[1] Covariance and invariance in physics A mathematical concept became a physical principle by Einstein. Einstein’s talent was just this: to interpret physically the mathematical results. George Mpantes www.mpantes.gr The history of mathematical transformations it is connected in physics with the Newton’s principle of relativity: Uniform rectilinear mechanical experiments. 1 motion cannot be intrinsically determined by It has it’s mathematical foundations in the transformation of the mechanical laws, under the Galilean transformations which (Galilean transformations) are physical presuppositions for space and time of Newtonian Universe. From the mathematical point of view, this physical assumption refers to the frames of reference of classical mechanics, and the facts of immediate concern may be put as follows: it is assumed that a frame of reference (called 1 “The motion of bodies included in a given space are the same among themselves , whether the space is at rest , or moves uniformly in a right line motion” Newton Corollary V to the “Laws of motion” without circular [2] by the physicists in inertial frame) exist in which Newton’s laws of classical mechanics hold. Any frame in uniform rectilinear motion in relation to the original system can be shown to be an inertial frame. We associate a rectangular Cartesian coordinate system with each reference frame.. The close relationship of this principle for classifying geometries, to the demand that physical laws have invariant mathematical forms with respect to a given transformation group, was pointed out by Klein. The orthogonal linear transformations accomplishes the first unification of geometry and the Galilean transformations the first unification of physics. (classical mechanics). Abstract The Newtonian mechanics is invariant under the simple Galilean transformations. This invariance was the mathematical description of Galilean relativity of motion. The electromagnetic theory of Maxwell (the two of the four equations) failed in invariance by terms in frame velocity, under rectilinear transformations, because of the partial derivatives with respect to t. This invariance was achieved by the Hertz’s invariants equation of Maxwell, published in 1892. Finally physicists decided that electromagnetics was covariant (not invariant) under the Lorentz transformations of coordinates, and this covariance was the mathematical description of Einstein’s principle of relativity. This behavior of electromagnetic theory is due to the non-operationalist definition of the concept of the electromagnetic field. The transformations Invariance of Newton’s mechanics in frame velocity The covariance Objectivity and covariance The end of invariance [3] The covariance as a physical principle The operationalist point of view Sources The transformations …if the form of a law is not changed by certain coordinate transformations , that is, if it is the same law in terms of either set of coordinates , we shall call that law invariant or covariant with respect to the transformations considered..(P.G. Bergmann (Introduction to the theory of relativity , Prentice Hall, New York 1946 p.10-11) This verbal confusion about the terms “invariance” and “covariance” arises because both are forms of mathematical form preservation. But what is their distinction in their physical implications ? This disjoining in the above reference is referred to the same physical reality or represent deeper differences in the whole course of physics? I shall prove that these meanings, spring from mathematics, and particularly from the theory of transformations, but for physics, are related to the operational school of definition of physical concepts. …..the transformation idea has more than historical interest. It plays a major role in the present day study of physical laws. In fact , the use of vector analysis as a descriptive language for physical sciences is largely based on the invariant properties of vector relations under certain types of transformations. (Introduction of vector and tensor analysis Robert C. Wrede (Dover p. 35) But what is transformed? The history of mathematical transformations is connected in physics with the Newton’s principle of relativity: [4] Uniform rectilinear mechanical experiments. 2 motion cannot be intrinsically determined by It has it’s mathematical foundations in the transformation of the mechanical laws, under the Galilean transformations which (Galilean transformations) are physical presuppositions for space and time of Newtonian Universe. From the mathematical point of view, this physical assumption refers to the frames of reference of classical mechanics, and the facts of immediate concern may be put as follows: it is assumed that a frame of reference (called by the physicists in inertial frame) exist in which Newton’s laws of classical mechanics hold. Any frame in uniform rectilinear motion in relation to the original system can be shown to be an inertial frame. We associate a rectangular Cartesian coordinate system with each reference frame.. The close relationship of this principle for classifying geometries, to the demand that physical laws have invariant mathematical forms with respect to a given transformation group, was pointed out by Klein. The orthogonal linear transformations accomplishes the first unification of geometry and the Galilean transformations the first unification of physics. (classical mechanics). Invariance of Newton’s mechanics in frame velocity Example 1. In mathematics we say that the laws of mechanics3 are invariant under the simple Galilean transformations r΄ =r-υt t΄=t …………..(1) 2 “The motion of bodies included in a given space are the same among themselves , whether the space is at rest , or moves uniformly in a right line without circular motion” Newton Corollary V to the “Laws of motion” I. every body tends to remain in a state of rest of uniform rectilinear motion unless compelled to change it’s state by action of an impressed force. II . The “rate of change of motion”, that is the change of momentum , is proportional to the impressed force and occurs in the direction of the applied force. III. To every action there is an equal and opposite reaction. 3 [5] where we associate the coordinates r΄ and r with a particle P moving uniformly and rectilinearly with respect to either system, the axes of the two rectangular Cartesian coordinate systems are parallel (have the same orientation in space) and the O’ system is in uniform translatory motion with respect to O. Newton’s second law of motion is   d 2r m 2  F .......... (2) dt and it is a measure of force. So we must to prove that force in relation (2) is invariant. Since m is invariant in Newtonian mechanics we have to prove the invariance of acceleration. Differentiating (1) twice we get the left-hand term     dr dr ΄  d 2r d 2r΄    ,......... ... 2  dt dt dt dt΄ 2     d 2r d 2r So m 2  (m 2 )΄  F  F΄ viz dt dt (2) becomes   d 2r (m 2 )΄  F΄ .......... (2΄ ) in the primed system dt We see that the form of the equation is the same, and each term is unaltered by the Galilean transformation. (by term we mean any additive quantity). We call this equation Galilean invariant under this transformation. We say that the second law of motion (and hence all the three) are Galilean invariant. [6] Example 2. the covariance We shall study now the transformation of equation (2) (now in the form Fk  d (m k )......( 2΄ ) dt under coordinate axis rotation. The two systems are in uniform rectilinear relative motion but have different orientation in space. It is assumed as before that at t=0 the origins coincide. The path of the of motion of one frame with respect to the other is illustrated in figure 1 and the complete transformation is given x  c x  t i i j j i x i wit h c  .......... ..( 4) x j i j The transformation of (2) under (4) will result from the axis rotation, as the translation does not vary the equation (2). This rotation is given by the linear transformations  x1   c11  2  2  x    c1  x3   c3    1 c12 c 22 c 23 c31  x 1    c32  x 2 . c33  x 3  Multiplying (2΄) with x k and summating for κ we have x r x x k d d Fk  (m k  k )  Fr  (m r )......... .......( 5) x r dt x r dt Comparison (2΄) and (5) shows that the two terms of (2΄) undergo similar “scrambling” of components in the barred system with the same i coefficients c j [7] For example this scrambling (linear combination of all components) is written for one component F  x1 x 2 x 3 F1  F2  F3 .......... ...( 6) . x x x This scrambling is a consequence of the vector character of the Newtonian force and not of the fact that F satisfies the Newton equation of motion. So the individual terms of the Newtonian law of motion, transform under an axis rotation to the same rule (of linear combination). The form of equation (2΄) , is the same in (5), but it’s terms are altered by the transformation, in the same way. We call this equation Galilean covariant. The law’s form expressed in the new symbols is independent of the orientation of the frame of reference. The covariance above was a mathematical demonstration of the abilities of transformations, as the rotations of axes had no motion, so it has no relation with the Galilean relativity. i It might be added that if c j vary with time –so that a transformation to rotating axes is described-, equation (2) is neither invariant nor covariant. The physical explanation is that rotation induces extra “non-inertial” forces , so the simplest form of motional relativity fails. But such scrambling of components in (6) as the result of covariance we get in the formula of transformation of the electric and magnetic three vectors between two Galilean frames, under (now) the simple Lorentz transformation E1 '  E1 , H1 '  H1 , E 2 '   ( E 2  H 3 / c), H 2 '   ( H 2  E3 / c), E 3 '   ( E3  H 2 / c)....... H 3 '   ( H 3  E 2 / c)........( 7) It should be noted that the transformation given above is a consequence of the tensor character of Frs (electromagnetic tensor) as t Lorentz [8] transformations, and depends in no way on the fact that Frs satisfies Maxwell’s equations. (J. L.Synge: Relativity: the special theory p 317) Objectivity and covariance There is a well defined mathematical distinction that we find in vector and tensor entities. They are composed of components which transform in a change of the coordinate system, they co-vary with the system, but the laws of transformation are such that if all the components are zero in the initial system , they will be zero in every system. So a tensor equation holds in every system if it holds in one. This consideration expresses a basic idea of modern science. The system of the tensor components is covariant , i.e it has a different numerical composition for each coordinate system. Yet we express in this fashion a state that it is independent of the coordinate system, i.e. an invariant state. The tensor as a whole is an invariant magnitude. We can recognize this property from its representation by means of components , since the components can be calculated for every coordinate system , if they are known for one.. it is unfortunate that the physical terminology does not reflect this well-defined mathematical distinction. (the philosophy of space and time p.236 Hans Reichenbach) We saw that in systems with different orientations in space, we have a covariant form of the second law of Newton, where the description of the terms, can be represented differently in different coordinate systems. ……Each of these descriptions presents the objective state in a particular way. The totality of these descriptions , however, defines an invariant situation , so to speak, namely, whereas one description gives only one component of the situation, it’s projection on a particular coordinate system. Among these components there is no difference with regard in truth. (Reichenbach p.241) Here Reichenbach denotes that an objective state, here the Newtonian laws, is the totality of the covariant states, as we have seen that [9] mathematically, -in linear transformations4-, the covariant states are produced from the various orientations of the systems of coordinates, and of course, the orientations in space are infinite. But We can visualize this as analogous to the geometrical meanings of equality and similarity. Invariance is as equality, as the scheme of the idol (image) of an object in a flat mirror, and covariance is as similarity, as the geometrical similar scheme of this idol. We can study the properties of the object in either of the two schemes, the similarity does not affect the causal chains of geometry of the schemes, as covariance does not affect the Newtonian laws of physics in various systems of coordinates. These as for philosophy but in physics there are different opinions for invariance and covariance from Hamilton’s time till today, in Thomas Phipps book (HERETICAL VERITIES Thomas E. Phipps p.120) …Hamilton had an opportunity to make the opposite realization; namely that the covariance of the Newtonian form of the equations of mechanics (under rotations as we demonstrated)was not an asset but a clue to imperfection of the form. Working from this or other clues , Hamilton produced “canonical” forms of the mechanical equations that were genuinely invariant under a huge group of (so called “contact” ) transformations – a group containing coordinate transformations , rotations etc. as small subgroups. ..in consequence physics flowered into canonical mechanics , the Hamilton Jacobi equation and (a short step onward) the Schroedinger equation and quantum mechanics…… Beyond the Hamilton’s canonical forms of equations of mechanics, there is a basic denial of symmetry in the edifice of physics. Maxwell’s equations hold in it’s form, only in the inertia frame of the observer- for his era in the frame of the immovable ether – and nowhere else. In all others inertial frames their terms alter , because alter the transformations , which now express different 4 Einstein claimed that the coordinate transformations transformations (between two Lorentz charts with a common origin) must be linear “on account of the properties of homogeneity which we attribute to space and time. [10] ideas for space and time , than Newton’s. But the invariance was rejected from physics. It is like Lorentz’s transformations to impose a rotation of axes in inertial frames and this is represented in four-dimensional systems in Minkowski’s space-time. The covariance as a physical principle. The end of invariance Special relativity interpreted for physics, exactly these mathematical results. It took almost three centuries of development of physics, before the principle of relativity was extended by Albert Einstein to the case of optical and electromagnetic phenomena , as observed in a closed , uniformly moving cabin. It was Einstein’s principle of relativity: Uniform rectilinear motion cannot be intrinsically determined by any experiment, mechanical or electromagnetical or optical . But Maxwell’s equations were not invariant as were Newton’s, under the Galilean transformations. Now the equations of physics ought to be Lorentz covariant because this occurred for the equations of Maxwell. Newton’s equations had to change because this covariance was now the physical state of the equations. Why was impossible a Galilean invariance of the Maxwell’s equations, as with Newtonian equations of mechanics? because the partial derivative with respect to t, does not transform invariantly under such a rectilinear transformation. We see that applying (1)and the chain rule of differentiation, we have  x'  y '  z '  t '   .     t t x' t y ' t z ' t t '      ( x y z ) t΄ x΄ y΄ z΄ Or       .΄ .......... .......... (8) t t΄ [11] So partial derivative with respect to t, as two of the four Maxwell equations, fail of invariance by terms of first order in frame velocity. So there is not Galilean invariance for Maxwell equations. The non-invariance of Maxwell’s equations under Galilean transformation, was interpreted (a) that an ether existed for electromagnetism (though not for mechanics) (b) that this ether existed in a physically detectable state of motion. The invariance now became covariance, under a new system of transformations, the Lorentz transformations. The operationalist view of point What if, we replaced partial time derivatives, whenever they appeared in Maxwell’s equations , with total time derivatives? These were Hertz’s equations, which become rigorously invariant under Galilean transformation, just as Newton’s equations are. But we had to change our views for the field. What physically is a field? We read in Thomas Phipps (heretical verities p.109) Definition: A field is what is “measured” by (i.e produces quantifiable response in) a field detector. Definition: A “field detector is any object that causes the field to manifest its presence locally. Phipps claims that this definition is better than: “fields are mathematical vectors” (taught at a prominent east coast U.S. university) How Hertz’s equations rendered Galilean invariant? He replaced partial time derivatives with total time derivatives, where the total time derivative inserts (8) a term proportional to some kind of velocity -the velocity υd of the detector in the new-Hertzian interpretation-, and the Galilean velocity transformation generates a new term which cancels the velocity υd, that spoils the Galilean invariance of Maxwell’s equations. So the operationalist definition [12] of the field abolishes the necessity for the Lorentz transformations for the covariance of Maxwell’s equations, giving a new direction in the whole of physics. But as we Know today the Galilean invariance was converted to Lorentz covariance and the proper term for both is “invariance in form”. So we have the final formula ……Any relation between physical quantities must be expressed by means of form-invariant or covariant equations C.Moller (the theory of relativity Oxford 1972 p.96) Sources: HERETICAL VERITIES: MATHEMATICAL THEMES IN PHYSICAL DESCRIPTION (Thomas E .Phipps, Jr, classic non-fiction library, Urbana) Relativity: the special theory: (J.L.Synge North-Holland publishing company Amsterdam New York Oxford) Introduction to vector and tensor Analysis: (Robert C.Wrede, Dover) Relativity and geometry : (Roberto Torretti, Dover) The philosophy of space and time : (Hans Reichenbach, Dover) Space, time , matter: (Hermann Weyl, Dover) www. mpantes.gr END [13]