Invariant Relativistic Electrodynamics. Clifford Algebra

Invariant Relativistic Electrodynamics. Clifford Algebra Ap- proach Tomislav Ivezi�c Ruder Bosˇkovic´ Institute, P.O.B. 180, 10002 Zagreb, Croatia ivezic@irb.hr In the usual Cli�ord algebra formulation of electrodynamics the Fara- day bivector �eld F is decomposed into the observer dependent sum of a rela- tive vector E and a relative bivector e5B by making a space-time split, which depends on the observer velocity. (E corresponds to the three-dimensional electric �eld vector, B corresponds to the three-dimensional magnetic �eld vector and e5 is the (grade-4) pseudoscalar.) In this paper it is proved that the space-time split and the relative vectors are not relativistically correct, which means that the ordinary Maxwell equations with E and B and the �eld equations (FE) with F are not physically equivalent. Therefore we present the observer independent decomposition of F by using the 1-vectors of electric E and magnetic B �elds. The equivalent, invariant, formulations of relativistic electrodynamics (independent of the reference frame and of the chosen coordinatization for that frame) which use F; E and B; the real multivector Ψ = E− e5cB and the complex 1-vector Ψ = E− icB are devel- oped and presented here. The new observer independent FE are presented in formulations with E and B; with real and complex Ψ. When the sources are absent the FE with real and complex Ψ become Dirac like relativistic wave equations for the free photon. The expressions for the observer independent stress-energy vector T (v) (1-vector); energy density U (scalar), the Poynting vector S and the momentum density g (1-vectors), the angular momentum density M (bivector) and the Lorentz force K (1-vector) are directly derived from the FE. The local conservation laws are also directly derived from the FE and written in an invariant way. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows and only a kind of union of the two will preserve an independent reality. H. Minkowski I. INTRODUCTION In the usual Cli�ord algebra treatments, e.g. [1− 3], of electrodynam- ics the Maxwell equations (ME) are written as a single equation using the 1 electromagnetic �eld strength F (a bivector) and the gradient operator @ (1- vector). (As expressed in [3] (Found. Phys. 23, 1295 (1993)) the reference [4] Clifford Algebra to Geometric Calculus is: "one of the most stimulating modern textbooks of applied mathematics, full of powerful formulas waiting for physical application.") In order to get the more familiar form the �eld bivector F is expressed in terms of the sum of a relative vector E (corre- sponds to the three-dimensional electric �eld vector) and a relative bivector e5B (B corresponds to the three-dimensional magnetic �eld vector, and e5 is the (grade-4) pseudoscalar) by making a space-time split, which depends on the observer velocity. It is considered in such formulation that the ME written in terms of F and of E and B are completely equivalent. The compo- nents of E and B are considered to de�ne in a unique way the components of F . Moreover in order to get the wave theory of electromagnetism the vector potential A is introduced and F is de�ned in terms of A. Thus such formu- lation with relative vectors E; B and with 1-vector A is not only observer dependent but also gauge dependent. However in the recent works [5− 7] an invariant formulation of special relativity (SR) is proposed (see also [8]) and compared with di�erent experi- ments, e.g., the "muon" experiment, the Michelson-Morley type experiments, the Kennedy-Thorndike type experiments and the Ives-Stilwell type exper- iments. In such invariant formulation of SR a physical quantity in the 4D spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geomet- ric quantity (CBGQ) comprising both components and a basis (when some basis has been introduced). This invariant formulation is independent of the reference frame and of the chosen coordinatization for that frame. The CBGQs representing some 4D physical quantity in di�erent relatively mov- ing inertial frames of reference (IFRs), or in di�erent coordinatizations of the chosen IFR, are all mathematically equal and thus they are the same quantity for di�erent observers, or in di�erent coordinatizations (this fact is the real cause for the name invariant SR). It is taken in the invariant SR that such 4D tensor quantities are well-defined not only mathematically but also experimentally, as measurable quantities with real physical meaning. The complete and well-de�ned measurement from this invariant SR viewpoint is such measurement in which all parts of some 4D quantity are measured. The invariant SR is compared with the usual covariant formulation, which mainly deals with the basis components of tensors in a speci�c, i.e., Einstein’s co- ordinatization (EC). In the EC the Einstein synchronization [9] of distant 2 clocks and cartesian space coordinates xi are used in the chosen IFR. Fur- ther the invariant SR is compared with the usual noncovariant approach to SR in which some quantities are not 4D tensor quantities, but rather quanti- ties from "3+1" space and time, e.g., the synchronously determined spatial length (the Lorentz contraction) [9]. It is shown in [6] that all the experi- ments (when they are complete from the viewpoint of the invariant SR) are in agreement with that formulation but not always with the usual covariant or noncovariant approaches to SR. It is also found in [5] that the usual trans- formations of the 3D vectors of the electric and magnetic �elds E and B are not relativistically correct. In this paper it is shown that the space-time split is not relativistically correct procedure and that the relative vectors are not well-de�ned quanti- ties from the SR viewpoint. This means that the ordinary ME with E and B are not physically equivalent with the observer independent FE with F . Further we write the Lorentz transformations (LT) in a coordinatization in- dependent way. Then we present the observer independent decomposition of F in terms of 1-vectors E and B: The new Cli�ord algebra formulations of relativistic electrodynamics with 1-vectors E and B and with the real mul- tivector Ψ = E − e5cB; or with the complex 1-vector Ψ = E − icB (i is the unit imaginary), which are completely equivalent to the formulation with the �eld bivector F , are developed and presented here. The expressions for the observer independent stress-energy vector T (v) (1-vector); energy density U (scalar, i.e., grade-0 multivector), the Poynting vector S (1-vector); the angu- lar momentum density M (bivector) and the Lorentz force K (1-vector) are directly derived from the FE and given in all four formulations. Consequently the principle of relativity is automatically included in such formulations with invariant quantities, whereas in the traditional formulation of SR this prin- ciple acts as the postulate established outside the mathematical formulation of the theory. The local charge-current density and local energy-momentum conservation laws are derived from the FE. It is also shown that in the real and the complex Ψ formulations the FE become a Dirac like relativistic wave equation for the free photon. The expressions for such geometric 4D quanti- ties are compared with the familiar ones from the 3D space considering our de�nitions in the standard basis fγµg and in the R frame (the frame of "�du- cial" observers) in which E0 = B0 = 0. This formalism does not make use of the intermediate electromagnetic 4-potential A; and thus dispenses with the need for the gauge conditions. The main idea for the whole approach is the same as for the invariant SR with true tensors [5− 8], i.e., that the 3 physical meaning is attributed, both theoretically and experimentally, only to the observer independent 4D quantities. We also remark that the observer independent quantities introduced here and the FE written in terms of them are of the same form both in the flat and curved spacetimes. II. SHORT REVIEW OF GEOMETRIC ALGEBRA. SPACE- TIME SPLIT. LORENTZ TRANSFORMATIONS A. A brief summary of geometric algebra First we provide a brief summary of geometric algebra. We write Clif- ford vectors in lower case (a) and general multivectors (Cli�ord aggregate) in upper case (A). The space of multivectors is graded and multivectors containing elements of a single grade, r, are termed homogeneous and writ- ten Ar: The geometric (Cli�ord) product is written by simply juxtaposing multivectors AB. A basic operation on multivectors is the degree projection hAir which selects from the multivector A its r− vector part (0 = scalar, 1 = vector, 2 = bivector ....). We write the scalar (grade-0) part simply as hAi : The geometric product of a grade-r multivector Ar with a grade-s mul- tivector Bs decomposes into ArBs = hABi r+s + hABi r+s−2 :::+ hABi jr−sj : The inner and outer (or exterior) products are the lowest-grade and the highest-grade terms respectively of the above series Ar � Bs � hABi jr−sj ; and Ar ^ Bs � hABi r+s : For vectors a and b we have ab = a � b + a ^ b; where a � b � (1=2)(ab + ba); and a ^ b � (1=2)(ab − ba): Reversion is an invariant kind of conjugation, which is de�ned by A˜B = B˜A˜; a˜ = a; for any vector a, and it reverses the order of vectors in any given expression. Also we shall need the operation called the complex reversion (for example, when working with complex 1-vector Ψ = E − ciB). The complex reversion of, e.g., Ψ, is denoted by an overbar Ψ: It takes the complex conjugate of the scalar (complex) coe�cient of each of the 16 elements in the algebra, and reverses the order of multiplication of vectors in each multivector. B. Standard basis, non-standard bases, and the space-time split In the treatments, e.g., [1− 3], one usualy introduces the standard basis. The generators of the spacetime algebra (STA) (the Cli�ord algebra gener- ated by Minkowski spacetime) are taken to be four basis vectors fγµg ; � = 0:::3; satisfying γµ � γν = �µν = diag(+− −−): This basis is a right-handed orthonormal frame of vectors in the Minkowski spacetime M4 with γ0 in the 4 forward light cone. The γk (k = 1; 2; 3) are spacelike vectors. This alge- bra is often called the Dirac algebra D and the elements of D are called d−numbers. The γµ generate by multiplication a complete basis, the stan- dard basis, for STA: 1; γµ; γµ ^ γν ; γµγ5,γ5 (24 = 16 independent elements). γ5 is the pseudoscalar for the frame fγµg : We remark that the standard basis corresponds, in fact, to the spec�c co- ordinatization, i.e., the EC, of the chosen IFR. However di�erent coordinati- zations of an IFR are allowed and they are all equivalent in the description of physical phenomena. For example, in [5] two very di�erent, but completely equivalent coordinatizations, the EC and "radio" ("r") coordinatization, are exposed and exploited throughout the paper. For more detail about the "r" cordinatization see, e.g., [5] and references therein. The next step in the usual treatments, e.g., [1− 3], is the introduction of a space-time split and the relative vectors. Since the usual STA deals exclusively with the EC it is possible to say that a given IFR is completely characterized by a single future-pointing, timelike unit vector γ0 (γ0 is tangent to the world line of an observer at rest in the γ0-system). By singling out a particular time-like direction γ0 we can get a unique mapping of spacetime into the even subalgebra of STA (the Pauli subalgebra). For each spacetime point (or event) x this mapping is speci�ed by xγ0 = ct+ x; ct = x � γ0; x = x ^ γ0: (1) To each event x the equation (1) assigns a unique time and position in the γ0- system. The set of all position vectors x is the 3-dimensional position space of the observer γ0 and it is designated by P 3 = P 3(γ0) = fx = x ^ γ0g : The elements of P 3 are all spacetime bivectors with γ0 as a common factor (x^γ0): They are called the relative vectors (relative to γ0) and they will be designated in boldface. Then a standard basis f�k; k = 1; 2; 3g for P 3; which corresponds to a standard basis fγµg for M4 is given as �k = γk^γ0 = γkγ0: The invariant distance x2 then decomposes as x2 = (xγ0)(γ0x) = (ct−x)(ct+x) = c2t2−x2: The explicit appearance of γ0 in (1) imply that the space-time split is observer dependent, i.e., it is dependent on the chosen IFR. It has to be noted that in the EC the space-time split of the position 1-vector x (1) gives separately the space and time components of x with their usual meaning, i.e., as in the prerelativistic physics, and (as shown above) in the invariant distance x2 the spatial and temporal parts are also separated. (In the "r" cordinatization there is no space-time split and also in x2 the spatial and temporal parts 5 are not separated, see [5].) This does not mean that the EC does have some advantage relative to other coordinatizations and that the quantities in the EC are more physical than, e.g., those in the "r" cordinatization. Di�erent coordinatizations refer to the same IFR, say the S frame. But if we consider the geometric quantity, the position 1-vector, x in another relatively moving IFR S 0; which is characterized by γ00; then the space-time split in S 0 and in the EC is xγ00 = ct 0 +x0; and this xγ00 is not obtained by the LT (or any other coordinate transformations) from xγ0: (The hypersurface t0 = const: is not connected in any way with the hypersurface t = const:) Thus the customary Cli�ord algebra approaches to SR start with the geomet- ric, i.e., coordinate-free, quantities, e.g., x; x2; etc.; which are physically well- de�ned. However the use of the space-time split introduces in the customary approaches such coordinate-dependent quantities which are not physically well-de�ned since they cannot be connected by the LT. The main di�erence between our invariant approach to SR (by the use of the Cli�ord algebra) and the other Cli�ord algebra approaches is that in our approach, as already said, the physical meaning is attributed, both theoretically and experimentally, only to the geometric 4D quantities, and not to their parts. Thus there is no need and moreover it is not physical from the viewpoint of the invariant SR to introduce the space-time split of the geometric 4D quantity. We consider, in the same way as H. Minkowski (the motto in this paper), that the spatial and the temporal components (e.g., x and t; respectively) of some geometric 4D quantity (e.g., x) are not physically well-de�ned quantities. Only their union is physically well-de�ned and only such quantity does have an independent reality. Thus instead of the standard basis fγµg ; � = 0:::3; for M4 we can use some basis feµg (the metric tensor of M4 is then de�ned as gµν = eµ � eν) and its dual basis feµg ; where the set of base vectors eµ is related to the eµ by the conditions eµ � eν = �νµ. The pseudoscalar e5 of a frame feµg is de�ned by e5 = e0 ^ e1 ^ ^e2 ^ e3: Then, e.g., the position 1-vector x can be decomposed in the S and S 0 frames and in the standard basis fγµg and some non-standard basis feµg as x = xµγµ = xµ′γµ′ = :::: = xµ′e eµ′ : The primed quantities are the Lorentz transforms of the unprimed ones. Similarly any multivector A can be written as an invariant quantity with the components and the basis, i.e., as the CBGQ. In such interpretation the LT are considered as passive transformations; both the components and the base vectors are transformed but the whole geometric quantity remains unchanged. Thus we see that under the passive LT a well-de�ned quantity on 4D spacetime, i.e., 6 a CBGQ, is an invariant quantity. This doesn’t hold for the relative vectors and thence they are not well-de�ned quantities from the SR viewpoint. C. Lorentz transformations In the usual Cli�ord algebra formalism, e.g., [1− 4], the LT are con- sidered as active transformations; the components of, e.g., some 1-vector relative to a given IFR (with the standard basis fγµg) are transformed into the components of a new 1-vector relative to the same frame (the basis fγµg is not changed). Furthermore the LT are described with rotors R; RR˜ = 1; in the usual way as p ! p0 = RpR˜ = pµ′γµ: But every rotor in spacetime can be written in terms of a bivector as R = eθ/2: For boosts in arbitrary direction eθ/2 = (1 + γ + γ�γ0n)=(2(1 + γ)) 1/2; � = �γ0n; � is the scalar velocity in units of c, γ = (1 − �2)−1/2, or in terms of an ‘angle’ � we have tanh� = �; cosh� = γ; sinh� = �γ; and n is not the basis vector but any unit space-like vector orthogonal to γ0; e θ = cosh� + γ0n sinh�: (One can also express the relationship between the two relatively moving frames S and S 0 in terms of rotor as γµ′ = RγµR˜:) The above explicit form for R = eθ/2 is frame independent but it is coordinatization dependent since it refers to the EC. Here a coordinatization independent form for the LT is introduced and it can be used both in an active way (when there is no basis) or in a passive way (when some basis is introduced). The main step is the introduction of the 1-vector u = cn; which represents the proper velocity of the frame S with respect to itself. Then taking that v is 1-vector of the velocity of S 0 relative to S we write the component form of L in some basis feµg which; as already said, does not need to be the standard basis, as Lµν = g µ ν + 2u µvνc −2 − (uµ + vµ)(uν + vν)=c 2(1 + u � v=c2); (2) or with the components and the basis, i.e., as the CBGQ, L = Lµνeµe ν ; see the second paper in [8] and [5] ; actually this form of the LT is a generalization to arbitrary coordinatization of the covariant form of the LT in the EC given in [10]. The rotor connected with such L is R = L=(L˜L)1/2 = Lµνeµe ν=(L˜L)1/2; L˜L = 8(γ + 1); γ = u � v=c2; (3) It can be also written as R = hRi+ hRi2 = cosh�=2 + ((u ^ v)= ju ^ vj) sinh�=2 = 7 exp((�=2)(u ^ v)= ju ^ vj); (4) R = ((1 + u � v)=2)1/2 + ((u ^ v)= ju ^ vj)((−1 + u � v)=2)1/2: One also can solve Lµν in terms of R as Lµν = 〈 eνR˜e µR 〉 : (5) The usual results are recovered when the standard basis fγµg ; i.e., the EC is used. But these results for L and R hold also for other bases, i.e., coor- dinatizations. (Thus one can easily �nd the LT in the "r" coordinatization, Lµν,r; and compare it with the corresponding result in [5].) III. THE F FORMULATION OF ELECTRODYNAMICS AND THE PROOF THAT THE SPACE-TIME SPLIT AND THE TRANS- FORMATIONS OF RELATIVE VECTORS E AND B ARE NOT RELATIVISTICALLY CORRECT A. The F formulation of electrodynamics We start the exposition of electrodynamics writing the FE in terms of F [1− 3]; an electromagnetic �eld is represented by a bivector-valued function F = F (x) on spacetime. The source of the �eld is the electromagnetic current j which is a 1-vector �eld. Then using that the gradient operator @ is a 1-vector FE can be written as a single equation @F = j="0c; @ � F + @ ^ F = j="0c: (6) The trivector part is identically zero in the absence of magnetic charge. No- tice that in [1− 3] the FE (6) are considered to encode all of the ME, i.e., that the FE (6) and the usual ME with E and B are physically equivalent. Our discussion will show that this is not true. The �eld bivector F yields the complete description of the electromagnetic �eld and, in fact, there is no need to introduce either the �eld vectors or the potentials. For the given sources the Cli�ord algebra formalism enables one to �nd in a simple way the electromagnetic �eld F: Namely the gradient operator @ is invertible and (6) can be solved for F = @−1(j="0c); see, e.g., [1− 3] : In the Cli�ord algebra formalism one can easily derive the expressions for the stress-energy vector T (v) and the Lorentz force K directly from FE (6) and from the equation for F˜ ; the reverse of F; F˜ @˜ = j˜="0c (@˜ di�erentiates 8 to the left instead of to the right). Indeed, us