If there is no universal frame of reference then how can time dilation occur?

[Iggy]

Your response, when asked if acceleration was relative, was to say "you can measure your acceleration as a purely local experiment". That's the thing I'm responding to. It isn't true. How would you propose it be done? You're in a box with no windows and an accelerometer measures 9.8 m/s^2 -- are you accelerating?

Yes.

An observer on earth's surface with an accelerometer will get that reading. A static observer using the Schwarzschild metric, for example (standing on the surface of a planet), will make that measurement. You would have to explain to me how a change in velocity of 9.8 m/s² is invariant when it is also zero in a valid coordinate system.

 

Well, to be blunt, you're wrong.

 

Theories in physics have to follow three simple metarules.

1. The theory has to have some logically consistent underpinning. This can range from a simple ad hoc equation whose only logically consistent underpinning is that the equation is a mathematically well-formed formula to a much deeper set of precepts from which one can derive mathematical predictions.
   
2. More important: It has to describe some testable behavior. No matter how internally consistent it might be, a "just-so" story that is not testable is not a physical theory.
   
3. Most important: It has to agree with observable reality.

The only theory on which I've relied is general relativity. The only interpretation I've given it is exactly the same as Einstein's 1918 -- as I've quoted:

 

"It is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' [the younger twin is at rest throughout] as coordinate system K [the older twin is at rest throughout]... it means for a man who maintains consistency of thought a great satisfaction to see that the concept of absolute motion, to which kinematically no meaning can be attributed, does not have to enter physics... We are not dealing here with two different, mutually exclusive hypotheses about the seat of the motion, rather with two ways, equally valid in principle, of representing the same factual situation."

 

It is possible that Einstein is misinterpreting his own theory, but simply telling him "you're wrong", no matter how bluntly you say it, isn't going to convince me.

 

Your concept is wrong for the simple reason that it does not agree with observable reality. The Hafele–Keating experiment, for example.

 

...You are ignoring that no matter which explanation one uses / which point of view one takes in the twin paradox, the answers are always the same. The traveling twin ages less.

I have maintained, from my very first post, that one twin ages less. I can't believe that I have been misinterpreted so badly. Either twin can consider himself at rest throughout the thought experiment and get the correct answer: the twin who accelerates relative to the bulk mass of the cosmos will age less. I showed how it could be calculated from either perspective in post 44.

 

If you think I've been arguing that both twins age less than the other because the other is moving from each perspective then I have seriously failed to communicate.

 

To argue against general relativity you need to construct some experiment and show how relativity fails that test. General relativity, at least so far, is consistent with every experiment constructed to test it.

My argument relies on GR being valid.

 

You appear to be misconstruing the concept of covariance in general relativity, which says that all reference frames are equally valid, to say something contrary to relativity theory.

To say that "acceleration" is absolute is to say that the laws of physics are different in different systems of reference... it is to reject the general principle of relativity.

 

It is, in other words, to assert that only the coordinate systems which have some particular observer accelerating at 9.8 m/s² are valid. If that acceleration is absolute then all correct systems of reference will have that value.

 

General covariance, the principle of relativity, and every reasonable interpretation of GR that I have heard (including Einstein's) does indeed reject that.

[Delta1212]

Here's the problem with using "bulk mass of the cosmos" as the sort of default reference for acceleration.

 

If you take the twin paradox, and instead of using twins, make it you vs the rest of the universe (let's say you're an immortal being with a lot of time on his hands who has managed to travel all over the universe and set up a system of engines on every significant piece of matter that will accelerate in the same direction).

 

You then drift out into space and have the entire rest of the universe accelerate in one direction until it comes close to the speed of light, then stop, turn around, and accelerate back in the opposite direction until it has reached the same position relative to you.

 

If the, I don't know, default coordinate system? in the experiment is defined as the one which the bulk mass of the cosmos sees as being at rest, and the accelerated frame is the one which the bulk mass of the cosmos sees as changing velocity with respect to it, then you should have aged less than the rest of the universe during that stretch of acceleration. In fact, the universe would have aged less than you, which means that you can't use "the reference frame containing the most mass" as any sort of default coordinate system against which you measure acceleration.

[DrRocket]

Man is this post derailed.

 

There is a universal frame of reference just that they don't have an accurate one yet.

 

 

 

wrong

 

 

 

 

My argument relies on GR being valid.

 

 

To say that "acceleration" is absolute is to say that the laws of physics are different in different systems of reference... it is to reject the general principle of relativity.

 

 

Your argument relies on your misconceptions of general relativity being valid.

 

It also relies on rejection of a mountain of experimental evidence. You are essentially arguing that when a car in which you are riding accelerates that you do not notice that fact.

[Iggy]

You are essentially arguing that when a car in which you are riding accelerates that you do not notice that fact.

I wouldn't argue that because it is meaningless. You didn't specify relative to what coordinate system and in what manner the car is accelerating so there is no way to say what kind of 'notice' one would take.

 

I should have the right to say that it is the other twin who is changing velocity. Does GR not give me the freedom to say that?

no

 

It is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' as coordinate system K...

 

-Einstein

 

If you ever find that you've said the exact opposite of Einstein concerning the theory of general relativity, it is certainly time to put the strawmen and the preaching aside and consider the issue

[JohnStu]

What I said is not wrong, just not fit your method. You guys probably misinterpret my actual age.

 

 

[zapatos]

What I said is not wrong, just not fit your method. You guys probably misinterpret my actual age.

I am quite sure that no one would change their view of the statements you made if they knew your actual age. In fact, I seriously doubt anyone has considered your age at all. You do however seem to be making an astounding number of inaccurate statements for someone who has been on this site for such a short period of time.

[DrRocket]

I wouldn't argue that because it is meaningless. You didn't specify relative to what coordinate system and in what manner the car is accelerating so there is no way to say what kind of 'notice' one would take.

 

If you need someone to hand you a coordinate system to notice that when that dragster takes off you are pushed back into the seat, then I can't help you.

 

The whole point of general relativity is what Einstein called "general covariance", which in modern parlance means that the equations are formulated without reference to any coordinate system (physicists embody this notion in the terminology of tensors). Coordinates are only invoked for specific calculations, but the physics is invariant.

 

 

 

 

If you ever find that you've said the exact opposite of Einstein concerning the theory of general relativity, it is certainly time to put the strawmen and the preaching aside and consider the issue

 

It is time that you actually studied general relativity. Gravitation by Misner, Thorne and Wheeler would be the place to start. There you will find real not only general relativity but an introduction to differential geometry and the theory of manifolds. Those subjects are necessary to accurately explain general relativity.

 

What you are missing is that in general relativity spacetime is a manifold, not an affine space and that there is no such thing as a reference frame.

 

The coordinate systems that you see in general relativity are "charts" in the language of manifold theory and there is no meaning whatever to a comparison of charts based at two different spacetime points. There is no global notion of space. There is no global notion of time. "Time here" vs "time there" makes no sense. This is a major difference between general relativity and special relativity. It is more profound than the difference between special relativity and Newtonian mechanics.

Edited March 1, 2012 by DrRocket

[Iggy]

If you need someone to hand you a coordinate system to notice that when that dragster takes off you are pushed back into the seat, then I can't help you.

"Takes off" relative to what? In what direction?

 

A car that accelerates toward and relative to the center of the earth at 9.8 m/s^2 isn't going to "push you back into the seat". A car that accelerates relative to another car may itself be sitting at rest relative to the road. How fast do you think you're accelerating relative to Halley's comet right now?

 

You're conflating the notion that the road is at rest and the car accelerates with the notion that the road doesn't feel the inertial force that the car does. The latter doesn't imply the former in GR.

 

Either twin can consider himself at rest while the other accelerates and solve the 'paradox' with GR correctly. If you still disagree with that or you don't follow what I mean then please read the link I've given.

[DrRocket]

"Takes off" relative to what? In what direction?

 

 

A car that accelerates toward and relative to the center of the earth at 9.8 m/s^2 isn't going to "push you back into the seat". A car that accelerates relative to another car may itself be sitting at rest relative to the road. How fast do you think you're accelerating relative to Halley's comet right now?

 

You're conflating the notion that the road is at rest and the car accelerates with the notion that the road doesn't feel the inertial force that the car does. The latter doesn't imply the former in GR.

 

Either twin can consider himself at rest while the other accelerates and solve the 'paradox' with GR correctly. If you still disagree with that or you don't follow what I mean then please read the link I've given.

 

There is no such thing as a reference frame in general relativitiy.

 

There is no such thing as a gravitational field in general relativity.

 

I showed you how to resolve the twin paradox in terms of general relativity.

 

I am not conflating anything.

 

You are not only conflating, but confusing, mixing, dicing and chopping the entire theory.

 

The Schwzrzchild metric applies to a radially symmetric spacetime. It has NOTHING to do with the twin paradox.

 

Go read the book that I suggested, a real book, not some popularization, not some simplificatioin, but the real theory.

 

I give up.

Edited March 1, 2012 by DrRocket

[JohnStu]

I am quite sure that no one would change their view of the statements you made if they knew your actual age. In fact, I seriously doubt anyone has considered your age at all. You do however seem to be making an astounding number of inaccurate statements for someone who has been on this site for such a short period of time.

You mean quick, concise, to the point. Not tedious and loopy. How do you know if I been on this site for such a short period of time. I been on the other science forum for ages. Also, I been extremely accurate, but brief.

[doG]

There is a universal frame of reference just that they don't have an accurate one yet.

I'll bet you can't support that claim.

[DrRocket]

I'll bet you can't support that claim.

 

Bet large. You can't lose.

[JohnStu]

I guess observing the movement of galaxies could lead to a better estimation of universal frame of reference. Trace back the route of nebula that gave birth to the solar system, then trace back which star gave birth to the nebula. That star should be the frame of refence as it had much lesser speed (relative to universal frame of refence) than solar system. Also, leaving radioactive substance onto various astroids of different regions. Leave them for a few years. Retrieve the substance and see which one decayed the fastest. Take gravity and speed history of their life into calculation, then the astroid whose radioactive substance decayed the fastest is the better universal frame of refence. *shrugs* Keep doing that thousands of times on different astroids. The frame of refence can be estimated better. It can never be exact, but will be more and more accurate.

 

But, really, one does not need a universal frame of reference. Just use sea level of Earth at some location as frame of refence for things around Earth. Use centre of sun as frame of reference for things around the sun. And so on. Plus, much beyond the Earth is utter estimation, so the universal frame of refence doesn't need to be much more precise.

[Cap'n Refsmmat]

I guess observing the movement of galaxies could lead to a better estimation of universal frame of reference. Trace back the route of nebula that gave birth to the solar system, then trace back which star gave birth to the nebula. That star should be the frame of refence as it had much lesser speed (relative to universal frame of refence) than solar system. Also, leaving radioactive substance onto various astroids of different regions. Leave them for a few years. Retrieve the substance and see which one decayed the fastest. Take gravity and speed history of their life into calculation, then the astroid whose radioactive substance decayed the fastest is the better universal frame of refence. *shrugs* Keep doing that thousands of times on different astroids. The frame of refence can be estimated better. It can never be exact, but will be more and more accurate.

 

But, really, one does not need a universal frame of reference. Just use sea level of Earth at some location as frame of refence for things around Earth. Use centre of sun as frame of reference for things around the sun. And so on. Plus, much beyond the Earth is utter estimation, so the universal frame of refence doesn't need to be much more precise.

JohnStu, please be mindful of our rules and do not post your own speculations in the mainstream science sections. This particular speculation doesn't make any sense in modern cosmology.

[A Tripolation]

You mean quick, concise, to the point. Not tedious and loopy...Also, I been extremely accurate, but brief.

But, really, one does not need a universal frame of reference. Just use sea level of Earth at some location as frame of refence for things around Earth. Use centre of sun as frame of reference for things around the sun. And so on. Plus, much beyond the Earth is utter estimation, so the universal frame of refence doesn't need to be much more precise.

 

 

No. Just...no. There's so much wrong with what you think that I don't know where to begin.

[Iggy]

There is no such thing as a reference frame in general relativitiy.

frame is shorthand for coordinate system

 

There is no such thing as a gravitational field in general relativity.

The book that you've twice now told me to read uses the phrase "gravitational field" over 100 times. In section 16.5 "the measurement of the gravitational field" it loosely defines the phrase as the "geometry of spacetime", and that is certainly on point with what I'm saying. When I say that the younger twin experiences a "gravitational field" from his perspective, it is the physical interpretation of the field equations (curved spacetime) to which I'm referring.

 

I don't mean a classical vector field, and it's on point for that reason in fact...

 

Had one tried to explain to Newton the equality of inertial and gravitational mass from the equivalence principle, he would necessarily have had to reply with the following objection: it is indeed true that relative to an accelerated coordinate system bodies experience the same accelerations as they do relative to a gravitating celestial body close to its surface. But where are, in the former case, the masses that produce the accelerations?

 

It is clear that the theory of relativity presupposes the independence of the field concept.

 

-The Meaning of Relativity -- Einstein -- 1922

 

I showed you how to resolve the twin paradox in terms of general relativity.

I don't recall that, but I do recall posting Einstein's GR solution to the twin paradox in post 44. If it is conceptually flawed is the question.

 

I am not conflating anything.

I think everyone who has responded to my query has conflated "acceleration relative to geodesic motion" with "acceleration". You did this by saying that an accelerating vehicle must push the occupants back in their seat. You clearly meant a vehicle accelerating relative to geodesic motion, which I could agree, and have, agreed with -- but you are vague and equivocate.

 

The Schwzrzchild metric applies to a radially symmetric spacetime. It has NOTHING to do with the twin paradox.

golly gee, you don't say!

 

DH said that observers must accelerate by the amount measured by their accelerometer (same conflation). I mentioned the schwarzschild metric because it was the easiest counterexample. I wasn't trying to solve the twin paradox with it, or trying to build you a strawman with it, or anything else like that.

 

Go read the book that I suggested, a real book, not some popularization, not some simplificatioin, but the real theory.

 

I give up.

Thank you for the literature suggestion.

 

If anyone would like to respond to the proposition, here is another quote from Einstein from a later publication (again, in case I'm communicating it horrifically),

 

If the motion of the carriage is now changed into a non-uniform motion, as for instance by a powerful application of the brakes, then the occupant of the carriage experiences a correspondingly powerful jerk forwards. The retarded motion is manifested in the mechanical behaviour of bodies relative to the person in the railway carriage. The mechanical behaviour is different from that of the case previously considered, and for this reason it would appear to be impossible that the same mechanical laws hold relatively to the non-uniformly moving carriage, as hold with reference to the carriage when at rest or in uniform motion. At all events it is clear that the Galileian law does not hold with respect to the non-uniformly moving carriage. Because of this, we feel compelled at the present juncture to grant a kind of absolute physical reality to non-uniform motion, in opposition to the general principle of relativity. But in what follows we shall soon see that this conclusion cannot be maintained.

 

The General Theory of Relativity -- s18.8

 

We can now appreciate why that argument is not convincing, which we brought forward against the general principle of relativity at the end of Section XVIII. It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognizes in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a “real” acceleration (retardation) of the carriage. He might also interpret his experience thus: “My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under the influence of this field, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced.”

 

The General Theory of Relativity -- s20.9

 

This makes such good sense to me that if it is indeed not a valid interpretation of his theory I would very much like to know why that is the case... or how it was determined not to be the case.

Edited March 3, 2012 by Iggy

[Iggy]

I should have the right to say that it is the other twin who is changing velocity. Does GR not give me the freedom to say that?

no

 

In light of a neg rep, , here are a couple more sources refuting the claim and supporting my position:

 

 

The clock paradox in a static homogeneous gravitational field

 

abstract:

 

The gedanken experiment of the clock paradox is solved exactly using the general relativistic equations for a static homogeneous gravitational field. We demonstrate that the general and special relativistic clock paradox solutions are identical and in particular that they are identical for finite acceleration. Practical expressions are obtained for proper time and coordinate time by using the destination distance as the key observable parameter. This solution provides a formal demonstration of the identity between the special and general relativistic clock paradox with finite acceleration and where proper time is assumed to be the same in both formalisms. By solving the equations of motion for a freely falling clock in a static homogeneous field elapsed times are calculated for realistic journeys to the stars.

 

arxiv:0604025

 

Viewpoint of the traveling twin

 

During the turnaround, the traveling twin is in an accelerated reference frame. According to the equivalence principle, the traveling twin may analyze the turnaround phase as if the stay-at-home twin were freely falling in a gravitational field and as if the traveling twin were stationary. A 1918 paper by Einstein presents a conceptual sketch of the idea. From the viewpoint of the traveler, a calculation for each separate leg, ignoring the turnaround, leads to a result in which the Earth clocks age less than the traveler. For example, if the Earth clocks age 1 day less on each leg, the amount that the Earth clocks will lag behind amounts to 2 days. The physical description of what happens at turnaround has to produce a contrary effect of double that amount: 4 days' advancing of the Earth clocks. Then the traveler's clock will end up with a net 2-day delay on the Earth clocks, in agreement with calculations done in the frame of the stay-at-home twin.

 

The mechanism for the advancing of the stay-at-home twin's clock is gravitational time dilation. When an observer finds that inertially moving objects are being accelerated with respect to themselves, those objects are in a gravitational field insofar as relativity is concerned. For the traveling twin at turnaround, this gravitational field fills the universe. In a weak field approximation, clocks tick at a rate of [math]t' = t (1 + \Phi / c^2)[/math] where [math]\Phi[/math] is the difference in gravitational potential. In this case, [math]\Phi = gh[/math] where ''g'' is the acceleration of the traveling observer during turnaround and ''h'' is the distance to the stay-at-home twin. The rocket is firing towards the stay-at-home twin, thereby placing that twin at a higher gravitational potential. Due to the large distance between the twins, the stay-at-home twin's clocks will appear to be sped up enough to account for the difference in proper times experienced by the twins. It is no accident that this speed-up is enough to account for the simultaneity shift described above. The general relativity solution for a static homogeneous gravitational field and the special relativity solution for finite acceleration produce identical results.

 

Wikipedia -- Twin paradox -- Viewpoint of the traveling twin

 

perhaps they are misguided, but they make such good sense to me that I would very much appreciate being shown why they must be mistaken... yada yada...

Edited March 4, 2012 by Iggy

[DrRocket]

In light of a neg rep, , here are a couple more sources refuting the claim and supporting my position:

 

 

Neg rep countered. You don't deserve a neg rep, as you are sincere. (And those rep points don't mean much anyway). But you are still wrong.

 

I know what a frame is, but your are a bit confused as to what that means in the context of a manifold and hence in the context of general relativity. A coordinate system in general relativity and a chart in the mathematical theory of manifolds are the same thing. There is no such thing as a global chart or a global coordinate system, and there is no meaning to a comparison of charts based at different points -- there is no meaning to "time here" vs "time there".

 

That applies as well to so-called "gravitational time dilation", which is measured in a single local coordinate system and is a coordinate effect, which locally approximates a difference in world liines (.ie. proper time).

 

Strictly speaking the only time in general relativity is proper time. Proper time is (in units in which c=1) just the length of the (time-like) world line of a physical body and for the twin paradox the issue at hand is the proper time of the two twins measured between two spacetime points at which their world lines intercept. I have not read the Wiki article in detail but it seems to be dealing with a calculation of proper time. You can make that calculation as difficult and obtuse as you want, but the beauty of general relativity is that the metric and arc length are completely independent of even local choices of coordinates.

 

The "twin paradox" is stated qualitatively (no definte trip length, no velocity vs time history, etc,) and it can be solved, as I showed you, completely and rigorously by means of merely comparing a geodesic world line with any other world line in terms of the spacetime metric. No laborious calculations are needed.

[swansont]

How do you know if I been on this site for such a short period of time. I been on the other science forum for ages.

 

!

Moderator Note

It's in your profile. You joined on Feb 28, 2012. This forum ≠ the other forum. How about we not further derail the thread with posts involving this sort of trivia?

[JohnStu]

!

Moderator Note

It's in your profile. You joined on Feb 28, 2012. This forum ≠ the other forum. How about we not further derail the thread with posts involving this sort of trivia?

 

Yes, I was on another account for 2 years.

Edited March 4, 2012 by JohnStu

[Iggy]

Neg rep countered. You don't deserve a neg rep, as you are sincere. (And those rep points don't mean much anyway).

ah, thank you. I almost wish my first [-1] would have been for something with a bit more drama

 

 

But you are still wrong.

At this point I would almost welcome being wrong if I could just get "you are wrong about X because Y proves X wrong".

 

 

A coordinate system in general relativity and a chart in the mathematical theory of manifolds are the same thing.

Yep.

 

 

There is no such thing as a global chart or a global coordinate system

uh... the global coordinate system doesn't have direct metrical meaning (if it's curved), but that's what the metric is for... unless you mean there is no truly global coordinate system covering the whole cosmos... I can't be sure if this is meant as an objection and if so how.

 

 

and there is no meaning to a comparison of charts based at different points -- there is no meaning to "time here" vs "time there".

 

That applies as well to so-called "gravitational time dilation", which is measured in a single local coordinate system and is a coordinate effect, which locally approximates a difference in world liines (.ie. proper time).

Ok, but my contention is that GR does not prevent us from analyzing the 'paradox' from either twin's perspective -- that the non-inertial twin can consider himself at rest and the other guy accelerating.

 

It's analogous (just an analogy) to analyzing two world-lines in the vicinity of a spherically symmetric mass, first from Schwarzschild coordinates then from free-fall coordinates (or one of several other similar choices). Proper time, the reading on an accelerometer, and other invariant quantities don't depend on the coordinate choice. Whether the observer is "at rest" or "changing velocity" does.

 

Hence, an observer who can model the world around him in a coordinate system where he is stationary while his friend is changing velocity can -- being just as correct -- model the world around him in a coordinate system where he is changing velocity and his friend is stationary. In terms of the twin 'paradox'... The Clock Paradox in a Static Homogeneous Gravitational Field (2006)

 

Differences in proper time being a "coordinate effect, which locally approximates a difference in world lines" is not an objection to this -- it rather relies on it.

 

 

Strictly speaking the only time in general relativity is proper time. Proper time is (in units in which c=1) just the length of the (time-like) world line of a physical body and for the twin paradox the issue at hand is the proper time of the two twins measured between two spacetime points at which their world lines intercept.

And the principle of extremal aging establishes that the inertial particle always ages more between events. Why would you think I disagree?

 

 

I have not read the Wiki article in detail but it seems to be dealing with a calculation of proper time. You can make that calculation as difficult and obtuse as you want, but the beauty of general relativity is that the metric and arc length are completely independent of even local choices of coordinates.

No, the metric does indeed depend on the coordinate choice. You wouldn't expect, for example again, Schwarzschild coordinates to have the same metric as Lemaitre coordinates even in the same physical situation. That would be like having the same scale on different style maps (different projections) of the same earth.

 

The length, I agree, must be the same -- all frames must agree on the ages of the reunited twins. I agree also it's more complicated to treat the non-inertial twin at rest than not.

 

The "twin paradox" is stated qualitatively (no definte trip length, no velocity vs time history, etc,) and it can be solved, as I showed you, completely and rigorously by means of merely comparing a geodesic world line with any other world line in terms of the spacetime metric. No laborious calculations are needed.

No, I don't recall -- I'm quite sure you never did -- show me how to compare arc lengths. Nor would you need to, or how to numerically evaluate line integrals for proper time. I do have a pretty good working knowledge of relativity to solve problems of that sort.

 

Your response, as helpful as you may mean it, doesn't address the issue. You said GR doesn't allow the non-inertial twin to consider himself at rest. This would mean he can't transform to a coordinate system with a line element where he's at rest and compare arc lengths.

[Klaynos]

Yes, I was on another account for 2 years.

 

Pray tell, which account?

[DrRocket]

 

 

Your response, as helpful as you may mean it, doesn't address the issue. You said GR doesn't allow the non-inertial twin to consider himself at rest. This would mean he can't transform to a coordinate system with a line element where he's at rest and compare arc lengths.

[/quot

 

The thing that you are missing is that general relativity, unlike special relativity, is not rooted in "reference frames". The metric of spacetime is absolute. It has nothing to do with reference frames.

 

That does not mean that there is not some representation of the metric in each chart. But the metric has an existence independent of any particular chart.

 

Comparison of arc lengths is done with clocks. The traveling twin measures less proper time than the non-traveling twin. That is absolute. It has nothing to do with "coordinate systems".

 

The spacetime manifold is a Lorentzian 4-manifold. That means that the atlas is composed of Lorentzian charts. A Lorentzian chart corresponds to a body in freefall. So if you work with some other chart, you must relate it to Lorentzian chart. "At rest" refers to Lorentzian charts. The accelerating twin is not "at rest".

 

There is simply no way to avoid the mathematics of manifold theory. You need to read something like Gravitation by Misner, Thorne and Wheeler. You have managed to confuse yourself into thinking that you understand something that you do not really understand. A great many texts sacrifice understanding of what is going on in general relativity for the ability to push symbols and do calculations in specific situations. It is quite easy to loose the forest for the trees. It is also quite easy to reach erroneous conclusions -- for instance the event horizon of a black hole was initially thought to be singular, but much later it was realized that this is just a limitation of a particular local chart.

 

BTW this is MUCH better understood now than it was by Einstein when he invented general relativity. That ought to be expected. People have had nearly 100 years to refine the theory and its presentation. Modern treatments are much more clean and ultimately understandable than are the early treatments. That is no slight to the genius of Einstein, which shines through in the simple fact that he discovered this marvelous theory. But to think that the legions of scientists who have worked on and studied the theory over the last century or so have not gained greated insight and clarity woud be to slight them. Any good graduate student today understands the essence of general relativity as well or better than did Einstein. He ought to, since he has the shoulders of giants such as Einstein on which to stand.

Edited March 5, 2012 by DrRocket

[Iggy]

The thing that you are missing is that general relativity, unlike special relativity, is not rooted in "reference frames".

I'd be delighted if you stopped putting quotes around "reference frame" as if it is a phrase that I used in this thread. I didn't. Like I said before, when I say "frame" it is shorthand for coordinate system.

 

I'm talking about coordinate systems because my claim is that acceleration is coordinate dependent.

 

The metric of spacetime is absolute...

 

That does not mean that there is not some representation of the metric in each chart. But the metric has an existence independent of any particular chart.

The field equations do not determine the metric uniquely. The components of the metric change with coordinate choice because the field equations leave four degrees of freedom associated with the choice of four coordinates.

 

Simply put, you get one metric if one twin solves saying "I'm at rest" (I'm at rest is a coordinate choice) and you get another if the other does the same. The solution is valid and you get the correct answer either way. I do hope this is clear.

 

 

Comparison of arc lengths is done with clocks. The traveling twin measures less proper time than the non-traveling twin. That is absolute. It has nothing to do with "coordinate systems".

I would also be pleased if you stopped repeating things that I said in my last post as if you are teaching me something. I don't want to be insensitive, but it could give the unfortunate impression that you don't read your replies.

 

 

A Lorentzian chart corresponds to a body in freefall.

Agreed.

 

 

So if you work with some other chart, you must relate it to Lorentzian chart.

I don't believe that follows.

 

"At rest" refers to Lorentzian charts.

Nah, I'm sure "at rest" can refer to Schwarzschild charts just the same. GR is not so picky as SR.

 

 

The accelerating twin is not "at rest".

He's not at rest in one coordinate system, and he's not accelerating in another. It's a riddle that only GR can solve

[mindless]

You do not need GR to get a twin paradox. All you need is for travellers A and B to arrange themselves either side of earth so that B is at least twice as far away as A. They then both start travelling towards earth at the same high speed.

 

As A passes Earth he synchronises his clocks with Earth time. He then travels on until he meets B coming the other way. B then synchronises her clocks with A as they pass and carries on to Earth. B will find that her clocks read an earlier time than those on earth. The total elapsed time of the entire journey to a distant point and back to earth will be found to be less than the time recorded for the journey on Earth.

 

No GR, no accelerations needed. The whole point of the thought experiment is to analyze the difference in times and this does not require the return of the outbound traveller.

 

Most importantly it is understanding the relativistic phase, ie: the progressive phase difference between clocks along the direction of motion, that is essential for understanding the twin paradox.

 

See http://en.wikibooks.org/wiki/Special_Relativity/Simultaneity,_time_dilation_and_length_contraction#The_twin_paradox

 

A quick note on relativistic phase: suppose the people on Earth had left clocks all over space that were synchronised with Earth time. The effect of relativistic phase is that even when A passes Earth he will be able to work out that, for him, the clock where he will meet B is already reading a time in the future of the clocks on Earth (even though the people on Earth calculate that they are still synchronised with that clock and both they and A see the same clock reading). To figure out how this can be the case you have either got to do the maths by calculating the distances and times taken for the synchronising light or just look at the Wikibook example. The net effect is that A is heading for a point in spacetime that is already in the future of the people on Earth. "A" gets a head start on the journey and it is this head start that means that his clocks show less elapsed time than might be expected.

 

Time dilation is just the effect of travelling into someone else's future. It does not need a universal frame of reference. If you and I were moving at 0.999 c relative to the people on this forum and I travelled away from you at 0.8c I would move the same number of seconds into your future with every light-second travelled as I would if we were both sitting on earth when I travelled away at 0.8c.