md65536

But the geometric lengths of the world lines OP describes between AB (where A and B pass) and BC is 4 years, between BC and AC is 4 years, and between AB and AC is 10 years, calculated using special relativity. Are you getting a different answer, or not able to get that answer, or are you proposing a different cause other than relativity that is giving you that answer?

But what we're talking about extremizing here is the proper time over the whole world line, which is the integral you posted. The "given curve" that we're extremizing is all the possible proper times of nearby world lines. If the freefall world line passes through sections where proper time is maximized, and sections where proper time is minimized, is the whole world line a maximum or minimum? If you can nudge the world line in one section and increase the proper time of that section, then the whole is not a maximum (neither local nor global). If you can nudge it in another section and decrease the proper time of that section, then the whole is not a minimum.

OP's description still has 3 clocks passing by each other. What physically changes? It's not cheating to set initial conditions to make measurements easier. Only one scenario is described, ie. 4 years of proper time measured by C at a speed of .6 c relative to A. Whether clock C reads 0 or 4 years or 500 when it meets B, it will read 4 years later when it meets A. You get the same answer either way; the path from AB to BC to CA is 8 years long, regardless of how the clocks are set. Also regardless of how many clocks are used to measure sections of that path. Can you show how to apply turnaround acceleration using the experiment described by OP?

This statement seems to be causing confusion and is unnecessary. Wikipedia instead calls it 'at the point corresponding to "turnaround" of a single traveller,' which emphasizes that nothing turns around. "there is a change in the frame of reference" sounds like a description of something physical happening. Nothing physical changes frame of reference. There's only a change in the frame of reference that we're considering, and it doesn't have to be described as a change. Instead maybe something like, "for the inbound leg, we're considering a different reference frame." That doesn't change the experiment that you described. What changes reference frame? Nothing physical that you've described. Another way to emphasize that nothing changes at the BC meeting point, is to not set clock C there at all. One way to do this is to set C beforehand so that it will already have the time you want (ie. same time as B) when it passes B. Another way is to just add up the proper times of clocks B and C clock between the events, without even mentioning or caring what they were set to. I see it as using 3 clocks to describe the differential aging along two different world lines, which is the heart of the twin paradox. You could have a single physical object follow the same path, and it would be a true twin paradox experiment. If you run OP's experiment alongside that, OP's is a way to measure a twin paradox experiment. It *is* a measure of the ageing in a twin paradox experiment.

I read it again and I think it's exactly what OP posted. What difference do you see in our descriptions?

3 clocks pass by each other at 3 events. What changes frames?

This all looks okay to me. This has been discussed in the past and someone referenced a paper written in the 1900s that describes the same experiment, but I can't find it quickly. Wikipedia has references, see https://en.wikipedia.org/wiki/Twin_paradox#Role_of_acceleration I don't like using the word "synchronized", but it's used in wikipedia. You must be careful because none of the clocks remain synchronized for any length of time, which I think does not satisfy the definition of synchronized. I'd just say the clocks are set the same. The path length that you describe, between meetings of AB, BC, and CA, has an invariant length of 8 years. This is true whether or not a real object follows it. If you compare path lengths, you'll find that what you're describing is true; one path length between AB and CA is 8 years, and another is 10 years. Describing things in terms of path length is incontrovertible. However even if you do this, people will disagree.

What is accelerating? For how long? Is it an infinite acceleration? Where is the acceleration accounted for in formulas? Do you have acceleration every time you consider a new frame of reference? This has been done countless times with the relevant formulas. This description is sufficient because it's a description of the result of those formulas. Here v = .6c and gamma = 1.25, anyone who properly applies the formulas will find that the description checks out. The proper time between when A and B meet, and B and C meet, plus the proper time between when B and C meet, and C and A meet, is 8 years.

I'm not sure if that's helpful, relating to maximal ageing. It's a very simple example but it's obvious the math becomes very complicated quickly with other examples. If you're using x as a representation of time so that you can speak of world lines, and then proper time would be an integral of the square root of dx^2 - dy^2, so that curved lines have a lower value than comparable straight lines, and line slope is less than 1 (analogous to speed of light) so that only positive proper times are considered, then that all makes sense. I just noticed that the Taylor/Wheeler reference above has more than one definition of the principle of maximum ageing for GR, so I'll give a later one: "Gravitation" mentions geodesics analogous to "mountain passes" as an example of a path of minimal aging, where those analogous to mountain peaks are maximal. That it's a minimum is due to the spacetime geometry, not due to how you choose the signs in your metric. As a very rough analogy, if you're making a straight-line (on a map) trip across a mountain, the longest possible trip is over the peak. If you're making a straight-line trip between two mountains, the shortest possible path is over the saddle-point of the mountain pass. I think a real-world example relating to extremal ageing, is a free-fall path that passes straight through the middle of two identical gravitational masses. If that's an example of a saddle point, then it's possible that this is a path of least ageing. (Edit: I don't think that's an example of minimal ageing. Instead just assume there's some path of minimum ageing.) Nudging a stone off this path, and then back onto the original world line, would only increase the ageing of the pushed stone. If the principle of maximal ageing still holds, it might do so only because there is no other path along pairs of adjacent local frames. I'd guess that means, if it's a path of least ageing and also a path of max ageing along pairs of local frames, then any deviation from the path involves effects of spacetime curvature that make it not a local inertial frame. But that seems quite suspicious. I think their definition simply doesn't apply to all free-fall world lines. As well, since the world line can be extended into other regions of spacetime apart from the "saddle-point", it must be possible for the same world line to pass through sections of max ageing and sections of min ageing. This seems like a problem for both Taylor/Wheeler's definition and Misner/Thorne/Wheeler's: "tau = [math] = (an extremum for timelike world line that is straight in each local Lorentz frame along its path, as compared to any 'nearby' variant of this line)," (Gravitation, p. 316), because both define the proper time over the entire world line as a maximum or extremum. Here I'm posting before understanding it enough, but I don't see how that problem can be resolved with the given definitions. Yes, in M/T/W's example, a star falling through the plane of a disc galaxy ("Gravitation", Fig. 13.2), vs being thrown up from the plane, could meet later. Both are geodesics but with different initial conditions. In my example, a stone falling straight between two masses could be met by a stone orbiting around them, both in freefall with different initial conditions.

The thrown stone is in freefall and ages the most among any nearby paths. What I meant by the first statement is that you can find other arbitrary non-freefall paths that age less than the stone sitting on top of the mountain, even one that includes sections of freefall as per OP's example. Eg. if you dropped a stone off the mountain in freefall, but then brought it back up so that you could make a definite comparison of their proper times, it could have aged less than the stone on the mountain, depending on how you do it (but this example involves a non-freefall path and doesn't violate the principle of max ageing). Also... technically nothing's stopping you from comparing two distant (in space and/or time) worldlines any way you want to, and coming up with different answers, but that's not going to violate the principle either. If it's worth it for you; exposure to more maths would help me. One thing I was stuck on is the idea and meaning of globally non-unique geodesics. A wiki page cites Misner/Thorne/Wheeler's Gravitation, p. 316 and I looked it up and it seems to contradict a couple of things from this thread. However I don't want to bring up the details until I understand it better, which I might not do. I doubt it affects the basic understanding of all this.

Alright, I have no further dispute and I think I'm getting hung up on details I don't understand. But back to the original problem, which is that a stone fixed on the top of a mountain will age more than some others, even though it isn't in freefall. However, another stone thrown straight upwards from the top of the mountain, with any velocity that keeps it nearby (no escape velocity etc.), will return to the same spot. Having a freefall path, it will have aged more than the fixed stone did between the throw and the landing. So you can vary the initial conditions and get different geodesics and let that determine the events A and B between which you're comparing different path lengths. Or you can fix A and B and get a locally unique geodesic and unique initial conditions. Another thing the principle doesn't mention is the notion of the two events A and B. All of these details are not aspects of the principle, they're just things we're using to properly specify a particular case that we're applying the principle to.

Feel free to bring us back to the original examples etc., I've kind of gone off into possibly irrelevant details. No. I don't know if you're trying to say, paraphrased, "A given free particle follows a path of maximum ageing, and a given particle has initial conditions." Which is obviously redundant. Or, are you saying "A given free particle follows a path of maximum ageing among all possible nearby paths with the same initial conditions." I can very roughly prove that this is redundant if you doubt it. Either way, the principle doesn't say anything about initial conditions and it doesn't imply that its application depends on initial conditions. I feel like we're having a conversation like this: P1: Given some initial conditions, the velocity of a particle is dx/dt. P2: That doesn't depend on initial conditions. P1: Do you understand that a specific particle that has velocity will have initial conditions? Otherwise, I'm missing the point of why you're talking about initial conditions, especially since limiting the initial conditions will exclude some of the applications of the principle already discussed in this thread.

My experience has been that yes, experts tend to correct their own errors when they're realized, and that it requires an expert to correct an expert (even if it's the same person) because expert replies are far more trusted and accepted than others, and that experts tend not to acknowledge errors in other experts' posts. I agree that a good way to disagree is to ask a question (like, aren't the number of photons sent and received the same?) because even when you're wrong, you're not wrong in asking! But too much stuff like "I need time to understand" says you think the problem is with yourself and that there's no reason for the other person to reconsider what they wrote. When other people read that, and compare a reply labelled "Expert" to one that says "I don't understand", it's easy to brush off the latter.

That means if you're talking about the galaxy as an object moving away from you through flat spacetime, then the light energy it emits and the light energy you receive will be the same in a given frame of reference? But if you're talking about spacetime expansion, you can't treat the source and receiver as having a common frame of reference (or can you speak of a global frame of reference for other laws etc.)?

Digest it? You mean the answer given was wrong and you're pretty sure of that? Is there some unwritten rule on this site that it's poor etiquette to disagree with a resident expert? If people ask questions and see a wrong answer from a resident expert, and no one authoritative corrects it, they tend to trust the wrong answer. 1. They don't, which you know. 2. That's right. 3. I don't know, hopefully someone else does. Photons have kinetic energy, which is different in different frames. There's no conservation of an object's energy between different frames. Where conservation of energy comes in is... in either frame, the galaxy loses the energy of each photon that leaves (ie. different in the 2 frames), and I guess is pushed in the opposite direction of each photon a tiny amount (different in each frame). The red-shift also lowers the intensity of the light (or the number of photons received per your second), so the energy you receive is less than 100 joules (I think you receive 100/n^2 watts for n seconds, 100/n joules total?).

"Rest" and acceleration are frame-dependent. A stone thrown upward momentarily comes to rest without proper acceleration. I disagree with the general characterization of the principle you're using. It's not something that applies only in the simplest cases, it always applies. So that includes a free-falling particle with a world line billions of years long, falling past countless moving masses. It can come to rest many times. The principle doesn't say anything about initial conditions, and it doesn't have to because it still applies in all cases. The only restriction is it can't be applied to "distant" (non-adjoining) spacetime patches. To try to paraphrase Taylor/Wheeler ("The Principle of Maximal Aging says that a free stone follows a worldline through spacetime (flat or curved) such that its wristwatch time (aging) is a maximum across every pair of adjoining spacetime patches."): The principle only applies without restriction in "flat enough" spacetime, but it can be applied to an arbitrarily complicated (curved?) free-fall world line by dividing the world line into small enough sections that pass through flat-enough spacetime patches, and applying the principle to each of those sections. If it's complicated enough, there may be other paths that involve greater aging (such as the multiple orbits examples I've given above), but those necessarily involve paths across spacetime patches that are not adjoining a patch through which the world line in question passes. (I think that's what it's saying.)

I'm describing cases where you have two freefall paths that pass through the same pair of events, A and B. A trivial example would be two particles in similar circumpolar orbits leaving together above the north pole and meeting again above the south pole. A more useful example is two particles in eccentric orbits of different sizes, and they meet at one's perihelion and the other's aphelion, and the particle in the smaller orbit makes two orbits for every one of the larger. Since the particle in the larger orbit makes an orbit at lower speeds (in the gravitational mass's reference frame, say) than the one in the smaller orbit, and is also at a higher gravitational potential, it must age more than the one in the smaller orbit. Therefore the principle of maximal aging cannot truthfully say "If a particle traveling between events A and B is in free fall, then its aging is greater than any other path between A and B." I've tried finding the actual definition of the principle, and found several variations, including many like the above which I think are false. I've also seen, "the aging is greater than any other nearby path" which is true, and "The path of maximal aging between A and B is a geodesic", which is true. A possible problem is they're assuming that a geodesic between A and B is unique, when really they can only assume that it is locally unique? The closest to definitive I can find is from Taylor and Wheeler's "EXPLORING BLACK HOLES Introduction to General Relativity Second Edition": That certainly excludes my example. It seems that free fall aging is maximal among "nearby" paths, and the caveat is necessary. A lot of web pages describing the principle of maximal aging are leaving it out and mislead me to the incorrect conclusion that any freefall paths between events A and B will have maximal aging among all possible paths between A and B. Yes they must pass through the same events to compare them, but they definitely don't have to be at rest. If two world lines intersect, that's a single event, regardless of the objects' velocities. If a world line passes through a given event, it does so in every frame of reference.

How is more time accumulated near the massive object? You've just stated that the far away clock appears to go faster. "Accumulation of time" would refer to proper time. Does the principle imply that two objects in different freefall orbits that intersect at two events, must age the same amount between the two intersections? One could not have maximal aging along one freefall path between the two events, yet have the other age more, right? If so, then you could have one clock orbiting a massive object several times at a fixed radius, while another clock orbits once, starting at the same radius but traveling far away from the mass before returning. Both are in freefall, both can start and end together. The "escaping" clock would need a faster initial speed, and would "age less" due to SR time dilation, but would also "age more" while having higher gravitational potential. Do they necessarily age the same between events where they meet? Or are there other caveats or restrictions to the principle? My intuition is that you could make the eccentric orbit so far away and so slow at aphelion that it would have to age more, but the principle seems to say that's wrong.

The principle concerns the path from A to B (two given events on a world line), it doesn't tell you where B must be. Other things tell you that. Eg. drop a stone from rest while standing on Earth, and it will fall downward. Or throw it upward, it will fall upward for some time. B will generally be different in these cases. Either way, if you take two points A and B on the stone's world line, the freefall path between A and B has the greatest proper time of all possible (including non-freefall) paths between A and B. For example, if A is some point on Earth, and B is the same location a few seconds later, a stone thrown upward so it passes through A and B in freefall will age the most. A fly that takes off from A and lands again at B will age less than the stone. A clock sitting on the ground at A (and B) will age less than the stone.

This being a relativity thread, it could, and generally would, be greater than that. How do you figure it would be greater? You must have conservation of mass+energy. Since we're talking about only mass, the energy of an object or a system of "constituents" wouldn't matter (and is unspecified anyway). I thought maybe you meant there is some new particle that only exists when other particles are combined, like maybe gluons, but those are likely massless. For your statement to be true, I figure either you'd have to have different constituents when an object is considered as a whole vs as parts, or you'd have to change the mass of individual constituents??? What makes up the difference in mass?

No, I never said anything was stationary (I said "a mass" in a stationary system doesn't emit gravitational waves). I went on to explain how Earth contributes to radiated changes in the gravitational field. My point is you can't look at the Earth all by itself and assume it must be making changes. It seems like you're ignoring parts of answers that go against what you say 'you'd have thought', and just repeating your original position. Do you understand that it's asymmetric acceleration that emits gravitational waves? Why not? All it means is putting all of spacetime into an ordered set of hypersurfaces, right? The problem (if I understand it) is that you can do it, in an infinite number of ways, basically requiring an arbitrary choice of decomposition that's not universally meaningful. (Or to try to put it in simple terms, you can arbitrarily define single moments throughout the universe that each include only events that are space-like separated, but your choice won't meaningfully represent a single moment elsewhere.) If it wasn't possible, what would that mean? That a foliation can't include all of spacetime? Or that some space-like hypersurfaces must intersect (which seems to go against causality)?

Sure, why not? A mass doesn't emit gravitational energy just by being there, though, so it doesn't "evolve" in a stationary system. The Earth's not perfectly symmetric, so by simply spinning, it emits a tiny amount of energy as gravitational waves (negligible?). Changes in the gravitational field are more due to Earth's interaction with other masses. For example, according to the gravitational wave wiki, the total energy of the Earth orbiting the Sun loses about 200 watts emitted as gravitational waves. This is an extremely small fraction of the total energy. Regarding your original question, there isn't a decrease in mass in this case, because the energy comes from the kinetic energy and gravitational potential energy of the system, ie. decay in orbit.

See https://en.wikipedia.org/wiki/Gravitational_wave Read the section "Sources", as that explains some cases where gravitational waves are emitted (or "in most cases" aren't). In particular, in the case of black hole mergers, a lot of mass can be lost as gravitational waves. This comes from the extreme acceleration of the masses rotating around each other. That's about 5% of the mass, radiated away. But once merged, the mass stops accelerating and losing mass to gravitational waves, as the animations and charts show.

There are lots of consistent ways to describe what's going on here. See http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html (your video is the second half of a twin paradox experiment so all the analyses apply). Your A and B are symmetrical so describing one describes both. Using Doppler shift analysis, what they observe is: The other's clock appears far behind at the start, and when I move toward it, their clock appears to run faster than mine the entire time until it catches up only when we meet. If you include relativity of simultaneity in your calculation of the other's rate of time, then depending on your acceleration you can describe it as above: the other runs faster during your acceleration, then slower while at higher velocity. Or, you can separate time dilation and relativity of simultaneity and say that the other's clock only runs slower the entire time they're moving (as the video suggests), but the acceleration involves a change in inertial reference frame and corresponding change in relative simultaneity, and the clocks are no longer in sync when they start moving. In this case, our clocks are sync'd while we're at rest, but when I accelerate towards you, events where I am are now simultaneous with events at the other clock where the other clock is far ahead of mine, and I spend the rest of the trip catching up to it. Any analysis works and they're all consistent, but to see it (and believe it) you're probably going to have to do the maths. You haven't calculated anything of what A and B would observe, and yet you're finding imagined paradoxes that aren't in the calculations. So start calculating!, and you'll find exactly where the problem lies (where the results don't add up) and then you'll be able to learn the missing pieces that solves the paradox. If it's your first time with the maths, keep it simple and consider only two periods of inertial motion (when they're at rest, and when they're approaching each other after a quick acceleration). I'm sure you'll get lots of help at every step. Just remember there are a lot of different ways to describe it, and you might not believe they're consistent until you see it in your calculations.

I don't understand why you would write out all those equations when they don't even relate to the case described. Even the classic twin paradox is typically presented with negligible acceleration time, or equivalently considering only 2 legs of inertial motion for the "traveling" twin. Adding in an acceleration phase makes it more complicated, and doesn't help in figuring out the simpler version of the paradox. Why complicate it if a simpler version is specified?