INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS ******************************************************************************** I. Basic Principles ...................................................................... 1 II. Three Dimensional Spaces ............................................................. 4 III. Physical Vectors ...................................................................... 8 IV. Examples: Cylindrical and Spherical Coordinates .................................. 9 V. Application: Special Relativity, including Electromagnetism ......................... 10 VI. Covariant Differentiation ............................................................. 17 VII. Geodesics and Lagrangians ............................................................. 21 ******************************************************************************** I. Basic Principles We shall treat only the basic ideas, which will suffice for much of physics. The objective is to analyze problems in any coordinate system, the variables of which are expressed as qj(xi) or q'j(qi) where xi : Cartesian coordinates, i = 1,2,3, ....N

for any dimension N. Often N=3, but in special relativity, N=4, and the results apply in any dimension. Any well-defined set of qj will do. Some explicit requirements will be specified later. An invariant is the same in any system of coordinates. A vector, however, has components which depend upon the system chosen. To determine how the components change (transform) with system, we choose a prototypical vector, a small displacement dx i. (Of course, a vector is a geometrical object which is, in some sense, independent of coordinate system, but since it can be prescribed or quantified only as components in each particular coordinate system, the approach here is the most straightforward.) By the chain rule, dqi = ( ∂qi / ∂xj ) dxj , where we use the famous summation convention of tensor calculus: each repeated index in an expression, here j, is to be summed from 1 to N. The relation above gives a prescription for transforming the (contravariant) vector dxi to another system. This establishes the rule for transforming any contravariant vector from one system to another. ∂qi Ai (q) = ( j ) Aj (x) ∂x Ai(q') = ( ∂q'i ∂qj ∂q' i ∂q'i ) Aj(q) = ( j ) ( k ) Ak (x) ≡ ( k ) Ak(x) ∂q ∂x ∂x ∂qj

i ∂qi Contravariant vector transform Λj (q,x) ≡ ∂xj The (contravariant) vector is a mathematical object whose representation in terms of components transforms according to this rule. The conventional notation represents only the object, Ak, without indicating the coordinate system. To clarify this discussion of transformations, the coordinate system will be indicated by Ak(x), but this should not be misunderstood as implying that the components in the "x" system are actually expressed as functions of the xi. (The choice of variables to be used to express the results is totally independent of the choice coordinate system in which to express the components Ak . The Ak (q) might still be expressed in terms of the xi, or Ak (x) might be more conveniently expressed in terms of some qi.) Distance is the prototypical invariant. In Cartesian coordinates, ds2 = δij dxi dxj , where δij is the Kroneker delta: unity if i=j, 0 otherwise. Using the chain rule, 1

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

∂xi dxi = ( j ) dqj ∂q ds2 = δij ( gkl (q) ≡ ( ∂xi ∂xj ) ( l ) dqk dql = gkl (q) dqk dql ∂qk ∂q

∂xj ∂xi k ) ( ∂ql ) δij (definition of the metric tensor) ∂q One is thus led to a new object, the metric tensor, a (covariant) tensor, and by analogy, the covariant transform coefficients: j ∂xj Λi(q,x) ≡ ( i ) ∂q Covariant vector transform

{More generally, one can introduce an arbitrary measure (a generalized notion of 'distance') in a chosen reference coordinate system by ds2 = gkl (0) dqk dql , and that measure will be invariant if gkl transforms as a covariant tensor. A space having a measure is a metric space.} Unfortunately, the preservation of an invariant has required two different transformation rules, and thus two types of vectors, covariant and contravariant, which transform by definition according to the rules above. (The root of the problem is that our naive notion of 'vector' is simple and welldefined only in simple coordinate systems. The appropriate generalizations will all be developed in due course here.) Further, we define tensors as objects with arbitrary covariant and contravariant indices which transform in the manner of vectors with each index. For example, l mn ij i j Tk(q) ≡ Λm (q,x) Λ n(q,x) Λ k(q,x) T l (x) The metric tensor is a special tensor. First, note that distance is indeed invariant: ds2(q') = gkl (q') dq'k dq'l = ( ∂q j ∂q'k ∂q' l ∂q i ) ( ) gij (q) ( s ) dqs ( t ) dqt ∂q' l ∂q ∂q ∂q' k

∂qi ∂q'k ∂q j ∂q' l = gij (q) ( k )( s ) ( )( t ) dqs dqt ∂q' ∂q ∂q' l ∂q ⇓ ∂qi = δis ∂qs = gij (q) dqi dqj ≡ ds2(q) ⇓ δjt

There is also a consistent and unique relation between the covariant and contravariant components of a vector. (There is indeed a single 'object' with two representations in each coordinate system.) dqj ≡ gji dqi 2

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

∂qk ∂q l ∂q'i p dq'j ≡ gji (q') dq'i = gkl (q) ( j ) ( ∂q' i ) (∂qp ) dq ∂q' ⇓ δlp = ( ∂qk ∂qk ) gkl (q) dql = ( j ) dqk ∂q' ∂q' j

Thus it transforms properly as a covariant vector. These results are quite general; summing on an index (contraction) produces a new object which is a tensor of lower rank (fewer indices). ij k ij Tk Gl = R l The use of the metric tensor to convert contravariant to covariant indices can be generalized to 'raise' and 'lower' indices in all cases. Since gij = δij in Cartesian coordinates, dxi =dxi ; there is no difference between co- and contra-variant. Hence gij = δij , too, and one can thus define gij in other coordinates. {More generally, if an arbitrary measure and metric have been defined, the components of the contravariant metric tensor may be found by inverting the [N(N+1)/2] equations (symmetric g) of gij (0) gik(0) gnj(0) = gkn(0). The matrices are inverses.} Ai(q) ≡ gij (q) Aj(q) ∂qk ∂xr ∂xs i ∂qi gj = gik gkj = ( m ) ( n ) δmn ( k ) ( j ) δrs ∂x ∂q ∂q ∂x | | ⇓ δrn s i ∂x i ∂q = ( s ) ( j ) = δij = δj ∂q ∂x i Thus gj is a unique tensor which is the same in all coordinates, and the Kroneker delta is sometimes i written as δj to indicate that it can indeed be regarded as a tensor itself. Contraction of a pair of vectors leaves a tensor of rank 0, an invariant. Such a scalar invariant is indeed the same in all coordinates: Ai(q')Bi(q') = ( ∂qk ∂q' i j j j ) A (q) ( ∂q' i ) Bk(q) = δjk A (q) Bk(q) ∂q = Aj(q) Bj(q)

It is therefore a suitable definition and generalization of the dot or scalar product of vectors. Unfortunately, many of the other operations of vector calculus are not so easily generalized. The usual definitions and implementations have been developed for much less arbitrary coordinate systems than the general ones allowed here. For example, consider the gradient of a scalar. One can define the (covariant) derivative of a scalar as 3

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

Ø(x),i ≡

∂Ø ∂xi

Ø(q),i ≡

∂Ø ∂Ø ∂xj =( j)( i) ∂qi ∂x ∂q

The (covariant) derivative thus defined does indeed transform as a covariant vector. The comma notation is a conventional shorthand. {However, it does not provide a direct generalization of the gradient operator. The gradient has special properties as a directional derivative which presuppose orthogonal coordinates and use a measure of physical length along each (perpendicular) direction. We shall return later to treat the restricted case of orthogonal coordinates and provide specialized results for such systems. All the usual formulas for generalized curvilinear coordinates are easily recovered in this limit.} A (covariant) derivative may be defined more generally in tensor calculus; the comma notation is employed to indicate such an operator, which adds an index to the object operated upon, but the operation is more complicated than simple differentiation if the object is not a scalar. We shall not treat the more general object in this section, but we shall examine a few special cases below. II. Three Dimensional Spaces For many physical applications, measures of area and volume are required, not only the basic measure of distance or length introduced above. Much of conventional vector calculus is concerned with such matters. Although it is quite possible to develop these notions generally for an N-dimensional space, it is much easier and quite sufficient to restrict ourselves to three dimensions. The appropriate generalizations are straightforward, fairly easy to perceive, and readily found in mathematics texts, but rather cumbersome to treat. For writing compact expressions for determinants and various other quantities, we introduce the permutation symbol, which in three dimensions is eijk = 1 for i,j,k=1,2,3 or an even permutation thereof, i.e. 2,3,1 or 3,1,2 -1 for i,j,k= an odd permutation, i.e. 1,3,2 or 2,1,3 or 3,2,1 0 otherwise, i.e. there is a repeated index: 1,1,3 etc. The determinant of a 3x3 matrix can be written as |a| = eijk a1i a2j a3k Another useful relation for permutation symbols is eijk eilm = δjl δkm - δjm δkl Furthermore, ijk ijk δlmn = eijk elmn and δijk = 3! ijk where δlmn is a multidimensional form of the Kroneker delta which is 0 except when ijk and lmn are each distinct triplets. Then it is +1 if lmn is an even permutation of ijk, -1 if it is an odd permutation. These symbols and conventions may seem awkward at first, but after some practice they become extremely useful tools for manipulations. Fairly complicated vector identities and rearrangements, as one often encounters in electromagnetism texts, are made comparatively simple. Although the permutation symbol is not a tensor, two related objects are: 1 εijk =  g eijk and εijk = eijk √ √g  where g ≡ | gij |

with absolute value understood if the determinant in negative. This surprising result may be confirmed by noting that the expression for the determinant given above may also be written as 4

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

elmn |a| = eijk ali amj ank which is certainly true for l,m,n=1,2,3, and a little thought will show it to be true in all cases. The transformation law for g may then be obtained as elmn |g(q')| =eijk g(q')li g(q')mj g(q')nk = eijk ( ∂qp ∂q q ∂q r ∂q s ∂q u ∂q v )( )( )( )( )( ) ∂q' l ∂q' i ∂q'm ∂q' j ∂q' n ∂q' k (tensor transform of metric tensors)

gpq(q) grs(q) guv (q)

∂q ∂q p ∂q r ∂q u )( )( )g g g = eqsv  ∂q'  (   ∂q' l ∂q'm ∂q' n pq rs uv (considering the terms with indices i,j,k) ∂q = epru g(q)  ∂q' ( ∂qp / ∂q'l )( ∂qr / ∂q'm )( ∂qu / ∂q'n )   (considering the terms with indices q,s,v in constituting a determinant as above) ∂q = elmn g(q)  ∂q'    2 (forming another determinant as above)

thus establishing that g transforms with the square of the Jacobian determinant. For the putatively covariant form of the permutation tensor, εijk(q') =  √ g(q) erst ( ∂qr ∂q s ∂q t i ) ( ∂q' j ) ( ∂q' k ) ∂q'

∂q = eijk  ∂q'   = eijk  the form desired. √ g(q) √ g(q'),   Raising indices in the usual way will produce the contravariant form by arguments similar to those applied above. The permutation tensors enable one to construct true vectors analogous to the familiar ones. The vector or cross product becomes Ai = εijk Bj Ck although again we have both co and contravariant forms.

5

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

The invariant measure of volume is easily constructed as ∆V = εijk dqi dqj dqk (3!)

which is explicitly an invariant by construction and can be identified as volume in Cartesian coordinates. ( This is a general method of argument in tensor calculus. If a result is stated as an equation between tensors [or vectors or scalars], if it can be proven or interpreted in any coordinate system, it is true for all. That is the power of tensor calculus and its general properties of transformation between coordinates.) Note that the application of this relation for ∆V in terms of dqi and transforming directly from Cartesian dxi gives immediately the familiar relation ∆V= J dq1dq2dq3 ∂x J =  ∂q  the Jacobian.  

For the volume integrals of interest, note that ∫ I ε ijk dqi dqj dqk , for I invariant, is invariant, but ∫ Tv εijk dqi dqj dqk is not a vector, because the transformation law for Tv in general changes over the volume. The operators of divergence and curl require more care. Just as the gradient has a direct physical significance, these operators are constructed to satisfy certain Green's theorems, Gauss' and Stokes law. These must be preserved if their utility is to continue. One can prove a beautiful general theorem in spaces of arbitrary dimension, from which all common vector theorems are simple corollaries, but the proof requires extensive formal preparation. Instead, we shall provide straightforward, if lengthy, proofs of the two specific results desired. For Gauss' law, we require a relation which is a proper equation between invariants and further reduces to the usual result in Cartesian coordinates,

∫ div(Tm)

εijk

dqi dqj dqk = 3!

∫ Ti

dSi

the choice dSi = εijk dqj dqk is explicitly a (covariant) vector, making the right integral invariant, and it gives the correct result in Cartesians. On the left, we require a suitable operator. We shall next prove that  1 ∂[(√ g )Ti] ∂qi √g  is such an invariant. It certainly gives the usual Cartesian divergence, but the inspiration for this guess must remain obscure, for it is deep in the development of general covariant differentiation and Christofel symbols. Fortunately, that need not concern us. Proof that this expression is indeed invariant requires proving that the form is the same in any two systems: ∂[(√ g ' )T' i]  1 = ∂qi √ g'   1 ∂[(√ g )Ti] ∂xi √g  ∂qi ∂  (√ g ' ) T k ( k )    ∂x  1 = ∂qi √ g'  ∂x where g' = ∂q  g ≡ J  g √ √ √   6

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

as shown above, introducing J for the Jacobian determinant. The expression in the new coordinates can then be written  ∂qi Tk 1  ∂ [ (√ g )Tk]   J( k)+ J ∂qi ∂x  J√ g    ∂ J ∂qi     ∂xk   ∂q   i 

where the first term is simply the desired expression in xi by the chain rule, and we must show that the second term, the portion in brackets [], is then zero. That term may be written ∂x ∂∂q   ∂qi ∂ 2 q i ∂xl ( k) + J k l ( i) ∂ q i ∂x ∂x ∂x ∂q ∂q and the first term converted using J' = ∂x = 1/J to   ∂q ∂∂x   ∂q ∂ 2 q i ∂xl -J2 = -J2 ∂x ( ) ∂xk   ∂x k ∂x l ∂qi thereby canceling the second term and proving the assertion. The last step requires some algebra to confirm, but it is straightforward using the methods used above for writing a determinant, considering all the terms present, and inserting a i ∂qi ∂xs δ j= ( s ) ( j ) ∂x ∂q ' (with appropriate choice of indices), the inverse of the usual procedure. The 'tensorial' form of the divergence theorem is therefore an equality of invariants: ∂ J'-1 = ∂xk

∫

 1 ∂[(√ g )Tm] dq i dq j dq k ε ijk = 3! ∂qm √g 

∫ Ti εijk dqj dqk

Furthermore, the familiar result, div(ØA) = Ødiv(A) + ∇Ø⋅A , remains as ∂Ø div(ØA) = Ødiv(A) + ∂q Ai i Fortunately, Stokes theorem is somewhat easier; there is only one subtlety. The naive generalization is

∫εijk ∂Tk ∂qj

εist dqs dqt =

∫ Ti

dqi

which again obviously reduces to the usual result in Cartesian coordinates and would be explicitly a good 'tensor' equation between invariants if ∂Tk/∂qj were indeed a covariant tensor of rank two. It is not, but the portion used in the equation above is. In general, 7

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

R ij =

R ij +R ji 2

+

R ij - R ji 2

the sum of a symmetric and antisymmetric part. For contractions with the anti-symmetric permutation symbol as used above, only the anti-symmetric part contributes; replacing ∂Tj ∂T  k  ∂Tk  ∂qj ∂qk ∂qj = 2 is equivalent and gives the identical Cartesian reduction. The antisymmetric expression is easily shown to be a tensor as follows: ∂T ∂T ∂T' ∂T' Rij ≡ ∂x i - ∂x j and R'ij ≡ ∂q i - ∂q j j i j i but by the laws of tensor transformation, this should also be ∂xk ∂  Tk( ∂q )  i  R' ij = ∂qj ∂xk ∂  Tk( ∂q )  j  ∂xk ∂xk = Rkl ( ∂q ) ( ∂q ) ∂qi i j ∂2xk - Tk ∂q ∂q i j

-

∂Tk ∂xk ∂T ∂xk ∂2xk = ( ∂qk ) ( ∂q ) - ( ∂q ) ( ∂q ) + Tk ∂q ∂q i i j j i j

where the last two terms cancel and the first two, using the chain rule (∂/∂qi)=(∂/∂xk)(∂xk/∂qi), give the required tensor transform of Rij . We therefore have the desired tensor form of the divergence and curl operators and the corresponding integral theorems. Note also that the important results curl ( grad Ø) = 0 and div ( curl Ai) = 0 both follow easily from these forms by symmetry ∂2 eijk ∂q ∂q = 0. i j III. Physical Vectors

The distinction between covariant and contravariant vectors is essential to tensor analysis, but it is a complication which is unnecessary for elementary vector calculus. In fact, the usual formulation of vector calculus can be obtained from tensor calculus as a special case, that being one in which the coordinate system is orthogonal. Most practical coordinate systems are of this type, for which tensor analysis is not really necessary, but a few are not. (For example, in plasma physics, the natural coordinates may be ones determined by the magnetic geometry and not be orthogonal.) In orthogonal systems with positive metric, one can define 'physical' vectors, which are neither covariant nor contravariant. Nevertheless, they have well-defined transformation properties among orthogonal systems, and they have simple physical significance. For example, all components of a displacement vector have the dimensions of length. They are the vectors of traditional vector calculus. For orthogonal systems of this type, gij = h2i δij (hi is not a vector; no summation)

8

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

A A(i) ≡ hi Ai = h i (no summation) i for the components of the 'physical' vector. The usual dot or scalar product is simply A(i)A(i) and produces the same result as given above. (In this special case, the metric tensor can be 'put into' the vector in a natural manner.) All the usual vector formulas can be obtained from the preceding tensor expressions by consistently converting to physical vectors. Note that g = (h1h2h3)2 and εijk = h eijk , using h = (h1h2h3). C(i) = A(j) X B(k) = eijk A(j) B(k) (grad Ø)(i) =(1/hi )(∂ Ø/∂qi) div A = (1/h){∂[hA(i)/hi]/∂qi} (curl A)(i) = (hi/h) eijk ∂[hkA(k)]/∂qj Volume: (d3v) = h eijk dqidqjdqk = d3l = eijk dlidljdlk Integrations are over physical volumes, areas, and lengths. If the integrals are set up in coordinates like dq, the necessary factors must be inserted to give the physical units as illustrated here for volume. IV. Examples

Cylindrical coordinates A simple example to illustrate the ideas is provided by cylindrical coordinates: x = r cos θ y = r sin θ z=z r = √  x2 + y2 θ = tan−1 (y/x) i\j= i ∂qi Λj ≡ j = ∂x i\j j ∂xj Λi ≡ i = ∂q  1 gij =  0  0 g = r2 0 r2 0  cos θ  -r(sin θ)   0 0  0  1  hi = (1,r,1) 9 sin θ r(cos θ) 0 0 0 1     0 r-2 0 0  0  1      1 cos θ -(sin θ)/r 0 2 sin θ (cos θ)/r 0 3 0 0 1    

 1 gij =  0  0

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

Spherical Coordinates A second example of broad utility is spherical coordinates: x = r sin θ cos φ y = r sin θ sin φ z = r cos θ r = √ 2  x2 + y2 + z  x 2 + y 2  θ = tan-1 √  z   φ = tan -1(y/x)

i ∂qi Λj ≡ j = ∂x

    i\j sin θ cos φ (cos θ cos φ)/r -sin φ r sin θ

sin θ sin φ (cos θ sin φ)/r cos φ r sin θ

cos θ -(sin θ)/r 0

      
   

j ∂xj Λi ≡ i = ∂q  1 gij =  0  0  g = r4 sin2 θ V. 0 r2 0

  

sin θ cos φ r cos θ cos φ -r sin θ sin φ 0 0 2sin2θ r    

sin θ sin φ r cos θ sin φ r sin θ cos φ  1 ij =  0 g  0  0 r-2 0

cos θ -r sin θ 0 0 0 -2sin-2θ r

hi = (1,r,r sin θ)

h = r2sin θ

Application: Special Relativity

Special relativity is generally introduced without tensor calculus, but the results often seem rather ad hoc. Einstein used the ideas of tensor calculus to develop the theory, and it certainly assumes its most natural and elegant formulation using tensors. The arguments are easily stated. The use of tensors is natural, for it guarantees that if the laws of physics are properly formulated as equations between scalars, vectors, or tensors, a result or equality in one coordinate system will be true in any. Special relativity is based on only two postulates. The first is that all coordinate systems moving uniformly with respect to one another are equivalent, i.e. indistinguishable from one another. The second is that the speed of light is constant in all such systems. (The first was a long-standing principle. The second was the implication of the Michelson-Morley experiment.) These are easily phrased in tensor calculus. The first implies that the metric tensor must be the same in all equivalent systems, otherwise the differences would provide a basis for distinguishing among them. The second is achieved by introducing a space of four dimensions with Cartesian coordinates (x,y,z,ct) and choosing the metric tensor to be -1 0 0 0  0 -1 0 0  gµν =  0 0 -1 0     0 0 0 1  [This is one of many equivalent choices, none of which has become standard. Sometimes the time is placed first, the indices may run from 0-3 instead of 1-4, and the factors of c can be put into g instead of into the coordinates.] 10

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

The resulting invariant measure "length" is d2σ = gµν dxµdxν = - d2s + c2d2t , introducing the usual convention that Greek indices range 1-4, whereas Latin indices range only over 1-3, the spatial dimensions: d2s = dxidxj; xµ = (x,y,z,ct) = (xi ,ct). It is this measure of "length", sometimes called 'proper distance', no better a choice of words, which makes c a unique constant. (You may be more familiar with this invariant called 'proper time' dτ = dσ/c.) Specifically, a disturbance propagating at c in one system (ds/dt=c in that system) will produce events in that system for which d2σ = 0. Since this "length" is invariant, it will be the same in all systems: d2σ = 0 for the events transformed to any other system, and they will thus also appear to move at ds'/dt'=c. For all equivalent uniformly moving systems, which have the metric above, a speed of c will be invariant. (This argument is carefully phrased to avoid "the speed of light", although "the speed of light in vacuum" would suffice. If light is observed in a medium, which is difficult to avoid, the medium introduces a preferred reference frame and the speed is no longer strictly invariant.) It remains only to obtain the transformation law between uniformly moving coordinate systems which will preserve the metric. Let the origins coincide at t=0 and the origin of one system (0,ct) move with velocity v in the other along x. If one looks for the simplest (covariant) transform which could accomplish this A α 0 Λ = 0 µ  C g'µν = 0 B 0 0  1 0 0 D Β2 − Α2 0 0 BD - AC 0 1 0 0 g'µν = Λ 0 -1 0 0 0 0 -1 0 BD - AC 0 0 D 2 - C2 α β Λ gαβ µ ν

  

  

where one must be careful if one does the tensor contraction as matrix multiplication; transposes must sometimes be used to obtain the proper index matching. The requirements are thus AC = BD Β2 − Α2 = -1 D2 - C2 = 1

(0,ct) → (- Bct,0,0,Dct) ⇒ B/D = v/c ≡β, where the signs come from using covariant displacements to employ the transform law above, but one is not concerned about the sign of v. Note that co and contravariant vectors differ, but only in sign of the spatial part.) The unique solution to these four equations in four unknowns is

       

γ 0 0 βγ γ 0 0 -βγ

0 1 0 0 0 1 0 0

0 0 1 0 0 0 1 0

βγ 0 0 γ -βγ 0 0 γ

       

= Λαµ

= Λα µ

11

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

γ ≡

1

 1-β √ 2

which give the rules for transforming tensors between uniformly moving systems. (Note that the metric is not positive definite here. The notion of physical vectors introduced in Section III cannot be employed to disguise a difference between co and contravariant. An attempt to do so introduces √(-1), the origin of the ubiquitous i's which permeate non-tensor treatments of special relativity. It is ironic that the attempt to "hide" the metric by introducing "physical" vectors should result in the rather unphysical appearance of imaginary dimensions.) Because the metric does not depend upon position, we have the useful generalization, already employed above, that not only is the displacement, dxµ, a contravariant vector, as it always must be, but the coordinates or vector position of a point, x µ , is also a vector, which is not true in general and constitutes a major conceptual subtlety in tensor calculus. This is a great simplification for special relativity, and it means that the law above for transformation of contravariant vectors is also the law for coordinate transformations. Finally, note that gµν = gµν, which can be confirmed by direct calculation. (As noted earlier, the two must be matrix inverses of one another.) All the usual relativistic effects follow in a straightforward manner from these equations. An event at xo, cto occurs at γ (xo- βcto ), γ(cto - βxo) in the moving system. The origin of the initial coordinates appears to be moving at -v in the new system, whereas the origin in the new system appears to be moving at v in the initial system. Events at the point xo but separated by ∆to occur at different points and different times, the time difference being γ ∆to, the well-known time dilation. A stationary bar with ends 0,cto and L,ct1 appears at −βγcto , γ cto and γ (L-βct1 ),γ(ct1- βL) and t' = γ(t1- βL/c) (L/γ) -βct', ct'

Expressed in terms of a new t', t' =γ to

−βct' , ct' and which implies that the ends appear separated by a distance L/γ, the contraction of length, if they are observed (measured) simultaneously in the new system. The velocity addition formula follows simply by applying two successive transformations:  (1+ββ')γγ' 0 0 -(β+β')γγ'   γ" 0 0 -β"γ"  0 1 0 0   =  0 1 0 0  0 0 1 0    0γ 0 1 γ0  γγ ' γγ '  0 0 (1+ββ') "   -(β+β')  -β" " 0 0 β+β' γ" = (1 + ββ')γγ' = γ"(β") 1+ββ' but note that the addition of two velocities in different directions gives much more complicated results; the transformations do not even commute. β" ≡ If the physical laws are expressed in terms of relativistic vectors and tensors, they will transform properly with coordinate system and have the same form in any system, as desired. The analog of velocity is vµ ≡ dxµ/dτ dσ = cdτ 12

INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS

vµ = (γ vi,γc)

pµ = mvµ =( pi, E/c)

These relations for the four-velocity follow directly if xi =vit, d2xi = v2d2t d2τ = d2t - d2xi /c2 = (1-β2)d2t= d2t /γ2 This is a well-formed vector which reduces to the usual velocity for v