Markus Hanke

To give the twin scenario more “twist”, you can allow spacetime to not be flat (GR twin scenario). For even more twist, allow spacetimes that are topologically non-trivial

As KJW has pointed out, the equivalence principle is a purely local statement, meaning it applies only to small spacetime regions - meaning small volumes over short periods of time. Within such regions, there’s no purely local experiment you can perform to distinguish between the two. For the twin scenario, the twin can at every individual instant (or short enough time period) be considered to be subject to a uniform gravitational field, where the gravitational potential may vary from instant to instant. This kind of foliation procedure produces the correct results - though I still don’t think it’s necessarily the best way to analyse the twin scenario.

You are of course always free to pick different coordinates to describe the same spacetime - which is one of the central insights in GR. However, when you do this you also change the physical meaning of those coordinates. In the standard FLRW metric, the time coordinate is chosen such that it corresponds to a clock that is co-moving with the cosmological medium, meaning it fits well with our own physical clocks here on Earth, and thus the “phenomenology” of the metric corresponds to what we actually observe, without any need for complicated transformations. You are free to choose a coordinate system where eg tick rates aren’t constant, but then you need to be very careful how you relate the metric to real-world observations, since the t-coordinate no longer corresponds to Earth-bound clocks. Ultimately it is best to describe the spacetime in terms of geometric properties that are independent of coordinate choice; in the case of FLRW for example, we can say the spacetime is conformally flat, meaning during free fall angles are preserved, but not volumes. These aren’t different “theories”, but simply coordinate choices. You’re describing the same spacetime in different coordinates. KJW has given an example how a “time-only” expansion metric could look like. Ultimately you want to choose coordinates that make your calculations as simple as possible, and that’s often ones based on the cosmological medium. But in principle, the choice is yours, so long as they’re related by valid transformations.

You might want to check the Dictionary of Obscure Sorrows

Technically yes, there will be some amount of frequency shift, though in practice the effect is quite small for a weak field such as the Earth’s. That’s hard to answer, since whether something is considered intuitive or not depends on the person. I kind of like the paths lengths way of looking at it, since most people can relate to it. Total accumulated proper time equals the geometric length of the path through spacetime, as I’ve mentioned already. The crucial point is that the two paths are not of equal lengths. I think you should read the article more carefully. I said there’s no rest frame to light, so it makes no sense to speak of “speed relative to waves”. There’s no tidal gravity in an accelerated frame, meaning that \[R_{\mu\nu}=0; W{^{\mu}}{_{\nu\alpha\beta}}=0\] and therefore \[R{^{\mu}}{_{\nu\alpha\beta}}=0\] However, there is a homogenous gravitational field due to proper acceleration locally, according to the equivalence principle. The Riemann tensor vanishes for such homogenous fields, so spacetime remains of course flat as expected. The metric in the accelerating frame, which now contains a term which is equivalent to a gravitational potential, is isomorphic to the Minkowski metric, also as expected.

The explanation is exactly what the maths says - pick a different path, and you’ll walk a different distance. There’s nothing else to it. No, I showed you that there’s no paradox that needs resolving. It seems to me that you’re wilfully refusing to “get” this. There’s no such thing as “relative to waves”, because light has no rest frame. The speed is always between emitter and receiver. Because it’s he who experiences acceleration locally in his frame. The equivalence principle tells us that uniform acceleration is locally equivalent to a uniform gravitational field; differently put, the accelerated twin sits at a different gravitational potential, which implies frequency shift.

I never mentioned any U-turn - I made it explicitly clear that I made no assumptions about what the path actually looks like, other than it being light-like (thus differentiable everywhere). Why? Because that’s irrelevant, since the difference in clock readings only depends on the total lengths of the two paths. It’s a global measure along the entire journey. Others here have repeatedly pointed this out too. And since both path length and proper acceleration are invariant measures, both twins agree on the outcome. That’s a meaningless statement. I used Einsteinian SR to show how the twin scenario requires no “resolution”. Really? How do you physically realise an instantaneous U-turn with infinite acceleration? Once again, the clock times are integral measures along the entire journey. In relation to the emitter, not the signal. There’s no rest frame for light.

You didn’t answer my question, you’re just making another unsubstantiated claim. It’s invariant, not constant. There’s a difference here. But yes, since the speed of light is finite and the frames are separated, there’s of course relativity of simultaneity; and since acceleration is a change of velocity, the relationship between the frames is time-dependent. I’ve already pointed out that the metric of a relativistically rotating disk is not the Minkowski metric. SR is of course “just” a mathematical model, same as any other model in physics. However, the lengths of paths through spacetime correspond to what clocks physically display, so that’s pretty “real” to me. Like I said, if you connect the same two points in any territory along different paths, it’s hardly a surprise that, in general, these come out at different lengths. What deeper mechanism do you need for that? That’s no one’s ‘interpretation’, it’s what the standard SR maths say, as I have shown you earlier. There are no interpretations involved in this. Only one of the frames experiences proper acceleration, which is not a relative measure - both frames agree on who’s accelerated and who isn’t. So there’s not any symmetry in this situation, except during those times when both frames are inertial. But that symmetry concerns the instantaneous rate at which the clocks tick, not their readings - if one of the clocks has first undergone non-inertial motion, and then becomes inertial, their tick rates are symmetrical, but nonetheless one clock displays a different total proper time. The effect of non-inertial motion is accumulative, which is what I pointed out above with line integrals - you have to account for the entire journey. I don’t know what you mean by this. No it isn’t - frequency shift is due to the fact that energy is a frame-dependent quantity, it has nothing to do with changes in c. Light propagates at the same speed in both frames, but they don’t agree on its energy.

Really? Please provide references to peer-reviewed experiments that unambiguously (ie not just in your “interpretation”) detect the ether. What is it made of? What are its equations of motion? You really need to stop repeating things that have already been shown to be wrong. You’re not doing yourself any favours. What exactly do you want explained? One of the frames measures acceleration (using a local accelerometer), and the frames are related by the transformations given in my link, instead of Lorentz transformations. This concerns the rotation of rigid objects at relativistic speeds - it’s been known for a long time that this involves a metric that isn’t Minkowski, so that’s hardly a “problem with SR”, but falls outside its scope. No. These integrals concern total accumulated proper time; this is an invariant quantity that’s not relative to anything. I deliberately did not use relative quantities, but invariant line integrals. What the equations say is that (in this sign convention) it is always the inertial clock that accumulates the most proper time between a given pair of events in spacetime. IOW, any clock that doesn’t trace out a geodesic between these events will record less proper time in comparison - and we know of course that there’s only one such geodesic for any given pair of events in Minkowski spacetime. Therefore, there’s no paradox, and nothing needs resolving. It’s simply that, if you choose two different paths, you can’t in general expect them to be of equal lengths. The dilation between clocks is an integral measure - it concerns a comparison between total geometric lengths of world lines, so one must take into account the entire journey. Thus in general you can’t reduce this to a single instant. The most we can say is that the accumulated times begin to diverge the instant the travelling clock ceases to be at rest relative to the Earth-bound clock. Also, he never gets “younger” - he just ages less. So again - SR very much does resolve this, contrary to your claim. PS. I remind you again to bear in mind what the twin scenario is fundamentally about - it’s a comparison between total accumulated proper times on two clocks that connect the same two events along different paths. And this is precisely what I mathematically described, not more and not less.

The discussion seems to be going off on a lot of tangents now, so I propose we bring it back to the above original claim. Here’s how I would approach this personally: The twin scenario concerns two clocks (E-Earth, T-Travelling) which start together at rest at some event A; twin T then separates for a period and eventually they reunite again at rest at another event B. In SR, the total accumulated time on a clock travelling between a pair of events is defined to equal the geometric length of the worldline traced out by that clock through spacetime: \[\tau=\int_{C}ds\] where C(t,x) is the path taken. Since in SR we have \[ds=\sqrt{\eta_{\mu \nu } dx^{\mu}dx^{\nu}}\] we get, for the Earth-bound twin (no change in spatial location!): \[\tau_{E}=\int_{A}^{B}\sqrt{\eta_{\mu \nu } dx^{\mu}dx^{\nu}}=\int_{A}^{B}\sqrt{\eta_{tt}}=c\int_{A}^{B}dt\] For the travelling twin, on the other hand, we get: \[\tau_{T}=\int_{C}\sqrt{\eta_{\mu \nu } dx^{\mu}dx^{\nu}}=\int_{C}\sqrt{c^2dt^2-dx^2-dy^2-dz^2}\] wherein we don’t assume any specifics about the path C, other than that it connects A and B, and remains light-like everywhere. I invite you to verify yourself that \[\tau_{T} < \tau_{E}\] by choosing any light-like path C and running the numbers in these line integrals. What this means is that, in SR, an inertial clock will always trace out the longest path through spacetime between events, ie it will accumulate the most time. Any path deviating from being a geodesic will necessarily be shorter (or vice versa, depending on your sign convention). Since the travelling twin starts out at rest wrt to its partner, but then travels away, it cannot remain on a geodesic - thus SR guarantees that its clock accumulates less time than the Earth-bound twin. Far from being a paradox, it is a natural consequence of the geometry of spacetime. Note that I made no assumptions here about the specific form of the path C, only that it is light-like and runs through A and B; thus it is realisable by any test particle with mass. So yes, SR very much can resolve this, contrary to your claim. It should also be explicitly noted that the total time dilation does not depend on the magnitude of acceleration at any one instant, but only on the total path length of the world line. Lastly, an experimental test of these predictions can be made by comparing the decay rates of unstable particles in accelerators against stationary reference samples. See eg Bailey et al (1977) with regards to CERNs muon storage ring.

I was talking about something you claimed - that SR cannot handle acceleration. This is manifestly wrong, because it very much can. What interpretations? The ether in LET is by design undetectable, and its presence has no physical consequences, thus both SR and LET make identical predictions. This isn’t a matter of interpretation, but belief - do you choose to believe and assume the presence of an entity that cannot be detected, has no physical consequences, and is not actually needed to obtain the correct dynamics? Not only has SR been exhaustively tested over the past ~120 years and thus shown to be consistent with the physical world, but we also use it directly in many aspects of modern technology. A lot of parts of the very machine you are using to make your posts here have been designed based directly or indirectly on SR, and it evidently works well. Also, SR kinematics are classical, just like Newtonian kinematics are. But APS is just based on a subgroup of the full Cl(1,3) Clifford spacetime algebra…? It is a convenient and very useful formalism, but hardly new physics. RG is a social network, not a peer review journal. Some stuff there is useful, but one has to use caution. Personally, I stopped reading when the author first talked about twins in Minkowski spacetime, and then gave a scenario as example where one twin is in orbit around a star. Not an especially convincing paper, and the author clearly has an agenda.

This is a misconception which is as common as it false. SR is a model of Minkowski spacetime - it describes the relationship between any set of frames within this paradigm, irrespective of what their states of relative motion and acceleration are. In the special case of inertial motion, this relationship is simply a hyperbolic rotation in spacetime (=Lorentz transformation); if acceleration is involved, the relationship is a little more complicated, but nonetheless well defined: https://en.m.wikipedia.org/wiki/Acceleration_(special_relativity) There’s no “paradox” in the twins scenario that somehow needs resolution, it’s simply a straightforward consequence of the geometry of Minkowski spacetime, which has to do with the lengths of world-lines.

Given that you assume the theory of relativity to be valid, this will imply that an absence of time would also mean an absence of gravity between bodies. Clearly, this is not what we observe.

I understand Just as a side remark though, it is interesting to note that the EFE also admits cosmological vacuum solutions. An example is the Kasner metric; this kind of universe is completely empty (no energy-momentum and no cosmological constant), yet still not Riemann-flat in general, and metrically expands in an anisotropic manner. One can contrast this against the case of FLRW, and finds that it is precisely the presence of matter/radiation that enables an isotropic expansion. Obviously this is purely academic, but nonetheless instructive.

Of course I was very fortunate that I grew up in a part of the world were an excellent basic education is freely available to everyone. I consider myself lucky.