Markus Hanke

This is definitely part of a possible solution. However, I don’t think this gives the full picture, because it seems to me - and that’s just a personal observation - that at the foundation of every conspiracy belief lies the desire to condense down an inherently complex and unpredictable world that is full of “grey zones” (morally, politically, philosophically etc) into a simple “good” vs “bad” narrative that is easy to grasp and understand. All such theories I can think of, irrespective of specific details, always and ultimately boil down to this - the idea that there is some nebulous “them” who do everything in their power to hide “the truth” from “us” for some nefarious purpose or another. Structuring the world in this simplistic manner gives one a sense of empowerment, since it feels like one sees through “their” deception and can actively resist “evil” by not buying into the alleged lies. To give just one random example - Flat Earth is ultimately not really about the shape of the earth at all, but about the fact that there is a “them” who have been hiding the “true” shape for their own evil ends. This tendency to want to simplify things in this manner stems from a deep-seated sense of powerlessness in the face of an increasingly complex world that is harder and harder to grasp and understand for the common Joe-on-the-street. It’s really very difficult to address this - but yes, education and critical thinking skills are definitely a large part of the answer.

This is an Internet meme (“Free Wifi password”) that for some reason or other seems to have gone viral - though, as studiot has correctly point out - the picture you have attached here has a typo in it, and the dx should be outside the square root. Anyway, here’s the link to a step-by-step solution, which simply involves recognising the meaning of the those integrals, and a view basic calculus facts: https://medium.com/@ericphamhung_76823/solving-the-free-wifi-equation-no-integrals-needed-b74930f18d93

There’s no way to be sure that this is what he was actually working on, but it might be interesting for the reader here to know that we now do have a fully worked-out tetrad formalism for General Relativity. This is, in fact, a more general description of GR than the usual tensor formalism, because it contains additional degrees of freedom - physically, these allow for gravity to couple directly to intrinsic angular momentum (ie spin). So long as we deal with purely bosonic matter fields, this doesn’t make any difference, and the physical predictions are the same. However, in situations where we are dealing with predominantly fermionic matter fields (or a mix of bosonic and fermionic matter), this leads to the appearance of new gravitational phenomena such as spin-spin and spin-orbit coupling. Since some of these mechanisms have repulsive effects, under some circumstances this leads to predictions that differ from those of ordinary GR in major ways - for example, given the right initial and boundary conditions, it is possible to avoid the formation of singularities during gravitational collapse processes. The trouble here is that, even though the tetrad formalism of GR is still purely classical, the extra degrees of freedom it introduces will also modify the fermionic wave equations in relativistic QM, notably the Dirac equation and the Rarita-Schwinger equation, making them non-linear. In principle at least this should be testable. In the case of the Dirac equation, there is currently no experimental evidence (that I’m aware of) to suggest such non-linear contributions do in fact exist. Of course it could just be the case that these effects are too small to be detectable yet…but still, the evidence is currently absent.

I think you aren’t too far off with that. The only thing I’d point out though is that ‘behaviours’ require interactions - so again, we arrive at the view that systems aren’t actually defined by stand-alone ‘things with properties’ that are real even in isolation from everything else, but rather by interactions and relationships with other systems. Thus, ‘reality’ might be contextual, and it becomes an essentially meaningless concept if you try to divorce it from relationships to other systems; something can be said to be real precisely to the extent to which it relates to its environment in specific ways.

I completely agree. This is why I’m personally biased towards looking at the wave function not as a description of the system itself, but rather as a description of how it relates to other systems. Which is essentially Rovelli’s approach to the whole issue. It’s a bit like a tree - it has many branches, but the bird can only land on one branch at a time. That doesn’t mean the act of it landing somehow makes all the other branches magically go away - it means only that for that particular bird, only that particular branch matters at that time, since this is where it interacts with the tree. Once it sits perched there, the other branches have become irrelevant, but not any less real; to the bird it just might look as if the tree has been reduced to that one branch it is sitting on. Prior to landing, there were many branches, each with a certain probability to become the landing spot; but the actual contact itself is always made with only one specific branch, which is determined by how the bird flies his final approach. This changes nothing about either the tree nor the bird, and requires no unknown mechanisms to work. This is of course an imperfect analogy, but you might get my drift. But I would even go a step further - I question the assertion that “a system exposes a causal interface to its environment; therefore, there must be something that possesses such an interface (ie the “real” system)”. I think this does not necessarily follow at all - I see no reason why the causal relationships cannot be the system. There doesn’t have to be anything “underlying” it - reality can just be a network of relationships, rather than “things”.

As Genady quite correctly said - the geometries in exterior vacuum and in the interior of bodies are not the same (in fact, they are quite different), so the trajectories will not in general be the same. What will be the same though is the situation at some distance from the central body - for some exterior and sufficiently distant observer who measures the gravitational effect of the body, there will be no difference between the body being extended and “Gruyère”, or melted down and small and compact (provided the entire situation is spherically symmetric).

It doesn’t have to be on the scale of the BB. In practice, any binary pair of massive enough compact objects will emit (currently) detectable G-radiation - examples would be binary neutron stars, or binary black holes. Also the merger of such objects (and, in the case of black holes, the ring-down phase afterwards) will be a G-wave emitter. I would hazard a guess and say that our G-wave detectors will become more sensitive as time goes by, so eventually we should also be able to detect the gravitational signature of less massive things, like ordinary binary stars. But I think we are very far away from being able to detect anything much smaller than that (like eg oddly-shaped planetary bodies).

Perhaps an illustration would help to clarify the difference between “dipole” and “quadrupole”. In the following, the two black dots in the middle move only vertically, i.e. up-down - so there’s only two “poles” (hence dipole), being the top and the bottom. Like so: https://en.wikipedia.org/wiki/Dipole#/media/File:Electric_dipole_radiation.gif On the other hand, the following shows a situation where the system oscillates up-down and left-right. Here you have four “poles”, being top, bottom, left, right. Hence quadrupole. It’s essentially a combined system of two dipoles. https://en.wikipedia.org/wiki/Gravitational_wave#/media/File:GravitationalWave_PlusPolarization.gif EM radiation is at least dipole, e.g. electrons oscillating up and down in an antenna will generate electromagnetic waves. Gravity requires at least a quadrupole in order to generate gravitational radiation - a system that has only a dipole moment is not enough. Does this make sense?

I wasn’t able to find a visualisation of what you’d see if you were to look out the front window of a relativistic rocket, but I found a visualisation of how your visual field gets distorted at a constant v=0.9c: The accelerated version would be similar, just…well…accelerated. The angular size of your visual field would shrink more and more as you keep accelerating (“tunnel vision”). You would also notice something strange happening behind the rocket - things would at first seem to recede fast, but then they’d appear to slow down and eventually “freeze” into place at some apparent distance, while all the way being redshifted away into invisibility. It’s as if a horizon forms behind your rocket - this is called a Rindler horizon. Looking at the above animation, I don’t know if you would consider the trampoline to be a good analogy for this. Mathematically the coordinates used to model accelerating frames are those that describe hyperbolas, so I guess there is some justification for it.

@Genady: Stephani’s Exact Solutions to the Einstein Field Equations is an absolute must if you want to go above and beyond the basics - not only does it classify the known exact solutions according to different schemes, it also explains the general features of these classes of spacetimes, and their mathematical treatment. The book also gives an overview over which general methods are available to find solutions to the equations, and what forms these solutions may take. So it’s definitely much more than “just” a reference catalogue. But be warned - this is definitely not a beginner’s text, it’s mathematically fairly demanding in places.

Something similar happens in written Thai as well, though to a much more limited extent - there’s specifically two short vowels that are not always written, but implied. That actually caused me a lot of grief when I first started learning.

Yes, good point Yes, with “necessarily” being the key term. It does work sometimes and gives the right intuition; but at other times it doesn’t work. The problem is that generally speaking there is no easy way to tell which is which - that’s why I think it’s dangerous to mix Newton with GR. That’s a difficult to answer question, unfortunately. My own GR understanding came about as the sum total of a large number of different texts, and there’s really no single book that has all the subtleties in it. One text that was instrumental for me personally is one that is unfortunately not available in English (to the best of my knowledge), which is T Fliessbach, Allgemeine Relativitätstheorie. It’s the only text I have seen that explicitly and step-by-step goes through the entire procedure of solving the Einstein equations for Schwarzschild and FLRW from scratch - you can see exactly where sources come into this, how boundary conditions appear, where things like the mass term in the metric originate etc. It found that very illuminating, because it shows how many of our Newtonian intuitions about gravity just don’t hold. But like I said, unfortunately there seems to be no English version. Then there’s of course MTW - ignoring the introductory parts, it’s the only text I know of that goes into the question of why the field equations look like they do. It goes into the topological considerations and conservation laws that underlie the entire machinery of GR, and explains the actual meaning of things like the Bianchi identities very well. I have not seen some of this material in any other text I know of. This is a 1500 pages tome, and evenly divided between introductory topics and more advanced stuff, but I think anyone interested in GR will take something away from this book, irrespective of what level they are on. For the formal maths the gold standard is of course Wald, though to be honest as being an amateur I found it to be above my pay grade. I can see this would be of great benefit to someone who actually has a background in maths, or at least has studied the entire subject matter at university level. It’s a great reference for theorems, proofs etc though. If you can give me some time, I will have to think about your question some more, and have a look through my personal library. It’s been years since I’ve really gone through these books - I’ve taken the key points out and compiled extensive notes for myself, which is what I am mainly working off these days. I might add some more recommendations here later

I think this statement is both false and ultimately meaningless. First of all, the distinction between “simple” and “complex” is not objective, but very much contextual - what’s simple to me might be complex for you, and vice versa. It depends on our respective backgrounds. Secondly, even if the distinction was objective, the statement that you haven’t really understood something unless you can explain it to your child/grandmother etc is clearly misleading - it was never meant to be taken this literally. In my native language, we have a reasonably complicated system of declining nouns and conjugating verbs - according to grammatical case, gender, number, aspect, conditionality, tense etc etc we add prefixes, suffixes, prepositions and postpositions, and sometimes we change the middle part of the word as well, depending on what type of word it is and how it is used. There is also a large number of personal pronouns, never even mind all the irregular verbs, plural forms and so on. There is no way I could quickly and easily explain these things to someone who comes from a different language background, irrespective of what age they are, because some of the basic concepts (e.g. grammatical gender of nouns, or explicit declination by case) simply may not exist in other languages. Of course anyone can - at least in principle - learn the language, but it is generally going to take years of study and practice (unless you are lucky enough to have a mother tongue that is closely related), and not so many foreigners ever become completely fluent in it. It’s just a grammatically complex language, to the extent that even native speakers occasionally have trouble with complex grammatical structures. Does my inability to easily explain these things mean I do not understand the concepts of my native language? Of course not - I understand them perfectly well, to the extent that they are intuitive and self-evident to me. But that doesn’t mean I can easily explain to someone from a different linguistic background why “girl” should be of neutral gender, or why the “me” in “give me it”<>”give it to me” requires different pronouns. Some of these things don’t evenhave explanations, they are just conventions that have organically grown over time, so they have to be acquired, not understood. It’s no different in science - e.g. I understand the concepts of differentiation and integration well enough, I consider these basic operations just like addition and subtraction. But could I explain them to a 5-year old, who has no concept of curves, functions, variables, tangents, limits etc etc, in a way that he’ll actually end up understanding it all? Probably not, unless I’m dealing with an unusually gifted child. But my inability to explain does not mean that I don’t understand, it just means that the child doesn’t have the necessary prerequisites yet to benefit from my explanations. Crucially, my inability to explain it also doesn’t mean that the 5-year old cannot acquire this understanding over time - if you teach him the requisite concepts, there’s no reason why he couldn’t understand differentiation and integration once he’s a bit older. You just start with the basics, and then build onto them. That’s how people acquire new skills. How do you define “fundamental” and “derived”? To me, the most fundamental and most broadly-applicable principle, which underlies a guess-timated 70% or so of all know physics from the Standard Model right up to General Relativity, is the principle of extremal action: \[\delta S=0\] This statement is as simple in form and function as it is powerful, and as close to a “theory of everything” as we have at this point in time. It also does not rely on any particular choice of units or spacetime embedding. To me this is pretty much the bedrock of much of currently known physics, though my feeling is that you probably wouldn’t consider it “fundamental”.

I don’t know, to be honest. In GR, the n-body problem is notoriously complicated, because you cannot ignore the non-linear effects; and once you add angular momentum to the situation, I wouldn’t dare to make any predictions based on intuition. This is a scenario that would require a numerical simulation. Yes, this is true at some distance from the geon - in fact, this spacetime is asymptotically flat, so sufficiently far from the geon the situation becomes completely Newtonian, it looks just like any other gravitating body. But my point was that the geon itself cannot exist in Newtonian theory - it’s just a region of non-trivial spacetime curvature that is held stable purely by non-linear self-interaction effects (there are no other forms of energy-momentum). Newtonian gravity has no such effects, nor does it have any concept of spacetime curvature, so the concept of a “geon” would be meaningless within that theory. You couldn’t describe the geon itself using Newtonian gravity - only its effects on external test particles that are sufficiently distant. It’s even more than that - it’s the question “how much energy-momentum in total does a given region of spacetime contain” that hasn’t got a straightforward answer that all observers can agree on. Fundamentally this is because you need to account for contributions from gravity itself, but in GR these contributions are not localisable, so they are difficult to work with. But in GR this isn’t really a big problem, because this concept of mass is not really used anywhere. You don’t necessarily need a concept of mass in order to solve the field equations, and obtain particle trajectories. This is unlike in Newtonian theory, where you have no choice but to use “mass”, because it explicitly appears in all the equations. The other option is to not look at this in terms of mass at all, but treat it purely as a geometric problem. This is what I tend to do. The advantage of such an approach is that it is pure GR, so it works in all cases, because there is a set procedure you can go through to work out the maths. The disadvantage is of course that even in some seemingly simple scenarios you may not be able to develop a good intuition of what happens, because the geometry is just too complicated to be sure. Your V-shaped arrangement of test particles above is a good example for this. So perhaps we can say that a mass-based Newtonian intuition will work fine in some cases, but some other cases require a fully geometric GR approach. The question is of course how do you know the difference, and unfortunately there’s no general answer to that. For me personally, any spacetime that contains degrees of freedom such as angular momentum, charge, multipole moments, stresses, strains etc etc, I wouldn’t treat using Newtonian-based intuition, because I have been stung with this too many times in the past. This is why - and you might have noticed that over the years - when I answer people’s GR questions on this forum, I am very careful about giving intuition-based predictions about what happens in specific scenarios; in fact I try to avoid doing so altogether, unless I have the mathematical abilities to actually check the prediction, or the scenario is already known and has been analysed in the literature. GR is just a very subtle beast, and it’s easy to be wrong about things.

I can’t really meaningfully comment since this isn’t my area of expertise. But I would like to offer a simple observation that may be relevant - I spent the first few years (up to the end of primary school) of my life in one of the former communist Eastern Bloc countries. Communist ideology was being drilled into people relentlessly and at all times, from kindergarten to primary, secondary and tertiary levels, as well as all public media etc. Outwardly, the vast majority of the populace toed the party line, shouted the right slogans, joined the right political committees, sang along with extra gusto when the national anthem was played, and so on. This is what ensured you were left alone, and could live a quiet life in relative comfort. But behind the scenes it was a different story - I can tell you for a fact that very few people actually genuinely believed all the nonsense the party tried to indoctrinate its citizen with. Nearly everyone we knew back then had hidden antennas in their attic to pick up western TV/radio; nearly everyone got clandestine “care packages” with well-concealed goods from relatives in the West; nearly everyone managed to get their hands on books that were technically blacklisted. It was pretty much an open secret. People outwardly played along out of fear, or simply in order to be left alone and live in comfort, but very few were “actual” fervent communists (though of course these did exist). In the end, how many genuine (=not organised by the party) mass protests were there in favour of keeping the Iron Curtain drawn shut? And that was before Internet and means of mass communication. So what I would suggest is that just because people are exposed to attempts at indoctrination of some kind of another does not at all mean that they will actually pick up these doctrines and make them their personal belief systems. They might outwardly play along with them, but that’s not the same thing at all. We’re not just unthinking sponges that unquestioningly soak up everything we are confronted with - human beings are much more complex than that, and that’s true even at a young and impressionable age. I think that’s too often forgotten. Hence, especially in the age of the Internet and easy access to information, banning books from schools and library is a nonsensical and ineffective waste of time, in my opinion. I would conjecture that the influence of upbringing and general social environment is far greater than anything that might (or might not) be taught in schools.

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.