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Natural Science Forum / Physics / General Physics / August 2007General Covariance vs Lorentz Invariance
Before this month. I thought that General Covariance is the General Relativity Theory version of Lorentz Invariance where the relativity principle in Special Relativity is extended to accelerated motion. But I found out it was not. So what is the "relativity" principle in General Relativity? Some believe "general relativity" is a misnomer and it should be called "Einstein's theory of gravitation". For years I thought General Relativity being "General" means lorentz invariance is extended to acceleration motion with Special Relativity being "Special" in that it works in limited case such as inertial reference frame (with constant velocity) but it wasn't correct. Lorentz Invariance in GR is only in the infinitesimal sense. So what actually is the "relativity" principle in General Relativity? Hope someone can shed some light into all this. Tnx Rex. : Before this month. I thought that General Covariance is : the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 14 lines] : : Rex. Actually the "relativity" principle in General Relativity is... bullshit. Or, in words of one syllable -- bull sh.t. 'we establish by definition that the "time" required by light to travel from A to B equals the "time" it requires to travel from B to A' because I SAY SO and you have to agree because I'm the great genius, STOOOPID, don't you dare question it. -- Albert Einstein, who in 1895 failed an examination that would have allowed him to study for a diploma as an electrical engineer at the Eidgenössische Technische Hochschule in Zurich (couldn't even pass the SATs). > Before this month. I thought that General Covariance is > the General Relativity Theory version of Lorentz Invariance > where the relativity principle in Special Relativity is extended > to accelerated motion. But I found out it was not. Yes, they definitely have different natures. Lorentz invariance is a constraint on physical laws: Newtonian mechanics doesn't satisfy it, for example. General covariance is just a way of writing down physical laws, and not a constraint. You can make a general relativistic version of any Lorentz invariant theory, though it helps if it has a conserved four-vector quantity in it. > So what is the "relativity" principle in General Relativity? It's something like this. Special relativity says that you're only allowed to write down laws of physics that are Lorentz invariant, i.e. you have to be able to write them in terms of variables x,y,z,t such that they're also true of any x',y',z',t' which are related to x,y,z,t by a Lorentz transformation. This is philosophically problematical, as even Newton realized, because the coordinates themselves can only be given meaning through the dynamics. To put it another way, Newton's law of inertia is circular: it says objects move uniformly in the absence of external forces, but the only way to measure force is by the deviation of objects from uniform motion. In practice, the first law seems to be saying something meaningful and important about the world, but it's difficult to pin down what that is. You can make this philosophical problem mathematically precise by writing down your laws of physics in a form which is unchanged under any (smooth) coodinate reparameterization at all. This is possible for any physical laws, for purely mathematical reasons. You do it by introducing what's effectively a new field (the metric tensor field) which transforms in a certain way under coordinate changes and whose job is to compensate for whatever havoc you've wreaked by the change of coordinates. The field has ten independent components, and you get a bunch of new terms in your equations involving those components. You also get 20 new partial differential equations which express the fact that the rewritten theory still satisfies global Lorentz invariance. These all have the form (nasty second-order partial differentials of metric field components) = 0. So now we have a clearer idea of what it means for a theory to be Lorentz invariant: instead of talking about coordinates, we can talk about these coordinate-independent constraints on the metric field. General relativity is what you get when you drop these 20 constraints. The right hand sides of ten of the equations, no longer constrained to be zero, describe a conserved four-vector quantity, the gravitational charge, which is identified with the four-momentum of the original special relativistic theory. The other ten right hand sides describe the free gravitational field, including gravitational waves. The interesting thing is that all of the forces of the Standard Model can be derived by the same general argument: you start with a global symmetry, extend it to an arbitrary local transformation, write down what it means to still satisfy the global symmetry, and then replace the right hand sides with charges and field components. So it's extremely tempting to suppose that the Standard Model forces have a geometric interpretation like gravity does. But no one has managed to make this work, except for the classical electromagnetic field in Kaluza-Klein theory. Warning: I don't really understand any of this as well as it may sound like I do. -- Ben [snip good stuff] > You can make this philosophical problem mathematically precise by writing > down your laws of physics in a form which is unchanged under any (smooth) > coordinate reparameterization at all. This is possible for any physical laws, > for purely mathematical reasons. [snip rest of good stuff] Therein resides the fun part! SMOOTH. Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. An irreducibly discrete coordinate transformation can violate General Relativity without contradiction. [Euclid does not work on the geoid (large scale navigation or land surveying) for there are no parallel lines (lines being great circles) on a sphere.] Required are chemically identical opposite geometric parity mass distributions whose parity emergence scale is vastly smaller than the observation scale. Consider a parity pair of millimeter- to centimeter-radiused solid spheres carved from opposite parity single crystals. The positions of their atomic nuclei in space (fractional nanometers, 10^7 length ratio) are absolutely incommensurable and outside the Einstein group. Their statics (geometric parity destruction through melting, solution, vaporization) and dynamics (vacuum free fall) will be anomalous. http://www.mazepath.com/uncleal/lajos.htm#a2 We're going to look, Christmas 2007. The worst it can do is null. No, wait - the worst it can do is *not* null!  SignatureUncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/lajos.htm#a2 : [snip good stuff] : [quoted text clipped - 3 lines] : > for purely mathematical reasons. : [snip rest of good stuff] [snip rest of sh.t] You can make this philosophical bullcrap mathematically precise by writing down your laws of physics in a form which is Newtonian. This is possible for any physical laws, for purely mathematical reasons. OR you can assume and then run into paradox. Signature'we establish by definition that the "time" required by light to travel from A to B equals the "time" it requires to travel from B to A' because I SAY SO and you have to agree because I'm the great genius, STOOOPID, don't you dare question it. -- Albert Einstein, who in 1895 failed an examination that would have allowed him to study for a diploma as an electrical engineer at the Eidgenössische Technische Hochschule in Zurich (couldn't even pass the SATs). "Counterfactual assumptions yield nonsense. If such a thing were actually observed, reliably and reproducibly, then relativity would immediately need a major overhaul if not a complete replacement." -- Tom Roberts. > > Before this month. I thought that General Covariance is > > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 61 lines] > > -- Ben Super post Ben, LOL at "havoc you've wreaked" I'm still giggling, sounds like a big fart or something. (Also enjoyed Uncle Als, and T.Roberts). My shot on Relativity is captured in a very brief statement "the vanishing of absolute motion", defined in Eq.(6) here, http://physics.trak4.com/modern-spacetime.pdf I'm a student of the application of the Einstein Field Equation's, and respect it's classical application to gravitation, but that's an application of using General Covariance that is derived from the General Principles of Relativity, that imports assumptions that may not be necessary, and possibly - as it appears - confusing, even to skilled GRist's. The more fundamental a theory like GR becomes, the more it embraces, and soon it inhabits all of physics, mysteriously lurking, rather like a strict teacher who will smack your knuckes if you don't dot or "i's" or cross or "t's", hard to ignore. Regards Ken S. Tucker >> Before this month. I thought that General Covariance is >> the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 22 lines] > something meaningful and important about the world, but it's difficult to > pin down what that is. I am with Lev Landau who defined inertial in terms of a frames symmetry. To me physics has honed in for quirte a while now on what its foundations are - symmetry. > You can make this philosophical problem mathematically precise by writing > down your laws of physics in a form which is unchanged under any (smooth) [quoted text clipped - 33 lines] > Warning: I don't really understand any of this as well as it may sound > like I do. No one does - which is why it is best done in math. BTW - excellent post. Thanks Bill > -- Ben > > So what is the "relativity" principle in General Relativity? > [quoted text clipped - 10 lines] > meaningful and important about the world, but it's difficult to pin down > what that is. I don't think it's all that circular, although it has the quality of many constructions that it could be prefaced by the claim "it's possible to find a self-consistent set of objects, such that...". We might for example resort to operational definitions, and, starting with our old friends, clocks and meter sticks, piece together the sticks in a frame with clocks at the intersections, creating our famous and often carelessly invoked "reference frame". If we are really obsessive we may worry about operational meanings of straight lines and right angles. But I feel we can overcome this objection, tediously, with (operational) geometric constructions. Anyway, piecing together our physical sticks and clocks this way in suitable regions of space we may find we have an "inertial frame": one in which the first law obtains, and is _not_ circular, because it is a statement about possible experiment. It only seems circular if we jump directly to our abstract "coordinate system", which hides its physical roots. I reserve the right to digest the remainder of your, as always, thoughtful and erudite post. > I thought that General Covariance is > the General Relativity Theory version of Lorentz Invariance > where the relativity principle in Special Relativity is extended > to accelerated motion. But I found out it was not. So what > is the "relativity" principle in General Relativity? There isn't one, really. SR had the Principle of Relativity, which (loosely) states that one can write the laws of physics relative to any inertial frame and they will be the same equations. In GR the analogous postulate is general covariance, which replaces "inertial frame" with "arbitrary coordinates" (a seemingly minor change with profound implications). > Some > believe "general relativity" is a misnomer and it should > be called "Einstein's theory of gravitation". It is actually rather more than just gravitation. It is about geometry. The difference between SR and GR is best described geometrically: GR describes any geometry, as long as it is consistent with the energy-momentum density of the world being modeled. SR describes only a geometry that has the 10 Killing vectors of the Lorentz group; this requires a flat manifold with the topology of R^4. Ben Rudiak-Gould wrote: >(nasty second-order partial differentials of metric field components) = 0. Those turn out to be just the statement that the 10 Killing vectors exist; equivalently: that the manifold is flat. Note that these equations do not constrain the topology, while I believe the requirement that all 10 Killing vectors exist does constrain it. > General relativity is what you get when you drop these 20 > constraints. [...] It's a bit more complicated that that -- one must do so in a way that retains both general covariance and the equivalence principle. With knowledge of tensor analysis the first is easy to accommodate; the latter is quite restrictive, and is actually the determining factor that gets you GR and not some other theory. Well, one gets a generalization of GR, with potentially higher orders of derivatives of the metric (i.e. other terms in the Lagrangian). > The interesting thing is that all of the forces of the Standard Model can be derived by the same general argument: you start with a global symmetry, extend it to an arbitrary local transformation, write down what it means to still satisfy the global symmetry, and then replace the right hand sides with charges and field components. This is known as a "gauge field theory". > Warning: I don't really understand any of this as well as it may sound like I do. Ditto. I understand what I wrote above, but not all the implications.... Tom Roberts > > I thought that General Covariance is > > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 8 lines] > "arbitrary coordinates" (a seemingly minor change with profound > implications). I have been reading this paper http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf (General covariance and the foundations of general relativity: eight decades of dispute) for a week and re-reading it two or three times. I got it at the bottom of wikipedia entry on "General Covariance" At the conclusion of the paper there is these words: "The second viewpoint has been developed by Anderson and is based on his distinction between absolute and dynamical objects. His "principle of general invariance" entails that a spacetime theory can have no no-trivial absolute objects. Anderson argues that the principle is a relativity principle, since it is a symmetry principle, and that it is what Einstein really intended with his principle of general covariance. In this approach, general relativity is able to extend the symmetry group of special relativity from the Lorentz group to the general group. This extension depends on the metric being a dynamical object, which is no longer required to be preserved by the symmetry transformations of the theory's relativity principle". Do you agree or not agree with the above, and why? Rex > > Some > > believe "general relativity" is a misnomer and it should [quoted text clipped - 38 lines] > > Tom Roberts > > > I thought that General Covariance is > > > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 79 lines] > > > Tom Roberts I like piezoelectric crystals. Start with one that is surveyed to be perfectly square in free-fall, aka an inertial frame. Next, subject the crystal to either of these 3 effects, 1) gravitational-field 2) acceleration 3) an electric field and you'll find the crystal becomes a parallelogram. In the cases (1) and (2) these are thought to create the voltage across the crystal, by mechanic stress, to form the parallelogram, and thus create the voltage. OTOH, a voltage (3) applied across the crystal will create an exactly equivalent geometry to the crystal...a parallelogram, with all the geometry's and voltages the same, except we swapped the gravitational or acceleration field effects with an electric field. >From that example, I find General Covariance easily able to unify gravity and electricity, mind you, the math is challenging. Best Regards Ken S. Tucker >> > I thought that General Covariance is >> > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 34 lines] > > Do you agree or not agree with the above, and why? Without pre-empting what Tom has to say, you will find a discussion of the issue in Gravitation and Space-time by Ohanian and Ruffini - page 370. Shortly after Einstein published his theory Kretchmann did a critique where he showed it basically had no physical content. Einstein eventually agreed - but clamed it had heuristic value. In modern times it is taken to mean terms in an equation can be divided into absolute (things like the speed of light and planks constant) and dynamical (things like fields). The physical content is that when in covariant form absolute and dynamical terms must remain absolute or dynamical. This immediately implies, for example, the space-time metric is dynamical - which is one of the key elements of GR. Thanks Bill > Rex > [quoted text clipped - 46 lines] >> >> Tom Roberts > >> > I thought that General Covariance is > >> > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 48 lines] > Thanks > Bill I think the relativity principle in General Relativity is simply that momentum energy is what defines spacetime (or curves it). Therefore there is no absolute fixed space and time. Space and time or spacetime is relative to matter (or the stress energy tensor). This is I think is the "relativity" principle in GR plain and simple. Agree or disagree, and why? rex > > Rex > [quoted text clipped - 50 lines] > > - Show quoted text - >> >> > I thought that General Covariance is >> >> > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 60 lines] > I think the relativity principle in General Relativity is simply that > momentum energy is what defines spacetime (or curves it). Then you think wrong. What you write is a consequence of something much more fundamental - no prior geometry which implies geometry itself is dynamical. No prior geometry is what the Principle of General Invariance in the modern sense basically is getting at - it implies the metric is dynamical since it can not be absolute. > Therefore > there is no absolute fixed space and time. The POR already implies that. > Space and time or spacetime > is relative to matter (or the stress energy tensor). Space-time is what is measured with rulers and clocks - it is not relative to anything, or, more precisely, it has the same relative nature of something to anything that measures what that something is. Understanding what the terms of a model mean is fundamental to understanding the model. Tautological discussions on the meaning of words is a philosophers game - not science. Bill > This is I think > is the [quoted text clipped - 65 lines] >> >> - Show quoted text - > >> "Rex" <relativitexcali...@yahoo.com> wrote in message > [quoted text clipped - 87 lines] > > Bill On Aug 27, 3:28 am, Tom Roberts <tjroberts...@sbcglobal.net> wrote: > > I have been reading this paper > >http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 83 lines] > > - Show quoted text - In other words. In SR, inertial frames are relative while accelerated frames are not relative. In GR. The spacetime was made curved so that acceleration would become coordinate acceleration and because of background independence (no prior geometry due to dynamical metric). General Covariance was born unto the world (where inertial and non-inertial frames became both relative in contrast to SR). rex > > This is I think > > is the [quoted text clipped - 69 lines] > > - Show quoted text - > I have been reading this paper > http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 15 lines] > > Do you agree or not agree with the above, and why? I'm not sure. IMHO the word "absolute" has so many different meanings and connotations to different readers that it should be avoided by careful authors. While it is possible that Norton defined what he means by it, I doubt it, and have not spent the time to look. To me, a constant like c or G is not "absolute" in any meaningful sense. In that vein, I see nothing "absolute" in GR at all. But note that in GR, once one has solved the field equation for a particular problem, the resulting spacetime manifold _IS_ "absolute" in every common meaning of the word, and is quite clearly non-trivial. So one must distinguish between the theory and the solutions and models it constructs. Yes, once one has a physical theory in one coordinate system, it is essentially trivial to extend it into a generally covariant theory -- simply apply elementary calculus (or more abstract geometrical techniques) to determine how the equations of the theory appear in any other coordinate system. The (unstated) problem with this is that one must start with a COMPLETE theory in the initial coordinate system; invariably one starts with Minkowski coordinates (or equivalent), and they are valid only in a flat manifold, so one has no clue whether or not the curvature of the manifold should be part of the theory or not. > I think the relativity principle in General Relativity is simply that > momentum energy is what defines spacetime (or curves it). Therefore > there is no absolute fixed space and time. Space and time or spacetime > is relative to matter (or the stress energy tensor). This is I think > is the "relativity" principle in GR plain and simple. Hmmm. Just the momentum-energy tensor is INSUFFICIENT to obtain a solution of the field equation, one must also impose boundary conditions (or stipulate a topology without boundaries). This is an aspect of the failure of GR to actually include Mach's principle. Moreover, the metric invariably appears in the equations defining the energy-momentum tensor -- while the techniques of differential equations easily accommodate that, it does remove the ontological basis you are trying to give to the energy-momentum tensor. IOW: you cannot define "momentum energy" without already knowing the metric. For instance, in classical mechanics the energy of a particle involves v.v; in classical electrodynamics the energy of the field involves E.E+B.B -- such dot products inherently involve the metric. > In other words, General Relativity is all about laws remaining > unchanged > under an arbitrary transformation of spacetime coordinates. This > however doesn't mean extending the relativity of motion to accelerated > motion. Hmmm. The motion of a given object is always specified relative to some coordinate system, and in GR neither of those quantities is constrained in any way. In particular, either the object or the coordinates can be "accelerated", and the equations of GR remain valid. One must be careful what one means by these words.... > Since last year and before this month. I thought General > Relativity was all about extending lorentz invariance of SR and the > relativity of motion to accelerated motion. I guess some newbies or > layman would also have this long standing misconception. Hmmm. IMHO the key aspects of GR are not related to "motion", accelerated or not. They are related to GEOMETRY. Tom Roberts >> I have been reading this paper >> http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 19 lines] > connotations to different readers that it should be avoided by careful > authors. For sure. The authors whose reference I gave do define it, but the definition they use is basically what I wrote - which is rather vague. They claim another reference defines it fully -- Anderson - Principle Of Relativity Physics. Section 8 of the following gives the full technical details of Andersons definition: http://www.pitt.edu/~jdnorton/papers/decades.pdf I prefer basing GR on 'no prior geometry', as Wheeler puts it, rather than recourse to absolute and dynamical terms. Wald says it succinctly - The principle of General Covariance states that the metric of space is the only quantity pertaining to space that can appear in the laws of physics - Wald - General Relativity page 57. To me that says is it all. But there is no doubt 'absolute' and 'dynamical' terms are what some authors use. To the original poster - please be aware that the exact principles of what General Relativity is based upon does vary from author to author and different time periods. The absolute and dynamical idea seems to be from textbooks in the 1960's. The definition from Wald which is a much more modern source. Again if you are interested do read the link above, and the reference I gave previously, where the issue is discussed. Like a lot of issues at the foundations of any subject there is often no right or wrong answer - simply different views that give a better insight. My view is GR is based on no prior geometry which implies the metric is dynamical. Thanks Bill > While it is possible that Norton defined what he means by it, I doubt it, > and have not spent the time to look. To me, a constant like c or G is not [quoted text clipped - 59 lines] > > Tom Roberts >>> I have been reading this paper >>> http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 110 lines] >> Hmmm. IMHO the key aspects of GR are not related to "motion", accelerated >> or not. They are related to GEOMETRY. I forgot to post the conclusion of the article I linked to which I believe sums it up rather well: 'The debate over the significance of general covariance in Einstein's general theory of relativity is far from settled. There are essentially three view points now current. First is the viewpoint routinely attributed to Einstein. It holds that the achievement of general covariance automatically implements a generalized principle of relativity. In view of the considerable weight of criticism, this view is no longer tenable. Relativity principles are symmetry principles; the requirement of general covariance is not a symmetry principle. The requirement of general covariance, taken by itself, is even devoid of physical content. It can be salvaged as a physical principle by supplementing it with further requirements. The most popular are a restriction to simple law forms and a restriction on the additional structures that may be used to achieve general covariance. However neither supplement condition has been developed systematically beyond the stage of fairly casual remarks. The second viewpoint has been developed by Anderson and is based on his distinction between absolute and dynamical objects. His 'principle of general invariance' entails that a space-time theory can have no non-trivial absolute objects. Anderson argues that the principle is a relativity principle, since it is a symmetry principle, and that it is what Einstein really intended with his principle of general covariance. In this approach, general relativity is able to extend the symmetry group of special relativity from the Lorentz group to the general group. This extension depends on the metric being a dynamical object, which is no longer required to be preserved by the symmetry transformations of the theory's relativity principle. The third viewpoint holds that the dynamical character of the metric is irrelevant in this context and that the metric must be preserved under the theory's symmetry group, if that group is to be associated with a relativity principle. Since the metrics of general relativistic space-time have, in general, no non-trivial symmetries, there is no non-trivial relativity principle in general relativity. Whatever may have been its role and place historically, general covariance is now automatically achieved by routine methods in the formulation of all seriously considered space-time theories. The foundations of general relativity do not lie in one or other principle advanced by Einstein. Rather, they lie in the simple assertion that space-time is semi-Riemannian, with gravity represented by its curvature and its metric tensor governed by the Einstein field equations.' My view lies mostly in in the third camp - although I suppose it also includes some elements of the second group. GR is based on 'no prior geometry' - which I take to mean 'the metric must be preserved under the theory's symmetry group'. Symmetry, like it is in much of physics, is the real key here - the symmetry implied by 'no prior geometry', which dictates the metric should have its own lagrangian. That is 'almost' enough to uniquely determine the field equations. Thanks Bill >> Tom Roberts > > I have been reading this paper > >http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 83 lines] > > - Show quoted text - In other words. In SR, inertial frames are relative while accelerated frames are not relative. In GR. The spacetime was made curved so that acceleration would become coordinate acceleration and because of background independence (no prior geometry due to dynamical metric). General Covariance was born unto the world (where inertial and non-inertial frames became both relative in contrast to SR). Agree? rex >> > I have been reading this paper >> >http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 90 lines] > independence (no prior geometry due to dynamical > metric). No. >General Covariance was born unto the world > (where inertial and non-inertial frames became both relative > in contrast to SR). Agree? No. Bill > rex > >> > I have been reading this paper > >> >http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 100 lines] > > Bill I was connecting what you are saying with Baez. In the article http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html He wrote: "The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is treated as absolute. In general relativity all motion is relative. To accommodate this change general relativity has to use curved space-time. In special relativity space-time is always flat." What I described is taken from his article connecting your description and uniting it all into general.. So Baez is wrong? Why and why not? rex > > rex- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - >> >> > I have been reading this paper >> >> >http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf [quoted text clipped - 134 lines] > your description and uniting it all into general.. > So Baez is wrong? No > Why and why not? Why is your comprehension ability not up to understanding what Baez said and able to explain it in a post, and instead write near gibberish like 'The space-time was made curved so that acceleration would become coordinate acceleration'? Beats me. He never used the words 'coordinate acceleration' - whatever the hell that is supposed to mean - little alone acceleration becoming coordinate acceleration - which is doubly obscure. This time please please read it more carefully. Bill > rex > [quoted text clipped - 3 lines] >> >> - Show quoted text - > >> "Rex" <relativitexcali...@yahoo.com> wrote in message > [quoted text clipped - 152 lines] > > Bill Oh. I just thought all accelarations in GR are coordinate accelerations. But realized coordinate accelerations is like free fall and separate from plain accelerations. In GR, we are eternally in movement in the geodesic path because of time. So I just can't imagine how to model plain acceleration of an object in the GR spacetime continuum. But I'd try to imagine it before going to sleep tomorrow. GR is really logic defying indeed :) rex > > rex > [quoted text clipped - 7 lines] > > - Show quoted text - > > "Rex" <relativitexcali...@yahoo.com> wrote in message > [quoted text clipped - 167 lines] > > rex Another. In SR. There is still fixed space and time background where object moves.. only they are united as spacetime. In GR. It's no prior geometry or background independent.. so I just can't imagine how object can move when the metric is dynamic itself and so have difficulty understanding acceleration of an object in the dynamical spacetime where there isn't even a background. Gee. If someone can give an intuitive insight or two about this. Pls. do so because I'm already getting dizzy thinking of all this :) Tnx. rex > > > rex > [quoted text clipped - 11 lines] > > - Show quoted text - >> >> "Rex" <relativitexcali...@yahoo.com> wrote in message >> [quoted text clipped - 171 lines] > Oh. I just thought all accelarations in GR are coordinate > accelerations. Where in the literature have you come across the term 'coordinate acceleration'? Or is this something you just made up? You know what they say - if you don't stop it you will ......' Bill > But realized coordinate accelerations is like free fall and separate > from plain accelerations. In GR, we are eternally in movement in [quoted text clipped - 16 lines] >> >> - Show quoted text - > >> "Rex" <relativitexcali...@yahoo.com> wrote in message > [quoted text clipped - 181 lines] > > Bill Learn it from Tom Roberts. See: http://groups.google.com/group/sci.physics/browse_thread/thread/418a378f219de225 /450903795d04d64b#450903795d04d64b You mean he is using an non-standard term? rex > > But realized coordinate accelerations is like free fall and separate > > from plain accelerations. In GR, we are eternally in movement in [quoted text clipped - 20 lines] > > - Show quoted text - Rex says... >> Where in the literature have you come across the term 'coordinate >> acceleration'? Or is this something you just made up? You know what they [quoted text clipped - 7 lines] > >You mean he is using an non-standard term? I think you misunderstood what Tom was talking about. There are two slightly different intuitive notions of acceleration. First is a physical notion of acceleration. If you are in a vehicle that is accelerating, then you are pressed against your seat. If you are in an elevator accelerating upwards, you are pressed against the floor. You can measure physical acceleration with an "accelerometer". The simplest type of accelerometer is a heavy metal ball connected to the end of a stiff spring. If the spring is compressed or stretched, then you are accelerating in the direction of the spring axis. If the spring is relaxed, you are not accelerating. The second notion of acceleration is in terms of coordinates: If you plot an object's position x as a function of time t, then if you get a straight line, then the object is not accelerating (in the x-direction). If you get a curved line, then the object is accelerating. These two notions are the same only in very special circumstances: 1. If there are no gravitational fields, and 2. If you are using an intertial Cartesian coordinate system. Why does gravity change things? Because sitting at rest on the Earth "feels" the same as accelerating upwards. Your feet are pressed against the ground, the spring in your accelerometer is compressed. It's exactly as if you were accelerating. But if you plot your height as a function of time, you will get a straight line. So you are accelerating in the physical sense, but not in the coordinate sense. Why does it matter whether you are using Cartesian coordinates? Cartesian coordinates are the usual x, and y coordinates that people are first introduced to. An example of non-Cartesian coordinates is polar coordinates: instead of x and y, you can describe an object's location in terms of r (its distance from the center) and theta (the direction of the object from the center, measured as a angle between 0 and 2 pi). Cartesian versus non-Cartesian makes a difference because whether the path of an object as a function of time is a "straight line" or not depends on what coordinate system you are using. For example, if an object is traveling in a circle of radius 10 meters at the rate of one complete circuit per second, then in terms of Cartesian coordinates, we have x = 10 cosine(2 pi * t) y = 10 sine(2 pi * t) If you plot x versus t or y versus t, you will *not* get a straight line. On the other hand, if you switch to polar coordinates, you have r = 10 theta = 2 pi * t If you plot r versus t and theta versus t, you will get a straight line. Only for Cartesian coordinates in the absence of gravity is it the case that a straight line means no physical acceleration. Now, you seem to be confused about what General Relativity has to say about acceleration. What it DOESN'T say is that "...all accelarations in GR are coordinate accelerations". What it says is exactly what I've just said: Physically measurable acceleration is not the same as coordinate acceleration in the presence of large massive gravitating bodies. An object if freefall experiences no physically measurable acceleration (an accelerometer in freefall isn't compressed; if you are standing on an elevator that is dropping in freefall, you aren't pressed against the floor). But if you plot its position versus time, you will not get a straight line. So physically measurable acceleration is zero while the coordinate acceleration is nonzero. On the other hand, if something is sitting at rest on the surface of the Earth, then the physically measurable acceleration is *not* zero (you feel your feet pressed against the floor of an elevator, an accelerometer will be compressed), but the coordinate acceleration (plotting height versus time) is zero. -- Daryl McCullough Ithaca, NY Bill Hobba says... >Where in the literature have you come across the term 'coordinate >acceleration'? By "coordinate acceleration", I think he just means: a_x = d/dt v_x a_y = d/dt v_y a_z = d/dt v_z where v_x = dx/dt v_y = dy/dt v_z = dz/dt Somewhere along the line, Rex must have heard someone say something like the following: An observer in freefall near the Earth feels no acceleration, but his coordinate acceleration is nonzero. From such a statement, he made the leap to: "In General Relativity, there is no real acceleration, there is only coordinate acceleration." Which is completely wrong, but I can sort of understand how he came to the conclusion. -- Daryl McCullough Ithaca, NY On Aug 31, 12:30 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Bill Hobba says... > [quoted text clipped - 23 lines] > > -- That's right. I was focusing much on the physics of free fall and gravity and understood and could visualize how it happens in terms of coordinate acceleration where the curveness of spacetime causes something to converge which is what gravity is all about. So fixated on coordinate acceleration I ignored normal acceleration of a car and how to model it in the spacetime or worldline diagram. But I guess it is easy to do that by just extending the acceleration in a normal newtonian diagram to a diagram where time is moving and the worldline is ever going upwards. In minkowski diagram, things occured as "events" compared to newtonian way of presenting it. Do you know of a web site that compares the two and illustrates it clearly? Right now I just want to visualize what happens in the spacetime continnum of GR when an object like a car is accelerating on the surface. I have already visualized clearly the mechanics of free fall but not something that is stuck to the surface and moving in between points on the surface (like an accelerating car). Maybe you can explain how this differs to free fall in a curved non-euclidean fashion? Tnx. rex > Daryl McCullough > Ithaca, NY > On Aug 31, 12:30 am, stevendaryl3...@yahoo.com (Daryl McCullough) > wrote: [quoted text clipped - 50 lines] > > rex Or more specifically. In free fall. We are in geodesic path with no force acting on us. How about in a car accelerating on the surface of the earth. What's the spacetime diagram for it? Is it something like two geodesic paths touching (the car and the surface of the earth). But what makes it confusing is time is also moving. So even if the car is parked on a location of the earth. The spacetime diagram still has time moving, now when the car accelerates, what happens to the worldline, maybe it moves to the x, y, and z coordinates while time increases? rex > > Daryl McCullough > > Ithaca, NY- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - > > On Aug 31, 12:30 am, stevendaryl3...@yahoo.com (Daryl McCullough) > > wrote: [quoted text clipped - 62 lines] > > rex Btw.. I have reviewed and understood already the difference between a car accelerating on a surface versus coordinate acceleration. In the worldline, if an object is considered at rest with respect to the three space coordinates, it is still travelling through the dimension of time. Its world line will be a straight line that is parallel with the time axis of the graph. If the object moves through space with uniform motion, its world line will still be straight, but no longer parallel with the time axis. If the object moves with nonuniform motion, its world line becomes curved. This is in flat minkowski spacetime. In case the flatness becomes curved. The the relationships still hold.. only there is possibility of coordinate acceleration or free fall or gravity. Now the $10 question:... by curving spacetime, General Covariance is possible, something flat spacetime can't do. Why is curving spacetime the answer to General Covariance? A simple reply or intuitive description will do without incredible hard mathematics and equations. rex > > > Daryl McCullough > > > Ithaca, NY- Hide quoted text - [quoted text clipped - 6 lines] > > - Show quoted text - > > > On Aug 31, 12:30 am, stevendaryl3...@yahoo.com (Daryl McCullough) > > > wrote: [quoted text clipped - 83 lines] > > rex Btw.. I'll try answering my own question after the insight of the difference between car acceleratiion and coordinate acceleration (free fall) and after re-reading this entire thread (many good stuff here so I'll print them for future reference. Thanks for all guys). So I think the the answer why curving spacetime is related to General Covariance is because spacetime is not a priori flat (which is limiting, it's like saying a woman entire body is flat in 2D). So the greatest flexibility is for the the curvature to happen. Then general covariance can be made to work esp. in light of the background indepedent nature. rex > > > > Daryl McCullough > > > > Ithaca, NY- Hide quoted text - [quoted text clipped - 10 lines] > > - Show quoted text - > SR had the Principle of Relativity, which > (loosely) states that one can write the laws of physics relative to any > inertial frame and they will be the same equations. There is your inertial frame of reference again. All frames of references should be the same. There are no special frames of reference such as these so called inertial frames. <shrug> > In GR the analogous > postulate is general covariance, which replaces "inertial frame" with > "arbitrary coordinates" (a seemingly minor change with profound > implications). This is a total nonsense. In GR, what matters is the mathematics itself. The field equations represent GR. Invention of phrases such as general covariance is to hide physicists's lack of understanding in the field equations. <shrug> GR is completely mathematical in which you sometimes referred to as geometrical. <shrug> > > Some > > believe "general relativity" is a misnomer and it should > > be called "Einstein's theory of gravitation". > > It is actually rather more than just gravitation. It is about geometry. GR is actually a reflection of Hilbert's ever capable BS skills. Isn't one of his problems calls out to extend the calculus of variations? To justify his BS Lagrangian that results in the field equations and thus GR. <shrug> > The difference between SR and GR is best described geometrically: Both GR and SR can be described geometrically. <shrug> The can also be described as a 4-dimensional extension to Riemann's 3-dimensional concept of curved space. <shrug> > GR describes any geometry, as long as it is consistent with the > energy-momentum density of the world being modeled. This "energy-momentum" tensor is zero (in vacuum) which covers 99.9% of the study. Since energy and momentum are both observer dependent, it absolutely ludicrous to have such an observer dependent quantity to manifest a curvature in spacetime. There goes your general covariant concept. <shrug> > SR describes only a > geometry that has the 10 Killing vectors of the Lorentz group; this > requires a flat manifold with the topology of R^4. This is by far the most complicated I have seen to describe flat space. <shrug> > Ben Rudiak-Gould wrote: > > General relativity is what you get when you drop these 20 > > constraints. [...] [quoted text clipped - 8 lines] > higher orders of derivatives of the metric (i.e. other > terms in the Lagrangian). With the field equations, you get exactly second order in the partial differential equations. I don't know what you mean by other than second order partial derivatives. > > The interesting thing is that all of the forces of the Standard Model can be derived by the same general argument: you start with a global symmetry, extend it to an arbitrary local transformation, write down what it means to still satisfy the global symmetry, and then replace the right hand sides with charges and field components. > [quoted text clipped - 3 lines] > > Ditto. I understand what I wrote above, but not all the implications.... I don't understand most of your word salad, but I can call your bluff any day. <shrug> >> SR had the Principle of Relativity, which >> (loosely) states that one can write the laws of physics relative to any [quoted text clipped - 3 lines] >references should be the same. There are no special frames of >reference such as these so called inertial frames. <shrug> A course in classical mechanics would be highly instructive for you. [...] > Before this month. I thought that General Covariance is > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 12 lines] > > Hope someone can shed some light into all this. Tnx Lorentz invariance is like the god transfrom to E-M cranks, so there is no way to explain it, it's just there to keep the AT&T und IBM stooges neo-nano-ing. General Covariance is an actual mathematical transformation. > Rex. > Before this month. I thought that General Covariance is > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 14 lines] > > Rex. The Principle of Relativity just says no absolute reference frame is necessary to describe nature. A consequence is that nature can be described successfully without reference to a coordinate system. An equation which describes something without referencing a coordinate system is "generally covariant". That's all "General Covariance" really means. Tensor calculus is a mathematical system designed (arguably) for that explicit purpose - to describe the mathematical structures which remain invariant (i.e. do not change with coordinates). On Aug 26, 6:19 pm, schoenfeld....@gmail.com wrote: > > Before this month. I thought that General Covariance is > > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 30 lines] > > - Show quoted text - In other words, General Relativity is all about laws remaining unchanged under an arbitrary transformation of spacetime coordinates. This however doesn't mean extending the relativity of motion to accelerated motion. Since last year and before this month. I thought General Relativity was all about extending lorentz invariance of SR and the relativity of motion to accelerated motion. I guess some newbies or layman would also have this long standing misconception. Rex > On Aug 26, 6:19 pm, schoenfeld....@gmail.com wrote: >> [quoted text clipped - 37 lines] > unchanged > under an arbitrary transformation of spacetime coordinates. Are you sure about that? This has been one of the things that has bothered me for years. Its obvious that the laws of electrodynamics change under a Galiliean transformation so this tells me that not all transformations are valid. Isn't it more accurate to say "General Covariance is all about laws of physics remaining unchanged under an arbitrary but legitimate, transformation of spacetime coordinates." Pete > This > however doesn't mean extending the relativity of motion to accelerated [quoted text clipped - 4 lines] > > Rex > > On Aug 26, 6:19 pm, schoenfeld....@gmail.com wrote: > [quoted text clipped - 47 lines] > > Pete Well. Since I mentioned "spacetime" then Galilean is not meant or being referred because it began when Minkowski unified or united space and time into spacetime. Therefore spacetime automatically implies relativistic. It is wrong to say "Galilean spacetime" because spacetime is like a new word that meant realtivistic. So "Galilean spacetime" should instead be "Galilean space/time". rex > > This > > however doesn't mean extending the relativity of motion to accelerated [quoted text clipped - 8 lines] > > - Show quoted text - >> > On Aug 26, 6:19 pm, schoenfeld....@gmail.com wrote: >> [quoted text clipped - 52 lines] > > Well. Since I mentioned "spacetime" then Galilean is not meant... Actually there is such a thing called Galilean spacetime and there is a transformation for it from one set of spaceteime coordinates to anotehr where t' = t.. E.g. See - http://physics.nmt.edu/~raymond/classes/ph13xbook/node39.html But if you disagreewith the name then lets forget about the name since that's not really the point I'm trying to make. Let us say we construct a spacetime transformation in which we start with the Lorentz transformation and set gamma = 1 and t' = t. If you were to check an SR text such as "Special Relativity: A Modern Introduction," by Hans C. Ohanian. Its on page 50 and reads t' = t x' = x - Vt y' = y z' = z Since this is a map from the 4-d spacetime manifold onto itself it is a spacetime transformation. The physical meaning being that it is an invalid transformation and only works for v << c and for "small" distances. Pete >> In other words, General Relativity is all about laws remaining >> unchanged >> under an arbitrary transformation of spacetime coordinates. > > Are you sure about that? It is one aspect of GR, but is not "all" of GR. > This has been one of the things that has bothered > me for years. Its obvious that the laws of electrodynamics change under a > Galiliean transformation so this tells me that not all transformations are > valid. You got it backwards: what that is really saying is that the laws of electrodynamics you used are not correct. It should be obvious that nature uses no coordinates, so valid models of nature cannot possibly depend on one's choice of coordinates. So you need to re-write your "laws of electrodynamics" so they are independent of coordinates. Here's one way of writing Maxwell's equations in coordinate-independent form: dF = 0 *d*F = J where d is the exterior derivative, F is the Maxwell 2-form, * is the Hodge dual, and J is the current 1-form. Note F includes both E and B fields, and these two equations include all 4 of the traditional set. No quantity in the above equations references any coordinate in any way, but when projected onto Minkowski coordinates they reduce to the traditional set of Maxwell's equations. > Isn't it more accurate to say > "General Covariance is all about laws of physics remaining unchanged under > an arbitrary but legitimate, transformation of spacetime coordinates." As long as one selects valid coordinate systems, any transformations between them are "legitimate", because ARBITRARY human choices cannot possibly affect physical phenomena. IOW: coordinate systems are merely systematic methods of applying labels to the points of a manifold, and how one applies labels does not affect anything (except descriptions that use the labels). The equations of GR, and the Maxwell's equations above, are invariant under a Galilean transform, a Lorentz transform, and any other transform between valid coordinate systems. Tom Roberts > >> In other words, General Relativity is all about laws remaining > >> unchanged [quoted text clipped - 40 lines] > under a Galilean transform, a Lorentz transform, and any other transform > between valid coordinate systems. Ok, so the caution is an equation like Guv = Tuv , is solved to provide the guv into the Guv to solve the Tuv. That renders the guv from the Tuv, and is ok with me. A simple example is the Schwarzschild solution, wherein the metrics accurately reflect the effects of a material particle on the metrics guv, such that "Euclidian" CS's are *excluded*, so statements like the CS doesn't matter are misleading and unscientific. Excerise for Tom, wrap a piece of graph paper around a ball without crinking it. Regards Ken >> The equations of GR, and the Maxwell's equations above, are invariant >> under a Galilean transform, a Lorentz transform, and any other transform [quoted text clipped - 4 lines] > effects of a material particle on the metrics > guv, such that "Euclidian" CS's are *excluded*, Sure. Note I said above "valid coordinate systems". > so statements like the CS doesn't matter are > misleading and unscientific. My statements involved only VALID coordinate systems (i.e. ones for which the points of the manifold in a given region can be put into 1-to-1 correspondence with N-tuples of the coordinates). Yes, there are lots of well-known coordinate systems that are invalid on this (and other) manifolds. But once one has a valid coordinate system, one can apply any valid coordinate transform to it and the equations remain valid in the new coordinates. One can certainly apply Galilean or Lorentz transforms to any coordinates (the result may not make any sense, but that's not the point; the point it that it is POSSIBLE, and such transforms, while useless in their own right, leave the equations of GR unchanged). Yes, I did not include all the necessary caveats and conditions -- in a finite time that simply is not possible. The important one here is that coordinates are in general valid only over some REGION of the manifold, not the entire manifold. > Excerise for Tom, wrap a piece of graph > paper around a ball without crinking it. Of course that's not possible -- they have incompatible metrics. Tom Roberts > On Aug 26, 6:19 pm, schoenfeld....@gmail.com wrote: > [quoted text clipped - 36 lines] > unchanged > under an arbitrary transformation of spacetime coordinates. Yes, the first postulate says that. The second says gravitation is the curvature of your coordinate system. The law of warpage (how a body curves your coordinates) is the second postulate taken to the maximal logical conclusion. Note that the first postulate does imply you can use ANY arbitrary coordinate system to describe nature. You are still shackled by the constraints of the other laws of physics (i.e. SR, EM, etc). Only out of the admissable coordinate systems which can describe nature must ALL THE LAWS take the same form, which they mostly do (see generally covariant formulations of EM). The laws of quantum physics do not succumb to this regime. >This > however doesn't mean extending the relativity of motion to accelerated > motion. Since last year and before this month. I thought General > Relativity was all about extending lorentz invariance of SR and the > relativity of motion to accelerated motion. I guess some newbies or > layman would also have this long standing misconception. SR can describe accelerated observers. In SR, Lorentz covariance applies to the entire spacetime. In GR, Lorentz covariance only applies to infinitessimally small regions of it because you need to account for curvature of your coordinates (and other affine properties where torsion is present). > Rex On Aug 26, 8:56 pm, schoenfeld....@gmail.com wrote: > > On Aug 26, 6:19 pm, schoenfeld....@gmail.com wrote: > [quoted text clipped - 41 lines] > curves your coordinates) is the second postulate taken to the maximal > logical conclusion. There is also a first and second postulates in GR? I know that in SR, the first postulate is that the laws of physics is the same in all inertial frames, second postulate is that light travels the same speed in all inertial frames. Why did you say the second is about gravitation. What is GR first and second postulate, or third if there is more? Of course we Relativists know gravitation is the curvature of the coordinate system. > Note that the first postulate does imply you can use ANY arbitrary > coordinate system to describe nature. You are still shackled by the [quoted text clipped - 12 lines] > > SR can describe accelerated observers. But without gravity. So in faraway location in the universe such as in the Void recently detected, SR can be used to model accelerated motions between the ships.. is this what you mean, and our GPS uses GR because of gravity. > In SR, Lorentz covariance applies to the entire spacetime. > > In GR, Lorentz covariance only applies to infinitessimally small > regions of it because you need to account for curvature of your > coordinates (and other affine properties where torsion is present). Hmm.. you seem to be mixing "covariance" and "invariance". They have different meanings. In our everyday worlds. We seldom have constant motion. Acceleration motion is our norm. So this means lorentz transformation can't be used in GR. Instead coordinates manipulation is used. What do you say? Also I think we must change the words "Special Relativity" and "General Relativity" to more precise words because it can confuse things a bit and create more anti-relativists in decades and centuries to come esp. when quantum gravity arrives. rex > > Rex- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - [...] > There is also a first and second postulates in GR? I know that in > SR, the first postulate is that the laws of physics is the same > in all inertial frames, second postulate is that light travels the > same speed in all inertial frames. Why did you say the second is > about gravitation. What is GR first and second postulate, or third > if there is more? The second postulate of GR is that which asserts gravitational mass is inertial mass. It's called the "strong equivalence principle". It does not follow from other assumptions so therefore it is a postulate. Physics is not like mathematics in that it is all a self-contained axiomatic framework. If you look hard enough there's plenty of implicit postulates everywhere (i.e. universe is a manifold M associated with a set of objects S, an object is a set of properties charge, spin, etc,). Such a logical decomposition cannot account for all observational phenomenon as branches physics in their current states cannot all be reconciled with each other. Therefore it is useless and thinking in such terms is transitively, useless (not that you were, just saying no point arguing about whats a postulate and whats not unless one clearly but ad-hoc implication of the other). > > SR can describe accelerated observers. > > But without gravity. So in faraway location in the universe such > as in the Void recently detected, SR can be used to model > accelerated motions between the ships.. is this what you > mean, and our GPS uses GR because of gravity. No, I mean that you can still model an accelerated observer in SR. Just doa google serach on it. > > In SR, Lorentz covariance applies to the entire spacetime. > [quoted text clipped - 4 lines] > Hmm.. you seem to be mixing "covariance" and "invariance". > They have different meanings. I thought you meant covariance. I dont think in these terms of Lorentz invariance or whatever, the only things invariant that matter in SR are Minkowski spacetime intervals from which all other invariants follow (via manipulation of 4-vectors). > In our everyday worlds. We seldom have constant motion. > Acceleration motion is our norm. So this means lorentz > transformation can't be used in GR. Instead coordinates > manipulation is used. What do you say? No you can't just use a Lorentz transform in GR because curvature (and torsion in Einstein-Cartan) affect the coordinates. Forget about the Lorentz transform, that's not important. It's the geometry of spacetime that's important. A Lorentz transformation is just a rotation in Minkowski spacetime (hyperbolic along the three time-space 2-planes and circular along the three spatial 2-planes). The Poincare group is just the set of rotations and translations in Minkowski spacetime. In GR you first need a solution to the Einstein-Hilbert field equation for some distribution of mass and energy which requires the simultaneous solving of the stress-energy and Einstein tensors. Once you have that you get your metric and you can construct coordinate transformations with it. > Also I think we must change the words "Special Relativity" > and "General Relativity" to more precise words because > it can confuse things a bit and create more anti-relativists in > decades and centuries to come esp. when quantum gravity > arrives. There is no axiomatic foundation and trying to think in such terms will lead to irrelevancy. > rex > [quoted text clipped - 3 lines] > > > - Show quoted text - > Before this month. I thought that General Covariance is > the General Relativity Theory version of Lorentz Invariance > where the relativity principle in Special Relativity is extended > to accelerated motion. It's often presented that way, in my experience. > But I found out it was not. So what > is the "relativity" principle in General Relativity? Some [quoted text clipped - 9 lines] > > Hope someone can shed some light into all this. Tnx Ben Rudiak-Gould have given you an erudite answer: maybe I can give the idiot's version. "General Relativity" indeed asserts the equivalence of a wider class of coordinate systems than special relativity, but, as Ben suggests, does so in a way with less physical content. We simply develop (well, maybe not so simply :O-) a mathematical machinery which allows physical laws to be expressed equivalently in a class of coordinate systems related by smooth transformations. This machinery was needed to state the physical content of GR, but is not itself equivalent to the physical content... You know, every time you think you are going to go off and give the two bit idiot's version of something, you wind up getting just as complicated as any other version. Sigh. Anyway, there is some very strong physical content to the assertion of "Lorentz invariance". We may take as an experimental given that there is at least one coordinate system wherein a formally simple set of physical laws obtains (this eliminates some of the circularity Ben was worried about). Now, we conceive a family of coordinate systems related to this first by "Lorentz transformations". Now comes the physical content: we discover that physical laws written in terms of these _new_ coordinates are all precisely the same form as those written in the original coordinates. Which, sadly, is a set of words which can be applied to the machinery of GR too! Oops. There is a precise way to state this, but it's not cheap to come by. I'd say we put less into the machinery of SR, so the statement of equivalence in various coordinate systems is physically stronger. We put more into the machinery of GR, so that much of the equivalence is simply tautological. Of course, buried in this tautology, is plain old Lorentz invariance at a point... Well, you've hit upon a good point, and one which I also stumbled on myself -- I'm not saying it is unknown, but it seems to be widely unappreciated among those who fancy that they appreciate such things. I just can't capture it succinctly at this moment. > Before this month. I thought that General Covariance is > the General Relativity Theory version of Lorentz Invariance > where the relativity principle in Special Relativity is extended > to accelerated motion. General Relativity is *not* Special Relativity "extended to accelerated motion". General covariance is not General Relativity; but pertains to ANY (and all) physical foundations that pertain to events that take place in space and time. This includes, also, Newtonian Physics (and its modern formulation: Newton-Cartan gravity); and even Aristotlean Physics, such as could be defined, or ever was. What distinguishes General Relativity from Newton-Cartan or any other curved spacetime formalism is NOT the principle of covariance, but the CAUSAL structure; i.e. the distinction between locally Lorentz or locally Galilei (i.e. SO(3,0,1)) (vs. locally Aristotlean, etc.) Under general covariances, local frames have a GL(4) symmetry, which admits arbitrary transformations of one (non-singular) 4 frame vectors to any other (non-singular) 4-frame. Lorentzian symmetry imposes a SO(3,1) (the Lorentz group) on a subset of frames termed "orthonormal". A definition of orthonormality for Galilean frames distinguishes a different causal structure; one that makes the Galilei metric (g_{mn} which has signature (+,0,0,0) and dual metric g^{mn} (of signature (0,+,+,+)) invariant). The Lorentzian metric g_{mn} and dual g^{mn} both have signature (+,-,-,-) or (-,+,+, +), depending on whose convention you adopt (quantum theorists usually adopt the signature (+,-,-,-), while general relativists, the signature (-,+,+,+)). The analogue for Galilean symmetry to SO(3,1) is the homogeneous Galilei group. Aristotlean symmetry would be with respect to a metric g_{mn} of signature (0,+,+,+) and dual metric of signature (+,0,0,0). There is no (named) analogue to SO(3,1), no "Aristotlean group" that is studied by anyone. > So what actually is the "relativity" principle in General Relativity? The relativity principle of any theory cast in the language of general covariance is simply Galileo's principle of relativity: motion is relative. The consequence of that fact, as seemingly innocuous as it is, is far- reaching. For one, though it is not generally appreciated as such, already it implies, of necessity, that one has to replace "space" and "time" by "spacetime", because a consequence of motion's relativity is that there is no longer any cohesive concept of "point" nor any notion of spatial geometry! Instead, one has to adopt as the fundamental "undefined", in place of "point", the deeper notion of "event" -- i.e., that which has a location AT an instant in time -- i.e. a "point" in spacetime. This is not hard to see To make motion relative, the zero of velocity is no longer a meaningful concept. This means the velocities no longer comprise a vector space, but an AFFINE space; since that feature (the absence of any distinguished "zero") is precisely what distingishes an affine space. Accelerations are still vectors. One might say that a second principle of relativity exists to relativise accelerations. But this can only be done by bringing in the notion of a universal force that satisfies the Equivalence Priciple, since this is what acceleration would have to relativise to. Anyway... the consequence of velocities being affine is that positions are NO LONGER an affine space -- i.e. they do not constitute an affine geometry! Not Euclidean, not spherical, not hyperbolic, not any geometry at all of a classical variety. For the very notion of two points at different times being the "same" or "different", then, itself becomes relative. If the motion of the Earth is relative, then in a frame of reference where the Earth is moving, New York on 9/11 2001 is FAR from New York on 9/11 2007. To commemorate the event, one would have to go somewhere in the middle of outerspace where the "Earth used to be". But the "used to be" part, is itself, relative because motion is relative. So, any point counts as "used to be", as long as one could get to that point at that time from 9/11 2001/New York by some motion. Thus, "point" is no longer a cohesive concept, since the prerequisite for it to be "cohesive" (the notion of "sameness" of points at different times) is now relative. In its place, one has the deeper notion of point-instant, or event. All of this is a consequence of Galilei, not Einstein or Minkowski and pertains to ALL physics, be it Aristotlean, Galilei/Newton, Poincare/ Einstein/Lorentz, etc. Since velocities are affine, instead of vectors, this means points reside in what's called an affine bundle -- which is a topological manifold endowed with a structure whereby the vicinity of each point (that is, "event") has the appearance, approximately, as an affine space. At this point, you're dealing with (generally curved) spacetimes. Whether you restrict it further with the (logically unecessary) assumption that it also be "flat" is not relevant here. The issue at hand is that this is the arena of general covariance. > Lorentzian symmetry imposes a SO(3,1) (the Lorentz group) on a subset > of frames termed "orthonormal". A definition of orthonormality for > Galilean frames distinguishes a different causal structure; one that > makes the Galilei metric (g_{mn} which has signature (+,0,0,0) and > dual metric g^{mn} (of signature (0,+,+,+)) invariant). This makes no sense: one must have g^ij g_jk = d^i_k (the Kroenecker delta), but your claimed metric and its dual yield ZERO. I believe the usual approach is to not assign a metric to spacetime at all, only to space. > Aristotlean symmetry would be with respect to a metric g_{mn} of > signature (0,+,+,+) and dual metric of signature (+,0,0,0). Ditto. > [... velocities have affine structure] Yes. This is often/usually expressed as all timelike 4-velocities have norm=1, so one can treat them as vectors rather than an affine 4-tuple. Spacelike trajectories of course have norm=-1, and null trajectories have norm=0. Physicists generally ignore the subtleties here.... Tom Roberts > Before this month. I thought that General Covariance is > the General Relativity Theory version of Lorentz Invariance [quoted text clipped - 10 lines] > only in the infinitesimal sense. So what actually is the > "relativity" principle in General Relativity? OK. Here is a simple and possibly even accurate analogy. Scientists living in a plane have discovered a set of coordinates x, y in which the Law of Squares, which states that x^2 + y^2 = r^2, is experimentally verified. The Planarians call these coordinates "inertial". A Planarian scientist postulates that any coordinates u,v related to x,y coordinates by orthogonal transformation (rigid rotation) are also inertial, in that u^2 + v^2 = r^2. Orthogonal transformations then correspond to "Lorentz transformations", and the persistence of the law is the "formal equivalence of physical law under transformation of coordinates". Now some exceptionally clever fellow realizes that we don't have to restrict ourself to orthogonal transformations: he notices that we can write the law in the form r = z'Mz, where z == (x,y) is the "coordinate vector" and M is the "metric matrix". Of course the metric matrix is so simple in so-called inertial coordinates that we hardly notice it, being the identity matrix, but by allowing general symmetric matrices we extend the "formal equivalence of physical law" to skew transformations of coordinates. In what sense are the two "formal equivalences" equivalent, and in what sense are they inequivalent? They are equivalent in the sense that "a physical law invariant in form under transformation of coordinates" describes either case: apparently we have simply extended the scope of our existing principle. There _is_ a sense in which our two "equivalences" are not equivalent, however. The first law has a particularly simple experimental interpretation: we lay out meter sticks along two coordinate directions, we lay them out along the long leg of the triangle, we square the results and we add -- we can use the same meter sticks in every coordinate system and perform the identical operations. The second law requires us in general to lay out our meter sticks as before, but now to perform manipulations involving numbers specific to the specific coordinate system. The specific numbers are hidden in our generic "M", so we have camouflaged the loss of symmetry. There really was something special (preferred) about the first class of coordinate systems, something operational, now hidden by the extended machinery. |
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