Markus Hanke

It would probably be described by some variant of Cauchy’s equations of motion for inhomogenous media; see for example paragraph 3.1 in this article: https://www.mdpi.com/2624-599X/3/4/45

You don’t seem to appreciate just how fundamental SR is to modern science and engineering - it underpins everything from the behaviour and properties of elementary particles (the entire Standard Model is a relativistic quantum field theory), to chemistry and all its uses, electrodynamics and all its engineering applications (including whatever device you use to make your posts here), to classical mechanics in the high-velocity regime, to everything to do with gravity, to cosmology. Our everyday world is full of relativistic phenomena - CRT screens, the colour of metals such as gold, particle accelerators and the interactions they observe, MRI scanners, the chemical properties of the materials in your smart phone…the list is endless. So the suggestion that SR is somehow “wrong” is just silly - we use it every day, and have done so for some time, and hence know that it is a good model from experience. So you’ll have to excuse when people call you out on some of your more egregious statements.

There is no such implication at all - no exchange of information takes place here. There is no such incompatibility - the combination of SR and QM gives you quantum field theory, which is perfectly well established, and extensively tested too. There are also simpler relativistic generalisations of the QM wave equations, such as the Dirac equation.

It changes by a factor \(\gamma^{-1}\), due to transformation of 3-forces perpendicular to direction of motion.

I think it depends exactly what you mean by determinism. What is stochastic about QM is only the outcome of specific measurements - but given some quantum state of a system, plus necessary boundary conditions, its evolution is entirely deterministic.

That’s principally invariance - but so long as you specifically talk about c in vacuum, and so long as there is no gravity involved, then it is also constant. Just bear in mind that it won’t be constant if you go from vacuum into another medium. No, sound is different from light, it’s neither invariant nor constant. The thing is this - even if you didn’t know anything about the theory of relativity, and just worked off Maxwell’s equations alone, you would still find c to be invariant. You can derive the electromagnetic wave equation from Maxwell, and solve it for a fast moving emitter - the resulting wave still propagates at exactly c. Special relativity simply describes the logical consequences of this fact in a coherent and simple way - something which Newtonian mechanics fails to do. Of course, we now know that Maxwellian electrodynamics is essentially a relativistic phenomenon, but Maxwell himself didn’t know this.

Constancy means that c always has the same value under all circumstances - which it evidently does not, since its value depends on the permittivity and permeability of the underlying medium. For example, c is different in glass than in vacuum. This is a direct result of Maxwell’s equations. Invariance means that its value remains the same irrespective of the relative state of motion between emitter and receiver. For example, light emitted from high-velocity particles (e.g. northern lights) propagates at the same c as light emitted from a stationary flashlight. In other words, the form of the laws of physics do not change if you introduce relative motion. No, these are independent concepts. For example, kinetic energy in an inertial system is constant, but not invariant; mass of that same system is invariant, but not necessarily constant. You can’t conflate these terms. That’s not how it works. A model is considered valid and successful if it is able to produce correct and accurate predictions for the largest possible domain of applicability. Newtonian physics works just fine for classical, low-energy, low-velocity scenarios (which is why we all still learn it at school), but it fails miserably in the quantum realm, are for high-energy, high-velocity situations. Relativity has a much larger domain of applicability (ie it works for a much wider range of situations), which is why it has been adopted as a useful model. Remember, physics makes models, not ontological claims. You are welcome to disregard relativity and try to describe the world in Newtonian terms, if that’s what you wish - but you’ll find that you very quickly run into major problems with this.

It’s invariance of the speed of light, not constancy. That’s an important difference. All aspects of SR have, over the past 100+ years, extensively tested in hundreds, perhaps thousands, of different experiments - it is arguably among the most well-tested theories in all of physics. At the heart of SR lies the symmetry of Lorentz invariance, so ultimately the aim is to look whether this symmetry is ever violated or not: https://en.m.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation No such violations have ever been observed within the domains we are able to experimentally probe, so SR stands firm.

Neither is the speed of light - it explicitly depends on the permittivity and permeability of whatever medium it travels through. But that is irrelevant, since SR is only about its invariance, not any notion of constancy.

I don’t see how it would be possible to separate out just the non-linear contributions, so it is difficult to test this directly, other than to compare the exact solutions against the linear approximation.

The non-linearity of the model is encoded in the structure of the field equations themselves, and means simply that you can’t just add two valid solutions (metrics) together, and expect the result to also be a valid solution to the equations for the particular physical scenario you are interested in. For example, in the aforementioned case of a binary BH merger, the metric of the spacetime containing the in-spiralling black holes is thus not just the sum of two “ordinary” black hole metrics, but a new and different solution in its own right, which has to be obtained from scratch by solving the field equations (which in this case can only be done numerically). The degree by which solutions fail to be linear will increase the more you move into the strong field regime - e.g. when your binary BH are still very far from each other, the overall spacetime almost (depending on your required levels of accuracy) looks like two ordinary BH spacetimes joined together; but as they continue their in-spiral and get closer together, the error of the linearised approximation becomes very large very quickly. Since, in this scenario, the form of the gravitational wave field far away depends explicitly and directly on the geometry of the spacetime close to the in-spiralling black holes, the difference (relative to a linearised approximation) is directly observable here. In general though it is difficult to separate out the effects purely due to non-linearity, since this self-interaction is encoded in the structure of the equations themselves, and thus does not appear as a computational term that can be isolated and separately measured. The basic idea is this - you treat the gravitational metric as a small enough deviation from flat Minkowski spacetime, so you make an ansatz of the form \[g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}\] and demand that \(|h_{\mu \nu}|\ll 1\). Also introduce the convention that an upper bar means trace removal, ie \[\overline{h}_{\mu \nu } \equiv h_{\mu \nu } -\frac{1}{2} \eta _{\mu \nu } h\] Without loss of generality (this can be formally proven, but I’ll obmit that here), one can then impose a gauge condition to simplify the maths, such as \[\overline{h}{^{\mu \alpha }}{_{,\alpha}}=0\] Setting all of this into the original Einstein equations and working through the considerably cumbersome expressions, everything decouples and simplifies into \[-\overline{h}{_{\mu \nu ,\alpha}}^{\alpha}=16\pi T_{\mu \nu}\] Unlike the full original Einstein equations, this equation is fully linear, and obviously far easier to solve.

The non-linearity of GR only really shows up in the strong field regime, so there’s no real way for us to experimentally test it. We can, however, test it observationally - in particular, gravitational wave forms observed from collisions of black holes and other heavy objects are consistent with full strong-field non-linear GR, but not with linearised GR. Generally speaking, most (not all) weak-field regimes can be quite accurately modelled with linearised GR, but strong-field scenarios generally require the full non-linear theory. The non-linearity of the GR equations is very well understood mathematically - ref any text on systems of differential equations. As for perihelion precession, the non-linearity would only show up for very elliptical orbits (not the case for Mercury), or for orbits in highly curved spacetimes.

It’s a real shame, I have many fond memories of my years on TSF. But that’s how it goes sometimes. This here is a good place though. Lol yes, I copied this across when I first came here. I still think it is one of the most beautiful (and important) results in all of maths :)

@KJW Well, mark my words…I remember you from the good old days back on The Science Forum…great to see you again

Yes, but this isn’t the point here. Stellar black holes start off with ordinary stars, which are described by energy-momentum tensor fields that always satisfy the positive energy condition. Then they undergo collapse, which can be described by an appropriate interior solution to the EFE. None of these solutions lead to a geometry below the horizon such as the one you describe, at least not as far as I am aware. You might be able to manually construct such a geometry be glueing together patches of suitable spacetimes (though I doubt even that is possible), but whether such a construct is a valid solution to the field equations for a physically reasonable collapse process is an entirely different matter. I very much doubt this, but I would also encourage you to actually go and try to derive such a geometry - it would be very instructive. If you do find something, then please present it here, I would be curious to take a closer look. I don’t quite understand - you said you want to apply a full Lorentz transformation to the 4-vector, so this already affects all four components, and guarantees that the vector norm is preserved.

Any Lorentz transformation is simply a hyperbolic rotation (+boost) of your coordinate system - you are merely labelling the same physical events in your spacetime in a different way. You are always free to do this, since it has no physical consequences in the classical world - the energy-momentum tensor is the conserved Noether current associated with time translation invariance, so you can choose to do this either in the future direction, or into the past, the difference just being a sign convention. The actual dynamics of the system are the exact same, so no laws of physics change form. So of course you can describe classical anti-particles as propagating backwards in time (relative to their positive-energy counterparts), but that just means you’ve chosen a different sign convention in your description of the system. And again, all Lorentz transformations (antichronous and orthochronous) are diffeomorphisms, so applying them to a given metric does not change anything about the geometry of that spacetime. Sure. However, this isn’t the only thing it does - it also changes the spatial components of the 4-vector such that its overall norm remains conserved. This is why 4-vectors (more generally: objects that are representations of the Lorentz group) are covariant under all Lorentz transformations, and laws formulated with them retain their form. There is no change in particle trajectories, such a spacetime has the exact same geodesic structure, only future-directed time-like unit vectors are now sign-inverted, so all processes “run backwards”, and all energies are negative wrt to the ordinary case. All curvature tensors and their invariants remain unaffected, so this is the same geometry described simply in a sign-inverted way. To make a long story short, the point is just this - if you start off with a positive energy-momentum tensor and run through the maths, you don’t end up with a negative-energy region beyond the horizon during a collapse process - at least not in any solution that I’m aware of, since M is a global property of the entire spacetime.

Yes: Reissner-Nordström interior Kerr interior Kerr-Newman interior

Well, this is going to be a problem then. I don’t think you will get anywhere useful if you plan to abandon local Lorentz invariance, and thus necessarily also CPT invariance, as well as some fundamental properties such as spin. Clearly, the outcome of particle accelerator experiments are what they are (in terms of how different types of particles behave in specific circumstances etc), and whatever form you choose to write your laws of physics in, they must be able to reproduce these outcomes in some way, or else these formalisms will be of no use at all, because they wouldn’t relate to what we actually see and measure in the real world. But why don’t you keep working on it a bit more, and once you have an actual formalism to present, people will be happy to take a look at it.

For the same reason why no one considers what might be north of the North Pole - it’s a meaningless concept. Yes, that’s the basic idea.

Simply apply a full antichronous Lorentz transformation of your choice to the given metric tensor, of course. Since tensors are covariant objects under the full Lorentz group, and all Lorentz transformations are by definition diffeomorphisms, you simply end up with a different coordinate description of the same spacetime. You’re basically just inverting the metric signature and picking new matching coordinates, which you are always free to do. This has no physical significance, in that it doesn’t change anything about the geometry of your spacetime - all curvature tensors and their curvature invariants remain unaffected by this. Likewise, the form of all your physical laws - written in covariant form - remains unaffected. If you wish to obtain a global geometry like the one you have described for black holes, you will have to derive it as a valid solution to the field equations. My feeling is that, so long as you start with a positive-energy star to begin with, there is no mechanism of classical gravity that can change this to a negative-energy distribution. Looking at how the interior metric and the collapse process are (approximately) described in Schwarzschild spacetime, I certainly don’t see any way to mathematically make it so.

It already does - the GR equations are invariant under the full Lorentz group, as @joigus has correctly pointed out earlier.

Well, there’s nothing wrong with trying this, so long as the resulting model can replicate already known (and well-tested) results. I would be interested to see what this would look like. I don’t know how you would recover local Lorentz invariance from a Euclidean metric, but again, there’s nothing wrong with trying, if you can show that it provides correct predictions.

Can you show us the field equations for this model? It’s difficult to comment unless one sees the maths.

Yes, that’s right. You need at least a rank-2 tensor in there as well, in order to capture all relevant degrees of freedom. But regardless, I agree that the result - if it exists at all - would end up being mathematically very complicated and require extra fields, whereas standard GR has pretty simple field equations. So the question naturally arises: why bother? What actual problem in the existing models are you attempting to address? You can look at ratios (!) of radioactive decay products over long periods. This has been done using natural fission reactors, with the consensus being that the relevant constants of nature involved in these processes have not changed over at least the last ~2 billion years.

I agree with @joigus, that’s how I would look at it too. But I, too, must do some further reading when I’ve got the time. And yes, SupD is certainly far from being an accepted consensus, and very much a minority view.