Markus Hanke

Yes, it means that all inertial observers experience the same laws of physics. Is it? I see it simply as an empirical observation, which was part of what motivated the development of SR in the first place. We have never once seen any evidence whatsoever (within the domain we can experimentally probe) that would contradict this. The known constraints on violations of Lorentz invariance are indeed extraordinarily stringent. Clearly my intuition is very different from yours, since to me it is intuitively obvious that a Euclidean map with only spatial dimensions in it cannot and does not accord with what we actually observe in the world around us, except perhaps as an approximation in the classical non-relativistic regime (i.e. “high school physics”). Again, historically this was one of the motivating forces behind the development of SR in the first place. So “intuition” is a rather poor guide, since it is highly subjective. Maybe because “belief” is just as poor a guide as “intuition” is.

How about an interior metric, such as FLRW - under the right circumstances, the distance between points in such a spacetime will increase over time (metric expansion), whereas under Newton the interior of energy-momentum distributions will always contract, but never expand (assuming there’s no effects other than gravity of course). Or how about something like a gravitational geon - a topological construct that is held together purely by gravitational self-energy, without the presence of any other energy-momentum sources at all? This particular solution relies entirely on GR self-interaction effects - under Newton, a completely empty spacetime without any gravitational sources cannot contain (or maintain) gravity. Any type of radiative spacetime should qualify too, since Newtonian gravity has no radiative degrees of freedom. At most you can have varying gravitational forces, but the oscillations of the force vector would be “longitudinal” (ie in the radial direction) and of a dipole nature, whereas in GR the effects are transverse and quadrupole. Or anything with angular momentum, since Newton can’t model frame dragging effects, and thus will give wrong trajectories for free-falling bodies around rotating objects. You can also give angular momentum to an interior spacetime, and get something like the Gödel metric - it contains a number of peculiar effects that I don’t think Newton would be able to replicate, and certainly not based on just invariant mass. I’m not so sure about this - the energies and momenta are certainly frame-dependent, but their sums should never cancel. Since E=hf, there is no physically realisable frame in which either beam is seen to have f=0 by the other beam, so I think you will always get a non-vanishing net energy of each beam with respect to the other. So I think in Newton, the beams should always attract according to an inverse-square law, irrespective of their relative direction of motion. Note that what we are asking about is the attraction between the beams (ie of each beam to the other), not how an external test-particle is attracted to the two-beam system as a whole (which would involve the system’s invariant mass in Newtonian gravity).

I honestly don’t think that anything that works as well as SR requires excuses to be made for it - unless of course you demand something from it that it wasn’t ever designed to do. But regardless, this is a discussion forum, so we can agree to disagree on that. It’s kind of like using a topographical map. Prior to joining the monastery I used to be big into thru-hiking and long-distance backpacking, and would always carry a paper map when I went into the backcountry. You can pick two points on such a map, and decide by which route you want to connect them, and the map will then tell you how far you have to go along that route. Now, the map obviously isn’t the territory, so why does this work so well? It works because the relationship between points on the map is the same as the relationship between locations in the “real” territory - two map-units exactly map into two territory-units, using a known conversion constant. It’s a faithful representation of relationships between locations in the territory. It would be meaningless to ask here by what “mechanism” something “acts” on your ruler to “make it” measure different distances along different routes on your map. Once the map is open on your lap, you just pick a route and read off the answer - at that point it’s just geometry, and there are no mechanisms or effects that act on anything. What is meaningful though is to ask why the relationships between locations in your territory are what they are - the answer to this might include geological processes, erosion due to weather etc. But of course your map can’t provide you with such information, because it was never designed to do that; it just represents what’s there right now. That doesn’t make it any less useful for your hike, and you wouldn’t need to “make excuses” for the map on account of that omission, would you? SR/GR is no different - it’s a faithful map-like representation of the relationships between events (locations in space at instances of time) in the real world. Once you pick two events and decide which route you wish to take between them, it will tell you how long that route will be. Just like on an ordinary map, different routes will naturally be of different lengths, and just like on a real map, that requires no mechanisms that act on your measurement devices to make that so. The reverse is just as true - if you measure different lengths between the same events, you know that different paths were taken. So SR/GR is about relationships between events, not about things somehow “happening” to those events. And just like on the map, it is indeed meaningful to ask why these relationships are what they are, which is the closest you’d get to having a mechanism - no one knows the answer to this just yet, but one possible answer could be that classical spacetime emerges from something more fundamental, according to its own set of rules and dynamics, just like surface topography on Earth emerges from plate tectonics and other geological processes. There’s no guarantee that this is so, but it’s a possibility that is testable at least in principle. But whatever the case may be, the answer would be outside the paradigm of SR/GR, in the same way as plate tectonics is outside the paradigm of a topographical hiking map. So IMHO, answering the question as to differing readings of travelling clocks in terms of spacetime geometry is perfectly reasonable, in the same way as it is perfectly reasonable to answer the question as to differing route lengths in terms of the Euclidean geometry on a topographical hiking map. These measurement differences - again IMHO - require no other causative mechanism; once the map is in front of me, the distances are a foregone conclusion, I just need to read them off, and I can rely on the fact that the distance I actually have to walk in the real world will coincide with what the map tells me, without wondering what mechanisms might act on my feet to make these numbers match. I think we can all agree that if there is a deeper reason as to why these geometries are what they are, then of course we want to know about it and understand it. But that’s a different paradigm, and provides answers to a different set of questions, and it isn’t something we can reasonably expect SR/GR to be able to do. When I ask how long I will have to walk to get to tonight’s camp spot, then I don’t want to get an answer in terms of plate tectonics - even if that is the ultimate mechanism, the answer would be essentially useless to me. So that’s just my own two cents on this subject - no one is under any obligation to adopt this type of philosophy, but I find it reasonable and it works for me.

That’s a good and valid and very tricky question, md65536. Unfortunately it’s not possible to do this globally - there simply is no standard by which you can take an entire spacetime geometry and say “this spacetime contains more gravity than the other one”, because you can’t meaningfully compare tensor fields as “more/less” or “smaller/greater”. The only thing you can in fact do is check whether two different metrics might describe the same physical spacetime - there’s a standard procedure for that. What you can do though - and that’s how I would answer your question - is make the issue local. Pick a specific small region within that spacetime, and then evaluate your curvature tensors in that specific region - for example you can look at the scalar invariants of the tensors there, or go for broke and explicitly calculate the tidal forces between test particles within that local region. You can then vary the gravitational sources (the binary system, in this example), and check what physical consequences this has in your test region. So while a direct comparison is pretty much meaningless globally, it can easily be done locally around a specific point in your spacetime. Just remember that if gravity gets stronger in your test region, that doesn’t mean that the same happens everywhere else in your spacetime - perhaps at other points nothing changes, or gravity even gets weaker there. Curvature can “shift around”, or “radiate away”. This statement is based on a Newtonian intuition (where it is absolutely valid), but unfortunately it isn’t this straightforward in GR at all, because here the curvature arises from several distinct things: 1. The actual local source term in the field equations, which is the energy-momentum tensor 2. Gravitational self-interaction, which is encoded in the non-linear structure of the equations themselves (no explicit source term) 3. Boundary/initial conditions, which you must manually supply to obtain a specific solution for the field equations (this roughly represents distant sources) 4. Integration constants, which appear in the process of solving the equations, and are given a physical meaning (if applicable) by comparison with other known solutions in a given region. If they appear in the final metric, they are global properties of the entire spacetime. Counterintuitively, the “mass” we are familiar with does not appear as a source at all - it is not part of the energy-momentum tensor (which only contains densities, fluxes, and pressures for the interior of bodies/fields), nor is it part of the boundary conditions you supply. Unlike in Newtonian gravity, in GR the mass simply isn’t in the picture until you get to step (4) - here, integration constants appear, and these can often (but not always!) be interpreted as concepts of mass, charge, angular momentum. One very important difference to Newton is that these constants are properties of the entire spacetime, not just some isolated body within it - so they contain their Newtonian equivalents, but also contributions from initial/boundary conditions (such as motion), as well as gravitational self-interaction. One could nearly say that mass/charge/angular momentum are not inputs into the equation, but rather that they arise from the equation, once it has been properly set up. So this is very different from Newton. And then there’s of course the issue of what “greater curvature” even means, because that’s not a straightforward concept either. For example, at the EH of a BH with the size of the earth, tidal forces are so great that any ordinary material body would immediately get ripped to shreds. On the other hand, at the EH of a supermassive BH, tidal forces are vanishingly small, to the degree that it would be an engineering challenge to detect them at all. So which curvature is “greater”, and how does this relate to “mass”? It’s not a straightforward comparison. As a simply example of where the Newtonian intuition fails completely, consider a particular spacetime called the Bonner beam - it’s essentially two parallel, very long beams of light. One is free to give each of these beams arbitrarily much energy. Because they contain lots of energy, and energy is equivalent to mass, we should be able to ascribe some notion of mass to them - meaning these beam should gravitationally attract one another, because of curvature etc. Right? What you will actually find is that, if you shoot these beams parallel in the same direction, they will not attract at all; but if you shoot them in opposite directions (all other things equal), they will attract, but not according to Newtonian inverse square laws, but something much more complicated. So, mass alone won’t always work well in GR - it sometimes gives the right intuition, but in other circumstances it can fail really badly. Far away from the binary system you simply have a radiation field with a succession of wave fronts of a particular wavelength and amplitude - so if you were to place a bunch of test particles there, then their separations would oscillate at a certain frequency and with a certain amplitude, transverse to the direction of propagation of the waves. Note that you can’t replicate this finding with Newtonian gravity. This is an example of a spacetime that has three “hairs” - mass, angular momentum, and a quadrupole moment. It’s “source” in the field-theoretic sense. In GR, sources are the energy-momentum tensor, gravitational self-interaction, and boundary conditions. The first appears explicitly in the equations, whereas the other two are encoded in the structure and nature of the equations themselves. Do also note that tensors are local objects - so the energy-momentum tensor is non-zero only in the interior of objects and fields, and vanishes in vacuum. So for example, if you are looking for the curvature around the binary system, you are in fact solving the vacuum equations \(R_{\mu \nu}=0\), absent of any explicit source term. Yes, the angular momentum you are adding to one of the bodies is a source of gravity, in the sense that it changes the geometry of this spacetime. It definitely has an influence. It’s just that in GR it doesn’t simply appear as a contribution to the mass of the body, in the same way as you might do that in Newtonian theory. Rather, adding angular momentum takes away one of the symmetries of your spacetime - any free fall into such a body will not only have a radial, but also an angular component (frame dragging), so the overall geometry is no longer spherically symmetric. In practice, you would start with a different metric ansatz that reflects these symmetries when you set out to solve the field equations. Fewer symmetries in your spacetime generally leads to more free parameters in your final solution - here, you’d get a spacetime with two global properties instead of one, which can be identified with mass and angular momentum. Again, remember that these are properties of the entire spacetime, not just the isolated body within it. As to where your test particle will fall, I can only make an educated guess - in GR a free fall test particle will trace out that world line which maximises its proper time (principle of extremal ageing). If you add angular momentum to one the planets, the curvature tensors will take on additional terms that reflect this, and the particle will fall not just radially but also sideways. I think this should translate to geodesics with longer geometric length in spacetime, so the particle should fall towards the rotating body, rather than the non-rotating one. It should be clearly noted though that this is an example of where Newtonian gravity gets the actual free-fall trajectory wrong, because it would just predict a more rapid but purely radial in-fall - whereas in GR the in-fall takes on an angular component as well, so you’d get a segment of a spiral. In practice one would have to run the actual numbers to be sure of what happens - which would be very non-trivial, because the influence of the other planet cannot really be neglected here, and curvatures combine in non-linear ways. So I would not be at all surprised if it turned out that my educated guess is in fact wrong. There are just too many subtleties involved to be sure without doing the maths. The fundamental concepts are completely different. With Newton, you start with sources (distribution of mass densities), put them into the field equation, and get potentials/forces as a result. In GR, everything is geometry - you start with a metric ansatz that reflects the global symmetries of the spacetime you are looking for, and then you constrain that ansatz more and more until you obtain a specific metric: you first combine the ansatz with any local sources but putting it into the field equations; you then combine it with distant sources by supplying boundary conditions; and finally you solve the system and find physical interpretations for any remaining integration constants. As a result you get a specific metric, from which you can derive curvatures, geodesics etc etc. The fundamental difference is that in GR the gravitational field also interacts with itself, so even in weak-field scenarios you might get phenomena that run counter to Newtonian intuition. It also means that many of the fundamental concepts like mass, angular momentum etc don’t straightforwardly carry over from Newton, because they can’t account for the self-interaction of gravity, not even in principle. Sometimes such differences can be neglected, and sometimes not - it really depends on the scenario. But I think the most important thing is to not try and mix concepts, because that is almost guaranteed to go wrong in some way or another.

Spin networks are mathematical objects that are now used to construct certain classes of models of quantum gravity, such as LQG for example. The classical limit of such models must always be SR/GR. Right at the beginning of this thread I mentioned once or twice that it is just this - a model for the emergence of classical spacetime from quantum gravity - which I consider to be the “mechanism” for all relativistic effects. So it appears we have come full circle. Why you brought an ether into all this is strange to me - it has no explanatory power or utility, because its presence or absence has no physical consequences whatsoever. It cannot provide the mechanism you are looking for, which is why it never became part of established physics (as Eise has explained). This entire discussion has already been had a century ago. I wish you would refrain from misrepresenting what I actually said, in the context of when I said it. So far as I am concerned, you have, over the time you have been here, made very valuable contributions to the forum across many different threads and discussions - but this here, I’m sorry to say, is really beneath you. Disappointing 😕

P.S. I’ve forgotten to mention that not only mass, but also angular momentum in binary systems such as this one is a concept that does not easily generalise from Newtonian to GR physics, because angular momentum in GR radiation fields is subject to a mathematical issue called “supertranslation ambiguity”. This has in fact long been a very difficult problem, that has only recently been resolved. So not only is “mass” a problematic concept, but “angular momentum” is too - that’s why neither of these are used when solving the Einstein equations for such systems. You just work with initial and boundary conditions, and let the non-linearity of the equations themselves take care of the rest.

Nice question! The major ones that come to mind are (not an exhaustive list): - The ADM formalism - The tetrad formalism - The Spinor formalism - The Ashtekar formalism - Of course the Lagrangian formalism - The Plebanski formulation - The geometric algebra formulation It can also be written as a gauge theory, though I must admit that many of the details here are above my pay grade - there seem to be some unresolved issues. The above is definitely not exhaustive, but it’s all the ones I can think of OTOH.

With your clock. That’s just the point - the entire geometry is such that the geometric length of a path between two events equals the accumulated time physically recorded on a clock that travels on that path (remember that’s a path through spacetime, not just space), so there is a very direct link between the theoretical formalism, and what physically happens. Once we adopt the empirical finding that c=invariant for all observers, and hold start and end points fixed, then the operation of varying the path leaves you with only one degree of freedom - its length, which is the total accumulated clock time. That’s how the times between the twins differ - they logically can’t be the same, unless either the paths coincide, or c is not an invariant. The former case is trivial, and the latter is so highly constrained by observational data as to be practically ruled out within the domain of our experimental capabilities.

Well, what I want to know is specifically what the spacetime geometry around a binary system looks like, and on what physical parameters it depends. The metric that describes that geometry must necessarily be a valid solution to the Einstein equations, so it can depend only on quantities that appear in these equations, either as a source term, a boundary condition, or as an integration constant when solving them. So I think it is very relevant for what I wish to do here. Newtonian concepts do not come into this for me at all, because they do not correctly describe the external radiation field and thus the dynamics of these bodies, nor even the geometry close to the binary system. I’m thinking purely in GR terms, whereas you appear to be mixing GR and Newtonian terms, so it seems we are unfortunately not talking about the same thing. As a word of warning - it’s rarely a good idea to mix Newton with GR, because there are subtle but extremely important differences in basic concepts - most notably mass, since in GR this will need to incorporate non-linear contributions from gravitational self-energy, and is usually a global property of the entire spacetime, rather than something that an isolated objected “possesses”. See link further below for a general overview of this. My understanding of geordief’s original question was that it was about GR, so I don’t use Newtonian physics here. You can of course try to analyse this using Newtonian gravity, but that’s not my goal; you will also find that for this kind of scenario the predictions from the two models will differ substantially. That’s Newtonian physics (and correct in that context only), not GR. Invariant mass never appears anywhere in this GR calculation. It can’t, because it is a Newtonian concept that does not straightforwardly generalise to GR. The curvature in the exterior vacuum does not arise from any source terms. Counterintuitively, the masses (however defined) of the two bodies do not even enter as boundary conditions. All that happens is that, while solving the equations, you are left with two integration constants that can’t be eliminated, and these are - through some subtle argumentation - identified as the “masses” of these bodies, but not in the same sense as would be done in Newtonian physics. These are parameters in a 2-parameter family of metrics, and thus global properties of the entire spacetime. Unlike for the case of - say - Schwarzschild, this spacetime is not asymptotically flat, so even far from the binary system you can’t straightforward identify these parameters as mass in the Newtonian sense, as you would do for the Schwarzschild case. Also, the above statement I quoted you on, when taken as a general statement, is highly problematic - Newtonian gravity has no concept of curvature, and GR has no concept of invariant mass, so saying that the two are related strictly speaking doesn’t have any meaning. In exterior vacuum, the Einstein equations have no source term; and for interior spacetimes, the source is the energy-momentum tensor, which also does not contain invariant mass in the Newtonian sense. So either way, invariant mass never comes into this. Again, I am treating this as a pure GR problem. What I am essentially saying is that this is a scenario where you cannot mix GR and the Newtonian concept of invariant mass. You’ll end up in a mess. And why would you? You can either analyse this purely in GR terms (that’s what I am trying to do), or purely in Newtonian terms (if you are prepared to ignore the radiation field, and the evolution of the system). But don’t mix them. In Newtonian gravity you can ascribe a gravitational force to each point of the surrounding space (iff you can define a potential field, which, btw, you can’t do here), which you can then compare - so if you analyse this in Newtonian terms, then the answer is probably yes. But in GR the question itself is essentially meaningless - all you have to compare are two metric tensors, and there is no meaningful way to say that one is greater/equal/less than the other. They are just different. What I suggest you could do though is look at a small region somewhere outside the binary system, take two test particles which are initially at relative rest, and measure by how much their relative separation changes as each wave front passes. A GW detector, essentially. You’ll find that the faster the bodies orbit, the larger the local wave amplitudes - so in that very particular and local sense, you could say the gravitation gets “stronger” with increasing angular momentum, in that region where you perform the measurement.

That means it cannot be the cause of the twin’s dilated proper time. I stated in my analogy that they all fly at the same ground speed at all times - there is no difference in velocities-over-ground between these planes, nor is there any other physical difference between them. Thus air drag obviously isn’t the reason why they take differing amounts of time to reach their destination (you could perform the same experiment in vacuum, using rockets instead, with the same outcome) - the reason is that they take different routes, and thus have different distances (at the same ground speed) to travel. Therefore their flight times must necessarily differ. In that sense, the choice of route has causal efficacy so far as the total accumulated flight time is concerned, all other variables being equal. The exact same principle holds in SR/GR as well, it’s just you’re now considering a path that goes through space and time, while holding the norm of your velocity 4-vectors equal. So the choice of path through spacetime has causal efficacy so far as total proper time is concerned, since all other physical parameters remain exactly equal between the twins, so only the distance through spacetime can differ. And that’s by definition precisely the total time physically accumulated on that clock.

Where and how does the kinetic energy in the system’s COM appear in the field equations, exactly? The dynamics of a system of moving masses, as in this example, most definitely has a gravitational influence - evidently a stationary system and a non-stationary one have very different spacetime geometries. I’ve tried to explain this in some detail in my previous post. My point though was specifically that you won’t find any explicit “mass” or “kinetic energy” terms in the vacuum field equations that somehow increase when there’s more motion, and that act as gravitational sources. That’s how it works in Newtonian gravity, but in GR it’s much more subtle. In fact, if you want to work out the geometry of the vacuum region outside the orbiting bodies, you’ll find that there are in fact no source terms at all. All you have is a rather complicated system of coupled differential equations for the relevant components of the metric, but nowhere does this system contain any terms for either masses or momenta (or energies of any kind). This being a set of differential equations, you then have to supply boundary and initial conditions to obtain a specific solution - and it’s only here that a “description” of the 2-body system enters the picture. The form these boundary conditions will take will be the position of the bodies in question at a specific time (in your chosen coordinate system), two free parameters, and the velocity 4-vectors of each body at that time, plus perhaps some global constraint or another. Starting with this initial “freeze frame”, the solution to the system of DGLs then describes the future evolution of this system - this will be a 2-parameter family of metrics, and these parameters are just the rest masses of the two bodies. So yes, motion (in vacuum) most certainly has a gravitational influence - but it enters as a boundary/initial condition, and not as a source term containing energies of any kind. The difference is subtle but very important. Also, because the system of equations is coupled, you cannot mathematically isolate the effects of motion from the effects of the other boundary conditions; so you can’t write the overall metric as the sum of a purely static metric, plus a metric purely for motion (which is, I believe, what geordief’s question was originally about). The effects of mass and motion are both intertwined in the overall metric, you can’t neatly separate them.

So if I understand you right, you are looking for a mechanism in spacetime that somehow acts on a clock to make it go slower? If so, then we are not talking about the same thing, because I’m looking for a mechanism for spacetime, in the sense of some system of dynamics the classical limit of which can be written as a semi-Riemannian manifold endowed as with a connection and a metric (ie GR). SR’s Minkowski spacetime is then just the local limit of that. Euclidean geometry is also a mathematical abstraction - one that ignores time, and doesn’t gel very well at all with the Standard Model. Why is Euclidean geometry scientific and physical, but Minkowski geometry is not? So far as I am concerned, the idea of the world being Euclidean is pretty much explicitly ruled out by existing observational data. Sure, but that’s just as true for my IPU (Invisible Pink Unicorn). The point is that we have no reason to think that either of these things exist, or even need to exist. My understanding of the ether in LET is that is has no physically detectable consequences, ie there is no experiment - local or global - that you can perform to detect the presence of this ether. This seems to imply to me that it cannot provide the mechanism you are looking for. No, I don’t think there needs to be any “something” at all, other than the choice of how those twin clocks move. To give a Euclidean analogy - it’s like connecting the same two cities by a group of airplanes. Let everything be the exact same - departure time, type of plane, ground speed, weather conditions, cruising altitude, laws of aerodynamics etc etc - except the route they take. One of the planes flies along the shortest possible route (great circle segment), whereas the others choose different routes, no two of which shall coincide. What is the mechanism that “makes” the total flight time different for each of these planes? Is there an ether that “drags” on those planes (without being detectable!), thus making them arrive later than the shortest-route flight? Why would you expect planes that fly along different routes to always take the same amount of time, in the first place? The only physically relevant mechanism here is that we need to apply measurable acceleration at some point in order to not make all the flight path coincide - and once you do, the flight times must necessarily diverge. This is entirely irrespective of the kind of mathematical description you give this situation. Actually, the point I was trying to make was simply that this idea postulates an unnecessary mechanism that then requires an entire set of new mechanism to make it work.

Yes, the motion does of course “contribute” to the geometry of this spacetime, as I said in my post - but not really in an intuitive Newtonian way of making it “stronger”, which is an ill-defined concept in GR. This is the classic “does a body become a black hole if it moves fast enough” question, which I don’t think we need to go over again. The rest (!) mass of the orbiting bodies doesn’t change, irrespective of how they move, and in any case we are looking for a vacuum solution here, so \(T_{\mu \nu}=0\) everywhere outside these bodies. Thus the equation you are solving is simply \(R_{\mu \nu}=0\), without any source terms at all. The angular motion makes an appearance only as part of the initial/boundary conditions once you solve that system of DGLs. So in practical terms, what would be the difference between the spacetime surrounding two bodies momentarily at relative rest at some distance wrt to each other, and the same system with the two bodies orbiting around a common center of gravity? In the former case you have a stationary spacetime that looks more and more like Schwarzschild the further away from the two bodies you get, and eventually becomes Newtonian; in the latter case you get a gravitational radiation field, where frequency and amplitude depend in some way on the rest masses of the binary system, the distance between the orbiting bodies, and the angular frequency. In the full (non-linearised) description these dependencies are pretty non-trivial, especially in the region very close to and in between the orbiting bodies. When you are located further away from the binary system, the amplitude will fall off with distance, again in a specific way. So the crucial difference is that the former case is stationary (i.e. the components of the metric do not depend on time, thus there is a time-like Killing field on this spacetime), whereas the latter is not. It’s a difference in symmetries, more so than degrees. What is true though is that the amplitude of the wave field at a given distance from the binary system will depend on the angular frequency of the bodies - the faster the bodies orbit each other, the higher the amplitude will be at any given distance. So if you wanted to, I guess you could - in this very particular sense - say that the gravitation of the binary system gets “stronger” if they orbit faster, since you’ll have more powerful tidal forces at a given distance. Personally though I would say that such a concept isn’t helpful, because it very easily lends itself to misinterpretation and various misconceptions. In general there just isn’t any way to meaningfully say that one metric is somehow “stronger” than another metric, because you are comparing tensors, not numbers. They are just different. But I guess you can always pick out specific local measurements and compare those. What you can absolutely not do though is take a known metric in a specific coordinate system - such as Schwarzschild -, and simply replace rest mass by relativistic mass to reflect relative motion, while leaving everything else the same, and expect the result to be a valid metric again. That doesn’t work, because such an operation is not in general a valid diffeomorphism.

Ok, but in order to discuss this we need to clarify just what “mechanism” actually means here, or else we won’t know whether or not anyone is denying its existence. So how do you define this term, exactly? It isn’t a straw man, it was merely my interpretation of what it is you are trying to say, because that is not at all clear to me. Since I evidently missed the actual point you were trying to make, it would be helpful if you could summarise it clearly and concisely. They are a description of your physical scenario, and thus both mathematical and physical. The physical difference is the path these clocks are taking. I disagree. The ether you are referring to here is undetectable and has no physical consequences whatsoever. I might as well postulate an Invisible Pink Unicorn in my back garden, for all the difference it makes. It’s philosophically permissible precisely because it has no detectable consequences - but it’s not science, because it isn’t amenable to the scientific method. Besides, if you do this, you are going to have to explain a “mechanism” for why that ether has the properties it has, why it behaves the way it does, and where it came from - so in the end you haven’t actually explained anything, you have just kicked the can down the road. Again, I think you need to define for us exactly what you mean by “mechanism”.

Indeed!

Ok, let me explain. My understanding of what you are trying to say is that we are missing a deeper mechanism of how gravity works, and why it is the way it is. We have an equation that allows us to describe any given scenario of classical gravity, but we don’t know why the equation works, or why it looks the specific way it does. We can’t even be 100% sure yet that the equation we are using is indeed the best possible description of classical gravity - I think it is at least conceivable that one of the very many possible modifications of GR might eventually turn out to be better. I’m not prepared to categorically rule this out, though for various reasons I think it is unlikely. But regardless, I agree with you on the basic premise - we do not know yet what the underlying mechanism might be, so our understanding of gravity is still lacking in that regard. It’s an epistemic description, but not an ontological one. As you put it, there’s time varying metrics etc, but no underlying mechanism to “make that happen”. What I question though is why you seem to hold GR accountable for not providing such a mechanism. It was never designed to do this, since it is simply a generalisation of SR, which in itself proposes no mechanisms either - it simply arose from the empirical observation that Minkowski spacetime provides a model that fits very well to available observational data. Newton also never provided any “implementation” of how his forces work - he simply posited them as a convenient computational tool, not as an ontological description of reality. So I feel that pointing out that what you are looking for is outside the remit of GR is a valid criticism. I also don’t agree that current practices make no attempt at find such underlying mechanisms, because that is what the whole quest for quantum gravity is ultimately all about - and it is an area of research that is ongoing and very active. Naturally in such a quest there’s going to be very many dead ends, especially since the domain in question is beyond our technological capabilities to probe it directly, essentially making is fumble in the dark here. Within current constraints (technology, funding, politics etc etc) I actually think we are doing pretty much the best we can in that regard. Our efforts aren’t perfect, but they are all we can muster right now. I think you are demanding way too much from contemporary physics - you seem to basically say that “if it isn’t a fully fledged ontological explanation, then it’s not science”. I cannot agree with this. I think any epistemic description of an aspect of the world that allows us to make predictions by way of computation is valuable, at least as an intermediary step, even if it is not explicitly ontological in nature. Of course we want such models to approach the status of an ontological explanation over time, but that is not going to happen all at once. You start with something purely descriptive, and then keep refining it; occasionally you might need to change your paradigms; and in the end we might get to something that approaches ontology. I understand physics as being a process that will take time. My guess is that the day will come when GR will be understood as merely the semi-classical description of something much more fundamental, but we are not at that stage yet. Mind you, there is also no guarantee that there actually is a fundamental ontological explanation - perhaps spacetime just is what it is and can’t be further reduced, in the same way that the specific and irrational numerical value of pi can’t be reduced to any more fundamental “mechanism”. I sincerely hope for this not to be the case, but I think we are also not in a position to categorically rule that out yet. Obviously, because they are using the same laws, but different boundary conditions. Sharing the same laws does not mean that everyone measures the same thing irrespective of the situation they find themselves in - it means only that they agree on what each of them measures. To be specific, the law in question is the total amount of proper time physically accumulated on a clock that propagates from one event to another. This is not just some nebulous theoretical concept, but it is what you physically see accumulated on a clock that you are holding in your hand while travelling. That total time is calculated as \[\tau =\int _{C} ds=\int _{C}\sqrt{g_{\mu \nu } dx^{\mu } dx^{\nu }}\] as you probably know already. This law is the same for everyone, since it is written in covariant form. What is not the same for everyone is the path C that you need to put in as a boundary/initial condition in order to evaluate that integral - it naturally depends on the pair of events that are being connected, and it also depends on the spatial trajectory that is being travelled. So once you evaluate that line integral, you are (in general) going to obtain different numerical results depending on your choice of C, even though the same law was used by everyone involved. That is just how line integrals work. Once again you can ask why this is so - why does this law look the way it is, ie why is it a line integral and thus dependant on a choice of path? But SR/GR cannot answer this question, because it is outside their remit. Hopefully we can eventually come up with a more fundamental model that can explain why this law looks the way it does, but at present we don’t have that yet. Of course not. These are gauge symmetries we are talking about. If they were always global, then the entire rich phenomenology of the universe around us would disappear - there wouldn’t be the kind of particle zoo we find, and there wouldn’t be any gravity. PS. When I speak about a “deeper mechanism” in the context of context of gravity, what I mean specifically is why, in the classical limit, curvature and gravitational sources are locally related via an equation of the general form G=T, as opposed to some other relationship. Part of the reason is clearly mathematical consistency, since the properties of G and T themselves already rule out most alternative forms of the equation. Another already known constraint comes from topological considerations, to do with the conservation of certain topological quantities. So we are not completely clueless as to the “why”, it’s just that the bigger picture is incomplete. Note also that once the local relationship between sources and curvature is established, and boundary conditions are set, then what happens in all the rest of spacetime is a foregone conclusion on account of the basic requirements of continuity and differentiability. So once you know what the deeper link between sources and curvature is, you don’t actually need any other “mechanisms” - it’s then simply a matter of logical consistency, because you can’t randomly glue any old geometry onto any old source distribution. The only true mechanism here is that initial (purely local!) link from sources to geometry.

The problem I see with this statement is that relativity itself wasn’t ever meant to be a “metaphysical principle” - right from the outset it served a very practical purpose, namely being a descriptive model that doesn’t suffer from the internal contradictions and conflicts with observational data that Newtonian physics did. In that it has been pretty successful, and crucially it allowed us to reduce our reliance on some rather dubious concepts - such as for example Newtonian forces. I mean, think about it - a Newtonian “force” is a supposed thing that cannot be directly observed or detected (we only ever see its effects), that is entirely non-local, somehow acts instantaneously across arbitrary distances, and there is no underlying mechanism that might explain how it could possibly do all these things. Metaphysically speaking this is entirely ridiculous by anyone’s standards. Yet it works to some degree, and thus to this day we teach it in our schools. If you are looking for an underlying metaphysical principle in relativity, then it would be that of symmetries - turns out that the fundamental objects which SR deals with (Lorentz transformations, tensors, spinors etc) are representations of the Lorentz group, whereas GR is a gauge field theory with GL(4) as its fundamental symmetry group. Of course Einstein himself didn’t know that at the time. Symmetries are also the metaphysical principle underlying many other areas of physics, most notably HEPP. Seems to me these things are all pretty useful!

I don’t think there’s an easy answer to this. However, at least part of any possible approach should be to educate our youngsters about how to skilfully relate to digital media - I mean specifically to teach them skills that help to recognise and properly relate to misinformation. I’m talking about general media skills here, which is quite a separate thing from having expertise in any particular area. “Media Skills” should be part of any school curriculum, IMHO. Ultimately we’ve got to understand that misinformation and crackpottery has always been present, and will continue to always be present. So the question isn’t how to eradicate this, but how to help people relate to it properly.

Absolutely not my area of expertise, but my amateur-ish opinion on this is that you will never be able to come up with any kind of objective measure of a civilisation’s development, simply because it relies on values that are not universal, but contextual. In my opinion, the best measure of a civilisation’s development is in fact one that is explicitly subjective - people’s self-reported general sense of well-being. Note that this is not the same as happiness, wealth, or even “feeling good” - someone might be living in a democratic state of affluence and plenty, and yet not be well in themselves (this is in fact depressingly common). Conversely, someone may live in simple and basic conditions, yet still have a strong sense of general well-being in their circumstance. Being well is the culmination of all the many factors that contribute to your basic needs being met, and you still having time to pursue other things in life as well - it is the coming-together of material, intellectual, environmental, and spiritual balance. All the traditional concerns such as economy, politics, healthcare, education etc etc contribute to this, but not in ways that are easily measured and broken down. So if you want to know how a civilisation is doing, ask its citizens if they are well - not happy, rich, healthy etc. If you make their governing body explicitly responsible for the overall sense of well-being experienced by the people, then I think this would be much more conducive to a more balanced world overall.

Not reliably of course - I would hazard a guess and say that evolutionary pressures on other inhabitable worlds will be broadly similar to our own, so any sentient race that evolves there will likely evolve a reality-model that is also broadly similar to ours. Based on what we see here on Earth, nature tends to come up with similar solutions for similar problems. Nevertheless, even small differences might help us get a better understanding of our own concept of reality, and how it might relate to a possible ding-an-sich external reality. As an aside, I would also conjecture that the more different a species’ reality model is from ours, the harder it would be to establish mutual communication. Arguably, if the models are sufficiently different, there might come a point at which no meaningful communication is possible at all, because we’d share too few fundamental categories.

I can’t answer this, as evolutionary biology isn’t my area of expertise. It’s a difficult subject also because the autism spectrum is so broad - there are some like myself, with very few to no situational support needs, and then there’s a sliding scale of increasing severity right up to forms of autism that make independent living (never even mind independent survival) practically impossible. So it’s hard to generalise. I’m speaking only for myself now, and perhaps those with similar profiles and predispositions as me. I think there might be an evolutionary advantage precisely because people like me don’t fit into the mainstream. For example, my sense of purpose, meaning, and well-being is not contingent upon social acceptance and belonging - things like how many friends I have, social gatherings and occasions, belonging to a certain group (or not), being around other people etc etc are simply of very little importance to me. This might at first glance sound like an evolutionary disadvantage, but think about it - it frees up enormous amounts of time and energy that can then be re-invested into other pursuits. I don’t know if there are statistics about this, but I bet that, among people who have made important contributions in their fields - the arts, sciences, literature etc etc - a disproportionally large amount might be found to be on the spectrum, or at least have autism-like traits of some sort or another. This is because such people are more likely to engage deeply in pursuits not directly concerned with survival and procreation (which is what social preoccupations are ultimately geared towards). I think society benefits from this kind of archetype - the ones who can stand on the sidelines, look back onto the mainstream from a more neutral and wider external perspective, and pursue “higher” things and unusual ways of thinking. I think there’s an evolutionary advantage for the group as a whole in having such individuals, because they function like a mirror that reflects back the forest when all you yourself are usually able to see is the trees, due to your own day-to-day involvement. Such individuals are often simultaneously despised (because they don’t fit in), and valued (for their contributions, often only posthumously), and sometimes burned at the stake; but whatever the case may be, their perspective is an important one. These are just some of my own thoughts, I’m making no claim to any academic truths here. Yes, but it’s not just that - it’s a theory of the world, including the physics side of things. When we are building models in physics, then these are necessarily models of aspects of how the world appears to us. They are thus models of aspects of another model, namely the reality-model that our minds create for us. We all tend to agree on certain aspects of that generated reality simply because we all share a similar sensory apparatus (plus its extensions), and a roughly similar neurophysiological brain structure - thus the boundary conditions are similar, meaning the resulting reality-model is also roughly similar. The reality-model of an organism that evolved under sufficiently different boundary conditions may potentially be quite different from ours - an example from sci-fi literature that comes to mind are the heptapods in Ted Chiang’s “Story of Your Life”, whose minds do not employ the principle of temporal sequencing in constructing their reality-model. I know it’s just a story, but it’s an interesting example. So what happens if the boundary conditions vary? I wrote about autism and social “mind-blindness” above - so what is actual reality here? Are social relations and intuitions real, irreducible aspects of the world - or are these contingent add-ons that your neurotypical brain artificially generates, and it is actually my own mind-blind autistic self who sees things as they really are? Or how about this - in addition to being autistic I am also a synesthete. Words to me have colour, texture, size, spatial orientation, and sometimes temporal extension. These, to me, are not associations (e.g. sky=blue), but intrinsic properties of the words themselves (so for me sky=off-white, smooth and cold like marble, angled backwards and to the right), like spin and charge for an elementary particle. For me this is so intrinsically normal that I am pretty much unable to imagine what experience would be like without these attributes - I only know intellectually that most people can’t experience this the way I do. So who perceives “actual” reality here - is the concept “sky” really smooth and cold, and you are all just blind to that? Or does my brain adds this on randomly? Who’s right and who’s wrong? Or is the entire concept of “reality” just a constructed idea, the meaning of which is strictly contextual? Now think about the wider implications for physics - it makes models of a model. But how do we know, within how the world appears to us, what is an actual part of exterior reality, and what is an add-on by our brain? How can we distinguish, in the absence of having an external reference in the form of other reality-modellers against whom we can compare our reality? Do (e.g.) time and space really exist in the way we experience them, or are they just convenient representations to impose order onto a set of data, like the windows on the GUI of your computer? Are there other ways to structure that same information? Or are there aspects of exterior reality that are not being represented in our model at all, not even by deduction or induction, perhaps because they are irrelevant to our continued evolution? Does the way we do science thus say more about ourselves and how or brains make reality appear to us, than exterior reality? I think these are important questions to consider not just in philosophy, but also in the foundations of science - just focussing on the model, while ignoring what the model is actually about, and who constructs it, might be misguided and eventually come back to haunt us. I don’t feel this is spoken about enough in the physics community, or even taken seriously.

I don’t know about “fundamental”, but ultimately this reality-modelling machine is a result of the process of evolution. What this means is that its function is not at all to neutrally reflect “external reality as it is”, but rather to present us with a model of external reality that is specifically geared towards survival and procreation, and as such will be filtered, distorted, and pre-digested accordingly, with this goal in mind (pun fully intended). For example - out in the jungles and savannahs where we originally came from, if you encounter other members of your species with whom you compete for limited resources, it is advantageous for you to have available a model that allows you to (at least to some degree) predict their intentions, mind-states, and possible future actions. Likewise with the flight path of an arrow, the weather, the behaviour of water in a river etc etc. If you have good models available that take sensory inputs, processes them, and generates something that allows for predictions of how your environment will evolve into the immediate future, you’ll simply have much better odds to do well and thrive, evolutionary speaking. So it’s actually not a surprise at all that things are as they are. As a little aside: I, as being on the autism spectrum, am missing a part of this reality-modelling machine - when I encounter another human being, I am socially blind; I generally have no intuitive concept whatever about what kind of mind-state that individual might currently have, I might as well be looking at a stone statue. I don’t immediately know their intentions, nor can I easily tell how they will behave in the next few seconds. All I can do is make educated guesses based on experiences gathered during previous interactions I have had with people; but this takes an active and conscious effort, and sometimes I get it quite wrong. It’s called “mind-blindness”. This is part of the reason why autistic people often struggle with social interactions.

All you can really meaningfully say here is that the geometry of these two spacetimes (two stationary masses vs two masses in relative motion) will be different - in particular, in the latter case of relative motion, some or all of the components of the metric will be explicitly time-dependent; it’s essentially a GR 2-body problem (which, btw, can only be solved numerically unless one of the masses is very much smaller than the other). But I think what you are getting at is ultimately whether relative motion in vacuum is in itself a source of gravity, and the answer to that is no, it isn’t. Its presence does, however, have an impact on spacetime geometry, in the sense that it will make the situation less symmetric and thus more complicated. But since you can’t in general meaningfully compare tensors (“tensor X is greater than tensor Y…”), all you can really say is that the spacetimes are different. This “difference” is in itself a non-trivial concept, because you can’t as a rule of thumb tell if spacetimes are different just by looking at the metric - for example, the Schwarzschild metric and the Aichlburg-Sexl metric look very different, but they do in fact describe the same physical spacetime. So there are a lot of subtle issues here. When you are not in a vacuum, ie in the interior of some mass-energy distribution, the situation becomes more complicated, because now the energy-momentum tensor explicitly contains terms that can be interpreted as momentum density. So for example, the motion of plasma currents in the interior of a star will have a gravitational effect within that interior region, as compared to some otherwise identical star without moving currents. But since the field equations are non-linear, it is not possible to neatly separate out these effects and isolate them from the other source terms, such as pressure, strain etc. They all interact and interplay. It should be noted though that in the exterior of the bodies the energy-momentum tensor vanishes, so the masses of the bodies and their relative motion only enters the field equations in the form of boundary conditions.

I would say that the defining characteristic of quantum systems isn’t so much the discrete spectra of some observables, but rather the fact that there are pairs of observables that do not commute. That’s something we don’t find in the classical realm.

Lol I can’t speak for others here, but I found that, whilst the vast majority of crankhoods and crackpotteries out there leave me large cold, there are some things that just tend to grate my gears. Relativity denials are an example of something I find hard to ignore, for some obscure reason which I don’t understand myself; so yeah, I tend to get sucked into those threads in particular, for better or worse. This is in some sense a vulnerability, because the topic pushes my buttons, so sometimes it’s hard to walk away from threads where there’s no further benefit to be gained from continued participation for neither myself nor the OP. It’s a strange thing - you know it’s time to walk away, but there’s that childish ego-based impuls towards having to get the last word in. I very rarely regret things I post here, but when I do then it’s usually connected to not having walked away when I should have.