particle? Does it hit the body, or not? Remember you cannot use any concept of coordinates, you don’t have a field equation, you don’t have a metric, you don’t have a concept of geodesics, nor do you have access to tensors or any other type of covariant object. I’m interested to see how you encode the gravitational sources, and how you find the trajectory of the particle, as test whether or not it hits the solid body.

PS. One more thing to consider is that for an outside stationary observer lowering a rope, its loose end would never appear to reach the horizon, even if he lets go and allows it to free-fall. The loose end would just appear to move downwards more and more slowly, while getting dimmer and dimmer, until it eventually fades out. Actually hitting the horizon would take infinitely long on his own clock. For a clock moving downwards alongside the rope, on the other hand, the (freely falling) rope would cross the horizon in finite time. So nothing about this seemingly simple scenario is quite straightforward.

The crossing of the horizon would happen in a tiny fraction of a second, orders of magnitude shorter than even the electrochemical signals in your nerves could propagate. So no, you wouldn’t notice anything special. Below the horizon there are no stationary frames, so if you lower a rope through it, it would simply break. In practice it would in fact break quite a bit before it even reaches the horizon, since at the horizon itself only massless particles could remain (at least in principle) radially stationary under the right conditions, though this would not be an equilibrium. So no, you can’t pull it back out. What you pull back towards you will be the broken, shortened end that hadn’t crossed the horizon yet. Actually, your head and legs won’t share a common notion of simultaneity in this situation, so it’s rather more complicated than you think. What’s more, for your head the horizon is below, whereas for your legs the horizon is in the past.

You’re right, the two are not necessarily the same - you can eg have “false vacuums”. So long as you assume that the laws of nature also apply to a “perfect void” of the type you describe, then yes, the void must have these properties.

Of the top of my head, so probably not an exhaustive list, and assuming you mean actual physical space: Topology, ie properties such as dimensionality, connectedness, orientability etc Geometry. Pick any set of points in your space, and there will be well-defined notions of distances, angles, volumes etc. IOW, a metric. Note that the absence of gravitational sources does not necessarily imply that this geometry is trivial. Permittivity and permeability, ie an ability to support EM fields, as well as a particular value of c. Yes. In an empty universe all quantum fields are in their ground (vacuum) state, and that state may be associated with non-zero vacuum energy. Could one conceive of a universe that does not contain any quantum fields? I don’t know. Possibly.

They could show themselves.

I think euthanising a living being for no good reason other than that is “unwanted” is ethically problematic. If you can’t or don’t want to keep your pet any longer, there are more appropriate options available. Also, keeping animals locked up in zoos just so we can gawk at them, is likewise ethically questionable.

Thanks for the recommendation! I haven’t heard of this story, but I do like Clarke in general, so I’ll try to find it somehow :)

Yes, and I still do. Well, what is “it” that you think is “going” someplace? I don’t know, it’s not something I’ve thought about much (and I haven’t been following developments in AI too closely) - but this is a good question. In my tradition, and at this point, it plays no role at all, and probably never will; but it may be different in other religions.

I’m not yet that old, but yes, I’ve found my way in life. That didn’t happen over night though, and took some doing.