Markus Hanke

I see, thanks for the replies. In that case, do we have some official procedure whereby one could request a display name change from whoever is able to actually make that change? I’ve been considering changing mine (for personal reasons)…but it isn’t a big deal, if it causes too much hassle.

Is there any way to change one’s display name on this forum? Maybe I’m blind, but I wasn’t able to find such a setting anywhere.

How do you define a “speed of time”? This seems like a fairly meaningless concept to me. Can you give an unambiguous mathematical definition for this? I should note here that in non-flat space times, energy-momentum is conserved everywhere locally, but there is no global law of conservation.

Another problem that occurred to me: as far as we can tell, all electrons are exactly alike; specifically their masses are the same, and those masses remain constant. Black holes don’t tend to behave that way, even if one completely discounts any thermodynamics.

He had a chance to do his measurement precisely because light has the properties we know it does. But that’s precisely what he did not do. He measured the relative differences in Io’s timings at different points of the Earth’s orbit, and thus inferred a finite speed starting from differences in distance to Jupiter. Changes in position are the whole point of this setup.

Yes, I concede that. The argument I gave implicitly relies on Hawking’s assumptions in deriving his results, and also on those assumptions still holding on small/quantum scales. So there is indeed a question mark here. BTW, how would one reconcile quantum mechanical spin (along with their respective spin statistics) with black holes?

But that isn’t the same as saying that there are “infinitesimal black holes at the centre of elementary particles”, as you did above. Also, my initial objection still stands - if elementary particles were black holes, none of them would be stable. For example, assuming an outgoing Vaidya black hole with mass equivalent to one electron, this would evaporate after something on the order of \(10^{-107}s\). Of course, this assumes a classical spacetime, which isn’t that plausible on those scales. But still. Also, as KJW has pointed out, some particles are massless, such as the photon and the gluon, which is not compatible with them being black holes at all.

The immediate problem I see with this idea is that black holes small enough to fit “into” (what does this actually mean?) a subatomic particle would evaporate almost instantaneously, and emit large amounts of radiation in the process. We don’t see this happening; many elementary particles are demonstrably stable.

I think a good way to think about this is to consider length contraction and time dilation (which always go together) as a relationship between frames, not as something that “happens” to clocks and rulers. You can specify how frames are related by stating their relative speed, or you can state the gamma factor (ie time dilation/length contraction) - these are entirely equivalent. So there is no physical force that squeezes rulers, any more than there would be if you observed your house from a distance and noted that it has gotten smaller; all that changes is how observer and observed relate to one another. In SR, that relationship is one in spacetime. Just be careful to not confuse length contraction with a mere optical illusion though - the flattened nucleus in the particle accelerator really does physically behave like a flattened disk in the lab frame, it doesn’t just “look” flattened.

In Kerr spacetime you have \[T^{\mu \nu}=0\] everywhere, so nothing there arises from the stress-energy tensor. Equivalently, one can say that the dynamics of the Kerr spacetime arise entirely from the requirement that all curvature be purely of the Weyl type: \[R^{\mu \nu}=0\] as well as appropriate boundary conditions.

In my post I responded to your comments on the energy-momentum tensor, your model wasn’t mentioned. The point was that it isn’t an ad-hoc invention, but arises from Noether’s theorem. No, that’s not my intention, I’m aware that what you’re trying to do is different. You do have to remember though that GR is a very thoroughly tested model of gravity, so whatever model you propose should match its predictions, at the very least those that have been verified. If you choose to ignore all source terms other than energy density, you’re off to a bad start. Did he? I seem to remember that he pointed out to you that the particular example at hand worked out only because it implicitly exploited a particular symmetry of the Schwarzschild metric; and also that he was of the opinion that the model was too simplistic to work for more complex scenarios.

You are deflecting from the actual point of my post, which was the energy-momentum tensor.

The energy-momentum tensor is the conserved quantity associated with spacetime translation-invariance under Noether’s theorem; it isn’t just an ad-hoc invention. But these nine components all themselves form sources of gravity, and are not derivable from just energy-density. If you ignore them, your model cannot recover all the dynamics of GR, which you claimed it is able to do.

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.