Appendix 1 Variational Derivation of Differential Equations

Let us recall the definition of a critical point for a function f belonging to the class C 2 ..a; b/ RI R/. Critical points are furnished by the roots of the equation f 0.x/ WD df.x/ D 0 in the interval .a; b/. Local (or relative) extrema are usually dx 00 0 00 given by a critical point x0 such that f .x0/ ¤ 0. In case f .x0/ D 0 and f .x0/> 0, the function f has a local (or relative) minimum. Similarly, for a critical point 00 x0 with f .x0/<0, f has a local (or relative) maximum. In case a critical point is neither a maximum nor a minimum, it is called a stationary point (or a point of inflection). We will now provide some simple examples. (1) Consider f.x/ WD 1 for x 2 .1; 1/. Every point in the interval .1; 1/ is a critical point. (2) Consider f.x/ WD x in the open interval .2;2/.Thereexistno critical points in the open interval. However, if we extend the function into the closed interval Œ2;2,there exists an extremum at the end points (which are not critical points). (3) The function f.x/ WD .x/3,forx 2 R, has a stationary point (or a point of inflection) at the origin. (4) The periodic function f.x/ WD sin x; x 2 R, has denumerably infinite number of local extrema, but no stationary points. (See, e.g., [32].) Now, we generalize these concepts to a function of N real variables x 1 N 2 N .x ; :::; x /. We assume that f 2 C .D R I R/. A local minimum x0 is defined by the property f.x/ f.x0/ for all x in the neighborhood Nı.x0/. Similarly, a local maximum is defined by the property f.x/ f.x0/ for all x in the neighborhood. Consider f as a 0-form (see Sect. 1.2). The critical points are provided by the roots of the 1-form equation:

df.x/ D 0Q.x/;

@i f D 0; i 2f1;2;:::;Ng: (A1.1)

A local (strong) minimum at x0 is implied by the inequality XN XN @2f.x/ xi xi xj xj ˇ >0: (A1.2) 0 0 @xi @xj ˇ iD1 j D1 x0

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 569 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 570 Appendix 1 Variational Derivation of Differential Equations

χ^

x χ

t1 t t2

Fig. A1.1 Two twice-differentiable parametrized curves into RN

Similarly, a local (strong) maximum is guaranteed by XN XN @2f.x/ xi xi xj xj ˇ <0: (A1.3) 0 0 @xi @xj ˇ iD1 j D1 x0 P P N N i j Example A1.1. Consider the quadratic polynomial f.x/ WD iD1 j D1 ıij x x N for x 2 R . There exists one critical point at the origin x0 WD .0;0;:::;0/.Now, XN XN @2f.x/ XN XN xi xj ˇ D 2 ı xi xj >0: @xi @xj ˇ ij iD1 j D1 .0;0;:::;0/ iD1 j D1 Therefore, the origin is a local, as well as a global (strong), minimum. 2 N Consider a parametrized curve X of class C Œt1;t2 RI R , as furnished in 0 R equation (1.19). A Lagrangian function L W Œt1;t2 DN DN ! is such that L.tI xI u/ is twice differentiable. The action function (or action functional) J 2 N is a mapping from the domain set C Œt1;t2 RI R into the range set R and is given by:

Zt2

X i J. / WD ŒL.tI xI u/j i DX i i D dX .t/ dt: (A1.4) x .t/; u dt t1 We assume that the function J is of class C 2, thus, totally differentiable in the abstract sense. (See (1.13).) b 2 N We consider a “slightly varied differentiable curve” X 2 C Œt1;t2 RI R . (See Fig. A1.1.) Two nearby curves are speciﬁed by the equations Appendix 1 Variational Derivation of Differential Equations 571

xi D X i .t/; bxi D Xbi .t/ WD X i .t/ C "hi .t/;

t 2 Œt1;t2 R: (A1.5)

Here, j"j >0is a small positive number and hi .t/ is arbitrary twice-differentiable function. The “slight variation” of the action function J in (A1.4) is furnished by J.X / WD J Xb J.X / ( ) Zt2 @L./ dhi .t/ @L./ D " hi .t/ ˇ C ˇ dt C 0."2/ @xi ˇ:: dt @ui ˇ:: t1 ( ) ( ) ˇ Zt2 ˇt2 @L./ @L./ ˇ D " hi .t/ ˇ dt C " hi .t/ ˇ ˇ i i ˇ @x ˇ:: @u ˇ:: t1 t1 ( ) Zt2 d @L./ " hi .t/ ˇ dt C 0."2/: (A1.6) dt @ui ˇ:: t1

The variational derivative of the function J is given by ıJ.X / J.X / WD lim ıX "!0 " ( ) Zt2 @L./ d @L./ D ˇ ˇ hi .t/ dt @xi ˇ:: dt @ui ˇ:: t1 ( ) ˇ ˇt2 @L./ ˇ C ˇ hi .t/ ˇ C 0: (A1.7) i ˇ @u ˇ:: t1

The critical functions or critical curves X.0/ are deﬁned by the solutions of the ıJ.X / equation ıX D 0. We can prove from (A1.7) (with some more work involving the Dubois-Reymond lemma [224]), that critical functions X.0/ must satisfy N EulerÐ Lagrange equations: ( ) @L./ˇ d @L./ ˇ ˇ D 0: (A1.8) @xi :: dt @ui ˇ:: Moreover, we must impose variationally permissible boundary conditions (con- sisting partly of prescribed conditions and partly of variationally natural boundary conditions [159]) so that the equations 572 Appendix 1 Variational Derivation of Differential Equations ( ) ˇ ( ) ˇt2 @L./ ˇ @L./ ˇ hi .t/ ˇ D ˇ hi .t/ i ˇ i ˇ @u ˇ:: @u ˇ:: ˇ t1 t2 ( ) @L./ i ˇ h .t/ ˇ D 0: (A1.9) @ui ˇ:: ˇ t1 must hold. Usually, 2N Dirichlet boundary conditions

X i i X i i .t1/ D x.1/ D prescribed consts.; .t2/ D x.2/ D prescribed consts.

i i are chosen. Such conditions imply, by (A1.5), that h .t1/ D h .t2/ 0. Therefore, (A1.9) will be validated. However, there are many other variationally permissible boundary conditions which imply (A1.9). Example A1.2. Consider the Newtonian mechanics of a free particle in a Cartesian coordinate chart. The Lagrangian is furnished by

m L.v/ WD ı v˛vˇ;m>0; 2 ˛ˇ @L.v/ D mı vˇ; @v˛ ˛ˇ @2L.v/ D mı : (A1.10) @v˛ @vˇ ˛ˇ

The EulerÐLagrange equations (A1.8), with a slight change of notation and ﬁxed boundary conditions X .t1/ D x.1/ and X .t2/ D x.2/, yield the unique critical curve x˛ x˛ X ˛ .2/ .1/ ˛ .0/.t/ D .t t1/ C x.1/: (A1.11) t2 t1

This is a portion of a straight line or geodesic. Substituting the above solution into the ﬁnite variation in (A1.6), we derive that Appendix 1 Variational Derivation of Differential Equations 573

F (r+s) y DN x

DN π (r+s)

N N(r+s)

y (r+s)

Fig. A1.2 The mappings corresponding to a tensor ﬁeld y.rCs/ D .rCs/.x/

J X.0/ C "h J X.0/ " # Zt2 ˇ Zt2 dh˛.t/ dX .t/ 1 dh˛.t/ dhˇ.t/ D "m ı .0/ dt C "2m ı dt C 0 ˛ˇ dt dt 2 ˛ˇ dt dt t1 t1

Zt2 1 dh˛.t/ dhˇ.t/ D 0 C "2m ı dt>0: (A1.12) 2 ˛ˇ dt dt t1

Therefore, the critical curve X.0/, which is a part of a geodesic in the Euclidean space, constitutes a strong minimum compared to other neighboring smooth curves. Remarks. (i) Local, strong minima provide stable equilibrium configurations. (ii) Timelike geodesics in a pseudo-Riemannian spaceÐtime manifold constitute strong, local maxima ! Now we shall consider .r C s/th order, differentiable tensor fields in an N - dimensional manifold MN , as discussed in (1.30). (More sophisticated treatments of tensor fields employ the concepts of tensor bundles [38,56].) For the sake of brevity, we denote an .r Cs/th order tensor field by the symbol .rCs/.x/ (in this appendix). Various mappings, which are necessary for the variational formalism, are shown in Fig. A1.2. 574 Appendix 1 Variational Derivation of Differential Equations

According to the mapping diagram, y.rCs/ D .rCs/ ı F.x/ DW .rCs/.x/; N x 2 DN R : (A1.13)

C C C .N /r s .N /r s 1 The Lagrangian function L W DN R R ! R is such that L .rCs/ .rCs/ .xI y I y i / is twice differentiable. The action function (or functional) is deﬁned by Z L .rCs/ .rCs/ ˇ N J.F/ WD xI y I y C d x: (A1.14) i ˇ .rCs/D .rCs/ .r s/D .rCs/ y .x/; y i @i DN A “slightly varied function” is deﬁned by

by .rCs/ D .rCs/ ı F.x/b D .rCs/.x/ C "h.rCs/.x/; (A1.15) where j"j is a small positive number. The variation in the totally differentiable action function is furnished by J.F / WD J Fb J.F/ 8 " # 9 Z < = @L./ @L./ D " h.rCs/.x/ ˇ C@ hrCs dN x C 0."2/: : @y .rCs/ ˇ i rCs ˇ ; :: @y i ˇ:: DN (A1.16) Here, indices .r C s/ are to be automatically summed appropriately. Now, we introduce the notion of a total-partial derivative by h i d C f.xI y.rCs/I y.r s// ˇ dxi j ˇ:: " # @f . / @f . / @f . / WD ˇ C @ .rCs/ ˇ C @ @ .rCs/ : @xi ˇ i @y .rCs/ ˇ i j .rCs/ ˇ :: :: @y j ˇ:: (A1.17)

By (A1.17), we can express (A1.16)as 8 Z < @L./ J.F / D " h.rCs/.x/ ˇ : .rCs/ @y ˇ:: DN 0 2 3 " # 19 L L = @ d 4 .rCs/ @ ./ 5 .rCs/ d @ ./ A N 2 C h .x/ C ˇ h .x/ C ˇ d xC0." /: dxi .r s/ ˇ dxi @y r s ˇ ; @y i :: i :: (A1.18) Appendix 1 Variational Derivation of Differential Equations 575

The variational derivative is provided by ıJ.F / J.F / WD lim ıF "!0 " 8 0 " # 19 Z < = @L./ d @L./ D h.rCs/.x/ @ ˇ A dN x : @y .rCs/ ˇ dxi rCs ˇ ; :: @y i ˇ:: DN 8 " # 9 Z < = @L./ C h.rCs/.x/ n dN 1x: (A1.19) : .rCs/ ˇ ; i @y i ˇ @DN :: (We have used Gauss’ theorem in (1.155) for the last term in (A1.19).) The critical .rCs/ ıJ.F / or stationary functions .0/ .x/ are given by the solution of the equations ıF D 0. We can derive, from (A1.19)[159], that critical functions must satisfy equations " # @L./ d @L./ C ˇ C ˇ D 0; (A1.20i) @y .r s/ ˇ dxi .r s/ ˇ :: @y i :: and 8 " # 9 Z < = @L./ h.rCs/.x/ n .x/ dN 1x D 0: (A1.20ii) : .rCs/ ˇ ; i @y i ˇ @DN :: Here, (A1.20i) yields EulerÐLagrange equations for critical tensor ﬁelds, whereas (A1.20ii) provides the variationally admissible boundary conditions. Example A1.3. Consider a background pseudo-Riemannian spaceÐtime and a scalar 4 ﬁeld y D .x/ for x 2 D4 R . The Lagrangian L,whichisa scalar density,is furnished by p p g.x/ g.x/ L.xI yI y / D L.xI yI y / WD gij .x/ y y ; i i 2 i j L L p @ ./ @ ./ kj 0; D g.x/ g .x/ yj : (A1.21) @y @yk

EulerÐLagrange equations (A1.21), using (A1.17), yield d @L./ p p 0 ˇ D @ @ g gkj C @ @ g gkj k ˇ j k j k dx @yk ::

p kj p D g g .x/rk rj D g D 0;

or D 0: (A1.22) 576 Appendix 1 Variational Derivation of Differential Equations

Thus, the critical functions .0/.x/ satisfy the wave equation in the prescribed curved spaceÐtime. Consider the wave equation (A1.22) in a domain D4 WD D3 .0; T /, 0

.0/.x;0/D p.x/; @4.0/j.x;0/ D q.x/; x 2 D3: Here, the prescribed functions p.x/ and q.x/ are assumed to be real-analytic. This initial value problem, by the Cauchy-Kowaleski Theorem 2.4.7, admits a unique, real-analytic solution .0/.x/ for a small positive number T . The “varied function” .0/.x/C"h.x/, which respects the initial values, must have h.x;0/D 0 D @4hj.x;0/. However, other than being real-analytic, there are no restrictions on h.x/ for 0< x4

@D4 Therefore, in general, initial value problems are not variationally admissible (due to the fact that on the final time hypersurface, the function h.x/ is completely arbitrary). Now, we shall investigate the variational derivation of the gravitational field equations. However, this process demands a slightly different mathematical approach. We explain the methodology by first exploring the following toy model.

Example A1.4. Consider the ﬂat spaceÐtime M4 and a Minkowskian coordinate chart. Therefore, the metric tensor ﬁeld is given by g .x/ D d D consts., and p ij ij g.x/ 1. We explore a Lagrangian function L i i ij i xI y; I yi ;j WD d @i @j W.x/ y i ; L xI y; i I y ;i ˇ D . W/ .x/ @ i : (A1.23) i j ˇ D D i ˇ y .x/; yi @i i D i i D i .x/; j @j

The slightly varied functions are denoted by by D .x/C "h.x/ and b i D i .x/ C "hi .x/.By(A1.16), we obtain that Z Z J.F / D W h.x/ d4x @ hi d4x " i D D Z4 Z4 4 i 3 D W h.x/ d x h .x/ ni d x:

D4 @D4 Appendix 1 Variational Derivation of Differential Equations 577

ıJ.F / Therefore, the critical functions, obeying ıF D 0, must satisfy: W.x/ D 0; Z i 3 and, h .x/ ni d x D 0: (A1.24)

@D4 Here, the coefﬁcient function W.x/need not satisfy any boundary condition and the i critical functions .0/.x/ need not satisfy any differential equation! Now, we shall take up the case of general relativity. Recall that in (2.146iÐiii) for relativistic mechanics, position variables and momentum variables are treated on the same footing. Similarly, in what is known as the Hilbert-Palatini approach ij ij k ofn variationo , metric functions y D g .x/ and connection coefﬁcients ij D k ij are both treated as independent variables. We choose the following invariant Lagrangian function L./1: L xI yij ;k I yij ;k hij k ij l i ij k k l k l k WD y kij ij k lk ij C ik lj ij c c DW y ij ab I abd ; ˇ ij k k l k L./ ij D g .x/ @j @k ˇyij Dgij .x/; y D@ gij ; ˇ ˚ k k ˚ ij ij lk ij ˇ k k k D ;k D@ ij ij ij l l ij l k C I ik lj

p ij k k L./j:: D g.x/ g .x/ @j @k ij ij l k l k C : (A1.25) lk ij ik lj

From the equations above, we can deduce the following partial derivatives: ab b @.y/ ab y y yac D y ı c; D y y ; @yab p p @ y 1 p @ y 1 p DC y yab; D y y ; ab ab @yab 2 @y 2

1 c Caution: Here, ab is not related to Ricci rotation coefﬁcients as such. 578 Appendix 1 Variational Derivation of Differential Equations @ p p 1 y yij D y ıi ıj y yij I @y ab a b 2 ab

@ij ./ b c b c l b c c b a Dı a ij ı i ı j la C ı i aj C ı j ia; @ bc

@ij ./ b c d d b c a D ı a ı i ı j ı a ı i ı j : (A1.26) @ bcd

bij ij ij bk Wen o denote the slightly varied functions by y D g .x/ C "h .x/ and ij D k C "hk .x/. (Note that hij .x/ and hk .x/ are .2 C 0/th order and .1 C 2/th ij ij ij n o k 2 order tensor ﬁelds, respectively, even though ij is not!) Using (A1.16)forthe Lagrangian in (A1.25), we derive that 8 9 Z " # < p ij = J.F / @ y y ab 4 D ij ./ ˇ h .x/ d x " : @y ab ˇ ; D4 :: Z ( p @ ./ C g.x/ gij .x/ ij ˇ ha .x/ a ˇ bc @ bc :: D4 ) @ ./ C ij ˇ @ ha d4x C 0."/: (A1.27) a ˇ d bc @ bcd ::

Therefore, for critical functions, (A1.25)Ð(A1.27), and a shortcut notation in regard to Gab.x/ yield Z ab p 4 0 D Gab.x/ h .x/ g.x/d x

D4 Z h i ji k ki j 4 C rj g .x/ h ki.x/ g .x/ h ki.x/ d v: (A1.28)

D4 n o ij k Thus, the critical functions g.0/.x/ and ij must satisfy

Gab.x/ D 0 in D4; (A1.29i)

p 2The popular way of writing one of the variations in (A1.26) is to put ı jgj gij D p j j ij 1 ij kl g ıg 2 gkl g ıg . (This equation holds in any Riemannian or pseudo-Riemannian manifold.) Appendix 1 Variational Derivation of Differential Equations 579 and (via the divergence theorem) Z h i ij k ki j 3 g .x/ h ki.x/ g .x/ h ki.x/ nj d v D 0: (A1.29ii)

@D4

p ij k i The variables yy and lij are analogous to the variables y and j respectively of Example A1.4. Equations (A1.29i)and(A1.29ii) are exactly analogous to the equations in (A1.4). Equation (A1.29i) is obviously equivalent to the vacuum equations of (2.160i). Moreover, (A1.29ii) provides implicitly the variationally admissible boundary conditions for the vacuum equations. There is an unsatisfactory aspect in the derivation of (A1.29i,ii).Inthat process, we have varied the metric tensor and connection coefﬁcients independently. However, the variational derivation does not reveal the actual relationship between these two. We can rectify this logical gap by augmenting the Lagrangian in (A1.25) ij with Lagrange multipliers k to incorporate the required constraints. Deﬁning the ij 1 unique entries yij WD y , we furnish the augmented Lagrangian LQ as the following function of .4/16 variables (without assuming any symmetry): Q ij k ij ij k L xI y ; ij ; kI y k; ij l h i ij k k l k l k WD y kij ij k lk ij C ik lj 1 C ij k ykl y C y y ; k ij 2 jli lij ij l

L.Q / ˇ ij D ij ij D ij ij D ij ˇy g .x/; y @kg ; .x/; ˇ 8 9 k 8k 9k ˇ

@LQ The EulerÐLagrange equation ij 0 D 0 yield the Christoffel symbols in terms @. k / j:: of metric tensor components. 580 Appendix 1 Variational Derivation of Differential Equations

Now, the boundary term in (A1.29ii) is related to the trace of the exterior curvature of the hypersurface @D4. To show this, consider the projection operator of p. 191, viz., Pi i i j .x/ WD ı j ".n/ n .x/nj .x/: We deﬁne the extended extrinsic curvature [56, 126]by 1 .x/ WD PkPl .r n Crn /; (A1.31i) ij 2 i j k l l k i j K .u/ D @ @ ij ..u// : (A1.31ii)

i (Here, the extrinsic curvature, K.u/, was deﬁned in (1.234). The vector was deﬁned in the same section.) We can derive the trace from (A1.31ii)as

i i i i .::/ Dri n rni : (A1.32)

i The variation of i .::/ is denoted by (Caution: Note that " is different from ".n/.): i i O i .::/ i .::/ DW "h.::/ : (A1.33) Now, (A1.32) yields two expressions: i i .::/ D @ ni C nj ; (A1.34i) i i ji k i .::/ D gij .::/ @ n n : (A1.34ii) i j i ij k

For a ﬁxed boundary @D4, the variations of ni are exactlyn o zero. In consideration of k the boundary term in (A1.29ii), only the variation of ij is allowed. The variation i of gij .::/ and n .::/ is set to zero for this boundary term. Therefore, (A1.34i,ii) provide

i j "h.::/ D 0 C "h ij .::/n .::/; (A1.35i) h i ij k "h.::/ D g .::/ 0 "h ij .::/nk .::/ : (A1.35ii)

Adding the two equations above, we obtain h i jk i ij k 2h.::/ D g .::/h ij .::/ g .::/h ij .::/ nk.::/: (A1.36) Appendix 1 Variational Derivation of Differential Equations 581

Therefore, the boundary term in (A1.29ii) is furnished by Z h i Z ij k ki j 3 3 g .x/h ki.x/ g .x/h ki.x/ nj d v D2 h.x/ d v: (A1.37) @D4 @D4 A popular way to write the above is to express it as Z Z ij k ki j 3 i 3 g .x/ı g .x/ı nj d v D2 ı i .x/ d v: (A1.38) @D4 ki ki @D4 (We have tacitly assumed here that ıgij .x/ 0 on the boundary.)3 Example A1.5. The invariant Lagrangians (A1.25)and(A1.30) contain second- order derivatives of the metric tensor components. In general, for Lagrangians involving second-order derivatives of the metric, we would have obtained fourth- order field equations as thep corresponding EulerÐLagrange equations. However, for the Lagrangian density g.x/R.x/, the higher order terms are transformable into a boundary term in (A1.29ii) and thus do not contribute to the field equations. Einstein investigated a Lagrangian [84] which has only first-order derivatives of the metric tensor components. It is furnished by h i p L./ WD yyij l k l k ik lj lk ij ij kl Y Fij Y m ; p Y ij WD yyij ; 8 9 L./ˇ < = k ˇ ij D ij k D y g .x/; ij : ; ij p l k l k D g.x/ gij .x/ : (A1.39) ik lj lk ij p Note that the above Lagrangian is only a part of g.x/ R.x/, and it is not a scalar density (of weight C1). We can compute the partial derivatives as L @ ./ˇ p p ij ˇ ij D ij ij D ij @Y Y gg ;Y k @k. gg / l k l k D ; lk ij ik lj L @ ./ k l k ˇ D ı : (A1.40) ij ˇ i jl ij @Y k ::

3In the metric variation approach, it is assumed that the connection is the metric connection and that only the metric is to be varied. In this scenario, the boundary term is much more complicated.R The generalR case (i.e., not assuming that ıgijR.x/ 0 on the boundary) yields 2 ı i .x/ d3v C r Pij .x/nk ıg .x/ d3v gik.x/r nj ıg .x/ d3v for the @D4 i @D4 i jk @D4 i jk boundary term. 582 Appendix 1 Variational Derivation of Differential Equations

(Consult [84].) The corresponding variational equations (A1.20i,ii) yield " # L L @ ./ˇ d @ ./ ˇ ˇ DRij .x/ D 0; (A1.41i) @Y ij :: dxk ij ˇ @Y k :: Z l k hkj .x/ hij .x/ n d3x D 0: (A1.41ii) jl ij k @D4

Thus, (A1.41i,ii) provide the gravitational ﬁeld equations and variationally admissible boundary conditions outside material sources. Example A1.6. We shall now derive the ArnowittÐDeserÐMisner (ADM) action integral [7], [184]. Assuming g44.x/ < 0, we express the metric tensor in the following equations:

˛ ˛ 4 ˇ ˇ 4 g::.x/ D g˛ˇ .x/ dx C N .x/ dx ˝ dx C N .x/ dx ŒN.x/2 x4 x4 " # d ˝ d I g˛ˇ g N gij D 2 ; g N g N N N " # g˛ˇ N 2N ˛N ˇ N 2N gij D I N 2N N 2 q q det gij D N det g˛ˇ : (A1.42)

Suppose that the spaceÐtime locally admits a one-parameter family of three- dimensional spacelike hypersurfaces. On the spacelike hypersurface characterized by x4 D T , the intrinsic metric is furnished as

˛ ˇ g::.x;T/D g˛ˇ .x;T/dx ˝ dx ˛ ˇ DW g˛ˇ .x;T/dx ˝ dx : (A1.43)

There is a slight notational difference between this equation and (1.223) because we choose here one member of the inﬁnitely many hypersurfaces. We show the ADM decomposition schematically in Fig. A1.3. Gauss’ equation (1.242i) for a hypersurface yields in this example, with the ˛ ˛ˇ (usual) notation A ./ WD g ./Aˇ./, Appendix 1 Variational Derivation of Differential Equations 583

T2 – Na(x) dx4 T 1 4 x =T2 x4 (x+Δx, T ) 2 x4

x 3 (x, T ) 1 1 x x2 4 4 x =T1

Fig. A1.3 Two representative spacelike hypersurfaces in an ADM decomposition of spaceÐtime

R.x;T/D K .x;T/K.x;T/ K.x;T/K.x;T/C R.x;T/;

R .x;T/D R .x;T/C K./ K ./ K ./ K ./;

ij ˛ˇ ˛4 R.x;T/D R ji.x;T/D R ˇ˛.x;T/C 2R 4˛.x;T/ h i 2 ˛ ˇ ˛ ˛4 D R.x;T/C K ˇ./ K ˛./ K ˛./ C 2R 4˛./: (A1.44)

j The last term in (A1.44) can be expressed as a divergence term analogous to rj A [184]. Therefore, the Hilbert action integral goes over into Z Z h i 2 p 4 ˛ ˇ ˛ R.x/ g.x/d x D R.x;T/C K ˇ./ K ˛./ K ˛./

D4 D4 p N.x;T/ g.x;T/ d3x dT C (boundary term): (A1.45) The above action integral is useful in several approaches to quantum gravity [10, 199, 221, 246]. Finally, we would like to make the following two mathematical comments. Firstly, consider a nonconstant Lagrangian density function L yij I yij k such that the action functional is given by Z L 4 J.F/ WD yij I yij k j:: d x;

yij D ij ı F.x/ D gij .x/; 0 yij k D ij k ı F .x/ D @k gij : (A1.46) 584 Appendix 1 Variational Derivation of Differential Equations

It can be rigorously proved [171]thatsuch an integral cannot be tensorially invariant. That is why, in Example A1.5, we had to deal with a noninvariant action integral. The second comment is about a Lagrangian scalar density of the type L yij I yij k I yij k l . It can be rigorously proved [170,171], that in a four-dimensional manifold, the most general Lagrangian density (made up of curvature invariants), which yields second-order EulerÐLagrange equations, must be of the form p p L./j:: WD c.1/ g.x/R.x/ C c.2/ g.x/ p ij k l mn C c.3/ " R ij .x/ Rmnkl.x/ C c.4/ g.x/ h i 2 j i kl ij .R.x// 4R i .x/ R j .x/ C R ij .x/ R kl.x/ : (A1.47)

2 Here, c.1/;c.2/;c.3/;c.4/ are four arbitrary constants with just one constraint Œc.1/ C 2 2 Œc.3/ C Œc.4/ >0. The ﬁrst term is simply proportional to the Einstein-Hilbert Lagrangian density, giving rise to the equations of motion of general relativity (2.161i). The second term gives rise to a cosmological constant (proportional to c.2/) in the equations of motion (see (2.158)). The third term has been shown to yield trivial equations of motion in four dimensions [157]. Finally, each term in the last expression in (A1.47) produces fourth-order equations.However,their combination yields second-order equations and, in four dimensions, its integral is simply a topological number proportional to the Euler characteristic. This expression is commonly referred to as the Gauss-Bonnet term. Outside this appendix, we shall refrain from adding subscripts .0/ to critical .rCs/ functions .0/ .x/ satisfying the EulerÐLagrange equations (A1.20i). Appendix 2 Partial Differential Equations

In general relativity, a considerable amount of effort is expended in attempting to solve partial differential equations. Therefore, we are including here an extremely brief review of the subject. (For extensive study, we suggest [43, 77, 94].) Consider the simplest partial differential equation known to mankind:

u D U.x;y/; .x;y/ 2 D R2 I

@U.x; y/ @ u D D 0: (A2.1) x @x

The general solution of equation above is furnished by

u D U.x;y/ D f.y/; y1

Here, f.y/ is an arbitrary differentiable function. Equation (A2.2) comprises of inﬁnitely many solutions, and it is aptly called the most general solution of the partial differential equation (p.d.e.) (A2.1). Remarks. (i) The function f.y/ in (A2.2) may be misconstrued as a function of a single variable. Strictly speaking, it is a shortcut notation for a function F.c;y/of two variables such that the ﬁrst variable is restricted to a constant c. (ii) The function f.y/ can be chosen to be discontinuous in the interval .y1;y2/, and still it will satisfy the p.d.e. (A2.1) exactly! (iii) In case the domain of validity D R2 in (A2.1)isnonconvex,thesetof solutions in (A2.2)isnot the most general.(See[112]forcounter-examples.)

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 585 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 586 Appendix 2 Partial Differential Equations

Now, let us consider the two-dimensional (one space and one time) wave equation given by the second order, linear p.d.e.: w D W.x;t/; .x;t/ 2 D R2;

.x; 0/ D for x1 coordinate system u D x t; v D x C t.(Compare with the Example 2.1.17.) The p.d.e. in (A2.3) goes over into w D W.b u; v/ WD W.x;y/; .u; v/ 2 Db R2 I

@2W.b u; v/ D 0: (A2.4) @u@v The most general solution is provided by

w D W.b u; v/ D f.u/ C g.v/; where f; g 2 C 2.Db R2I R/:

However, we notice that the condition of C 2 differentiability on f.u/ can be completely relaxed. Discontinuous functions f.u/ can yield exact solutions W.b u; v/ D @2bW.u; v/ f.u/Cg.v/ for the p.d.e. (A2.4), but not for the p.d.e. @v @u . Physically speaking, such solutions yield shock waves not revealed in the usual x t coordinate system. However, the deﬁnition of a solution can be generalized to a weak solution [43], which can extract rigorously discontinuous solutions in any coordinate system. Instead of the p.d.e. (A2.3), a weak solution has to satisfy the integral condition: Z

W.x; t/.@x @xf @t @t f/dx dt D 0; (A2.5) D for every C 1-function f with compact support in D. (Such functions are called distributions [270].) The second-order p.d.e. (A2.3) is exactly equivalent to the following linear, ﬁrst order system:

@xw D Px.x; t/;

@t w D Pt .x; t/;

@xPx @t Pt D 0;

@t Px @xPt D 0: (A2.6) Appendix 2 Partial Differential Equations 587

The last of the above equations is the integrability condition. (We have already discussed a system of .N 1/, ﬁrst-order, linear p.d.e.s in (1.236).) In a general system of p.d.e.s in D RN , the highest derivatives that occur determine the order of the system. In a nonlinear system, the highest power (or exponent) of highest derivatives, is called the degree of the system. An N -dimensional ﬁrst-order p.d.e. is called a linear equation if it has the form:

XN i N .x/ @i w C .x/w D h.x/; x 2 D R : (A2.7) iD1

Here, i .x/, .x/,andh.x/ are prescribed continuous functions. (For a homogeneous p.d.e., h.x/ 0.) An N -dimensional ﬁrst-order equation is called a semilinear p.d.e. provided it is given by XN i .x/ @i w C h.w;x/D 0: (A2.8) iD1 A ﬁrst-order p.d.e. is called quasilinear p.d.e. if it has the form:

XN i .w;x/@i w C h.w;x/D 0: (A2.9) iD1 A second-order p.d.e. in an N -dimensional domain is called a linear p.d.e. provided it is furnished by

XN XN XN ij i .x/ @i @j w C .x/ @i w C .x/w D h.x/: (A2.10) iD1 j D1 iD1

(The linear equation (A2.10) is homogeneous provided h.x/ 0.) A second-order p.d.e. is called a semilinear p.d.e. provided it is expressible as

XN XN ij .x/ @i @j w C h.@i w; w;x/D 0: (A2.11) iD1 j D1

A second-order p.d.e. is called a quasilinear p.d.e. if it has the form:

XN XN ij .@kw;x/ @i @j w C h.@i w; w;x/D 0: (A2.12) iD1 j D1

A system of p.d.e.s in an N -dimensional domain is a collection of coupled p.d.e.s. 588 Appendix 2 Partial Differential Equations

System of p.d.e.s

Linear system Non−linear system

Homogeneous Non−homogeneous Disguised Quasi−linear Non−linear linear of degree ≥ 2

Semi−linear

Fig. A2.1 Classiﬁcation diagram of p.d.e.s

We classify systems of p.d.e.s in an N -dimensional domain in Fig. A2.1. We shall now sketch very briefly some of the solution procedures for each of the type of systems mentioned in the figure. Linear systems with constant coefficients are soluble usually by employing Fourier series, Fourier integrals, or Laplace integrals.Q Special classes of linear p.d.e.s N i admit separable solutions of the type W.x/ D iD1 W.i/.x /. Consider a single quasilinear first-order p.d.e. (A2.9). Lagrange’s solution method involves .N C 1/ characteristic ordinary differential equations (o.d.e.s):

dx1 dx2 dxN dw D DD D D dt; 1.w;x/ 2.w;x/ N .w;x/ h.w;x/

X i W d .t/ i d .t/ or, D .w;x/j:: ; Dh.w;x/j:: ; dt dt W.t/ WD WŒX .t/ WŒX 1.t/; : : : ; X N .t/: (A2.13)

The general solution of (A2.13) consists of .N C 1/ arbitrary constants. Alterna- tively, the solution curve for (A2.13) may be furnished by the intersections of the following differentiable hypersurfaces:

.i/.xI w/ D c.i/ D const.; @ .1/;:::;.N / ¤ 0: (A2.14) @.x1;:::;xN /

1 N Consider an arbitrary nonconstant function F 2 C .D./ R I R/.Themost general solution of the quasilinear p.d.e. in (A2.9)isimplicitly provided by F .1/.xI w/;:::;.N /.xI w/ D c D const. (A2.15) Appendix 2 Partial Differential Equations 589

Remark. Lagrange’s method is valid for a linear or a semilinear p.d.e. also. Example A2.1. Consider the following ﬁrst-order, quasilinear p.d.e.:

w D W.x/; x 2 D R2 I

3 w.@1w C @2w/ .w/ D 0:

The characteristic o.d.e.s (A2.13) reduce to

dX 1.t/ dX 2.t/ dW.t/ D W.t/ D ; D ŒW.t/3 ; dt dt dt dX 1.t/ dW.t/ ŒW.t/2 0: dt dt Solving the above equations, we obtain d X 1.t/ X 2.t/ 0; dt 1 2 .1/.xI w/ WD x x D c.1/; 1 1 .2/.xI w/ WD x C .w/ D c.2/; @ .1/;.2/ D 1: @.x1;x2/ Therefore, by (A2.14), the most general solution of the p.d.e. is implicitly provided by F x1 x2;x1 C .w/1 D c D const.

(However, there is an additional singular solution, W.x1;x2/ 0.) In case @F ..1/;.2// ¤ 0, the above yields, by the implicit function theorem [32], @.2/

x1 C .w/1 D g.x1 x2I c/;

1 or, w D W.x1;x2/ D g.x1 x2I c/ x1 ;

3 3 0 0 3 or, w.@1w C @2w/ w D Œ Œ.g C 1/ C g Œ 0:

Here, g is an arbitrary differentiable function. The solution above furnishes a general class of inﬁnitely many explicit solutions. (The domain of validity D must avoid the curve given by g.x1 x2I c/ x1 D 0.) The quasilinear p.d.e. of this example is a disguised linear equation.Bythe transformation of the variable s WD .w/1, the p.d.e. is transformed into the linear equation:

@1s C @2s 1 D 0: 590 Appendix 2 Partial Differential Equations

The characteristic o.d.e.s are

dX 1.t/ dX 2.t/ dS.t/ D D 1 D : dt dt dt The most general solution is implicitly provided by (A2.15)as F x1 x2;x1 s D F x1 x2;x1 C .w/1 D c:

Here, F is an arbitrary, nonconstant function of class C 1. Now we consider the nonlinear ﬁrst-order p.d.e.: 1 N G x ;:::;x I wI p.1/;:::;p.N / D 0; @W .x/ (A2.16) p WD @ w D ;D RN R RN : .i/j:: i @xi Here, G is assumed to be a nonconstant differentiable function such that at least one n of p.i/ occurs as Œp.i/ .i/ ;n.i/ 2. The associated .2N C 1/ characteristic o.d.e.s are furnished by [43, 94]

dX i .t/ @G./ D ˇ ; dt @p.i/ ˇ:: P d i .t/ @G./ˇ @G./ D ˇ p.i/ ˇ ; dt @xi :: @w ˇ:: dW.t/ XN @G./ D p ˇ : (A2.17) dt .i/ @p ˇ iD1 .i/ :: Note that along each of the characteristic curves,

dŒG./ @G./ dX i .t/ @G./ dP .t/ ˇ D ˇ C ˇ i ˇ i ˇ dt :: @x :: dt @p.i/ ˇ:: dt @G./ dW.t/ C ˇ 0: (A2.18) @w ˇ:: dt

Therefore, the solution hypersurface for the p.d.e. (A2.16) is spanned by the congruence of characteristic curves. Example A2.2. Consider the (truly nonlinear) p.d.e.: 1 2 G x ;x I wI p.1/;p.2/ WD p.1/ p.2/ w D 0;

@1w @2w D w;

W.0;x2/ WD .x2/2: Appendix 2 Partial Differential Equations 591

(This is an initial value problem of a nonlinear p.d.e.) The characteristic o.d.e.s (A2.17) reduce to

dX 1.t/ dX 2.t/ D P .t/; D P .t/; dt .2/ dt .1/

dP .t/ dP .t/ dW.t/ .1/ D P .t/; .2/ D P .t/; D 2P .t/ P .t/: dt .1/ dt .2/ dt .1/ .2/ The general solutions of the system of o.d.e.s are given by

t t P.1/.t/ D c.1/e ; P.2/.t/ D c.2/e ;

1 t 2 t X .t/ D c.2/e C c.3/; X .t/ D c.1/e C c.4/;

2t W.t/ D c.1/c.2/e C c.5/:

Here, c.1/;:::;c.5/ are arbitrary constants of integration with c.2/ ¤ 0. Inserting 2 initial values X 1.0/ D 0; W.0/ D W X 1.0/; X 2.0/ D X 2.0/ , etc., we ﬁnally obtain 1 2 t 1 t X .t/; X .t/ D c.2/ .e 1/; .4/ c.2/ .e C 1/ ; 1 2 2 2 X .t/ C 4X .t/ W X 1.t/; X 2.t/ D W.t/ D .c =2/et D : .2/ 4

From equation above, we conclude that x1 C 4x2 2 w D W.x1;x2/ D 4 is the unique solution of the present p.d.e. under the imposed initial values. Example A2.3. Consider a domain of pseudo-Riemannian spaceÐtime and the relativistic HamiltonÐJacobi equation (for m>0) furnished by 1 2 3 4 1 ij 2 G x ;x ;x ;x I sI p1;p2;p3;p4 WD .2m/ g .x/pi pj C m D 0; (A2.19i) @S.x/ @S.x/ gij .x/ C m2 D 0 (A2.19ii) @xi @xj or ij 2 g .x/ ri S rj S C m D 0: (A2.19iii) Compare (A2.19i) with the constraint of the mass shell in (2.33). (Caution:Herethe parameter s is not the proper time parameter of (2.146iÐiii).) 592 Appendix 2 Partial Differential Equations

From (A2.19i), we deduce that

@G./ @G./ D .m/1 gij .x/ p ; D .2m/1 @ gkj p p ; j i i k j @pi @x

@G./ 0: @s Therefore, the characteristic equations (A2.17) yield

dX i .t/ D .m/1 gij .x/ p ; dt j j:: dP .t/ i D.2m/1 @ gjk p p ; dt i j kj:: S d .t/ 1 ij D m g .x/ p p j Dm: (A2.20) dt i j :: Identifying the parameter t with the proper time parameter, we have just re-derived the relativistic canonical equations (2.146iÐiii) for a timelike geodesic in spaceÐ time. Example A2.4. Consider the (truly) nonlinear, ﬁrst-order p.d.e.: 1 2 1 G x ;x I wI p.1/;p.2/ WD p.1/ p.2/ x D 0;

or,

1 1 2 2 @1w @2w x D 0; x D .x ;x/ 2 D R :

We shall solve the p.d.e. by a different technique. Note that the nonzero Hessian is furnished by @2G./ det D1 ¤ 0: @p.i/@p.j /

We make a Legendre transformation (as we have done in (2.138)), by

@W .x/ p D ; .i/ @xi @ @W .x/ xi p W.x/ D ıi p 0; @xk .i/ k .i/ @xk i 1 2 p.1/;p.2/ WD x p.i/ W.x ;x /; Appendix 2 Partial Differential Equations 593

@./ D xi ; @p.i/

1 2 i W.x ;x / D x p.i/ .p.1/;p.2//:

The p.d.e. under consideration reduces to @./ D p.1/ p.2/: @p.1/ The most general solution of the above is provided by 2 .p.1/;p.2// D .1=2/ p.1/ p.2/ C h p.2/ :

Here, h.p.2// is an arbitrary differentiable function. Thus, the most general solution of the original p.d.e. is implicitly given by

1 @./ x D D p.1/ p.2/; @p.1/ 2 @./ 2 0 x D D .1=2/ p.1/ C h p.2/ ; @p.2/

1 2 @./ @./ W.x ;x / D p.1/ C p.2/ ./ @p.1/ @p.2/ 2 0 D p.1/ p.2/ h p.2/ C p.2/ h p.2/ : p 1 2 1 2 Choosing h p.2/ 0, we obtain a particular solution W.x ;x / D x 2x for x2 >0. Remark. The technique of Legendre transformations is often employed in attempts at the canonical quantization of gravitational ﬁelds [10, 221, 246]. We shall now investigate second-order p.d.e.s in R2. We start with a second-order, semilinear p.d.e. (in a two-dimensional domain) expressed as

X2 X2 ij .x/ @i @j w C h.xI wI @i w/ D 0; iD1 j D1

11 12 22 .x/ @1@1w C 2 .x/ @1@2w C .x/ @2@2w C h.xI wI @i w/ D 0; 2 2 2 11.x/ C 12.x/ C 22.x/ >0 x 2 D R2: (A2.21) 594 Appendix 2 Partial Differential Equations

The coefﬁcients ij .x/ are of class C 1. The classiﬁcation of the p.d.e. (A2.21)is based on the determinant .x/ WD det ij .x/ : (A2.22)

The p.d.e. (A2.21)issaidtobe I. Hyperbolic p.d.e. in D, provided .x/ < 0 II. Elliptic p.d.e. in D, provided .x/ > 0 III. Parabolic p.d.e. in D, provided .x/ D 0 Example A2.5. Consider the Tricomi’s p.d.e. [43]:

1 @1@1w C x @2@2w D 0:

The p.d.e. is elliptic in the half-plane x1 >0, and it is hyperbolic in the other half-plane x1 <0Š Now, we would like to simplify the p.d.e. (A2.21) for the purpose of solving it. We explore a possible coordinate transformation of class C 2 and (1.107i) to derive the following equations:

bx i D Xbi .x/ D Xbi .x1;x2/; (A2.23i)

w D W.x/ D W.b bx/; (A2.23ii)

X2 X2 @Xbi .x/ @Xbj .x/ bij .bx/ D kl.x/; (A2.23iii) @xk @xl kD1 lD1 @.bx1;bx2/ 2 .b bx/ D .x/; (A2.23iv) @.x1;x2/ h i sgn .b x/O D sgn Œ.x/ : (A2.23v)

By (A2.23ii)and(A2.23iv), it is clear that the classiﬁcation of a p.d.e. remains intact under a coordinate transformation. The p.d.e. (A2.21) goes over into b11 b b b12 b b b22 b b .bx/@1@1w C 2 .bx/@1@2w C .bx/@2@2w CD0; (A2.24i) X2 X2 @Xb1./ @Xb1./ b11.bx/ D kl.x/; (A2.24ii) @xk @xl kD1 lD1 Appendix 2 Partial Differential Equations 595

X2 X2 @Xb2./ @Xb2./ b22.bx/ D kl.x/; (A2.24iii) @xk @xl kD1 lD1

X2 X2 @Xb1./ @Xb2./ b12.bx/ D kl.x/ b21.bx/: (A2.24iv) @xk @xl kD1 lD1 The characteristic surface .x/ over D R2 of p.d.e. (A2.21) is deﬁned by the ﬁrst-order, nonlinear p.d.e. of degree two:

X2 X2 @.x/ @.x/ ij .x/ D 0: (A2.25) @xi @xj iD1 j D1

In case both the coordinate functions Xb1.x/ and Xb2.x/ satisfy the characteristic p.d.e. (A2.25), the original p.d.e. (A2.21) reduces to

b12 b b .bx/ @1@2w CD0; @2W.b bx1;bx2/ or, CD0: (A2.26) @bx1@bx2 The normal forms of the various p.d.e.s are listed below:

I(i). Hyperbolic p.d.e. :

@2W.b bx1;bx2/ Cbh./ D 0: (A2.27i) @bx1@bx2

I(ii). Hyperbolic p.d.e. :

@2W.b bx1;bx2/ @2W.b bx1;bx2/ Cbh./ D 0: (A2.27ii) .@bx1/2 .@bx2/2

II. Elliptic p.d.e. :

@2W.b bx1;bx2/ @2W.b bx1;bx2/ C Cbh./ D 0: (A2.27iii) .@bx1/2 .@bx2/2 596 Appendix 2 Partial Differential Equations

III. Parabolic p.d.e. :

2 b b1 b2 @ W.x ; x / b 1 2 b b C h bx ;bx I wI @1w; @2w D 0: (A2.27iv) .@bx1/2

Example A2.6. Consider the homogeneous, linear, second-order p.d.e.: 2 1 @1@1w c @2@2w C @1w c @2w D 0; .x/ D c2 <0: (A2.28)

The parameter c is assumed to be nonzero, and the p.d.e. (A2.28) is obviously hyperbolic. The characteristic surface of p.d.e. (A2.28) is governed by the nonlinear ﬁrst- order p.d.e.: 2 2 2 .@1/ c .@2/ D 0; (A2.29i) h i 1 2 2 2 2 G x ;x I I p.1/;p.2/ WD .1=2/ p.1/ c p.2/ D 0: (A2.29ii)

The characteristic curves for equations above (which are bicharacteristic curves for the p.d.e. (A2.28)) are furnished by (A2.17)as

X 1 X 2 d .t/ @G./ d .t/ 2 D ˇ D p.1/j:: ; Dc p.2/j:: ; dt @p.1/ ˇ:: dt

dP .t/ dP .t/ .1/ D .2/ 0: (A2.30) dt dt The general solutions of (A2.30)and(A2.29ii)aregivenby P P 2 2 2 1 .1/.t/ D c.1/; .2/.t/ D c.2/;c.1/ D c c.2/;c.1/ D˙c c.2/; 1 1 2 X .t/ Dc c.2/t C c.3/ D c X .t/ Cbc.3/;

1 1 2 or else, X .t/ D c c.2/t C c.3/ Dc X .t/ Cbc.4/;

1 2 1 2 thus, .1/.x/ WD x cx D bc.3/;.2/.x/ WD x C cx D bc.4/: (A2.31) Making a coordinate transformation

1 1 2 2 1 2 bx D .1/.x/ D x cx ; bx D .2/.x/ D x C cx ;

w D W.x/ D W.b bx/; Appendix 2 Partial Differential Equations 597 the p.d.e. (A2.28) reduces to h i b b @1 @2w C .1=2/w D 0: (A2.32)

The above second-order p.d.e. is equivalent to a pair of ﬁrst-order equations

b @2w C .1=2/w D bg.bx/; (A2.33) b @1bg D 0:

(Compare the above equations with (A2.6).) The most general solution of (A2.33) b2 0 and (A2.28), putting bg.bx2/ D ex =2g .bx2/, is provided by b2 b2 w D W.b bx/ D ex =2g bx2 C e .x =2/ f bx1

1 2 1 2 D e.x Ccx /=2g x1 C cx2 C e.x Ccx /=2 f x1 cx2 D W.x/: (A2.34)

Here, f and g are of class C 2 and otherwise arbitrary. Consider now a related nonhomogeneous, linear p.d.e.: 2 1 @1@1w c @2@2w C @1w c @2w D 4: (A2.35)

A particular solution of this equation is given by 1 2 w.p/ D 2 x cx :

Therefore, denoting solution (A2.34) of the homogeneous equation by w.h/, the most general solution of (A2.35) is furnished by the superposition: 1 2 .x1Ccx2/=2 1 2 w D w.p/ C w.h/ D 2 x cx C e g x C cx

1 2 Ce.x Ccx /=2 f x1 cx2 : Example A2.7. Consider the elliptic Liouville equation [191]:

2 2w 2 w r w D @1@1w C @2@2w D 4e ;x2 D R : (A2.36)

The above is a semilinear, second-order, elliptic p.d.e. in the normal form. The corresponding characteristic surface,asgivenby(A2.25), reduces to the p.d.e. 2 2 .@1/ C .@2/ D 0: (A2.37) 598 Appendix 2 Partial Differential Equations

Obviously, only real-valued solutions1 of (A2.37) are provided by constant-valued functions .x/! Therefore, nondegenerate, characteristic curves, analogous to the preceding hyperbolic example, do not exist in an elliptic case. However, there is an interesting device for treating elliptic equations by complex conjugate coordinates (which are formally analogous to the characteristic coordinates of hyperbolic cases). (See [191].) Let us introduce complex conjugate coordinates and complex derivatives by the following equations:

WD x1 C ix2; WD x1 ix2; x1 D Re ./ D .1=2/ C ;x2 D Im ./ D .1=2i/ ; Fb ; WD F x1;x2 ; @ @ @ @ Fb D Fb ; WD .1=2/ i F x1;x2 ; @ @x1 @x2 b @ b @ @ 1 2 @NF D F ; WD .1=2/ C i F x ;x ; @ @x1 @x2 " # 2b @ 2 @ 2 @ F ; F r2F D C F x1;x2 D 4 : @x1 @x2 @@ (A2.38)

A holomorphic function f./and a conjugate holomorphic function g./ satisfy respectively the p.d.e.s: @f . / @ @ D .1=2/ C i Re f x1 C ix2 C iIm f x1 C ix2 D 0; @ @x1 @x2 (A2.39i) h i @g./ @ @ D .1=2/ i Re g.x1 C ix2/ iIm g.x1 C ix2/ D 0: @ @x1 @x2 (A2.39ii) (The complex equation (A2.39i) is exactly equivalent to the Cauchy-Riemann equations.)

1A nonlinear (real) differential equation of degree greater than one may or may not admit any 2 C 2 C D solution. The p.d.e. .@1/ .@2/ coshŒ.x/ 0 does not admitp any real-valued solution 2 C 2 C D function .x/. However, the p.d.e.p .@1/ .@2/ coshŒ.x/ 2 1 admits a single, particular solution .x1;x2/ D 2. Appendix 2 Partial Differential Equations 599 Consider the real-valued harmonic function h x1;x2 D bh.; / satisfying

2 h r h D @1@1h C @2@2h D 0; b or, @ @Nh D 0: (A2.40)

It can be proved [191]thatthe most general solution of (A2.40)isgivenby h i bh ; D .1=2/ f./C f./ ; h x1;x2 D Re f x1 C ix2 : (A2.41)

Here, f./is an arbitrary holomorphic function in the domain of consideration. Now let us go back to the Liouville equation (A2.36)again.The most general solution of this equation is provided by [191] jf 0./j w D W x1;x2 D Wb ; D log ; 1 jf./j2 ˚ D WD 2 C W f 0./ ¤ 0; jf./j <1 : (A2.42)

Here, f 0./ ¤ 0,andjf./j <1, but the holomorphic function f./ is otherwise arbitrary. Now, we consider second-order, semilinear p.d.e.s in an N -dimensional domain. Such p.d.e.s need not be tensor ﬁeld equations. (That is why we suspended the summation convention in preceding discussions!) Let us now revert back to the usual summation convention for tensorial, as well as nontensorial p.d.e.s. Recall that the semilinear, second-order p.d.e. (A2.11), reinstating the summation convention, can be expressed as

ij N .x/ @i @j w C h.xI wI @i w/ D 0 I x 2 D R : (A2.43)

Here, the coefﬁcients ji.x/ ij .x/ are continuous functions such that at least ij one of them is nonzero. Therefore, the symmetric matrix .x0/ has N real (usual) eigenvalues so that at least one of them is nonzero. The eigenvalues can be always arranged in the order:

.1/ .x0/>0;:::;.p/ .x0/>0I .pC1/ .x0/<0;:::;.pCn/ .x0/<0I

.pCnC1/ .x0/ DD.pCnC/ .x0/ D 0: (A2.44) ij Here, p C n C D N; r WD p C n is the rank of the matrix .x 0/ ,and ij D N r is the nullity of the matrix. Thus, >0if and only if the matrix .x0/ 600 Appendix 2 Partial Differential Equations is singular. (Without loss of generality, we can always arrange p n 0. Compare with the metric equation (1.90).) According to Sylvester’s law of inertia [177, 240], the numbers p; n; remain invariant under a (real) coordinate transformation of the type (A2.23iii)in(A2.43). Thus, it is logical to choose the classiﬁcation of the general semilinear p.d.e. (A2.43) according to the following criteria: I. Elliptic p.d.e. if D 0 D n, p>0 II. Hyperbolic p.d.e. if D 0, n D 1, p D N 1 III. Ultrahyperbolic p.d.e. if D 0, 10 Note that the .N 1/-dimensional characteristic hypersurface of equation (A2.43) is governed by the ﬁrst-order, nonlinear p.d.e.:

ij .x/ @i @j D 0: (A2.45)

Example A2.8. Consider the following semilinear, second-order p.d.e.: h i 2w @ @ w C @ @ w @ @ w C .@ w/2 C .@ w/2 .@ w/2 D 0 I 1 1 2 2 3 3 1 w2 1 2 3

x 2 D R3 Ijwj <1: (A2.46)

In this case, N D 3; p D 2; n D 1; D 0; and r D 3. Therefore, the p.d.e. is hyperbolic in the domain of consideration. The corresponding characteristic surface is furnished by

2 2 2 .@1/ C .@2/ .@3/ D 0:

The bicharacteristic curves are (null ) straight lines in the three-dimensional domain. Now, we make the following transformation: ˇ ˇ ˇ ˇ 1 ˇ1 C wˇ g.w/ WD ln ˇ ˇ ; w D tanh.g/; jwj <1; 2 1 w W x1;x2;x3 WD tanh g x1;x2;x3 :

The p.d.e. (A2.46) reduces to the linear p.d.e.:

@1@1g C @2@2g @3@3g D 0: (A2.47)

Thus, the original p.d.e. (A2.46)isa disguised linear p.d.e. Appendix 2 Partial Differential Equations 601

A class of general solutions of (A2.47)and(A2.46) is provided by Z q 1 2 3 1 2 2 2 3 g x ;x ;x D f k1x C k2x C k1 C k2 x dk1 dk2; R2 2 3 Z q 1 2 3 4 1 2 2 2 3 5 W x ;x ;x D tanh f k1x C k2x C k1 C k2 x dk1 dk2 : R2

Here, f is a twice-differentiable function such that the integrals converge uniformly. The function f is otherwise arbitrary. (Uniform convergences are needed to commute differentiations and the integration [32].) Let us go back to the semilinear, second-order p.d.e. (A2.43). It is equivalent to the ﬁrst-order system:

@i w D p.i/;

@j p.i/ D @i p.j /; ij .x/ @i p.j / C h xI wI p.i/ D 0: (A2.48)

(Compare (A2.6)and(A2.33).) In general, a system of semilinear second-order p.d.e.s (and possibly another system of ﬁrst-order p.d.e.s) can be expressed equivalently as a single ﬁrst order system: i B AB .x/ @i w C hA.xI w/ D 0: (A2.49) Here, the summation convention is also carried on capital Roman indices which take values from f1;2;:::;dg. We construct the characteristic matrix [43] by the following: i Œ ./ ŒAB .x/ WD AB .x/ @i : (A2.50) dd

The characteristic hypersurface is furnished by the ﬁrst-order p.d.e.: i det Œ ./ D det AB .x/ @i D 0: (A2.51)

Example A2.9. Consider Maxwell’s equations for electromagnetic ﬁelds in ﬂat spaceÐtime. (See (2.54iÐiv).) With the notation w1; w2; w3 WD E1;E2;E3 ; w4; w5; w6 WD H 1;H2;H3 ; 602 Appendix 2 Partial Differential Equations six of the Maxwell’s equations can be expressed as

5 6 1 @3w @2w C @4w D 0; 6 4 2 @1w @3w C @4w D 0; 4 5 3 @2w @1w C @4w D 0; 2 3 4 @3w @2w @4w D 0; 3 1 5 @1w @3w @4w D 0; 1 2 6 @2w @1w @4w D 0: (A2.52) By (A2.50)and(A2.51), we derive 2 3 @ 0 0 0 @ @ 6 4 3 2 7 6 7 6 0@0 @ 0@ 7 6 4 3 1 7 6 7 6 7 6 00@4@2 @107 Œ D 6 7 ; (A2.53i) 6 7 66 6 0@3 @2 @40 07 6 7 6 7 4 @30@10@405

@2 @1000@4 h i 2 2 2 2 2 2 det Œ D.@4/ .@1/ C .@2/ C .@3/ .@4/ : (A2.53ii)

It is clear that for @4 ¤ 0, the system of p.d.e.s in (A2.52)ishyperbolic. (We have obtained classiﬁcations of p.d.e.s in general relativity on page 202.) Now, we shall very brieﬂy touch upon the topic of nonunique (or chaotic) solutions. Consider a system of o.d.e.s: X i d .t/ i D F .tI x/j j DX j ; dt x .t/ ˚ N D WD .tI x/ 2 R R Wjt t.0/j

(Compare the equation above with (1.75).) Suppose that the functions F i ./ are continuous over D WD D [ @D . However, the entries of the Jacobian matrix 2 3 @ F 1 @ F 1 @ F 1 6 1 2 N 7 6 7 6 : : : 7 4 : : : 5 N N N @1F @2F @N F are continuous in D,butnot continuous on @D . Then, the initial value problem Appendix 2 Partial Differential Equations 603

(1, 1)

(0, 0) t x=ε<0

Fig. A2.2 Graphs of nonunique solutions

X i .t/ X i i d ˇ i t.0/ D x.0/; ˇ D c.0/; dt tDt0 has nonunique (or chaotic) solutions2 [151]. Example A2.10. Consider the single o.d.e. :

dX .t/ p D F.tI x/j:: WD X .t/ ; dt

D D f.t; x/ 2 R R W 0

p The function F.tI x/ D x is continuous over D.However,@F . / D p1 is not @x 2 x continuous on @D . Therefore, the solution for the speciﬁc I.V.P.

X X d .t/ˇ .0/ D 0; ˇ D 0 dt tD0 has (inﬁnitely) many answers! We furnish these solutions explicitly in the following: ( 0 for t k<1I X .t/ WD .1=4/.t k/2 for 0

The functions X .t/ are shown graphically in the Fig. A2.2.

2Moreover,h i bifurcation points will satisfy simultaneous equations: F i .tI x/ D 0,and @F i ./ D det @xj 0.(See[138].) Appendix 3 Canonical Forms of Matrices

We shall start this appendix with some very simple models. Consider 2 2 matrices with real entries (or elements). (See [5].) 31 Example A3.1. Consider the symmetric matrix ŒS WD . The characteristic 13 polynomial is given by p./ WD det ŒS I D 2 6 C 8. Therefore, the usual eigenvalues are .1/ D 4 and .2/ D 2. The corresponding eigenvectors are " p # " p # 1= 2 1= 2 eE.1/ D p ; eE.2/ D p : 1= 2 1= 2

These column vectors are orthonormal in the usual Euclidean sense. We construct a matrix ŒP (with help of eigenvectors) as " p p # 1= 2 1= 2 ŒP WD p p : 1= 21=2

By the similarity transformation " # 40 ŒP 1 ŒS ŒP D ; 02 and the symmetric matrix is diagonalized. p 20 Example A3.2. Consider the symmetric matrix ŒS WD p . The usual 0 2 p eigenvalue .1/ D 2 has the multiplicity 2. Moreover, every nonzero 2 1 column vector is an eigenvector.

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 605 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 606 Appendix 3 Canonical Forms of Matrices 32 Example A3.3. Consider the matrix ŒM WD . The (usual) characteristic 21 2 polynomial is furnished by p./ D . C 1/ . The eigenvalue .1/ D1 is of t multiplicity 2. The eigenvectors are of the form ;t¤ 0. Therefore, there t is only one eigendirection and the matrix is nondiagonalizable. (This matrix has a 2 nonlinear or nonsimple elementary divisor E.2/./ D .C1/ according to (A3.3).) However, consider the matrix " # 11=2 ŒP WD : 11

By the similarity transformation " # " # " # 11 10 01 ŒP 1 ŒM ŒP D D C : 0 1 0 1 00

Thus, the matrix ŒM is reducible to an upper triangular form. Moreover, the triangular form is a diagonal matrix plus a nilpotent matrix. 0 1 Example A3.4. Consider the antisymmetric matrix ŒA WD . The character- 10 istic polynomial p./ D 2C1. Thus, there exist no real eigenvalues. Consequently, there are no (real) eigenvectors. However, extending into the (algebraically closed) complex ﬁeld, we have two complex-conjugate eigenvalues .1/ D i; .2/ Di D 11 .1/. Consider the complex matrix ŒP WD . The similarity transformation ii i0 ŒP 1 ŒA ŒP reduces ŒA into the complex diagonal matrix . 0 i

Example A3.5. Consider the 2 2matrix generated by the Lorentz metric (in 10 Example 1.3.2) ŒD WD d D . The Lorentz-invariant characteristic .i/.j / 0 1 polynomial of a matrix ŒM is deﬁned by p#./ WD detŒM D. The invariant # eigenvalues are furnished by the roots of p ./ D 0. (Compare with (1.213).) A 2a a C b symmetric matrix can be expressed as ŒS D . The corresponding a C b2b invariant eigenvalues are given by p#./ DŒ .a b/2 D 0. There exists one (real) invariant eigenvalue .1/ D .a b/ of multiplicity 2. There exists one t eigendirection .t ¤ 0/. The Lorentz-invariant separations (of (1.86)) are t 10 t provided by Œt; t 0. Thus, the eigendirection is along a double 0 1 t Appendix 3 Canonical Forms of Matrices 607 null vector. Moreover, this matrix is associated with nonsimple elementary divisor 2 E.2/./ D . a C b/ according to (A3.3). 01 Example A3.6. Consider the symmetric matrix ŒS WD . (This happens to be 10 one of the Pauli matrices.) The usual eigenvalues are given by .1/ D 1; .2/ D 1. The Lorentz-invariant eigenvalues are furnished by p#./ D.2 C 1/ D 0. Therefore, we have complex-conjugate invariant eigenvalues .1/ D i; .2/ Di D .1/. Therefore, a symmetric matrix, with real entries, can have complex, Lorentz- invariant eigenvalues! Now we shall investigate the canonical classiﬁcation of an N N matrix ŒM with real entries M.a/.b/. The (usual) characteristic polynomial is furnished by N N 1 p./ WD det M.a/.b/ ı.a/.b/ D ./ C c.1/ CCc.N / D 0; h i .a/.c/ .a/ or, det ı M.c/.b/ ı .b/ D p./ D 0:

The equation above admits 2s 0 complex roots and N 2s 0 real roots [21]. Now consider thesamerealN N matrix ŒM , together with an N N metric matrix ŒD of (1.90) (which is not positive deﬁnite). The corresponding invariant, characteristic polynomial equation is provided by # N # N 1 # p ./ WD det M.a/.b/ d.a/.b/ D ./ C c.1/ CCc.N / D 0; h i .a/.c/ .a/ or, det d M.c/.b/ ı .b/ D 0:

Suppose that the above equation admits 2s# >0complex roots and N 2s# >0 real roots. These invariant roots must differ from the roots of the usual characteristic equation p./ D 0. (Compare with Example A3.6.) However, the subsequent i general theorems involving any real N N matrix ŒM D ŒM j apply to both distinct cases p./ D 0 and p#./ D 0. In the case where there exist only real i i roots of detŒM j ıj D 0 as eigenvalues, the Jordan decomposition theorem is stated as follows: i Theorem A3.7. Let ŒM D ŒM j be an N N matrix with real entries and real i i eigenvalues as roots of detŒM j ıj D 0. Then the matrix can be transformed by a similarity transformation into the following block diagonal form: 608 Appendix 3 Canonical Forms of Matrices 2 3

6 A.1/ 0 7 6 7 6 7 6 A.2/ 7 6 : 7 6 :: 7 ŒM .J / D 6 :: 7 : (A3.1) 6 :: 7 6 :: 7 6 :: 7 4 : 5 0 A.k/

Here, A.l/ ,forl 2f1;:::;kg,isann.l/ n.l/ matrix given by 2 3 6 J .1/ 0 7 6 .l/ 7 6 7 6 .2/ 7 6 J 7 6 .l/ 7 6 :: 7 A.l/ WD 6 : : 7 ; (A3.2i) 6 :: 7 6 :: 7 6 :: 7 6 :: 7 4 5 0 .ql / J.l/

2 3 6 .l/ 100 7 6 7 h i 6 0 10 7 6 .l/ 7 .i/ 6 : : : : 7 J.l/ WD : : : : : ; (A3.2ii) 6 : : :: : 7 .i/ .i/ 6 : : : 7 n n 4 : : : 1 5 .l/ .l/ : 000.l/

Xql Xk .i/ i 2 f1;:::;ql g ; n.l/ DW n.l/; n.l/ D N: iD1 lD1 Proof of the above theorem is available in [133, 177].

.1/ .2/ .ql / Remarks. (i) It is usually assumed that n.l/ n.l/ h ni.l/ 1. .i/ .i/ (ii) In the case n.l/ D 1, the corresponding 1 1 matrix J.l/ has the single entry .l/. It stands for a diagonalh i element. .i/ (iii) Some authors write J.l/ as a lower triangular form. (iv) In relativity theory, after solving p#./ D 0, many authors use the inequalities in the reversed order as

.ql / .2/ .1/ n.l/ n.l/ n.l/ 1: Appendix 3 Canonical Forms of Matrices 609

(v) The Segre characteristic of the matrix ŒM .J / is denoted ([90, 210]) by the .1/ .2/ .1/ .2/ symbol Œ.n.1/;n.1/; : : :/; .n.2/;n.2/;:::/;:::.

The matrices ŒM .J / in (A3.1)and(A3.2i,ii) deﬁne a hierarchy of elementary divisors [113]. It is provided by the following string of equations:

.1/ .1/ n.1/ n.2/ E.N /./ WD .1/ .2/ ;

.2/ .2/ n.1/ n.2/ E.N 1/./ WD .1/ .2/ ;

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: : (A3.3) Example A3.8. We shall discuss an example from the general theory of relativity. Consider the orthonormal components t.a/.b/ WD T.a/.b/.x0/ D t.b/.a/ of the energyÐ 4 momentumÐstress tensor in (2.45) and (2.161ii) (at a particular event x0 2 D R ). The usual eigenvalues of the symmetric matrix t.a/.b/ are all real, and the matrix is always diagonalizable. However, the Lorentz invariant eigenvalues of t.a/.b/ are, physically speaking, much more relevant. Since detŒt.a/.b/ d.a/.b/ D 0 , .a/.c/ .a/ detŒd t.c/.b/ ı .b/ D 0, these invariant eigenvalues are identical with the .a/.c/ usual eigenvalues of the 4 4 matrix Œ WD d t.c/.b/ which need not be symmetric.(SeeExamplesA3.5 and A3.6.) Assuming that the invariant eigenvalues are all real, we classify the allowable types of canonical forms of Œ in the following: 2 3 I: Type-I.a/: .1/ 00 0 6 7 6 7 6 0.2/ 007 6 7 Œ.J / D 6 7 : (A3.4) 4 00.3/ 0 5

000.4/

Here, if .1/;.2/;.3/; .4/ are all distinct. The Segre characteristic in (A3.4)is speciﬁed by Œ1;1;1;1.

Type-I.b/:If.1/ D .2/ and .1/;.3/; .4/ are distinct. The Segre characteristic is speciﬁed by Œ.1; 1/; 1; 1.

Type-I.c/:If.1/ D .2/;.3/ D.4/ and .1/ ¤ .3/. The Segre characteristic is provided by Œ.1; 1/; .1; 1/.

Type-I.d/:If.1/ D .2/ D .3/ and .1/ ¤.4/. The Segre characteristic is furnished by Œ.1; 1; 1/; 1. 610 Appendix 3 Canonical Forms of Matrices

Type-I.e/:If.1/ D .2/ D .3/ D.4/. The Segre characteristic is speciﬁed by Œ.1; 1; 1; 1/.

2 3 II: Type-II.a/: .1/ 000 6 7 6 7 6 0.2/ 007 6 7 Œ.J / D 6 7 : (A3.5) 4 00.3/ 1 5

000.3/

If .1/;.2/,and.3/ are all distinct. The Segre characteristic is given by Œ1;1;2.

Type-II.b/:If.1/ D .2/ and .1/ ¤ .3/ in (A3.5). The Segre characteristic is provided by Œ.1; 1/; 2.

Type-II.c/: Here, .1/ ¤ .2/ and .2/ D .3/. The Segre characteristic is furnished by Œ1; .1; 2/.

Type-II.d/:If.1/ D .2/ D .3/. The Segre characteristic is speciﬁed by Œ.1; 1; 2/.

2 3 III: Type-III.a/: .1/ 100 6 7 6 7 6 0.1/ 007 6 7 Œ.J / D 6 7 : (A3.6) 4 00.2/ 1 5

000.2/

Here, we assume .1/ ¤ .2/. The Segre characteristic is Œ2; 2.

Type-III.b/:If.1/ D .2/ in (A3.6). The Segre characteristic is provided by Œ.2; 2/.

2 3 IV: Type-IV.a/: .1/ 000 6 7 6 7 6 0.2/ 107 6 7 Œ.J / D 6 7 : (A3.7) 4 00.2/ 1 5

000.2/

Here, we assume that .1/ ¤ .2/. The Segre characteristic is Œ1; 3. Appendix 3 Canonical Forms of Matrices 611

Type-IV.b/: Here, we assume that .1/ D .2/. The corresponding Segre characteristic is Œ.1; 3/. 2 3 V: .1/ 100 6 7 6 7 6 0.1/ 107 6 7 Œ.J / D 6 7 : (A3.8) 4 00.1/ 1 5

000.1/

The Segre characteristic is furnished by Œ4. The corresponding eigenvectors are quadrupole null vectors along a single null eigendirection. Since the signature of the Lorentz metric Œd.a/.b/ is C2 and t.b/.a/ D t.a/.b/, the 14 cases in the equations above in this example contain all possible Segre characteristics of the energyÐmomentumÐstress tensor (with real Lorentz-invariant eigenvalues). (See [90, 210].) Furthermore, it can be noted that an eigenvector associated with a nonsimple, elementary divisor is null. i Now we shall deal with a real N N matrix ŒM D ŒM j which possesses only 2s D N complex conjugate roots of the characteristic equation. (Thus, in this case, there are no real roots.) The following theorem elaborates the canonical form of such a matrix: Theorem A3.9. Let ŒM be a real N N matrix such that the (usual ) characteristic equation admits 2s D N complex conjugate roots .l/ D a.l/ C ib.l/, .l/ D a.l/ ib.l/;l2f1;:::;sg. Then, the Jordan canonical form of the matrix is furnished by: 2 3

6 B.1/ 7 6 7 6 7 6 B.2/ 7 6 : 7 6 :: 7 ŒM .J / D 6 :: 7 ; 6 :: 7 6 :: 7 6 :: 7 4 : 5 B.r/

2 3 6 .1/ 7 6 C.l/ 7 6 7 6 7 6 C .2/ 7 B 6 .l/ 7 .l/ 6 :: 7 WD 6 : : 7 ; 6 :: 7 2n.l/ 2n.l/ 6 : 7 6 :: 7 6 :: 7 6 : 7 4 .sl / 5 C.l/ 612 Appendix 3 Canonical Forms of Matrices 2 3 6 a b 10 7 6 .l/ .l/ 7 6b.l/ a.l/ 01 7 6 7 6 a b 10 7 h i 6 .l/ .l/ 7 6 b.l/ a.l/ 01 7 C .i/ 6 7 .l/ 6 :: :: 7 WD 6 : : :: 7 ; 6 :: :: 7 .i/ .i/ 6 : 7 2n.l/ 2n.l/ 6 :: 10 7 6 :: 7 6 : 01 7 6 7 6 7 4 a.l/ b.l/5 b.l/ a.l/

Xsl Xr .i/ i 2 f1;:::;sl g ; n.l/ DW n.l/; n.l/ D s D .N=2/: (A3.9) iD1 lD1 The proof of this theorem can be found in [133,177]. The Segre characteristic of the matrix ŒM .J /,in(A3.9)isgivenby .1/ .2/ .1/ .2/ .1/ .2/ .1/ .2/ n.1/;n.1/;::: ; n.1/;n.1/;::: I n.2/;n.2/;::: ; n.2/;n.2/;::: I ::: :

Finally, we consider an N N real matrix ŒM which admits both real and complex conjugate (usual) eigenvalues. The canonical or block diagonal form is provided by 2 3 j 6 A j 7 6 .1/ 7 6 j 7 6 A.2/ j 7 6 j 7 6 : 7 6 :: j 7 6 j 7 6 7 A.k/ j 6 j 7 ŒM .J / D 6 7 : (A3.10) 6 j 7 6 B 7 6 j .1/ 7 6 j 7 6 j B 7 6 .2/ 7 6 j : 7 6 j :: 7 4 5 j B j .r/

Here, matrices A.l/ and B.l/ are furnished by (A3.2i)and(A3.9), respectively. Appendix 3 Canonical Forms of Matrices 613

Example A3.10. Consider the 4 4 matrix 2 3 1=2 1=2 0 0 6 7 6 7 6 1=2 3=2 0 0 7 ŒM WD 6 7 : 4 0021 5 0012

The usual characteristic polynomial equation is furnished by

p./ D . C 1/2 .2 4 C 5/ D . C 1/2 . 2 i/ . 2 C i/ D 0:

Thus, the distinct eigenvalues are

.1/ D1; a.1/ C ib.1/ D 2 C i; a.1/ ib.1/ D 2 i with multiplicities 2, 1,and1, respectively. Therefore, the Jordan canonical form is given by the block diagonal form 2 3 " # 11 6 7 A.1/ 6 0 1 7 ŒM .J / D DW 4 5 : B.1/ 21 12 The Segre characteristic is 2I 1; 1 . The elementary divisor for this matrix is 2 E.4/./ D . C 1/ . 2 i/ . 2 C i/. Example A3.11. Consider a domain of the spaceÐtime manifold and orthonormal components of the energyÐmomentumÐstress tensor ﬁeld provided by [126]: 2 3 .x/ 0 0 0 6 .1/ 7 6 7 6 0.2/.x/ 0 0 7 T.a/.b/.x/ WD 6 7 ; (A3.11i) 4 00˙.x/15 0010 2 3 000 6 .1/ 7 6 7 6 0.2/ 007 t.a/.b/ WD T.a/.b/.x0/ D 6 7 ; (A3.11ii) 4 0015 0010 614 Appendix 3 Canonical Forms of Matrices 2 3 000 6 .1/ 7 6 7 .a/.c/ 6 0.2/ 007 T Œ WD d t.c/.b/ D 6 7 ¤ Œ : (A3.11iii) 4 0015 0010 Four eigenvalues (invariant values for the matrix in (A3.11ii) or usual values of h p i 2 the matrix in (A3.11iii)) are furnished by .1/;.2/;.3/ D .1=2/ C 4 , h p i 2 and .4/ D .1=2/ 4 . Here, .1/ and .2/ are real and .3/;.4/ are real provided jj2. However, the eigenvectors in this case can be spacelike! Therefore, for the case .1/ ¤ .2/ and jj >2, we have the Segre characteristic Œ1;1;1;1. In the case .1/ ¤ .2/ and jjD2, the Segre characteristic is Œ1;1;.1;1/.Butfor.1/ ¤ .2/ and jj <2, the Segre characteristic is 1; 1I 1; 1 . Moreover, there are corresponding Segre characteristics for the coincidence .1/ D .2/. This example illustrates that the energyÐmomentumÐstress tensor of an exotic material can change Segre characteristics from domain to domain! Example A3.12. In Example A3.8, we considered invariant eigenvalues of the symmetric energyÐmomentumÐstress matrix Œt.a/.b/ WD ŒT.a/.b/.x0/. In that example, we restricted the classiﬁcation only to the various cases of real, invariant eigenvalues. In this example, we extend the mathematical investigation to cases of complex, invariant eigenvalues of Œt.a/.b/.(Recall that these eigenvalues are .a/ identical to the usual eigenvalues of the nonsymmetric matrix Œ D Œ .b/ WD .a/.c/ Œd t.c/.b/.) The Jordan canonical forms of various types follow from (A3.9). The type VI is furnished below: 2 3 VI: Type-VI.a/: .1/ 000 6 7 6 7 6 0.2/ 007 6 7 Œ.J / D 6 7 : (A3.12) 4 00a.1/ b.1/ 5

00b.1/ a.1/

Here, we assume that real eigenvalues .1/ ¤ .2/ and the real number b.1/ ¤ 0. The corresponding Segre characteristic is provided by Œ1; 1I 1; 1. Type-VI.b/:If.1/ D .2/ and b.1/ ¤ 0, the Segre characteristic is given by Œ.1; 1/I 1; 1. 2 3 VII: Type-VII.a/: .1/ 100 6 7 6 7 6 0.1/ 007 6 7 Œ.J / D 6 7 : (A3.13) 4 00a.1/ b.1/ 5

00b.1/ a.1/ Appendix 3 Canonical Forms of Matrices 615

Here, we take the real number b.1/ ¤ 0. The Segre characteristic is furnished by Œ2I 1; 1. 2 3 VIII: Type-VIII.a/: a.1/ b.1/ 00 6 7 6 7 6 b.1/ a.1/ 007 6 7 Œ.J / D 6 7 : (A3.14) 4 00a.2/ b.2/ 5

00b.2/ a.2/

Here,wehavetakenb.1/ ¤ 0, b.2/ ¤ b.1/,andb.2/ ¤ 0. The Segre characteristic is Œ1; 1I 1; 1. Type-VIII.b/: Here, we assume that real numbers a.2/ D a.1/ and b.2/ D b.1/ ¤ 0. The corresponding Segre characteristic is Œ.1; 1I 1; 1/.

Every type of Œ.J / exhibited in the previous example (involving energyÐ momentumÐstress tensor components T.a/.b/.x0/) violates energy conditions in (2.190)Ð(2.192). Therefore, in the usual conditions of the macroscopic universe, these types of energyÐmomentumÐstress tensor components are usually deemed unphysical. However, in the quantum realm, it may be possible to violate energy conditions. An example of this is an effect known as the Casimir effect where quantum ﬁelds subject to certain boundary conditions may violate energy conditions (see, e.g., [236, 254] and references therein). Exotic solutions in general relativity usually possess energy condition violation, as may be seen in Appendix 6 and Sect. 5.3, as well as [68] where energy condition violation is shown to result inside a black hole in a certain class of gravitational collapse models. Appendix 4 Conformally Flat Space–Times and “the Fifth Force”

Let us recall the criterion for conformal flatness of a Riemannian (or a pseudo- Riemannian) manifold discussed in Theorem 1.3.33. We reiterate the same theorem with different notations in the following theorem. Theorem A4.1. Let D RN (with N>3) be a domain corresponding to an open subset of an N -dimensional Riemannian (or a pseudo-Riemannian) manifold of differentiability class C 3. Then, the domain D RN is conformally flat i @ j k l if and only if the tensor field C jkl.x/ @xi ˝ dx ˝ dx ˝ dx D O:::.x/ in D. Proof of the above theorem is provided in [56, 90, 171]. Consider a conformally flat domain of a pseudo-Riemannian (or a Riemannian) manifold MN . It is endowed with a coordinate chart such that

gij .x/ D exp Œ2.x/ dij : (A4.1)

We consider another chart, intersecting the preceding one, such that

bgij .bx/ D exp Œ2b.bx/ dij : (A4.2)

Let us consider the following set of coordinate transformations furnished by

bx i D xi ;¤ 0; (A4.3i)

bx i D xi C ci ; (A4.3ii)

bi i i i j x D l j x ;dij l k l m D djm; (A4.3iii) i i j k bx D x = djk x x ; (A4.3iv)

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 617 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 618 Appendix 4 Conformally Flat SpaceÐTimes and “the Fifth Force”

i i i l j k m 1 bx D x b xl x 1 2.bj x / C .bk b / .xm x / : (A4.3v)

i i i j Here, , c , l j ,andb are constant-valued parameters. Moreover, xi WD dij x ,and i xi x is assumed to be nonzero. The set of coordinate transformations presented above constitutes a group.Itis called the conformal group C .p; nI R/ (with p C n D N ). This group involves .1=2/ .N C 1/ .N C 2/ independent parameters. The following theorem provides the importance of this group in regard to conformally flat manifolds. Theorem A4.2. A conformally flat metric g::.x/ D expŒ2.x/ d:: goes over into another conformally flat metricbg:: .bx/ D exp Œ2b.bx/ d:: for N 3 if and only if the coordinate transformation belongs to the conformal group C .p; nI R/. Partial proof of the above theorem is provided in [56]. We shall now state Willmore’s theorem [266] for a conformally flat domain of a pseudo-Riemannian (or a Riemannian) manifold. Theorem A4.3. Let a domain D RN with N>3, corresponding to a domain of a pseudo-Riemannian (or a Riemannian) manifold of differentiability class C 3, admit a tensor field S::.x/ of differentiability class C 2. Moreover, let the following tensor field equations hold:

Rhijk.x/ D ghk.x/ Sij .x/ C gij .x/ Shk.x/

ghj .x/ Sik.x/ gik.x/ Shj .x/: (A4.4) (i) The conformal tensor C:::.x/ D O:::.x/ in D. (ii) Moreover, Sji.x/ Sij .x/ in D. Proof of the above theorem is skipped here. However, the next theorem due to Das [51] incorporates the proof of the preceding theorem. Theorem A4.4. Let the curvature tensor R:::.x/ of a pseudo-Riemannian (or a Riemannian) manifold of differentiability class C 3 and dimension N>3admit a twice-differentiable tensor ﬁeld T::.x/ in the domain of consideration. Moreover, let the following tensor ﬁeld equations hold: n 1 Rlijk.x/ D .N 2/ glj .x/ Tik.x/ C gik.x/ Tlj .x/ glk.x/ Tij .x/ gij .x/ Tlk.x/ o 1 C 2.N 1/ T.x/ glk.x/ gij .x/ glj .x/ gik.x/ ;

ij T.x/WD g .x/ Tij .x/: (A4.5) Appendix 4 Conformally Flat SpaceÐTimes and “the Fifth Force” 619

(Here, is an arbitrary non-zero constant ). Then,

.i/Gij .x/ D Tij .x/; (A4.6i)

ij .ii/Tji.x/ Tij .x/ Iri T D 0; (A4.6ii) 1 .iii/ rk Tij rj Tik C .N 1/ gik.x/rj T gij .x/rkT D 0; (A4.6iii)

l .iv/Cij k .x/ 0; (A4.6iv)

.v/ Moreover,

1 l Rij k .x/ WD .N 2/ .N 3/ rl C ij k D 0: (A4.6v)

Proof. (i) By the single contraction in (A4.5), it follows that 1 Rij .x/ D .N 2/ T.x/ gij .x/ Tij .x/ ; (A4.7i)

R.x/ D 2 .N 2/1 T.x/; (A4.7ii)

Gij .x/ D Tij .x/: (A4.7iii)

(ii) By the algebraic symmetry of Einstein’s tensor and the differential identities ij ri G 0,(A4.6ii) follow. (iii) By the ﬁrst contracted Bianchi’s identities (1.150i), (A4.6ii), and (A4.5), the proof of (A4.6iii) follows. (iv) Substituting Rlijk.x/ from (A4.5), using (A4.7i,ii), (1.169i) yields (A4.6iv). l (v) The covariant differentiation of C ij k .x/ in (1.169i), together with (A4.6iv), implies (A4.6v).

In case we would like to apply the preceding theorem to general relativity, we must choose p D 3; n D 1,andN D 4 for the pseudo-Riemannian spaceÐtime manifold. The theorem, that is suited to general relativity is stated and proved in the following: Theorem A4.5. Let the curvature tensor R:::.x/ of the pseudo-Riemannian spaceÐ time manifold of differentiability class C 3 admit a twice differentiable tensor ﬁeld T::.x/ in the domain D R4 of consideration. Moreover, let the following tensor ﬁeld equation hold: 620 Appendix 4 Conformally Flat SpaceÐTimes and “the Fifth Force”

n Rlijk.x/ D .=2/ glj .x/ Tik.x/ C gik.x/ Tlj .x/ glk.x/ Tij .x/ gij .x/ Tlk.x/ o C .2=3/ T.x/ glk.x/ gij .x/ glj .x/ gik.x/ : (A4.8)

2 ij e Then, the following implications for gij D dij , WD d @i @j ,andT .0/ WD ij e d T ij hold true: 1 kl .i/@i @j dij 2 @i @j .1=4/ dij d @k @l

D.=2/ Tij .x/ .x/: (A4.9)

.ii/R.x/D 63 : (A4.10)

1 1 2 .iii/Rij .x/ D 2 @i @j C dij 4 @i @j (A4.11) 2 kl e kl C dij d @k @l D T ij DW Tij .1=2/ dij d Tkl :

e .iv/ C .=6/ T .0/.x/ .x/ D 0: (A4.12) .v/ Field equations (A4.11) are covariant under the conformal group C .3; 1I R/ involving 15 independent parameters.

2 Proof. (i) The proof follows by noting that gij .x/ D Œ.x/ dij and inserting this metric into the usual expression for Rij .x/. Then, using (A4.7iii) and (1.149ii), the following equations emerge:

1 2 Gij .x/ D 2 @i @j 4 @i @j 1 2 kl dij 2 d @k @l D Tij .x/: (A4.13)

Next, from Einstein’s ﬁeld equations (A4.7iii), (A4.9) is derived. (ii) Double contraction of (A4.13) leads the (A4.10). (iii) By the equality Rij .x/ D Gij .x/ C .1=2/ gij .x/ R.x/,and(A4.9), (A4.10) and deﬁnitions

e k T ij .x/ WD Tij .x/ .1=2/gij .x/ T k.x/; (A4.14i)

e ij e ij T .0/.x/ WD d T ij .x/ DT.0/.x/ DW d Tij .x/; (A4.14ii)

(A4.11) is deduced. (iv) Double contractions of equations (A4.11) (with d ij ) lead to (A4.12). (v) This statement can be proved by using Theorem A4.2. Appendix 4 Conformally Flat SpaceÐTimes and “the Fifth Force” 621

Remarks. (i) It is clear from (A4.5)and(A4.8)thatTij .x/ 0 implies that Rlijk.x/ 0. In other words, mathematically, the support of R::::.x/ the support of T::.x/ in conformally flat spaceÐtimes. Therefore, the class of gravitational phenomena governed by field equation (A4.8)or(A4.9)is nontrivial only in the presence of material sources. Outside material sources, this class of gravitational forces just “switches off”! That is why we interpret gravitational force, arising from field equation (A4.9)asthe“fifth force”.1 (See [101, 216].) This effect can equivalently be seen utilizing (1.169i) and field equation (2.163i). The trace of the latter implies that R.x/ D 0 and Rij .x/ D 0 when Tij .x/ D 0 holds. Then, condition (A4.6iv) for conformally flat spaceÐ times for N>3, when used in (1.169i), implies that Rij k l .x/ D 0 whenever Tij .x/ D 0. (ii) Field equation (A4.9), governing the “fifth force”, does not admit general covariance. However, this equation does admit covariance under the 15- parameter conformal group C .3; 1I R/. (iii) The system of field equations (A4.9)ishighly overdetermined.(See[52].) In spite of the fact that the system of field equations is overdetermined, many exact solutions of the system do exist. In fact, many of these exact solutions turn out to be extremely important for understanding of the cosmological universe. We shall furnish some of these exact solutions in the following examples. Example A4.6. In this example, the following choice is made:

Tij .x/ WD .3K0=/ gij .x/; 1 T.x/D12 K0: (A4.15)

(Here, K0 is a constant.) Substituting (A4.15)into(A4.8), we obtain Rhijk.x/ D K0 ghj .x/ gki.x/ ghk.x/ gji.x/ : (A4.16) Therefore, by (1.164i), the metric is that of a spaceÐtime of constant curvature. Such a spaceÐtime is also called the de Sitter universe (for K0 >0) and anti-de Sitter universe (for K0 <0). (See #1 of Exercise 6.1. Also see [126].) By Theorem 1.3.30, we can transform this metric locally to the conformally ﬂat form: 2 k l 2 i j ds D 1 C .K0=4/ dkl x x dij dx dx : (A4.17)

2 By the equation gij .x/ D Œ.x/ dij , we obtain in this example i j 1 .x/ D 1 C .K0=4/ dij x x : (A4.18)

Thus, an explicit expression for .x/ is furnished.

1It should be stressed here that this is nomenclature only. This phenomenon is still a special class of gravitational effect, and not due to some new force. 622 Appendix 4 Conformally Flat SpaceÐTimes and “the Fifth Force”

The next example deals with FÐLÐRÐW cosmological metrics of (6.7i) and (6.41). These are extremely important metrics in regard to the proper understanding of the cosmological universe. Example A4.7. In this example, we choose the energyÐmomentumÐstress tensor for a perfect ﬂuid from Theorem 6.1.3 and (6.46). It is provided by Tij .x/ D .x/ C p.x/ Ui .x/ Uj .x/ C p.x/ gij .x/; (A4.19i)

i U .x/ Ui .x/ 1: (A4.19ii)

Recall the FÐLÐRÐW metric from (6.41) as

2 4 2 ˛ ˇ 2 4 2 ds D a.x / 1 C .k0=4/ ı˛ˇ x x ı dx dx dx ; ˚ D WD x W x 2 D R3I x4 >0 ;

k0 2 f0; 1; 1g : (A4.20)

The corresponding ﬂuid velocity components, in the comoving frame (derivable E˛ from ﬁeld equations 4./ D 0)aregivenby

U ˛.x/ 0; U 4.x/ 1: (A4.21)

Einstein’s ﬁeld equations from (6.47iÐiii) are furnished by " # 2aR .a/P 2 C k E1 ./ E2 ./ E3 ./ D C 0 C p.x4/ D 0; (A4.22i) 1 2 3 a a2 " # .a/P 2 C k E4 ./ D 3 0 .x4/ D 0; (A4.22ii) 4 a2 aP T 4./ DP C 3 . C p/ D 0; (A4.22iii) a

da.x4/ aP WD ; etc. (A4.22iv) dx4

In the case of k0 D 0, the metric from (A4.20) goes over into 2 4 2 ˛ ˇ 4 2 ds D Œa.x / ı˛ˇ dx dx .dx / : (A4.23) Appendix 4 Conformally Flat SpaceÐTimes and “the Fifth Force” 623

(Consult Example 7.2.2.) Now we make a transformation,

bx˛ D x˛; Z dx4 bx4 D ; a.x4/ ba.bx4/ WD a.x4/>0: (A4.24)

Themetricin(A4.23) yields the conformally ﬂat form: 2 4 2 ˛ ˇ 4 2 ds D Œba.bx / ı˛ˇ dbx dbx .dbx / ;

b 2 i j DW Œ.bx/ dij dbx dbx : (A4.25)

Therefore, the corresponding b.bx/ ﬁeld is explicitly furnished by

b.bx/ D ba.bx4/: (A4.26)

For the case of k0 D 1, (6.7i), (A4.20), and the answer to # 2i of Exercise 6.1 yield n h i o 2 ds2 D ba.bx4/ .db/2 C sin2 b .db/2 C sin2 .db/2 .dbx4/2 : (A4.27)

Now we make another coordinate transformation by 2 sinb r# D ; cosb C cosbx4

2 sinbx4 bx4 D ; cosb C cosbx4 #;# D b; b ;

cosb C cosbx4 ¤ 0I

a#.x#4/ WD ba.bx4/>0: (A4.28)

The corresponding conformally ﬂat metric from (A4.27) emerges as

1 2 2 ds2 D a#.x#4/ cos # C cos x#4 4 ˚ .d#/2 C .r#/2 .d #/2 C sin2 # .d#/2 .dx#4/2 ˚ DW Œ#.x#/2 .dr#/2 C .r#/2 .d #/2 C sin2 # .d#/2 .dx#4/2 : (A4.29) 624 Appendix 4 Conformally Flat SpaceÐTimes and “the Fifth Force”

Therefore, the corresponding ﬁeld #.x#/ is analytically furnished by

1 #.x#/ D a#.x#4/ cos # C cos x#4 : (A4.30) 2

Thus, in an important model of cosmology, an analytic expression for the #.x#/ ﬁeld is explicitly provided. Appendix 5 Linearized Theory and Gravitational Waves

In this appendix we briefly review the linearized theory of gravitation and discuss a class of solutions known as gravitational waves. There are currently several major experiments in progress which hope to detect gravitational waves in the near future. If they are successful, a new arena, known as gravitational wave astronomy, will potentially yield insight into many interesting astrophysical phenomena. The subject of gravitational waves and the detection of gravitational waves is vast, and we can only touch upon the subject here. The interested reader is referred to [33, 103, 172, 230], and references therein. Although wave solutions also exist in the full nonlinear theory, as mentioned in Chap. 7, we shall study here the more physically relevant scenario where the gravitational field is weak. In such a case the metric may be written as a perturbation about the flat spaceÐtime metric:

2 gij .x/ D dij C "hij .x/ C O." /; (A5.1) with ", a small parameter such that terms of order "2 or higher may be ignored.1 (We will use equalities below instead of approximate equalities, and it is understood that this implies “equal to order ".”) The flat spaceÐtime metric is known as the background metric. Of course, one could also perturb a non-flat solution to the Einstein field equations (in fact, many interesting phenomena do indeed consider non-flat backgrounds); however, (A5.1) will suffice for our purpose. In the linear approximation, the Einstein equations are computed utilizing metric (A5.1), and terms to order " are retained. The conjugate metric tensor components gij .x/ from (A5.1) are furnished by

1 (1) Here, hij .x/ will be used to denote the perturbation of the metric. This notation should not be confused with the one of hij .x/ in Appendix 1, where it represented the variation of the metric tensor components. (2) Weak gravitational ﬁelds, up to an arbitrary order O."n/, have been investigated in [64].

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 625 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 626 Appendix 5 Linearized Theory and Gravitational Waves

" gij .x/ D d ij d ik d jl C d jk d il h .x/: (A5.2) 2 kl Therefore, we obtain by raising indices,

kl km ln h .x/ D g .x/ g .x/ hmn.x/ n " D d km d ln d ln d kp d mq C d mp d kq 2 o km lp nq np lq C d d d C d d hpq.x/ hmn.x/; (A5.3i)

ij i dij h .x/ DW h i .x/ n o " D d ij d ik d jl C d jk d il h .x/ h .x/; (A5.3ii) 2 kl ij k det gij .x/ D 1 C "h k.x/ : (A5.3iii)

Further, we shall require the Christoffel symbols, which are given as i " D d il @ h .x/ C @ h .x/ @ h .x/ : (A5.4) jk 2 k lj j lk l jk

In addition, to form the Einstein tensor in linearized theory, we shall require the Riemann tensor and its contractions to order " which may be computed from (A5.4)as: i i Ri .x/ D @ @ jkl k lj l kj

" D @ @ hi .x/ C @ @i h .x/ @ @i h .x/ @ @ hi .x/ ; (A5.5i) 2 k j l l jk k jl l j k " R .x/ D @ @ hk .x/ C @ @kh .x/ @k @ h .x/ @ @ hk .x/ ; (A5.5ii) ij 2 i j k k ij i jk k j i k i i k R.x/ D " @ @kh i .x/ @k@ h i .x/ : (A5.5iii)

(Compare the above with (2.172).) From these, the Einstein tensor is furnished as

1 G D R g R ij ij 2 ij " D @ @ hk .x/ C @ @k h .x/ C d @ @l hk .x/ 2 i j k k ij ij k l k l k k dij @ @kh l .x/ @ @i hjk.x/ @k@j h i .x/ : (A5.6) Appendix 5 Linearized Theory and Gravitational Waves 627

At this point it is worthwhile noting that if we introduce the trace-reversed perturbation tensor,definedby 1 h .x/ WD h .x/ d hk .x/; (A5.7) ij ij 2 ij k the expression (A5.6) simplifies greatly: h i " k k G D @ @k h .x/ C d @ @l h .x/ @k@ h .x/ @ @ h .x/ : (A5.8) ij 2 k ij ij k l i jk k j i One may simplify this expression even further by an appropriate choice of gauge (which is equivalent to a coordinate transformation in general relativity theory). From (A5.8) it can be seen that considerable simplification would occur if we could choose a coordinate system such that

k @ hkj .x/ D 0: (A5.9) The condition in (A5.9) is known as the Lorentz gauge condition (also known as the harmonic, deDonder,orFock-deDonder gauge condition). Establishing such a condition is always possible under an inﬁnitesimal (and differentiable) coordinate transformation of the form xi ! x0 i D xi C "i .x/. Under such a transformation, the metric perturbation becomes

0 hij .x/ !hij .x/ D hij .x/ @i j .x/ @j i .x/; (A5.10i) 0 1 h .x/ !h .x/ D h0 .x/ d h0k .x/ D h .x/@ .x/@ .x/Cd @ k .x/: ij ij ij 2 ij k ij i j j i ij k (A5.10ii)

(Note that "i .x/ is of order ", and therefore, its index is raised or lowered with the background metric.) 0 By taking the four-divergence of (A5.10ii) it can be seen that hij .x/ respects the Lorentz gauge condition if

k j @k@ i .x/ D @ hji.x/: (A5.11)

Therefore, under this gauge condition, the Einstein tensor may be written as

" 0 G D @ @kh .x/ ; ij 2 k ij so that the linearized Einstein ﬁeld equations read

0 k "@k@ hij .x/ D2"Tij .x/: (A5.12)

Note that since the background space-time is ﬂat, Tij .x/ is of order ". 628 Appendix 5 Linearized Theory and Gravitational Waves

Fig. A5.1 An illustration of the quantities in (A5.13)in ∂ B T the three-dimensional spatial ij 0 submanifold. The coordinates 123 xs, known as the source (x s, x s, x s) points, span the entire source (shaded region). O represents an arbitrary origin of the xs coordinate system x−xs

O x (x1, x2, x3)

Remarks. (i) From physics, (A5.12) is the equation for a spin-two tensor field propagating on flat spaceÐtime with a source Tij .x/ [98]. Gravitons (hypotheti- cal quanta mediating gravitational interactions) should therefore carry a spin of two in the corresponding quantum theory. (ii) In the case where Tij .x/ D 0,(A5.12) is simply the wave equation for a second- 0 rank tensor field hij .x/ with speed equal to unity (the speed of light, or massless particles). Therefore, in the linearized theory, it can be argued that gravitational interactions propagate at the speed of light.

If Tij .x/ is known and of compact support, one may ﬁnd a solution to (A5.12) utilizing the ﬂat spaceÐtime retarded Green function for the wave operator: Z 0 4 4 Tij .x;E x jEx Exs j/ 1 2 3 hij .x;E x / D dxs dxs dxs ; (A5.13) 2 B jEx Exs j where the domain of integration, B, is over the source (see Fig. A5.1). For the remainder of this appendix, we concentrate on the case where Tij .x/ D 0 (i.e., away from sources). In this scenario, (A5.12) reduces to the wave equation:

0 k @k@ hij .x/ D 0: (A5.14)

The Lorentz gauge condition is actually a class of gauges, as i .x/ is not completely ﬁxed. Within the Lorentz gauge it is possible to add to i .x/ a vector ﬁeld, i .x/, such that the condition

k @k @ i .x/ D 0 (A5.15) holds, as this will not spoil (A5.11). The Lorentz gauge condition, along with the conditions (A5.15), allows us to reduce the number of components of hji.x/ (omitting primes from now on) from ten to two. At this stage, all coordinate freedom has been exhausted, and the system of equations contains two physical (i.e., not Appendix 5 Linearized Theory and Gravitational Waves 629 arising from coordinate artifacts which do not affect the curvature) degrees of freedom. k For physical considerations, it is convenient to choose 4.x/ such that h k D 0 and choose .x/ such that h4.x/ D 0. This choice constitutes what is known as the traceless-transverse gauge. Note that in this coordinate system there is no TT distinction between hij .x/ and hij .x/. We therefore use the notation hij .x/ to denote metric perturbations in the traceless-transverse gauge below. A class of important solutions to (A5.14) is comprised of a superposition of plane waves. Let us consider a single wave h i l TT ikl x hij .x/ D Re Aij e ; (A5.16) with Aij , the components of a symmetric, constant complex tensor (called the polarization tensor)andkl , the components of a null vector (called the wave i TT vector). The traceless condition implies that A i D 0, and the condition h4.x/ D 0 implies A4 D 0. Without loss of generality, let us consider the wave in (A5.16) traveling in the x3 direction. The wave vector then possesses the form Œki D Œ0;0;!;!with ! representing the frequency of the wave. The conditions (A5.9) and A4 D 0 dictate that Ai3 D 0 and A44 D 0. Assembling all the information we have gathered on the tensor, we see that the two degrees of freedom mentioned previously are encoded in the two components A11 DA22 and A12 D A21 (the ﬁrst equality arising from the traceless condition and the second equality from the symmetric property of hij .x/). To determine the physical effects of these gravitational waves, let us consider two nearby test particles, located at x2 D x3 D 0 in the spaceÐtime of a gravitational wave. Consider a vector connecting these two particles, whose components, at i some initial time, t0, are furnished by Œ jt0 D Œ; 0; 0; 0. To the linear order considered in this appendix, the geodesic deviation equations (1.191) yield, after some calculation:

@21.x/ @2 D TTh .x/ ; (A5.17i) .@x4/2 2 .@x4/2 11

@22.x/ @2 D TTh .x/ : (A5.17ii) .@x4/2 2 .@x4/2 12

(Note that since the Riemann tensor is already of order ", the components of the 4- 0 i i velocity ( in the notation of (1.191)) are given by ı 4.) On the other hand, if two 630 Appendix 5 Linearized Theory and Gravitational Waves a b

time

Fig. A5.2 The C (top)and (bottom) polarizations of gravitational waves. A loop of string is deformed as shown over time as a gravitational wave passes out of the page. Inset: a superposition of the two most extreme deformations of the string for the C and polarizations particles are located at x1 D x3 D 0 and the initial separation vector components i are given by Œ jt0 D Œ0;;0;0, then the separation vector satisﬁes the equations:

@22.x/ @2 D TTh .x/ ; (A5.18i) .@x4/2 2 .@x4/2 22

@21.x/ @2 D TTh .x/ : (A5.18ii) .@x4/2 2 .@x4/2 12

TT From the form of hij .x/ in (A5.16), it is obvious that the geodesic separation vector components, i .x/, are oscillatory. There exist two polarization states of gravitational waves. One state affects particle motion, as shown in Fig. A5.2aand TT TT TT TT is characterized by h11.x/ D h22.x/ ¤ 0 and h12.x/ D 0 D h21.x/. TT This state is known as the C-state. The other state is characterized by h12.x/ D TT TT TT h21.x/ ¤ 0 and h22.x/ D 0 D h11.x/ and is known as the -state. This motion is depicted in Fig. A5.2b. These two states correspond to two independent polarizations since A11 (or equivalently A22 DA11)andA12 (D A21)are independent of each other. The two polarizations are orthogonal to each other in ij the sense that A jCAij j D 0 (the bar here representing complex conjugation). There are currently a number of gravitational wave detectors in operation throughout the world, with several others in the advanced planning and design stages. The largest land-based detector, Laser Interferometer Gravitational Wave Observatory (LIGO), consists of two interferometers: one in Hanford, Washington, and one in Livingston, Louisiana, which have characteristic lengths of approximately 4 km. The LISA detector (Laser Interferometer Space Antenna), originally scheduled for launch in the next decade, would consist of three orbiting satellites, each separated by approximately 5 million kilometers. These mammoth devices possess the sensitivity to detect gravitational waves from various astrophysical sources (see Fig. A5.3) with amplitudes as small as 109 cm. (See, for example, [182].) The current status of the LISA project is under assessment. We have treated here only the very simplest type of gravitational wave, as a more in-depth treatment is beyond the scope of this text. A more thorough review would Appendix 5 Linearized Theory and Gravitational Waves 631

Fig. A5.3 The sensitivity of the LISA and LIGO detectors. The dark regions indicate the likely amplitudes (vertical axis, denoting change in length divided by mean length of detector) and frequencies (horizontal axis, in cycles per second) of astrophysical sources of gravitational waves. The approximately “U”-shaped lines indicate the extreme sensitivity levels of the LISA (left)and LIGO (right) detectors. BH D black hole, NS D neutron star, SN D supernova (Figure courtesy of NASA)

include solutions with sources (Tij .x/ ¤ 0, such as many of the phenomena in Fig. A5.3), which are astrophysically relevant, as well as estimations of the power emitted via gravitational wave emission for various astrophysical processes such as black hole collisions and binary neutron star orbits. (In closing this appendix we note that the 1993 Nobel Prize in Physics was awarded to Russell A. Hulse and Joseph H. Taylor, Jr., for their discovery of a binary pulsar system which is observed to be losing energy at a rate in agreement with gravitational wave calculations [137].) Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines

This appendix deals with several interesting classes of exotic solutions to the gravitational field equations. The purpose of studying such solutions is several-fold: On the one hand, they are interesting in their own right, as they illustrate just how rich the arena of solutions to the field equations can be. There are no analogs of these solutions in pregeneral relativistic theories of gravity. It is therefore of great interest in gravitational field research to study just what is possible within general relativity theory under extreme situations. Secondly, there is pedagogical value in studying such solutions. As we will show below, these solutions concretely illustrate the utility of many of the topics and techniques discussed in this text in a very simple and clear fashion. Finally, the types of solutions mentioned here actually have application in specialized physical topics within general relativity theory. Wormholes, for example, provide a simplistic picture of “spaceÐtime foam,” a potential model for the vacuum in theories of quantum gravity, depicted in Fig. A6.1. (Whether this topology is allowed to fluctuate is not clear at the moment, as some arguments against topology change are indifferent to whether one is considering classical or quantum effects [126].) An in-depth review of exotic solutions of the type discussed here may be found in [168].

A6.1 Wormholes

The ﬁrst exotic solution we will discuss is the wormhole. This type of object has already been mentioned previously. (See Fig. 2.17c.) Qualitatively, the wormhole represents a handle (or shortcut) in spaceÐtime, which introduces a nontrivial topology. The most often studied topology is R2 S 2. The wormhole handle may connect two otherwise disconnected universes (an interuniverse wormhole) or else connect two regions of the same universe (an intrauniverse wormhole). The two cases are depicted in Fig. A6.2. It should be noted that the near-throat region (the “narrowest” section of the handle) has an uncanny resemblance to the manifold depicted in Fig. 3.1 and, in fact, the part of the Schwarzschild manifold depicted in

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 633 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 634 Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines

Fig. A6.1 A possible picture for the spaceÐtime foam. Space-time that seems smooth on large scales (left) may actually be endowed with a sea of nontrivial topologies (represented by handles on the right) due to quantum gravity effects. (Note that, as discussed in the main text, this topology is not necessarily changing.) One of the simplest models for such a handle is the wormhole

identify identify

Fig. A6.2 A qualitative representation of an interuniverse wormhole (top) and an intra-universe wormhole (bottom). In the second scenario, the wormhole could provide a shortcut to otherwise distant parts of the universe

Fig. 3.1 can actually represent one half of a type of wormhole known as an Einstein- Rosen bridge. The shading in the throat region here represents the presence of some matter field, which is the source of the gravitational field supporting the wormhole. Let us now quantify the geometry represented in Fig. A6.2 . We shall be limiting our attention to the case where the wormhole exhibits spherical symmetry and, for simplicity, we will assume the spaceÐtime is static. In such a case, we may take the line element to be that of (3.1) and utilize the field equations and conservation equations given by (3.44iÐvii). Note that here, r1 is equal to the coordinate radius of the throat. To avoid confusion, below we shall use coordinate indices .r;;';t/ instead of .1;2;3;4/as was used in Chap. 3. Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines 635

x 3 x 3

ϕ ϕ

x 3=P (r) + rotate

x 1 x 1 throat throat x 3=P (r) x 2 x 2 −

Fig. A6.3 Left: A cross-section of the wormhole proﬁle curve near the throat region. Right:The wormhole is generated by rotating the proﬁle curve about the x3-axis

Since we have a specific geometry in mind, the g or mixed methods of solving the field equations will be most useful here. To mathematically construct the wormhole, we consider Fig. A6.3 which illustrates a cross-section or profile curve of a two- dimensional section, t D constant, D =2, of the wormhole near the throat (see also Fig. 1.18). The wormhole itself is constructed via the creation of a surface of revolution when one rotates this profile curve about the x3-axis. The surface of revolution may be parameterized as follows: ˛ ˛ 1 2 3 xj:: D .r; '/; .r; '/; .r; '/; .r; '/ WD .r cos '; r sin '; P.r//; (A6.1) q where r D .x1/2 C .x2/2 and ' is shown in Fig. A6.3. Therefore, the induced metric on the surface of revolution is given by n o 2 1 2 2 2 3 2 2 2 2 2 dj:: D .dx / C .dx / C .dx / j:: D 1 C Œ@r P.r/ dr C r d' : (A6.2)

Note that the corresponding four-dimensional spaceÐtime metric, (3.1), may now be written as n o 2 2 2 2 2 2 2 2 ˙.r/ 2 ds˙ D 1 C Œ@r P˙.r/ dr C r d C r sin ./ d' e dt ; (A6.3) with coordinate ranges

For the case of a wormhole throat, the coordinate chart of (A6.3)isinsufficient to describe the entire wormhole. Instead, two spherical charts are required, one for 636 Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines the upper half of the wormhole (the “+” domain) and one for the lower half (the “” domain). (See [74, 75, 107, 185, 254] for various charts which cover both the + and domains.) Hence, we use the ˙ subscript in the above line element. The constant m0 in (3.45i) can be set by the requirement that @r P˙.r/ !˙1 as r ! r1 or, equivalently, that (3.45i) vanish in the limit r ! r1. This condition implies that the constant 2m0 D r1. We are now in a position to analyze the geometry. For now we limit the analysis to the upper half of the profile curve (the “+” region) in Fig. A6.3 since the bottom half is easily obtained once we have the upper half (for example, via reflection symmetry through the x1 x2 plane if one wants a wormhole symmetric on either side of the throat, although this is not required). It should also be noted that r D 0 is not part of the manifold, unlike in the case of stars studied in earlier chapters. Therefore, divergences as r tends to zero are not an issue here. From the profile curve in Fig. A6.3, it may be seen that a function P.r/is required with the following properties:

1. The derivative, @r P.r/, must be “infinite” at the throat and finite elsewhere. 2. The derivative, @r PC.r/, must be positive at least near the throat (negative for the “” domain). 3. The function PC.r/ must possess a negative second derivative at least somewhere near the throat region (positive for the “” domain). 4. Since there will be a cutoff at some r D r2, where the solution is joined to vacuum (Schwarzschild or Kottler solution), no specification needs to be made as r tends to infinity. From now on, we often drop the explicit r dependence in functions for brevity. In terms of the function P.r/, the field equations read

2 .@h r P/ .@ri/r r DT r ; (A6.4i) 2 2 1 C .@r P/ r

h i n 1 2 1 C .@ P/2 4.@ P/ .@ @ P/C 2.@ /C 2.@ / .@ P/2 4r r r r r r r r 2 2 2 2 Cr.@r / C 2r.@r @r / .@r P/ C 2r.@r @r /C r.@r / .@r P/

' 2r.@r P/ .@r @r P/ .@r /g DT T ' ; (A6.4ii) h i 3 .@r P/ .@r P/ C 2.@r @r P/r C @r P h i t 2 DT t ; (A6.4iii) 2 2 r 1 C .@r P/ where it is understood that these equations must hold in both the C and domain. A natural question to ask at this stage is what are the properties a matter ﬁeld must possess in order to support such a wormhole spaceÐtime? Analyzing (A6.4iii) Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines 637 by performing a careful limit as r ! r1, and taking into account the properties t 1 of P.r/ listed above, we see that G t .r1/ D .r1/ 2 , .r/ being the local r1 energy density as measured by static observers. (We are assuming here that the stress-tensor components are nonsingular.) To make the arguments more physical, we shall employ a mixed method as opposed to the purely geometric g-method, and r therefore we will treat T r .r/ as a function that is to be prescribed. The orthonormal Riemann tensor components are furnished by 1 b˙ 1 2 R ˙ D 1 @ @ ˙ C .@ ˙/ .t/.r/.t/.r/ 2 r r r 2 r # b˙ ˙ 1 @r b r @r ˙ ; (A6.5i) 2 r b˙ 1 b˙ R ˙ D @ ˙ 1 D R ˙ ; (A6.5ii) .t/./.t/./ 2r r r .t/.'/.t/.'/ 1 b˙ R ˙ D @ b˙ D R ˙ ; (A6.5iii) .r/./.r/./ 2r2 r r .r/.'/.r/.'/

b˙ R ˙ D ; (A6.5iv) ./.'/./.'/ r3

2 .@r P˙/ where we use the notation b˙ WD r 2 to simplify the expressions. 1C.@r P˙/ To continue the analysis, it is noted that to make the the orthonormal Riemann components finite in this scenario, @r .r/ must be finite. (See [74] for full details as to why this renders all components finite.) From the field equations, it can be shown that this quantity is given by h i r 1 2 1 @ ˙ D r T C 1 C .@ P˙/ ; (A6.6) r r˙ r2 r r

r r from which it may be seen that a prescription of T r compatible with T r .r1/ D 1 pr .r1/ D2 must be applied. Therefore, a negative pressure (i.e., atension) r1 must be present at the wormhole throat in order to support it. Finally, we consider the combination h i ˙ 1 t r e 2 .˙ C p ˙/ D G G D @ e ˙ 1 C .@ P˙/ (A6.7) r t ˙ r ˙ r r r in the immediate vicinity of the wormhole throat. This quantity is directly related to the energy conditions discussed in Chap. 2. From the discussions above, it can be seen that at the throat, nonnegativity of this combination can (barely) be met (i.e., .r1/ C pr .r1/ 0). However, as one moves away from the throat, note that the quantity in braces in (A6.7) must go from a value of zero at the throat 638 Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines

(where @r P !1) to a positive value away from the throat. Therefore, around the throat region, the derivative of the expression in braces is positive and thus (A6.7) is negative in a neighborhood of the throat. That is, a static wormhole throat violates the weak/null and strong energy conditions! (See, e.g., [146].) The reader interested in this topic is referred to Visser’s book on the subject of Lorentzian wormholes [254].

A6.2 Warp-Drive Space–Times

We consider here briefly another interesting class of solutions to the field equations known as warp-drive space-times. The first such solution was put forward by Miguel Alcubierre in 1994 [4], and since then much analysis has been done on these types of solutions. We will present here the original Alcubierre warp-drive. The basic idea is simple. One wishes to construct a spaceÐtime in which an object (popularly, a spaceship is chosen) can be propelled between two points, such that the travel time (as measured by both the proper time of the object and external observers) is much less than the distance separating the points divided by the speed of light. Ideally, one also wishes to demand that the interval of proper time of the traveling observers is the same as the proper time interval for static external observers (so that the two observers age by the same amount!). Before beginning the analysis, it should also be noted that no object will be traveling faster than the local speed of light. It is useful to employ the ADM formalism here (see Appendix 1) utilizing the metric (A1.42). An ADM form of the metric is useful, as a spaceÐtime domain constructed from the ADM formalism will not contain any closed causal curves (to be discussed in the next section) unless one introduces a topological identification. The original warp-drive used the following quantities:

N.x/ D 1; (A6.8i)

˛ 4 ˛ N .x/ Dsv.x /f .s r.x//ı 1 ; (A6.8ii)

g˛ˇ .x/ D ı˛ˇ ; (A6.8iii) where we are employing a Cartesian-like coordinate system in the three-dimensional spatial hypersurface. The line element therefore is the following: 2 1 4 4 2 2 2 3 2 4 2 ds D dx sv.x /f .sr.x//dx C dx C dx dx : (A6.9)

X 1 4 Here, v.x4/ WD d s .x / represents the coordinate velocity of the spaceship and s h dx4 i 1=2 1 1 4 2 2 2 3 2 sr.x/ WD x sX .x / C .x / C .x / . The coordinate position of the Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines 639

Fig. A6.4 A “top-hat” function for the warp-drive spaceÐtime with one direction (x3) suppressed. The center of the ship is located at the center of the top hat, corresponding to sr D 0

1 1 4 2 3 ship is x Ds X .x /; x D 0; x D 0 (note that we are assuming spatial motion 1 in the x direction only) and therefore s r.x/ represents the coordinate distance from the ship. For the system to function as desired, the function f.s r.x// should approximate a “top-hat” function. In the original warp-drive solution, the following function was chosen (the explicit x dependence is henceforth dropped):

tanh Œ . r C R/ tanh Œ . r R/ f. r/ WD s s ; (A6.10) s 2 tanh .R/ with R>0and 0 but otherwise arbitrary constants. This function is displayed in Fig. A6.4 with the x3 coordinate suppressed. This spaceÐtime possesses several interesting properties. Note from (A6.9)that 4 2 x D constant hypersurfaces are always flat. Also, for distances sr R ,the full four-dimensional spaceÐtime is flat, as is the spaceÐtime anywhere where the first two derivatives of f.sr/ are zero. As well, it is not difficult to show that i observers possessing 4-velocity components u D Œs vf.s r/;0;0;1are traveling 1 along geodesics. As a concrete example, consider an observer located at xO D 1 4 2 3 sX .x /; xO D xO D 0 and therefore traveling with the ship. It may immediately be seen that, for such an observer, the invariant infinitesimal line element (A6.9) 2 4 2 yields the condition dsO D.dx /O. (Note that f.sr/ D 1 in the immediate vicinity of the ship.) Recalling that ds2 along a world line yields minus the infinitesimal proper time squared along that world line tells us that the lapse of proper time for an observer traveling with the ship is equal to the lapse of proper time for a static observer located at some distance >Rfrom the ship, for whom ds2 D.dx4/2 also holds. Therefore, there is no relative time dilation between such observers. (We are assuming that the function f.sr/ is very close to a true top-hat function, which is a good approximation as long as is large.) Qualitatively, the top hat, often called the warp bubble, is “propelled” through the external spaceÐtime with a coordinate velocity sv, and any object enclosed in the bubble moves along with it. This velocity can be very large, the important point being that objects inside the bubble always possess 4-velocities that are timelike. This emphasizes an important point in general relativity that the structure of the light cones is a local phenomenon. As long as 4-velocities remain within their local light cone, the velocity is timelike or subluminal (in the local sense). 640 Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines

4 2 –K 0 –2 –4 –10 –10 –5 –5 0 0 5 5 ^ x^2 10 10 x 1

Fig. A6.5 The expansion of spatial volume elements, (A6.12), for the warp-drive spaceÐtime with the x3 coordinate suppressed. Note that, in this model, there is contraction of volume elements in front of the ship and an expansion of volume elements behind the ship. The ship, however, is located in a region with no expansion nor compression

To glean further properties of this spaceÐtime, let us next consider the extrinsic ˛ curvature components, K ˇ, of the spatial slices. A small amount of calculation reveals that

d f. r/ @ r K˛ Dv s s ı˛ ı1 C : ˇ s 1 1 ˇ off-diagonal terms (A6.11) d sr @x

The negative trace of this quantity yields an expression for the expansion of volume elements in this space: 1 X 1 4 ˛ d f.sr/ x s .x / K ˛ D sv : (A6.12) d sr sr

This function is plotted (with the x3 coordinate suppressed) in Fig. A6.5. It should be stressed that the contraction of volume elements in front of the spaceship is simply a “by-product” phenomenon due to the type of metric chosen. The space ship does not travel at apparent superluminal speed by moving through the compressed space. The space ship is always located inside the warp bubble, where the space is approximately flat. Finally, we briefly analyze the properties of the matter field required to support a warp bubble such as the one described here. The energy density, as measured by observers with 4-velocity components ui D Œ0; 0; 0; 1 (geodesic observers), is given by

1 T .x/ui .x/ uj .x/ D G .x/ui .x/ uj .x/ ij ij " # 1 @f . r/ 2 @f . r/ 2 D . v/2 s C s : (A6.13) 4 s @x2 @x3 Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines 641

Fig. A6.6 Aplotofthe distribution of negative energy density in a plane (x3 D 0) containing the ship 0 for a warp-drive spaceÐtime –2 –4 rho –6 –8 –10 –10 –10 –5 –5 0 0 5 x^1 x^2 5 10 10

From this expression, it can be noted that a negative energy density is required in order to generate this exotic geometry. The distribution of this negative energy density is plotted in Fig. A6.6. Again, energy conditions are violated.

A6.3 Time Machines

We conclude our survey of exotic solutions by considering some space-times which contain closed causal (null or timelike) or closed timelike curves (CTCs). We have already briefly touched upon such phenomena when discussing the Kerr spaceÐtime. We shall present a few more such solutions here. CTCs are associated with space-times which possess some property which allows a continuous timelike curve to intersect with itself. That is, some observer may travel into his or her own past. (These definitions may be extended to include null curves as well.) Two qualitative examples of CTCs are displayed in Fig. A6.7,where “forward light cones” have been drawn to indicate the timelike nature of the curves. It should be noted that the space-times in the figures may possess an everywhere nonvanishing vector field which is timelike and continuous (and thus may be time- orientable). Also, locally, one can define a “past” and a “future” in the figures. However, globally this is not possible, as a forward-pointing timelike vector tangent to a timelike curve may be FermiÐWalker transported along the curve in the forward direction (see Fig. 2.13), only to wind up in the past of its point of origin. Causality violation refers to the violation of the premise that cause precedes effect, as would be possible in a spaceÐtime with CTCs. We define the causality violating region of a spaceÐtime point p, C.p/,as

C.p/ WD C.p/ \ CC.p/; 642 Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines

identify

Fig. A6.7 Two examples of closed timelike curves. In (a) the closure of the timelike curve is introduced by topological identification. In (b) the time coordinate is periodic where C.p/ and CC.p/ represent the causal past and causal future of p,respectively (see Fig. 2.2). For causally well-behaved space-times, this intersection is empty for all p. The causality violating region of a spaceÐtime M is [ C.M / D C.p/; p2M and a spaceÐtime whose C.M / D;is known as a causal spaceÐtime. It is interesting to note that the source of causality violation in many space-times with closed timelike curves is the presence of large angular momentum. Several of the examples below shall reveal this phenomenon, and it will become clear that the coupling of a time coordinate with an angular coordinate (caused by the presence of rotation) can cause the angular coordinate to become timelike, and thus generate causality violating regions of the type in part b) of Fig. A6.7. We consider first the anti-de Sitter (AdS) spaceÐtime (see Exercise 10 of Sect. 2.3). AdS spaceÐtime is a constant (negative) curvature solution to the field equations which is supported by a negative cosmological constant, . The solution may be obtained from the induced metric on the hyper-hyperboloid defined by

1 U 2 V 2 C X 2 C Y 2 C Z2 D ; (A6.14) 2jj embedded in the ﬁve-dimensional ﬂat space admitting a line element of

2 2 2 2 2 2 ds5 DdU dV C dX C dY C dZ : (A6.15)

We show a schematic of this embedding (with 2 dimensions suppressed) in Fig. A6.8. (See Exercise #1iii of Sect. 6.1 for a discussion of the analogous diagram for de Sitter spaceÐtime.) Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines 643

Fig. A6.8 The embedding of U anti-de Sitter spaceÐtime in a ﬁve-dimensional ﬂat “space–time” which possesses two timelike coordinates, U and V (two dimensions suppressed) X V

The topology of the AdS manifold is therefore S 1 R3. Referring to Fig. A6.8,it can be shown that, for example, curves represented by circles orbiting around the X- axis on the hyperbolic manifold are everywhere timelike. This spaceÐtime therefore exhibits closed timelike curves of the type displayed in Fig. A6.7a).Usually,when dealing with AdS spaceÐtime, one deals with the universal covering space of anti- de Sitter spaceÐtime, which involves unwrapping the S 1 part of the topology, thus yielding R4 topology. Next, in our brief review of causality-violating solutions, we present the G¬odel universe [116]. The Godel¬ universe represents a rotating universe supported by a negative cosmological constant, , and a dust matter source and possessing R4 topology. The rotation is somewhat peculiar in the sense that all points in this universe may be seen as equivalent. That is, there is no preferred point about which the Godel¬ universe is rotating. (See [126] for details.) In one popular coordinate system, the Godel¬ universe metric admits a line element of h i 2 p ds2 D d%2 C dz2 C sinh2 % sinh4 % d'2 C 2 2 sinh2 % d' dt dt 2 ; jj (A6.16) with <0, 1 ln.1 C 2/, this “angle” is timelike. Although not a geodesic, an observer in the Godel¬ spaceÐtime at %>% could travel along a trajectory in the ' direction and meet

1One has to be cautious when applying this loose criterion. A more rigorous discussion is based on the eigenvalue structure of the metric tensor as, for example, presented in various preceding chapters. 644 Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines

Fig. A6.9 The light-cone structure about an axis % D 0 in the Godel¬ spaceÐtime. On thepleft,the light cones tip forward, and on the right, they tip backward. Note that at % D ln.1 C 2/ the light cones are sufficiently tipped over that the ' direction is null. At greater %,the' direction is timelike, indicating the presence of a closed timelike curve up with him/herself after one complete “revolution”! This type of causal structure is similar to that depicted in Fig. A6.7b. A figure of this peculiar light-cone structure is displayed in Fig. A6.9 (also see figure caption). The final solution we discuss is the Van Stockum spaceÐtime [251]. This solution to the field equations was one of the first to exhibit the behavior of CTCs. Physically, the solution describes an infinitely long cylinder of rotating dust with a vacuum exterior, the angular momentum preventing the gravitational collapse of the dust. The interior solution admits the following line element in corotating cylindrical coordinates: 2 2 2 ds2 D e! % d%2 C dz2 C %2.d'/2 dt !%2 d' ; (A6.17) with 1 1 the periodic ' coordinate becomes timelike. One could argue that by placing the boundary of the 1 dust cylinder at some % D %0 < ! , the problem of CTCs would be cured, as the above interior metric is no longer valid at !% D 1. However, it has been shown that 1 if !%0 > 2 CTCs still appear in the vacuum exterior [248]. But causality violation 1 is avoided if !%0 2 , although there may exist other pathologies. As can be seen Appendix 6 Exotic Solutions: Wormholes, Warp-Drives, and Time Machines 645 here, strong rotation also causes the light cones to tip and a large enough value of ! will cause CTCs to occur somewhere in the spaceÐtime. This is analogous to the situation shown in Fig. A6.7b. Before closing this section, we mention that the wormholes described earlier in this appendix may also be used as a time machine. One process involves the time dilation effect when two mouths of a wormhole move with respect to each other. The interested reader is referred to [247]. Appendix 7 Gravitational Instantons

Let us go back to Tricomi’s p.d.e. of Example A2.5.Itisgivenby

1 @1@1w C x @2@2w D 0:

The partial differential equation above originates in the theory of gas dynamics [43]. The p.d.e. is elliptic for the half-plane x1 >0, but hyperbolic for the other half-plane x1 <0. Thus, this equation motivates us to consider mathematically the following metric in order to introduce the subject of this appendix: @ @ @ @ g:: x1;x2 WD ˝ C x1 ˝ ; (A7.1i) @x1 @x1 @x2 @x2

1 g:: x1;x2 D dx1 ˝ dx1 C x1 dx2 ˝ dx2 ; (A7.1ii) ˚ ˚ 1 2 1 2 1 2 1 2 D2 WD x ;x W x >0;x 2 R [ x ;x W x <0;x 2 R : (A7.1iii)

The metric above is positive deﬁnite (or Riemannian) for x1 >0;but pseudo- Riemannian (of signature zero)forx1 <0:(A7.1iii) yields a union of apparently two disconnected two-dimensional differential manifolds. Let us consider another toy model of a four-dimensional metric furnished by: 1 4 ˛ ˇ 1 2 4 3 4 4 g::.x/ WD x ı˛ˇ dx ˝ dx x x dx ˝ dx ; 2 1 4 ˛ ˇ 1 2 4 3 4 2 ds D x ı˛ˇ dx dx x x dx ; ˚ ˚ D WD x W x1 >0;x2 2 R;x3 2 R;x4 >0 [ x W x1 >0;x2 2 R;x3 2 R;x4 <0 : (A7.2)

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 647 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 648 Appendix 7 Gravitational Instantons

In case x4 >0;the above metric is transformable to the Kasner metric of Example 4.2.1. However, in case x4 <0;the metric yields a Ricci-flat, positive definite metric. Two distinct sectors of the manifold have a common three-dimensional, positive definite hypersurface with the metric: 1 4 ˛ ˇ g::.x/ WD x ı˛ˇ dx ˝ dx ; ˚ D WD x W x1 >0;x2 2 R;x3 2 R :

The toy model in (A7.2) goes beyond the traditional definition of either a Rieman- nian or else a pseudo-Riemannian manifold. The common boundary D can be called an instanton-horizon. The static, vacuum, spaceÐtime metrics can generate positive definite, Ricci-flat metrics as in (A7.2). At this point, we should mention that a gravitational instanton metric is not synonymous with a Riemannian metric. That is, the presence of a positive definite metric is necessary but not sufficient to guarantee that the solution is a gravitational instanton.1 Theorem A7.1. Let the function w.x/ and the positive definite, three-dimensional ı 3 3 metric g::.x/ be of class C in D R : Moreover, let 2w.x/ ı 2w.x/ 4 4 g::.x/ WD e g::.x/ e dx ˝ dx satisfy vacuum field equations R::.x/ D O::.x/ in D WD D R: Then, the four- dimensional, positive definite metric 2w.x/ ı 2w.x/ 4 4 bg::.x/ WD e g::.x/ C e dx ˝ dx is Ricci flat in D R: Proof. For both of the metrics g::.x/ andbg::.x/; conditions of Ricci flatness reduce to the same equations (4.41i,ii). However, such an easy transition to the positive definite status is not possible from stationary, vacuum metrics (which are not static). Let us explore this situation in more detail. Consider the “stationary,” positive definite, four-dimensional metric: 2u.x/ ı 2u.x/ ˛ 4 ˇ 4 g::.x/ WD e g::.x/ C e w˛.x/ dx C dx ˝ wˇ.x/ dx C dx I

x D .x;x4/ 2 D R: (A7.3)

1A gravitational instanton metric is a C 3 class of vacuum metrics in a “space–time” with a positive deﬁnite C4 signature which is geodesically complete. Often in cases of interest, they possess a self-dual or anti-self dual Riemann tensor (to be discussed shortly). Appendix 7 Gravitational Instantons 649

@ (Obviously, the metric above admits the Killing vector @x4 : Needless to say, the word timelike is undeﬁned in this context.) Some of the Ricci-ﬂat conditions imply the existence of potential N .x/ such that N 4u ı ˇ @˛ D .1=2/ e ˛ @ wˇ @ˇw : (A7.4)

The function N .x/ is called the NUT potential. The remaining conditions of Ricci ﬂatness reduce to the following p.d.e.s:

ı 4u R .x/ C 2 @u @u .1=2/ e @N @N D 0; (A7.5i)

ı 2 4u ı r u .1=2/ e g @N @N D 0; (A7.5ii)

ı 2 ı r N 4 g @u @N D 0: (A7.5iii)

Now we shall derive some conclusions about potentials u.x/ and N .x/: Theorem A7.2. Let ﬁeld equations (A7.5ii,iii) hold in a star-shaped domain D R3: (i) Moreover, let N .x/ be continuous up to and on the boundary @D: In case N .x/ attains the extremum at an interior point x0 2 D; the metric (A7.3)is transformable to the “static form.” (ii) Let the function u.x/; satisfying (A7.5ii) attain the maximum at an interior point x0 2 D: Then, the four-dimensional curvature tensor R::::.x/ O::::.x/ for all x 2 D R: Proof. (i) Use Theorems 4.2.2 and 4.2.3 for the function N .x/. ı ı 2 4u (ii) Note that r u D .1=2/ e g @N @N 0: Use Hopf’s Theorem 4.2.2 for the function u.x/.

Now assume a functional relationship

e2u.x/ D fŒN .x/ > 0;

d fŒN f 0 WD ¤ 0: (A7.6) d N Substituting (A7.6)into(A7.5ii,iii), subtracting (A7.5iii) from (A7.5ii), and assuming @˛N 6 0; we deduce that

f 00 f 0 1 C D 0: (A7.7) f 0 f ff 0 650 Appendix 7 Gravitational Instantons

(Compare and contrast the equation above with (4.156.)) The general solution of (A7.7) is furnished by ˚ 2 2 fŒN D c0 C 2c1N C .N / ; (A7.8) 4u.x/ N 2 2 e D .x/ C c1 C c0 c1 >0:

Here, c0 and c1 are constants of integration. We now make special choices

c0 D c1 D 0; e2u.x/ D N .x/>0: (A7.9)

Field equations (A7.5i) yield ı R .x/ D 0: (A7.10)

ı Thus, the three-dimensional metric g::.x/ is (ﬂat) Euclidean. Choosing a Cartesian chart for D R3; (A7.5ii,iii) boil down to the Euclidean Laplace’s equation: ˚ ˚ 2 1 1 r ŒN .x/ WD ı @@ ŒN .x/ D 0: (A7.11)

Equations (A7.4) reduce to ˚ 1 @ˇw˛ @˛wˇ D"˛ˇ @ ŒN .x/ ; n o or, rE wE .x/ D rE ŒN .x/ 1 : ˚ or, curl wE D grad ŒN .x/1 : (A7.12)

The “stationary metric” reduces to 2 1 ˛ 4 2 ds D ŒN .x/ ı .dx dx / C ŒN .x/ w˛.x/ dx C dx : (A7.13)

The class of exact solutions given by (A7.13), (A7.11), and (A7.12) was discovered by Kloster, Som, and Das (KÐSÐD in short) in 1974 [149]. Hawking rediscovered the same metric in 1977 [125]. We shall call the metric in (A7.13)astheHÐKÐSÐD metric. Now recall the Hodge-dual mapping in (1.113). It implies for the curvature tensor

rs Rijkl.x/ D .1=2/ rskl.x/ Rij .x/: (A7.14)

In case Rij k l .x/ Rij k l .x/; (A7.15) Appendix 7 Gravitational Instantons 651 the curvature tensor R::::.x/ is called self-dual. On the other hand, if

rs Rij k l .x/ .1=2/ rskl.x/ Rij .x/ DRij k l .x/; (A7.16) the curvature tensor is called anti-self-dual.Both(A7.15)and(A7.16) can be combined into Rij k l .x/ " Rij k l .x/; " D˙1: (A7.17) It can be proved that in case of the HÐKÐSÐD metric of (A7.13), the condition in (A7.17)isidentically satisfied by the use of (A7.9)[110]. Instantons are regular, finite-action solutions of the field equations in a four- dimensional, positive definite, flat (or curved) “space–time”. For Yang-Mills type of gauge field theories, the construction of Atiyah, Drinfield, Hitchin, and Manin [11] allows one to compute all regular, self-dual (or anti-self-dual) solutions. However, in the case of gravitation, the situation is more complicated. We define gravitational instantons as positive definite metrics of class C 3; which are Ricci flat and geodesically complete. There exist gravitational instantons which are neither self-dual, nor anti-self-dual. A class of these are provided by the “static metrics” of Theorem A7.1. Fortunately, HÐKÐSÐD metrics supply infinitely many self-dual (or anti-self-dual) gravitational instantons which are useful for some path-integral approaches to quantum gravity. Example A7.3. Consider the multicenter HÐKÐSÐD gravitational instantons furnished by the function

XN 2m ŒN .x/ 1 D a C A >0: (A7.18) k x x k AD1 A

Here, a 0 and mA >0are constants. (Compare with the pseudo-Riemannian 4 Example 4.2.8.) If mA D m for all A 2f1;:::;Ng and if x is a periodic variable 4 with the fundamental period in 0 x . m=N /; then singularities at x D xA are removable. Such solutions qualify as gravitational instantons. In case a D 1 and N D 1; the metric is called the self-dual Taub-NUT metric [126]. In case a D 0 and N D 2; we obtain the Eguchi-Hansen gravitational instantons [85]. Appendix 8 Computational Symbolic Algebra Calculations

In this appendix, we present some calculations relevant to general relativity utilizing popular symbolic algebra computer programs. Calculations in general relativity are notoriously lengthy and complicated, and therefore, computational symbolic algebra programs are very useful in this ﬁeld. Two popular symbolic algebra programs are MapleTM and MathematicaTM , both of which we concentrate on here. For the sake of brevity, we do not discuss special packages, available for both programs, which are designed to supplement the ability of these programs to do general relativistic calculations. (For example, we do not discuss the use of the excellent add-on package available for both MapleTM and MathematicaTM called GRTensorII, which is available at http://grtensor.phy.queensu.ca/).

A8.1 Sample MapleTM Work Sheet

This work sheet deals with quantities related to the vacuum Schwarzschild metric:

> #MAPLE worksheet involving the vacuum Schwarzschild metric.

> restart: #The restart command clears all set variables.

> with(tensor): #This loads Maple’s tensor package.

> coord:=[r, theta, phi, t]; #The coord command defines˜the four coordinates. coord WD Œr;;;t

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 653 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 654 Appendix 8 Computational Symbolic Algebra Calculations

> g_compts:=array(symmetric, sparse, 1..4, 1..4); #create a 4x4 sparse array, to be used for metric components. The symmetric command means that if you have non-diagonal components, you only have to enter one of the symmetric pair, the other will automatically be assumed. g compts WD array.symmetric; sparse; 1::4; 1::4; Œ/

> g_compts[1,1]:=1/(1-2*m/r); #This is g_f11g. 1 g compts WD 1; 1 2m 1 r > g_compts[2,2]:=rˆ2; #This is g_f22g. 2 g compts2; 2 WD r > g_compts[3,3]:=rˆ2*(sin(theta))ˆ2; #This is g_f33g. 2 2 g compts3; 3 WD r sin./ > g_compts[4,4]:=-1/g_compts[1,1]; #This is g_f44g. 2m g compts WD 1 C 4; 4 r > g:=create([-1,-1], eval(g_compts)); #This command creates the metric. The [-1,-1] indicates covariant components on both indices. g WD table.Œ2index char D Œ1; 1; 3 1 6 00 07 6 2m 7 6 1 7 6 r 7 6 2 7 compts D 6 0r 007/ 6 2 2 7 4 00rsin./ 0 5 2m 00 01 C r > ginv:=invert(g,‘detg‘); #Creates the inverse metric. ginv WD table2 .Œindex char D Œ1; 1; 3 r C 2m 6 00 07 6 r 7 6 7 6 1 7 6 0 007 6 r2 7 compts D 6 7/ 6 1 7 6 00 0 7 6 2 2 7 4 r sin./ 5 r 000 r C 2m Appendix 8 Computational Symbolic Algebra Calculations 655

> D1g:=d1metric(g,coord): #Creates first partial derivatives of the metric. Note the colon (instead of semi-colon). This suppresses the output, which can be lengthy.

> D2g:=d2metric(D1g,coord): #Creates the second partial derivatives of the metric.

> Cf1:=Christoffel1(D1g): #Creates Christoffel symbols of the first kind.

> Cf2:=Christoffel2(ginv,Cf1): #Creates Christoffel symbols of the second kind.

> displayGR(Christoffel2, Cf2); #Display only the non-zero components of the Christoffel symbols of the second kind. The fa,bcg indicates that the first index is ‘‘contravariant’’ and the second and third indices are ‘‘covariant’’. The Christoffel Symbols of the Second Kind non zero components W m f1; 11gD .r C 2m/r f1; 22gDr C 2m f1; 33gD.r C 2m/sin./2 .r C 2m/m f1; 44gD r3 1 f2; 12gD r f2; 33gDsin./ cos./ 1 f3; 13gD r cos./ f3; 23gD sin./ m f4; 14gD .r C 2m/r > RMN:=Riemann(ginv,D2g,Cf1): #Creates the Riemann curvature tensor.

> displayGR(Riemann,RMN); #Display only the non-zero components of the Riemann tensor. The Riemann Tensor non zero components W 656 Appendix 8 Computational Symbolic Algebra Calculations

m R1212 D r C 2m m sin./2 R1313 D r C 2m 2m R1414 D r3 R2323 D 2rmsin./2 .r C 2m/m R2424 D r2 .r C 2m/msin./2 R3434 D r2 character W Œ1; 1; 1; 1 > RICCI:=Ricci(ginv,RMN); #Create the Ricci tensor. Maple contracts the first and fourth index of the Riemann tensor in order to create the Ricci tensor, compatible with the conventions in this book. It is zero here since we are dealing with a vacuum solution. 2 3 0000 6 00007 RICCI WD table.Œindex char D Œ1; 1; compts D 6 7/ 4 00005 0000

> RS:=Ricciscalar(ginv,RICCI); #The Ricci scalar. RS WD table.Œindex char D Œ; compts D 0/

> upRMN:=raise(ginv, RMN, 1,2,3,4): #Calculates the contravariant Riemann tensor by raising the first, second, third and fourth index using the inverse metric.

> Kret:=prod(RMN,upRMN,[1,1],[2,2],[3,3],[4,4]); #Creates the Kretschmann scalar by multiplying the covariant and contravariant Riemann tensor and contracting first index with first, second with second, third with third and fourth with fourth. 48 m2 Kret WD table.Œindex char D Œ; compts D / r6 > Estn:=Einstein(g,RICCI,RS): #Calculates the covariant Einstein tensor. Appendix 8 Computational Symbolic Algebra Calculations 657

> mixeinst:=raise(ginv,Estn,1): #Raises the first index of the covariant Einstein tensor, creating the mixed-character Einstein tensor.

> displayGR(Einstein,mixeinst); #Displays only the non-zero components of the mixed-character Einstein tensor. There are no non-zero components. Caution must be applied when using this command to display this tensor’s components; only components where the row-index is less than or equal to the column-index will be displayed. The Einstein Tensor non zero components W None character W Œ1; 1

A8.2 Sample MathematicaTM Notebook

This notebook deals with quantities related to the vacuum Schwarzschild metric. It is based on a notebook which originally supplemented the text Gravity: An introduction to Einstein’s general relativity by Hartle [124]. Mathematica notebook involving the vacuum Schwarzschild metric.

This next command clears various variables that might be set. Note that in Mathematica, you need to press shift - enter after you are finished typing a cell. ClearŒcoord; metric; ginv; riemann; ricci; rs; einstein; christoffel;r;theta; phi;t

This next command sets the coordinates to be used. coord Dfr; theta; phi;tg

Out[2]=fr; theta; phi;tg

This next command creates the metric. metric Dff1=.1 2 m=r/;0;0;0g; f0; r ^2; 0; 0g; f0; 0; r ^2 .SinŒtheta/^2; 0g; f0; 0; 0; .1 2 m=r/gg 658 Appendix 8 Computational Symbolic Algebra Calculations

nn o ˚ ˚ ˚ o 1 2 2 2 2m Out[3]= 2m ;0;0;0 ; 0; r ;0;0 ; 0; 0; r SinŒtheta ;0 ; 0; 0; 0; 1 C r 1 r

This next command will display the metric in matrix form. metric//MatrixForm 0 1 1 00 0 1 2m B r C B 2 C Out[4]//MatrixForm=B 0r 00C @ 00r2SinŒtheta2 0 A 2m 00 0 1 C r This next command will create the inverse metric. ginv1 D InverseŒmetric (( ) ( ) 2 2 2 2mrSinŒtheta r SinŒtheta 3 2 4 2 2m 2m 2mr SinŒtheta r SinŒtheta 1 r 1 r Out[5]= 3 2 4 2 ;0;0;0 ; 0; 3 2 4 2 ;0;0 ; 2mr SinŒtheta r SinŒtheta 2mr SinŒtheta r SinŒtheta 2m 2m 2m 2m ( 1 r 1 r) 8 1 r 1 r 99 2mr r2 < == 2m 2m 4 2 1 r 1 r r SinŒtheta 0; 0; 3 2 4 2 ;0 ; 0; 0; 0; 2mr SinŒtheta r SinŒtheta : 2mr3SinŒtheta2 r4SinŒtheta2 ;; 1 2m 1 2m 1 2m . r / 2m 2m r r 1 r 1 r The last output was messy. Let’s simplify it a bit with the Simplify command. ginv D SimplifyŒginv1 n˚ ˚ n o ˚ o 2m 1 CscŒtheta2 r Out[6]= 1 r ;0;0;0 ; 0; r2 ;0;0 ; 0; 0; r2 ;0 ; 0; 0; 0; 2mr

We can also display the inverse metric in matrix form. ginv//MatrixForm 0 1 1 2m 00 0 B r C B 1 C 0 r2 00 Out[7]//MatrixForm=B 2 C @ CscŒtheta A 00 r2 0 r 00 0 2mr We next calculate the Christoffel symbols of the second kind. The triplets such as fi;1;4g in the braces correspond to the index (i) and the range of the index (1 to 4). The operator DŒmetricŒŒs; j ; coordŒŒk calculates the partial derivative of the s j metric component with respect to the coordinate k. christoffel D SimplifyŒTableŒ.1=2/ SumŒ.ginvŒŒi; h/ .DŒmetricŒŒh; j ; coord ŒŒk C DŒmetricŒŒh; k; coordŒŒj DŒmetricŒŒj; k; coordŒŒh/; fh; 1; 4g; fi;1;4g; fj; 1; 4g; fk; 1; 4g Appendix 8 Computational Symbolic Algebra Calculations 659 ˚˚˚ ˚ m 2 n Out[8]= 2mroor2 ;0;0;0 ; f0; 2m r; 0; 0g; 0; 0; .2m r/SinŒtheta ;0 ; C ˚˚ ˚ 0; 0; 0; m. 2m r/ ; 0; 1 ;0;0 ; 1 ;0;0;0 ; f0; 0; CosŒthetaSinŒtheta; 0g; r˚˚3 r r ˚ 1 1 ˚˚f0; 0; 0; 0gg ; 0; 0; r ;0 ; f0; 0; CotŒtheta; 0˚g; r ; CotŒtheta; 0; 0 ; f0; 0; 0; 0g ; m m 0; 0; 0; 2mrr2 ; f0; 0; 0; 0g; f0; 0; 0; 0g; 2mrr2 ;0;0;0

Next the Riemann tensor (first index contravariant, second through fourth index covariant) is created. Note that this differs from the one calculated in the MapleTM worksheet which was completely covariant in character. The riemann:= statement supresses the output. riemann:=riemann D SimplifyŒTableŒ DŒchristoffelŒŒi; j; l; coordŒŒk DŒchristoffelŒŒi; j; k; coordŒŒlC SumŒchristoffelŒŒs; j; lchristoffelŒŒi; k; s christoffelŒŒs; j; kchristoffelŒŒi; l; s; fs; 1; 4g; fi;1;4g; fj; 1; 4g; fk; 1; 4g; fl;1;4g

This next command will display only the non - zero Riemann components. Note that these differ from the ones calculated in the MapleTM worksheet which was completely covariant in character. The output from the first command is supressed. riemannarray:=TableŒIfŒUnsameQŒriemannŒŒi;j;k;l;0;fToStringŒRŒi; j; k; l; riemannŒŒi;j;k;lg; fi;1;4g; fj; 1; 4g; fk; 1; 4g; fl;1;k 1g

TableFormŒPartitionŒDeleteCasesŒFlattenŒriemannarray; Null; 2

m R[1, 2, 2, 1] r mSinŒtheta2 R[1, 3, 3, 1] r 2m.2mCr/ R[1, 4, 4, 1] r4 m R[2, 1, 2, 1] .2mr/r2 2mSinŒtheta2 R[2, 3, 3, 2] r m.2mr/ R[2, 4, 4, 2] r4 Out[11]//TableForm= m R[3, 1, 3, 1] .2mr/r2 2m R[3, 2, 3, 2] r m.2mr/ R[3, 4, 4, 3] r4 2m R[4, 1, 4, 1] .2mr/r2 m R[4, 2, 4, 2] r mSinŒtheta2 R[4, 3, 4, 3] r This next command creates the Ricci tensor. Note the summation over the index i. ricci:=ricci D SimplifyŒTableŒSumŒriemannŒŒi;j;l;i;fi;1;4g; fj; 1; 4g; fl;1;4g 660 Appendix 8 Computational Symbolic Algebra Calculations

Next we display only the non - zero components of the Ricci tensor. There are no non-zero components. ricciarray:=TableŒIfŒUnsameQŒricciŒŒj; l; 0; fToStringŒRŒj; l; ricciŒŒj; lg; fj; 1; 4g; fl;1;jg

TableFormŒPartitionŒDeleteCasesŒFlattenŒricciarray; Null; 2 Out[14]//TableForm = fg

We next calculate the Ricci curvature scalar.

RS D SumŒginvŒŒi; j ricciŒŒi; j ; fi;1;4g; fj; 1; 4g Out[15]=0

We next calculate the Einstein tensor, by explicitly creating it with the Ricci tensor, the Ricci scalar and the metric. einstein:=einstein D ricci .1=2/RS metric

Display only the non - zero components of the Einstein tensor. There are no non-zero components, indicating a vacuum solution. einsteinarray:=TableŒIfŒUnsameQŒeinsteinŒŒj; l; 0; fToStringŒGŒj; l; einsteinŒŒj; lg; fj; 1; 4g; fl;1;jg

TableFormŒPartitionŒDeleteCasesŒFlattenŒeinsteinarray; Null; 2 Out[18]//TableForm=fg References

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Symbols Bianchi’s differential identities, 60 1-form, 9 complex-valued, 512 4-acceleration, 181 consequences of in NewmanÐPenrose 4-force, 115 formalism, 527Ð529 4-momentum, 116 first contracted, 63 total, 121 second contracted, 63 4-velocity, 113 bicharacteristic curves, 596 big crunch, 426 big-bang cosmological model, 423, 434 A Birkhoff’s theorem, 260 acceleration bivector space, 470 4-acceleration, 181 black hole, 351Ð418 Newtonian, 74 Bondi-Metzner-Sachs group, 231 action function or functional boost, 109 (see also Lagrangian), 570, 574 Born-Infeld (or tachyonic) scalar field, ADM action, 582 451Ð455 affine parameter, 77, 79 BoyerÐLindquist coordinate chart, 322, 384 alternating operation, antisymmetrization, 27 BrinkmanÐRobinsonÐTrautman metric, 495 angle field, 42 Buchdahl inequality, 252 anisotropic fluid, 212Ð214, 265 bugle, 69 collapse, 408Ð415 anti-de Sitter spaceÐtime, 192, 621, 642Ð643 anti-self-dual, 651 C antisymmetric oriented tensor, 48 canonical energy-momentum-stress tensor, 119 antisymmetric tensor, 27 canonical or normal forms, 492 antisymmetrization, 27 Cartesian chart, 43, 68 arc length parameter, 82 Casimir effect, 615 arc separation function, 85 Cauchy horizon, 388, 402 arc separation parameter, 77 Cauchy problem, 203 Arnowitt-Deser-Misner action integral, 582 Cauchy-Kowalewski theorem, 203 atlas, 3 causal cone, 107 complete, 3 causal spaceÐtime, 642 maximal, 3 causality violation, 641 characteristic hypersurface, 600 B characteristic matrix, 601 Betti number, 565 characteristic ordinary differential equations, Bianchi type-I models, 433 588

A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical 669 Exposition, DOI 10.1007/978-1-4614-3658-4, © Springer Science+Business Media New York 2012 670 Index characteristic polynomial equation, 174 normal or hypersurface orthogonal, 67 characteristic surface, 595 orthogonal, 61 charge-current, 124 PainleveÐGullstrand,« 357 charged dust, 125, 133, 220, 305 pseudo-Cartesian, 43, 68 chart, 1 Regge-Wheeler tortoise, 358 Christoffel symbol, 57 Riemann normal, 67 first kind, 57 Synge’s doubly-null, 359 second kind, 57 Weyl’s, 278 closed form, 33 Weyl-Lewis- Papapetrou (WLP), 317 Codazzi-Mainardi equations, 100 coordinate components, 20 commutator or Lie bracket, 39 coordinate conditions, 164 comoving coordinate chart, 193 cosmological constant, 162, 226, 420 complete atlas, 3 cosmological principle, 419 complex conjugate coordinates, 598 cotangent vector field, 9 complex electromagnetic potential, 342 cotangent vector space, 9 complex gravo-electromagnetic potential, 342 Cotton-Schouten-York tensor, 297 complex null tetrad field, 468 covariant, 51 complex potential, 319, 333 covariant components, 10 components covariant derivative, 50 coordinate, 20 of relative and oriented relative tensor covariant, 10 fields, 63 orthonormal, 46 covariant index, 50 physical, 120 covariant order, 17 conformal group, 618 covariant vector field, 9 conformal mapping, 70 critical energy density, 433 conformal tensor, 71 critical point, 569 conformally flat domain, 71 curvature invariant, 63 conformally flat spaceÐtime, 146, 617Ð624 curvature scalar, 63 conformastat metric, 297 curvature tensor, 54 conformastationary metric, 344 curve conjugate holomorphic function, 598 integral, 35 conjugate points, 81 non-null, 81 conjugate, contravariant, or inverse metric null, 111 tensor, 43 parameterized, 11 constant curvature, space, 68, 73 profile, 91 continuity equation, 122, 125, 183, 225 spacelike, 111 continuum mechanics, 117 timelike, 111 contraction operation, 21 curved domain, 64 contravariant index, 50 Curzon-Chazy metric, 288 contravariant order, 17 cusp, 16 coordinate chart or system BoyerÐLindquist, 322, 384 Cartesian, 43, 68 D comoving, 193 dark energy, 446 Doran, 400 de Sitter spaceÐtime, 416, 436, 621 doubly-null, 130 deceleration function, 433 EddingtonÐFinkelstein, 358, 370 deceleration parameter, 433 Gaussian normal, 67, 197 deDonder gauge, 627 geodesic normal, 67, 102, 197 deformable solid, 214Ð215, 265 Kerr-Schild, 400 degree (system of p.d.e.s), 587 KruskalÐSzekeres, 366Ð367 delta local, 1 Kronecker, 8 local Minkowskian, 151 derivative Minkowskian, 105 covariant, 50, 63 Index 671

covariant directional, 51 elementary divisors, 609 directional, 7, 51 elliptic p.d.e., 594 exterior, 31 embedded manifold, 93 Fermi, 145 Emden equation, 551 gauge covariant, 538 energy conditions, 178 Lie, 35Ð38 energy-momentum-stress tensor, 118 variational, 571, 575 equation of state, 211 derivative mapping, 22 equilibrium of a deformable body, 215 determinate system, 164 equivalence principle, 137 differentiable manifold, 3 ergosphere, 388 differential conservation of energy- Ernst equation, 320 momentum, 119 eternal black holes, 403 differential form, 30 eternal white holes, 403 directional derivative, 7 Euler equations for a perfect fluid, 225 domain Euler-Lagrange equations, 78, 157, 571 curved, 64 classical mechanics, 572 flat, 64 for particle in curved spaceÐtime, 155 dominant energy condition, 178 for particle in Schwarzschild spaceÐtime, Doran coordinate chart, 400 234, 235 dot product or inner-product, 40 for scalar field, 575 double Hodge-dual, 85 event horizon, 354 doubly-null coordinates, 130 exact form, 33 dual vector space, 9 expansion scalar, 181 dubiosity, 82 expansion tensor, 181 dust, 122, 125, 133, 182, 183, 210, 220, 305, extended extrinsic curvature, 580 420, 422, 424, 426, 433, 438, 455 extended real numbers, 394 collapse, 370Ð374 exterior derivative, 31 exterior product, 28 extrinsic curvature tensor, 88Ð104, 165 E extended, 580 eccentricity, 236 EddingtonÐFinkelstein coordinates, 358, 370 Eguchi-Hansen gravitational instantons, 651 F Einstein space, 69 FÐLÐRÐW metrics (or FriedmannÐLemaˆõtreÐ Einstein static universe, 419 RobertsonÐWalker), 421Ð456 Einstein summation convention, 6 F-W transport, 145 Einstein tensor, 63 Fermi derivative, 145 Einstein’s field equations, 162 Fermi-Walker transport, 145 complex-valued form, 511 fifth force, 621 EinsteinÐMaxwell equations, 253 fine-structure constant (electrodynamic), 559 EinsteinÐMaxwellÐKleinÐGordon equations, fine-structure parameter (eigenvalue of 537 charged scalar-wave gravitational orthonormal form, 541 condensate), 559 Einstein-Hilbert Lagrangian density, 166 Finsler metric, 156 Einstein-Maxwell equations, 218 first contracted Bianchi’s identities, 63 Einstein-Rosen bridge, 634 first curvature, 145 electrical charge density, 125 first fundamental form, 91 electro-vac universe, 253 Flamm’s paraboloid, 233 electromagnetic duality-rotation, 226 flat domain, 64 electromagnetic field tensor, 33, 50 flat manifold, 67 electromagnetic four-potential, 34, 218, 226, Fock-deDonder gauge, 627 486, 538 form electromagneto-vac domain, 218 1-form, 9 electrostatic potential, 307 closed, 33 672 Index form (cont.) Mobius,¬ 342 differential, 30 Poincare,« 108 exact, 33 symplectic, 171 Frenet-Serret formulas, 83 group of motion, 73 generalized, 82 frequency of a wave, 486 FriedmannÐLemaˆõtreÐRobertsonÐWalker H metrics (or FÐLÐRÐW), 421Ð456 HÐKÐSÐD metric (or Hawking-Kloster-Som- Das), 650 Hamilton-Jacobi equation, 591 G Hamiltonian Godel¬ universe, 643Ð644 for particle in a static gravitational gauge covariant derivatives, 538 potential, 313 gauge transformation, 34 for particle in an electromagnetic field, 158 Gauss’ equations, 100 relativistic, 156 Gauss’ theorem, generalized, 65 relativistic equations of motion, 158 Gauss-Bonnet term, 584 relativistic mechanics, 156 Gaussian curvature, 91 super, 157 Gaussian normal coordinate chart, 67, 197 harmonic coordinate conditions, 172 general covariance, 130 harmonic function, 296, 599 general relativity, 166 harmonic gauge, 172, 627 generalized D’Alembertian, 216 Hausdorff manifold, 1 generalized Frenet-Serret formulas, 82 Hawking-Kloster-Som-Das metric (or generalized Kronecker tensor, 29 HÐKÐSÐD), 650 generalized wave operator, 216 heat flow vector, 225 geodesic, 76 Heaviside-Lorentz units, 122 non-null, 77 Hessian, 592 null, 77 Higgs fields, 456 geodesic deviation equations, 80 Hilbert-Palatini approach for deriving field geodesic equations, 77 equations, 577 for dust, 183 Hodge star operation, 49 for particle in T -domain, 356 Hodge-dual operation, 479 for particle in Schwarzschild spaceÐtime, holomorphic function, 598 234 holonomic constraint, 157 for the Euclidean plane E2,78 homeomorphism, 1 geodesic normal coordinate chart, 67, 102, 197 homogeneity, 427 geodesically complete manifold, 168 homogeneous p.d.e., 587 geometrized units, 162 homogeneous state of the complex wave geons, 567 function, 547 Global Positioning System (G.P.S.), 323 Hopf’s theorem, 296 gravitational constant, G, 66, 160 Hubble function, 433 gravitational field equations, 162 Hubble parameter, 433 gravitational instantons, 337, 647Ð651 hybrid tensor field, 96Ð98 gravitational mass, 136 hyperbolic p.d.e., 594 gravitational redshift, 242 hypersurface, 354 gravitational wave astronomy, 625 null, 130 gravitational waves, 166, 625Ð631 hypersurface orthogonal coordinate, 67 gravitons, 628 gravo-electromagnetic potential, 342 group I Bondi-Metzner-Sachs, 231 I-S-L-D jump conditions, 165 conformal, 618 identity tensor, 45 Lie, 44 immersion, 93 Lorentz, 44, 108 incoherent dust, 182 Index 673 index KruskalÐSzekeres coordinates, 366Ð367 contravariant, 50 KruskalÐSzekeres metrics, 366Ð367 covariant, 50 index lowering, 47 index raising, 47 L inertial mass, 136 Lagrange multiplier, 157 inertial observer, 112 Lagrange’s solution method, 588 inflationary era, 431 Lagrangian, 570 initial value problem, 203 alternative Lagrangians, 159 inner-product or dot product, 40 augmented gravitational, 579 instanton, 337, 647Ð651 classical mechanics, 572 instanton-horizon, 648 EinsteinÐMaxwellÐKleinÐGordon, 541 integrability conditions, 98 Einstein-Hilbert, 577 integral conservation, 120 Einstein-Maxwell, 218 integral curve, 35 first-order gravitational, 581 intrinsic curvature for particle in T domain, 356 extended, 580 for particle in curved spaceÐtime, 154, 160 intrinsic metric, 90 for particle in electromagnetic field, 158 invariant eigenvalue problem, 91 for particle in Schwarzschild spaceÐtime, invariant eigenvalues, 174 233, 234 irrotational motions, 181 for scalar field, 575 isomorphism, 483 for static field equations, 292 on tensor space, 47 for stationary field equations, 339, 340 isotropic sectional curvature, 427 for tachyonic scalar field, 453 isotropy, 427 relation to super-Hamiltonian, 157 isotropy equation of static spherical symmetry, square-root particle, 453 250 Lagrangian density, 166 augmented gravitational, 579 EinsteinÐMaxwell, 346 J Einstein-Hilbert, 166, 577 Jacobian, 3 most general curvature yielding Jacobian mapping, 22 second-order equations, 584 Jordan decomposition theorem, 607 Lambert W-Function, 361 junction conditions Lane-Emden equation, 551 I-S-L-D, 165 Laplacian operator, 84 Synge’s, 164 Legendre transformation, 156 Lemaˆõtre metric, 269 lemma of Poincare,« 32 K length or norm, 42 Kaluza-Klein theory, 456, 464 Levi-Civita permutation symbol, 29 Kasner metric, 207, 294, 298, 311 Levi-Civita tensor, 48 Kerr metric, 322Ð326, 384Ð403 Lichnerowicz-Synge lemma, 204 higher dimensional generalization, 329 Lie bracket or commutator, 39 KerrÐNewman solution, 401 Lie derivative, 35Ð38 Kerr-Newman solution, 323 Lie group, 44 Kerr-Schild coordinate chart, 400 light cone, 107 Killing tensor, 399 linear equation, 587 Killing vector, 72 linear p.d.e., 587 KleinÐGordon field linearized theory of gravitation, 625Ð631 (see also scalar field, massive), 537Ð567 Liouville equation, 597 Kottler solution, 243 Lipschitz condition, 35 Kretschmann invariant, 86, 385 local coordinate system, 1 Kronecker delta, 8 local Minkowskian coordinates, 151 Kronecker tensor, generalized, 29 London equation, 547 674 Index

Lorentz gauge condition, 172, 216, 627 conjugate, contravariant, or inverse, 43 Lorentz group, 44, 108 Curzon-Chazy, 288 Lorentz matrix, 132 de Sitter, 416, 436, 621 Lorentz metric, 43, 64 Einstein static universe, 419 Lorentz signature, 42 Finsler, 156 Lorentz transformation, 44, 108 FriedmannÐLemaˆõtreÐRobertsonÐWalker orthochronus, 189 (or FÐLÐRÐW), 421Ð456 LorentzÐHeaviside units, 122 Godel,¬ 643 Hawking-Kloster-Som-Das (or HÐKÐSÐD), 650 M Kasner, 207, 294, 298, 311 magnetic potential, 303 Kerr, 322Ð326, 384Ð403 manifold higher dimensional generalization, 329 constant curvature, 68, 73 KerrÐNewman, 401 differentiable, 3 Kerr-Newman, 323 Einstein, 69 Kottler, 243 embedded, 93 KruskalÐSzekeres, 366Ð367 flat, 67 Lemaˆõtre, 269 geodesically complete, 168 Lorentz, 43, 64 Hausdorff, 1 positive-definite, 42, 64 orientable, 3 pseudo-Schwarzschild, 299 paracompact, 1 Reissner-Nordstrom-Jeffery,¬ 255, 381 pseudo-Riemannian, 64 RobinsonÐTrautman, 527 Ricci flat, 70, 162 Schwarzschild, 231Ð242, 299, 336, Riemannian, 64 351Ð369 Riemannian or pseudo-Riemannian, 56, 64 higher dimensional generalization, topological, 3 244 Maple(TM), 653 Schwarzschild’s interior, 253 mass signature, 42 gravitational, 136 Taub-NUT, 651 inertial, 136 tensor, 41 mass density, 117 Tolman-Bondi-Lemaˆõtre, 265 mass-shell, 116 Vaidya, 273 Mathematica(TM), 653 Van Stockum, 644 matrix warp-drive, 638 characteristic, 601 Weyl-Majumdar-Papapetrou-Das Lorentz, 132 (or W-M-P-D), 307 nilpotent, 606 wormhole, 635 rank of, 17 metric equations, 309 unit, 8 metric tensor, 41 matter-dominated era, 430 Minkowskian chart, 105 maximal atlas, 3 Minkowskian chronometry, 112 maximum-minimum principle, 296 Monge surface, 102 Maxwell’s equations, 33, 122 monogenic forces, 154 MaxwellÐLorentz equations, 124 most general solution, 585 mean curvature, 91 multilinearity conditions, 17 Meissner effect, 547 Mobius¬ group, 342 metric anti-de Sitter, 192, 621, 642 Bianchi type-I, 433 N BrinkmanÐRobinsonÐTrautman, 495 NewmanÐPenrose equations, 517, 523Ð525 components, 43 Newtonian acceleration, 74 conformastat, 297 Newtonian approximation, 236 conformastationary, 344 Newtonian gravitational constant, G, 66, 160 Index 675

Newtonian gravitational potential, 66, 136, parametric surface, 88 140, 160Ð161, 167Ð168, 198, 269, parametrized hypersurface, 93 279Ð283, 288, 292, 307 Penrose-Carter compactification, Newtonian mass, 66 394 nilpotent matrix, 606 perfect fluid, 177, 210Ð212 nilpotent operator, 33 perihelion, 236 no hair theorem, 398 perihelion shift, 237 non-degeneracy conditions, 12 permutation symbol, 29 non-linear eigenvalue problem for a theoretical Petrov-types-I, D, II, N and III, fine-structure constant, 559 493 non-null differentiable curve, 81 physical components, 120 non-null geodesic, 77 Planck length, 457 norm or length, 42 Planck mass, 567 normal coordinate, 67 Planck’s constant, 538 normal forms, 595 Poincare« group, 108 normal or canonical forms, 492 Poincare’s« lemma, 32 null cone, 107 Poisson’s equation, 161 null curve, 111 polarization states of gravitational waves, null electromagnetic field, 224, 485 630 null energy condition, 178 polarization tensor, 629 null hypersurface, 130 polytropic equation of state, 416 null vectors, 106 positive-definite metric, 42, 64 numerical tensor, 29, 110 potential NUT potential, 649 complex, 319, 333 complex electromagnetic, 342 complex gravo-electromagnetic, 342 O electromagnetic four, 34, 218, 226, 486, Oppenheimer-Volkoff equation, 250 538 order (system of p.d.e.s), 587 electrostatic, 307 order of a tensor, 16, 17 equation, 309, 311 orientable manifold, 3 function, 296 oriented relative tensor field, 25 gravo-electromagnetic, 342 oriented tensor field, 49 magnetic, 303 oriented volume, 64 Newtonian gravitational, 66, 136, 140, orthochronus Lorentz transformation, 189 160Ð161, 167Ð168, 198, 269, orthogonal coordinate chart, 61 279Ð283, 288, 292, 307 orthogonal vector fields, 41 NUT, 649 orthonormal basis, 41 scalar field, 446, 449 orthonormal components, 46 twist, 333, 340, 342 outer ergosurface, 388 potential equation, 309, 311 outer product, 18 potential function, 296 overdetermined system, 164 pressure, 177 principal curvature, 145 principal pressures, 179 P principal tensions, 179 PainleveÐGullstrand« coordinate chart, 357 principle of equivalence, 137 Palatini approach for deriving field equations, Proca equation, 547 577 profile curve, 91 Papapetrou-Ehlers class, 336 projection tensor, 180 parabola, semi-cubical, 16 proper time, 112 parabolic p.d.e., 594 pseudo-angle, 43 paracompact manifold, 1 pseudo-Cartesian chart, 43, 68 parallelly propagated, 74 pseudo-Riemannian manifold, 64 parameterized curve, 11 pseudo-Schwarzschild metric, 299 676 Index

Q massive, 443Ð446 quantum field theory in curved spaceÐtimes, massless, 440Ð443, 456Ð464, 575 538 quintessence, 446Ð450 quantum gravity, 583, 633, 651 scalar field potential, 446, 449 quasi-linear p.d.e., 587 Schwarzschild metric, 231Ð242, 299, 336, quintessence scalar field, 446Ð450 351Ð369 higher dimensional generalization, 244 Schwarzschild radius, 232, 351 R Schwarzschild’s interior solution, 253 radiation era, 431 second contracted Bianchi’s identities, 63 Rainich problem, 542 second fundamental form, 91 raising and lowering of indices, 47 Segre characteristic, 609Ð611 rank of energy-momentum-stress tensors, matrix, 17 175Ð177 tensor, 16, 17 of Ricci tensor, 498 Raychaudhuri-Landau equation, 181Ð182 self-dual, 651 redshift, 242 self-duality, 480 Regge-Wheeler tortoise coordinate, 358 semi-cubical parabola, 16 region, 395 semi-linear p.d.e., 587 regularity conditions, 12 semiclassical gravity, 538 ReissnerÐNordstromÐJeffery¬ metric, 381 separation of a vector field, 41 Reissner-Nordstrom-Jeffery¬ metric, 255 shear tensor, 181 relative tensor field, 25, 63 shock waves, 586 relativistic canonical equations of motion, 158 signature of the metric, 42 relativistic Hamiltonian equations of motion, singularity theorems, 417Ð418, 434Ð435 158 solid particle, 221 relativistic Hamiltonian mechanics, 156 space of constant curvature, 68, 73 relativistic Lorentz equations of motion, 126 spaceÐtime foam, 633, 634 relativistic total angular momentum, 121 spacelike curve, 111 reparameterization of a curve, 13, 14 spacelike vectors, 106 reparametrization, 94 spin coefficients, 517 Ricci flat space, 70, 162 spinor algebra, 525 Ricci identities, 61 spinor analysis, 525 Ricci rotation coefficients, 58 spinor fields, 110 complex-valued, 511 stationary limit surface, 388 Ricci tensor, 62 Stokes’ theorem, generalized, 34 Riemann curvature tensor, 54 stress density, 117 Riemann normal coordinate chart, 67 stress tensor field, 118 Riemann-Christoffel curvature tensor, 59, 62 string theory, 453, 456, 464, 537 algebraic identities, 59 strong energy condition, 178 differential identities, 60 summation convention, 6 Riemannian manifold, 64 surface Riemannian or pseudo-Riemannian manifold, Monge, 102 56, 64 parametric, 88 rigid motion, 39 ruled, 85, 96 rigid motions, 181 surface of revolution, 91 RobinsonÐTrautman metric, 527 Sylvester’s law, 600 Routh’s theorem, 234 symmetric tensor, 27 ruled surface, 85, 96 symmetrization operation, 26 symmetrized curvature tensor, 153, 170 symplectic group, 171 S Synge’s doubly-null coordinates, 359 scalar field, 19, 440Ð456 Synge’s junction conditions, 164 Born-Infeld or tachyonic, 451Ð455 Synge’s world-function, 86 Index 677

T vorticity, 181 T-domain, 269 Weyl conformal, 71 tachyon condensation, 453 Weyl’s projective, 71 tachyonic (or Born-Infeld) scalar field, tensor density field, 25 451Ð455 tensor field, 19 tangent vector, 6, 7 tensor product, 18 tangent vector space, 7 tetrad, 106, 465 Taub-NUT metric, 651 complex null, 468 tensor Theorema Egregium, 101 antisymmetric, 27 tidal forces, 141 canonical energy-momentum-stress, 119 time machine, 641Ð645 conformal, 71 time orientable space-times, 641 conjugate, contravariant, or inverse metric, timelike curve, 111 43 timelike vectors, 106 Cotton-Schouten-York, 297 Tolman-Bondi-Lemaˆõtre metric, 265 curvature TomimatsuÐSato solution, 329 Riemann, 54 topological manifold, 3 extended extrinsic, 580 topology, 1 extrinsic, 88Ð104, 165 torsion tensor, 54 Ricci, 62 total 4-momentum, 121 Riemann-Christoffel, 59, 62 total charge, 133 symmetrized, 153, 170 total conservation of energy-momentum, 120 density, 25 total mass, 133 Einstein, 63 total mass function, 261, 262 electromagnetic field, 33, 50 totally antisymmetric oriented tensor, 48 energy-momentum-stress, 118 totally antisymmetric tensor, 27 expansion, 181 totally differentiable function, 10 extended extrinsic curvature, 580 totally geodesic hypersurface, 313 extrinsic curvature, 88Ð104, 165 trace-reversed perturbation tensor, 627 field, 19 traceless-transverse gauge, 629 generalized Kronecker, 29 twist potential, 333, 340, 342 hybrid field, 96Ð98 identity, 45 Killing, 399 U Levi-Civita, 48 underdetermined system, 164 metric, 41 unit matrix, 8 numerical, 29, 110 unit vector field, 41 order of, 16, 17 upper triangular form, 606 oriented, 25, 49, 63 oriented antisymmetric, 48 polarization, 629 V projection, 180 vacuum field equations, 162 rank of, 16, 17 Vaidya metric, 273 relative, 25, 63 Van Stockum spaceÐtime, 644Ð645 Ricci, 62 variational derivative, 571, 575 Riemann, 54 variationally permissible boundary conditions, Riemann-Christoffel, 59, 62 571, 575 shear, 181 vector stress, 118 covariant, 9, 10 symmetric, 27 heat flow, 225 symmetrized curvature, 153, 170 Killing, 72 torsion, 54 norm or length, 42 totally antisymmetric, 27, 48 null, 106 trace-reversed perturbation, 627 separation, 41 678 Index vector (cont.) weight of relative tensor field, 25 spacelike, 106 Weingarten’s equations, 98 tangent, 6, 7 Weyl conformal tensor, 71 timelike, 106 Weyl’s coordinate chart, 278 unit, 41 Weyl’s projective tensor, 71 vector space Weyl-Lewis-Papapetrou (WLP) charts, 317 cotangent, 9 Weyl-Majumdar-Papapetrou-Das dual, 9 (or W-M-P-D) metric, 307 tangent, 7 white hole, 366 volume Willmore’s theorem, 618 oriented, 64 winding number, 14 vorticity tensor, 181 world line, 112 world tube, 117 world-function (Synge’s), 86 W wormhole, 169, 255, 633Ð638 warp-drive, 638Ð641 wave number, 486 weak energy condition, 178 weak solution, 586 Y wedge product, 28 Yukawa equation, 549