Re: [Vo]: FW: Einstein's Twin Paradox
Harry: The raison d'être of GR is to explain gravity. Stephen: That's right. But you don't need it to resolve the twins problem, which takes place in flat space. I am confused. In your first response to me you started off by saying the opposite: Harry: That works in SR, but the solution is inconsistent with GR. Stephen: Wrong. In fact the full solution can only be had using techniques commonly considered to be part of GR. Harry
Re: [Vo]: FW: Einstein's Twin Paradox
On 2/17/07, Stephen A. Lawrence [EMAIL PROTECTED] wrote: An accelerometer is a purely local instrument (which, of course, can't tell the difference between gravity and acceleration). Actually there is a way, or technically 2 ways at least. (besides the fact that experiments have shown that things don't all drop at the same speed meaning that there is a difference between inertial force and gravity) One way is to measure the difference at the floor and ceiling (typically this thought experiment takes place in an elevator).and measure the difference as gravity is of course going to be stronger at the bottom, where a constant acceleration will be equal at each end. The other way is to measures the curvature of the gravity field (measure it's convergence/divergence). But the more important hole is that in real world experiments it is found that things can drop at very different speeds, for instance an iron sphere and a carbon sphere both of the same weight, the carbon sphere will fall faster despite being much larger hence having greater drag. In another case Don A. Kelly, a Free Energy researcher made a device which consisted of a bread board with a bunch of magnets layed out somehow, this would drop something like 1/3rd slower that it should. On 2/17/07, Michel Jullian [EMAIL PROTECTED] wrote: But enough bickering. Talking about centrifugal force, you do know that by running around a bucket of water you incurve the water as if it was centrifuged don't you? :) Ok, I just tried it, I ran really really fast and you are wrong ;) Funny, now I know your a girl I feel bad about stuff I previously said ;) There are far far too few female interested in science, physics especially and alt science most of all.
Re: [Vo]: FW: Einstein's Twin Paradox
- Original Message - From: John Berry [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Saturday, February 17, 2007 9:18 PM Subject: Re: [Vo]: FW: Einstein's Twin Paradox ...Talking about centrifugal force, you do know that by running around a bucket of water you incurve the water as if it was centrifuged don't you? :) Ok, I just tried it, I ran really really fast and you are wrong ;) Anyway, the water did really curve through the effect of the gravitational waves you emitted, only the effect was too small for you to see. If your mass had been comparable to that of the universe, the effect would have been quite noticeable. In fact if you had been the universe itself, the bucket would have thought it was being centrifuged wrt you. Michel (without an e at the end, enough of this)
Re: [Vo]: FW: Einstein's Twin Paradox
Harry Veeder wrote: Harry: The raison d'être of GR is to explain gravity. Stephen: That's right. But you don't need it to resolve the twins problem, which takes place in flat space. I am confused. In your first response to me you started off by saying the opposite: Harry: That works in SR, but the solution is inconsistent with GR. Stephen: Wrong. In fact the full solution can only be had using techniques commonly considered to be part of GR. Err hmmph. Well. What I meant by that is this Gravity, and the attendant curved space, is unnecessary to the solution to the twins problem. However, if you want to solve the problem from the point of view of the traveling twin, without neglecting acceleration (i.e., without assuming velocity changes happen instantly), then you need to use techniques which are commonly treated as part of general relativity, rather than special relativity. If one is willing to do the actual calculations using the frame of reference of the stationary twin, however, then there's no need to integrate over a curved path with a non-Minkowski metric, and we can solve it, acceleration included, using techniques from SR. See, for example: http://www.physicsinsights.org/accelerating_twins.html In case that was all insufficiently murky, let me add to the confusion by explaining that the boundary between GR and SR has been a bit of a moving target over the years. Initially SR dealt only with inertial frames (no acceleration), and in fact the basic postulates of SR don't really say what happens during acceleration. However, if we add the _assumption_ that clocks are not affected by acceleration, then at the expense of some additional complexity in the math we can solve problems such as this one entirely within the extended version of SR, in either frame of reference. From that point of view, anything _except_ gravity is in the bailiwick of SR. Finally, I should mention that there is a somewhat surprising error on my accelerating twins page, linked to above. (I don't mean it's surprising that there's an error; rather, the particular error is surprising!) At the bottom of the page I mentioned that I hadn't yet done the porthole view from the ship but asserted that it contains no new surprises. That's utterly wrong -- the porthole view from the ship is extremely surprising, as I found when I started to work it out. Someday I may get around to putting together an integrated page on all this, showing how the pieces fit together; it's not as straightforward as it appears at first glance. Here's a partial writeup: http://www.physicsinsights.org/porthole_view_1.html As someone said when I mentioned the effects discussed on that page, Oh, that's just aberration!. Well, yeah, it is aberration -- but I wouldn't have said just aberration; I think it's highly weird... Harry
Re: [Vo]: FW: Einstein's Twin Paradox
- Original Message - From: Stephen A. Lawrence [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Friday, February 16, 2007 3:37 AM Subject: Re: [Vo]: FW: Einstein's Twin Paradox ... This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. ... To be more precise the traveling twin is the only one who accelerates _wrt the initial common frame of reference_, that's what breaks the symmetry (otherwise one could argue that they both accelerate wrt each other) No you could not. Acceleration is absolute, not relative. ... Not in the general sense Stephen. _Geometrically_, both twins accelerate wrt each other, agreed? It's _acceleration wrt an inertial frame of reference_ which is absolute of course, hence my point. I wasn't contradicting you, just highlighting a point which may not be obvious to everyone. Michel
Re: [Vo]: FW: Einstein's Twin Paradox
Michel Jullian wrote: - Original Message - From: Stephen A. Lawrence [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Friday, February 16, 2007 3:37 AM Subject: Re: [Vo]: FW: Einstein's Twin Paradox ... This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. ... To be more precise the traveling twin is the only one who accelerates _wrt the initial common frame of reference_, that's what breaks the symmetry (otherwise one could argue that they both accelerate wrt each other) No you could not. Acceleration is absolute, not relative. ... Not in the general sense Stephen. _Geometrically_, both twins accelerate wrt each other, agreed? You are talking about what we might call coordinate acceleration, which, I would claim, is a somewhat nonstandard use of the term acceleration. It's _acceleration wrt an inertial frame of reference_ which is absolute of course, hence my point. I wasn't contradicting you, just highlighting a point which may not be obvious to everyone. Acceleration, as I have generally seen the term used in casual conversation (and in discussions of the twins paradox), is that which is measured by an accelerometer. An accelerometer is a purely local instrument (which, of course, can't tell the difference between gravity and acceleration). (d/dt)(dq/dt) where q is an arbitrary general coordinate is not usually referred to simply as acceleration. And, when the word acceleration /is/ used that way, it often leads to interminable pointless arguments about the difference between a real force and a fictitious force, as well as lengthy discussion of the true meaning of centrifugal force :-) Michel
Re: [Vo]: FW: Einstein's Twin Paradox
You are talking about what we might call coordinate acceleration Yes indeed Stephen, we might call it thus, although just acceleration is better and simpler. Time derivative of velocity in an arbitrary frame, not just an inertial one. That's the most general definition of acceleration, that's why I said Not in the general sense: ac·cel·er·a·tion n. 1. a. The act of accelerating. b. The process of being accelerated. 2. Abbr. a Physics The rate of change of velocity with respect to time. But enough bickering. Talking about centrifugal force, you do know that by running around a bucket of water you incurve the water as if it was centrifuged don't you? :) Michel - Original Message - From: Stephen A. Lawrence [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Friday, February 16, 2007 4:09 PM Subject: Re: [Vo]: FW: Einstein's Twin Paradox Michel Jullian wrote: - Original Message - From: Stephen A. Lawrence [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Friday, February 16, 2007 3:37 AM Subject: Re: [Vo]: FW: Einstein's Twin Paradox ... This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. ... To be more precise the traveling twin is the only one who accelerates _wrt the initial common frame of reference_, that's what breaks the symmetry (otherwise one could argue that they both accelerate wrt each other) No you could not. Acceleration is absolute, not relative. ... Not in the general sense Stephen. _Geometrically_, both twins accelerate wrt each other, agreed? You are talking about what we might call coordinate acceleration, which, I would claim, is a somewhat nonstandard use of the term acceleration. It's _acceleration wrt an inertial frame of reference_ which is absolute of course, hence my point. I wasn't contradicting you, just highlighting a point which may not be obvious to everyone. Acceleration, as I have generally seen the term used in casual conversation (and in discussions of the twins paradox), is that which is measured by an accelerometer. An accelerometer is a purely local instrument (which, of course, can't tell the difference between gravity and acceleration). (d/dt)(dq/dt) where q is an arbitrary general coordinate is not usually referred to simply as acceleration. And, when the word acceleration /is/ used that way, it often leads to interminable pointless arguments about the difference between a real force and a fictitious force, as well as lengthy discussion of the true meaning of centrifugal force :-) Michel
Re: [Vo]: FW: Einstein's Twin Paradox
Harry Veeder wrote: Gotta love those probabilities. With them you can save relativity from obscurity. Harry Professor Resolves Einstein's Twin Paradox Science Daily http://www.sciencedaily.com/ — Subhash Kak, Delaune Distinguished Professor of Electrical and Computer Engineering at LSU, Specializing in relativity, which is a branch of physics? Tensor calculus doesn't see a lot of use in CS/EE departments, FWIW. recently resolved the twin paradox, known as one of the most enduring puzzles of modern-day physics. First suggested by Albert Einstein more than 100 years ago, the paradox deals with the effects of time in the context of travel at near the speed of light. Einstein originally used the example of two clocks -- one motionless, one in transit. He stated that, due to the laws of physics, clocks being transported near the speed of light would move more slowly than clocks that remained stationary. In more recent times, the paradox has been described using the analogy of twins. If one twin is placed on a space shuttle and travels near the speed of light while the remaining twin remains earthbound, the unmoved twin would have aged dramatically compared to his interstellar sibling, according to the paradox. This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. The paths close -- the twins meet again -- which can't happen unless at least one of the twins accelerates. Invoking Mach's principle to try to resolve this seems kind of silly, since the paradox (which is not a paradox, anyway) exists in the SR model of the world even when there are just two items in the model: two clocks. You don't need to bring in the fixed stars to state the problem, nor to resolve it. What's more, if you start with a planet that's in motion with respect to the fixed stars and a traveler who stops relative to the fixed stars, you find find yourself back where you started -- Mach's principle is kind of useless in general, IMHO, in that it never really explains anything, even when it seems to. If the twin aboard the spaceship went to the nearest star, which is 4.45 light years away at 86 percent of the speed of light, when he returned, he would have aged 5 years. But the earthbound twin would have aged more than 10 years! said Kak. The fact that time slows down on moving objects has been documented and verified over the years through repeated experimentation. But, in the previous scenario, the paradox is that the earthbound twin is the one who would be considered to be in motion -- in relation to the sibling -- and therefore should be the one aging more slowly. Einstein and other scientists have attempted to resolve this problem before, but none of the formulas they presented proved satisfactory. This is false. Kak's findings were published online in the International Journal of Theoretical Science, and will appear in the upcoming print version of the publication. I solved the paradox by incorporating a new principle within the relativity framework that defines motion not in relation to individual objects, such as the two twins with respect to each other, but in relation to distant stars, said Kak. Using probabilistic relationships, Kak's solution assumes that the universe has the same general properties no matter where one might be within it. The implications of this resolution will be widespread, generally enhancing the scientific community's comprehension of relativity. I doubt that a whole lot. It may eventually even have some impact on quantum communications and computers, potentially making it possible to design more efficient and reliable communication systems for space applications. Note: This story has been adapted from a news release issued by Louisiana State University.
Re: [Vo]: FW: Einstein's Twin Paradox
Stephen A. Lawrence wrote: This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. That works in SR, but the solution is inconsistent with GR. harry
Re: [Vo]: FW: Einstein's Twin Paradox
Harry Veeder wrote: Stephen A. Lawrence wrote: This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. That works in SR, but the solution is inconsistent with GR. Wrong. In fact the full solution can only be had using techniques commonly considered to be part of GR. Acceleration is acceleration, in either SR or GR. In either case you integrate the path length, using the pseudo-Riemannian metric of Minkoski, in order to find the elapsed proper time of either twin, and in either case the path which includes the acceleration comes out shorter. A geodesic in GR is the longest distance between two points. Acceleration pushes you off the geodesic, as a result of which you follow a shorter path. If the two twins have worldlines which cross at two points, and if one accelerates while the other follows a geodesic, the one on the geodesic will age more. That's straight out of GR ... or SR, take your pick. If both accelerate, then neither follows a geodesic and you need to know the details of the problem to determine who ages more (if either). GR and SR only really differ when you introduce gravity, which doesn't enter into this problem. harry
Re: [Vo]: FW: Einstein's Twin Paradox
- Original Message - From: Stephen A. Lawrence [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Thursday, February 15, 2007 10:33 PM Subject: Re: [Vo]: FW: Einstein's Twin Paradox ... This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. ... To be more precise the traveling twin is the only one who accelerates _wrt the initial common frame of reference_, that's what breaks the symmetry (otherwise one could argue that they both accelerate wrt each other) Michel
Re: [Vo]: FW: Einstein's Twin Paradox
- Original Message - From: Robin van Spaandonk [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Thursday, February 15, 2007 4:22 PM Subject: Re: [Vo]: FW: Einstein's Twin Paradox In reply to Harry Veeder's message of Thu, 15 Feb 2007 15:43:03 -0500: Hi, [snip] I solved the paradox by incorporating a new principle within the relativity framework that defines motion not in relation to individual objects, such as the two twins with respect to each other, but in relation to distant stars, said Kak. [snip] ..IOW by throwing out the concept of relativity and introducing a special absolute frame of reference. :) Funny, everything always seems to come back around to that. But its not an absolute frame of reference...no, just the distant stars... What does distant stars _really_ mean? This is a nice case of, we have a problem that makes us unhappy, so lets just move the problem way away, and say that it is solved, since it is now too far away to trouble us...out of sight, out of mind. Even worse is the 9 billion names for vacuumDirac sea, Spacetime, Space, physical vacuum, ZPF, ether, aether (to use the archaic spelling), ad infinatum ad tedium ad nauseam. So whats for breakfast? --Kyle
Re: [Vo]: FW: Einstein's Twin Paradox
Distant stars are not out of sight fortunately :) Nothing wrong with the concept, except it is not needed to solve the problem at hand, so the alledged discovery is caput mortuum. Laplace : Sire, I had no need of that hypothesis -- Michel - Original Message - From: Kyle R. Mcallister [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Friday, February 16, 2007 1:43 AM Subject: Re: [Vo]: FW: Einstein's Twin Paradox - Original Message - From: Robin van Spaandonk [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Thursday, February 15, 2007 4:22 PM Subject: Re: [Vo]: FW: Einstein's Twin Paradox In reply to Harry Veeder's message of Thu, 15 Feb 2007 15:43:03 -0500: Hi, [snip] I solved the paradox by incorporating a new principle within the relativity framework that defines motion not in relation to individual objects, such as the two twins with respect to each other, but in relation to distant stars, said Kak. [snip] ..IOW by throwing out the concept of relativity and introducing a special absolute frame of reference. :) Funny, everything always seems to come back around to that. But its not an absolute frame of reference...no, just the distant stars... What does distant stars _really_ mean? This is a nice case of, we have a problem that makes us unhappy, so lets just move the problem way away, and say that it is solved, since it is now too far away to trouble us...out of sight, out of mind. Even worse is the 9 billion names for vacuumDirac sea, Spacetime, Space, physical vacuum, ZPF, ether, aether (to use the archaic spelling), ad infinatum ad tedium ad nauseam. So whats for breakfast? --Kyle
Re: [Vo]: FW: Einstein's Twin Paradox
- Original Message - From: Michel Jullian [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Thursday, February 15, 2007 8:10 PM Subject: Re: [Vo]: FW: Einstein's Twin Paradox Distant stars are not out of sight fortunately :) Depends on how close to the rather light pollutive city of Buffalo you are. ;) Nothing wrong with the concept, except it is not needed to solve the problem at hand, so the alledged discovery is caput mortuum. Laplace : Sire, I had no need of that hypothesis Hmmm...I also wonder what the alleged effect of this on quantum computers and communications will be, as the article stated. (I am being sarcastic...) It seems as if every time something is discovered it is said that it will have a big implication for quantum XYZ Quantum seems to be the new Ain't...its a word that is hard to define precisely, is used in many places where it doesn't belong, and in some circles you don't fit in with the crowd unless you use it. I'm going to go shovel the snow off my ~100 ft long driveway. I wonder if it will have important future implications for quantum computers? --Kyle
Re: [Vo]: FW: Einstein's Twin Paradox
Michel Jullian wrote: - Original Message - From: Stephen A. Lawrence [EMAIL PROTECTED] To: vortex-l@eskimo.com Sent: Thursday, February 15, 2007 10:33 PM Subject: Re: [Vo]: FW: Einstein's Twin Paradox ... This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. ... To be more precise the traveling twin is the only one who accelerates _wrt the initial common frame of reference_, that's what breaks the symmetry (otherwise one could argue that they both accelerate wrt each other) No you could not. Acceleration is absolute, not relative. This is where Mach's principle starts looking totally silly. In a situation without gravity (which is what SR deals with) drop a ball. Does it fall? If so, then you're accelerating. You can perform this test without looking out a window or examining anything outside your own laboratory. That is what I mean when I say it is absolute -- either you _are_ accelerating or you are _not_, and simple tests can determine the difference (to a specified level of accuracy, of course). When you transform to accelerated coordinates many things change, including the metric. Neither Galilean relativity nor Einstein's relativity ever tried to pretend that accelerating and inertial frames were in all ways identical. Inertial coordinates are special, and that's what puts the SPECIAL into special relativity -- it's the limited form of the theory that applies to inertial coordinates. As it happens it can be applied to accelerating frames as well just by extending the math a bit, but to take the jump to include gravitation you need to introduce curvature and the field equations, and that's when special relativity is left behind. Michel
Re: [Vo]: FW: Einstein's Twin Paradox
Stephen A. Lawrence wrote: Harry Veeder wrote: Stephen A. Lawrence wrote: This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. That works in SR, but the solution is inconsistent with GR. Wrong. In fact the full solution can only be had using techniques commonly considered to be part of GR. Acceleration is acceleration, in either SR or GR. In either case you integrate the path length, using the pseudo-Riemannian metric of Minkoski, in order to find the elapsed proper time of either twin, and in either case the path which includes the acceleration comes out shorter. A geodesic in GR is the longest distance between two points. No, it is the shortest distance between two points on a spacetime manifold. Acceleration pushes you off the geodesic, as a result of which you follow a shorter path. If the two twins have worldlines which cross at two points, and if one accelerates while the other follows a geodesic, the one on the geodesic will age more. That's straight out of GR ... or SR, take your pick. In GR, acceleration due to gravity is treated as indistinguishable from a manufactured acceleration. If both accelerate, then neither follows a geodesic and you need to know the details of the problem to determine who ages more (if either). GR and SR only really differ when you introduce gravity, which doesn't enter into this problem. You can't ignore gravity. The raison d'être of GR is to explain gravity. Ignore gravity and you are back in the flat spacetime of SR. Harry
Re: [Vo]: FW: Einstein's Twin Paradox
Kyle R. Mcallister wrote: I'm going to go shovel the snow off my ~100 ft long driveway. I wonder if it will have important future implications for quantum computers? --Kyle No way. You need to be shovelling sh*t to have that affect. ;-) Harry
Re: [Vo]: FW: Einstein's Twin Paradox
Twin paradox solved by a universal static aether adjustment to SR ;) SR is totally broken. And no inertial acceleration doesn't solve it, the twin at home is undergoing plenty of acceleration around the earth, around the sun, thermal and sound vibrations. Also the acceleration to light speed can be arbitrarily steep for a thought experiment so the accelerating, decelerating part of the trip could be no more than 1 sec total by anyones watch. You could also have two (very long) parallel trains with windows and clocks, this way you can see the rate of time of the other train as you might have left that carriage you started opposite a long way behind but a synchronized clock is always in view. Or what of the case of a near light speed orbit, you are always in view of the stationary mass you are orbiting and it can always see you with out any Doppler related time effects. It has never made sense and never will. Many experiments and observations show that the speed of light isn't always constant either, it's all bunk and obviously so once you see the holes. On 2/16/07, Harry Veeder [EMAIL PROTECTED] wrote: Kyle R. Mcallister wrote: I'm going to go shovel the snow off my ~100 ft long driveway. I wonder if it will have important future implications for quantum computers? --Kyle No way. You need to be shovelling sh*t to have that affect. ;-) Harry
Re: [Vo]: FW: Einstein's Twin Paradox
Harry Veeder wrote: Stephen A. Lawrence wrote: Harry Veeder wrote: Stephen A. Lawrence wrote: This is not a paradox, and the paradoxical nature of the problem was in fact resolved something on the order of a century ago. The traveling twin accelerates; the stay-at-home twin does not; thus, the symmetry is broken. That works in SR, but the solution is inconsistent with GR. Wrong. In fact the full solution can only be had using techniques commonly considered to be part of GR. Acceleration is acceleration, in either SR or GR. In either case you integrate the path length, using the pseudo-Riemannian metric of Minkoski, in order to find the elapsed proper time of either twin, and in either case the path which includes the acceleration comes out shorter. A geodesic in GR is the longest distance between two points. No, it is the shortest distance between two points on a spacetime manifold. Work it out or look it up if you don't believe me. It's the path with maximal magnitude, not minimal. It's commonly referred to as extremal but unlike most extremal paths we deal with it's (locally) maximal, not (locally) minimal. Take a simple example: The metric, with c=1 and signature diag(1,-1,-1,-1), in 1 space dimension, can be written as dt^2 - dx^2. If I go directly from point (0,0), to point (5,3), the squared path length is 25 - 9 = 16 and elapsed proper time is 4. If, instead, I go from (0,0) to (2,1) and then from (2,1) to (5,3) (with a kink in the path), the squared path length is (4-1) + (9-4) = 8 which is smaller; the elapsed proper time is 2*sqrt(2), which is also smaller than 4. Again, the geodesic is the path which (locally) maximizes the proper time. (I can't cough up the geodesic equation to prove this in general without some time dredging around in my memory or some time looking at a book; maybe tomorrow.) Acceleration pushes you off the geodesic, as a result of which you follow a shorter path. If the two twins have worldlines which cross at two points, and if one accelerates while the other follows a geodesic, the one on the geodesic will age more. That's straight out of GR ... or SR, take your pick. In GR, acceleration due to gravity is treated as indistinguishable from a manufactured acceleration. In GR acceleration due to gravity appears as nonzero connection coefficients in the metric, which is also where you see the effects of a uniform acceleration field. In that sense, they're indistinguishable. However, gravity due to real bodies also has tidal effects, which manifest themselves as nonzero curvature; that is _not_ the same as acceleration. Curvature can be tested for locally, and cannot be transformed away by careful choice of coordinates, unlike uniform acceleration. Curvature -- tidal effects -- cannot be handled in special relativity. If both accelerate, then neither follows a geodesic and you need to know the details of the problem to determine who ages more (if either). GR and SR only really differ when you introduce gravity, which doesn't enter into this problem. You can't ignore gravity. You can in the twins problem; there isn't any. But in any case, gravity doesn't affect the fact that geodesics are paths which maximize the proper time -- that's true regardless of the presence of gravity. The raison d'être of GR is to explain gravity. That's right. But you don't need it to resolve the twins problem, which takes place in flat space. Ignore gravity and you are back in the flat spacetime of SR. Harry
