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Implications of the statement Acceleration is not relative

  1. Feb 9, 2013 #1
    Implications of the statement "Acceleration is not relative"

    As a result of the discussion which ensues from this post I hope to understand the implications of this statement: "Acceleration is not relative."

    First, I think it will be helpful to establish some context.

    Context point 1: I am not an "objector" looking for a paradox, or hoping to catch relativists in a contradiction. I am asking my question in good faith, ready to learn from those who know more than I. (I reserve the right to evaluate responses, as any intellectually honest person must do, but I promise to give a fair and open-minded hearing to all.)

    Context point 2: Since my last post, in which I made a dumb mistake in the interpretation of the spacetime diagram (or rather, failed to check my mental image of a pole-in-barn episode against the spacetime diagram), I've worked through the problem in sufficient detail to understand my mistake, and hopefully avoid similar mistakes in future. So, you are not (I hope) wasting your time as you respond to my question. [As a side note, I have not succeeded in forming a mental picture of the "one reality represented by the spacetime diagram", as I had hoped to do. I've come to the conclusion, for now at least, that there is no way to form such a mental image from the spacetime diagram. The Lorentz transformation provides a means of predicting how an episode (collection of events) will be observed in any chosen inertial reference frame. If an extra-frame view of the episode is to be had, it is not going to come from the spacetime diagram by itself.]

    Context point 3: Having come to an understanding of the pole-in-barn paradox, I was naturally drawn to consider the last remaining paradox that is unresolved for me. This is the twin paradox. One might say that there are two aspects to the paradox. The first is that the twins would be of different ages when they meet at the end of the episode. That is not really a paradox; it is explained by the notion of proper time, and well illustrated by a spacetime diagram in which the earth is considered to be at rest and the rocket twin is moving. The second aspect of the twin paradox comes about when one considers the rocket twin to be at rest and the earth to be moving. In that case, it is the earth twin who will be younger, a contradiction in that both twins cannot be younger. That paradox I have not resolved, though I have read multiple explanations of it. Yesterday I decided to try again.

    I began with Taylor and Wheeler. The twin paradox is dealt with in section 4.6 of Spacetime Physics. The proper time for each twin is calculated, showing the age difference on return of the traveler. Then, the (always rude and unreasonable) objector says, "If there is any justice, if relativity makes any sense at all, it should be equally possible to regard you [the earthbound twin] as the stay-at-home." There follows a detailed explanation that ends with, "notice that the traveler is unique in changing frames, only the traveler suffers the terrible jolt of reversing direction of motion." But this explanation does not address the objection, for the objection is that the rocket twin should be considered at rest. A resting twin cannot reverse motion. The jolt that the resting rocket twin feels must come from some other cause than reversal of motion.

    I went to Born, in Einstein's Theory of Relativity. In VI-5, he deals with the objection of the resting rocket twin by asserting that only the rocket twin accelerates. Once again, it seems to me that this objection sidesteps the issue, for by definition the resting rocket twin does not accelerate.

    So this morning I searched on this forum, wishing to avoid being the 9,488th person to ask about the twin paradox. I found this thread. Quite quickly I saw what must be at the root of the explanation of the paradox, but which I do not recall ever seeing stated explicitly: "Acceleration is not relative." (This statement was not challenged in the first twelve pages; and anyway I think it merits its own thread.)

    I say "at the root of the explanation" because it is the immediate implication of the statement which avoids the paradox: The rocket twin cannot be considered at rest. (More precisely, the rocket twin cannot be considered at rest while he is accelerating. This is implicit in ghwells statement in post #161.)

    This statement, to one who began the study of relativity with Einstein's Relativity, is nothing short of shocking. In a religious context (which this is not, of course, but the analogy is too striking to omit mention), this would be tantamount to heresy.

    So I went to Einstein's book to see if I had missed something. No, I don't think so. He begins by expressing his desire to bring acceleration into the realm of the principle of relativity. In doing so, his development of the equivalence of inertial and gravitational mass is premised on the example of a man in an accelerating chest who considers himself to be at rest.

    I then reread Einstein's own explanation of the twin paradox. Again, the rocket twin is unambiguously at rest when the rocket frame is the reference frame.

    It is quite clear to me that Einstein considered acceleration to be relative, and that he considered it perfectly justifiable to consider an observer who accelerates with respect to an inertial frame to be at rest.

    I will take a moment here to object to the reason given for the claim that acceleration is not relative. The reason given (post #10 and following) is that acceleration may be independently measured, or felt, without reference to some other frame. But what one measures or feels is force, not acceleration. This distinction, together with the necessary equivalence of inertial mass and gravitational mass in all frames, is the basis for Einstein's claim that an observer in an accelerating frame may consider himself to be at rest in a gravitational field.

    What are the broader implications of the statement that acceleration is not relative? Does this mean, as it certainly would appear to mean, that modern relativity is in this very important respect not Einsteinian relativity? Are there other implications as to the meaning of the principle of relativity?
     
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  3. Feb 9, 2013 #2

    Dale

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    Re: Implications of the statement "Acceleration is not relative"

    If you will go back to post #10 you will see that I already discussed this objection in the * comments.

    Do you understand the difference between proper acceleration (the kind of acceleration measured by accelerometers independently of any reference frame) and coordinate acceleration (the kind of acceleration relative to some reference frame)?
     
  4. Feb 9, 2013 #3
    Re: Implications of the statement "Acceleration is not relative"

    My objection is that accelerometers do not measure acceleration, but force. Force does not imply acceleration--at least, not according to Einstein.
     
  5. Feb 9, 2013 #4
    Re: Implications of the statement "Acceleration is not relative"

    I should probably mention why I have had difficulty accepting Einstein's explanation of the paradox. In it, the gravitational field is posed as the result of induction from the distant stars. I don't see how a gravitational signal (which as I understand travels at the speed of light) can cause the force felt by the resting rocket traveler. It is true that Einstein first argues that the gravitational field is just as real as kinetic energy, which also "disappears" with the appropriate choice of reference frame. But he does not seem convinced by this explanation, and looks to the massive stars as the source.
     
  6. Feb 9, 2013 #5

    Dale

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    Re: Implications of the statement "Acceleration is not relative"

    They measure proper acceleration. Proper acceleration does not imply coordinate acceleration.

    I get the impression that you may not be familiar with the distinction between the two concepts. Here are some places to start:

    http://en.wikipedia.org/wiki/Accelerometer
    http://en.wikipedia.org/wiki/Proper_acceleration
     
    Last edited: Feb 9, 2013
  7. Feb 9, 2013 #6

    PeterDonis

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    Re: Implications of the statement "Acceleration is not relative"

    [Edit: I see I didn't type fast enough. :wink: However, I still think the following is worth consideration.]

    The first thing to do, IMO, is to state it properly (pun intended, as you will see in a moment):

    Proper acceleration is not relative. (It is a direct observable.)

    Coordinate acceleration is relative. (It depends on the coordinates you adopt.)

    This distinction is crucial; it's only a very slight exaggeration to say that every time I've seen someone confused about "acceleration", it's because they're confusing the two types of acceleration given above.

    Yes, that's correct, and it's good that you recognize it; if only all the people who have been posting twin paradox threads recently would do so... :wink: However, you don't appear to fully understand *why* it is true. See below.

    No, this is not correct. The calculation of the two proper times, which is illustrated in a spacetime diagram in which the stay-at-home twin is at rest, as you say, assumes that the stay-at-home twin is at rest in a single inertial frame (the frame in which the diagram is drawn) for the entire scenario. The traveling twin does not satisfy that condition; there is no single inertial frame in which he is at rest for the entire scenario. So you can't run the same argument for the traveling twin.

    If you define "motion" as "inertial motion", then this is true. But with this definition of "motion", you *cannot* simply declare by fiat that you are going to consider the rocket twin as being at rest. "Being at rest", on this definition of "motion", is not a convention; it's a physical condition that can be objectively tested--just test whether the observer feels acceleration. The traveling twin does; the stay-at-home twin doesn't. So the stay-at-home twin can be considered to be "at rest", but the traveling twin can't.

    Alternatively, you could define "motion" in such a way that you can legitimately say that the traveling twin does not "reverse motion", so he can be considered "at rest" during the entire scenario. But if you do *that*, then you can't simply declare by fiat that the twin who is "at rest" is the one whose proper time is greater. You have defined "at rest" so that it no longer always corresponds to maximal proper time; an observer "at rest" may feel acceleration, and if he does, you will be able to find some other observer who experiences more proper time between two given events than the observer "at rest" does.

    Notice that I have basically just re-stated what I said at the start of this post, that proper acceleration is not relative. The traveling twin has a nonzero proper acceleration for at least some portion of his trip; the stay-at-home twin has zero proper acceleration during the entire scenario. That is an invariant physical difference between them.

    It does; it comes from his firing the rocket. There's no need to stipulate that his motion "reverses". The fact of his feeling acceleration, where the stay-at-home twin does not, is an objective physical difference between them that doesn't depend on whether or not he "reverses motion".

    Here you are, once again, confusing proper acceleration with coordinate acceleration. The "resting" twin does not have any coordinate acceleration; but that doesn't mean he has no proper acceleration. And it's proper acceleration that is relevant for determining elapsed proper time, because proper acceleration is the direct observable.

    Again, see my clarification at the start of this post. Which someone probably gave in the thread you linked to; certainly I've given it in plenty of twin paradox threads lately, in more or less the form I've given it here.

    This is true if you define "at rest" as "at rest in an inertial frame". Which I believe was the implicit definition of "at rest" that was being used in that thread. The reason it's a common definition is that, as I said above, inertial motion has special properties, physically, because it corresponds to zero proper acceleration and therefore maximal proper time. If you define "at rest" to allow observers with nonzero proper acceleration to be "at rest", as I noted above, you lose that key property.

    Note that the man feels acceleration; he feels his own weight and can stand on the "floor" of the chest as he would stand on the Earth's surface. So he has nonzero proper acceleration, and if he is considered to be "at rest" then we have adopted a definition of "at rest" which does not guarantee that "at rest" corresponds to "maximal proper time".

    Again, don't confuse coordinate acceleration with proper acceleration. Einstein did consider coordinate acceleration to be relative; the observer who accelerates with respect to an inertial frame has nonzero coordinate acceleration with respect to that inertial frame, but zero coordinate acceleration with respect to his own "rest frame" (which is not an inertial frame). But he has nonzero proper acceleration regardless of which frame you choose; proper acceleration is not relative, and Einstein agreed with that too.

    This is a distinction without a difference, because in order to make this claim, you have to define "force" in such a way that only "proper force" (force that corresponds to nonzero proper acceleration) is defined as a force. So gravity is *not* a force on this definition; a person at rest on Earth's surface, for example, does *not* feel the "force of gravity"; he feels the force of the Earth's surface pushing up on him. A person who only moves under the "force of gravity", such as an observer in orbit about the Earth, feels *no* force; he is moving inertially, in free fall, weightless. Similarly, the traveling twin feels the force of his rocket engine pushing on him, whereas the stay-at-home twin never feels any force at all. Just substitute "feels force" for "feels acceleration" in everything I said above and all my arguments still go through just fine.

    It means you have to pay attention to the crucial distinction between proper acceleration (or "feeling force", if you like that term better) and coordinate acceleration (which may or may not correspond to a felt force). The latter is relative; the former is not.

    No. The distinction was always there in relativity, and Einstein was well aware of it.

    Only that, once again, the difference between inertial and non-inertial motion (feeling no force vs. feeling a force, in the terms you appear to prefer) is an observable, invariant physical difference; it's not relative.
     
  8. Feb 9, 2013 #7
    Re: Implications of the statement "Acceleration is not relative"

    I will read them. The premise of Einstein's approach, as I understand it, is that there is no acceleration at all in the reference frame of the resting rocket twin--proper or otherwise.
     
  9. Feb 9, 2013 #8

    PeterDonis

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    Re: Implications of the statement "Acceleration is not relative"

    No, that's not the premise. There is no coordinate acceleration, but there is proper acceleration. You can't make proper acceleration disappear by changing frames; it's an invariant. Einstein knew that.
     
  10. Feb 9, 2013 #9

    Dale

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    Re: Implications of the statement "Acceleration is not relative"

    [Now it is my turn to not type fast enough :smile:]

    I am not sure when the term "proper acceleration" was coined, but it is safe to say that if Einstein made such a premise then he was refering to coordinate acceleration.

    Proper acceleration is the measurement on an accelerometer, so it is necessarily a frame invariant quantity. All coordinates will agree on the reading on an accelerometer even though they may not agree that the reading represents (coordinate) acceleration.

    Perhaps you can see the parallel between proper time and coordinate time. Proper time is the measurement on a clock, so it is necessarily a frame invariant quantity. All coordinates will agree on the reading on a clock even though they may not agree that the reading represents (coordinate) time.
     
  11. Feb 9, 2013 #10
    Re: Implications of the statement "Acceleration is not relative"

    This is an inference.

    Again, the bold text is an inference. Einstein interprets the behavior of the instrument in two ways. Observed from the inertial reference frame, it is indeed acceleration that causes the displacement and counteracting force. Observed from the non-inertial frame, it is a gravitational field and the forcible restraint from acceleration that displaces the mechanism. There is no acceleration in the non-inertial frame, according to Einstein's interpretation.


    The bold text is contrary to Einstein's explicit statement that there is no gravitational field in the inertial frame. What does not exist in the frame cannot act on the observer in the frame. He explicitly and repeatedly states that the gravitational field exists only in the non-inertial frame. I'm not saying that I agree or disagree with his interpretation; just that this is his interpretation.
     
  12. Feb 9, 2013 #11

    Nugatory

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    Re: Implications of the statement "Acceleration is not relative"

    All it means is that Einstein was sometimes careless about distinguishing between coordinate and proper acceleration when it was clear from the context which was intended.... And there's no broader implication beyond that. Coordinate acceleration is relative and can be produced without applying any force, simply by choosing non-inertial coordinates. Proper acceleration is not relative, doesn't depend on the coordinates, and happens if and only if a force is applied.

    You are right that it is force rather than acceleration that we measure directly, but if we have net force we can infer (proper) acceleration; we don't get one without the other.
     
  13. Feb 9, 2013 #12
    Re: Implications of the statement "Acceleration is not relative"

    The Wikipedia page on the twin paradox, in the section on the "viewpoint of the traveling twin", explains the use of "gravitational time dilation" (via the "equivalence principle") to resolve the paradox from the traveler's viewpoint. The result is that, according to the traveler, the home twin's age increases a lot during the traveler's turnaround, enough to more than make up for the home twin's slower aging when the traveler isn't turning around. They also show how to get that same result, by using accelerated motion instead of a fictitious gravitational field.
     
  14. Feb 9, 2013 #13

    Nugatory

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    Re: Implications of the statement "Acceleration is not relative"

    Those are ways of calculating the differential aging in the traveler's coordinate time. They do not change the coordinate-independent facts: the traveler follows a path of shorter proper time; the traveler clearly is not inertial for his entire journey.
     
  15. Feb 9, 2013 #14

    PeterDonis

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    Re: Implications of the statement "Acceleration is not relative"

    This is a matter of terminology, not physics. The acceleration--or force, if you prefer--that is felt by the mechanism is the same regardless of which frame you use. That's the physics. Similarly, in the twin paradox, the traveling twin feels a force; the stay-at-home twin does not. That's a physical difference, and it's there regardless of which frame you use to describe the scenario.
     
    Last edited: Feb 9, 2013
  16. Feb 9, 2013 #15

    Dale

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    Re: Implications of the statement "Acceleration is not relative"

    I would call it a definition, but your manner makes me curious. What is wrong with inference? It seems like you are using it as a perjorative, but I don't understand why.

    Therefore, the proper acceleration is frame invariant.

    There is no coordinate acceleration in the non inertial frame.
     
  17. Feb 10, 2013 #16

    ghwellsjr

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    Re: Implications of the statement "Acceleration is not relative"

    What statement of mine are you referring to in post #161?
     
  18. Feb 10, 2013 #17

    stevendaryl

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    Re: Implications of the statement "Acceleration is not relative"

    Why do you say that? Conceptually, you could think of an accelerometer as just a box with a metal ball in the center held in place by springs. When you accelerate the box, the position of the ball within the box is altered. By measuring the position of the ball, you can determine the acceleration of the box.
     
  19. Feb 10, 2013 #18

    stevendaryl

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    Re: Implications of the statement "Acceleration is not relative"

    That depends on how you define "acceleration". As people have pointed out, you're right that in the noninertial frame coordinate acceleration is zero. But that is an artifact of the coordinate system you are using, and doesn't have any physical meaning. The physically meaningful notion of acceleration is proper acceleration, which is nonzero for an accelerating rocket, no matter what coordinate system you use.

    You're getting things all mixed up by saying there is a gravitational field in the noninertial frame. Why do you believe that? Because of Einstein's Equivalence Principle? If so, then you're mixing up two different theories, if you're trying to understand the twin paradox. The twin paradox is pure Special Relativity (according to the modern view of the distinction between General and Special Relativity), and the Equivalence Principle has no relevance to Special Relativity, and therefore to the twin paradox. Special Relativity has no gravitational fields.
     
  20. Feb 10, 2013 #19
    Re: Implications of the statement "Acceleration is not relative"

    It was this one: "So what I did was transform from the IRF in which the black inertial twin is at rest to the IRF in which the blue traveling twin is at rest during the outbound portion of his trip."
    However, reading it again I see that it does not necessarily imply that the traveling twin is not at rest while accelerating wtr to the stay-at-home twin. I inferred that from the context. My apologies if I attributed a position to you which you do not hold.
     
  21. Feb 10, 2013 #20

    Dale

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    Re: Implications of the statement "Acceleration is not relative"

    GregAshmore, I think that it is pretty clear from your comments that you do not get the difference between proper acceleration and coordinate acceleration. As PeterDonis mentioned, this distinction is critical and is the source of almost all confusion about acceleration. Let's use the following as the definitions of proper and coordinate acceleration:

    Proper acceleration is the acceleration that would be measured by an ideal accelerometer.

    Coordinate acceleration is the second time derivative of the coordinate position.

    From those definitions, do you understand why proper acceleration must be frame invariant and coordinate acceleration must be frame variant?
     
  22. Feb 10, 2013 #21
    Re: Implications of the statement "Acceleration is not relative"

    I have now read all your comments. My responses to this point (with the exception of my answer to ghwells) were to comments made by DaleSpam, as I had not yet seen comments from anyone else. I probably needed to refresh my browser.

    I understand that the distinction between proper acceleration and coordinate acceleration is important to this discussion. I don't get that distinction yet--I'm not able to point to one example that I am sure is coordinate acceleration and another that I am sure is proper acceleration. I will study it.

    [I see that while writing this, DaleSpam has provided definitions for proper acceleration and coordinate acceleration. I'll have to think about what they mean. What follows in this post is unaffected.]

    In his book Relativity, Einstein talks about different kinds of gravitational fields. He points out that only gravitational fields of a "quite special form" (his words) can be made to "go away" (my words) by the choice of reference frame. Perhaps there is a correspondence between the various kinds of gravitational fields and proper acceleration versus coordinate acceleration.

    However, whatever the kind of acceleration that we are dealing with in the twin paradox, Einstein states unequivocally that it is relative.

    At the end of section XVIII in Relativity he says,
    A few pages later, after developing the principle of equivalence of inertial mass and gravitational mass, he says,
    According to Einstein, the rocket twin may be considered at rest throughout the episode. The rest frame is not inertial, of course. But it is a rest frame nonetheless. In that rest frame, the acceleration of the rocket is not "real"; there is no acceleration. What the rocket twin feels is not acceleration, but the force (transmitted through the seat) that holds the rocket still in a gravitational field.

    According to Einstein, the acceleration of the rocket is relative. Taken at face value, the statement, "Acceleration is not relative", is at odds with Einstein's statement.
     
  23. Feb 10, 2013 #22

    Dale

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    Re: Implications of the statement "Acceleration is not relative"

    This is where we should start then. Please study and come up with questions that we can discuss about the two concepts.

    Once you understand the concepts of proper and coordinate acceleration then understanding his writings is easy, but going the other way is not so easy. So let's concentrate on the concepts first.
     
  24. Feb 10, 2013 #23

    PeterDonis

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    Re: Implications of the statement "Acceleration is not relative"

    An object in free fall has zero proper acceleration. So if you are standing at rest on the surface of the Earth, and use coordinates in which you are at rest, then if you drop a rock, the rock has zero proper acceleration and nonzero coordinate acceleration; but you have zero coordinate acceleration and nonzero proper acceleration.

    Kind of. Here's a restatement of what Einstein was saying that may help: in flat spacetime, we can choose coordinates so that objects in free fall have nonzero coordinate acceleration; but we are never forced to. In flat spacetime, we can always find coordinates where objects in free fall have zero coordinate acceleration, and objects not in free fall don't. In other words, we can always find coordinates in which coordinate acceleration is present if and only if proper acceleration is present.

    In curved spacetime, however, we can never find such coordinates; no matter which coordinates we choose, there will be objects in free fall that have nonzero coordinate acceleration. So we can never find coordinates in curved spacetime that will allow us to clearly distinguish, using coordinates alone, between free fall and proper acceleration.

    What Einstein meant by "gravitational fields of a special form" was the apparent "field" produced by choosing coordinates in flat spacetime so that objects in free fall have nonzero coordinate acceleration. You can make them go away by choosing coordinates where objects in free fall never have nonzero coordinate acceleration; but as above, you can only do that in flat spacetime.

    I'm not sure how you are inferring this from the passages you quote. He is saying that coordinate acceleration is relative, because we can make it disappear by choosing coordinates in which the railway carriage is at rest. But that doesn't make the jerk that the passengers in the carriage feel disappear, and the felt jerk is what corresponds to nonzero proper acceleration. Einstein never says the jerk is only there in one set of coordinates.

    Similarly, in the twin paradox, the traveling twin feels a jerk when he fires his rocket to turn around. He feels it regardless of the coordinates we choose.

    That's why you shouldn't take it at face value. You need to look at what the terms being used actually *mean*. In the statement "acceleration is not relative", "acceleration" means proper acceleration, and the statement is correct.

    It's true that Einstein did not use the term "acceleration" to refer to what we have been calling proper acceleration. But as I said before, that's a matter of terminology, not physics. You can't change the physics by changing terminology, any more than you can change it by changing coordinates. By "proper acceleration" we are talking about "the real thing that corresponds to the felt jerk". If you want to know why we use the term "acceleration" for that, we can go into that, but it seems to me to be a separate question.
     
  25. Feb 10, 2013 #24

    PeterDonis

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    Re: Implications of the statement "Acceleration is not relative"

    On second thought, perhaps it isn't a separate question. Here's a quick answer: proper acceleration is the derivative of proper velocity (usually called 4-velocity) with respect to proper time. This is obviously analogous to coordinate acceleration, which is the derivative of coordinate velocity with respect to coordinate time.

    The key difference, though, is that proper acceleration is independent of coordinates, because proper velocity and proper time are independent of coordinates. In fact, you don't even need coordinates at all to define any of them. And in relativity, the physical content of the theory is entirely contained in quantities that are independent of coordinates and can be defined without needing coordinates. (Einstein said that, too; I'm pretty sure he said it, or something close to it, in the relativity book that has been quoted from.)
     
  26. Feb 10, 2013 #25
    Re: Implications of the statement "Acceleration is not relative"

    I'll respond to this one first.
    I don't know what Einstein may have said on the subject of coordinate systems outside of the two documents I have cited, so I do not claim that what follows is the sum total of his views on the subject. It's probably not important at this time to be concerned with the totality of his thought on that subject. I make the following comments mostly out of general interest.

    In Relativity, Einstein is content to speak strictly in terms of the coordinates of a reference body while discussing special relativity. When he gets into general relativity, he dispenses with the reference body and introduces the idea of Gaussian coordinates. These coordinates, he points out, have no physical meaning in themselves. Even so, the principle of general relativity is stated in terms of coordinates: "All Gaussian coordinate systems are essentially equivalent for the formulation of the general laws of nature."

    In the magazine article in which he explains the twin paradox, Einstein says that while it is desirable to divorce the laws of physics from coordinate systems, the effort to do so has failed. The relevant text follows. The first paragraph sets the context for the discussion of coordinate systems. The meat of the matter begins with the bold text. The "money quote" is: "Only certain, generally quite complicated expressions, that are constructed out of field components and coordinates [my emphasis], correspond to coordinate-independent, measurable (that is, real) quantities."

    From the magazine article:

    It should be kept in mind that in the left and in the right section exactly the same proceedings are described, it is just that the description on the left relates to the coordinate system K, the description on the right relates to the coordinate system K'. According to both descriptions the clock U2 is running a certain amount behind clock U1 at the end of the observed process. When relating to the coordinate system K' the behaviour explains itself as follows: During the partial processes 2 and 4 the clock U1, going at a velocity v, runs indeed at a slower pace than the resting clock U2. However, this is more than compensated by a faster pace of U1 during partial process 3. According to the general theory of relativity, a clock will go faster the higher the gravitational potential of the location where it is located, and during partial process 3 U2 happens to be located at a higher gravitational potential than U1. The calculation shows that this speeding ahead constitutes exactly twice as much as the lagging behind during the partial processes 2 and 4. This consideration completely clears up the paradox that you brought up.
    Critic:
    I do see that you have cleverly pulled away from the noose, but I would be lying if I would declare myself fully satisfied. The stumbling stone has not been removed; it has been relocated. You see, your consideration only shows the connection of the difficulty that was just discussed with another difficulty, that has also often been presented. You have solved the paradox, by taking the influence on the clocks into account of a gravitational field relative to K'. But isn't this gravitational field merely fictitious? Its existence is conjured up by a mere choice of coordinate system. Surely, real gravitational fields are brought forth by mass, and cannot be made to disappear by a suitable choice of coordinate system. How are we supposed to believe that a merely fictitious field could have such an influence on the pace of a clock?
    Relativist:
    In the first place I must point out that the distinction real - unreal is hardly helpful. In relation to K' the gravitational field "exists" in the same sense as any other physical entity that can only be defined with reference to a coordinate system, even though it is not present in relation to the system K. No special peculiarity resides here, as can easily be seen from the following example from classical mechanics. Nobody doubts the "reality" of kinetic energy, otherwise the very reality of energy would have to be denied. But it is clear that the kinetic energy of a body is dependent on the state of motion of the coordinate system, with a suitable choice of the latter one can arrange for the kinetic energy of the continuous motion of a body to assume a given positive value or the value of zero. In the special case where all the masses have a velocity in the same direction and of the same magnitude, a suitable choice of coordinate system can adjust the collective kinetic energy to zero. To me it appears that the analogy is complete.
    Rather than distinguishing between "real" and "unreal" we want to more clearly distinguish between quantities that are inherent in the physical system as such (independent from the choice of coordinate system), and quantities that depend on the coordinate system. The next step would be to demand that only quantities of the first kind enter the laws of physics. However, it has been found that this objective cannot be realized in practice, as has already been demonstrated clearly by the development of classical mechanics. One could for instance consider, and this has actually been attempted, to enter into the laws of classical mechanics not the coordinates, but instead just the distances between the material points; a priori one could expect that in this way the goal of the theory of relativity would be reached most easily. The scientific development has however not confirmed this expectation. She cannot dispense with the coordinate system, and therefore has to use in the coordinates quantities that cannot be construed as results of definite measurements. According to the general theory of relativity the four coordinates of the space-time continuum are entirely arbitrary choosable parameters, devoid of any independent physical meaning. This arbitrariness partially affects also those quantities (field components) that are instrumental in describing the physical reality. Only certain, generally quite complicated expressions, that are constructed out of field components and coordinates, correspond to coordinate-independent, measurable (that is, real) quantities. For example, the component of the gravitational field in a space-time point is still not a quantity that is independent of coordinate choice; thus the gravitational field at a certain place does not correspond to something "physically real", but in connection with other data it does. Therefore one can neither say, that the gravitational field in a certain place is something "real', nor that it is "merely fictitious".
    The circumstance that according to the general theory of relativity the connection between the quantities that occur in the equations and the measurable quantities is much more indirect than in terms of the usual theories, probably constitutes the main difficulty that one encounters when studying this theory. Also your last objection was based on the fact that you did not keep this circumstance constantly in mind.
    You declared the fields that were called for in the clock example also as merely fictitious, only because the field lines of actual gravitational fields are necessarily brought forth by mass; in the discussed examples no mass that could bring forth those fields was present. This can be elaborated upon in two ways. Firstly, it is not an a priori necessity that the particular concept of the Newtonian theory, according to which every gravitational field is conceived as being brought forth by mass, should be retained in the general theory of relativity. This question is interconnected with the circumstance pointed out previously, that the meaning of the field components is much less directly defined as in the Newtonian theory. Secondly, it cannot be maintained that there are no masses present, that can be attributed with bringing forth the fields. 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. Approximate integration of the gravitational equations has in fact yielded the result that induction effects must occur when masses are in accelerated motion. This consideration makes it clear that a complete clarification of the questions you have raised can only be attained if one envisions for the geometric-mechanical constitution of the Universe a representation that complies with the theory. I have attempted to do so last year, and I have reached a conception that - to my mind - is completely satisfactory; going into this would however take us too far.
     
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