OPERATIONS WITH GENERAL TENSORS 193 Similarly, the covariant derivative of a contravariant vector is defined as (10.208) The covariant derivative is also shown as a,, that is, ajui = ui;j. The covariant derivative of a higher-rank tensor is obtained by treating each index at a time as (10.209) Covariant derivatives distribute like ordinary derivatives, that is, (AB)., = A,;B + AB,i (10.210) and (uA + bB);i = uA;~ + bB,i (10.211) where A and B are tensors of arbitrary rank and a and b are scalars. 10.9.8 Some Covariant Derivatives In the following we also show equivalent ways of writing these operations commonly encountered in the literature. 1. Using definition Equation (10.123) we can write the covariant derivative of a scalar function @ as an ordinary derivative: This is also the covariant component of the gradient (9.); . (10.212) (10.213) 2. Using the symmetry of Christoffel symbols, the curl of a vector field 3 can be defined as the second-rank tensor (bx 3). , = aiv, - a,vj = vi;j - vj;i (10.214) 23 (10.215) 194 COORDINATES AND TENSORS Note that because we have used the symmetry of the Christoffel symbols, the curl operation can only be performed on the covariant components of a vector. 3. The covariant derivative of the metric tensor is zero: akgij = &;k = 0, (10.216) with Equation (10.209) and the definition of Christoffel symbols the proof is straightforward. 4. A frequently used property of the Christoffel symbol of the second kind is In the derivation we use the result from the theory of matrices, where g = det gij. 5. We can now define covariant divergence as 7.3 = diUi = u:. 7% (10.217) (10.218) (10.219) (10.220) (10.221) If vz is a tensor density of weight +1, divergence becomes V.v+==U!Z (= did), (10.222) which is again a scalar density of weight fl. 6. Using Equation (10.213) we write the contravariant component of the gradient of a scalar function as We can now define the Laplacian as a scalar field: (10.223) (10.224) OPERATIONS WITH GENERAL TENSORS 195 10.9.9 Riemann Curvature Tensor Let us take the covariant derivative of ui twice. The difference ui;jk - Ui;kj can be written as (10.226) where Rijk is the fourth-rank Riemann curvature tensor, which plays a central role in the structure of Riemann spaces: Three of the symmetries of the Riemann curvature tensor can be summarized as (10.229) (10.230) (10.231) Actually, there is one more symmetry that we will not discuss. The signifi- cance of the Riemann curvature tensor is, that all of its components vanish only in flat space, that is we cannot find a coordinate system where Rijkl = 0 (10.232) unless the space is truly flat. which is obtained from Rijkl by contracting its indices as An important scalar in Riemann spaces is the Riemann curvature scalar, R == y'lgikR ajkl = jLRi 221 = Ra.,3 32 . (10.233) Note that &jkl = 0 implies R = 0, but not vice versa. Example 10.1. Laphcian as a scalar field: We consider the line element ds2 = dr2 + r2d2 + r2 sin2 Bd$, (10.234) where x1 = r, x2 = 8, x3 = 4 (10.235) and 911 = 1, g22 = r2, 933 = r 2 sin28. (10.236) 196 COORDINATES AND TENSORS Contravariant components 9'3 are: (10.237) Using Equation (10.225) and g = r4 sin2 0, we can write the Laplacian as (10.238) - - 1 [" (r2sinog) +ae d (TaB) r2sinOaW +a (WE)] After simplifying, the Laplacian is obtained as r2sin6 dr 34 r2sin28 a4 ' (10.239) Here we have obtained a well-known formula in a rather straightfor- ward manner, demonstrating the advantages of the tensor formalism. Note that even though the components of the metric tensor depend on position [Eq. (10.236)], the curvature tensor is zero, Rijki = 0; (10.240) thus the space of the line element [Eq. (10.234)] is flat. However, for the metric it can be shown that not all the components of Rijkl vanish. In fact, this line element gives the distance between two infinitesimally close points on the surface of a hypersphere (S-3) with constant radius &. 10.9.10 Geodesics Geodesics are defined as the shortest paths between two points in a given geometry. In flat space they are naturally the straight lines. We can gener- alize the concept of straight lines as curves whose tangents remain constant along the curve. However, the constancy is now with respect to the covariant derivative. If we parametrize an arbitrary curve in terms of arclength s as SPACETIME AND FOUR-TENSORS 197 its tangent vector will be given as . dxi ta = -_ ds (10.243) For geodesics the covariant equation of geodesics as dX-3 t"- J ds or as derivative of ti must be zero; thus we obtain the = [-$ + {31i}t"] g = 0 - &xi + { }dxj dx" = 0. ds2 jk ds ds (10.244) (10.245) 10.9.11 lnvariance and Covariance We have seen that scalars preserve their value under general coordinate trans formations. Certain other properties like the magnitude of a vector and the trace of a second-rank tensor also do not change under general coordinate transformations. Such properties are called invariants. They are very im- portant in the study of the coordinate-independent properties of a system. An important property of tensors is that tensor equations preserve their form under coordinate transformations. For example, the tensor equation transforms into (10.247) This is called covariance. Under coordinate transformations individual com- ponents of tensors change; however, the form of the tensor equation remains the same. One of the early uses for tensors in physics was in searching and expressing the coordinate independent properties of crystals. However, the covariance of tensor equations reaches its full potential only wit$h the intro- duction of the spacetime concept and the special and the general theories of relativity. 10.10 SPACETIME AND FOUR-TENSORS 10.10.1 Minkowski Spacetime In Newton's theory, the energy of a freely moving particle is given by the well-known expression for kinetic energy: E=-mu. 12 (10.248) 2 198 COORDINATES AND TENSORS fig. 10.8 Minkowski spacetime Because there is no limit to the energy that one could pump into a system, this formula implies that in principle one could accelerate particles to any desired velocity. In classical physics this makes it possible to construct infinitely fast signals to communicate with the other parts of the universe. Another property of Newton’s theory is that time is universal (or absolute), that is, identical clocks carried by moving observers, uniform or accelerated, run at the same rate. Thus once two observers synchronize their clocks, they will remain synchronized for ever. In Newton’s theory this allows us to study systems with moving parts in terms of a single (universal) time parameter. With the discovery of the special theory of relativity it became clear that clocks carried by moving observers run at different rates; thus using a single time parameter for all observers is not possible. After Einstein’s introduction of the special theory of relativity another remarkable contribution toward the understanding of time came with the introduction of the spacetime concept by Minkowski. Spacetime not only strengthened the mathematical foundations of special relativity but also paved the way to Einstein’s theory of gravitation . Minkowski spacetime is obtained by simply adding a time axis orthogonal to the Cartesian axis, thus treating time as another coordinate (Fig. 10.8). A point in spacetime corresponds to an event. However, space and time are also fundamentally different and cannot be treated symmetrically. For example, it is possible to be present at the same place at two different times; however, if we reverse the roles of space and time, and if space and time were symmetric, then it would also mean that we could be present at two different places at the SPACETIME AND FOUR-TENSORS 199 same time. So far there is no evidence for this, neither in the micro- nor in the macro-realm. Thus, in relativity even though space and time are treated on equal footing as independent coordinates, they are not treated symmetrically. This is evident in the Minkowski line element: ds)’ = c2dt2 - dx)’ - dy2 - dZ2, (10.249) where the signs of the spatial and the time coordinates are different. It is for this reason that Minkowski spacetime is called pseudo-Euclidean. In this line element c is the speed of light representing the maximum velocity in nature. An interesting property of the Minkowski spacetime is that two events connected by light rays, like the emission of a photon from one galaxy and its subsequent absorption in another, have zero distance between them even though they are widely separated in spacetime. 10.10.2 In Minkowski spacetime there are many different ways to chotlse the orienta- tion of the coordinate axis. However, a particular group of coordinate systems, which are related to each other by linear transformations of the form Lorentz Transformation and the Theory of Special Relativity 9 = ugzo + ay + a&)’ + 4.3 z1 = u;zo + a;21 + up + 4x3 2)’ = + uqd + a;x2 + a3r3 z3 = u;.o + a;z1 + u;2 + 4.3 (10.250) and which also preserve the quadratic form (2))’ - (d))’ - (2))’- (23))’, (10.251) have been extremely useful in special relativity. In these equations we have written zo = ct to emphasize the fact that time is treated as another coordi- nate. In 1905 Einstein published his celebrated paper on the special theory of relativity, which is based on two postulates: First postulate of relativity: It is impossible to detect or measure uniform translatory motion of a system in free space. Second postulate of relativity: The speed of light in free space is the maximum velocity in the universe, and it is the same for all uniformly moving observers. In special relativity two inertial observers K and E, where is moving uniformly with the velocity 21 along the common direction of the d- and 200 COORDINATES AND TENSORS x3 23 fig. 10.9 Lorenta transformations ?$-axes are related by the Lorentz transformation (Fig. 10.9): 1 V 2J = J [." - (,) x'] 1 - u2/c2 1 V -1 z - - J [-(,)."+x'] 1 - u2/c2 (10.252) (10.253) (10.254) (10.255) -2 - 2 x -x 2=x. 3 Inverse transformation is obtained by replacing u with -v as V zo = d- [ TO+ (;) 2'1 1 - v2/c2 z 1 = d- 1 [( )3?+"1] U 1 - v2/c2 If the axis in K and R remain parallel but the velocity ?i' of (10.256) (10.257) (10.258) (10.259) ame ?T in frame K is arbitrary in direction, then the Lorentz transformation is generalized as (10.260) (10.261) 8 = y [xo- (3.34 SPACETIME AND FOUR-TENSORS 201 We have written y = l/Jm and 3 = 3/c. 10.10.3 Two immediate and important consequences of the Lorentz transformation equations [Eqs. (10.252-10.255)] are the time dilation and length contraction formulas, which are given as Time Dilation and Length Contraction (10.262) 212 C2 AT = At(1- -)1/2 and respectively. These formulas relate the time and the space intervals measured by two inertial observers ?7 and K. The second formula is also known as the Lorentz contraction. The time dilation formula indicates that clocks carried by moving observers run slower compared to the clocks of the observer at rest. Similarly, the Lorentz contraction indicates that meter sticks carried by a moving observers appear shorter to the observer at rest. 10.10.4 Addition of Velocities Another important consequence of the Lorentz transformation is the formula for the addition of velocities, which relates the velocities measured in the K and 1T frames by the formula (10.264) dx where u1 = - and a' = are the velocities measured in the K and the - dt dt K frames, respectively. In the limit as c f 03, this formula reduces to the well-known Galilean result u1 =ti' +v. (10.265) It is clear from Equation (10.264) that even if we go to a frame moving with the speed of light, it is not possible to send signals faster than c. If the axes in K and ?? remain parallel, but the velocity 3 of frame 1T in frame K is arbitrary in direction, then the parallel and the perpendicular components of velocity transform as (10.266) (10.267) 202 COORDINATES AND TENSORS In this notation UII and 31 refer to the parallel and perpendicular components with respect to d and y = (1 - Y~/c~)-~/~. 10.10.5 Four-Tensors in Minkowski Spacetime From the second postulate of relativity, the invariance of the speed of light means 3 3 (10.268) This can also be written as 7japdEadZp = gapdxadxB = 0, ( 10.269) where the metric of the Minkowski spacetime is 10 0 - ga(3 = gap = [ i ; [I i]- We use the notation where the Greek indices take the values 0,1,2,3 and the Latin indices run through 1,2,3. Note that even though the Minkowski space time is flat, because of the reversal of sign for the spatial components it is not Euclidean; thus the covariant and the contravariant indices differ in space time. Contravariant metric components can be obtained using (Gantmacher) (10.270) (10.271) as 10 0 0 gap= [ 1. (10.272) Similar to the position vector in Cartesian coordinates we can define a position vector r in Minkowski spacetime as r = xa = (xo,x',x2,x3) (10.273) = (xO,T), where r defines the time and the position of an event. In terms of linear transformations [Eq. (10.250)] xa transforms as (10.274) -a x =a;xP. [...]... equations are (10. 371 ) (10. 372 ) + = where y = I/( 1 - P2)'I2 and obtained by interchanging with 10.10.14 3 ' transformations are easily , Inverse ~ -d Maxwell's Equations in Terms of Potentials The Electric and magnetic fields can also be expressed in terms of the potentials -A' and 4 as (10. 373 ) (10. 374 ) In the Lorentz gauge (10. 375 ) 2 and 4 satisfy (10. 376 ) and (10. 377 ) respectively Defining a four-potential... Given the values of Fys in an inertial frame K , we can find it in another inertial frame 17 as F a@ = a a aB Fy6 y 6 (10.364) If IT corresponds t o an inertial frame moving with respect to I with velocity ( + along the common Z1- and 21-axes, the new components of 3 and B are 21 (10.365) (10.366) (10.3 67) 214 COORDINATES AND TENSORS and ( 10.368) (10.369) (10. 370 ) If is moving with respect to K with... polar coordinates 10.22 Write the divergence operator in spherical polar coordinates 10.23 Write the Laplace operator in cylindrical coordinates 10.24 In four dimensions spherical polar coordinates (r, 6,Q,) are defined x, as x y z = rsinxsinBcosQ, = rsinXsinBsinq5 = rsinxcos6 w = r cos x 220 COORDINATES AND TENSORS i) Write the line element ds2 = dx2 + dy2 + dz2 + dw2 x, 6,') ii) What are the ranges of...SPACETIME AND FOUR-TENSORS 203 For the Lorentz transformations [Eqs (10.252-10.255) and (10.260-10.261)], a$ are given respectively as (10. 275 ) and a; = (10. 276 ) For the general linear transformation [Eq (10.250)] matrix elements a$ can be obtained by using dZa a; = - dxp (10. 277 ) In Minkowski spacetime the distance between two infinitesimally close points (events) can be written... Abelian 226 CONTINUOUS GROUPS AND REPRESENTATIONS 11.2 INFINITESIMAL RING OR LIE ALGEBRA For a continuous (Lie) group G if A(t) E G, we have seen that its generator [Eq (10 .76 )] given as X = A’(0).The ensemble (A’(0)) of transformations is is called the infinitesimal ring or Lie algebra of G, and it is denoted by ‘G Differentiating A(at) E G we get A’(&)= aA’(at),where a is a constant Substituting t = 0... is defined as is the permutation symbol 1 8 If corresponds to an inertial frame moving with respect to K w i s 02 velocity v along the 21-axis, show that the new components of 3 and B become 22 2 COORDINATES AND TENSORS and 10.29 Show that the field-strength tensor can also be written as Fa@ = d"AP - where the four-potential is defined as A" = (4,T) and , CONTINUOUS GROUPS and REPRESENTATIONS In everyday... X (11.6) Also, if A(t) and B(t) are any two elements of G, then C ( t )= A(t)B(t) E G (11 .7) Differentiating this and substituting t = 0 and using the fact that A(0) = B(0) = I, we obtain C’(0) = A’(0) + B’(O), (11.8) [Xi, = xixj - x x Xj] ji + =X+Y (11.9) Hence, X Y E ‘G if X,Y E ‘G Lie has proven some very interesting theorems about the relations between continuous groups and their generators One... should be covariant with respect t o it In this regard we also need to write Newton's dynamical equation as a four-tensor equation in spacetime Using the definition of four-momentum pa = m& = ( E / c , p2) (10.3 87) and differentiating it with respect to invariant proper time, we can write the Newton's dynamical equation in covariant form as (10.388) 216 COORDINATES AND TENSORS where F a is now the four-force... is 10.3 Using the properties of the permutation symbol and the Kronecker delta, prove the following identities in tensor notation: i) ii) [ Z X [Sxb]]+ [73 'x [ r l x 7 f ] ] + [ c x [Z.3]]=0, iii) Z x [3xd]=S(7f.d)-d(;i'.B*) 1 04 The trace of a second-rank tensor is defined as t r ( A ) = A: Show that trace is invariant under general coordinate transformations 10.5 ment Under general coordinate transformations... Find the expressions for the div and the grad operators for the following metrics: i) ds2 = dr2 ii) ds2 = [ ] 1 - kr2/R2 + p2dB2 + dz2 dr2 where k takes the values k = 0,1, -1 + r 2 a 2+ r 2sin2Odd2, PROBLEMS 10.16 i) where ii) 219 Write the Laplacian operator for the following metrics: x E [0,T I , x + ds2 = R2(dX2 sin2 xd6' + sin2xsin2SdQ,'), 6 E [O,T], Q, E [0,27r] + ds2 = R2(dx2 sinh2xd02 + sinh2Xsin26d42), . however, the form of the tensor equation remains the same. One of the early uses for tensors in physics was in searching and expressing the coordinate independent properties of crystals. However,. covariant and the contravariant indices differ in space time. Contravariant metric components can be obtained using (Gantmacher) (10. 270 ) (10. 271 ) as 10 0 0 gap= [ 1. (10. 272 ) Similar. position vector in Cartesian coordinates we can define a position vector r in Minkowski spacetime as r = xa = (xo,x',x2,x3) (10. 273 ) = (xO,T), where r defines the time and the position