Markus Hanke

Sounds like ordinary metric expansion to me, then.

I don’t know what this is supposed to show, to be honest. One wire is infinite, the other one is not (it returns to the same starting point). In both cases they look locally flat. But globally these situations differ. In GR, any small enough patch of spacetime can locally be considered Minkowski; but this doesn’t at all imply that the entire spacetime is globally without curvature, or that curvature is somehow “hidden”, it just means that curvature effects are small enough to become negligible, but they are still there. For example, longitudinal lines on Earth still converge at the poles, even if locally on a small enough scale they appear to be parallel. This is where the connection (and thus the Riemann tensor) come in, because it tells is how these local patches are globally related to one another.

This is not what you said earlier - you explicitly stated that matter shrinks relative to some fixed absolute background. That is a rescaling of atoms.

The question then is why we only detect that one signal from the civilisation in question, and not all the other radio signals that would have emanated (largely unintentionally) from the same location before. If a civilisation possesses radio technology, it’s unlikely they’d only employ it one single time.

Sure it will, because eventually the ant will end up again at the same point where it started, and thus realise that it wasn’t on a flat surface. While its trajectory was everywhere locally flat, the global geodesic it followed was a circle. Sure you can use these to prove the Bianchi identity, but you don’t have to. As KJW has pointed out already, you must remember that the Christoffel symbols aren’t tensors, so you can always make coordinate choices such that they locally vanish. However, Riemann is a tensor, so an a curved surface it can never be made to vanish. It measures the commutativity of the covariant derivative. On curved surfaces, covariant derivatives do not commute, and the extent to which they fail to do so is measured by the Riemann tensor.

No, here’s where the issue is - specifically the strong and weak interactions do operate on an absolute size scale. For example, the upper limit for binding between quarks is at around 0.8 femtometers, and the residual strong force between nucleons operates within ~3 femtometers. These are very much absolute values at ordinary energies. This is why the rescaling of atoms does not work.

Yes, so long as the manifold is flat. No they don’t, unless the manifold is flat. All tensors are purely local objects, and the metric tensor in particular allows you to define the notion of inner product at a point. However, the numerical values of the components of the tensor depend on where they are evaluated (thus the Christoffel symbols don’t automatically vanish), unless the manifold is flat. And then the metric tensor becomes the Minkowski metric, since we are working in GR. The Riemann tensor is defined at a point, like all tensors, but it is made up of derivatives of the metric, which don’t vanish on a curved manifold. All these tensors - the metric, Riemann etc - are in fact tensor fields, ie they are not isolated objects, but defined at all points of the manifold. So there’s no such thing as “hidden curvature” - a manifold is either curved, so that Riemann doesn’t vanish, or it isn’t, in which case Riemann is zero. There are no other options here. Note that all of this is very precisely defined using calculus on manifolds, and well understood.

Ok, so you’re working on a flat manifold. But what do you mean by “loop Lie bracket”? What are you taking a Lie bracket of, exactly? I don’t know what “uniformity of information” means. If the manifold is flat, as you stated above, then the vectors will coincide after parallel transport; but of course that means you don’t have any gravity in this situation.

But this is not what we are doing. And what you’ll find is that they don’t coincide, just as the maths say. There’s no such thing as “hidden curvature”. If the manifold is curved, the vectors can’t coincide - that’s all there is to it. I don’t know where you are getting this from, but if makes no sense. Given the Levi-Civita connection on a manifold with curvature, the Lie bracket of two vector fields measures the extent to which differential operators associated with these fields fail to commute; specifically in this case, it’s related to the commutator of covariant derivatives, which on a Riemann manifold is not zero, unless the manifold is perfectly flat. In other words, parallel transport is path-dependent in a curved spacetime. I’m afraid this doesn’t make any sense.

No it won’t. On a curved manifold the initial and final vectors can’t coincide. This result is easily shown, and entirely independent of any physics models such as GR. You need to remember that differential (Riemann) geometry was already well established long before Einstein - GR simply uses this discipline of mathematics to formulate its framework. There is no notion of “duration” associated with this; you simply compare the effects of the connection on your manifold to the original vector. Another, perhaps simpler, way to state the same thing is that on curved manifolds, covariant derivatives do not commute, which is likewise easily shown. GR is by design a purely classical model, it describes no quantum effects. The symmetry group associated with GR (cosmological constant or not) is non-compact and infinite-dimensional; any spin-2 quantum field theory based on such a symmetry group will be non-renormalisable.

That’s exactly the problem. You’re assuming that these fundamental interactions don’t change, but at the same time you’re saying that atoms “shrink” over time relative to some absolute background. This doesn’t work, since the interactions don’t scale - if you try to shrink atoms, you break the physics in the process.

Yes. Or else you can also calculate it directly from the metric. This is one way to define the Riemann tensor, though it doesn’t have to be a parallelogram…any closed curved will do. Firstly, the space the parallel transport is performed on is already 4-dimensional spacetime, so in general you’re transporting in space and time. Secondly, the crucial point is whether the transported vector, once it returns to its starting point, coincides with its original version before parallel transport, or not. In a curved spacetime, it generally won’t, so the Riemann tensor will allow you to find the difference between the two (ie a new vector that points from the tip of the first vector before parallel transport to the tip of that same vector after parallel transport). All of these vectors are 4-vectors in spacetime. In GR, the connection on the manifold is given, being the Levi-Civita connection. Since this connection is torsion free, the only dynamics can come from curvature, so the Riemann tensor captures the entire geometry on your spacetime. Note though that the Einstein equations in GR do not by themselves completely fix the Riemann tensor (or even the metric). You always need to supply extra boundary conditions to uniquely determine the geometry.

That doesn’t work, since neither the weak nor the strong interaction (ie their respective Lagrangian) are invariant under rescaling, irrespective of how you fudge any constants.

All atomic clocks tick at exactly “one second per second”, and all ideal rulers measure exactly “one meter per meter”, in their own local frames. What changes is only the relationship between local frames across spacetime. This is what time dilation and length contraction are - they are relationships between frames, not something that physically “happens” to the clocks and rulers themselves in their own frames. The consequences of such relationships between frames are just as real and physical, but it’s nonetheless crucially important to understand the difference.

The job of physics is exclusively to develop descriptive models of aspects of how the universe works on a fundamental level - it is simply about knowledge and understanding. What people do with this knowledge is a whole different question, which lies outside the domain of physics itself. For example, quantum physics has given us the MRI machine at your local hospital, but also the nuclear bomb. As for the specific model on this thread, it’s too early to ask about potential implications, because no final fully renormalised version exists yet. Only once the mathematical groundwork has been done, can we judge whether this is worth investigating further, and what the model actually tells us about the world, if anything.

I think it’s an interesting approach, that may very well turn out to be quite viable. The huge advantage here is of course that this model directly integrates into the Standard Model, since it’s build on the same paradigm from the ground up. The basic idea here is that you start with flat Minkowski spacetime, and then define a suitable gauge field on it that has the same degrees of freedom as ordinary GR, so that the observables cleanly map into each other. It turns out that this works if you use a collection of spinors as the fundamental mathematical object. You can then simply apply all the well established techniques of quantisation and renormalisation, since we’re just working with a field on ordinary Minkowski space. The authors have shown that the resulting model is renormalisable to first order, which is a great start. Much of the technical details are kind of over my head too, but I get the main ideas, and I think it’s very promising. I don’t see any obvious reason why the renormalisation shouldn’t work to higher orders too, but we’ll have to see. It’s also interesting to note that this model contains no new free parameters, it works entirely with already known fundamental constants.

So then you need to impose extra boundary conditions to establish a unique relationship between your scalar field and the metric (or the Riemann tensor). Fundamentally the issue is that a rank-0 scalar field does not contain the same amount of physical information as a rank-2 tensor field.

But as @KJW has already pointed out, gravity is not just time dilation. You cannot in general reduce the degrees of freedom of gravity to a single scalar field; you need at least a rank-2 tensor for this.

I’ve just tried it in the sandbox, but it doesn’t seem to work for me, at least not using the “\ [“ syntax. What is the correct way to use it now? \[R_{\mu \nu}-\frac{1}{2} g_{\mu \nu}R=\kappa T_{{}\mu \nu}\] PS. Never mind, it seems to be working now.

\[math]R_{\mu \nu}-\frac{1}{2} g_{\mu \nu}R=\kappa T_{{}\mu \nu}\[math]

If you don’t switch, you can’t compare it to real world observations. The number of degrees of freedom need to be the same, or else any system of equations in the formalism is either overdetermined, or unsolvable. Yes, like I said, you need ordinary GR first in order to actually determine the gravitational environment. It can’t come out of your formalism. Only after you know all corrections can you formulate things. The comment was about the form of the laws in your formalism, not the physics themselves. If you change the meaning of clocks and rulers, the form of all laws changes too. I’m sorry to say I don’t share your enthusiasm. To me this is at most a mildly curious intellectual exercise, but like I said, such a formalism would be a complete nightmare to actually work with for any kind of practical application. I think the best will be for you too keep working on it, and see for yourself what comes out of it. If you can come up with something of value, then great, I’ll be the first to congratulate 👍 What I predict will happen though is that you’ll very quickly find yourself in a world of mathematical pain, while at the same time loosing all physical intuition, since none of the quantities you are working with directly correspond to ordinary measurements in the real world. But as an intellectual exercise at least, this project can be very instructive. (Highlight is mine) Ok, I get you now. That’s an interesting question, I’ll have to think about that for a bit. My immediate guess would be no, not every arbitrary connection necessarily admits a metric tensor field with vanishing derivative - I think if you mix time-like and space-like parts in the definition of the covariant derivative in the right way, the inner product will no longer vanish under parallel transport. This is to say that connections should exist that never preserve any metric. If I have time, I’ll try and construct an explicit example of such a connection.

From the looks of it we have enough equations to determine all unknowns, but only if the connection is of type Levi-Civita. If the connection is not guaranteed to be torsion-free, then we need at least one additional constraint to solve this system of equations. So the answer looks to be yes, so long as we know it’s an LC connection. Whether the metric thus obtained is unique is a different question again - my feeling is that it might only be determined up to an isometry, but I might be wrong; I haven’t actually sat down and attempted a formal solution.

I’m afraid it’s much more subtle than this. You have to remember that on your u-spacetime not only time and space are redefined, but also any derivatives taken with respect to them, and any quantities integrated from them. Thus, while it probably still is possible to define the concept of a “Lagrangian”, this won’t have the same physical meaning any longer. In addition, the Euler-Langrange equation will be of a different form too; in fact, I think its form would be different at every event on your manifold. And it gets worse still. Your u-manifold still formally admits a notion of invariance with respect to translations/rotations in u-space as well as translations in u-time. It should also be possible to formally find an corresponding notion of Noether’s theorem, so you might have some form of conserved Noether currents. In particular, u-time translation invariance will lead to some concept of “u-energy-momentum”. Unfortunately this object bears no relation to the energy-momentum tensor in ordinary physics, so you cannot use it to describe distributions of energy-momentum in the real world. Translating that into your formalism will yield something that has a different form at every point in your spacetime. I’m afraid not, because these “gravity corrections” cannot be calculated from your formalism. Since the form of all physical laws is explicitly dependent on the gravitational background, you cannot describe any real-world physical situation in terms of your formalism, unless you already know the gravitational corrections. Thus it is not possible to write down a gravitational field equation with this formalism, as any such attempt would be self-referential. You would need to start with prior knowledge of the ordinary gravitational metric, and, based on this, you can then figure out “corrections” and use those to write down equations in your formalism. So you’re effectively multiplying your workload - first you have to use GR to find a metric, then you use it to model things in your formalism, and in the end you have to translate the result back again to actually relate it to real world measurements. Note also that finding the correct form of physical laws for a given set of gravitational corrections is not a trivial task (how would you even approach that problem, since it affects all laws of physics?). Also think about what this implies - not only would the form of physical laws be different for different observers in different gravitational environments, but that form might also vary for the same observer over time. While it may be formally possible to do such a thing, actually working with such a formalism in a practical sense would be an absolute nightmare. You asked earlier why no one seems to consider this type of formalism - well, this is part of the answer. We really need one set of physical laws that has the same form for all observers, irrespective of their states of relative motion or their location in space and time; anything else just isn’t very useful when it comes to describing the real world, except perhaps in highly idealised scenarios. That is why general covariance is such a fundamental part of contemporary physics.

Well yes, essentially you are understanding the problem - namely that the outcome of experiments performed in u-time depend on where (and also when) they are performed. So it becomes very difficult to relate predictions from the model to physical outcomes in the real world. In u-time, no two clocks can be dilated wrt one another, so you always have \(g_{\mu 0}=g_{0\nu}=\pm 1\). This creates another issue, consider the following: Suppose you have two u-clocks that start off together at the same place near some very massive, rotating object, like a pulsar. They are initially synchronised and at relative rest. Now these clocks travel along a closed trajectory around the pulsar’s equatorial plane, and come to rest again at the same place afterwards. Both clocks travel along the exact same spatial trajectory with the exact same speed profile, but in opposite directions. Because of the way you defined your u-time, at the end of the experiment both clocks read the exact same amount of total accumulated u-time. Unfortunately, in the real world this isn’t what happens. If you perform this experiment with ordinary t-clocks, their readings will differ when they come together again. So in order to relate predictions calculated from your model to the real world, you need to account not just for location and time, but also for the history of the physical system in question. The spacetime isn’t flat, unless you want to demand that your metrics must be isometric to the Minkowski metric. Is that what your doing? Furthermore I suggest we stick to established classical physics for now, and not introduce unnecessary complications and speculations. Im afraid this makes no sense. Like I have said several times now, it’s not the connection itself that changes, only the connection coefficients.

Yes, this is what I said in my post (I’m quoting my self): But my main point was rather that once a metric is established to be of type Levi-Civita, then all its characteristics are uniquely determined, so you can’t have two “different” connections that are both LC. What changes according to the metric are only the connection coefficients, not the connection itself. That’s an important difference. Yes, I get what you are trying to do. Unfortunately I don’t think you have grasped the concerns I have tried to level at this idea, perhaps because they got buried in technical arguments. So let me try a more practical approach. In the first instance, consider this simple scenario - let’s say you have a box that contains a quantity of muons (a bit contrived, I know, but bear with me). The box is locally in an inertial frame, and otherwise isolated from any external influences. There’s no spatial motion in the frame of the observer, the box just sits there and ages in time. I’d like to use your own earlier example of a metric here, where the 00-component is unity, and the notion of time is your own adapted “new time”, not SI seconds; I will be using the letter u for this, to distinguish it from ordinary time t. In this spacetime, the geometric length of the muons’ world lines between two events A and B then is \[s\prime =\int\limits_{B}^{A} ds=\int_{B}^{A} \sqrt{g_{\mu \nu}dx^{\mu}dx^{\nu}}=\int\limits_{B}^{A} \sqrt{g_{00}} du=\int\limits_{B}^{A} du=\bigtriangleup u+C\] so it is just simply the difference in u’s (we can choose C=0 for simplicity). Let’s say the two events are 1 second u-time (not t-time!) apart, and at u=A the box contains X muons. My question is: how many muons are left at u=B, ie after 1 second u-time? All I’m after is a percentage of the original number of particles X, so nothing to do with any units. I’m interested to see how you go about solving this - which, in ordinary physics, would be an almost trivially simple problem. Like so: \ [ Latex code \ ] just without the space between backslash and angle bracket.