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## Implications of the statement "Acceleration is not relative"

 Quote by stevendaryl it seemed to me that other people were saying that Langevin's derivation required some kind of theoretical justification.
I was not trying to say Langevin's derivation required extra justification at the time he made it; he was basically making a bet that a theory of gravity of the sort he was thinking of would be consistently possible.

I was only saying that that does not allow us, today, knowing that the kind of theory of gravity he was thinking of is *not* consistently possible, to say that his version of the twin paradox, with the traveling twin not feeling acceleration, can be explained "just by using SR". He thought it could, but today we know it can't.

 Quote by stevendaryl Before GR, people used SR to explain the Michelson-Morley experiment, which certainly took place in a gravitational field. If the presence of a gravitational field makes SR inapplicable, then it was never applicable.
This is a different case. The MM experiment can be analyzed entirely in a single inertial frame that covers the entire experiment. Langevin's twin paradox scenario cannot. So knowing that Langevin's scenario can't be analyzed just using SR, in the light of today's knowledge, does not imply that the MM experiment can't be analyzed just using SR, in the light of today's knowledge.

 Quote by DaleSpam []SR predicts a very large accelerometer reading during the turnaround, and real free falling accelerometers read 0. Not every measurement in every scenario is sensitive to gravity. This one is. I am not making a claim that SR is inapplicable in every scenario where there is any gravity present. It is inapplicable in the twin paradox, for the reasons I stated above.
I am "getting" very little of the discussion concerning the relationship between SR, Langevin's scenario, and GR. That's not surprising, as I understand only the most basic principles of SR (and for all I know that understanding may need fine tuning), have only a vague conception of GR as a theory in which space and time curve to produce relative motion of massive objects without applied force in the presence of a gravitational field--and absolutely no knowledge of Langevin's ideas.

But this much I believe to be undeniably true of a purely SR treatment of a scenario in which two bodies, one inertial and the other non-inertial, separate from each other and then approach to reunion: the non-inertial body must experience unbalanced force at the transition from separation to approach. There is no other way for the period of separation to end. Therefore I agree with DaleSpam's statement in [my] bold, above.

I think I understand the point that even if one posits that the non-inertial twin reverses direction by "swinging around" a star, there must still be an unbalanced force--a non-zero reading on an accelerometer. The unbalanced force is due to the change of gravitational potential during the flyby. However, at this time I am unable to verify my understanding by calculation, so I have no actual opinion in the matter.

I'm about ready to sign off this thread, as the question in the OP has been answered to the extent possible with my current knowledge. My response to George's concerns will be in a new thread, as it pertains specifically to the explanation of the twin paradox, rather than to the more general question of the relativity of acceleration.

What have I learned?

1. Coordinate acceleration is relative; proper acceleration is not.

2. Proper acceleration may be experienced while at rest in a coordinate system. (This follows from 1.)

3. Loosely speaking, the experience of proper acceleration corresponds to the experience of an unbalanced force. I think this is in agreement with the definition of proper acceleration as the phenomenon that occurs when there is a non-zero reading on an accelerometer. However, I personally am not a fan of a definition of a fundamental physical phenomenon that requires the use of a mechanism. It seems to me that this leads to getting bogged down in the details of the design of the mechanism. I'd rather talk about the underlying phenomenon that the mechanism is intended to measure. In engineering, we are constantly aware of the difference between theory (the ideal) and practice (the inability to make actual conditions to correspond to the ideal). Defining proper acceleration as the reading on an instrument blurs that distinction, in my opinion.

4. Formally, proper acceleration is the derivative of proper velocity with respect to proper time. I have no idea how proper acceleration can ever be non-zero, because I cannot understand how proper velocity can ever be non-zero, if one defines proper time as the interval between two events at the same location. However, at this point in my education I am content to let this alone (for now).

5. From 3, only non-inertial bodies experience proper acceleration.

6. In the twin paradox, only the rocket twin is non-inertial. Therefore, the earth twin must have a straight world line in a spacetime diagram, and the rocket twin must have a bent worldline. By spacetime diagram I mean a diagram that charts the coordinate (Lorentz) transformation between inertial frames. I believe this is the same thing as saying Minkowski diagram. The design of the diagram does not allow a non-inertial body to be represented by a straight worldline, nor does it allow an inertial body to be represented by a bent worldline.

7. Also from 5, and illustrated in 6, the rocket twin must experience less elapsed proper time than the earth twin; there is no treatment of the episode in SR that can result in the earth twin being younger than the rocket twin.

8. From all the foregoing (with special emphasis on 2), the "absoluteness" of proper acceleration does not contradict the claim of the rocket twin to be at rest throughout the episode. Therefore, the statement that proper acceleration is absolute does not have any "shocking" implications with respect to the general principle of relativity.

9. The case of the rocket twin at rest is treated in the Minkowski diagram. The typical explanation of the twin paradox does not draw attention to this fact, leaving some good-faith objectors unsatisfied with the conclusion that the earth twin cannot be younger than the rocket twin. Further elaboration on this point will be given in the new thread that I intend to open; this will also be my response to George's concerns.

10. The discussion above is limited to the kinematics of SR. The essentially dynamic state of being non-inertial is recognized in the solution of the problem, but it is not analyzed with respect to the laws of dynamics.

11. [edited for clarity] In my mind, 10 leads to a question. In the intuitive understanding of the universe, the Earth is absolutely at rest. The Earth, as it were, is anchored in place. The impression one gets from popular books on relativity is that the intuitive understanding of the universe may legitimately be claimed by any observer: Every observer may consider himself to be anchored in place.

What are the implications of the rocket twin being anchored in place? Simply this: How is it that a force applied to the rocket causes the Earth and all the stars to move? Einstein's proposal is that a gravitational field is the cause. Granting that point for the sake of discussion, one must still ask how the rocket produces enough energy to accelerate the immense mass of the Earth and stars at the observed rate.

[Side note: This objection was alluded to by harrylin at one point in this discussion. I believe it is at the root of his claim that few physicists these days accept the idea that the rocket is "really in rest". I find it interesting in this regard (without drawing any conclusions) that DaleSpam says that most physicists these days tend to leave the question of the gravitational field in SR alone.]

Please understand that I am making no claim regarding the validity of the principle of relativity. I am merely stating the question that I wish to be able to answer, and wish (eventually) to be able to verify by calculation.

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 Quote by GregAshmore 3. Loosely speaking, the experience of proper acceleration corresponds to the experience of an unbalanced force. I think this is in agreement with the definition of proper acceleration as the phenomenon that occurs when there is a non-zero reading on an accelerometer. However, I personally am not a fan of a definition of a fundamental physical phenomenon that requires the use of a mechanism. It seems to me that this leads to getting bogged down in the details of the design of the mechanism. I'd rather talk about the underlying phenomenon that the mechanism is intended to measure. In engineering, we are constantly aware of the difference between theory (the ideal) and practice (the inability to make actual conditions to correspond to the ideal). Defining proper acceleration as the reading on an instrument blurs that distinction, in my opinion.
As is common in physics, there are multiple equivalent definitions. You may prefer the definition in terms of what is called the covariant derivative. Specifically, the proper acceleration can be defined as the covariant derivative of the tangent vector to an object's worldline along the worldline.

Here is a link on covariant derivatives:
http://en.wikipedia.org/wiki/Covaria...ve_along_curve

It is closely related to the concept of parallel transport:
http://en.wikipedia.org/wiki/Parallel_transport

And the concept of a connection:
http://en.wikipedia.org/wiki/Levi-Civita_connection

Sorry about the hard-to-digest math. It is the price you pay for getting rid of the accelerometer definition. It doesn't add anything new (so feel free to skip it until you are ready for GR); it just defines it mathematically instead of physically.

Personally, I prefer the accelerometer one for precisely reasons that you find objectionable. One problem with defining terms in general is that since there are always a finite number of terms you must always either wind up having circular definitions or undefined terms. In physics, we get around that by defining some terms experimentally. Proper time is the thing measured by a clock, distance is the thing measured by a rod, proper acceleration is the thing measured by an acclerometer. That accomplishes two things, first, it makes the link between the mathematical theory and the physical world more clear, and second it avoids the problem of leaving those things undefined. So, I personally prefer those kinds of "measurement based" definitions of fundamental quantities, but I recongnize that is a personal preference and alternative equivalent definitions are possible which hide the problem by pushing the measurements further away or embrace the problem by leaving some things completely undefined.

 Quote by GregAshmore How is it that a force applied to the rocket causes the Earth and all the stars to move? Einstein's proposal is that a gravitational field is the cause. Granting that point for the sake of discussion, one must still ask how the rocket produces enough energy to accelerate the immense mass of the Earth and stars at the observed rate.
As I explained to harrylin, it doesn't. If you say "A causes B" then that means that the presence of A implies B. So, if we say that "a force applied to the rocket causes the Earth and all the stars to move" that means that a force applied to the rocket implies that the Earth and all the stars must move. In an inertial frame, there may be a force on the rocket without movement of the Earth, so the force on the rocket does not imply movement of the Earth. Therefore the force on the rocket does not cause the Earth to move.

So what does cause the Earth to move? The answer is that specific choice of non-inertial coordinates. That choice of coordinates implies that the Earth moves, regardless of the presence or absence of any rockets with any forces. Every time you use that choice of coordinates the Earth moves. So the choice of coordinates causes the Earth to move, not the rocket.

 Quote by DaleSpam As is common in physics, there are multiple equivalent definitions. You may prefer the definition in terms of what is called the covariant derivative. Specifically, the proper acceleration can be defined as the covariant derivative of the tangent vector to an object's worldline along the worldline. Here is a link on covariant derivatives: http://en.wikipedia.org/wiki/Covaria...ve_along_curve It is closely related to the concept of parallel transport: http://en.wikipedia.org/wiki/Parallel_transport And the concept of a connection: http://en.wikipedia.org/wiki/Levi-Civita_connection Sorry about the hard-to-digest math. It is the price you pay for getting rid of the accelerometer definition. It doesn't add anything new (so feel free to skip it until you are ready for GR); it just defines it mathematically instead of physically. Personally, I prefer the accelerometer one for precisely reasons that you find objectionable. One problem with defining terms in general is that since there are always a finite number of terms you must always either wind up having circular definitions or undefined terms.
I hadn't thought about definitions enough to realize this.

 Quote by DaleSpam In physics, we get around that by defining some terms experimentally. Proper time is the thing measured by a clock, distance is the thing measured by a rod, proper acceleration is the thing measured by an acclerometer. That accomplishes two things, first, it makes the link between the mathematical theory and the physical world more clear, and second it avoids the problem of leaving those things undefined. So, I personally prefer those kinds of "measurement based" definitions of fundamental quantities, but I recongnize that is a personal preference and alternative equivalent definitions are possible which hide the problem by pushing the measurements further away or embrace the problem by leaving some things completely undefined.
Fair enough. But this may also lead to problems. For example, if proper time is measured by a clock, what is the proper time for the life of an individual particle? What clock do we read to measure its proper life span? This is of particular importance with regard to SR, as experiments with high speed particles are offered as evidence in support of the theory. We do not send a clock to accompany the particle on its journey in the accelerator. It seems to me that one is reduced to claiming that the particle is itself the clock. But if the particle is itself the clock, then there is no independent measure of the proper time that the particle existed, and thus no verification of the theory. There is no question that high speed particles live longer, as measured from our perspective. The question would be whether time in the rest frame of the particle is the same regardless of the speed of the particle measured in some other inertial frame, as the theory of SR requires. (I need to think about this some more; perhaps my logic is not entirely sound.)

 Quote by DaleSpam As I explained to harrylin, it doesn't. If you say "A causes B" then that means that the presence of A implies B. So, if we say that "a force applied to the rocket causes the Earth and all the stars to move" that means that a force applied to the rocket implies that the Earth and all the stars must move. In an inertial frame, there may be a force on the rocket without movement of the Earth, so the force on the rocket does not imply movement of the Earth. Therefore the force on the rocket does not cause the Earth to move. So what does cause the Earth to move? The answer is that specific choice of non-inertial coordinates. That choice of coordinates implies that the Earth moves, regardless of the presence or absence of any rockets with any forces. Every time you use that choice of coordinates the Earth moves. So the choice of coordinates causes the Earth to move, not the rocket.
Before I give you my initial reaction, I will tell that I intend to think carefully about what you say. It may be that my initial reaction is merely the expression of prejudice.

My initial reaction is: Nonsense. I'm sitting at rest in my rocket the whole time. Don't tell me about choosing coordinate frames--there is only one coordinate frame that matters: mine. (Isn't that the meaning of "absolute space", or "anchored in place"?) When I throw a ball, its acceleration (with respect to the only coordinate system that matters) is determined by its mass and the magnitude of the applied force. When the earth and the stars move, the same law should apply. {Edit: Not exactly the same law. I realize that gravity will cause coordinate acceleration without applied force. But the moving Earth and stars have acquired kinetic energy with respect to the rocket. That energy must have come from somewhere.}

A secondary (and less emotional) reaction is to ask the original question in a more precise way. What causes the spacial displacement between the rocket and the Earth to change?

 Quote by ghwellsjr What statement of mine are you referring to in post #161?

It seems to me you guys are just playing with words - proper, real, coordinate.
Try defining them before hitting one another on the head with them!

I always thought position was x,y,z - whatever they are, they are relative.
And velocity is their first differential with respect to time - so is relative.
And acceleration is the second differential of relative things - so is also relative.

Yes you can invent a special acceleration and use the word "proper" for it.
But how can you MEASURE it in an experiment?
As for "force" it can never be applied to anything without that thing witstanding it (unless it fractures) Hence "action and reaction are equal and opposite" whether acceleration results or not. So the net force at an SURFACE sums to zero!
As for the idea of force "applied at the centre of an object" there is no way to measure it except by the ASSUMPTION that force is mass times "acceleration"

When I stand here on the floor, my acceleration is 32 ft/sec^2 and it is as simple as that!
No need to dream up "force" at all. All we need is the upward acceleration required to cancel my downward acceleration. Fortunately my brain is well used to providing this acceleration.

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Quote by Drmarshall
 Quote by ghwellsjr What statement of mine are you referring to in post #161?
It seems to me you guys are just playing with words - proper, real, coordinate.
Try defining them before hitting one another on the head with them!

I always thought position was x,y,z - whatever they are, they are relative.
And velocity is their first differential with respect to time - so is relative.
And acceleration is the second differential of relative things - so is also relative.

Yes you can invent a special acceleration and use the word "proper" for it.
But how can you MEASURE it in an experiment?
As for "force" it can never be applied to anything without that thing witstanding it (unless it fractures) Hence "action and reaction are equal and opposite" whether acceleration results or not. So the net force at an SURFACE sums to zero!
As for the idea of force "applied at the centre of an object" there is no way to measure it except by the ASSUMPTION that force is mass times "acceleration"

When I stand here on the floor, my acceleration is 32 ft/sec^2 and it is as simple as that!
No need to dream up "force" at all. All we need is the upward acceleration required to cancel my downward acceleration. Fortunately my brain is well used to providing this acceleration.
Why are you dragging me into this? What did I say?

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 Quote by Drmarshall It seems to me you guys are just playing with words - proper, real, coordinate. Try defining them before hitting one another on the head with them! I always thought position was x,y,z - whatever they are, they are relative. And velocity is their first differential with respect to time - so is relative. And acceleration is the second differential of relative things - so is also relative. Yes you can invent a special acceleration and use the word "proper" for it. But how can you MEASURE it in an experiment?.
Your last statement gets at exactly why relativity required new definitions. Precisely when proper acceleration, defined as covariant derivative by proper time along a world line, differs from derivative if (x,y,z) by t, then experiments (using accelerometers) measure proper acceleration and DO NOT measure what you define as acceleration. Similarly, proper time is what clocks measure, NOT the time coordinate difference in some coordinate system.

 Quote by Drmarshall It seems to me you guys are just playing with words - proper, real, coordinate. Try defining them before hitting one another on the head with them! I always thought position was x,y,z - whatever they are, they are relative.
The modern way of thinking about it is that a position, such as a location on Earth, is absolute. The top of the Eiffel Tower is a definite spot; there is no ambiguity, or relativism involved. But there are infinitely many coordinate systems that can be used to specify a position.

In relativity, the primary thing is not a position, but an event, a point in space and time. So "the top of the Eiffel tower when Michelle Obama went up it" is an event, and it's absolute. But if I try to describe it using 4 numbers, for example, (latitude, longitude, altitude in meters, time in seconds since 1900), its description is relative to a coordinate system.

A spacetime path, giving the events that a traveler passes through, as a function of the time on his watch, is an absolute thing, because each event is absolute. But to describe the path as a set of 4 functions $x(\tau), y(\tau), z(\tau), t(\tau)$ is relative to a choice of a coordinate system.

The proper velocity of a path is again an absolute thing, while the components of the proper velocity are relative to a coordinate system. Proper acceleration is an absolute thing, while its components are relative to a coordinate system.

 Yes you can invent a special acceleration and use the word "proper" for it. But how can you MEASURE it in an experiment?
Yes, with the notion of "proper acceleration" used in General Relativity, one can measure its magnitude with an accelerometer. A simple accelerometer can be constructed by just taking a cubic box, putting a metal ball in the center, and then connecting the ball to the sides of the box using 6 identical springs. If the ball is exactly in the center, then the box has no proper acceleration. If the ball is closer to one wall, then the box is accelerating in the direction of the opposite wall.

 Quote by stevendaryl The modern way of thinking about it is that a position, such as a location on Earth, is absolute. The top of the Eiffel Tower is a definite spot; there is no ambiguity, or relativism involved. But there are infinitely many coordinate systems that can be used to specify a position.
That is not the modern way, that is Newton's way, unless you consider Newton's the modern way of thinking (but we're not in the eighteenth century anymore, remember?). Anyway the rest of your post gets it right that the more modern relativist thinking considers events in space time rather that position in space as absolute, so I don't know what you meant by this introduction.

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 Quote by TrickyDicky That is not the modern way, that is Newton's way, unless you consider Newton's the modern way of thinking (but we're not in the eighteenth century anymore, remember?). Anyway the rest of your post gets it right that the more modern relativist thinking considers events in space time rather that position in space as absolute, so I don't know what you meant by this introduction.
I must confess that I'm not reading Stevendaryl's point the way you are. The top of the Eiffel tower absolutely and unambiguously identifies a particular absolute coordinate-independent timelike worldline - and you'll notice that Stevendaryl carefully avoided identifying a "position" with a point in classical three-space.

 Quote by TrickyDicky That is not the modern way, that is Newton's way, unless you consider Newton's the modern way of thinking (but we're not in the eighteenth century anymore, remember?). Anyway the rest of your post gets it right that the more modern relativist thinking considers events in space time rather that position in space as absolute, so I don't know what you meant by this introduction.
A position on the earth is absolute. A position in space isn't.

The earth is a 3D object, while space is 4D in the modern way of looking at it.

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 Quote by GregAshmore When I throw a ball, its acceleration (with respect to the only coordinate system that matters) is determined by its mass and the magnitude of the applied force.
Really? When you, standing on the surface of the Earth, throw a ball upward, its motion is determined purely by its mass and the force you apply? Then why does it come back down?

 Quote by GregAshmore When the earth and the stars move, the same law should apply. {Edit: Not exactly the same law. I realize that gravity will cause coordinate acceleration without applied force. But the moving Earth and stars have acquired kinetic energy with respect to the rocket. That energy must have come from somewhere.}
The ball changes its kinetic energy with respect to you even though you didn't exert any additional force on it; at some point in its trajectory, it is momentarily motionless with respect to you (up in the air at the instant it stops rising and starts falling back). Where did the kinetic energy you gave the ball go?

You give away the problem with the position you are trying to take when you say "not exactly the same law". That's just the point: if you want "the laws of physics" to be "the same" in all reference frames, so that you can always view yourself "at rest", then the laws of physics have to include counterintuitive things like the Earth and the stars changing direction and speed just because you fired your rocket engine. If you want the laws of physics to always look simple, then you have to restrict yourself to frames in which they look simple (because all the counterintuitive stuff cancels out in those frames). You can't have it both ways; you can't have both simple-looking laws *and* a free choice of frames; your choice of frames determines how simple the laws look in the frames you choose.

 Quote by stevendaryl The modern way of thinking about it is that a position, such as a location on Earth, is absolute. The top of the Eiffel Tower is a definite spot; there is no ambiguity, or relativism involved. But there are infinitely many coordinate systems that can be used to specify a position. In relativity, the primary thing is not a position, but an event, a point in space and time. So "the top of the Eiffel tower when Michelle Obama went up it" is an event, and it's absolute. But if I try to describe it using 4 numbers, for example, (latitude, longitude, altitude in meters, time in seconds since 1900), its description is relative to a coordinate system. A spacetime path, giving the events that a traveler passes through, as a function of the time on his watch, is an absolute thing, because each event is absolute. But to describe the path as a set of 4 functions $x(\tau), y(\tau), z(\tau), t(\tau)$ is relative to a choice of a coordinate system.
I agree in principle with what you've said, but I question its practical utility. I read it this way: "A spacetime event (position and time) has a real existence apart from any coordinate system, yet can only be described in terms of some coordinate system." It would seem that the absoluteness of a spacetime event is metaphysical, because it cannot be verified empirically.

Furthermore, I believe that the rocket twin will deny what you say about "path", and all that follows from it. See below.

 Quote by stevendaryl The proper velocity of a path is again an absolute thing, while the components of the proper velocity are relative to a coordinate system. Proper acceleration is an absolute thing, while its components are relative to a coordinate system.
I assume that you develop the absolute path of a particle in this way. An arbitrary coordinate system whose origin is at the object under scrutiny is chosen. Using myself as an example, X is to my right, Y is straight ahead, Z is out the "top" of my head. My path through spacetime is marked by placing a monument in space at regular time intervals (by my clock).

At each iteration of my clock, I place a monument. I inscribe on the monument the time as read from my clock. I also consult my accelerometer to determine (some calculation is necessary) the change in my orientation since the previous iteration. I inscribe the differential change in orientation on the monument that was placed at the previous iteration. The change in orientation is necessarily expressed as rotations about the axes of the arbitrarily chosen coordinate system. Finally, I take my measuring rod and place its end against the previously placed monument; I then read directly the distance traveled since the previous iteration. I write that distance on the previous monument. Thus, my friend can follow my path without a map (coordinate system) if he starts at the first monument, adjusts his orientation as directed, and travels the distance indicated. At each monument, he repeats the process.

All of that is well and good, if one accepts the premise that I am moving through space. But, if you will recall, I am that very obstinate occupant of the rocket who insists that he is not moving at all. In my world, there is only one monument, and my orientation does not change.

 Quote by GregAshmore I agree in principle with what you've said, but I question its practical utility. I read it this way: "A spacetime event (position and time) has a real existence apart from any coordinate system, yet can only be described in terms of some coordinate system." It would seem that the absoluteness of a spacetime event is metaphysical, because it cannot be verified empirically.
There's nothing metaphysical about it--it's very concrete. A meteor crashes to the Earth. That marks a unique event. You don't need to have coordinates for it. George Washington is born. That marks a unique event. A star goes supernova. That's a unique event.

On a piece of paper, you draw a dot. That dot is a unique location on the piece of paper. You don't need coordinates to know that it's unique. You don't need coordinates to know whether the dot is at the same location as the X that someone else drew on the paper.

 I assume that you develop the absolute path of a particle in this way. An arbitrary coordinate system whose origin is at the object under scrutiny is chosen. Using myself as an example, X is to my right, Y is straight ahead, Z is out the "top" of my head. My path through spacetime is marked by placing a monument in space at regular time intervals (by my clock). At each iteration of my clock, I place a monument.
Specifying the initial location of the monument isn't good enough. You have to also specify it's initial velocity.

A path through spacetime is a 4D analogue of a curve drawn a piece of paper. An event in spacetime corresponds to a point on the paper. A velocity of a path corresponds to the slope of the tangent line drawn through a curve.

 I inscribe on the monument the time as read from my clock. I also consult my accelerometer to determine (some calculation is necessary) the change in my orientation since the previous iteration. I inscribe the differential change in orientation on the monument that was placed at the previous iteration. The change in orientation is necessarily expressed as rotations about the axes of the arbitrarily chosen coordinate system. Finally, I take my measuring rod and place its end against the previously placed monument;
Once again, an event is a single moment. You can't place a monument at a single moment, and you can't return to an earlier moment. The monument is going to follow its own path through spacetime, and when and if you get back to the same monument, it's not the same point in spacetime. Both you and the monument have moved since then.

 All of that is well and good, if one accepts the premise that I am moving through space. But, if you will recall, I am that very obstinate occupant of the rocket who insists that he is not moving at all.
EVERYONE moves at all times. If you look at your watch, then wait a while and look at your watch again, the second look is a different event from the first event. You've traveled from one event to another. You've "moved" through spacetime.

Now, you can certainly choose a coordinate system so that the spatial coordinates of the second event are the same as the spatial coordinates of the first event. But there is no way to choose coordinates so that all coordinates are the same. There is no way to avoid having motion in spacetime.

 Quote by PeterDonis Really? When you, standing on the surface of the Earth, throw a ball upward, its motion is determined purely by its mass and the force you apply? Then why does it come back down?
I perhaps should have been even more careful in my wording. The "gravity" under discussion here is not the gravity of Earth or any massive body. For the purposes of the twin paradox problem, the gravitational field due to the mass of the Earth is ignored. The gravity under discussion is the gravity of unspecified origin that Einstein posits to explain the motion of the Earth when the rocket engine is fired. This gravity is purely the result of the choice of coordinate system, as I understand DaleSpam.

When I throw a ball in SR, its motion is indeed determined purely by its mass and the force I apply. It does not return. It continues to move forever at some constant speed.

 Quote by PeterDonis The ball changes its kinetic energy with respect to you even though you didn't exert any additional force on it; at some point in its trajectory, it is momentarily motionless with respect to you (up in the air at the instant it stops rising and starts falling back). Where did the kinetic energy you gave the ball go?
This paragraph does not apply; the ball does not reverse in SR.

 Quote by PeterDonis You give away the problem with the position you are trying to take when you say "not exactly the same law". That's just the point: if you want "the laws of physics" to be "the same" in all reference frames, so that you can always view yourself "at rest", then the laws of physics have to include counterintuitive things like the Earth and the stars changing direction and speed just because you fired your rocket engine. If you want the laws of physics to always look simple, then you have to restrict yourself to frames in which they look simple (because all the counterintuitive stuff cancels out in those frames). You can't have it both ways; you can't have both simple-looking laws *and* a free choice of frames; your choice of frames determines how simple the laws look in the frames you choose.
The problem is not that the law of physics proposed by DaleSpam to explain the sudden movement of the Earth and stars at the firing of the rocket is not simple, or is not intuitive. The problem is that he has not proposed any law at all. Or at least, I do not recognize the statement "the movement of the Earth and stars was not caused by the firing of the rocket; it was caused by my choice of a certain set of coordinates" as a law of physics; certainly no other law of physics that I have learned looks like that. Furthermore, the statement borders on the delusional (I tried to find a neutral word; I could not; sorry) in that it denies the obvious causal connection between the firing of the rocket and the movement of the earth and stars (which I made bold in the quote of your post).

That is how it seems to me. I don't really have the right to speak on the matter because I do not know anything about Christoffel symbols, and therefore cannot understand the line of reasoning taken by DaleSpam. It is much better for me to leave this alone for the time being. I only mentioned it in my summary because it is an outstanding issue that must eventually be addressed.

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 Quote by GregAshmore The gravity under discussion is the gravity of unspecified origin that Einstein posits to explain the motion of the Earth when the rocket engine is fired.
Ah, ok, so the rocket is floating in flat spacetime. That clarifies things. But my comments still apply. See below.

 Quote by GregAshmore This gravity is purely the result of the choice of coordinate system, as I understand DaleSpam.
If spacetime is flat, yes.

 Quote by GregAshmore The problem is not that the law of physics proposed by DaleSpam to explain the sudden movement of the Earth and stars at the firing of the rocket is not simple, or is not intuitive. The problem is that he has not proposed any law at all.
No, the "problem" is that you have picked a scenario with a particular kind of simplicity, but then you want to choose a frame that doesn't match up with that simplicity. You have set your scenario in flat spacetime; in flat spacetime the laws of physics looks simplest in a global inertial frame. If you pick a non-inertial frame, like your "rest frame" when you fire your rocket, the laws of physics won't look as simple; they will have counterintuitive stuff in them like the Earth and the stars moving just because you fired your rocket engine. Once again, you can't have it both ways.

 Quote by GregAshmore Or at least, I do not recognize the statement "the movement of the Earth and stars was not caused by the firing of the rocket; it was caused by my choice of a certain set of coordinates" as a law of physics; certainly no other law of physics that I have learned looks like that.
How about "you picked a reference frame that doesn't match up with the special properties of the spacetime you are in". Does that help?

The laws in question are the simple laws of flat spacetime. You already know them in an inertial frame. The talk about a "gravitational field" that appears when you choose non-inertial coordinates, or about the movement of the Earth and stars being caused by the choice of coordinates, is just a way of describing the fact that non-inertial coordinates make the laws look more complicated.

 Quote by GregAshmore Furthermore, the statement borders on the delusional (I tried to find a neutral word; I could not; sorry) in that it denies the obvious causal connection between the firing of the rocket and the movement of the earth and stars (which I made bold in the quote of your post).
What is this causal connection? How does the firing of your rocket make stars that are light years away suddenly move? It doesn't. It can't. Causal influences can only propagate at the speed of light; there's no way your firing your rocket engine here and now can make a star that is light years away move "right now".

This is one way that trying to choose a frame in which you are always at rest, when your motion is non-inertial, makes the laws of physics look more complicated: the laws of physics now have to include the possibility of "motions" that violate the usual rules of causality. The example Einstein used was rotation: if I consider myself, sitting here on the surface of the Earth, to be "at rest", then the stars must be moving around me faster than the speed of light.

But nothing can move faster than light! you say. Correct: but the "motion" of the stars due to my rotation is not a "real motion" that is subject to that law. The complete laws of physics in my "rest frame" now have to include the possibility of "fictitious motions" like the motion of the stars around me, or the motion of the stars in response to you firing your rocket engine, which can be faster than the speed of light and which can stop and start "instantly" if I change my state of motion, even though that "violates" causality.

Once more, you can't have it both ways. If you want simple, intuitive laws of physics, where there are no "fictitious motions" or "fictitious forces", you have to pick a reference frame that allows the laws to look that simple. If you insist on picking a frame where you are always at rest, even when you move non-inertially, the laws will not look simple in that frame. You can't avoid that trade-off.

Mentor
 Quote by GregAshmore Fair enough. But this may also lead to problems. For example, if proper time is measured by a clock, what is the proper time for the life of an individual particle? What clock do we read to measure its proper life span? This is of particular importance with regard to SR, as experiments with high speed particles are offered as evidence in support of the theory. We do not send a clock to accompany the particle on its journey in the accelerator. It seems to me that one is reduced to claiming that the particle is itself the clock. But if the particle is itself the clock, then there is no independent measure of the proper time that the particle existed, and thus no verification of the theory. There is no question that high speed particles live longer, as measured from our perspective. The question would be whether time in the rest frame of the particle is the same regardless of the speed of the particle measured in some other inertial frame, as the theory of SR requires. (I need to think about this some more; perhaps my logic is not entirely sound.)
This is merely a current technological limitation, not an in-principle limitation. In principle you could accelerate a regular clock up to .9999c and use it to measure the lifetime of particles that we shoot alongside it as it passes by. There will always be experiments that we would like to do but cannot currently accomplish.

However, what we can do with current tecnhology is to take modern clocks and make them so incredibly stable and accurate that we can measure relativistic effects with ordinary velocities. I.e. whether or not a velocity is "relativistic" or not depends on your sensitivity, and modern clocks are so exquisitely sensitive that we can measure relativistic effects at walking speeds.

 Quote by GregAshmore Nonsense. I'm sitting at rest in my rocket the whole time. Don't tell me about choosing coordinate frames--there is only one coordinate frame that matters: mine.
This is the nonsense statement. It is nonsense for two reasons. First, because it uses an undefined concept. "Your frame" is non-inertial and there is no standard definition of a non-inertial object's frame. Second, because it is false. All coordinate systems have equal validity and yours is not particularly important and doesn't "matter" any more than any other coordinates.

 Quote by GregAshmore When I throw a ball, its acceleration (with respect to the only coordinate system that matters) is determined by its mass and the magnitude of the applied force.
This statement is wrong. The acceleration in that frame is not only determined by the mass and magnitude of the applied force, but also by the fictitious force (gravity) acting on it in that frame.

 Quote by GregAshmore When the earth and the stars move, the same law should apply.
The same laws do apply to both, you just made a mistake in the case of the ball.

 Quote by GregAshmore {Edit: Not exactly the same law. I realize that gravity will cause coordinate acceleration without applied force. But the moving Earth and stars have acquired kinetic energy with respect to the rocket. That energy must have come from somewhere.}
Actually, according to Noether's theorem, energy is NOT conserved in a non-inertial frame like that of the rocket. Energy is also frame variant.

 Quote by GregAshmore A secondary (and less emotional) reaction is to ask the original question in a more precise way. What causes the spacial displacement between the rocket and the Earth to change?
What do you mean by "spacial displacement"? Do you just mean the coordinate displacement or are you thinking of some physical measure of displacement? If the latter, then exactly what measure are you thinking of?

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