Markus Hanke

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.

I’ve had a look at these papers, and none of the authors here claim any kind of superluminal propagation (!) velocities, never even mind instantaneous propagation. This seems to be purely about certain quantum phenomena around tunneling and uncertainty (as one would expect in the near field), as well as phase velocities. Such apparent “superluminal” phenomena have already been known for some time (as is mentioned in Zhang), but do not constitute violations of local Lorentz invariance. So far as I can see, nothing in these references even remotely supports your claim.

That’s just a particular choice of coordinate basis, which you are always free to make. You can even choose coordinates that don’t correspond to any physical clocks and/or rulers at all (eg Kruskal-Szekeres coordinates), and still obtain useful solutions. The metric in GR essentially represents your chosen way of how you label physical events in your spacetime, and allows you to define measurements (angles, distances, volumes etc) in terms of those labels. It basically tells you how events are related, given a particular choice of coordinate basis. It also relates vectors to 1-forms and vice versa, and allows for the definition of certain important operations, like the Hodge dual. If you picture a street map of your local city, the metric would be the scheme by which you assign street names, as well as define distances and routes between addresses. Just like you are free to re-name streets without affecting the physical layout of your city, so you can choose different coordinates on the same spacetime without affecting its geometry. To be honest, I don’t see how such a formulation would be helpful, given that local Lorentz invariance is an extremely well established experimental result. c simply isn’t seen to locally vary in any meaningful way, so I don’t see why one would want to write a model based on this. Is your basic idea to try and replace the rank-2 tensor description of GR with some kind of scalar field theory?

This is the bit I don’t get - what exactly do you mean by “alternative metric structure”? You can always just pick a new metric tensor that isn’t related to the old one by any diffeomorphism, which gives you a new spacetime that is not isomorphic to the old one. But that’s probably not what you have in mind? And why would you want to do this at all - what is the advantage? In that case you need to go away from metric structures altogether, and consider non-metric approaches. The aforementioned Gauge Theory Gravity is an example for this. More fundamentally, it is probably gauge theory in general that you should take a closer look at.

You are always free to choose a connection other than Levi-Civita, but if you do that, you will have to adjust your physical laws accordingly, since their form might differ now. Ok, so we are in agreement on this. The link between them is given by the connection coefficients (Christoffel symbols), which allow one to express the effects of the chosen connection in terms of the metric and its derivatives in a consistent way, should the manifold be endowed with a metric. So there is never any “conflict” between them. Changing the metric simply changes the connection coefficients, it has no bearing on the connection itself. For example, GR allows for infinitely many different metrics on spacetime, but it always uses the Levi-Civita connection. Ok, so now you need to define for us just exactly what it is you mean by “geometry”. In standard GR, two given spacetimes are said to have the same geometry if they pass the Cartan-Karlhede algorithm, meaning “geometry” is given by curvature tensors, their curvature invariants, and the functional relationships between them. But I don’t think this is what you have in mind - it sounds more like you wish to model spacetimes without reference to any metric at all, and thus express “geometry” in terms of different dynamical variables. If so, gauge theory gravity might be an example of what you are looking for.

Yes, it’s a connection for which torsion vanishes - this is what I wrote above. It also makes the covariant derivative of the metric identically vanish, which I likewise mentioned in a previous post. As I explained already, this is not its fundamental definition, because this applies only to those manifolds that are endowed with a metric. The actual definition is a curve that parallel-transports its own tangent vector - you don’t need a metric for this. If you do have a metric (as in GR), then geodesics extremize (either minimise or maximise) the separation of events, but doing this extremisation procedure requires a metric. Spacetime manifolds in GR are pseudo-Riemannian (ie locally Lorentzian), so they are always endowed with a connection. Without that there wouldn’t be a notion of parallel transport, and thus curvature, and so it would be useless for the purposes of the model. Of course we can - this is the basis of differential topology and affine geometry. You use a connection for these, but no metric. It’s rather the other way around. There are some aspects of geometry and topology that are entirely independent of any metric. Parallel transport is one of them. Parallel transport doesn’t have anything to do with angles, it is defined purely in terms of the connection. However, if you do have a metric as well, then the two are related via the connection coefficients and the metric compatibility condition, so of course there is never any conflict. I am still not sure if I understand what you are actually trying to do. But I’ll wait for further comments first.

Nicely put +1

Yes, that’s fine, you can always do this. Several such reformulations already exist. Proper time is defined to equal the geometric length of the clock’s world line through spacetime - this is very convenient, and greatly simplifies much of the maths. While it is certainly possible to make other choices (affine parametrisation), I don’t know why you would. But so far as I can see this doesn’t lead to any different geometries - you’re still dealing with the same geodesics on the same manifold, you’re just parametrising them differently. Connections are not derived from metrics, they are separate and more fundamental structures. The LC connection is a very specific one, namely that for which torsion vanishes. Without intending any disrespect, but it seems like there’s some confusion here about the meaning of “metric” and “connection” in differential geometry, because what you write above doesn’t make any sense. The connection exists quite independently from metrics and coordinates; its purpose is to relate tangent spaces at different points of the manifold to one another, so that a covariant derivative can be defined. A metric provides a way to define an inner product for vectors and forms, and thus a notion of lengths, angles, volumes etc. These are different things, and you can have a connection without a metric on your manifold - this allows you to do a certain amount of topology, define parallel transport, as well as tensor fields and some operations between them (excepting index raising/lowering, which requires a metric). Choosing a different metric thus has no bearing on your connection at all, it only changes the measurements of lengths and angles. Standard GR uses curvature on a semi-Riemannian manifold to model gravity. An example of an alternative approach is teleparallel gravity - here you use a parallelizable manifold and endow it with a Weizenböck connection, which yields a situation where you have no curvature at all, but only torsion. So gravity here is described solely through torsion on parallel geodesics, with the field equations adapted accordingly. If I understand you correctly, that’s an example of what you mean by “different geometry”. A second example would be Einstein-Cartan gravity - here you choose a connection that allows both curvature and torsion, and adapt your field equations accordingly. A third example is the ADM formalism - you replace your manifold with a foliation of 3D hypersurfaces, and wrap all your dynamics into how these surfaces are related to one another, using the Hamiltonian formulation. And there are many more such formalisms. Do note that these are all specific examples of gauge theories - which is kind of the overarching framework when it comes to “different geometries”. Is this helpful?

No, it has a very precise meaning, but clearly it doesn’t correspond to whatever it is you have in mind here. Can you explain precisely what you actually mean by “changeable”? You seem to be using the term in a non-standard way, so far as tensor calculus is concerned. Or are you referring to the covariant derivative, which of course vanishes for the metric tensor? I don’t know what you mean by this - the Einstein equations relate energy-momentum to average curvature, so they place a local constraint on the metric. Along with boundary conditions they provide a unique geometry for any given distribution of energy-momentum. If you change that geometry, you have to also change that distribution, so you are no longer dealing with the same physical situation. If you want to keep your physical situation, but use a different geometry to describe it, then you have to change the form of your physical laws. Steven mentions this too in the screenshot you posted. NC gravity is much more general than standard Newtonian gravity - they are equivalent only if you impose certain external constraints on NC that don’t follow from the theory itself. Otherwise you get a large collection of different models that aren’t necessarily equivalent to Newton. Here is a good overview. Do you mean the screenshot of Steven’s post from PhysicsForums? I’m sorry, I can’t make any sense of what you are saying here - first you say the connection will be different, then you seem to say we are using LC? So if I understand all this correctly, you want to reformulate GR by using a geometry different from a semi-Riemannian manifold endowed with LC connection and metric to model gravity, right? Note that many of such alternatives already exist.

I’m having trouble following you here. A covariant tensor equation is precisely this - covariant. This means that its form does not change at all when you choose a different metric. In GR in particular, you are free to choose any suitable system of coordinates you like to describe your spacetime, and this has no consequences for what form the Einstein equations take - they will always look the same. What does have real consequences is choosing a different connection on your manifold. GR assumes the Levi-Civita connection, and if you choose a different one, you will in general get a different model that might make different predictions. Is this what you mean, perhaps? PS. To reiterate again, connection and metric are very different things. You need both to meaningfully speak of lengths, angles, volumes etc, but general tensors and their relationships (though not all index manipulations) are defined without recourse to a metric, and you can still make some statements about the topology and geometry of your manifold (“affine geometry”) even without any metric at all.

Yes, very good point +1

I disagree. Relativity is just a particular application of (much more general) differential geometry, which is extremely well understood and worked out.

I think it suffers from the same problem as the opposing claim - you cannot ever disprove the existence of a designer. The best we can do is show that the laws and processes of physics as we see them arise without need for outside intervention - but at the moment we can’t really do that yet. But even if we can do this, the mere absence of such a need still does not necessarily rule out a designer - it could have been designed even though there wasn’t a need for a designer. So I think looking for evidence for either claim is ultimately a waste of time, unless the alleged designer chooses to reveal himself in indisputable and unambiguous ways.

The actual definition of a geodesic is a curve that parallel-transports its own tangent vector. This requires a connection, but not necessarily a metric - IOW, a manifold that is endowed with a connection but not a metric, will exhibit a geodesic structure. Of course, if you have both a connection and a metric (as is the case in GR), then the geodesic equation can be written in terms of derivatives of the metric tensor. But that is not its fundamental definition. I don’t know what you mean by this. A metric is a structure on a manifold that has a very precise definition, which has to do with fiber bundles and tangent spaces, but not with any particular coordinate choices. What, exactly, do you want to replace here? The Einstein equations are a covariant tensor equation, so its form is the same irrespective of what geometry the manifold has. That’s the entire point of general covariance. IOW, you are quite free to use a different concept of time (coordinate basis) to describe your scenarios, but that doesn’t change the laws of physics, and thus all tensor equations remain unaffected. In that case you will obtain incorrect predictions for the polarisation states of gravitational waves, which cannot be modelled by any rank-1 model. You need at least a rank-2 model to correctly account for all relevant degrees of freedom, so gravity cannot be a force in the Newtonian sense. It’s not absolute, it’s just a convenient choice of coordinate basis that makes certain astrophysical calculations easier. You are always free to choose your coordinates and units as is convenient - that doesn’t change anything about the laws of physics, in particular not their form when written as tensor equations.

No, it’s a religious belief you have chosen to adopt. As I have shown you, patterns can arise independently of any intentional creator, so this is not evidence for your claim.

Actually, I’m hoping that there might be some convincing reason to once and for all rule out superdeterminism (I don’t like the idea) - but I can’t find any, and a number of quite esteemed physicists seem to pursue this line of research.

A pattern is not evidence of intent. Consider the action of wind blowing across sand (beach, desert etc) - the motion of individual grains of sand is chaotic and cannot be predicted past the characteristic Lyapunov time of that system, yet after a while a well-ordered wave pattern emerges on large scales from the chaotic motion of many billion grains of sand. No intentional creator is required for this. And that’s just one random example.

I’m not trying to convince you, since I myself am not convinced that superdeterminism is a viable way to go. I’m saying only that it shouldn’t be readily dismissed, since it does reproduce the same results as ordinary QM. I’ll have to do more study myself regarding the precise details, though.