Markus Hanke

Any time And your questions are good and valid ones, it’s just that they’re kind of difficult to address without lots of maths references. The key concept in this is diffeomorphism invariance - that one can describe the same spacetime/geometry in many different coordinates, without affecting any of the physics. Thus, having purely spatial expansion, and a mix of spatial and temporal expansion in the metric, really can be two different descriptions of the exact same physical situation. This is not intuitively obvious, but mathematically rigorous.

I’m afraid I still don’t really know what you mean by this. The FLRW solution explicitly depends on the energy-momentum tensor being non-zero - an empty universe where it vanishes won’t have the same geometry as FLRW. You can’t get the FLRW solution from the vacuum field equations. The cause isn’t the presence of energy-momentum, since there also exist vacuum solutions that metrically expand (eg the Kasner metric); in my earlier comment I meant that the presence of energy-momentum implies that the spacetime cannot be flat. And by this I meant Riemann flat, which perhaps I should have stated explicitly. Dark energy is also not the cause of metric expansion; it merely influences the rate at which the expansion happens. The expansion itself happens whether there is DE or not. But in a spacetime that is globally Minkowski, there is no expansion, and thus no redshift. So I struggle to make sense of your question. By curvature I meant Riemann curvature - sorry, I should have made that explicit.

It’s not flat because when solving the EFE, you start with a universe filled with energy-momentum, which necessarily has a gravitational effect. The scale factor arises as part of the solution; it’s not a source term that appears in the initial setup. If you remove all gravitational sources, you no longer get an FLRW solution - you’d be in a different spacetime.

Such a coordinate transformation wouldn’t be a diffeomorphism, meaning you are dealing with two different spacetime geometries, and thus two completely different solutions to the EFE. IOW, an expanding-time-only metric would not just be an FLRW spacetime written in different coordinates. A metric where only the time part is “expanding” describes a completely flat spacetime, only from the perspective of an observer that undergoes some form of accelerated motion. This is not a good description of our universe, since that contains matter and radiation, and thus can’t be a flat spacetime. So far as I can tell (someone correct me if I’m wrong), a metric where both the time and space parts are expanding could be the natural description of an observer who undergoes non-uniformly accelerated motion with respect to the cosmological medium, provided the time part has a suitable mathematical form (if not, it won’t correspond to any physical observer). For such an observer, measurements of both spatial distances and time durations would explicitly depend on when they are performed. He’d see an expanding universe, but also detect proper acceleration, which would play the role of the distinct time component. The FLRW spacetime is not Riemann-flat, so I’m not sure how to answer this. Given the time component has the right form, the distinction would be in the presence of proper acceleration in the motion of the observer. The details of the spacetime geometry - curvature, geodesic structure etc - are independent of coordinate choices, so I don’t think we’re missing anything. It’s like having an electric charge - in the rest frame of that charge, you detect only an E-field. An observer who moves past that same charge sees both an E-field and a B-field. In both cases, you have the same electromagnetic field. It’s the same physical situation, seen from different vantage points. But no observer under these circumstances will ever see just a B-field alone, since that would be a physically different situation. The expanding space situation is similar - you can re-distribute the expansion among the components of the metric by a suitable coordinate transformation (same spacetime, different vantage point), but no observer will ever see a time-only expansion, since that would imply being in a flat spacetime. FLRW spacetime is not flat, so I don’t quite understand what you’re trying to ask…?

Yes, the Lentz warp drive is potentially interesting. However, as the author of this review paper mentions, unless someone can demonstrate the feasibility of each stage in the entire life cycle - and I would add steerability to this list -, we still don’t know whether this is actually physically realisable or not.

I don’t think it was. Life back in hunter-gatherer days would have been much simpler, but also brutal, painful, and generally short. There’s nothing wrong per se with technology and civilisation, it’s alleviated a lot of unnecessary suffering, and - broadly speaking - freed up resources that enabled us as a species to pursue things other than immediate survival and procreation, at least potentially. I for one wouldn’t want to go back to the dark times. The problem is only that one has to have a certain degree of wisdom with it, and that’s where we seem to be lacking. We’ve become completely dependent on our own creations, and in some sense relinquished our freedom to them. That’s problematic, but also inevitable I suppose, since our basic instinctual-psychological patterns have remained the same.

What you’re referring to is the instant that the camera crosses the event horizon. The reading on the bottom clock (“External time”) refers to what an external observer who is at rest far away from the BH would see. But the problem is that from such an observer’s point of view, the falling camera never reaches the horizon at all; just before it gets there, it will appear to fall slower and slower, and will visually appear redder and dimmer, until it fades into invisibility. There is no instant on the distant observer’s clock at which the camera is reckoned to have crossed the horizon - the entire region of the horizon and below cannot be mapped into the external observer’s coordinate system at all, because from his point of view such a region cannot be accessed (and vice versa, the falling camera cannot get back out either). Thus, once the horizon is reached, the times on the falling camera’s clock no longer correspond to any times on the external clock, even though the camera continues to fall and continues to accumulate time normally on its own clock. There simply is no longer any meaningful notion of simultaneity at all, whether relativistic or not. So you see, time on its own in GR is a purely local concept, specific to a specific observer. It may not be shared by others. In order to ensure agreement between observers, you must now use covariant quantities instead.

I came across another good video, concerning in-fall into a Kerr BH (mass, angular momentum). This one contains explanations, and also readings on two reference clocks; I thought some readers here might like this:

Let’s further assume that clock (1) is far from the BH, and orbits slowly. This clock will see (2) to be still ticking, but at a much slower rate (compared to itself). It will see (3) to initially fall at increasing speed, while its tick rate gradually slows; as it approaches the horizon its descent will appear to slow more and more, and its tick rate appears slower and slower. Its visual appearance becomes dimmer and redder, and it eventually just fades into invisibility. But it will never be reckoned to have reached the horizon. (2) sees (1) to be ticking at a much faster rate, and it will visually appear blue-shifted. (3) will appear similar as described above, just at a different rate; it will also never be reckoned to have reached the horizon. For (3) itself, the time it takes to reach the horizon is finite; the fall time from horizon to singularity is also finite (for a 10 solar mass BH this will be on the order of ~150 microseconds). What tick rates it will see on clocks (1) and (2) will depend on where along its free fall trajectory the clock is - this is a bit of a balancing act between its own position in the gravitational well, and the degree of relative motion between the clocks. However, once it has fallen below the horizon, it should see both (1) and (2) to be ticking faster wrt itself. After the initial free fall period, they would all be seen to be asymptotically slowing towards the same region of space, while gradually fading from view. But they would never quite catch up. This is not simple, since it’s a mathematical model in four dimensions. Spacetime is not a substance, so resist the temptation to think about it that way. The best I can offer is to think about it as a network of relationships - it tells you how clocks and rulers at different places and times are related to one another. Each nexus within the network corresponds to a physical event (meaning: a specific point in space at a specific instant in time), and has attached to it an object (called the metric tensor) which, if you tell it a direction, will give you the spacetime distance to its closest neighbouring nexus in that direction. So it’s a network of separations between events. In special relativity that separation is the same wherever and whenever you are (the metric tensor is just a matrix of constants), but in general relativity it may explicitly depend on where and when you are. For events that are more widely separated (not neighbouring), you add up all the individual separations between the nexus that are in between, so you perform an integration. The global properties of that network as a whole influence local relationships, and the form of local relationships puts constraints on how the global network can look like. Also, if something changes locally, the effects of that will propagate outwards and “ripple along” the network. It’s a bit like a spiderweb. Be careful about the trampoline analogy - it is just a visual plot that tells you how certain length measurements (rulers) are related along a specific coordinate in a specific geometry. It’s not a complete picture of what this spacetime would look like. Technically speaking, it’s the limit of scalar curvature that diverges - the point r=0 isn’t part of the manifold, so no curvature tensors are definable there. This is why the technical definition uses geodesic incompleteness, and not curvature. That just as an aside

The problem with this is that - at least in ordinary GR - once something is below the horizon, a complete collapse is inevitable, irrespective of what that “something” is made of. This is due to the geometry of spacetime itself - below the horizon, the singularity is in the future of every particle that finds itself there. It is thus not possible to have stable, stationary objects there. In the classical realm, it is possible to consider alternative theories of gravity that avoid singularities by postulating different types of geometries, such as for example Einstein-Cartan gravity. In the quantum realm all bets are off right now, since we don’t have a working model of quantum gravity. Time dilation is a relationship between at least two clocks, so you have to specify which clocks you wish to compare. Generally speaking, the answer depends on what type of black hole you consider (ie the geometry of spacetime), where the clocks are, and how they move. Here’s what you would see if you fell into a Schwarzschild BH (only mass): And here the same thing for a Kerr-Newman BH (massive, rotating, and electrically charged): As you can see, what you’d observe depends on the specifics of the black hole. For a Schwarzschild BH, an external stationary observer would not even see the test particle reach the horizon in finite time (on his own clock).

Because the author of that thread explicitly asked about frequency shift, so I offered one particular way to look at that. Personally though I would altogether avoid any analysis of what happens at every instant in different frames, and simply compare the lengths of the two world lines. That works irrespective of the details of how the twins move, and irrespective of what spacetime they find themselves in.

Yes. FLRW spacetime is a “dust solution” - a universe homogeneously and isotropically filled with energy-momentum that interacts only gravitationally. The choice of coordinate system is arbitrary, it represents no physical assumption. You are basically just picking an observer on whose point of view you base your labelling. You can take the ordinary FLRW metric (usually written in what is called Gauss coordinates) and just perform a valid coordinate transformation to arrive at a different point of view; this can be done directly, and has nothing to do with the field equations or the physics. For example, you could choose an observer that is accelerated at all times - you would get a metric that at first glance looks very different, but still describes the same spacetime. As I said, the choice of coordinates is arbitrary. For example, if you were to base your coordinate system on a clock that is not comoving with the cosmological medium (eg one that is accelerated at all times, possibly non-uniformly), you would get a metric where both the time and space parts explicitly depend on the t-coordinate. So long as the coordinate transformation is a valid diffeomorphism, this is perfectly allowed, though probably an algebraic nightmare to actually work with. It’s important again to realise that this describes the same spacetime, just in terms of different coordinates. What is not possible though is to try and have only the time part expand - there’s no valid transformation that yields this. See above - you could “distribute” the expansion across both time and space parts of the metric by a suitable coordinate transformation, which has no physical consequences. It’s the same spacetime, you’d just label events in it differently. The question is why you would want to do this - it would greatly complicate most calculations relevant to us, since such coordinates wouldn’t straightforwardly correspond to our own clocks and rulers. But of course you can do this, if you really wanted to.

Indeed. The same is true also with power - all the major decisions which affect humanity globally (eg starting wars etc) are made by an extremely small group of individuals. Who has decided that this would be a great system?

Well, the fundamental assumptions underlying this solution are homogeneity and isotropy - if you feed this kind of energy-momentum distribution into the field equations, you get as solution a spacetime that expands. You are free to choose yourself what kind of coordinate system you wish to use to describe this, but obviously it is smart to use a system where your intended calculations are easy. I understand what you are trying to say. The FLRW metric does rely on the cosmological principle, that’s an assumption we make - that on large scales the universe is homogenous and the same in all directions. Since there’s an observational horizon past which we can’t see, it’s possible at least in principle that perhaps one of these doesn’t actually hold. The underlying premise is really the laws of gravity, meaning Einstein’s equations. If you start off with a distribution of energy-momentum that interacts (approx) only gravitationally, then it’s actually difficult to avoid solutions that metrically expand in some way. FLRW is by no means a unique thing, it’s just a particular example of a large number of such solutions. This is not just due to coordinate choices. Indeed. And you are correct - you need to pick some boundary conditions to solve the EFE, which in this case is the cosmological principle. But in GR, the choice of coordinate system has no physical consequences, so it’s not due to that. The coordinates are chosen such that they correspond to an observer co-moving with the medium; this seems to apply to Earth too, since we don’t observe anything different. We remain in our galaxy, which is part of a local cluster, which co-moves along with everything else. In physics you are always restricted by the set of available data - our task is to find a model that best fits this currently available data. If the data set changes, then sometimes the model needs to change too. There’s many possible objections to the Lambda-CDM model, but honestly, right now there’s nothing else that fits all available data better. Let’s just consider this a work in progress. Physics would be boring if all the last words had been spoken already.

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.