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Natural Science Forum / Physics / Research / November 2007General Covariance
Hello Can someone please explain the principle of general covariance in simple terms. Thanks > Hello > Can someone please explain the principle of general covariance in > simple terms. Consider two cartographers, each mapping a certain region of the land. Each one uses his or her conventions (scale, orientation, distortion, units, etc.). As each one is entitled to make individual choices, different conventions are not a problem. However, suppose that the regions they are mapping overlap. Then both maps have to agree on the set of geographic features in the overlapping region. This comparison involves translating from one set of cartographic conventions to another and may be quite involved. To make things simpler the two cartographers agree on a set of rules for describing geographic features, which make the dependence on individual cartographic conventions explicit and in a form ready for conversion from one set of conventions to another. The principle of general covariance, applied to cartography, is the agreement to use the mentioned set of rules for describing geographic features. In physics, the same phenomena may be described quite differently by different laboratories, since each one makes different choices for measurement apparatus, units of measurement, geometry of experiments, etc. The principle of general covariance states that the equations of physics should be expressed in a way that makes the dependence on these individual choices explicit and makes conversion between different conventions simple. There are precise mathematical ways of stating the above, however it's not clear whether you are familiar with the relevant mathematics. Hope this helps. Igor > > Hello > > Can someone please explain the principle of general covariance in [quoted text clipped - 30 lines] > > Igor Thanks Igor. Your explanation definitely helps. I understand that using tensors to express the laws of physics (in general relativity for example or electromagnetism) is one way to keep the expression of these laws independent of the specific coordinate systems used. Can you give me an example of a law of physics that is not in accordance with the principle of general covariance? Also, I would assume that a law not in accordance with the principle must be inherently flawed, would that be correct? Finally, I'm having trouble with the term covariance. To me, it hints at 2 processes changing together. It would seem to indicate the equations change as the coordinate system changes. This, however, is the opposite of what general covariance is about. Any insight into that? Perhaps, I'm simply not looking at it from the right angle. Thus spake Hayssam <hayssam.hajar@gmail.com> >Thanks Igor. Your explanation definitely helps. >I understand that using tensors to express the laws of physics (in >general relativity for example or electromagnetism) is one way to keep >the expression of these laws independent of the specific coordinate >systems used. Can you give me an example of a law of physics that is >not in accordance with the principle of general covariance? Newton's first law springs to mind, being phrased with respect to absolute space > Also, I >would assume that a law not in accordance with the principle must be >inherently flawed, would that be correct? Indeed, inertial objects do not continue indefinitely on straight lines. We retain a local form of the law, that inertial objects move locally on straight lines with respect to each other - local here means local in time as well as space, and motion is necessarily relative motion. >Finally, I'm having trouble with the term covariance. To me, it hints >at 2 processes changing together. It would seem to indicate the >equations change as the coordinate system changes. This, however, is >the opposite of what general covariance is about. Any insight into >that? Perhaps, I'm simply not looking at it from the right angle. Vectors in gtr come in two forms, contravariant, with an index at the top, and covariant, with an index at the bottom. Indices of these types automatically get built into tensors also. As the names suggest, under coordinate transformation contravariant and covariant indices are transformed in opposite ways such that the change cancels from the inner product - which is essentially a product between a covariant and contravariant vector. For example we might express a vector law in a particular coordinate system. When we change coordinates, the vectors described in the law change in a particular way. However, in the new coordinate system the law has the same form, up to the choice of coordinate axes. The important point is that if you take the scalar product between the vector and a given axis you get an invariant, because the description of the axis has also changed, and in such a way that the scalar product is invariant. Regards  SignatureCharles Francis moderator sci.physics.foundations. substitute charles for NotI to email > Can you give me an example of a law of physics that is > not in accordance with the principle of general covariance? The most famous is Newton's F=ma (or F=dp/dt), as commonly taught in elementary physics. It does not work at all in an accelerated coordinate system (c.f. "centrifugal and Coriolis forces" for rotating coordinate systems, which are symptoms of this more general disease). [Note that it can be fixed up by carefully setting up a system of tensors on 3-space. These are not the 3-vectors of your youth (except when projected onto inertial coordinates), and this is much more subtle and complicated than in the spacetime of relativity. This is not elementary; MTW devotes a section to it.] The only other examples I can think of easily are various approximations to GR or other theories (indeed the above is at base such an example). One often makes an approximation to the covariant derivative in such computations (e.g. omitting spatial connection components, as they are often smaller by a factor of 1/c or 1/c^2), and that destroys the underlying invariance of the equations. > Also, I > would assume that a law not in accordance with the principle must be > inherently flawed, would that be correct? Yes, assuming that physics itself is at all possible in the world we inhabit. Fortunately, direct experience shows that physics is indeed possible. If phenomena actually depended on how they were described, then chaos in our understanding of the world would be complete, and that is what a violation of general covariance in a valid theory would imply. > Finally, I'm having trouble with the term covariance. To me, it hints > at 2 processes changing together. It would seem to indicate the > equations change as the coordinate system changes. This, however, is > the opposite of what general covariance is about. Any insight into > that? Perhaps, I'm simply not looking at it from the right angle. The term "covariance" (in this context) was coined by physicists with their usual disdain for rigor [#]. The usual example is an equation written in tensor components, such as: F^i = DP^i/d\tau [this is a set of 4 equations written compactly] and "covariance" means that under a change of coordinates, the set of components of the 4-force {F^i} vary among themselves in precisely the same manner as the set of components of the proper-time covariant derivative of 4-momentum {DP^i/d\tau}. That identity in behavior of these two sets of numbers is what you called "2 processes changing together". So while the VALUES of the components change when one changes coordinates, the EQUATIONS themselves do not; the EQUATIONS are said to be "covariant" (the term really only applies to equations). At base the difficulty is attempting to describe tensors in terms of their components. "General covariance" is merely the requirement that equations among tensor components (with indexes properly balanced) look the same no matter what coordinate system one uses. Or equivalently, that valid equations always have properly balanced indexes. When one uses the tensors themselves, one instead has a "principle of invariance" (i.e. tensors are invariant under changes of coordinates), which is a more valuable statement, and is one that can easily be made rigorous (unlike "general covariance"). Indeed this invariance is part and parcel of the definition of tensors on a manifold. [#] Possibly by misconstruing a mathematical term with the same name and related meaning. Note the reversal in many/most physics books: the components of a covariant vector are themselves a set of contravariant quantities, and when one conflates a vector with its components, one applies the term "contravariant" to a "vector" that actually behaves in a covariant manner under diffeomorphism. This is just one more unfortunate stake in the stout fence that unites mathematics and theoretical physics.... Tom Roberts Thus spake Tom Roberts <tjroberts137@sbcglobal.net> >> Can you give me an example of a law of physics that is >> not in accordance with the principle of general covariance? [quoted text clipped - 3 lines] >system (c.f. "centrifugal and Coriolis forces" for rotating coordinate >systems, which are symptoms of this more general disease). I wouldn't go along with that. Einstein pointed out that the implication of acceleration being relative is that one cannot treat inertial forces separately from active forces in the second law, not that the law is abandoned. It is rewritten in terms of four vectors, and then obeys general covariance. As a result coordinate transforms to rotating frames introduce centrifugal and Coriolis forces, which are then treated on an equal footing with other forces, including, of course, gravity. >At base the difficulty is attempting to describe tensors in terms of >their components. I don't go along with that either. Empirically, what we measure, is always components in a given reference frame. I do not think it is justified in logic to claim that there is anything else. We have laws which are true whatever coordinates we choose. It does not follow from that that something can be described without coordinates. Even from a pure mathematical point of view, the moment one has a (finite dimensional) vector space one has an infinite choice of bases. One does not have something which exists in the absence of a basis. I question the "insights" which come from coordinate free notations. It seems to me that they come nearer to metaphysics based on misinterpretation of what the mathematics really does. Regards SignatureCharles Francis moderator sci.physics.foundations. substitute charles for NotI to email > Can you give me an example of a law of physics that is > not in accordance with the principle of general covariance? Sure: any set of field equations on a fixed spacetime (manifold & metric). For example, the Maxwell equations on flat spacetime. A succinct (albeit somewhat highbrow) way to define "general covariance" is that the physical law (usually involving some DEs) should only depend upon the spacetime manifold, with no other fixed structures (e.g., a given metric or other fixed tensor fields, a preferred foliation, etc.). See, e.g., the textbook by Wald. charlie torre Thus spake Hayssam <hayssam.hajar@gmail.com> >Can you give me an example of a law of physics that is not in >accordance with the principle of general covariance? Also, I would >assume that a law not in accordance with the principle must be >inherently flawed, would that be correct? I just want to add to my prior response by considering the meaning of N2 and N3. For N2 we have Force = dp/dt Now, if we replace the three vector force with a four vector, 3 momentum with 4 momentum and t with proper time, s, we have a covariant law Force = dp/ds A specific example is the Lorentz force law dp^i/ds = F^ij J_j where F is the Faraday tensor, and J is 4-current. http://en.wikipedia.org/wiki/Electromagnetic_tensor It is straightforward to check that this gives you the Lorentz law 3-Force = e(E + v x B) dp^i/ds = F^ij J_j is covariant, because when you change reference frame you get an identical law expressed in terms of the new coordinate axes, but it is not invariant, because it is expressed in terms of coordinate axes. If we choose rotating or accelerating coordinates, then it is seen that the law includes inertial forces, e.g. centrifugal, Coriolis, and as Einstein realised in the principle of equivalence, gravity. In relativity we cannot describe acceleration with respect to a background as in Newtonian mechanics, but only with respect to other matter, so there is no way of excluding forces due to the choice of the reference frame. Newton's third law, otoh, discusses only active forces. Inertial forces like centrifugal and Coriolis do not produce a reaction, and break the third law. As the covariant description of a force includes inertial force as a result of the choice of reference frame, N3 cannot be put directly into a covariant form. Instead we have to replace N3 with an equivalent law, namely conservation of momentum. This is expressed in terms of 3-vectors, which can be replaced by 4-vectors and is built into Einstein's field equation through the statement that G satisfies the Bianchi identity. Regards SignatureCharles Francis moderator sci.physics.foundations. substitute charles for NotI to email > Can someone please explain the principle of general covariance in > simple terms. My understanding of the principle of general covariance is that it is the same as saying the following: for something to be a physical quantity, it must transform under a representation of the diffeomorphism group. A physical quantity X, can be transformed to a quantity X'=T(g)X where T(g) is the transformation and g is an element of the diffeomorphism group. Now we can transform it again, X''=T(g')X' and so X''=T(g')T(g)X. However, we could also get X'' directly from X by X''=T(g'g)X and so for consistency we have to demand that the two ways of getting X'' are the same and so T(g'g)X=T(g')T(g)X. This last equation says that X transforms as a representation of the diffeo group. In other words, the principle of general covariance is the same as requiring the transformations of a physical quantity are mutually consistent. Hayssam also wrote: > Also, I would assume that a law not in accordance with the principle must be >inherently flawed, would that be correct? Yes, because if a quantity in a law does not transform as a representation of the diffeo group it would simply not be consistent and would not qualify as a physical quantity. Hayssam then asked: >Can you give me an example of a law of physics that is >not in accordance with the principle of general covariance? No, there are none; they would simply be inconsistent. However, Oh No and Tom Roberts replied to Hayssam with examples of laws which they claimed are not in accordance with the principle. Oh No >Newton's first law springs to mind, being phrased with respect to >absolute space Tom Roberts > The most famous is Newton's F=ma (or F=dp/dt), it does not work at > all in an accelerated coordinate system. The answers of Oh No and Tom Roberts contradict my answer to Hayssam's third question because I think that they are using a different definition of the principle. I think they mean: The principle of general covariance means a physical quantity transforms under a tensor representation of the diffeomorphism group. Their definition is too restrictive; if one only considered physical quantities which transform as tensor reps, then, for example, it would rule out spinors. The laws of physics are always in the form L(x)=y where L is some sort of operator, x is a coordinate or a field and y is a source or driving force. If x transforms under the rep T1(g) of the diffeo group, and y transforms under another diffeo rep T2(g) then the operator must transform under the diffeo rep, L'(.)=T2(g)L(T1(g^-1).) where the dot (.) is the empty slot for the thing the operator acts upon. Now if we transform everything in Lx=y we get L'x'=y' and this is the law in the transformed frame. This law in the transformed frame is consistent with the law in the original frame: L'x'=y' T2(g)L(T1(g^-1)T1(g)x)=T2(g)y T2(g)L(x)=T2(g)y and so applying T2(g^-1) on the left of both sides gives the original law, L(x)=y. In other words, the principle in the first form (general covariance means physical quantities are reps of diffeo group) simply ensures that the laws of physics are consistent across frames. Here is an example which constructs the reps which ensure that Newton's F=ma transforms under representations of a subgroup of the diffeo group. Tom Roberts said that F=ma does not work in an accelerated coordinate system, so suppose we consider a set of accelerated reference frames. I'll need two inequivalent reps in this argument, so I'll denote the transformation of the cordinates by T1(g) so that, (x',t')=T1(g)(x,t)=(x-gt^2/2,t). T1(g) transforms to a (x',t') frame with acceleration g along the positive x axis as seen from the (x,t) frame. Now transform (x',t') to (x'',t''), x''=T1(g')x'=x'-g't'^2/2. The combined transformation (x,t) to (x'',t'') is, x''=x-gt^2/2-g't^2/2=x-(g+g')t^2/2=T1(g'g)x This shows that the accelerated reference frames are a 1-dimensional Abelian additive subgroup of the diffeo group because the group element g'g means the element g+g' formed by adding the accelerations. Furthermore, the above equation shows that the coordinate x transforms under a representation of the subgroup and the time t transforms under the trivial rep. I'll write Newton's law in the (x,t) frame as Lx=F/m where the differential operator L=d^2/dt^2. Newton claims that L,x and F/m are physical quantities so they must transform as reps of the acceleration subgroup of the diffeo group. The coordinates (x,t) transform as our first rep T1(g), (x',t')=T1(g)(x,t)=(x-gt^2/2,t). The physical quantity force per unit mass y=F/m transforms under another rep of the acceleration subgroup, which I'll call T2(g), y'=T2(g)y=y-g=F/m-g. The differential operator must transform as the rep, L'(.)=T2(g)L(T1(g^-1).) When the operator L' in the (x',t') frames acts on some function x(t), L'(x)=T2(g)L(T1(g^-1)x)=T2(g)L(x+gt^2/2)=L(x+gt^2/2)-g=L(x)+g-g=L(x) so that the operator transforms under the trivial rep L'(.)=L(.). In the (x,t) frame, Newton's law is L(x)=y and in the (x',t') frame it is L'(x')=y' but L'=L so L(x')=y' and this is, d^2x'(t')/dt'^2=F'/m=F/m-g. In the (x',t') frame there is a acceleration -g which appears as an impressed gravitational field. stebla http://www.stebla.pwp.blueyonder.co.uk Thus spake Stephen Blake <stebla@blueyonder.co.uk> >Oh No >>Newton's first law springs to mind, being phrased with respect to [quoted text clipped - 14 lines] >quantities which transform as tensor reps, then, for example, it would >rule out spinors. Certainly, this is true. I have only attempted to explain the principle of general covariance in the form in which it is often stated in gtr, "the equations of physics have tensorial form". Spare a thought for the poor O.P., however. He is just learning gtr for the first time. Tensors are quite enough for him to be getting along with. I don't think he really wants quantum theory thrown in on top. :-) Regards SignatureCharles Francis moderator sci.physics.foundations. substitute charles for NotI to email > > > Hello > > > Can someone please explain the principle of general covariance in > > > simple terms. > > Consider two cartographers, each mapping a certain region of the land. > > Each one uses his or her conventions (scale, orientation, distortion, [quoted text clipped - 26 lines] > > > > Igor > Thanks Igor. Your explanation definitely helps. > I understand that using tensors to express the laws of physics (in [quoted text clipped - 9 lines] > the opposite of what general covariance is about. Any insight into > that? Perhaps, I'm simply not looking at it from the right angle. Hello Hayssam, Let me extend Igor's analogue - which I like very much - extend, or complement in a hopefully not too abstract manner, starting from your question along the litteral meaning of the words. For your feeling is right, that those who coined the term covariance had a quite sensible approach to the meaning of the words to be used. Thus, '*co*variance' means to vary together with something, indeed, while '*contra*variant' means to vary contrarily/oppositely to something, where - in this context - "something" means the coordinates. Here, "vary" does not refer to changes caused by motion, but to changes related to the change of measures. For simplicity, consider a rod within prerelativistic mechanics. When we say that its length equals 1 m[eter], we actually refer to coordinate systems, where this holds true. In case you have a gauge with one mark per meter, then the length of the rod equals 1 mark. In case you have a gauge with one mark per centimeter, then the length of the rod equals 100 marks. Now, the distance between two marks corresponds to the length of a base vector along the direction of the rod, while the number of marks corresponds to the value of the corresponding component of a vector. In physics, it is common, however, to use basis vectors of unit length and to assign both dimension and real value of a vector-valued variable (position, force, etc.) to the vector itself. This makes it necessary to account for gauge changes in the vectors such that any change of gauge does *not* affect the description of the physical objects (above, the rod), because, the universe is independent of how we describe it. As a result, "covariant" (say, x_u) and "contravariant" vector indices (x^u) were introduced such, that the combination of both: x_u x^u is *invariant* against any change (transformation) in the base vectors. Here, you may contradict and say, that this is most artificial! Indeed, for the time being, I have no argument against that. Looking forward, Peter > Hello > Can someone please explain the principle of general covariance in > simple terms. > Thanks General covariance is symmetry under all smooth coordinate transformations (going beyond the scale independence of conformal symmetry). SignatureUncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/lajos.htm#a2 > Can someone please explain the principle of general covariance in > simple terms. The general principle is: Physical phenomena can not depend on how a human observer chooses to describe them. When applied to physical theories intended to describe physical phenomena, and the coordinate systems in which they can be written, one has the principle of general covariance: the laws of physics are independent of coordinate choice. This provides a constraint on the mathematics used in valid physical theories. The simplest way to satisfy the constraint is to write all physical theories as tensor equations -- tensors on a manifold are inherently independent of any coordinates on the manifold, and are thus good candidates to model physical phenomena in a theory that uses a manifold to model spatio-temporal relationships. Historically, that is the route used in all of our current fundamental theories of physics. But the difficulties of quantum gravity imply that this route may not work in the future, and mathematics other than manifolds and tensors may be required. Tom Roberts > difficulties of quantum gravity imply that this route may not > work in the future, and mathematics other than manifolds and tensors may > be required. > > Tom Roberts Any suggestions as to 'other' mathematics. Richard > > Can someone please explain the principle of general covariance in > > simple terms. [quoted text clipped - 3 lines] > describe them. > Tom Roberts This is exactly the argument developed by Einstein in the introduction in his (long) fondamental paper on GR "Die Grundlage der allgemeinen Relativitatstheorie" Annalen der Physik 1916 vol XLIX p 769-882. A frame is just a practical way for describing (measuring parameters of) physical law, but the form of physical laws should be independant of the frame. He used his favorite argument for justifying this argument , relying on the fact that a measurement is no more than performing a "coincidence" between "marks" on the mesured parameters of the objet and marks on the measurement devices (rules, clocks) , some kind of topological argument independant of the frame. Jacques >> Can someone please explain the principle of general covariance in >> simple terms. [quoted text clipped - 20 lines] > > Tom Roberts TR: So GCoV is a "sociological" assertion- kinda like Kants synthetic_apriori - rather than a physical assertion like coulombs Law ? nss ***** Hello: > >Can you give me an example of a law of physics that is not in > >accordance with the principle of general covariance? Also, I would > >assume that a law not in accordance with the principle must be > >inherently flawed, would that be correct? Here was my reading of Sean Carroll's notes on this topic. Covariance is not about changing coordinates in the sense that Newton's laws of physics work fine in spherical coordinates as well as Euclidean ones. Instead it is about the ability to change the metric, yet still describe exactly the same physical situation. In Newtonian mechanics, time is completely separate from space, so there will be no changing of the manifold. In special relativity, the metric is nailed in place by inertial observers. In general relativity, one moves about spacetime, and the manifold changes in a smooth and continuous way. Far away from a mass, the Minkowski metric is used to describe a pair of events. Closer to a source mass, the Schwarzschild metric is used to describe the same pair of events. The metric has changed in a continuous way, but not the events. I would prefer to say that classical mechanics and special relativity have known limitations. doug > I understand that using tensors to express the laws of physics (in > general relativity for example or electromagnetism) is one way to keep [quoted text clipped - 8 lines] > the opposite of what general covariance is about. Any insight into > that? Perhaps, I'm simply not looking at it from the right angle. Several posters have already explained what the term "covariance" refers to. Let me say a few more words about general covariance. There is an argument due to Kretschmann, contemporary to Einstein's work on GR, which roughly states that any physical theory can be cast into general covariant form (that is, be expressed in terms of tensor equations). You can search the details in the group archives (try Kretschmann or Kretschmannization). This argument mostly nullifies the usefulness of general covariance as a selection criterion for physical theories. However, it does not destroy the wisdom of this principle when it comes to *communicating* physical theories. The reason is that expressing laws of physics in tensor form often reveals various hidden or implicit assumptions, which once recognized are amenable to generalization or further scrutiny. Let me give an explicit example. Supose you have a physical theory with one vector v and one covector u, whose components are related by the following equation: u_1 = v^2 . This is a basic example of an equation that violates general covariance: you are equating components of a vector and a covector, which transform very differently under changes of coordinates. However, at the point, Kretschmann might come in and say, Wait a second, what you really want to write is u_a = T_ab v^b . The covariant 2-tensor T_ab comes in to save the day. Your previous equality can be simply interpreted as specifying the components of T_ab in some special coordinate system. The expression for T_ab, and hence the form of the same equality, in any coordinate system can be deduced through standard tensor transformations. Having made the presence of T_ab explicit, several questions automatically appear on your mind: Is T_ab always the same? Does T_ab depend on v or u, or any other relevant quantity? Does T_ab obey its own dynamics? And so on. This particular example may not be as academic as it seems. Almost precisely the same situation occurs in continuum mechanics. There, normal directions are converted into shear forces through the stress tensor. Once the stress tensor is identified, its importance in the formulation of continuum mechanics becomes paramount. A similar story follows from Minkowski's formulation of special relativity. Once the presence of the flat space-time metric (aka Minkowski metric) becomes plain, it is only a short step to considering the metric as a dynamical variable, that is, GR. Another useful aspect of expressing physical laws in tensor form (aka the principle of general covariance) is an algorithm for translating equations from one coordinate system to a different one. I touched on this point some time ago: news:1165470519.941369.314830@80g2000cwy.googlegroups.com http://groups.google.com/group/sci.physics.research/msg/da13e3aa5b13d4f2 Hope this helps. Igor > There is an argument due to Kretschmann, contemporary to Einstein's > work on GR, which roughly states that any physical theory can be cast > into general covariant form (that is, be expressed in terms of tensor > equations). You can search the details in the group archives (try > Kretschmann or Kretschmannization). Good point. An example of this idea is the "parametrized field theory". You take any field theory on a fixed spacetime background and by adding the set of diffeomorphisms to the configuration space of fields in a particular way (see, e.g., hep-th/9204055, JMP 3:3802-3812,1992 and references therein) you end up with a field theory which admits the diffeomorphism group as a symmetry group and is equivalent to the original field theory modulo diffeos. This is a field theoretic version of the "paramterized particle", familiar from classical mechanics, where the time variable is promoted to a configuration space variable giving a time reparametrization invariant version of the original dynamical system. > This argument mostly nullifies the usefulness of general covariance as > a selection criterion for physical theories. Maybe. Maybe not. Earlier in this thread I mentioned a particularly elegant way to define general covariance in terms of absence of any a priori, fixed structures except the spacetime manifold. This definition has the advantage of providing a way to distinguish, say, general relativity from a theory obtained via Kretschmann's ideas (e.g., parametrized field theory). The Kretschmann type theories always have some fixed structures beyond the manifold. For example, in a parametrized field theory there is a fixed spacetime metric. Nowadays this version of "general covariance" is usually called "background independence". charlie torre |
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