Implications of the statement "Acceleration is not relative"

my_wan

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

harrylin

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

PAllen

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

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.

Correct.

Correct.
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.

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

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

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

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

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.

Attached Thumbnails

   

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_Ev...Space_and_Time

GregAshmore

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.

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

GregAshmore

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

It does.

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

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.



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.

GregAshmore

my_wan said,
I take it that the bold text above is what harrylin refers to as the modern argument...

No, I was not disagreeing with Einstein's contention that it is a matter of free opinion as to whether the rocket accelerates or not.

My understanding is that, even according to modern ideas, it is indeed a matter of free opinion as to whether the rocket accelerates--if one considers acceleration in the usual sense of "rate of change in velocity as measured with respect to a set of coordinates". The statement was, "Coordinate acceleration is relative". (Edit to clarify.)

It is proper acceleration which is absolute; but one may be at rest in a coordinate system while experiencing proper acceleration.

There is at least one person in the universe who will disagree with the claim that the rocket is accelerating: the observer in the rocket who is convinced that he is at rest. (This, I think, is along the lines of another comment on the claim, posted by someone else.)

Still, I don't know anything about how the dynamics of the "resting while accelerating rocket" work, so I'm not making any statement of my own opinion on this issue. I'm only giving my understanding of what I have been told.

DaleSpam

Yes.

Correct.

It seems like you get the distinction between coordinate and proper acceleration.

harrylin

I somewhat agree with that; I quickly looked around for definitions and if I see it correctly the definition by Smoot of "proper acceleration" relates to the coordinate acceleration with respect to an instantaneously co-moving inertial frame (thus there is a change of velocity from zero to non-zero) and that makes sense to me.
In contrast, for me the definition of "proper acceleration" as given in Wikipedia is a misnomer for what I would call apparent gravitation. - http://en.wikipedia.org/wiki/Proper_acceleration
It may be that such different definitions bugged you (they did bug me in earlier discussions).