Implications of the statement Acceleration is not relative

[stevendaryl]

In my previous post, I asked for a SR solution of the problem in which the non-inertial rocket is always at rest. I don't recall anyone presenting such a solution. My sense is that my calls for such a solution have been rebuffed.

Once you understand how things work in inertial coordinates, then how they work in any other coordinate system (such as one in which the traveling twin is always at "rest") is just an application of calculus. So you are demanding that someone demonstrate a calculus exercise to you?

Okay, if it really will make you happy, I will post such a demonstration.

 

[stevendaryl]

Accelerated Rocket in Inertial Coordinates

Let's choose the zero for x and t to make the mathematics as simple as possible.

So assume that one twin is at rest at some location x=L, in inertial coordinates. The other twin starts off at location x=L_1 at time t = -t_1, travels in the -x direction until he reaches x=L_0 at time t=0, and then travels in the +x direction until he reaches x=L again at time t=+t_1. The time origin is chosen so that his journey is symmetric in time about the point t=0. The mathematical description of the traveling twin's path is:

x(t) = \sqrt{(L_0)^2 + c^2 t^2}

Time t_1 is chosen so that (t_1)^2 = \dfrac{L_1^2 - L_0^2}{c^2}

The velocity of the traveling twin at any time is given by:

v(t) = \dfrac{c^2 t}{\sqrt{(L_0)^2 + c^2 t^2}}

and the time-dilation factor \sqrt{1-\dfrac{v^2}{c^2}} is given by:

\sqrt{1-\dfrac{v^2}{c^2}} = \dfrac{L_0}{\sqrt{(L_0)^2 + c^2 t^2}}

The elapsed time for the traveling twin is given by:
\tau = \int \sqrt{1-\dfrac{v^2}{c^2}} dt

I'm not going to do the integral, but you can see that the integrand is less than 1, so the result will certainly be less than \int dt = 2 t_1. So the traveling twin will be younger.

I'm going to make another post where I describe this same situation from the point of view of the traveling twin.

 

[stevendaryl]

Accelerated Rocket in Accelerated Coordinates

In inertial coordinates, the path of the traveling twin is given by:
x(t) = \sqrt{L_0^2 + c^2 t^2}

Now, let's switch to a coordinate system in which the traveling twin is at rest, by making the transformation:

X = \sqrt{x^2 - c^2 t^2}
T = \dfrac{L_0}{c} arctanh(\dfrac{ct}{x})

where arctanh is the inverse of the hyberbolic tangent function.

In terms of these coordinates, the path of the stay-at-home twin is given by:

X(T) = L_1\ sech(\dfrac{cT}{L_0})

where sech is the hyperbolic secant function.

To see that this is reasonable, you can look at the Taylor series expansion: sech(\theta) = 1 - \frac{1}{2} \theta^2 + \ldots. So for small values of T, we have:

X(T) = L_1(1 - \frac{1}{2} \dfrac{c^2T^2}{L_0^2} + \ldots)
= L_1 - \frac{1}{2} g_1 T^2 + \ldots

where g_1 = \dfrac{c^2 L_1}{L_0^2}

That is the path of an object that starts off at X = L_1 at time T=0 and falls at the acceleration rate of g_1.

So in these coordinates, the traveling twin is always at the location X = L_0, while the stay-at-home is at X = L_0 at some point (at some time prior to T=0, rises to X=L_1 at time T=0, and then falls back to X=L_0 at some later time.

In terms of the coordinates X,T, the elapsed time d\tau for a traveling clock is given by:

d\tau = \sqrt{\dfrac{X^2}{L_0^2} - \dfrac{V^2}{c^2}} dT

where V = \dfrac{dX}{dT}

Note the difference with the formula for an inertial frame: d\tau not only on the velocity, but on the position X.

For the traveling twin, who is always at X = L_0, his proper time is:
d\tau = dT

For the stay-at-home twin, whose position X is always greater than or equal to L_0, the first term \dfrac{X^2}{L_0^2} > 1. If you work out the details, you will find that d\tau = \sqrt{\dfrac{X^2}{L_0^2} - \dfrac{V^2}{c^2}} dT > dT

So in accelerated coordinates, it's also the case that the stay-at-home twin ages more than traveling twin.

 

[PeterDonis]

In Langevin's "twin" scenario, the space capsule feels no force during the voyage.

IIRC this is only because spacetime in Langevin's scenario has a different topology than standard Minkowski spacetime; it is spatially a cylinder in the x direction instead of an infinite line. The "traveling" twin goes around the cylinder whereas the "stay at home" twin does not. The two paths belong to different topological classes; you can't continuously deform one into the other.

In this kind of situation you can have multiple free-fall paths between the same pair of events with different proper times; but each free-fall path has maximal proper time compared to all non-free-fall paths within the same topological class. (There are other topological classes possible as well; for example, there could be another free-fall twin that went around the cylinder twice, etc.) For example, an accelerating "twin" that didn't go around the cylinder would have less elapsed proper time than the free-fall stay-at-home twin (but not necessarily less than the free-fall traveling twin); and an accelerating "twin" that *did* go around the cylinder would have less elapsed proper time than the free-fall traveling twin.

 

[Dale]

I'll have to work on this to understand it.

I think that this is probably one of the key topics of modern relativity.

The key concepts of relativity, both special and general, are geometrical (Minkowski geometry for SR and pseudo-Riemannian geometry for GR).

Just like you can take a piece of paper and draw geometrical figures and discuss many things, such as lengths and angles, without ever setting up a coordinate system. The same thing is possible in relativity. The "piece of paper" is spacetime which has the geometrical structure of a manifold. The "geometrical figures" are worldlines, events, vectors, etc. that represent the motion of objects, collisions, energy-momentum, etc.

In this geometrical approach, the twin scenario is simply a triangle, and the fact that the traveling twin is younger is simply the triangle inequality for Minkowski geometry. In a coordinate-independent sense, the traveler's worldline is bent, and that in turn implies that his worldline is necessarily shorter as a direct consequence of the Minkowski geometry.

Now, on top of that underlying geometry, you can optionally add coordinates. Coordinates are simply a mapping between points in the manifold (events in spacetime) and points in R4. The mapping must be smooth and invertible, but little else, so there is considerable freedom in choosing the mapping. It is possible to choose a mapping which maps straight lines in spacetime to straight lines in R4, such mappings are called inertial frames.

It is also possible to choose a mapping which maps bent lines in spacetime to straight lines in R4, a non-inertial frame. Such a mapping does nothing to alter the underlying geometry. The bent lines are still bent in a coordinate-independent geometrical sense, but because it simplifies the representation in R4 it can still be useful on occasion in order to simplify calculations.

Because the mapping is invertible, in many ways it doesn't matter if you are talking about points in the manifold or points in R4. So you talk about things being "at rest" based on R4, and things happening "simultaneously" based on R4, and many other things. However, it is occasionally important to remember the underlying geometry.

I hope this helps.

Well, no, it hasn't been made clear. It ought to be clear, I'm sure.

It has, in fact, been made clear to you that SR can handle the twins paradox. What obviously hasn't been made clear to you is why. Your repetition of bald assertions that have already been contradicted is unhelpful. It wastes your time in repeating it and it wastes our time in repeating our responses. It also irritates those (maybe only me) who feel like their well-considered and helpful responses have been completely ignored by you.

You have been provided explanations and references addressing the topic, which clearly didn't "do it" for you. Read the explanations and references and point out the specific things that you don't understand or don't agree with. Then we can help clarify and make some progress.

 

[TrickyDicky]

Just like you can take a piece of paper and draw geometrical figures and discuss many things, such as lengths and angles, without ever setting up a coordinate system. The same thing is possible in relativity. The "piece of paper" is spacetime which has the geometrical structure of a manifold.

If this is so I wonder why rigorous definitions of manifolds are based on charts (that define local coordinate systems). IOW and simplifying: any object (that is of course also a topological space with certain topological features that make it well behaved) that can be "charted" is a manifold.
So the "piece of paper" must have the property that you can set up a coordinate system on it, that is its defining property if you want to call it a manifold. The fact that "you don't have to" is obviously relying on the fact that it is implicit in the definition, so I always have trouble understanding the insistence on banning charts.

 

[TrickyDicky]

Addressing the OP more specifically, maybe he is finding difficult to understand why if there is no two notions of velocity coexisting (an absolute and a relative velocity) in relativity, one does have both absolute and relative acceleration. While one justifies easily the first case because in relativity we have a spacetime rather than separate space and time and therefore one doesn't have any absolute space wrt which define an absolute velocity, this doesn't work in the case of acceleration and ultimately it seems one has to settle with "because this is just the way it is".
This isn't something that is very often clarified.

 

[stevendaryl]

Addressing the OP more specifically, maybe he is finding difficult to understand why if there is no two notions of velocity coexisting (an absolute and a relative velocity) in relativity, one does have both absolute and relative acceleration. While one justifies easily the first case because in relativity we have a spacetime rather than separate space and time and therefore one doesn't have any absolute space wrt which define an absolute velocity, this doesn't work in the case of acceleration and ultimately it seems one has to settle with "because this is just the way it is".
This isn't something that is very often clarified.

I don't think that things are that dissimilar when it comes to velocity and acceleration.

You can define a 4-velocity V^\mu by V^\mu = \dfrac{d}{d \tau} x^\mu and you can similarly define a 4-acceleration A^\mu. Neither is more absolute than the other. However, you can always choose coordinates so that the spatial components of V^\mu are all zero, but you can't always do that for the spatial components of A^\mu

 

[TrickyDicky]

However, you can always choose coordinates so that the spatial components of V^\mu are all zero, but you can't always do that for the spatial components of A^\mu

Right there, that's what I mean. This is the asymetry that is not so easy to explain. And maybe the OP naively thinks that if there is an absolute acceleration it would imply a rate of change of absolute velocity but that can't be because there is no such a thing as absolute velocity in relativity.
At this point I guess I should wait for the OP to confirm if this gets any close to his line of thought.

 

[harrylin]

[..] My point remains, that the "gravitational field" he refers to by "A gravitational field appears, that is directed towards the negative x-axis. Clock U1 is accelerated in free fall, until it has reached velocity v. An external force acts upon clock U2, preventing it from being set in motion by the gravitational field. When the clock U1 has reached velocity v the gravitational field disappears" is the field from the Christoffel symbols and it is entirely determined by the choice of coordinates. [..]

Once more, Einstein's comment related to your point was:
"To be sure, the accelerated coordinate systems cannot be called upon as real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion. But all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during the accelerated phases of the coordinate system K' they are accelerated relative to the latter and thereby can induce a gravitational field, similar to how electric charges in accelerated motion can induce an electric field."

please be explicit about what mathematical term you think he intends and why.

I think that Einstein was clear enough, and I was explicit in my clarification - he gave a physical explanation which I don't copy and to which you turned a blind eye. That shows that you also don't copy it; please stop trying to turn your agreement with me in an argument about something else.

 

[harrylin]

[..] In my previous post, I asked for a SR solution of the problem in which the non-inertial rocket is always at rest. I don't recall anyone presenting such a solution. My sense is that my calls for such a solution have been rebuffed. [..] The article discussing acceleration in special relativity also speaks of the accelerating object as in motion.

Indeed, SR presents solutions for the physical consideration of a traveler who changes velocity; SR isn't made for the view of a traveler who is constantly at rest, such that the universe is bouncing around while the traveler doesn't accelerate.

I, in my rocket, claim that I am not in motion. I have been sitting in my rocket in the same position throughout the episode. You claim that I will be younger than my twin at the end of the episode. As proof, you present to me a diagram that shows me in motion. I reject it. I categorically deny that the diagram applies to me. The diagram shows me in motion; I have not moved. [..]

An SR diagram does not address your issue. Einstein's 1918 paper does; however few people accept his Machian solution.

I would like to see a SR solution which has me in one position throughout the episode.

A non-moving traveler isn't addressed in SR. SR can only provide a mapping to rocket, such that the description is from the view of the accelerating rocket.

 

[Dale]

I think that Einstein was clear enough, and I was explicit in my clarification - he gave a physical explanation which I don't copy and to which you turned a blind eye.

I wasn't asking for a physical explanation, I was asking for clarification about its definition. What is meant by the term "gravitational field"? Until it is known what is meant by the term it is nonsense to even talk about providing a physical explanation for it. If I give you a physical explanation for a flubnubitz without defining the term, have I actually told you anything? E.g. "All the stars that are in the universe, can be conceived as taking part in bringing forth the flubnubitz."

You certainly were not explicit at all about what you believe he means by the term "gravitational field" even when directly asked for clarification, and you seem to disagree with Einstein's use of the term although you quote him. If you wish to clarify what specifically you believe Einsetin refers to by the term "gravitational field" then we can continue the discussion.

I don't understand your reluctance to clarify your position. Surely by now you realize how easily misunderstandings can arise in online forums. A request for clarification should always be taken seriously and complied with willingly.

 

[stevendaryl]

Once more, Einstein's comment related to your point was:
"To be sure, the accelerated coordinate systems cannot be called upon as real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion. But all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during the accelerated phases of the coordinate system K' they are accelerated relative to the latter and thereby can induce a gravitational field, similar to how electric charges in accelerated motion can induce an electric field."

I would say that there are several aspects of gravity:

1. For each point in spacetime, and for each possible initial velocity, there is a unique inertial path, the freefall path, or geodesic. These geodesics are physical, and are affected by the presence of matter and energy.
2. In general, geodesics that start close together and parallel do not remain close together and parallel as you follow them. This is a manifestation of spacetime curvature.
3. If one takes a path through spacetime that does not follow a geodesic, there will be resistance--it requires the application of a physical force to depart from geodesic motion.
4. If one uses a coordinate system in which coordinate axes are not geodesics, then the path of an object with no physical forces acting on it will appear "curved", meaning that the components of velocity in this coordinate system are not constant.

The first 3 aspects of gravity don't involve coordinates at all, and only mention physical forces, not fictitious forces. The 4th aspect to me is what the meaning of "fictitious forces" are, which is that the coordinate flows are not geodesics. The terminology "coordinate flow" is made up by me; I'm not sure what the technical term is, or if there is one. But in a similar way that specifying an initial point in spacetime, together with an initial velocity, determines a path through spacetime--the geodesic, specifying an initial point in spacetime, together with an initial velocity, determines a different path through spacetime, the coordinate flow, which is the solution to the component equation:

\dfrac{dV^\mu}{d\lambda} = 0

(where \lambda is the affine parameter for the path).

This would be a geodesic, if it were flat spacetime and inertial cartesian coordinates were used.

I think that Einstein was clear enough

Actually, I don't think he was very clear, mainly because he is using the words "gravitational field" (or something in German, more likely) without giving a precise definition of what it means.

 

[Dale]

The 4th aspect to me is what the meaning of "fictitious forces" are, which is that the coordinate flows are not geodesics. The terminology "coordinate flow" is made up by me; I'm not sure what the technical term is, or if there is one.

I think that the technical term would be "integral curves of the coordinate basis", but it certainly isn't a commonly used term. "Coordinate flows" sounds nice.

 

[my_wan]

I just wanted to add a few words for GregAshmore concerning the distinction between coordinate and proper acceleration in the form of a thought experiment, in the hopes it might help make it more obvious.

I take it you understand the relative nature of velocity, but when two inertial objects have a constant velocity with respect to each other it cannot be said which is 'really' moving and which is at rest. It depends on which coordinates you choose. If one hits the gas to speed up, not only does the occupant feel the force, but no matter which coordinates you choose every observer will agree that its speed is changing. Hence anybody with a good understanding of relativity can calculate the force felt by the occupants. This is the consequence of proper acceleration. However, depending on the coordinate choice of the observer, not everybody will agree on how much or in what direction it is accelerating with respect to their coordinates. The acceleration is absolute, but how much acceleration occurs in relation to a particular coordinate choice is relative. Some observers will even say they are slowing down, which also requires acceleration. This is referred to as coordinate acceleration.

To illustrate imagine sitting in a seat and tossing a rock straight up and catching it when it falls. That rock traces out a straight up and down line in your coordinates. Now imagine your seat is the back seat of a car moving at a constant 100 kph. The guy on the side of the road sees you toss the rock down the road and and the car carries you down with it to catch it. As far as the laws of physics are concerned the straight up and down path and the curved path up over the road are just as real. There are no 'real' paths in that sense, and the classical aether failed because it essentially sought to establish which path was real. It is effectively like an American arguing with the Chinese over which way is really up. The laws of physics allows everybody to agree on what is accelerating, though not necessarily by how much or what direction, but does not allow everybody to agree on what is moving at some constant velocity verses at rest.

Gravity turns this relationship on its head. Note the force felt when accelerating. When on the surface of a gravitational mass, like Earth, the principle of equivalence tells us this weight we feel is the same force we feel when accelerating. Now if you jump off a roof then while accelerating toward the ground you feel no force, effectively weightless. Hence, in your frame of reference you are at rest, i.e., not accelerating, but the Earth is accelerating toward you. Yet everybody in the Universe can agree that your speed is changing, even if not by how much. In this case proper gravity is absolute, while how much gravity and coordinate paths are relative. So in this case gravity is absolute, but the coordinate acceleration due to gravity is relative.

Now reconsider the twin paradox. If two observers experience the same amount of acceleration X time, then neither one will age any faster than the other. If two spaceship pass each other at a constant velocity, such that t=0 is mutually defined at the point of closest approach, then each will observe the others clock going slower than their own. To resolve this you accelerate your ship to catch up to the other, which requires applying an absolute force, i.e., proper acceleration. It doesn't matter whether you accelerate a little and take longer to catch up, or a lot to catch up faster, the effect is the same. You are the one that experienced a proper acceleration, hence your clock will be the one that appears to have slowed when you catch up. If you both apply the same proper acceleration to catch up to each other then no time dilation will be apparent under the definition given by t=0. Time is as fluid as the path of the rock in the back seat of the car, but like the rocks coordinate path it must transform from one coordinate choice to another by a well defined set of rules. Others have done an excellent job of articulating these quantitative rules here.

 

[Dale]

If one hits the gas to speed up, not only does the occupant feel the force, but no matter which coordinates you choose every observer will agree that its speed is changing.

Just a point of clarification. This should be "no matter which inertial coordinates you choose".

 

[my_wan]

Just a point of clarification. This should be "no matter which inertial coordinates you choose".

Yep, it even looks awkward stated the way I did now that you pointed it out.

 

[harrylin]

I just wanted to add a few words for GregAshmore [..]
Note the force felt when accelerating. When on the surface of a gravitational mass, like Earth, the principle of equivalence tells us this weight we feel is the same force we feel when accelerating. Now if you jump off a roof then while accelerating toward the ground you feel no force, effectively weightless. Hence, in your frame of reference you are at rest, i.e., not accelerating, but the Earth is accelerating toward you. Yet everybody in the Universe can agree that your speed is changing, even if not by how much. [..] So in this case gravity is absolute, but the coordinate acceleration due to gravity is relative.

Thanks for bringing that up, as it is exactly that modern argument that 1916GR denies; and I had the impression that GregAshmore noticed that point, that it's basically that issue that he discovered. Einstein tried to relativise acceleration by relativising gravitation, so that it's a matter of free opinion if a rocket accelerates or not. Nowadays few people accept that view.

Now reconsider the twin paradox. If two observers experience the same amount of acceleration X time, then neither one will age any faster than the other.

That argument fails in the first version by Langevin, see my earlier remarks as well as elaborations by others.

 

[PAllen]

Gravity turns this relationship on its head. Note the force felt when accelerating. When on the surface of a gravitational mass, like Earth, the principle of equivalence tells us this weight we feel is the same force we feel when accelerating. Now if you jump off a roof then while accelerating toward the ground you feel no force, effectively weightless. Hence, in your frame of reference you are at rest, i.e., not accelerating, but the Earth is accelerating toward you. Yet everybody in the Universe can agree that your speed is changing, even if not by how much. In this case proper gravity is absolute, while how much gravity and coordinate paths are relative. So in this case gravity is absolute, but the coordinate acceleration due to gravity is relative.

Not quite everyone in the universe: someone who jumped off the roof next to you would not agree. As for someone far away, in GR, the whole concept of relative velocity at a distance is fundamentally ambiguous because you can't compare vectors at a distance in curved spacetime. So this argument is not so clear cut.

I would agree that gravity is absolute for a different reason: tidal 'forces', physically; curvature geometrically. Tidal forces are detectable in a small region.

 

[GregAshmore]

I decided that I ought to do the calculations that I should have done Friday night, before reading any of the posts or papers referenced since then. Here's my best shot at the Twin Paradox. Later this evening, or maybe tomorrow night, I'll see how what I did compares with your suggestions.

I can't say that I did this without any outside influences since Friday night. I did see that the image that George referenced looks like a spacetime diagram. That may have triggered some thoughts about coordinates.

I do all the calculations for the spacetime diagram but did not include an image of it. You all know what it looks like.

Solution of the Twin Paradox - to the degree possible knowing only the Lorentz transform and the usage of the spacetime diagram.

Given:
G1. Earth and rocket are both at rest at same position. Earth clock and rocket clock are synchronised.
G2. At time 0.0, rocket fires a pulse.
G3. Earth and rocket separate at relative velocity 0.8c.
G4. At distance 10 units from Earth, as measured in Earth frame, rocket fires a pulse.
G5. Earth and rocket approach at relative velocity -0.8c.
G6. Upon reaching Earth, rocket fires a pulse, coming to rest on Earth.
G7. Gravitational effects of mass are to be ignored.

Questions:
Q1. Both the Earth and the rocket claim to be at rest throughout the episode. Can both make good on this claim?

Q2. What are the clock readings on Earth and rocket at G6?

Q3. Are the clock readings calculated for Q2 unambiguously unique?

Solution:
The questions are with regard to kinematics only: positions and times. With one exception noted later, the dynamics of the episode need not be considered.

It will also be assumed that acceleration does not affect clocks. This assumption, together with G7, allows special relativity to be used in the attempt at a solution.

It will also be assumed that the acceleration from rest to velocity V is instantaneous. Any effect assumption this might have on the calculated clock readings will be ignored for the purposes of this exercise.

The Earth is at the origin of its own coordinate system. Likewise, the rocket is at the origin of its own coordinate system. By G1, the origins are coincident at the start of the episode. For convenience, the X axes of the two systems are colinear, and the relative velocity is along that axis, with positive V in the positive X direction.

Four spacetime events will be considered.
Event A: Corresponds to G2. Time and position are zero in both the Earth frame and the rocket frame. Velocity of the rocket frame is V. For convenience, the axes are set so that positive V is along coordinates ; positive V is made to be along the for convenience . For the moment, the Earth frame will be shown with orthogonal the velocity will be shown "to the right" on the spacetime diagram,

Event B: Corresponds to G4, on the worldline of the Earth.

Event C: Corresponds to G4, on the worldline of the rocket.

Event D: Corresponds to G6. The worldlines of the Earth and rocket meet here, and become colinear.

Notation:
T represents time.
X represents position.

Events and frames are represented by lower case letters; event followed by frame.
e represents Earth frame.
r represents rocket frame.

Example: Tbe represents the time at event B in the Earth frame.

Times are given as the distance that light travels in one unit of time. T = ct.
With this unit of time, and with velocity given as constant factor of c (v = Vc), the Lorentz transforms have the form:
X' = g(X - VT)
T' = g(T - VX)
where g = 1 / Sqrt(1 - V^2)

With V = 0.8, g = 1.667


Calculate time in Earth frame at Events B and C.
By the statement of G4, events B and C are simultaneous in the Earth frame. Measurements of distance in a frame are by definition taken at a single instant in the frame.

Time in the Earth frame at Events B and C is distance from Earth to rocket (as measured in Earth frame) divided by relative velocity.
Tbe = Tce = 10 / V = 10 / 0.8 = 12.5.


Calculate coordinates at Event B.
Xbe = 0.0 (Earth is inertial; Xae = 0.0; position in an inertial frame does not change with time.)
Tbe = 12.5 (As calculated above.)

Xbr = 1.667 * (0.0 - (0.8 * 12.5)) = -16.67
Tbr = 1.667 * (12.5 - (0.8 * 0.0)) = 20.83


Calculate coordinates at Event C.
Xce = 10.0 (By G4)
Tce = 12.5 (As calculated above.)

Xcr = 1.667 * (10.0 - (0.8 * 12.5)) = 0.0
Tcr = 1.667 * (12.5 - (0.8 * 10.0)) = 7.5

Note that calculation of Xcr confirms what is already known. Xcr must be zero because Xar = 0.0 and the rocket is inertial to this point.


Earth or rocket must change frames at velocity reversal.
The reversal of velocity in G4 must be represented by a change of frame in the spacetime diagram. Without a change of frame, the worldlines of Earth and rocket can never meet.
Either the Earth or the rocket, or both, must change frames.
The Earth cannot change frames: No unbalanced force acts on it; it is inertial.
The rocket must change frames: It is acted on by an unbalanced force; it is not inertial.


Setting up the new rocket frame.
The rocket will be in its new frame during the approach in G5. The rocket approach frame will be represented by the addition of the lower case 'p' to the notation.

The rocket must be assigned position and time coordinates in its approach frame. At the start of an exercise, coordinate values may assigned at will, due to the linearity of the Lorentz transform. In this case, the approach frame comes into play at an event in an ongoing episode, at Event C. To maintain correspondence with the physical reality, and taking into account the assumption of instantaneous acceleration, the coordinates of the rocket in the approach frame at Event C must match the coordinates of the rocket in the original separation frame at Event C.


Transformation from the rocket approach frame to the Earth frame.
The Lorentz transformation equations were derived with the origins of the two frames coincident. Therefore, Event C must be treated as a local origin for the purposes of transformation from frame to frame. Event coordinates relative to the local origin are transformed from frame to frame, as shown in the following equations.

In these equations, replace the underscore with the symbol for the event to be transformed.

To transform from the Earth frame to the rocket approach frame:
X_rp = g((X_e - Xce) - V(T_e - Tce)) + Xcrp
T_rp = g((T_e - Tce) - V(X_e - Xce)) + Tcrp

To transform from the rocket approach frame to the Earth frame:
X_e = g((X_rp - Xcrp) + V(T_rp - Tcrp)) + Xce
T_e = g((T_rp - Tcrp) + V(X_rp - Xcrp)) + Tce

As discussed above,
Xce = 10.0
Tce = 12.5

Xcrp = 0.0
Tcrp = 7.5

Calculate the coordinates of Event B in the rocket approach frame.
Xbrp = g((Xbe - Xce) - V(Tbe - Tce)) + Xcrp
Xbrp = 1.667((0.0 - 10.0) - (-0.8)(12.5 - 12.5)) + 0
Xbrp = -16.67 (Same as Xbr)

Tbrp = g((Tbe - Tce) - V(Xbe - Xce)) + Tcrp
Tbrp = 1.667((12.5 - 12.5) - (-0.8)(0.0 - 10.0)) + 7.5
Tbrp = 1.667(0 - (-0.8)(-10.0)) + 7.5
Tbrp = 1.667(0 - 8.0) + 7.5
Tbrp = 1.667(-8.0) + 7.5
Tbrp = -5.83 (Compare 20.83 for Tbr)

Calculate the coordinates of Event D.
Time for approach is same as time for separation. (Relative velocity is the same.)

Xde = 0.0
Tde = Tbe + Tbe = 25.0

Xdrp = g((Xde - Xce) - V(Tde - Tce)) + Xcrp
Xdrp = 1.667((0.0 - 10.0) - (-0.8)(25.0 - 12.5)) + 0.0
Xdrp = 1.667(-10.0 - (-10.0)) + 0.0
Xdrp = 0.0 (Confirms inertial behavior of rocket from Event C to Event D: Xcrp = 0.0)

Tdrp = g((Tde - Tce) - V(Xde - Xce)) + Tcrp
Tdrp = 1.667((25.0 - 12.5) - (-0.8)(0.0 - 10.0)) + 7.5
Tdrp = 1.667(12.5 - (-0.8)(-10.0)) + 7.5
Tdrp = 1.667(12.5 - 8.0) + 7.5
Tdrp = 7.5 + 7.5
Tdrp = 15.0 (Confirms approach time equals separation time in rocket frame.)


Answers to questions:
Q1. Both the Earth and the rocket claim to be at rest throughout the episode. Can both make good on this claim?
Yes, kinematically. Earth and rocket X coordinates are 0.0 throughout.
Whether this makes physical sense dynamically is unknown, given limited knowledge noted above.

Q2. What are the clock readings on Earth and rocket at G6?
The Earth clock reads 25.0.
The rocket clock reads 15.0.

Q3. Are the clock readings calculated for Q2 unambiguously unique?
Yes. There is only one way to construct the spacetime diagram, due to the unique non-inertial behavior of the rocket.
Visually, the Earth and rocket experiences are symmetric. Each sees the other move away and return. Nevertheless, the rocket is unambiguously younger than the Earth at reunion.

 

[ghwellsjr]

Answers to questions:
Q1. Both the Earth and the rocket claim to be at rest throughout the episode. Can both make good on this claim?
Yes, kinematically. Earth and rocket X coordinates are 0.0 throughout.
Whether this makes physical sense dynamically is unknown, given limited knowledge noted above.

The Earth can claim to be at rest in an inertial frame. The rocket can claim to be at rest in a non-inertial frame.

Q2. What are the clock readings on Earth and rocket at G6?
The Earth clock reads 25.0.
The rocket clock reads 15.0.

Correct.

Q3. Are the clock readings calculated for Q2 unambiguously unique?
Yes.

Correct.

There is only one way to construct the spacetime diagram, due to the unique non-inertial behavior of the rocket.

I don't know why you would say this. Are you referring to the Earth's inertial rest frame? There are an infinite number of ways to construct spacetime diagrams for your scenario, all with different velocities with respect to each other and all just as valid and all producing the same final clock readings and the same things that each observer actually sees.

Visually, the Earth and rocket experiences are symmetric. Each sees the other move away and return. Nevertheless, the rocket is unambiguously younger than the Earth at reunion.

No, they are not symmetric. The rocket sees the Earth move away at 0.8c for a distance of 3.333 units and then remain at that distance for some time and then come back to the rocket. But the Earth sees the rocket move away at 0.8c to a distance of 10 units and then immediately start coming back. It doesn't matter what spacetime diagram you use to depict your scenario, they are all just as valid and produce the same results. The only thing that is different in them are the values of the coordinates (and the geometric shapes of the worldlines).

 

[my_wan]

Einstein tried to relativise acceleration by relativising gravitation, so that it's a matter of free opinion if a rocket accelerates or not. Nowadays few people accept that view.

In a sense I would say Einstein succeeded via the principle of equivalence, with caveats. The first problem is as PAllen stated in his objection to my use of the term "everyone". A gravitational field cannot be globally transformed away unless it is itself globally uniform. The second problem is that, even if you could, any time two inertial observers are accelerated with respect to each other a gravitational field must be involved, such that these two observer cannot be at rest with respect to each other and still be inertial. Everyone can agree that a gravitational field exist even if they may not agree on where the gravitational field is located, its geometry, etc. This issue is the reason energy conservation became so controversial in GR, but it's really more a localization issue than a conservation issue.

The break in the symmetry under both special and general relativity occurs whenever an observer enters a non-inertial state. Sitting motionless on the surface of the Earth is a non-inertial state, which is why you feel weight. By the principle of equivalence you feel this same g-force when you accelerate under special relativity, which breaks the symmetry GregAshmore is wanting to absolutely maintain, leading to his difficulties.

You cannot break this symmetry in either special or general relativity and pretend it is not broken. Once this symmetry, which applies only to inertial observers, is broken it is no longer "matter of free opinion" as to whether it is broken or not. GregAshmore is assuming the symmetry wasn't broken when his spaceship was accelerated. Whether this symmetry is broken by a rocket engine (SR) or a gravitational field (GR) makes no difference, though both break it in a different or inverse manner, breaking this symmetry requires one or the other.

 

[harrylin]

[..] The break in the symmetry under both special and general relativity occurs whenever an observer enters a non-inertial state. [..]
You cannot break this symmetry in either special or general relativity and pretend it is not broken. Once this symmetry, which applies only to inertial observers, is broken it is no longer "matter of free opinion" as to whether it is broken or not. GregAshmore is assuming the symmetry wasn't broken when his spaceship was accelerated. Whether this symmetry is broken by a rocket engine (SR) or a gravitational field (GR) makes no difference, though both break it in a different or inverse manner, breaking this symmetry requires one or the other.

That's correct of course; perhaps I read too much in GregAshmore's issues and is it only a matter of problems with the calculation methods. If so, then that should be easy to fix. So I'll also look into his last attempt.

 

[harrylin]

[..] Here's my best shot at the Twin Paradox. Later this evening, or maybe tomorrow night, I'll see how what I did compares with your suggestions.[..]
Solution of the Twin Paradox - to the degree possible knowing only the Lorentz transform and the usage of the spacetime diagram.

OK - that implies purely SR. As you seem to have solved the equations without issues, I'll skip those.

Given:
G1. Earth and rocket are both at rest at same position. Earth clock and rocket clock are synchronised.
G2. At time 0.0, rocket fires a pulse.
G3. Earth and rocket separate at relative velocity 0.8c.
G4. At distance 10 units from Earth, as measured in Earth frame, rocket fires a pulse.
G5. Earth and rocket approach at relative velocity -0.8c.
G6. Upon reaching Earth, rocket fires a pulse, coming to rest on Earth.
G7. Gravitational effects of mass are to be ignored.

Questions:
Q1. Both the Earth and the rocket claim to be at rest throughout the episode. Can both make good on this claim?

It depends on what they supposedly mean with that. In normal use of the word "in rest", in the context of SR calculations, the rocket cannot claim to be all the time in rest. This is because it's not all the time in rest in any inertial frame. Precision: "inertial frame" in SR means a set of coordinate systems that is in rectilinear uniform motion according to Newton's mechanics; also called by Einstein a "Gallilean" reference system.

Q2. What are the clock readings on Earth and rocket at G6?

Q3. Are the clock readings calculated for Q2 unambiguously unique?

Solution:
The questions are with regard to kinematics only: positions and times. With one exception noted later, the dynamics of the episode need not be considered.

While that is often said, it is only half true, and therefore misleading. The dynamics is inherent by the prescription of reference to inertial frames for the Lorentz transformations. If one purely considered kinematics only then the situation would be symmetrical.

[..]
Earth or rocket must change frames at velocity reversal.
The reversal of velocity in G4 must be represented by a change of frame in the spacetime diagram. Without a change of frame, the worldlines of Earth and rocket can never meet.
Either the Earth or the rocket, or both, must change frames.
The Earth cannot change frames: No unbalanced force acts on it; it is inertial.
The rocket must change frames: It is acted on by an unbalanced force; it is not inertial.

Note my earlier clarifications why such kind of reasoning does not generally hold. What matters for your SR calculation is that the rocket is not all the time at rest in an inertial frame. Also the Earth is not at rest in an inertial frame as it is in an orbit around the Sun; however the effect is small compared to the rocket. That is another simplification of the calculation.

[..] Visually, the Earth and rocket experiences are symmetric. Each sees the other move away and return. Nevertheless, the rocket is unambiguously younger than the Earth at reunion.

Right.
[addendum: kinematically the situation looks symmetrical (which is what I supposed you meant); however next George correctly highlights that of course there are visual differences that can be observed. That is pertinent for understanding the physics. This difference in observations has also been elaborated by Langevin in the article that I linked earlier.]

 

[ghwellsjr]

Given:
G1. Earth and rocket are both at rest at same position. Earth clock and rocket clock are synchronised.
G2. At time 0.0, rocket fires a pulse.
G3. Earth and rocket separate at relative velocity 0.8c.
G4. At distance 10 units from Earth, as measured in Earth frame, rocket fires a pulse.
G5. Earth and rocket approach at relative velocity -0.8c.
G6. Upon reaching Earth, rocket fires a pulse, coming to rest on Earth.
G7. Gravitational effects of mass are to be ignored.

Here is a spacetime diagram for Earth's inertial rest frame:

Earth is shown as the wide blue line and the rocket as a wide red line. Dots along each path indicate one unit of elapsed Proper Time and I have marked most of them. I have also marked the four events that you indicated.

Here is a diagram to show the rocket at rest:

Both Earth and the rocket send a signal to the other one every unit of Proper Time. These signals provide the information that accounts for the visualization that you mentioned at the end of your post. The diagrams make it obvious that the situation is not symmetrical between the Earth and the rocket. They also make it clear that either diagram will provide all the information to determine the visualization of either observer.

You should track a few of the signals, noting the Proper Time (according to the dots) they were sent and received and then go to the other diagram and confirm the same information.

I didn't necessarily use the same coordinates that you used but, again, this will have no bearing on any outcome. The Coordinate Times are not significant when comparing between frames, only the Proper Times matter.

 

[harrylin]

It may be useful to elaborate a little on Langevin's discussion about the fact that acceleration has an absolute sense, as he meant it in a slightly different way than those people in this forum to which your first post relates; however Langevin gave the "twin" example for exactly this purpose, to illustrate the "absolute" effects of acceleration. The way he meant it is made clear by his description (as well as by the text that precedes it, but that's too long to cite here):

Only a uniform velocity relative to it cannot be detected, but any change of velocity, or any acceleration has an absolute sense. [..]
the laws of electromagnetism are not the same in respect to axes attached to this [accelerated] material system as in respect to axes in collective uniform motion of translation.
We will see the appearance of this absolute character of acceleration in another form. [..]
For [..] observers in uniform motion [..]l the proper time [..] will be shorter than for any other group of observers associated with a reference system in arbitrary uniform motion. [..] We can [..] say that it is sufficient to be agitated or to undergo accelerations, to age more slowly, [..]

Giving concrete examples: [..]
This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further [in time].
For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so.
[etc.]

- starting p.47 of http://en.wikisource.org/wiki/The_Evolution_of_Space_and_Time

 

[GregAshmore]

I don't know why you would say this. Are you referring to the Earth's inertial rest frame? There are an infinite number of ways to construct spacetime diagrams for your scenario, all with different velocities with respect to each other and all just as valid and all producing the same final clock readings and the same things that each observer actually sees.

I was too general in my wording. I only meant that Events A, B, and C have been placed, there is only one way to place Event D.

No, they are not symmetric. The rocket sees the Earth move away at 0.8c for a distance of 3.333 units and then remain at that distance for some time and then come back to the rocket. But the Earth sees the rocket move away at 0.8c to a distance of 10 units and then immediately start coming back. It doesn't matter what spacetime diagram you use to depict your scenario, they are all just as valid and produce the same results. The only thing that is different in them are the values of the coordinates (and the geometric shapes of the worldlines).

Again, poor choice of words. "Similar" would have been better than "symmetry".

 

[GregAshmore]

Right there, that's what I mean. This is the asymetry that is not so easy to explain. And maybe the OP naively thinks that if there is an absolute acceleration it would imply a rate of change of absolute velocity but that can't be because there is no such a thing as absolute velocity in relativity.
At this point I guess I should wait for the OP to confirm if this gets any close to his line of thought.

Well, I did have a thought that might resolve to something like what you say. As I recall, someone defined proper acceleration as the derivative of proper velocity with respect to proper time. I don't understand how there can be proper velocity at all, because proper time is the elapsed time at "the same place". If position never changes with respect to time, it would seem that proper velocity must be zero. I did not say anything about it because my objection was much more basic than that, and I figured there would be a logical explanation for it if and when I get that far.

 

[GregAshmore]

I think that this is probably one of the key topics of modern relativity.

The key concepts of relativity, both special and general, are geometrical (Minkowski geometry for SR and pseudo-Riemannian geometry for GR).

Just like you can take a piece of paper and draw geometrical figures and discuss many things, such as lengths and angles, without ever setting up a coordinate system. The same thing is possible in relativity. The "piece of paper" is spacetime which has the geometrical structure of a manifold. The "geometrical figures" are worldlines, events, vectors, etc. that represent the motion of objects, collisions, energy-momentum, etc.

In this geometrical approach, the twin scenario is simply a triangle, and the fact that the traveling twin is younger is simply the triangle inequality for Minkowski geometry. In a coordinate-independent sense, the traveler's worldline is bent, and that in turn implies that his worldline is necessarily shorter as a direct consequence of the Minkowski geometry.

Now, on top of that underlying geometry, you can optionally add coordinates. Coordinates are simply a mapping between points in the manifold (events in spacetime) and points in R4. The mapping must be smooth and invertible, but little else, so there is considerable freedom in choosing the mapping. It is possible to choose a mapping which maps straight lines in spacetime to straight lines in R4, such mappings are called inertial frames.

It is also possible to choose a mapping which maps bent lines in spacetime to straight lines in R4, a non-inertial frame. Such a mapping does nothing to alter the underlying geometry. The bent lines are still bent in a coordinate-independent geometrical sense, but because it simplifies the representation in R4 it can still be useful on occasion in order to simplify calculations.

Because the mapping is invertible, in many ways it doesn't matter if you are talking about points in the manifold or points in R4. So you talk about things being "at rest" based on R4, and things happening "simultaneously" based on R4, and many other things. However, it is occasionally important to remember the underlying geometry.

I hope this helps.

It does.

It has, in fact, been made clear to you that SR can handle the twins paradox. What obviously hasn't been made clear to you is why. Your repetition of bald assertions that have already been contradicted is unhelpful. It wastes your time in repeating it and it wastes our time in repeating our responses. It also irritates those (maybe only me) who feel like their well-considered and helpful responses have been completely ignored by you.

Not ignored; not understood. Annoying either way, when the reason for not understanding is a failure to work out misunderstandings on paper before making statements.

Another factor on my side was that I thought you did not understand exactly what I was troubled by. Working through the twin paradox, looking for an answer to what troubled me, also led me to understand why SR is valid for solving the problem, at least with respect to kinematics. (I don't say SR isn't valid with respect to dynamics, only that I don't know enough to say it is.) I know that what I did wrt the twin paradox is at the most elementary level. But for me, it was like the transition from saying "ga ga, goo goo" to standing up on two feet and taking a step or two (before stumbling). Hopefully I will be less annoying in future.

 

[GregAshmore]

It depends on what they supposedly mean with that. In normal use of the word "in rest", in the context of SR calculations, the rocket cannot claim to be all the time in rest. This is because it's not all the time in rest in any inertial frame.

I struggled with the issue of whether the rocket is at rest throughout, or not. Part of the struggle has to do with the definition of "frame". If one defines frame to be an inertial frame, the rocket is not at rest while accelerating, while changing inertial frames. However, if one recognizes non-inertial frames, then the rocket is at rest throughout. If a frame is the same thing as the coordinate system whose origin is coincident with an observer, then there is indeed such a thing (in the abstract) as a non-inertial frame. You will note that I hadn't fully thought this through when I posted the calculations: I made sure to say "coordinate system" instead of "frame" when describing the setup. Edit: And it was George's clarification which helped me see this.

I don't say that I fully understand the concept of "absolute proper acceleration" being compatible with "no absolute space". However, it seems to me that this is something that needs to be considered with the dynamics of SR. Kinematically, the spacetime diagram shows that the rocket is at rest in its non-inertial frame.

That, by the way, is the objection that I felt was being dismissed--the call for consideration of the case in which the rocket is at rest. If I (now, or finally) understand George correctly, that objection is intentionally dismissed by Taylor and Wheeler. Indeed, when the objection is raised, it could be shown, using the spacetime diagram that is already under consideration, that the rocket is at rest throughout the episode. What the objector wants (or at least, what I wanted) was for the symmetrical diagram to be drawn, because he thinks that this is the only spacetime diagram in which the rocket is at rest. That thought is based on a misconception of the spacetime diagram--a misconception which I began to perceive as I thought more about the spacetime diagrams I drew for the pole-in-barn paradox.

While that is often said, it is only half true, and therefore misleading. The dynamics is inherent by the prescription of reference to inertial frames for the Lorentz transformations. If one purely considered kinematics only then the situation would be symmetrical.

The "one exception" (which I did not explicitly point out later, I realize now) is the rocket being non-inertial while accelerating. That is a dynamic phenomenon.