Markus Hanke

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.

But you were explicitly saying that, I quote, we should “forget the old world entirely”, with old world referring to real-world measurements with clocks and rulers. The choice of units has no impact at all on the physical outcome of experiments, or on the form of physical laws that are written in covariant notation. You can measure lengths and angles in meters and radians, or you can use fingers and degrees, but the apple will always fall when it is ripe, and it will do so radially downwards. No redefinition you do changes this physical process. So you might as well work with the simplest mathematical description of it, and safe yourself unnecessary complications. Of course you can - like a grandfather clock for example, which is influenced by local gravity conditions. Nothing wrong with that, but the question is how useful that is. No it isn’t. It’s a reflection of the fact that relative uniform motion has no influence on how electromagnetism works. Your laptop works in your living room just the same as it would in a rocket at 99% light speed wrt Earth. Of course you can always write down a model where this is not so, I just fail to see the point why you would do that, as you then aren’t describing what actually happens in the real world. Are you talking about free fall here? There is no proper acceleration in the rest frame of a free falling particle. If you change the standard definition of time in a manner that makes it explicitly dependent on location, then \(a\left( t \right) \neq \ddot{s} \left( t \right)\), and thus Newton’s laws are no longer valid in their usual form F=ma. All laws of physics will take on a different form in this case. As stated previously, you can of course do this if you really want, I’m questioning only the point and usefulness in that. For the connection to be of type Levi-Civita, it needs to preserve the metric and be torsion free, otherwise it can’t be said to be Levi-Civita (there are infinitely many possible types). But take careful note that you can have a connection (of any type) on a manifold without there being any metric, it’s just that you then can’t formally say that that connection is of type Levi-Civita. So it doesn’t make much sense to say the connection is explicitly defined via the metric, only that the connection is of that specific type in the presence of a metric. If there’s only one smooth manifold, then there’s only one LC connection, though you can have as many metrics as you want. But you can’t have two “different” connections that are both Levi-Civita, that makes no sense. If your formal manifolds are both Riemann, an LC connection must preserve any metric on either one of these manifolds. If it doesn’t do that, then it’s either not an LC connection, or one of the manifolds is not Riemann, or it’s not a valid metric in the first place (not every notion of inner product is automatically a valid metric, there are conditions here too).

Not it isn’t the same set of points, because in order to flatten it, you had to cut a piece out of the surface (as you correctly stated yourself), so you have lost information in the process. This depends on what exactly you mean by “geometry”. In the context of GR, this means the various tensor fields that describe the distribution of energy-momentum and the associated effects this has on the world lines of test particles. And these very much are invariant under diffeomorphisms, which is to say you can label the same physical events in different ways. If you change neither the physical meaning of coordinates nor the form of equations expressed with them, then you can’t have a different geometry, unless you describe a different physical situation. You can’t have it both ways. If that is what you want to do, you should take a look a teleparallel gravity. This model has no curvature (ie all geodesics remain parallel even globally), and all the information about gravity is contained in the form of torsion along world lines. Einstein himself investigated this in some detail. But then this model is entirely useless, because you cannot compare anything calculated from it against quantities physically measured with real-world instruments. You might as well be talking about invisible pink unicorns, for all the use it has in modelling the real world. Physics makes models that describe aspects of the physical world around us - as such we must be able to extract predictions from those models and compare them to real-world measurements. If you ask us to just forget about the real world, then I’m sorry to say we’re not interested, because it’s of no use to us when solving practical issues like eg calculating the orbit of a satellite around a gravitating body, and knowing how a clock on that satellite relates to a clock on the surface. In standard GR, this comes right out of the model, because what clocks physically read is always identical to the geometric (mathematical) length of the world line it traces out in spacetime, so the problem is rather straightforward, if not always simple. In what you propose that is not so, and you’re asking us to just ignore this…? Im sorry, but I’m still completely failing to see the actual point in all of this.

No they won’t. Same manifold, same connection, same metric even - just expressed in a different coordinate system. No, I’m not talking about connections. I’m talking about a situation where you express the same physical situation (ie spacetime) in a different coordinate system, so that all laws of physics and all tensorial quantities remain the same - including the metric tensor. This is just the usual diffeomorphism invariance of GR. Yes, it’s enough. You end up with a set of parameters that don’t correspond to what clocks and rulers actually read, so for every quantity you calculate from such a model you need to apply a mapping that takes it back to real-world measurements - and that map is just precisely the inverse of the “correction” you applied in the first place. Like I said, lots of extra work for no discernible benefit. Gravity isn’t a force, and can’t be modelled as one - this is precisely the difference between Newtonian gravity and GR. A rank-1 theory such as a vector field model cannot capture all relevant degrees of freedom of gravity. For example, the polarisation modes of gravitational radiation in any force-based model will be inclined by 90°, whereas in reality these modes are at 45°. You really do not at least a rank-2 tensor model, such as GR. No, it’s a lot more than that. As other posters here have correctly pointed out, for there to be radiation at all, the second derivatives wrt time and space of your “waving quantity” need to be related via a very specific form: \[\frac{\partial^{2}}{\partial t^{2}} =c^{2}\frac{\partial^{2}}{\partial x^{2}}\] In the case of light, the relevant quantity is the electromagnetic 4-potential, and the equation thus becomes (in Lorentz gauge) \[\square A^{\mu}=0\] If c isn’t a constant, this relationship is violated - there is no electromagnetic radiation in such a universe, at least not of the form we see in the real world. This has nothing to do with measurements or conventions.

I use it in the formal sense as defined in differential geometry, ie as a structure that allows you to meaningfully define the inner product of tangent vectors at points on the manifold, which in turn gives a meaningful notion of lengths, angles, areas and volumes. Yes. You need to be careful here - the Christoffel symbols and the connection are not the same thing. A connection allows you to relate tangent spaces at different points on the manifold to one another, ie it provides a notion of parallel transport. This is quite independent of any metric, which is to say you can meaningfully have a manifold that is endowed with a connection, but not a metric. The Christoffel symbols then give you the connection coefficients, ie they tell you what effects your connection has in a particular coordinate basis. They do this by describing what happens to basis vectors as you transport them between neighbouring points, which is something you can calculate from the metric and its derivatives. Without a metric you can still do parallel transport, but you can’t tell what happens to lengths and angles when you do it. Long story short - you can have a connection without a metric. See above. Having a different metric changes the Christoffel symbols (they are not tensors!), but not the connection. Ok, but in the context of physics (SR/GR) the term “metric” is most often used in the differential geometry sense. Physically speaking, equivalence then means a diffeomorphism, so that both metrics describe the same spacetime and thus physical situation. But here’s the thing - as explained above, you’re still on the same manifold endowed with the Levi-Civita connection. By changing the metric like this, you’re doing one of two things: 1. You’re describing a different spacetime, ie a different physical situation, since the two metrics aren’t related by any valid diffeomorphism; or 2. You’re describing the same physical situation, but the coordinates you are using no longer have the same physical meaning. I think what you are trying to do is (2). But the thing is that now measurements on your mathematical manifold (ie in the model) no longer correspond to measurements in the real world, so anything you calculate from this - eg the length of a world line - must first be mapped back into suitable physical coordinates to compare them to real-world measurements. Such a mathematical map may or may not exist, depending on the specifics of the setup. This will also change the form of physical laws, so all the various equations etc will be different for each choice of transformation you make. In either case, this creates a lot of additional work and confusion, for no discernible benefit. It would look for differences in the outcomes of experiments if you vary direction of relative motion, as mentioned previously. For example, if a uranium atom decays if you move it in one direction, but doesn’t decay if you move it at a 90° angle to that direction (everything else remains the same), then you have anisotropic space. This has nothing to do with conventions.

Just a few corrections here. The basic object of this framework is a differential manifold. This can initially be “bare”, ie without additional structure, but, as you say, there’s not a whole lot one can do with that. So we can endow the manifold with additional structures - firstly, we can endow it with a connection, which allows us to relate tangent spaces at different points. This is thus equivalent to having a notion of covariant derivative. Given a connection (but no metric yet), you can define things like curvature and torsion (these can be defined purely in terms of the connection), parallel transport, and tensor fields - IOW, you can do differential topology. But what you don’t have yet is a notion of lengths and angles, and you also don’t have a relation between tangent and dual spaces, so you can’t raise or lower indices on tensors. For these things you need to endow the manifold with a metric, in addition to a connection. Now you can use the full machinery of differential geometry. If your metric is positive-definite, it’s called a Riemann metric; if the metric tensor is everywhere non-degenerate and symmetric, it’s a semi-Riemannian metric, which is what is used to model spacetime. Smooth manifolds, connections and metrics are their own independent concepts, they are not defined in terms of each other. I have difficulty making sense of this - see also what I wrote above. I think what you mean is that you have one differentiable manifold endowed with the Levi-Civita metric, as well as to different metrics on that manifold, each of which uses its own notion of time, but both describe the same physical situation? What do you mean by “equivalent” in this context, exactly? Usually, metrics that are equivalent are those related via a diffeomorphism. My understanding so far is that we have only one manifold, which is endowed with the Levi-Civita connection plus two metrics, so the above makes no sense to me. Whether the metrics are equivalent or not, they are always preserved under the Levi-Civita connection; this is one of the defining characteristics of this connection. But you have so far explicitly stated that what we are using is the Levi-Civita connection…? But then you’re directly contradicting experiment, which clearly shows that space is isotopic, at least within the domain we can experimentally probe. So what is the point in all this?

(Bold/italic are mine) You are really contradicting yourself here - so are we working on one and the same manifold, or not? This makes no sense at all - if the connection is Levi-Civita, it always is torsion-free by definition, and it always preserves the metric; those are not observer-dependent. If it doesn’t do those things, it’s not a Levi-Civita connection…but then you explicitly state that it is, so I don’t know what you’re actually trying to say here. No, because c is the conversion factor between time-like and space-like parts of the line element, it remains locally constant irrespective of connection or metric or observer. I don’t even know what you’d have to do to make it appear non-constant…you’d maybe have to parametrise world lines not by proper time, but by some other non-trivial affine parameter that somehow varies in some sense along the curve. I’ve never seen that done, so not sure if that is even mathematically meaningful ( @studiot?). Across an extended region you can then maybe get a “speed” that varies without acceleration. I’m beginning to suspect that what are you referring to is in fact the scheme by which we parametrise world lines. Ordinarily this is done by using proper time, since that way the geometric length of world lines in the mathematical model directly corresponds to accumulated times on a physical clock. But of course you can use other parametrisations too, such as is done for example with null geodesics (where you otherwise would have ds=0). This in effect introduces a new concept of “time” that is not based on what physical clocks actually read. The trouble is that what you have verbally posted is contradictory and ambiguous. It would be much better if you could present your thoughts in mathematical form, so we all understand what it actually is you are talking about. The reality is also that in all experiments we have ever conducted, the laws of physics have never been seen to vary between inertial frames, which implies that c must be invariant at least within that experimental domain, irrespective of its precise numerical value. I therefore don’t understand why you would try to construct a model where this is not the case - at best it creates additional computational work, at worst it will be just plain wrong.

Just to elaborate a bit more. When we speak of the invariance (not constancy!) of the speed of light, what this physically means is that the outcome of experiments is always the same in all inertial frames, ie uniform relative motion has no bearing on the outcome of experiments. This has nothing much to do with units or numerical values. Yes, it is always possible to describe the same physical situation in terms of different “geometries”, if you so will. You can eg forego any reference to curvature completely by choosing a different connection on your spacetime - the geometry is now curvature-flat, and instead contains all information about gravity in the form of torsion. But all this is saying is that one can draw different types of maps over the same territory, like having a topographical map vs a road map over the same region. That way you emphasise different information, but the actual experience of physically crossing that terrain is always the same, irrespective of what map you use to navigate. This is not revolutionary or mysterious, and reveals nothing new about the world. It’s “kind of trivial” as the poster in your screenshot correctly said. So I think if you put enough thought into it, it may perhaps be possible to come up with a mathematical description of spacetime in which c is explicitly a function of something. The reason why no one uses such a description is that any measurements of space and time obtained from this description won’t directly correspond to what clocks and rulers physically measure in the real world - you’d have to first map them into real-world measurements, which means additional work and complications without any discernible benefit. Irrespective of what description you use, the outcome of experiments will still be the same in all inertial frames, and this is what we actually observe in the real world.