Appendix A Vectors and Vector Analysis

Transcription

1 Appendix A Vectors and Vector Analysis A.1 Vector Algebra A.1.1. Let a i, i = 1, 2, 3 be the components of a vector a in the orthonormal basis u i of an Euclidean three-dimensional space. Using Einstein s summation convention, the analytical expression of a is a = a i u i. (A.1) The analytical expression of the radius-vector is then r = x i u i. (A.2) A.1.2. Let a and b be two arbitrary vectors. Skipping addition and subtraction, one can define 1. The scalar product or dot product of the two vectors: since a b = (a i u i ) (b k u k ) = a i b i = ab cos(â, b), u i u k = δ ik. (A.3) (A.4) 2. The vector product, orcross product of the two vectors: a b = b a = ɛ ijk a j b k u i, a b =ab sin(â, b), (A.5) as well as u i u j = ɛ ijk u k, u s = 1 2 ɛ siju i u j, (A.6) Springer-Verlag Berlin Heidelberg 2016 M. Chaichian et al., Electrodynamics, DOI /

2 592 Appendix A: Vectors and Vector Analysis where +1, ifi, j, k are an even permutation of 1, 2, 3, ɛ ijk = 1, ifi, j, k are an odd permutation of 1, 2, 3, 0, if any two indices are equal, (A.7) is the Levi-Civita permutation symbol (see Sect. A.5). One can easily verify the property δ il δ im δ in ɛ ijk ɛ lmn = δ jl δ jm δ jn δ kl δ km δ kn. (A.8) For i = l, (A.8) becomes ɛ ijk ɛ imn = δ jm δ kn δ jn δ km, (A.9) while for i = l, j = m it becomes ɛ ijk ɛ ijn = 2δ kn. (A.10) Obviously, ɛ ijk ɛ ijk = 3! =6. A.1.3. Let a, b, c be three arbitrary vectors. With these vectors one can define the following two types of product: 1. The mixed product in particular, 2. The double cross product a (b c) = ɛ ijk a i b j c k, u i (u j u k ) = ɛ ijk. (A.11) (A.12) a (b c) = (a c)b (a b)c. (A.13) A.2 Orthogonal Coordinate Transformations The transformation from a three-dimensional orthogonal Euclidean coordinate system S(Oxyz) to another system S (O x y z ) can be accomplished by three main procedures: i) translation of axes; ii) rotation of axes; iii) mirror reflection.

3 Appendix A: Vectors and Vector Analysis 593 The first two types of transformations do not change the orientation of the axes of S with respect to S (proper transformations), while under mirror reflection (i.e. x = x, y = y, z = z) a right-handed coordinate system transforms into a left-handed coordinate system (improper transformation). There are also possible combinations of these transformations, with the observation that a succession of two proper/improper transformations gives a proper transformation, while a proper transformation followed by an improper transformation leads to an improper transformation. Suppose that S and S have the same origin, and let u i, u i, i = 1, 2, 3bethe orthonormal associated bases. Since r = x i u i = x i u i, (A.14) we have x i = a ik x k, a ik = u i u k, x i = a ki x k, a ki = u k u i, i, k = 1, 2, 3, (A.15) as well as u i = a ik u k, u i = a ki u k. (A.16) The coefficients a ik form the matrix of the orthogonal transformation (A.15) 1, and a ki thematrix of the inverse transformation (A.15) 2. The invariance of the distance between two points, yields the orthogonality condition r 2 = x i x i = x k x k, a ik a im = δ km, i, k, m = 1, 2, 3. (A.17) Using the rule of the product of determinants, we find det(a ik ) =±1. (A.18) On the other hand, det(a ik ) = ɛ ijk a 1i a 2 j a 3k, i, j, k = 1, 2, 3, (A.19) that is ɛ ijk a 1i a 2 j a 3k =±1, (A.20)

4 594 Appendix A: Vectors and Vector Analysis or ɛ ijk a li a mj a nk =±ɛ lmn = ɛ lmn det(a ik ), l, m, n = 1, 2, 3. (A.21) Here the sign + corresponds to the case when the frames have the same orientation, and to the case when the orientations are different. Under a change of orthonormal basis, we have p + q = p i u i + q i u i = p i a mi u m + q i a mi u m = p m u m + q m u m = p + q, p q = p i q i = (a mi p m )(a siq s ) = δ ms p m q s = p s q s = p q, p q = ɛ ijk p j q k u i = ɛ ijk a mj a sk a li p m q s u l = (p q ) det(a ik ), p (q r) = ɛ ijk p i q j r k = ɛ ijk a li a mj a nk p l q m r n = p (q r ) det(a ik ). (A.22) Consequently, the addition (subtraction) and dot product of two vectors do not change when changing the basis, while the cross product and the mixed product (of polar vectors) change sign when the bases have different orientations. We call scalars of the first kind or scalar invariants those quantities whose sign does not depend on the basis orientation (e.g., temperature, mass, mechanical work, electric charge, etc.), and scalars of the second kind or pseudoscalars those quantities which change sign when the basis changes its orientation (e.g., magnetic flux, dφ = B ds, the moment of a force F with respect to an axis Δ, M Δ = u Δ (r F), etc.). We call polar vector or proper vector a vector which transforms to its negative under the inversion of its coordinate axes (electric field intensity, velocity of a particle, gradient of a scalar, etc.), and axial vector or pseudovector a vector which is invariant under inversion of coordinate axes (magnetic induction, angular velocity, etc). A.3 Elements of Vector Analysis A.3.1 Scalar and Vector Fields If to each point P of a domain D of the Euclidean space E 3 one can associate a value of a scalar ϕ(p), ind is defined a scalar field. If to each point P one can associate a vector quantity A(P), ind is defined a vector field. A scalar (or vector) field is called stationary if ϕ (or A) do not explicitly depend on time. In the opposite case, the field is non-stationary.

5 Appendix A: Vectors and Vector Analysis 595 A.3.2 Frequently Occurring Integrals In classical/phenomenological electrodynamics one encounters three types of integrals: 1. Line integral P2 P 1 a ds along a curve C, taken between the points P 1 and P 2, where a is a vector with its origin on the curve, and ds is a vector element of C. This integral is called circulation of the vector a along the curve C, between the points P 1 and P 2.If the curve C is closed, the circulation is denoted by a ds. C 2. Double integral a ds = a ds = a n ds, S S where a is a vector with its origin on the surface S, ds is a vector surface element, and n is the unit vector of the external normal to ds. This integral is called the flux of the vector a through the surface S. IfS is a closed surface, the integral is denoted by a ds. S 3. Triple integral a dτ = a dτ = a dr = a dv, V V V where V is the volume of the three-dimensional domain D E 3, and a has its origin at some point of D. A.3.3 First-Order Vector Differential Operators The nabla or del operator is defined as = x 1 u 1 + x 2 u 2 + x 3 u 3. (A.23)

6 596 Appendix A: Vectors and Vector Analysis By applying it in different ways to scalars and vectors, we obtain the gradient, divergence, and curl. 1. Gradient. 1 Let ϕ(r) be a scalar field, with ϕ(r) a continuous function with continuous derivative in D E 3. The vector field A = grad ϕ = ϕ = ϕ x i u i, i = 1, 2, 3 (A.24) is the gradient of the scalar field ϕ(r). The vector field A defined by (A.24) is called conservative. Equipotential surfaces. Consider the fixed surface ϕ(x, y, z) = C(= const.). (A.25) Then dϕ = ϕ dr = 0 shows that at every point of the surface (A.25) the vector ϕ is oriented along the normal to the surface. Giving values to C, one obtains a family of surfaces called equipotential surfaces, orlevel surfaces. Field lines. LetA(r) be a stationary vector field, and C a curve given by its parametric equations x i = x i (s), i = 1, 2, 3. If at every point the field A is tangent to the curve C, then the curve is a line of the vector field A. The differential equations of the field lines are obtained by projecting on axes the vector relation A dr = 0, where dr is a vector element of the field line. Directional derivative. Let us project the vector ϕ unto the direction defined by the unit vector u. Since dr = u dr =u ds, where ds is a curve element in the direction u, wehave ( ϕ) u = ( ϕ) u = dϕ ds. (A.26) If, in particular, u is the unit vector n of the external normal to the surface (A.25), then ϕ n = dϕ dn 0, (A.27) meaning that the gradient is oriented along the normal to the equipotential surfaces, and points in the direction of the greatest rate of increase of the scalar field. 1 As a matter of fact, gradient, divergence, andcurl are not bona fide first-order vector differential operators, but the results of the nabla operator (Hamilton s operator) action, in different ways, upon scalar and/or vector fields. Note that the Laplacian = = 2 and the d Alembertian = 1 2 are true differential operators, but of the second order. v 2 t 2

7 Appendix A: Vectors and Vector Analysis Divergence. Consider a continuously differentiable vector field A(r). The scalarvalued function div A = A = A i x i = i A i = A i,i, i = 1, 2, 3 (A.28) is called the divergence of the vector field A(r). If the vector field A(r) satisfies the condition A = 0, (A.29) then it is called source-free or solenoidal. The lines of such a field are closed curves (e.g. the magnetostatic field). 3. Curl. The curl (or rotor) of a vector field A(r) is another vector field B(r) defined as B = A = u i ɛ ijk j A k. (A.30) If the field A(r) has the property A = 0, (A.31) then it is called irrotational or curl-free (e.g. the velocity field of a fluid that moves in laminar flow regime). A.3.4 A Fundamental Theorems Divergence theorem Consider a spatial domain D bounded by the surface S, and a vector field A(r) of class C 1 in D and C 0 in D (the closure of the domain D, consisting of all the points of D and its boundary surface S). It can be shown that A n ds = Adτ, (A.32) S which is the mathematical expression of the divergence theorem. It is also called the Green Gauss Ostrogradsky theorem, after Johann Carl Friedrich Gauss ( ), George Green ( ), and Mikhail Ostrogradsky ( ). Here n is the unit vector of the external normal to S. Let us now diminish the surface S, so that the volume V becomes smaller and smaller. In the limit, we have 1 A = lim A n ds. (A.33) Δτ 0 Δτ S V

8 598 Appendix A: Vectors and Vector Analysis This formula can be considered as the definition of the divergence at a point. It is useful because it expresses divergence independently of any coordinate system (intrinsic relation). If at some point we have A > 0, we say that at that point there is a source;if A < 0, that point is a sink ;if A = 0, the point is a node. A Stokes Theorem Let C be any closed curve in the three-dimensional Euclidean space, and let S be any surface bounded by C.IfA is a vector field of class C 1 in C S, then A dl = ( A) ds = ( A) n ds. (A.34) C This is the Stokes theorem, named after George Stokes ( ) (sometimes called the Stokes Ampère theorem). If n is the unit vector of the external normal to S, then the circulation on C is given by the right-hand screw rule. In particular, if A = ϕ, wehave A dl = dϕ = 0. (A.35) C Contracting the surface S until it becomes an infinitesimal quantity, in the limit we may write 1 ( A) n = n ( A) = lim A dl, (A.36) ΔS 0 ΔS C which is the intrinsic definition of curl A. C S A.3.5 Some Consequences of the Divergence Theorem and Stokes Theorem Let A(r) = ϕ(r) e, where ϕ(r) is a scalar field of class C 1 in D E 3, and e is a constant vector. Formula (A.32) then yields ϕ ds = ϕ dτ, (A.37) S V while (A.34) leads to ϕ dl = (n ϕ) ds. (A.38) C S

9 Appendix A: Vectors and Vector Analysis 599 If in (A.32) we choose A = e B, where B is a vector field of class C 1 in D,the result is n B ds = Bdτ. (A.39) S Consider, now, the vector field A of the form A = ψ ϕ, where ψ C 1 (D) and ϕ C 2 (D).Formula(A.32) leads to V ( ψ ϕ + ψδϕ) dτ = V S ψ ϕ n ds. (A.40) Interchanging ϕ and ψ in (A.40), then subtracting the result from (A.40), we obtain ( (ψδϕ ϕδψ) dτ = ψ ϕ ) n ϕ ψ ds, (A.41) n V where (ϕ, ψ) C 2 (D) C 1 (D). Relations (A.40) and (A.41) are known as Green s identities. S A.3.6 Useful Formulas The nabla operator,, is sometimes applied to products of (scalar or vector) functions, or can be found in successive operations ( ϕ, etc.). Here are some useful formulas, frequently encountered in electrodynamics: (ϕψ) = ϕ ψ + ψ ϕ, (A.42) (ϕa) = ϕ A + A ϕ, (A.43) (ϕa) = ϕ A + ( ϕ) A, (A.44) (A B) = B A A B, (A.45) (A B) = A ( B) + B ( A) + (A )B + (B )A, (A.46) (A B) = A B B A + (B )A (A )B, (A.47) ( A) = 0, (A.48) ( ϕ) = 0, (A.49) ϕ = 2 ϕ = Δϕ; Δ = 2 = i i, x i x i (A.50) ( A) = ( A) ΔA. (A.51)

10 600 Appendix A: Vectors and Vector Analysis If r is the radius-vector of some point P E 3 with respect to the origin of the Cartesian coordinate system Oxyz, then r = r r = u r, r =1, r = 3, r = 0, ( ) 1 = r r r, ( ) 3 1 Δ = 0, (A.52) r r =0 ( ) 1 Δ = 4πδ(r), r ( ) 1 Δ = 4πδ(r r ). r r If A is a constant vector, then (A.45) (A.47) and (A.52) yield (A r) = 0, (A r) = A, (A r) = 2A. (A.53) (A.54) (A.55) Given the fields ϕ(r) and A(r), with r = r, one can show that ϕ = ϕ u r, A(r) = u r A, A(r) = u r A. (A.56) where ϕ = dϕ dr, A = da dr, u r = r r. A.4 Second-Order Cartesian Tensors A system of three quantities A i, i = 1, 2, 3 which transform according to (A.15) upon a change of basis, that is A i = a ik A k, i, k = 1, 2, 3, (A.57)

11 Appendix A: Vectors and Vector Analysis 601 where a ik satisfy the orthogonality condition (A.17), form a first-order orthogonal affine tensor, oranorthogonal affine vector. Asystemof3 2 = 9 quantities T ik, i, k = 1, 2, 3 which transform like the product A i B k, that is according to the rule T ik = a ija km T jm, i, j, k, m = 1, 2, 3 (A.58) is a second-order orthogonal affine tensor. If in(a.58) we set i = k and use the orthogonality condition (A.17), we have T ii = a ij a im T jm = δ jm T jm = T mm. (A.59) The sum T ii = T 11 + T 22 + T 33 is called trace (Tr) or spur (Sp) of the tensor T ik. The relation (A.59) shows that the trace is invariant under the coordinate change (A.15). A tensor T ik is called symmetric if T ik = T ki, and antisymmetric if T ik = T ki. In an n-dimensional Euclidean space, E n, a symmetric tensor has (Cn 2) = n(n + 1)/2 distinct components, and an antisymmetric tensor has Cn 2 = n(n 1)/2 distinct components. Therefore, in E 3 a symmetric tensor has 6 independent components, while an antisymmetric tensor has 3 distinct components. An antisymmetric tensor is characterized by T ii = 0 (no summation), i.e. the elements on the principal diagonal are zero. A tensor with the property T ik = 0 (i = k) is called diagonal. Such a tensor is, for example, the Kronecker symbol, also named the second-order symmetric unit tensor δ ik, i, k = 1, 2, 3. Its components do not change upon a change of coordinates. Indeed, δ ik = a ila km δ lm = a il a kl = δ ik. Given the antisymmetric tensor A ik, let us denote or, in a condensed form, A ij = ɛ ijk A k, A 12 = A 3, A 23 = A 1, A 31 = A 2, A i = 1 2 ɛ ijka jk, i, j, k = 1, 2, 3. (A.60) The ordered system of objects (quantities, numbers, etc.) A i form a pseudovector, called the pseudovector associated with the antisymmetric tensor A ik. Such a situation is encountered in the case of the cross product of two polar vectors. Taking into

12 602 Appendix A: Vectors and Vector Analysis account that any second-order tensor can be written as a sum of a symmetric and an antisymmetric tensor, A ik = 1 2 (A ik + A ki ) (A ik A ki ), (A.61) we can write c i = (a b) i = ɛ ijk a j b k = 1 2 ɛ ijkc jk, (A.62) where c jk = a j b k a k b j. (A.63) Consequently, the vector associated with a second-order antisymmetric tensor is a pseudovector. The antisymmetric tensor A ik and the pseudovector A i = 1 2 ɛ ijka jk are said to be dual to each other. A.5 Cartesian Tensors of Higher Order An order-p tensor or tensor of type p in E 3 is a system of 3 p components which, under an orthogonal transformation of coordinates, transform according to T i 1 i 2...i p = a i1 j 1 a i2 j 2...a i p j p T j1 j 2... j p, i 1,...,i p, j 1,..., j p = 1, 2, 3. (A.64) An order-p pseudotensor in E 3 is a system of 3 p components which, under an orthogonal transformation of coordinates, transform according to T i 1 i 2...i p = [det(a ik )] a i1 j 1 a i2 j 2...a i p j p T j 1 j 2... j p, i 1,...,i p, j 1,..., j p = 1, 2, 3. (A.65) In other words, under a proper orthogonal transformation, characterized by det(a ik ) = +1, pseudotensors transform like tensors, while under an improper orthogonal transformation, for which det(a ik ) = 1, there appears a change of sign. Comparing (A.65) and (A.21), we conclude that the Levi-Civita permutation symbol (A.7) is a pseudotensor. It is called the third-order totally antisymmetric unit pseudotensor. It will therefore transform according to the rule ɛ ijk = [det(a ik)] a il a jm a kn ɛ lmn. (A.66) In view of (A.21), we then have ɛ ijk = [det(a ik)] 2 ɛ ijk = ɛ ijk, (A.67) which says that the components of ɛ ijk do not depend on the choice of orthonormal basis.

13 Appendix B Tensors B.1 n-dimensional Spaces An n-dimensional space S n is a set of elements, called points, which are in biunivocal and bicontinuous correspondence with n real variables x 1, x 2,...,x n.the variables x 1, x 2,...,x n are called coordinates. To a set of values of the coordinates corresponds a single point and vice-versa. The number of coordinates defines the dimension of the space. Let x 1,...,x n be another set of coordinates in S n and consider the transformation x ν = h(x μ ), μ, ν = 1, 2,...,n, (B.1) where h is a function mapping S n to itself. If the Jacobian of the transformation (B.3) is different from zero, J = (x 1,...,x n ) = 0, (x 1,...,x n ) (B.2) then the transformation (B.1) is (at least locally) reversible, or biunivocal, and we also have x μ = h 1 (x ν ), μ, ν = 1, 2,...,n, (B.3) meaning that x 1,...,x n are also a system of coordinates in S n. A transformation like (B.1) or(b.3) is called coordinate transformation. The transformations (B.3) and (B.1) are inverse to each other. If the variables x μ are linear with respect to x ν, then the transformation (B.1)iscalledlinear of affine.(an affine transformation is any transformation that preserves collinearity and ratios of distances.) If in S n one defines the notion of distance between two points (called a metric), the space S n is called metric space. Springer-Verlag Berlin Heidelberg 2016 M. Chaichian et al., Electrodynamics, DOI /

14 604 Appendix B: Tensors Let f (x 1,...,x n ) be a function of coordinates. Under a coordinate transformation (B.1), the functional form of f is changed: f (x μ ) = f (h 1 (x μ )) = ( f h 1 )(x μ ) = f (x μ ), (B.4) with the notation f = ( f h 1 ). A function which satisfies f (x 1,...,x n ) = f (x 1,...,x n ), (B.5) is an invariant. Functions of coordinates describing physical quantities can have more complicated transformation properties under coordinate transformations. If the coordinate transformations form a continuous group, we say that those functions transform under a representation of that group. In that case, the system is covariant under the transformation of coordinates, i.e. the equations of motion characterizing the system keep their form upon a coordinate transformation. In that case, invariants are formed also from combinations of different representations of the group of transformations. In Chap. 7 there are many examples of Lorentz invariants formed by combinations of vectors and tensors. B.2 Contravariant and Covariant Vectors Differentiating relation (B.1), we have dx ν = x ν x μ dxμ = x ν μ dxμ, μ, ν = 1, 2,...,n, (B.6) where we denoted x ν μ = x ν x μ, (B.7) and used Einstein s summation convention over repeated indices. Asystemofn quantities (objects) A 1,...,A n which transform according to (B.6), i.e. A ν = x ν μ Aμ, (B.8) formafirst-order contravariant tensor, or a tensor of type (1, 0), orann-dimensional contravariant vector. The quantities A μ, μ = 1, 2,...,n are the vector components in x μ coordinates, while A μ are the components of the same vector in x μ coordinates. Observing that dx μ = xμ x ν dx ν = x μ ν dx ν, (B.9)

15 Appendix B: Tensors 605 where x μ ν = xμ x ν, (B.10) one can define the inverse (relative to (B.8)) transformation A μ = x μ ν A ν. (B.11) Consider now the scalar function f (x μ ), μ = 1, 2,...,n and take its partial derivatives with respect to x μ : f x = f x ν μ x ν x = x ν f μ μ, μ, ν = 1, 2,...,n. (B.12) x ν A set of n quantities B μ, μ = 1, 2,...,n which transform according to (B.12), that is B μ = x ν μ B ν, (B.13) define a first-order covariant tensor, or a tensor of type (0, 1), or an n-dimensional covariant vector. The inverse transformation is B ν = x μ ν B μ. (B.14) In the Euclidean space with Cartesian orthogonal coordinates, x μ ν = x ν μ, hence there is no distinction between contravariant and covariant vectors. It can be easily shown that the contracted product of a contravariant and a covariant vector is an invariant. Indeed, A ν B ν = x ν μ x λ ν Aμ B λ = δ λ μ Aμ B λ = A μ B μ = A μ B μ. (B.15) Observation: In tensor analysis, covariance and contravariance describe how the quantitative description of certain geometrical or physical entities changes when passing from one coordinate system to another. To be coordinate system invariant, vectors like radius-vector, velocity, acceleration, their derivatives with respect to time, etc., must contra-vary with the change of basis to compensate. That is, the components must vary oppositely to the change of basis. For a dual vector (covector), like the gradient, to be coordinate system invariant, its components must co-vary with the change of basis, that is its components must vary by the same transformation as the basis. If, for example, vectors have units of distance (radius vector), the covectors have units of inverse of distance. This distinction becomes very important in general relativity.

16 606 Appendix B: Tensors B.3 Second-Order Tensors A set of n 2 quantities T μν form a second-order contravariant tensor if, upon a coordinate change (B.1), they transform as the product A μ B ν, that is according to T μν = x μ λ x ν ρ T λρ, T μν = x μ λ x ν ρ T λρ. (B.16) A set of n 2 quantities U μν form a second-order covariant tensor if, upon a coordinate change (B.1), they transform as the product A μ B ν, that is according to U μν = x λ μ x ρ ν U λρ, U μν = x λ μ x ρ ν U λρ. (B.17) A set of n 2 quantities Vν μ form a second-order mixed tensor if, upon a coordinate change (B.1), they transform as the product A μ B ν, that is according to V ν μ = x μ λ x ρ ν V ρ λ, Vν μ = x μ λ x ρ ν V ρ λ. (B.18) Such a tensor is the Kronecker symbol: δ μ ν = x μ λ x ρ ν δλ ρ = x μ λ x λ ν = x μ x λ x μ = x λ x ν x = ν δμ ν. (B.19) A contravariant index of a tensor can be lowered using the metric tensor g μν, and a covariant index can be raised using the inverse metric tensor g μν.onemusttake care of the order of the indices. If the mixed tensor is symmetric, indices are written on the same vertical line, one below the other. One can show that the contracted product T μν A μ B ν μ, ν = 1, 2,...,n is an invariant. Indeed, T μν A μ B ν = x μ λ x ν ρ x σ μ x κ ν T λρ A σ B κ = δ σ λ δκ ρ T λρ A σ B κ = T λρ A λ B ρ. (B.20) The covariant second-order tensor T μν is symmetric if T μν = T νμ, and antisymmetric if T μν = T νμ. These definitions are analogous for contravariant tensors. The property of symmetry (antisymmetry) is invariant under coordinate transformations. A second-order (contravariant or covariant) tensor can always be decomposed into a sum of symmetric and antisymmetric parts, T μν = 1 ( ) 1 ( ) Tμν + T νμ + Tμν T νμ = Sμν + A μν, (B.21) 2 2 where S μν = S νμ and A μν = A νμ.

17 Appendix B: Tensors 607 B.4 The Metric Tensor Consider an m-dimensional Euclidean space E m and let y 1,...,y m be the Cartesian coordinates of a point P. The squared distance between P and an infinitely closed point P is ds 2 = dy J dy J, J = 1, 2,...,m. (B.22) Take now in E m an embedded submanifold R n (n < m) and let x 1,...,x n be the coordinates of a point in R n. Obviously, y J = y J (x 1,...,x n ), J = 1, 2,...,m. (B.23) The squared distance between two infinitely closed points in R n is ds 2 = y J x μ y J x ν dxμ dx ν = g μν dx μ dx ν, (B.24) where we denoted g μν (x 1,...,x n ) = y J y J, J = 1, 2,...,m; μ, ν = 1, 2,...,n. (B.25) x μ x ν The bilinear form (B.24) ispositive definite and, according to (B.20), it is invariant under coordinate changes. Since dx μ and dx ν are contravariant vectors, it results that g μν is a second-order covariant symmetric tensor, called the metric tensor.(the terms metric and line element are often used interchangeably.) Since dx μ = g μν dx ν, (B.26) we may write ds 2 = dx μ dx μ. But in this case relation (B.26) is also true for the components of any vector A μ : A μ = g μν A ν, μ, ν = 1, 2,...,n. (B.27) Relations (B.27) can be considered as a system of n algebraic equations with n unknowns A 1,...,A n. Solving the system by Cramer s rule, we obtain A ν = g νλ A λ, ν, λ = 1, 2,...,n, (B.28) where g νλ = Gνλ g (B.29)

18 608 Appendix B: Tensors are the components of the contravariant metric tensor. HereG νλ is the algebraic complement of the element g νλ in the determinant g = det(g νλ ). Since G νλ = G λν,wehaveg νλ = g λν. Relation (B.27) also shows that A μ = g μν A ν = g μν g νλ A λ, which yields g μν g νλ = δ λ μ. (B.30) Using the metric tensor, one can lower or raise the indices of any tensor. In an Euclidean space holds the relation g μν = δ μν, so that A μ = δ μν A ν = A μ. Consequently, on such a manifold there is no distinction between contravariant and covariant indices. B.5 Higher Order Tensors In an analogous way can be defined tensors of any variance. For example, the system of n α+β real quantities T j 1... j β k 1...k α form a tensor of order (α + β), or a type (α, β) tensor, i.e. α-times covariant and β-times contravariant, if its components transform according to T j 1... j β k 1...k α = x p 1 k 1...x p α k α x j 1 i 1...x j β i β T i 1...i β p 1...p α. (B.31) B.6 Tensor Operations B.6.1 Addition Two tensors can be added (subtracted) only if they have the same order and the same variance. For example, the tensors U μ νλ and V μ νλ can be added to give T μ νλ = U μ νλ + V μ νλ. (B.32) B.6.2 Multiplication Let U i 1...i α j 1... j β and V k 1...k γ m 1...m δ be two tensors of arbitrary order and variance. Their product is defined by T i 1...i α k 1...k γ j 1... j β m 1...m δ = U i 1...i α j 1... j β V k 1...k γ m 1...m δ, (B.33)

19 Appendix B: Tensors 609 being a tensor of order (α + β + γ + δ), (α + γ)-times contravariant and (β + δ)- times covariant. B.6.3 Contraction Consider a mixed tensor (T ) of order p 2. The operation of contraction consists in setting equal one contravariant index and one covariant index, and summing over them. By one contraction, the order of the tensor reduces by two units. As an example, consider the tensor T νλ, and take ν = μ. The result is μ T μλ μ = x μ ρ x λ κ x σ μ T ρκ σ = δ σ ρ x λ κ T ρκ σ = x λ κ T ρκ ρ, (B.34) which is the transformation of a contravariant vector. In general, by setting equal the first γ covariance indices with the first γ contravariance indices of the tensor T i 1...i α j 1... j β, one obtains the tensor U i γ+1...i α j γ+1... j β, of type (α γ, β γ). B.6.4 Raising and Lowering Indices Consider the tensor T μ νλρ and perform the product g μσ T μ νλρ = T σνλρ, (B.35) which is a covariant tensor. We call this operation index lowering (μ in our case). In the same manner, an index can be raised: g νσ T μ νλρ = T μσ λρ. (B.36) To lower (raise) n indices, this operation has to be accomplished n times. The operations of lowering and raising indices do not modify the order of a tensor, but only its variance. B.6.5 Symmetric and Antisymmetric Tensors Consider the tensor T i 1...i α j 1... j β (α, β 2) (B.37)

20 610 Appendix B: Tensors on an n-dimensional Euclidean space. If its components do not change under a permutation of a group of indices, either of contravariance or of covariance, we call the tensor symmetric in that group of indices. Let us consider, for the beginning, a totally symmetric tensor of type (α, 0).Due to the symmetry, its number of free parameters is naturally lower than that of an arbitrary type (α, 0) tensor. Namely, the number of distinct components is given by the number of combinations with repetitions, into groups of α indices, formed with 1, 2,...,n. This number is given by ( ) C α n = C α n(n + 1)...(n + α 1) n+α 1 =. (B.38) α! An analogous formula is obtained for the number of free parameters of a totally symmetric tensor of type (0, α). If we consider the general tensor (B.37), symmetric in the first γ contravariance indices i 1,...i γ, then the number of its distinct components is ( ) C γ n n α+β γ = C γ n+γ 1 nα+β γ. (B.39) As an example, let us take the second-order symmetric tensor T μν. According to (B.38), it has distinct n(n + 1)/2 components (6 components in three-dimensional space, 10 in a four-dimensional space, etc). If the tensor (B.37) changes its sign when permuting any two indices out of a group of indices, either of contravariance or of covariance, the tensor is called antisymmetric in that group of indices. Suppose that (T ) is antisymmetric in the first γ covariance indices, and denote by [ j 1... j γ ] the group of these indices. Then we have T i 1...i α [ = ( 1) I T i 1...i α, (B.40) j 1... j γ] j γ+1... j β ( j 1... j γ) j γ+1... j β where ( j 1... j γ) is an arbitrary permutation of indices j1... j γ, and I the number of inversions of indices needed in order to achieve the final ordering. In this category falls, for example, the Levi-Civita permutation symbol ɛ μνλρ. The number of distinct components of the tensor (B.40) isc γ n n α+β γ, i.e. C γ n nα+β γ = n(n 1)(n 2)...(n γ + 1) n α+β γ. γ! (B.41) For example, the second-order antisymmetric tensor A μν has n(n 1)/2 distinct components (3 in a three dimensional space, 6 in a four-dimensional space, etc.). Since A μν = A νμ, it results that A μμ = 0 (no summation). In general, if a pair of indices are equal in the group of antisymmetry indices, the corresponding tensor component is zero. A tensor with the property Tν μ the Kronecker symbol δν μ. = 0 (μ = ν) is called diagonal. Such a tensor is

21 Appendix B: Tensors 611 Observation: The number of distinct components of a tensor diminishes if among components there are supplementary relations. For example, if the symmetric tensor T μν in Euclidean three-dimensional space has zero trace, the number of distinct components is 6 1 = 5. In general, each independent relation between the components diminishes the number of independent components by one. B.7 Tensor Variance: An Intuitive Image As we have seen, the variance of tensors can be of two types, covariance and contravariance. Strictly speaking, one way of defining a co- or contravariant first-order tensor is the one presented in Sect. B.2; in the following, we wish to provide a more intuitive image about these concepts, and at the same time emphasize the necessity of introducing them. For an easier presentation, we shall refer in the following to a three-dimensional space. Between the simplest, Cartesian type of coordinate system and the most general coordinate system, there are two types of intermediate systems, namely: 1) orthogonal curvilinear coordinate systems (the coordinate axes are curvilinear, but at each point the tangent vectors to the axes form an orthogonal trihedron see Fig. B.1c); 2) non-orthogonal rectilinear coordinate systems (the coordinate axes are straight lines, but they form angles different from π/2 seefig.b.1d). The orthogonal curvilinear coordinate systems are discussed in Appendix D, where it is shown that the principal effect of the curving of axes is the appearance of the Lamé coefficients. As we shall see, the non-orthogonality of the coordinate axes brings about the necessity of introducing the variance of tensors. To facilitate the graphical presentation, we shall use mainly two-dimensional spaces (the generalization to three dimensions is trivial). Let us then consider on the Euclidean plane a vector a, and express its components with respect to an orthogonal and a non-orthogonal coordinate system (Fig. B.2). In the first system (Fig. B.2a) we write a 1 = a u 1, a 2 = a u 2, (B.42) and if we denote a 1 = a 1 u 1, a 2 = a 2 u 2, then we have a = a 1 + a 2, (B.43)

22 612 Appendix B: Tensors Fig. B.1 Four types of coordinate system: (a) Cartesian (orthogonal rectilinear), (b) general curvilinear, (c) orthogonal curvilinear, and (d) non-orthogonal rectilinear. Fig. B.2 Two types of rectilinear/straight coordinate axes: (a) orthogonal and (b) non-orthogonal. where u 1 and u 2 are the versors of the two axes, while a 1 and a 2 are the components of the vector a in this basis (in other words, the orthogonal projections of the vector a on the coordinate axes).

23 Appendix B: Tensors 613 In the second system (Fig. B.2b), we notice that there are two possibilities of defining the components of the vector a: 1) by orthogonal projection on the axes, leading to the components denoted by a 1 and a 2 ; 2) by drawing parallels to the axes through the tip of the vector a, leading to the components denoted by a 1 and a 2. Thus, in the first case we write a 1 = a u 1, a 2 = a u 2, (B.44) but, with the notation a 1 = a 1 u 1 and a 2 = a 2 u 2, a relation like (B.43) is not valid anymore, i.e. a = a 1 + a 2. (B.45) In the second case, denoting a 1 = a 1 u 1, a 2 = a 2 u 2, (B.46) we find that a relation of the type (B.43) remains valid, i.e. a = a 1 + a 2, (B.47) but we cannot write anymore a relation of the type (B.42), since in this case a 1 = a u 1, a 2 = a u 2. (B.48) In other words, the components a 1 and a 2 are not the scalar products of the vector a with the versors of the coordinate axes. If we wish to have simultaneously valid relations of the type (B.44) and (B.45), we have to introduce a new basis, called the dual basis. Let u, v, and w be three linearly independent vectors in E 3. By definition, they form a basis. Then, any vector a in E 3 can be written as a = λu + μv + νw, (B.49) where λ, μ, and ν are three scalars, which are called the components of the vector a in the basis {u, v, w}.

24 614 Appendix B: Tensors Let us introduce three more vectors, denoted by u, v, and w, satisfying the following conditions: u u = 1, v u = 0, w u = 0, u v = 0, v v = 1, w v = 0, (B.50) u w = 0, v w = 0, w w = 1. For instance, the vectors u, v, and w can be given by the relations u = v w (u, v, w), v = w u (u, v, w), w = u v (u, v, w), (B.51) where (u, v, w) = u (v w) is the mixed product of the vectors. From the definition, it follows that the vectors u, v, and w are also linearly independent, and thus they form a basis in E 3. Such vectors are called dual to the vectors u, v, and w, respectively, while their basis is called dual basis. Moreover, one can easily check that the dual of a dual vector is the original vector, i.e. ( u ) = u, ( v ) = v, (B.52) ( w ) = w. Taking the scalar product of the vector (B.49) with the vectors u, v, and w of the dual basis and using (B.50), one can express the components of the vector in the original basis as λ = a u, μ = a v, ν = a w. (B.53) Returning to our problems, we notice that, using the vectors of the dual basis, we can write i) for the components a 1 and a 2 defined by (B.44): where a = a 1 + a 2, a 1 = a 1 u 1, with a 1 = a u 1 = a (u 1), a 2 = a 2 u 2, with a 2 = a u 2 = a (u 2). (B.54)

25 Appendix B: Tensors 615 ii) for the components a 1 and a 2 defined by (B.46): a 1 = a u 1, a 2 = a u 2, (B.55) as well as a = a 1 + a 2, with a 1 = a 1 u 1, a 2 = a 2 u 2. The customary notations are different from the above, namely, { a 1 = a 1, a 2 = a2, { u 1 = u 1, u 2 = u2. (B.56) Thus, the vector a has two sets of components: one with lower indices, a 1 = a u 1, a 2 = a u 2, (B.57) and another with upper indices, a 1 = a u 1, a 2 = a u 2. (B.58) The components with lower indices are called covariant, while the ones with upper indices contravariant. The original basis is formed of covariant versors, while the dual basis is formed with contravariant versors. Moreover, in Fig. B.2b one notices that, if the angle between the axes becomes π/2 (the system becomes orthogonal), then the two types of components coincide. To conclude, in a non-orthogonal coordinate system, any vector has two sets of components: covariant components, by means of which one writes the vector in the dual, or contravariant, basis, and contravariant components, used to write the vector in the original, or covariant, basis. For example, in the case of a three-dimensional non-orthogonal frame, the radius vector of a point can be written as follows: 1) in dual (contravariant) basis, using the covariant components x 1, x 2, and x 3 : r = x 1 u 1 + x 2 u 2 + x 3 u 3 ; 2) in the original (covariant) basis, using the contravariant components x 1, x 2, and x 3 : r = x 1 u 1 + x 2 u 2 + x 3 u 3.

26 616 Appendix B: Tensors Clearly, dr = dx 1 u 1 + dx 2 u 2 + dx 3 u 3, dr = dx 1 u 1 + dx 2 u 2 + dx 3 u 3, therefore the metric of the space can be written in the following ways: ds 2 = dr 2 = dr dr ( dx1 u 1 + dx 2 u 2 + dx 3 u 3) (dx ( ) ( 1 u 1 + dx 2 u 2 + dx 3 u 3) ), = dx 1 u 1 + dx 2 u 2 + dx 3 u 3 dx 1 u 1 + dx 2 u 2 + dx 3 u 3, ( dx1 u 1 + dx 2 u 2 + dx 3 u 3) (dx ) 1 u 1 + dx 2 u 2 + dx 3 u 3. With the new notations, relations (B.50) can be written in a condensed manner as u i u j = δ i j, hence ( dxi u i) (dx j u j) = ( ds 2 = dr 2 ( ) ( ) ( u i u j) ) dx i dx j, = dr dr = dx i u i dx j u j = ui u j dx i dx j, ( dxi u i) (dx ) ( ) j u j = ui u j dxi dx j. (B.59) (B.60) Usually, one denotes the scalar products of the contravariant and covariant versors by ( ui u j) = g ij, ( ) ui u j = gij, (B.61) leading to ds 2 = g ij dx i dx j = g ij dx i dx j = dx i dx i. (B.62) Thus, we have arrived at the fundamental metric tensor, with contravariant components, g ij, or covariant components, g ij. Relations (B.61) provide an intuitive interpretation of the elements of the metric tensor. Finally, from the last equality in (B.62) it follows that dx i = g ij dx j, (B.63) which shows that the lowering of indices by means of the metric tensor appears naturally. The equality g ij dx i dx j = dx i dx i from (B.62) leads to dx i = g ij dx j, (B.64) i.e. the raising of indices.

27 Appendix C Representations of Minkowski Space C.1 Euclidean-Complex Representation Let x 1 = x, x 2 = y, x 3 = z, x 4 = ict be the coordinates of an event in Minkowski space. The metric then is ds 2 = dx μ dx μ = g μν dx μ dx ν = dx dx2 2 + dx2 3 + dx2 4 = dx 2 + dy 2 + dz 2 c 2 dt 2, (C.1) which means g μν = δ μν, (C.2) corresponding to a pseudo-euclidean space. Such a representation of Minkowski space is called Euclidean-complex representation. The coordinates x μ, μ = 1, 2, 3, 4 of an event form a four-vector. Thespace components x 1, x 2, x 3 are denoted by x i, i = 1, 2, 3, and the time component by x 4. Under a change of coordinates the components of the position (or radius) four-vector transform according to (see (A.57)): x μ = a μνx ν, μ, ν = 1, 2, 3, 4. (C.3) If x μ are the coordinates of the same event, but determined in the inertial frame S, moving along Ox O x axis, with velocity V with respect to S, then a μν are the elements of the Lorentz transformation matrix A = Γ 00i V c Γ i V c, (C.4) Springer-Verlag Berlin Heidelberg 2016 M. Chaichian et al., Electrodynamics, DOI /

28 618 Appendix C: Representations of Minkowski Space while (C.3) represents, in condensed form, the Lorentz transformation t = Γ (t Vc 2 x ), x = Γ (x Vt), y = y, z = z. (C.5) A system of 4 quantities A μ, μ = 0, 1, 2, 3 which transform like the coordinates, that is according to A μ = a μν A ν, μ, ν = 1, 2, 3, 4, (C.6) form a four-vector. In Euclidean-complex representation the space components of a four-vector are real, and the time component is imaginary. A second-order four-tensor transforms as the product A μ B ν, that is according to T μν = a μλa νρ T λρ, λ, ρ, μ, ν = 1, 2, 3, 4. (C.7) In the same way, one can define four-tensors of order three, four, etc. In the theory of relativity, a special role is played by the totally antisymmetric fourth-order unit pseudotensor ɛ μνλρ. It is defined as being +1, 1, 0, as the indices are even, odd, or repeated-index permutation of 1, 2, 3, 4. The quantities ɛ μνλρ form a pseudotensor (sometimes called axial tensor) because they exhibit a tensor behaviour under rotations and Lorentz boosts, but are not invariant under parity inversions. It can be shown that δ μσ δ μκ δ μγ δ μζ ɛ μνλρ ɛ σκξζ = δ νσ δ νκ δ νξ δ νζ δ λσ δ λκ δ λξ δ λζ. (C.8) δ ρσ δ ρκ δ ρξ δ ρζ It then results ɛ μνλρ ɛ μνσκ = 2! (δ λσ δ ρκ δ λκ δ ρσ ), ɛ μνλρ ɛ μνλσ = 3! δ ρσ = 6 δ ρσ, ɛ μνλρ ɛ μνλρ = 4! =24. (C.9) With the help of ɛ μνλρ one can define the dual of an antisymmetric tensor. If A μν is an antisymmetric tensor, then A μν and the pseudotensor à μν = 1 2 ɛ μνλρ A λρ are called dual to each other. Similarly, the third-order antisymmetric pseudotensor A μνλ = ɛ μνλρ A ρ and the four-vector A ρ are dual to each other.

29 Appendix C: Representations of Minkowski Space 619 There are four possible types of manifolds that can be embedded in the four-space, which means that there exist four types of integrals: 1) Line integral, when the integration is performed along a curve. 2) Integral over a two-dimensional surface.ine 3, as surface element, one uses d S i. This is the integration differential and it is the dual of the antisymmetric tensor ds ik (see (A.60)): d S i = 1 2! ɛ ijkds jk. From the geometric point of view, d S i is a vector orthogonal to the surface element and having the same modulus as the elementary area. In the four-dimensional space, d S μν = 1 2 ɛ μνλρds λρ, μ, ν, λ, ρ = 0, 1, 2, 3, (C.10) therefore the dual of the tensor ds μν is also a second-order tensor and, geometrically, represents a surface element equal to ds μν and orthogonal to it. 3) Integral over a three-dimensional hypersurface. In three dimensions, the volume element is constructed as the mixed product of three arc elements corresponding to three coordinate directions which intersect at a point. In four-dimensions, as hypersurface element one defines the antisymmetric tensor ds μνλ, together with its dual d S μ : d S μ = 1 3! ɛ μνλρ ds νλρ, ds μνλ = ɛ μνλρ d S ρ. (C.11) Geometrically, the four-vector d S μ is orthogonal to the hypersurface element, and has the modulus equal to the area of this element. In particular, ds 0 = dx dydz is the projection of the hypersurface element on the hyperplane x 0 = const. 4) Integral over the four-dimensional hypervolume. The volume element in this case is dω = dx 0 dx 1 dx 2 dx 3 = ds μ dx μ, (no summation over μ), (C.12) where the hypersurface element is orthogonal to the arc element dx μ. Using these notions, one can generalize the divergence theorem and the Stokes theorem in Minkowski space. In view of (C.12), we have A μ ds μ = Aμ x μ dω, (C.13)

30 620 Appendix C: Representations of Minkowski Space which generalizes in four dimensions the divergence theorem. Formally, the integral extended over a hypersurface can be transformed into an integral over the four-domain enclosed by the hypersurface by substituting ds μ dω x μ, μ = 0, 1, 2, 3. (C.14) In a similar way, an integral over a two-dimensional surface, of element ds μν = dx μ dx ν, can be transformed according to Aμ Aμ ds μν = dx μ dx ν = A μ dx μ, (C.15) x ν x ν meaning that the circulation along a closed curve in four-dimensional space can be transformed into an integral over the two-dimensional surface bounded by the curve, by substituting dx μ ds μν. (C.16) x ν After some index manipulation in (C.15), one obtains A μ dx μ = 1 ( Aν A ) μ ds μν. 2 x μ x ν (C.17) One can also establish a formula connecting an integral over a two-dimensional surface, and the boundary three-dimensional surface. As an example, if A μν is an antisymmetric tensor, we may write ( Aμν Aμν ds μ + A ) νμ ds μ = 1 x ν 2 x ν = 1 ( Aμν 2 x ν x μ ds ν ds μ A μν x μ ds ν ). If one denotes d S μν 1 ( ) ds μ ds ν, (C.18) 2 x ν x μ if follows that Aμν ds μ = x ν A μν d S μν. (C.19)

31 Appendix C: Representations of Minkowski Space 621 C.2 Hyperbolic Representation An event in Minkowski space can be also defined by the choice x 0 = ct, x 1 = x, x 2 = y, x 3 = z, in which case the metric ds 2 = dx μ dx μ = g μν dx μ dx ν = c 2 dt 2 dx 2 dy 2 dz 2 (C.20) gives g 00 =+1, g 11 = g 22 = g 33 = 1, g μν (μ = ν) = 0. (C.21) A system of coordinates in a pseudo-euclidean (e.g. Minkowski) space in which the line element has the form: ds 2 = eμ 2dx2 μ, where e μ =±1, is a Galilean coordinate system. We have, therefore, above a Galilean coordinate system, while this representation of Minkowski space is called hyperbolic. Relations (C.21) show that in such a representation one makes distinction between contravariance and covariance indices. For example, in case of a four vector, A μ = g μν A ν { A0 = g 00 A 0 = A 0, A i = g ii A i = A i (no summation). (C.22) The square of a four-vector is A μ A μ = A 0 A 0 + A i A i = A 0 A 0 A i A i = invariant. (C.23) In Minkowski space, the components of a contravariant four-vector transform accordingto(b.8), where the transformation matrix is Γ V c (x ν μ ) Λν μ = Γ 00 V c Γ Γ (C.24) If, for example, the relative motion of frames S and S takes place along Ox O x - axis, then the contravariant components of A μ transform according to A 0 = Γ (A 0 Vc ) A1, A 1 = Γ ( Vc ) A0 + A 1, A 2 = A 2, A 3 = A 3, (C.25)

32 622 Appendix C: Representations of Minkowski Space while the covariant components obey the rule A 0 (A = Γ 0 + V ) c A 1, ( ) V A 1 = Γ c A 0 + A 1, A 2 = A 2, A 3 = A 3. (C.26) In view of (C.21) and (B.30), we can write g μν = g μν, μ, ν = 0, 1, 2, 3. (C.27) In Galilean coordinates, both the contravariant and covariant components of a fourvector are real. Let us have a look over the relations written in the previous section in Euclidean complex representation. At the beginning, we choose the contravariant Levi-Civita permutation symbol as ɛ 0123 =+1. (C.28) It then follows that ɛ μνλρ = g μσ g νκ g λυ g ρζ ɛ σκυζ = ɛ μνλρ, (C.29) because irrespective of the order of the four different indices, the product of the four metric tensor components is 1. Then we have δσ μ δμ κ δμ υ δμ θ ɛ μνλρ ɛ σκυθ = δσ ν δν κ δν υ δν θ δσ λ δλ κ δλ υ δλ. (C.30) θ δ ρ σ δρ κ δρ υ δρ θ Summation over two, three, and four pairs of indices gives ɛ μνλρ ɛ σκλρ = 2! δσκ μν = 2 ( δσ μ δν κ δμ κ σ) δν, ɛ μνλρ ɛ σνλρ = 3! δσ μ = 6δμ σ, ɛ μνλρ ɛ μνλρ = 4! = 24. (C.31) If A μν is an antisymmetric tensor, its dual is the pseudotensor à μν = 1 2 ɛμνλρ A λρ. The product à μν A μν is a pseudoscalar. In the same way, we observe that the antisymmetric pseudotensor ɛ μνλρ A ρ and the four-vector A μ are dual to each other. x μ The symbol of partial derivative x μ is a contravariant four-vector. is a covariant four-vector, while the symbol

33 Appendix C: Representations of Minkowski Space 623 Consider now, as we already did in the Euclidean representation of Minkowski space, the four possible types of integrals and the relations between them: 1) Line integral, with the arc element dx μ. 2) Integral over a two-dimensional surface, with the surface element d S μν = 1 2 ɛμνλρ ds λρ (C.32) which, geometrically, is an area element orthogonal (and quantitatively equal) to ds μν. 3) Integral over a three-dimensional hypersurface, of surface element d S μ = 1 3! ɛμνλρ ds νλρ, such as ds μνλ = ɛ μνλρ ds ρ, d S 0 = ds 123 = ds 123, etc. (C.33) (C.34) The four-vector d S μ has its modulus equal to the area of the hypersurface element, being orthogonal to it. 4) Integral over a four-dimensional domain, the elementary hypervolume being dω = dx 0 dx 1 dx 2 dx 3 = dx μ ds μ (no summation), (C.35) where the line element dx μ and the hypersurface element ds μ are orthogonal. The divergence theorem in this representation is A A μ μ ds μ = x dω, μ while the Stokes theorem takes the form A μ dx μ = 1 2 ( Aν x A μ μ x ν ) ds μν. (C.36) (C.37) Finally, the generalization of (C.19) in the hyperbolic representation of Minkowski space is A μν d S μν = 1 2 ( A μν x ds ν μ ) Aμν x ds μ ν A μν = x ds μ. ν (C.38)

34 624 Appendix C: Representations of Minkowski Space C.3 Representation in General Curvilinear Coordinates One can represent the Minkowski space in the most general manner in a system of general curvilinear coordinates. These considerations are especially useful in the study of the gravitational field which, according to general relativity, manifests itself through the curvature of space-time. However, locally, the space-time is flat (of Minkowski type) and the coordinate systems are in their turn locally defined. In general curvilinear coordinates, the metric tensor g μν depends on the coordinates. Let us first express the law of transformation of the Levi-Civita symbol when one passes from the Galilean coordinates x μ to an arbitrary set of general curvilinear coordinates, x μ = x μ (x ν ), μ, ν = 0, 1, 2, 3. According to the rule of transformation, we have ɛ μνλρ = xσ x κ x ξ x υ x μ x ν x λ x ɛ σκξυ, (C.39) ρ where ɛ σκξυ is defined in the Galilean coordinates x μ, and ɛ μνλρ in the curvilinear coordinates x μ. If A ν μ, μ, ν = 0, 1, 2, 3 is an arbitrary second-order mixed tensor, it can be shown that A σ μ Aκ ν Aξ λ Aυ ρ ɛ σκξυ = A ɛ μνλρ, (C.40) where A = det(a ν μ ). Relation (C.40) is a generalization in four dimensions of (A.21). Then we may write ɛ μνλρ = det ( x μ x ν ) ɛ μνλρ = 1 J ɛ μνλρ, where J = (x 0, x 1, x 2, x 3 ) (x 0, x 1, x 2, x 3 ) (C.41) is the functional determinant of the transformation x μ x μ. Using the transformation rule, we have also g μν = xλ x ρ x μ x η λρ, ν where η λρ = diag(1, 1, 1, 1) is the Minkowski metric tensor. If we take the determinant of the above relation, we find g = 1 det(η J 2 μν ), where g = det(g μν ). Since det(η μν ) = 1, we have J = 1. (C.42) g We then define the antisymmetric unit tensor in curvilinear coordinates by δ μνλρ = gɛ μνλρ. (C.43)

35 Appendix C: Representations of Minkowski Space 625 The transformation rule of the contravariant components ɛ μνλρ is found in a similar way: ɛ μνλρ = x μ x ν x λ x ρ x σ x κ x ξ x υ ɛσκξυ = J ɛ μνλρ, that is δ μνλρ = 1 g ɛ μνλρ, (C.44) with ɛ μνλρ = δ μνλρ.inviewof(c.43) and (C.44), relation (C.29) yields δ μνλρ = gδ μνλρ. (C.45) If g = 1, we find the Galilean formula (C.29). Let us now write the transformation rule of the four-dimensional elementary volume. In Galilean coordinates, dω = dx 0 dx 1 dx 2 dx 3 is an invariant. In the curvilinear coordinates x μ the element of volume is dω = JdΩ. Since the four-volume must be an invariant, in the new coordinates x μ not dω,but g dω has to be used as integration (hyper)volume element: dω 1 J dω = g dω. (C.46) If, as a result of integration over Ω of the quantity gϕ, with ϕ a scalar, one obtains an invariant, then gϕis called a scalar density. In the same way are defined the notions of vector density g A μ and tensor density g T μν, respectively. The elementary three-dimensional surface is g dsμ = 1 3! ɛ μνλρ g ds νλρ = 1 3! δ μνλρ ds νλρ. (C.47) Similarly, the two-dimensional surface element is g d S μν = 1 gɛμνλρ ds λρ = 1 2! 2! δ μνλρ ds λρ. (C.48) C.4 Differential Operators in General Curvilinear Coordinates In special relativity the fundamental equations of conservation involve the vector or tensor four-divergence operators, written in terms of the usual derivatives. On curved space-times the usual partial derivatives with respect to coordinates have to be replaced by covariant derivatives, as we have explained in Sect Here are the most important differential operators, expressed in curvilinear coordinates.