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Thread: Equivalence Principle

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    Equivalence Principle

    Hi, I am new here, and thought that someone could help me with this query:

    A while back, I was reading a book written by the late Professor Robert Mills (co-inventor of Yang-Mills Theory), when I encountered this passage:

    Another way of stating the principle of equivalence, a way that better reflects its name, is to say that all reference frames, including accelerated reference frames, are equivalent, that the laws of Physics take the same form in any reference frame…. And it is also correct to say that the Copernican view (with the sun at the centre) and the Ptolemaic view (with the earth at the centre) are equally valid and equally consistent! [Mills (1994), pp.182-83.]
    I later read this from Max Born (the Nobel Laureate):

    Thus from Einstein's point of view Ptolemy and Copernicus are equally right. What point of view is chosen is a matter of expediency. For the mechanics of the planetary system the view of Copernicus is certainly the more convenient. But it is meaningless to call the gravitational fields that occur when a different system of reference is chosen 'fictitious' in contrast with the 'real' fields produced by near masses: it is just as meaningless as the question of the 'real' length of a rod...in the special theory of relativity. A gravitational field is neither 'real' nor 'fictitious' in itself. It has no meaning at all independent of the choice of coordinates, just as in the case of the length of a rod. [Born (1965), p.345.]
    Mills, R. (1994), Space, Time And Quanta (W H Freeman).

    Born, M. (1965), Einstein's Theory Of Relativity (Dover, 2nd ed.).

    Now, I'm a mathematician, and have enough maths to be able to grasp the basics of Relativity Theory, but not enough physics to answer this question:

    If, say, Ptolemaic astronomy is equally valid as the Copernican system -- according to the Equivalence Principle [EP] --, then isn't the fact that the stars would have to orbit the earth once a day (making them travel many times faster than light) an effective refutation of this Principle? Not all frames of reference can be equally valid if at least one of them has this crazy consequence.

    Now, I am not interested in Geocentrism (I know you have discussed this before); my concern is over the fact that I do not know enough physics to see where this goes wrong. One consequence of the EP is that Special Relativity is false!

    Of course, I might be making a 'rookie mistake' here, and would appreciate some guidance.

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    i think it is just that in any system you can choose your centre and work from there. In any two bodies example in space, for example, even when one is tiny wrt the other, if you place youself on one body, the other appears to orbit you.

    The equivalence princple is different in that Einstein said and showed that acceleration forces, i.e. inertial forces cannot be distinguished from gravity forces.

    He went on to explain that gravity is a distortion of space time by mass. it is a field.

    You can map any rotations to a different centre so the choice of centre is arbitrary for convenience, the sun after all orbits within our galaxy. but we don't usually choose the galactic centre for solar system understanding
    sicut vis videre esto
    When we realize that patterns don't exist in the universe, they are a template that we hold to the universe to make sense of it, it all makes a lot more sense.
    Originally Posted by Ken G

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    Quote Originally Posted by Rosa Lichtenstein View Post
    Hi, I am new here, and thought that someone could help me with this query:

    A while back, I was reading a book written by the late Professor Robert Mills (co-inventor of Yang-Mills Theory), when I encountered this passage:



    I later read this from Max Born (the Nobel Laureate):



    Mills, R. (1994), Space, Time And Quanta (W H Freeman).

    Born, M. (1965), Einstein's Theory Of Relativity (Dover, 2nd ed.).

    Now, I'm a mathematician, and have enough maths to be able to grasp the basics of Relativity Theory, but not enough physics to answer this question:

    If, say, Ptolemaic astronomy is equally valid as the Copernican system -- according to the Equivalence Principle [EP] --, then isn't the fact that the stars would have to orbit the earth once a day (making them travel many times faster than light) an effective refutation of this Principle? Not all frames of reference can be equally valid if at least one of them has this crazy consequence.

    Now, I am not interested in Geocentrism (I know you have discussed this before); my concern is over the fact that I do not know enough physics to see where this goes wrong. One consequence of the EP is that Special Relativity is false!

    Of course, I might be making a 'rookie mistake' here, and would appreciate some guidance.
    Just because it is true in minkowski space doesn't mean it must be true in a generalization of it. Over the distance from you to that far-away star the space will not be approximated well with minkowski space, and hence there is no expectation for an invariant of a transformation in minkowski space to remain an invariant.

  4. 2013-Nov-20, 08:29 PM

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    Caveman, thanks for that, but I didn't understand a word of it, and I'm a mathematician!

    Profloater:

    i think it is just that in any system you can choose your centre and work from there. In any two bodies example in space, for example, even when one is tiny wrt the other, if you place youself on one body, the other appears to orbit you.

    The equivalence princple is different in that Einstein said and showed that acceleration forces, i.e. inertial forces cannot be distinguished from gravity forces.

    He went on to explain that gravity is a distortion of space time by mass. it is a field.

    You can map any rotations to a different centre so the choice of centre is arbitrary for convenience, the sun after all orbits within our galaxy. but we don't usually choose the galactic centre for solar system understanding
    Thanks, but I know all this.

    My question was, if this is so, then, if we take the earth as our reference frame, the stars will have to exceed the speed of light. How can it be a valid consequence of one part of Relativity that it falsifies another?

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    Can the mod who is vetting my posts, forget about the first one, and only consider the second one for publishing? Thanks.

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    Quote Originally Posted by Rosa Lichtenstein View Post
    Can the mod who is vetting my posts, forget about the first one, and only consider the second one for publishing? Thanks.
    Done
    At night the stars put on a show for free (Carole King)

    All moderation in purple - The rules

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    Quote Originally Posted by Rosa Lichtenstein View Post
    Caveman, thanks for that, but I didn't understand a word of it, and I'm a mathematician!
    You know that special relativity is formulated in minkowski space? General relativity is not formulated in minkowski space but in a more general pseudo-riemann space, such that the tangent space at each point of that pseudo-riemann space is minkowski space. Why would you expect a theorem in minkowski space (for an inertial reference frame there is nothing with velocity greater than c) to still apply in a more general space?

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    Is rotation relative then, rather than an absolute motion. The idea of the entire universe orbiting the earth seems very strange physically. Do we claim that in a frame attached to the earth some forces exist that cause the entire universe to revolve around the earth???

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    I personally can't explain relative rotation. If rotation was relative, then spinning up a space station would not induce gravity, since a frame spinning with the station would have no 'forces'. Therefore, all frames are not strictly equivalent.

    Unlike linear momentum, angular momentum seems to have a special place. You see this in the 'spin' of a particle, and the angular momentum of a black hole.

    Now, can you work out the laws of physics in any frame? Apparently, yes you can in general relativity. It would seem we need to be careful when using the word 'valid' to mean 'easily interchangeable'.

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    Quote Originally Posted by ShinAce View Post
    I personally can't explain relative rotation. If rotation was relative, then spinning up a space station would not induce gravity, since a frame spinning with the station would have no 'forces'. Therefore, all frames are not strictly equivalent.
    I always thought it was relative to the axis, that the difference in motion between the outer rim and the center constituted a changing "frame" and accounted for the (pseudo-)acceleration. But that may be my misunderstanding of reference frames.
    "I'm planning to live forever. So far, that's working perfectly." Steven Wright
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    Can I put the distant stars another way? So we want to use Earth as a reference and now the stars are going round us as in the ancient view. But we know now it is only their light reaching us that we see, so the idea that they are travelling at some vast speed is an illusion. Once you add the vast distance and the speed of light, the idea of a sky sphere with holes through to heaven drops away.
    Of course you then replace your model with that of "fixed" stars and a spinning Earth. The conundrum of spin is really quite basic, if an object is spinning or two points are rotating about each other the mysterious property inertia comes into play and inertial forces operate as calculated by Newton. The mystery of inertia is directly linked to mass and must therefore have something to do with Higgs field. The interesting question remains of whether there is a point spin at the smallest size limit. i.E. does spin of a particle have the same meaning as spin of a group of particles.?
    sicut vis videre esto
    When we realize that patterns don't exist in the universe, they are a template that we hold to the universe to make sense of it, it all makes a lot more sense.
    Originally Posted by Ken G

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    Rosa, my guess would be that you are confusing movement (trajectories) of the stars through spacetime with movement of spacetime: Even if we take the Earth as stationary at the center of a spinning universe we don't have to take it that the universe is orbiting Earth in the conventional sense. Indeed, this would cause a lot of problems and contradictions, as you've noted ion the OP. It makes more sense to have all of space-time revolving about the central axis of Earth (it helps me to imagine Earth as a point) - so you could say its the co-ordinate system that is spinning, not the objects within the co-ordinate system. I guess this would be indestinguisahable from a rotating Earth in a static univerese, which is kinda the point of equivilence. As the speed of light rule says the speed of light cannot be exceeded in the local spacetime we get to a work around for the rotating universe: The distant stars aren't moving FTL compared to their local spacetime, spacetime itself is moving and carrying them around, so the stars aren't breaking any rules. They aren't moving relative to each other at speeds greater than light, and any calculation we do using the rotating co-ordinates will return exactly the same answer as for a static universe with a spinning Earth, so no information can travel between any point in the co-ordinates sytem at speeds greater than lightspeed, preserving causality. That the distant stars are exceeding c in Earths reference frame makes no difference since in their local co-ordinates they aren't moving at all. Earth's rotating reference frame can go hang, because in it space is also rotating.

    If that seems like a bit of a cheat, consider that astrophysicists are perfectly happy with the idea that the expansion of the universe could be carrying distant stars away from Earth at (apparent) speeds that are greater than lightspeed: From their point of view we're the ones moving away at greater than lightspeed, but it's all ok because what is really happening is that the space time between us is stretching at a rate greater than lightspeed, and neither observer is actually moving very fast at all. We cannot observe them now, but in the early universe they and Earth (or the points in space where they and earth now reside) would have been easily observable to each other using light beams, so it's very hard to say they haven't moved away at greater than lightspeed: They haven't moved, but the space between here and there has moved at greater than lightspeed. In the same way, the distant stars are changing position in Earths frame at a rate larger than c, but they aren't moving because space is moving with them.

    Err.. I think.
    Last edited by marsbug; 2013-Nov-21 at 02:15 PM.

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    The Ptolemaic view is valid each time I pick up my star charts and use them. It is valid as I try and locate my position from a boat or aircraft using a sextant and the changing positions of the stars and where their subpoint locations are on our planet at given moments in time. The point of view here is expedience. We are only in the same reference frame as the earth is as long as we are standing still or sitting still. Any motion we make along the ground takes us out of that reference frame. Those accelerating reference frames give us views that inertial reference frames have more difficulty giving.

    For hunters/gatherers it would have been easier to figure out that the earth rotates on an axis and moves around the sun. The tree rule states that a tree gets larger as you approach it while getting smaller as you leave it. But you can make it move to your view and not change its size by spinning around. As hunters/gatherers moved from east to west coasts of Africa and South America they could easily notice that the angular size of the sun never changes from dawn to dusk. Without knowing the actual size of the sun and its distance in ancient times, one would expect the sun to obey the tree rule and to be as obvious. Secondly, the self catch rule states that when someone throws a rock straight up, it will reach its zenith right above the rock thrower. But to another person walking from the rock thrower sees the zenith constantly changing. Therefore, when observing the sun's apparent motion, the accelerating reference frame of the hunter/gatherer could easily notice that he/she is "trapped in the middle", never seeing the zenith change, something that can only happen from spinning.

    The synodic motions can be reflected in spinning dances that couples made near firesides as they revolved around the fire, noticing that different background trees will appear along ones line of sight past the partner's head, while the entire forest's reference frame remained intact, similar to the changing positions of the stars as we revolve around our sun.

    Mostly, we need to remember that the laws of physics, meaning the laws of motion plus the laws of electromagnetism, do not change when measuring from either a Ptolemaic reference frame or a Copernican reference frame. And we need to also remember that one observer is not a reference frame. There has to be a shared motion by two or more observers, whether they are in uniform or accelerating motions.
    Last edited by blueshift; 2013-Nov-21 at 01:27 PM.

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    Quote Originally Posted by Rosa Lichtenstein View Post
    I later read this from Max Born (the Nobel Laureate):
    Thus from Einstein's point of view Ptolemy and Copernicus are equally right. What point of view is chosen is a matter of expediency. For the mechanics of the planetary system the view of Copernicus is certainly the more convenient. But it is meaningless to call the gravitational fields that occur when a different system of reference is chosen 'fictitious' in contrast with the 'real' fields produced by near masses: it is just as meaningless as the question of the 'real' length of a rod...in the special theory of relativity. A gravitational field is neither 'real' nor 'fictitious' in itself. It has no meaning at all independent of the choice of coordinates, just as in the case of the length of a rod. [Born (1965), p.345.]
    Mills, R. (1994), Space, Time And Quanta (W H Freeman).

    Born, M. (1965), Einstein's Theory Of Relativity (Dover, 2nd ed.).

    Now, I'm a mathematician, and have enough maths to be able to grasp the basics of Relativity Theory, but not enough physics to answer this question:

    If, say, Ptolemaic astronomy is equally valid as the Copernican system -- according to the Equivalence Principle [EP] --, then isn't the fact that the stars would have to orbit the earth once a day (making them travel many times faster than light) an effective refutation of this Principle? Not all frames of reference can be equally valid if at least one of them has this crazy consequence.

    Now, I am not interested in Geocentrism (I know you have discussed this before); my concern is over the fact that I do not know enough physics to see where this goes wrong. One consequence of the EP is that Special Relativity is false!

    Of course, I might be making a 'rookie mistake' here, and would appreciate some guidance.
    The "rookie mistake" is not reading far enough!

    Max Born's book has this passage less than a dozen pages later:
    A similar error lies at the root of the following objection, which is continually being brought forward, although the explanation is very simple.

    According to the general theory of relativity, a coordinate system which is rotating with respect to the fixed stars (i.e., which is rigidly connected with the earth) is fully equivalent to a system which is at rest with respect to the fixed stars. In such a system, however, the fixed stars themselves acquire enormous velocities.
    The explanation follows. He refers back to equation (99), which defines the speed of light in terms of the coefficients of the metric, and points out (in both places) that they are not all identically equal to 1 (and c is therefore not 299792458 m/s) in the presence of a gravitational field, the same reason that light is "bent" in that presence.

    Your edition may have the same typo mine does, or it may have been fixed in the second edition. Mine reads "As soon as the distance exceeds 1/2 pi x 365 light years, the velocity becomes greater than c." Of course, it should read "1/(2 x pi x 365)"

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    Quote Originally Posted by profloater View Post
    The conundrum of spin is really quite basic, if an object is spinning or two points are rotating about each other the mysterious property inertia comes into play and inertial forces operate as calculated by Newton.
    Bingo! We take it for granted, but this is very deep.

    It was also named Mach's principle by Einstein.

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    I think the main points have been brought out already-- that there's a difference between what a coordinate system is doing and what actual objects are physically doing. Key tenets of relativity are that there is no need for any concept of absolute motion, and the laws of physics (which we might regard as "what is really happening" if we are of a rationalistic bent) must be invariant for all observers, i.e., must be expressible in a coordinate-free way. This is important-- we invariably use coordinates to talk about and quantify motion, but what motion "is" in some deeper sense must transcend those coordinates. The same is true if we regard the Earth as stationary and the universe as rotating-- that rotation is not physically real, it is a manifestation of the chosen coordinates. However, Born's point is that the same holds if we take the Earth as rotating, it's still just another coordinate system. That is the main difference between GR and Newtonian gravity right there, and was Einstein's motivation in deriving GR.

    So we could reframe the equivalence between the Copernican and Ptolemaic models by saying that we should recognize these as nothing but different coordinate systems, rather than statements about absolute motion or the absolute state of reality. All the same, we should not conclude that Galileo's trial was much ado about nothing, if he was just being tried for using different coordinates. The real difference between the Copernican and Ptolemaic models is indeed a world view, not a coordinate system, because the Copernican view is that the Earth is just another cosmic body, like any other planet, and the Sun is like any other star. The Ptolemaic view is that the Earth is something totally different from the rest of the cosmos, and that is really what was on trial there. It is a totally revolutionary view of our place in the grand scheme, which is quite different from the coordinates we choose to carry out any given prediction of observed motions in space.

    By the way, this is all related to "Mach's principle," which is essentially the statement that motion under gravity (which GR regards as inertial motion, i.e., force-free motion) is a universal phenomenon, not a local phenomenon, by which I mean it is not a force due to local mass distributions, it is a kind of history of spacetime that takes into account the entire universe. So if you refer all inertial motions back to the universe as a whole, there is no meaning in asking whether or not the universe as a whole can move or rotate, it is the standard by which all other inertial motions are regarded. In a sense, no object "knows what to do" even when there are no forces on it without referring back to the history of the spacetime through which that object moves, and that history is a story of the history of the universe as a whole, i.e., Big Bang cosmology.

    However, it should be pointed out that experts in GR often debate whether GR successfully embodies Mach's principle or not! It was Einstein's motivation in deriving it (not explaining Mercury's orbit, that was very much a sidelight to Einstein), yet it is hard to make a clear and rigorous statement of just what Mach's principle really is, and it is also hard to say whether GR obeys it or not. So be warned that there are many quite technical issues that are way beyond me and the experts don't even agree what are the crucial issues there.

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    Quote Originally Posted by Rosa Lichtenstein View Post
    One consequence of the EP is that Special Relativity is false!
    I forgot to comment on this earlier, but that is correct.

    General relativity describes space as having intrinsic curvature. Curvature tells masses how to move, and masses tell space how to curve. In the case where the curvature=0, space is flat and special relativity is true again. This is curvature in the mathematical sense. http://en.wikipedia.org/wiki/Curvatu...nian_manifolds

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    It might be better to say that the EP implies special relativity is only true locally. In other words, for any amount of gravity, special relativity will only work on sufficiently small scales, just as any surface looks flat on small enough scales.

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    Quote Originally Posted by caveman1917 View Post
    You know that special relativity is formulated in minkowski space? General relativity is not formulated in minkowski space but in a more general pseudo-riemann space, such that the tangent space at each point of that pseudo-riemann space is minkowski space. Why would you expect a theorem in minkowski space (for an inertial reference frame there is nothing with velocity greater than c) to still apply in a more general space?
    Thanks for that, but I was asking a question about the universe out there, not abstract space.

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    Quote Originally Posted by profloater View Post
    Can I put the distant stars another way? So we want to use Earth as a reference and now the stars are going round us as in the ancient view. But we know now it is only their light reaching us that we see, so the idea that they are travelling at some vast speed is an illusion. Once you add the vast distance and the speed of light, the idea of a sky sphere with holes through to heaven drops away.
    Of course you then replace your model with that of "fixed" stars and a spinning Earth. The conundrum of spin is really quite basic, if an object is spinning or two points are rotating about each other the mysterious property inertia comes into play and inertial forces operate as calculated by Newton. The mystery of inertia is directly linked to mass and must therefore have something to do with Higgs field. The interesting question remains of whether there is a point spin at the smallest size limit. i.E. does spin of a particle have the same meaning as spin of a group of particles.?
    Thanks, for that, but I'm not sure how I addresses the question I asked.

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    Quote Originally Posted by grapes View Post
    The "rookie mistake" is not reading far enough!

    Max Born's book has this passage less than a dozen pages later:

    The explanation follows. He refers back to equation (99), which defines the speed of light in terms of the coefficients of the metric, and points out (in both places) that they are not all identically equal to 1 (and c is therefore not 299792458 m/s) in the presence of a gravitational field, the same reason that light is "bent" in that presence.

    Your edition may have the same typo mine does, or it may have been fixed in the second edition. Mine reads "As soon as the distance exceeds 1/2 pi x 365 light years, the velocity becomes greater than c." Of course, it should read "1/(2 x pi x 365)"
    Again, thanks for that, but I am far from sure how that answers my question.

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    Quote Originally Posted by Ken G View Post
    I think the main points have been brought out already-- that there's a difference between what a coordinate system is doing and what actual objects are physically doing. Key tenets of relativity are that there is no need for any concept of absolute motion, and the laws of physics (which we might regard as "what is really happening" if we are of a rationalistic bent) must be invariant for all observers, i.e., must be expressible in a coordinate-free way. This is important-- we invariably use coordinates to talk about and quantify motion, but what motion "is" in some deeper sense must transcend those coordinates. The same is true if we regard the Earth as stationary and the universe as rotating-- that rotation is not physically real, it is a manifestation of the chosen coordinates. However, Born's point is that the same holds if we take the Earth as rotating, it's still just another coordinate system. That is the main difference between GR and Newtonian gravity right there, and was Einstein's motivation in deriving GR.

    So we could reframe the equivalence between the Copernican and Ptolemaic models by saying that we should recognize these as nothing but different coordinate systems, rather than statements about absolute motion or the absolute state of reality. All the same, we should not conclude that Galileo's trial was much ado about nothing, if he was just being tried for using different coordinates. The real difference between the Copernican and Ptolemaic models is indeed a world view, not a coordinate system, because the Copernican view is that the Earth is just another cosmic body, like any other planet, and the Sun is like any other star. The Ptolemaic view is that the Earth is something totally different from the rest of the cosmos, and that is really what was on trial there. It is a totally revolutionary view of our place in the grand scheme, which is quite different from the coordinates we choose to carry out any given prediction of observed motions in space.

    By the way, this is all related to "Mach's principle," which is essentially the statement that motion under gravity (which GR regards as inertial motion, i.e., force-free motion) is a universal phenomenon, not a local phenomenon, by which I mean it is not a force due to local mass distributions, it is a kind of history of spacetime that takes into account the entire universe. So if you refer all inertial motions back to the universe as a whole, there is no meaning in asking whether or not the universe as a whole can move or rotate, it is the standard by which all other inertial motions are regarded. In a sense, no object "knows what to do" even when there are no forces on it without referring back to the history of the spacetime through which that object moves, and that history is a story of the history of the universe as a whole, i.e., Big Bang cosmology.

    However, it should be pointed out that experts in GR often debate whether GR successfully embodies Mach's principle or not! It was Einstein's motivation in deriving it (not explaining Mercury's orbit, that was very much a sidelight to Einstein), yet it is hard to make a clear and rigorous statement of just what Mach's principle really is, and it is also hard to say whether GR obeys it or not. So be warned that there are many quite technical issues that are way beyond me and the experts don't even agree what are the crucial issues there.
    Thanks for the detailed reply, but I am not sure what the expression 'but what motion "is" in some deeper sense must transcend those coordinates' actually means.

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    Quote Originally Posted by Ken G View Post
    I think the main points have been brought out already-- that there's a difference between what a coordinate system is doing and what actual objects are physically doing. Key tenets of relativity are that there is no need for any concept of absolute motion, and the laws of physics (which we might regard as "what is really happening" if we are of a rationalistic bent) must be invariant for all observers, i.e., must be expressible in a coordinate-free way. This is important-- we invariably use coordinates to talk about and quantify motion, but what motion "is" in some deeper sense must transcend those coordinates. The same is true if we regard the Earth as stationary and the universe as rotating-- that rotation is not physically real, it is a manifestation of the chosen coordinates. However, Born's point is that the same holds if we take the Earth as rotating, it's still just another coordinate system. That is the main difference between GR and Newtonian gravity right there, and was Einstein's motivation in deriving GR.

    So we could reframe the equivalence between the Copernican and Ptolemaic models by saying that we should recognize these as nothing but different coordinate systems, rather than statements about absolute motion or the absolute state of reality. All the same, we should not conclude that Galileo's trial was much ado about nothing, if he was just being tried for using different coordinates. The real difference between the Copernican and Ptolemaic models is indeed a world view, not a coordinate system, because the Copernican view is that the Earth is just another cosmic body, like any other planet, and the Sun is like any other star. The Ptolemaic view is that the Earth is something totally different from the rest of the cosmos, and that is really what was on trial there. It is a totally revolutionary view of our place in the grand scheme, which is quite different from the coordinates we choose to carry out any given prediction of observed motions in space.

    By the way, this is all related to "Mach's principle," which is essentially the statement that motion under gravity (which GR regards as inertial motion, i.e., force-free motion) is a universal phenomenon, not a local phenomenon, by which I mean it is not a force due to local mass distributions, it is a kind of history of spacetime that takes into account the entire universe. So if you refer all inertial motions back to the universe as a whole, there is no meaning in asking whether or not the universe as a whole can move or rotate, it is the standard by which all other inertial motions are regarded. In a sense, no object "knows what to do" even when there are no forces on it without referring back to the history of the spacetime through which that object moves, and that history is a story of the history of the universe as a whole, i.e., Big Bang cosmology.

    However, it should be pointed out that experts in GR often debate whether GR successfully embodies Mach's principle or not! It was Einstein's motivation in deriving it (not explaining Mercury's orbit, that was very much a sidelight to Einstein), yet it is hard to make a clear and rigorous statement of just what Mach's principle really is, and it is also hard to say whether GR obeys it or not. So be warned that there are many quite technical issues that are way beyond me and the experts don't even agree what are the crucial issues there.
    It is also worth pointing out that my question wasn't about the relative merits, or, indeed, the validity, of the Copernican and Ptolemaic systems (which themselves incorporate metaphysical assumptions that science can't answer). But, how can they be equivalent if the one has the stars orbiting the earth so fast? Or, rather, how one implication of the EP is that they must travel so fast?

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    Quote Originally Posted by ShinAce View Post
    I forgot to comment on this earlier, but that is correct.

    General relativity describes space as having intrinsic curvature. Curvature tells masses how to move, and masses tell space how to curve. In the case where the curvature=0, space is flat and special relativity is true again. This is curvature in the mathematical sense. http://en.wikipedia.org/wiki/Curvatu...nian_manifolds
    Ok, so how does space 'tell' anything to anything else if it isn't sentient? And why does matter 'obey'? Is it intelligent, too?

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    Quote Originally Posted by Ken G View Post
    It might be better to say that the EP implies special relativity is only true locally. In other words, for any amount of gravity, special relativity will only work on sufficiently small scales, just as any surface looks flat on small enough scales.
    Does this mean that someone is 'looking' at what happens here? If not, I don't get the analogy.

    I'm sorry to ask such seemingly basic questions, but it appears to me that several of you are trying to give me mathematical answers (spruced up with a few rather odd-looking metaphors and/or analogies that are in fact more mysterious than the original question, or which have untoward anthropomorphic implications build into them), when the actual physics involved is being ignored.

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    Quote Originally Posted by Rosa Lichtenstein View Post
    But, how can they be equivalent if the one has the stars orbiting the earth so fast? Or, rather, how one implication of the EP is that they must travel so fast?
    Traveling "fast" in a global sense (the speed of something over there relative to something over here) is not an absolute notion in relativity, which means it isn't something "real." It's just a coordinate, meaning it is just something that we are choosing to say is happening based on some perspective we have chosen to take. If I spin around in my chair right now, the world appears to spin around me. Does this mean the world is really spinning? No, but it also doesn't mean that I am really spinning. Neither is really spinning, there is a relative motion there that I can coordinatize either way-- and the laws of physics will be just fine with it. That is the triumph of GR, right there-- it's much more than a theory of gravity.

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    Quote Originally Posted by Rosa Lichtenstein View Post
    Does this mean that someone is 'looking' at what happens here?
    No, by "local" I mean that the only observers who actually observe any phenomenon are at the same place and time that the phenomenon occurs. In other words, if we see a distant supernova, we don't actually see the supernova (which happened somewhere else and at some other time, and was experienced by observers who are likely no longer with us), we see light entering our telescope. We can take the attributes of that light, and weave a story about something happening somewhere else, and indeed that's just what we want to use science to do, but it is not something we actually observe, it is a story we tell built from what we actually observe. Observations are local. Similarly, the rules of special relativity are local-- the speed limit that is c represents the greatest speed that any two objects can pass each other at a given point (as perceived by observers on either object). But the stories we tell about speed are something different, and suffer from no such speed limit. Such it is with the rotation of the universe around a stationary Earth. None of this is mathematics-- the mathematics are way more difficult. This is the physics of relativity.

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    Quote Originally Posted by Rosa Lichtenstein View Post
    Thanks for that, but I was asking a question about the universe out there, not abstract space.
    The only thing i can say about the universe is what is being said by the theories that describe it, i'm not sure how it could be done differently.

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    Quote Originally Posted by Rosa Lichtenstein View Post
    Quote Originally Posted by ShinAce View Post
    General relativity describes space as having intrinsic
    curvature. Curvature tells masses how to move, and
    masses tell space how to curve. In the case where the
    curvature=0, space is flat ...
    Ok, so how does space 'tell' anything to anything else
    if it isn't sentient? And why does matter 'obey'? Is it
    intelligent, too?
    Imagine the surface of the ground, with hills and valleys,
    bumps and depressions, ridges, peaks, slopes and level
    places.

    Imagine rain falling on it. When the rain lands on a high
    part of the ground, the water runs downhill, toward lower
    ground. Which direction the water runs and how fast it
    runs in that direction depends on the shape of the ground:
    the direction and steepness of the slope.

    The shape of the ground tells the rainwater how to move.

    Now imagine that instead of ground, we have a big rubber
    sheet, with heavy objects scattered about on it. The heavy
    objects weigh down on the rubber sheet, distorting it so
    that the surface is pushed down near each object.

    The weights of the objects tell the sheet how to curve

    This is an unusual and ironic analogy in that it employs
    gravity as an analogy to gravity! I hope that won't cause
    confusion. Most analogies use one thing to represent a
    different thing. In this case, gravity represents gravity.

    The distortions in the rubber sheet act just like the hills
    and dales of the ground. Any object placed on the sheet
    that can slide across the sheet will slide down into the
    hollows made by the heavy objects.

    Now imagine that the sheet is extremely slippery, so that
    all those heavy objects can slide around on the surface.

    Because of the depressions in the sheet, when one object
    moves past another, instead of moving in a straight line,
    its motion is redirected by the curvature of the sheet to a
    different direction. The weight of the objects on the sheet
    tell the sheet how to curve, and the curves in the sheet
    tell the objects how to move.

    No sentience or intelligence needed.

    -- Jeff, in Minneapolis

    .
    Last edited by Jeff Root; 2013-Nov-25 at 05:01 PM.
    http://www.FreeMars.org/jeff/

    "I find astronomy very interesting, but I wouldn't if I thought we
    were just going to sit here and look." -- "Van Rijn"

    "The other planets? Well, they just happen to be there, but the
    point of rockets is to explore them!" -- Kai Yeves

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    Quote Originally Posted by Rosa Lichtenstein View Post
    Again, thanks for that, but I am far from sure how that answers my question.
    Max Born answers your question in the book that you quoted. Were you reading the book?

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