Relativity: Absolute Acceleration & the Twin Paradox

[joey_m]

It seems to me that all of the talk about "absolute" acceleration is a complete non-issue, in terms of relativistic effects. For, if there is no such thing as absolute position, then there can be no such thing as absolute change in position, whether this change is understood in the sense of uniformity (non-accelerated) or non-uniformity (i.e. accelerated).

As far as relativity is concerned, then, isn't there always complete symmetricity as far as relatively moving observers are concerned, no matter the uniformity of this motion?

I always see references to g-forces being the determining factor as far as who is "truly" accelerating. But isn't this the precise reason why Einstein always used such things as "practically rigid rods" and "ideal clocks" in his thought experiments? In other words, things that are "practically rigid" or "ideal" are, by definition, impervious to the stresses caused by external forces, are they not?

It seems that people who invoke such things as the "twin paradox" are overlooking these essential considerations of Einstein.

But my real confusion arises when I see that wikipedia (in http://en.wikipedia.org/wiki/Twin_paradox) attributes this same kind of fallacious thinking about absolute acceleration to Einstein himself!

So what's the deal? Is wikipedia lying? Or did Einstein really contradict his earlier work in his later years? Or am I just totally nuts?

 

[yuiop]

In this example of the twin's paradox one twin accelerates away on his outward journey to 0.8c in an Easterly direction. If we call this "absolute" acceleration then we imply that we are absolutely sure that the accelerated twin's clock rate is slower than that of the Earth twin, but this is not true. To an observer that has always been moving at 0.8c in an Easterly direction relative to the Earth it looks like the accelerated twin has de-accelerated to a stop and therefore his clock rate should now be going faster than that of the Earth twin. As you can see there is nothing absolute here and it is impossible to determine whose clock rate has actually changed until the twins are brought together again. If the Earth twin that remained behind (Edward) decides to chase after his spacebound sibling (Adam) then it turns out that it is Edward that aged the least when he catches up with his brother. If Edward behaves himself and stays at home like he supposed to in the classical twins paradox then it is Adam that ages the least when they meet again on Earth. It is impossible to determine what the absolute clock rates are on the outward journey and clearly if one observer sees an object as accelerating while another sees it as de-accelerating then the notion of absolute acceleration is nonsense.

It is probably best to view the twins paradox in terms of the path length through spacetime. The twin that takes the shortest (straightest) path through spacetime ages the most and any observer will agree with this measurement.

However, it is only fair to point out that a notion of absolute rotation (which is a form of acceleration) is valid.

As far as "practically rigid rods" are concerned, it is well known that an absolutely infinitely rigid rod is not compatible with relativity.

 

[A.T.]

I always see references to g-forces being the determining factor as far as who is "truly" accelerating.

That is a way to define acceleration, which is independent of the observer. It's similar to the definition of proper time:
- Proper time is what a clock measures.
- Acceleration is what a accelerometer measures.

But isn't this the precise reason why Einstein always used such things as "practically rigid rods" and "ideal clocks" in his thought experiments?

I would say, "absolutely rigid rods" are used in thought experiments, to omit effects of the maximum signal velocity c, so you can examine the "relativistic effects" alone. Absolutely rigid rods don't exists, so in reality you have to consider the signal velocity in your observation, which is no problem if you know it. But for thought experiments you can omit this technical detail.

 

[Dale]

It is probably best to view the twins paradox in terms of the path length through spacetime. The twin that takes the shortest (straightest) path through spacetime ages the most and any observer will agree with this measurement.

I agree. The acceleration explanation is technically correct as far as breaking the symmetry goes, but it cannot be used to determine the magnitude of the difference in ages when the twins are reunited nor can it be used in complicated situations. The spacetime interval is clear, simple, and detailed. It also is applicable for arbitrary geometries, i.e. where both twins are accelerating along some complicated worldline.

 

[JesseM]

It seems to me that all of the talk about "absolute" acceleration is a complete non-issue, in terms of relativistic effects. For, if there is no such thing as absolute position, then there can be no such thing as absolute change in position, whether this change is understood in the sense of uniformity (non-accelerated) or non-uniformity (i.e. accelerated).

Consider a geometric analogy. Suppose you have a 2D plane with some lines drawn on it, and want to place an xy coordinate system on the plane to describe the lines. Since you are free to orient your coordinate axes any way you like, there is no absolute truth about the slope of the lines--i.e. the rate that their x-coordinate changes as you vary the y-coordinate. But do you think this means there can also be no absolute truth about whether the lines have constant slope or non-constant slope, i.e. whether they are straight lines or non-straight lines?

I always see references to g-forces being the determining factor as far as who is "truly" accelerating. But isn't this the precise reason why Einstein always used such things as "practically rigid rods" and "ideal clocks" in his thought experiments? In other words, things that are "practically rigid" or "ideal" are, by definition, impervious to the stresses caused by external forces, are they not?

It's impossible to have perfectly rigid objects in relativity, since this would imply that pushing one end of the object causes the other to move instantaneously, an FTL effect--I think Einstein used "practically" there to denote the idea that as long as you restrict the rods to move inertially, then they can behave as if they are rigid, although you would see they are not if you tried to accelerate them.

In any case, there would be numerous experimental ways you could determine if you were accelerating or not in flat spacetime. For example, you could hold out a ball in front of you so that it is still in your coordinate system, then let go of it; if it continues to stay at the same coordinates then you're moving inertially, if it begins to move then you're accelerating. Many laws of physics that hold in inertial coordinate systems would not hold in accelerating ones--the speed of light is not even constant in accelerating coordinate systems! So, there is really no ambiguity about whether one is accelerating or not.

 

[HallsofIvy]

It seems to me that all of the talk about "absolute" acceleration is a complete non-issue, in terms of relativistic effects. For, if there is no such thing as absolute position, then there can be no such thing as absolute change in position, whether this change is understood in the sense of uniformity (non-accelerated) or non-uniformity (i.e. accelerated).

As far as relativity is concerned, then, isn't there always complete symmetricity as far as relatively moving observers are concerned, no matter the uniformity of this motion?

No, acceleration is fundamentally different from either position or velocity. And you don't need (Einsteinian) relativity to see that- it goes back to Gallilean relativity. Gallileo argued that if you were riding in a closed coach going at a rigidly constant velocity (no bumps or anything that would change the velocity) you could not perform any experiment inside the coach that would tell you how fast you were going. Of course, Gallileo didn't know about electricity or magnetism. It was the fact, from Maxwell's equations, that electro-magnetic force was dependent on the speed of an electron, that seemed to violate that and led to experiments involving electromagnetism (i.e. light- the Michaelson-Morley experiment among others) and then to Einsteinian relativity to show why electro-magnetic force did not violate it.

But acceleration is different- Force= mass * acceleration so we can feel acceleration.

 

[joey_m]

All of you guys seem to be missing one crucial thing: general relativity. When I invoked the term, "acceleration," that should have clued you in. After all, what does the following statement mean...

... we wish to understand by the "general principle of relativity" the following statement: All bodies of reference K, K1, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion.

(Source: Chapter 18 of Relativity: The Special and General Theory)

In other words, special relativity is just an arbitrary case of the general theory. It is of no fundamental importance whatsoever.

 

[JesseM]

All of you guys seem to be missing one crucial thing: general relativity. When I invoked the term, "acceleration," that should have clued you in. After all, what does the following statement mean...



(Source: Chapter 18 of Relativity: The Special and General Theory)

In other words, special relativity is just an arbitrary case of the general theory. It is of no fundamental importance whatsoever.

This seems to be kind of a complicated issue...my understanding is that what Einstein called the "general principle of relativity" is now usually interpreted as something called "general covariance" or "diffeomorphism invariance", but it turns out that any law of physics can be stated in a generally covariant tensor form which will work for arbitrary coordinate systems, not just GR. As I said in post #29 on this thread:

General covariance is not really a physical feature of laws at all but more of a feature of the mathematical form the law is stated in, since any law whatsoever can be put into the appropriate tensor form and it will then be generally covariant, even Newtonian laws. See the last paragraph on http://www.bun.kyoto-u.ac.jp/~suchii/gen.GR7.html Julian Barbour writes:

In 1907, Einstein realized that the equivalence principle enabled him to extend the restricted relativity principle to include another ‘impotence’ – the inability to detect uniform acceleration. This insight was decisive. Some years earlier, Einstein had read Mach’s critique of Newton’s absolute space and time and was extremely keen to reformulate dynamics along the broad lines advocated by Mach. He wanted to show that absolute space did not correspond to anything in reality – that it could not be revealed by any experiment. Special relativity had shown that uniform motion – relative to the ether or to absolute space – could not be detected by any physical process. The equivalence principle suggested to him that it might be possible to extend his relativity principle further. If he could extend it so far as to show that the laws of nature could be expressed in identical form in all conceivable frames of reference, this requirement of general covariance would “[take] away from space and time the last remnant of physical objectivity”[6]. He would have achieved his Machian aim. Unfortunately, within two years Einstein had been forced by a critique of Kretschmann [7] to acknowledge that any physical theory must, if it is to have any content, be expressible in generally covariant form. He argued [8] that the principle nevertheless had great heuristic value. One should seek only those theories that are simple when expressed in generally covariant form. However, Einstein gave no definition of simplicity. Since then, and especially as a result of quantization attempts, there has been a vast amount of inconclusive discussion about the significance of general covariance [9] and its implications for quantization.

Barbour then discusses his own notion of what it means to say a theory is "simple" in general covariant form, but there doesn't seem to be any widely agreed-upon meaning.

Also see http://www.bun.kyoto-u.ac.jp/~suchii/gen.relativity.html for an even more detailed discussion, which concludes:

Thus, we have to be very careful, not to be misled by the similarity of names, "special relativity" and "general relativity"; in a nutshell, there is no "generalized principle of relativity" as Einstein once intended. Instead, we have a nice theory of gravity, with no priviledged frame of reference, except for local Lorentzian systems.

By "local Lorentzian systems" they mean locally inertial coordinate systems in a small patch of spacetime where the curvature is negligible. On the subject of why they call them "priviledged", they had earlier quoted Friedman saying:

The indistinguishability group of a space-time theory characterizes the laws of that theory: it determines which reference frames (states of motion) are distinguishable (by a "mechanical experiment") relative to those laws, and in well-behaved theories the indistinguishability group is contained in the symmetry group (indistinguishable models are identical). Covariance, on the other hand, is really a property of formulations of space-time theories rather than space-time theories themselves ... As we have seen, we cannot interpret the general principle of relativity as an indistinguishability requirement, for the indistinguishability group of the general theory is just the restricted group of transformations from one local inertial frame to another.

So even in GR, it seems that local inertial frames are picked out as "special" in a sense that global corodinate systems on curved spacetime (which can't be inertial), or local non-inertial frames, are not.

 

[Dale]

All of you guys seem to be missing one crucial thing: general relativity. When I invoked the term, "acceleration," that should have clued you in. After all, what does the following statement mean.

There is no reason to introduce GR into the twin paradox. SR can handle acceleration just fine. What it cannot handle is graviataion (or curved spacetime). You can do all the accelerating you want in SR as long as you do it far away from any massive body.

 

[my_wan]

I think the misunderstanding about acceleration being absolute here is more fundamental.

With motion we can't say who is moving and who is at rest. Any observer regardless off motion can rightfully claim to be at rest. However, if an object accelerates then ALL other observers will agree that that object accelerated. In that way it is absolute. Yet observers will disagree on how much and even the direction the object accelerated. In this way acceleration is not absolute.

When we talk about acceleration being absolute we are only talking about the fact that all observers can agree on what accelerated, not on how much or in what direction it accelerated. You are right that there is no absolute change of position.

 

[Ookke]

I'm quite unhappy with acceleration-solution of Twin Paradox. Let's modify Twin Paradox to a "chasing" form:

Twins A and B are initially at rest in the same place. A accelerates instantly to 0.8c and continues steadily thereafter. B waits some time and then accelerates instantly to 0.8c to the same direction, entering A's reference frame. A and B compare their clocks.

Both have experienced identical acceleration, but nevertheless B's clock is behind. Really?

 

[JesseM]

I'm quite unhappy with acceleration-solution of Twin Paradox. Let's modify Twin Paradox to a "chasing" form:

Twins A and B are initially at rest in the same place. A accelerates instantly to 0.8c and continues steadily thereafter. B waits some time and then accelerates instantly to 0.8c to the same direction, entering A's reference frame. A and B compare their clocks.

Both have experienced identical acceleration, but nevertheless B's clock is behind. Really?

The "acceleration solution" simply claims that if two twins depart from the same place and reunite later, and one accelerated while the other did not, it will always be the one who accelerated who has aged less. It doesn't say that identical accelerations produce identical differences in aging, even in this case. Similarly, in plane geometry if you have two paths which start from one point and end at another point, the non-straight path always has a greater length than the straight one. If you use an x-y coordinate system to describe these paths, the straight-line path has a constant slope dy/dx (akin to constant velocity), while the non-straight path will have at least one region where the slope is changing (akin to acceleration), but it's not as if the size of the changing-slope segment of the path uniquely determines the total difference in the lengths of the paths. I expanded on this geometric analogy in post #9 of this thread, if it's helpful.

 

[Dale]

If you have to use a geometric analogy to explain the acceleration solution then why not just go completely with geometry to begin with and use the spacetime interval solution?

 

[JesseM]

If you have to use a geometric analogy to explain the acceleration solution then why not just go completely with geometry to begin with and use the spacetime interval solution?

Well, because the point of analogies is to compare something hard to think about directly with a situation we're all familiar with and can visualize in a natural way. No one can actually directly visualize a geometry in which a straight worldline actually has a greater length than any other worldline between the same two points, and likewise no one can directly visualize a space where distance is given not by the pythogorean theorem $$\sqrt{dx^2 + dy^2}$$ but by the formula $$\sqrt{c^2 dt^2 - dx^2}$$.

 

[robphy]

Well, because the point of analogies is to compare something hard to think about directly with a situation we're all familiar with and can visualize in a natural way. No one can actually directly visualize a geometry in which a straight worldline actually has a greater length than any other worldline between the same two points, and likewise no one can directly visualize a space where distance is given not by the pythogorean theorem $$\sqrt{dx^2 + dy^2}$$ but by the formula $$\sqrt{c^2 dt^2 - dx^2}$$.

I think my approach comes close to visualizing intervals in spacetime geometry
http://www.phy.syr.edu/courses/modules/LIGHTCONE/LightClock/

 

[Dale]

Well, because the point of analogies is to compare something hard to think about directly with a situation we're all familiar with and can visualize in a natural way. No one can actually directly visualize a geometry in which a straight worldline actually has a greater length than any other worldline between the same two points, and likewise no one can directly visualize a space where distance is given not by the pythogorean theorem $$\sqrt{dx^2 + dy^2}$$ but by the formula $$\sqrt{c^2 dt^2 - dx^2}$$.

I disagree with this.

First, it is easy to visualize the geometry, it is just distance measured as a family of hyperbolas instead of a family of circles. But even if it is not easy to visualize it is easy to use and calculate.

Second, the point isn't to provide analogies, the point is to resolve the paradox. This exact same post comes up on this forum at least weekly, so obviously talking about acceleration is a poor way to resolve the twin paradox. Most students fail to understand that the acceleration only breaks the symmetry of the situation and instead come away mistakenly thinking that acceleration causes time dilation. Of the students who do grasp the symmetry-breaking explanation it still leaves them unable to make correct predictions about even slight variations in the paradox like Ookke brought up. It also generally leaves them unable to quantify how much time should have elapsed for each twin.

The spacetime interval explanation suffers from none of these problems. It is clear, quantitative, and generally applicable. The resulting tool provides a motivation for an understanding of four-vectors and Minkowski geometry. I think it is a travesty that the acceleration explanation is still used when it is demonstrably such a poor teaching tool.