PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY O ~ CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge, United KingdomCAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, UK40 West 20th Street, New York, NY 10011-4211, USA477 Williamstown Road, Port Melbourne, VIC 3207, AustraliaRlIiz de Alarcon 13, 2g014 Madrid, SpainDock House, The Waterfront, Cape Town 8(Xll, South Africahttp://www.cambridge.org 0, and similar expressions hold for P2 and P3. The weak energy conditiontherefore implies2.1 Energy conditions 31p ~ 0, P + Pi> O. (2.7)2.1.3 Null energy conditionThe null energy condition makes the same statement as the weak form, except thatv'" is replaced by an arbitrary, future-directed null vector k"'. Thus,(2.8)is the statement of the null energy condition. Substituting Eqs. (2.1) and (2.5) givesP +a'2pI + bt2p2 +C'2p3 ~ O.Choosing b' = cl= 0 enforces a' = 1, and we obtain P + PI ~ 0, with similarexpressions holding for P2 and P3. The null energy condition therefore impliesP +Pi ~ O.Notice that the weak energy condition implies the null form.2.1.4 Strong energy conditionThe statement of the strong energy condition is(2.9)(2.10)or TafJvavfJ ~ - ~ T , where va is any future-directed, normalized, timelike vector.Because TafJ - ~ Tg"'fJ = RafJ /8n by virtue of the Einstein field equations, thestrong energy condition is really a statement about the Ricci tensor. SubstitutingEqs. (2.1) and (2.4) gives2( 2 b2 2 1 ( )y P +a PI + P2 +c P3) ~ 2 P - PI - P2 - P3 .Choosing a = b = c = 0 enforces y = 1, and we obtain P + PI + P2 + P3 ~ O.Alternatively, choosing b = c = 0 implies y2 = I/O - a2), and after some sim-ple algebra we obtain P + PI + P2 + P3 ~ a2(p2 + P3 - P - pt). Because thismust hold for any a2< 1, we have P + PI ~ 0, with similar relations holding forP2 and P3. The strong energy condition therefore impliesP +PI + P2 + P3 ~ 0, p+ Pi ~ O. (2.11)It should be noted that the strong energy condition does not imply the weak form.32 Geodesic congruences2.1.5 Dominant energy conditionThe dominant energy condition embodies the notion that matter should flow alongtimelike or null world lines. Its precise statement is that if va is an arbitrary, future-directed, timelike vector field, then- T ~ vfJ is a future-directed, timelike or null, vector field. (2.12)The quantity - T ~ vfJ is the matter's momentum density as measured by an ob-server with four-velocity va, and this is required to be timelike or null. SubstitutingEqs. (2.1) and (2.4) and demanding that - T ~ vfJ not be spacelike givesp2 _ a2pl 2_ b2pl- C2p32:::: O.Choosing a = b = c = 0 gives p2 :::: 0, and demanding that - T ~ vfJ be future di-rected selects the positive branch: p :::: o. Alternatively, choosing b =c = 0 givesp2 :::: a2P12. Because this must hold for any a2< I, we have p :::: IplI, havingtaken the future direction for - TpvfJ . Similar relations hold for P2 and P3. Thedominant energy condition therefore impliesp:::: 0, (2.13)2.1.6 Violations o/the energy conditionsWhile the energy conditions typically hold for classical matter, they can be violatedby quantized matter fields. A well-known example is the Casimir vacuum energybetween two conducting plates separated by a distance d:Jr2 Iip = -no d4 'Although quantum effects allow for a localized violation of the energy conditions,recent work suggests that there is a limit to the extent by which the energy con-ditions can be violated globally. In this context it is useful to formulate averagedversions of the energy conditions. For example, the averaged null energy conditionstates that the integral of TafJkakfl along a null geodesic y must be nonnegative:i TafJkakfJdA :::: O.Such averaged energy conditions playa central role in the theory of traversablewormholes (see Section 2.6, Problem 1). The averaged null energy condition isknown to always hold in flat spacetime, for noninteracting scalar and electromag-netic fields in arbitrary quantum states; this is true in spite of the fact that TafJkakfJcan be negative somewhere along the geodesic. Its status in curved spacetimes2.2 Kinematics of a deformable medium 33is not yet fully settled. A complete discussion, as of 1994, can be found in MattVisser's book.2.2 Kinematics of a deformable medium2.2.1 Two-dimensional mediumAs a warm-up for what is to follow, consider, in a purely Newtonian context, theinternal motion of a two-dimensional deformable medium. (Picture this as a thinsheet of rubber; see Fig. 2.1.) How the medium actually moves depends on its in-ternal dynamics, which will remain unspecified for the purpose of this discussion.From a purely kinematical point of view, however, we may always write that for asufficiently small displacement ~ a about a reference point 0,for some tensor Bab. The time dependence of this tensor is determined by themedium's dynamics. For short time intervals,whereand b.t = t1 - ta. To describe the action of Bthwe will consider the simple figuredescribed by ~ a ( t a ) = ra(cos, sin); this is a circle of radius ra drawn in thetwo-dimensional medium.\ _/C---+-al--bFigure 2.1 Two-dimensional deformable medium.34 Geodesic congruences2.2.2 ExpansionSuppose first that Bab is proportional to the identity matrix, sO thatBab = ) ,where e == Then = sincf, which corresponds to achange in the circle's radius: n = ro + The corresponding change in areais then t,.A == Al - Ao = rrro2et,.t, so that1 t,.A8=--.Ao t,.tThe quantity eis therefore the fractional::hange of area per unit time; we shall callit the expansion parameter. This is actually a function, because emay depend ontime and on the choice of reference point O.2.2.3 ShearSuppose next that B'b is symmetric and tracefree, so thatThen = rot,.t (a+ cos cf> + ax sin cf>, -a+ sin cf> + ax coscf. The paramet-ric equation describing the new figure is rl (cf = ro(l + a+t,.t cos 2cf> +ax t,.t sin 2cf. If ax = 0, this represents an ellipse with major axis oriented alongthe cf> = 0 direction (Fig. 2.2). If, on the other hand, a+ = 0, then the ellipse's ma-jor axis is oriented along cf> =rr /4. The general situation is an ellipse oriented atan at'bitrary angle. It is easy to check that the area of the figure is not affected bythe transfonnation. What we have, therefore, is a shearing of the figure, and a+and ax are called the shear parameters. These may also vary over the medium... _--_ ..Figure 2.2 Effect of the shear tensor.2.2 Kinematics of a deformable medium2.2.4 RotationFinally, we suppose that Bab is antisymmetric, so that35Then = rowt,.t (sin , - cos ), and the new displacement vector is (tJ) =roecos ', sin '), where ' = - wt,.t. This clearly represents an overall rotationof the original figure, and this operation also leaves the area unchanged; w is calledthe rotation parameter.2.2.5 General caseThe most general matrix Bab has 2 x 2 = 4 components, and it may be expressedasax) (0-0'+ + -wThe action of this most general tensor is a linear combination of expansion, shear,and rotation. The tensor can also be expressed as1Bab = "2e Dab + aab +Wab,where e = (the expansion scalar) is the trace part of Bab, O'ab = B(ab) - !eDab(the shear tensor) is the symmetric-tracefree part of Bab, and Wab = B[abl (therotation tensor) is the antisymmetric part of Bab'2.2.6 Three-dimensional mediumIn three dimensions the tensor Bab would be decomposed as1Bab = "38 Dab + aab +Wab,Where 8 = Baa is the expansion scalar, aab = B(ab) - the shear tensor, andWab :;= B[ab] the rotation tensor. In the three-dimensional case, the expansion is thefractional change of volume per unit time:1 t,. V8=--.V t,.tfo see this, treat the three-dimensional relation.)u Geodesic con/?ruencesas a coordinate transformation from sa (to) to sa (tl). The Jacobian of this transfor-mation isJ = det[oab + Bab\t]= 1+Tr[BabM]= 1+ )\t.This implies that volumes at to and tl are related by VI = (1 +e\t) Va, so thatVa) = (VI - Vo)/ \t. This argument shows also that the volume is not affectedby the shear and rotation tensors.2.3 Congruence of timelike geodesicsLet 6' be an open region in spacetime. A congruence in 6' is a family of curvessuch that through each point in 6' there passes one and only one curve from thisfamily. (The curves do not intersect; picture this as a tight bundle of copper wires.)In this section we will be interested in congruences of timelike geodesics, whichmeans that each curve in the family is a timelike geodesic; congruences of nullgeodesics will be considered in the following section. We wish to determine howsuch a congruence evolves with time. More precisely stated, we want to determinethe behaviour of the deviation vector sa between two neighbouring geodesicsin the congIUence (Fig. 2.3), as a function of proper time r along the referencegeodesic. The geometric setup is the same as in Section 1.10, and the relationsa 1 a fi_OU Ua = -, U ;fiu - , a t:fi _ t: a fiU ;fi':i - ':i ;fiu , urtsa = 0,where urtis tangent to the geodesics, will be assumed to hold. Notice in particularthat sa is orthogonal to ua: the deviation vector points in the directions transverseto the flow of the congruence.Figure 2.3 Deviation vector between two neighbouring members of acongruence.2.3 Congruence of timelike geodesics 372.3.1 Transverse metricGiven the congruence and the associated timelike vector field uOl, the spacetimemetric gOlfJ can be decomposed into a longitudinal part -UOlUfJ and a transversepart hOlfJ given by(2.14)The transverse metric is purely 'spatial', in the sense that it is orthogonalto uOl: uahOlfJ = 0 = hOlfJufJ. It is effectively three-dimensional: in a comovingLorentz frame at some point P within the congruence, Ua ~ (-1,0,0,0), gOlfJ ~diag(-1, 1, 1, 1), and hOlfJ ~ diag(O, 1, 1, 1). We may also note the relations hOlOl =3 and hOl/Lhj = hOlfJ.2.3.2 KinematicsWe now introduce the tensor fieldBOIfJ = uOl;fJ (2.15)Like hOlfJ, this tensor is purely transverse, as uaBOIfJ = UOlUOl:fJ = ! (UOlUOl);fJ = 0and BOIfJufJ = uOI;fJUfJ = O. It determines the evolution of the deviation vector:from ~ ~ f J u f J = u ~ f J ~ f J we immediately obtain(2.16)and we see that Bf; measures the failure of ~ O I to be parallel transported along thecongruence.Equation (2.16) is directly analogous to the first equation of Section 2.2. Wemay decompose BOIfJ into trace, symmetric-tracefree, and antisymmetric parts.This gives1BOIfJ = -e hOlfJ +aOlfJ +waf!,3 (2.17)where e = Baa = U ~ O I is the expansion scalar, aOlfJ = B(OIfJ) -1e hOlf! the sheartensor, and WOIfJ = B[OIfJl the rotation tensor. These quantities come with the sameinterpretation as in Section 2.2. In particular, the congruence will be diverging(geodesics flying apart) if e > 0, and it will be converging (geodesics comingtogether) if e < o.Geodesic congruences}:3___-+__+-_-+ }:2}:!Figure 2.4 Family of hypersurfaces orthogonal to a congruence of timelikegeodesics.2.3.3 Frobenius' theoremSome congruences have a vanishing rotation tensor, wafJ = O. These are said to behypersurjace OIthogonal, meaning that the congruence is everywhere orthogonalto a family of spacelike hypersurfaces foliating 0' (Fig. 2.4). We now provide apartial proof of this statement.The congruence will be hypersurface orthogonal if uais everywhere propor-tional to na, the normal to the hypersurfaces. Supposing that these are describedby equations of the form wa.fJ = 0, (2.38)and this concludes the proof. (In Section 2.6, Problem 6 you will show that ifwa.fJ = 0 for a specific choice of auxiliary null vector Na., then wa.fJ = 0 for allpossible choices.)The congruence is hypersurface orthogonal if there exists a scalar field (xa.)Which is constant on the hypersurfaces and ka. = -M,a. for some scalar M. A50 Geodesic congruencesvector of this form automaticaHy satisfies the geodesic equation:ka;fJkfJ = -C/.L;afJ +,af.L,fJ)kfJ= -(f.L,fJ,fJ)ka,where we have used :afl,fJ = ;fJa,fJ = = 0, This is the gen-eral form of the geodesic equation, con'esponding to a parameterization that is notaffine. Affine parameterization is recovered when f.L,aka = 0, that is, when jJ., doesnot vary along the geodesics.2.4.4 Raychaudhuri's equationThe derivation of the null version of Raychaudhuri's equation proceeds much as inSection 2.3.4. In particular, the equationdB afJ a fJd)" = -B BfJa - RafJk kfollows from the same series of steps. It is then easy to check that BafJ BfJa =Bafl BfJa = +uafJuafJ - wafJwafJ' which givesdB 1 2 afJ afJ a fJ- = --B - u a: fJ +W W fJ - R fJk k (2.39)dA 2 a a a .. This is Raychaudhmi's equation for a congruence ofnul! geodesics. It should benoted that this equation is invariant under a change of auxiliary null vector Na;this is established in SectiOll 2.6, Problem 6. We also note that because the shearand rotation tensors are purely transverse, u afJuafJ 2: 0 and wafJwafJ 2: 0, with theequality sign holding if and only if the tensor vanishes.2.4.5 Focusing theoremThe nuH version of the focusing theorem goes as follows: Let a congruence of nullgeodesics be hypersurface orthogonal, so that wafl = 0, and let the null energycondition hold, so that (by virtue of the Einstein field equations) RafJkakfJ 2: O.Then the Raychaudhuri equatiO!l impliesdB 1 2 afl a fJ- = - -B - u a: fJ - R flk k < 0d)" 2 a a -'which means that the geodesics are focused during the evolution of the congruence.Integrating dB/d).. :s yieldsB-I()..) > e, -J + - 0 2'2.4 Congruence of null geodesics 51where 80 == B(O). This shows that if the congruence is initially converging (Bo : net Cf>.et > O. It iseasy to check that net is given by(3.5)be the normal vector; the sign is chosen so that ketis future-directed when Cf>increases toward the future. Because ketis orthogonal to itself (ketket =0), thisvector is also tangent to the null hypersurface E (Fig. 3.2). In fact, by comput-ing and showing that it is proportional to ket, we can prove that ketis tan-gent to null geodesics contained in I:. We have ket;fJkfJ = Cf>;etfJ$fJ = Cf>;fJet$,fJ = because Cf>.fJCf>,fJ is zero everywhere on E, its gradient must be di-rected along ket, and we have that (Cf>,fJ Cf>,fJ);et = 2Kket for some scalar K. We havefound that the normal vector satisfieskClkfJ = Kket;fJ 'the general form of the geodesic equation. The hypersurface is therefore gener-ated by null geodesics, and ketis tangent to the generators. The geodesics areparameterized by A, so that a displacement along each generator is described bydxet= ketdA. In general A is not an affine parameter, but in special situations inCf> = 0Figure 3.2 A null hypersurface and its generators.62 Hypersuifaceswhich the relations (x"') =constant describe a whole family of null hypersur-faces (so that ,f3,f3 is zero not only on I: but also in a neighbourhood aroundI:), K = 0 and Ais an affine parameter.When the hypersurface is null, it is advantageous to install on I: a coordinatesystem that is well adapted to the behaviour of the generators. We therefore let theparameter Abe one of the coordinates, and we introduce two additional coordinateseA(A = 2,3) to labellhe generators; these are constant on each generator, andthey span the two-dimensional space transverse to the generators. Thus, we shallset(3.6)(3.7)(3.8)when I: is null; varyipg A while keeping eAconstant produces a displacementalong a single generator, and varying eAproduces a displacement across genera-tors.3.1.3 Induced metricThe metric intrinsic to the hypersurface I: is obtained by restricting the line el-ement to displacements confined to the hypersmface. Recalling the parametricequations x'" = x'" (ya), we have that the vectorsax'"e'" =--a ayaare tangent to curves contained in I:. (This implies that e ~ n", = 0 in the non-nullcase, and e ~ k " , = 0 in the null case.) Now, for displacements within I: we haved s ~ = g"'f3 dx'" dxf3(ax'" a) (ax f3b)= g"'f3 aya dy ayb dy= hab dya dyb,where(3.9)is the induced metric, or first fundamental form, of the hypersurface. It is a scalarwith respect to transformations x'" -+ x"" of the spacetime coordinates, but it be-haves as a tensor under transformations ya -+ ya' of the hypersurface coordinates.We will refer to such objects as three-tensors.These relations simplify when the hypersllrface is null and we use thecoordinates of Eq. (3.6). Then e'i = (ax'" ;aA)eA == k'" and it follows that3.1 Description ofhypersuifaces 63hll =gafJkakfJ = 0 and hlA = = 0, because by construction eA ==(3xa/3e Ah is orthogonal to ka. In the null case, therefore, =o'ABdeAdeB, (3.10)where( 3Xa)eA= 3e A ).. (3.11)Here the induced metric is a two-tensor.We conclude by writing down completeness relations for the inverse metric. Inthe non-null case,(3.12)where hetbis the inverse of the induced metric. Equation (3.12) is verified by com-puting all inner products between naand and recovering the expected results.In the null case we must introduce, everywhere on I:, an auxiliary null vectorfield Nasatisfying Naka= -1 and NaeA = 0 (see Section 2.4). Then the inversemetric can be expressed asgafJ = _kaNfi - NakfJ (3.13)where 0' AB is the inverse of 0'AB. Equation (3.13) is verified by computing all innerproducts between ka, Na, and eA'3.1.4 Light cone in flat spacetimeAn example of a null hypersurface in flat spacetime is the future light coneof an event P, which we place at the origin of a Cartesian coordinate sys-tem xa. The defining relation for this hypersurface is Cf> == t - r = 0, wherer2= x2+i +Z2. The normal vector is ka = -3a (t - r) = (-1, x/r, y/r, z/r).A suitable set of parametric equations is t = A., x = A. sin ecos , y = A. sin esin ,and z = A. cose, in which yet = (A., e, ) are the intrinsic coordinates; A. is anaffine parameter on the light cone's null generators, which move with constantvalues of eA= (e, ).From the parametric equations we compute the hypersurface's tangent vectors, = 3xa= (1, sin ecos , sine sin , cos e) = ka,3A.3xaeli = ae = (0, A. cose cos, A. cose sin, -A. sine),e' = axa= (0, -A. sine sin, A. sine cos, 0).364 HypersuifacesYou may check that these vectors are all orthogonal to ket. Inner products betweenegand f!' define the two-metric (JAS, and we find(JAB de AdeB = A2(de2+ sin2e d2).Not surprisingly, the hypersurface has a spherical geometry, and A is the arealradius of the two-spheres.It is easy to check that the unique null vector Net that satisfies the relationsNetket= -1 and NeteA= 0 is Net = !O, - sine cos, - sine sin, - cos e).You may also verify that the vectors ket, Net, and e'A combine as in Eq. (3.13)to form the inverse Minkowski metric.3.2 Integration on hypersurfaces3.2.1 Surface element (non-null case)If I: is not null, then(3.14)where h == det[habL is an invariant three-dimensional volume element on the hy-persurface. To avoid confusing this with the four-dimensional volume elementA d4x, we shall refer to dI: as a surface element. The combination netdI: isa directed surface element that points in the direction of increasing O.As a first example of how this works, consider a hypersurface of constant tin Minkowski spacetime. If ~_(_get>\j1 ~ O. From all this we findthat 12~ gget>et>g\j1\j1, which we rewrite as f2 ~ cofactor(g... associated with A"!'l'''; Latin indices are lowered andraised with hab and hab, respectively. Equations (3.25) and (3.26) show that onecan easily go back and forth between a tangent tensor field A"!'l'" and its equivalentthree-tensor Aab". We emphasize that while Aab... behaves as a tensor under atransformation ya -r ya' of the coordinates intrinsic to L:, it is a scalar under atransformation x" -r x'" of the spacetime coordinates.3.4.2 Intrinsic covariant derivativeWe wish to examine how tangent tensor fields are differentiated. We want to relatethe covariant derivative of A"!'l'" (with respect to a connection that is compatiblewhh the spacetime metric g"f3) to the covariant derivative of Aab... , defined in termsof a connection that is compatible with the induced metric hab. For simplicity weshall restrict our attention to the case of a tangent vector field A", such thatGeneralization to three-tensors of higher ranks will be obvious.74 HypersurfacesWe define the intrinsic covariant derivative of a three-vector All to be the pro-jection of Aa;fJ onto the hypersurface:(3.27)We will show that Aa1b, as defined here, is nothing but the covariant derivative ofAa defined in the usual way in terms of a connection rllbc that is compatiblc withhabTo get started, let us express the right-hand side of Eq. (3.27) asA a fJ _ (A a) fJ A a fJa:fJeaeb - aea :fJeb - aea:fJebA fJ fJAc y= a,fJeb - eay;{3eb ecaAa axfJ y {3 c= --II --b - ec eaY'fJeb Aax/' ay ,= Aa,b - rca/JAc,where we have defined(3.28)Equation (3.27) then rehds(3.29)and this is the familiar expression for the covariant derivative.The connection used here is the one defined by Eq. (3.28), and we would like toshow that it is compatible with thc induced metric. In other words, we would liketo prove that r cah, as defined by Eg. (3.28), can also be expressed as1r cab = "2 (hca,b + hcb,a - hllb,c). (3.30)This could be done directly by working out the right-hand side of Eq. (3.28). It iseasier, however, to show that the connection is such that hablc == hafJ;y e[ =O. Indeed,I afJY_( ) afJYlafJ;yeaebec - gafJ -EnanfJ ;yeaebec= -E(na;ynfJ =0,because na = O. Intrinsic covariant differentiation is therefore the same opera-tion as straightforward covariant differentiation of a three-tensor.3.4 Differentiation of tangent vector fields 753.4.3 Extrinsic curvatureThe )uantities Aalb = et are the tangential components of the vectorA The question we would like to investigate now is whether this vector pos-sesses also a normal component.To answer this we re-express as and decompose the metricinto its normal and tangential parts, as in Eq. (3.12). This givesand we see that while the second term is tangent to the hypersurface, the first termis normal to it. We now use Eq. (3.27) and the fact that AJ1 is orthogonal to nJ1:At this point we introduce the three-tensorK - O'efJab = nO';fJ ea b' (3.31)called the extrinsic curvature, or secondfundamentalform, of the hypersurface L:.In terms of this we have(3.32)and we see that Aa1b gives the purely tangential part of the vector field, while-EAaKab represents the nonnal component. This answers our question: The nor-mal component vanishes if and only if the extrinsic curvature vanishes.We note that if is substituted in place of Aa, then AC = oCa and Eqs. (3.29),(3.32) imply(3.33)This is known as the Gauss-Weingarten equation.The extrinsic curvature is a very important quantity; we will encounter it oftenin the remaining sections of this book. We may prove that it is a symmetric tensor:(3.34)The proof is based on the properties that (i) the vectors and nO' are orthog-onal, and (ii) the basis vectors are Lie transported along one another, so that76 Hypersuljilces'" /3 '" fjn",;/3eaeb = -n",ea;/3eb'" fj= -n",eb;fjea'" /3= n",;/Jebea'and Eq. (3.34) follows. The symmeldc property of the extrinsic curvature givesrise to the relationsKab = = (3.35)and K"b is therefore intimately related to the normal dedvative of the metric lensor.We also note the relation(3.36)which shows that K is equal to the expansion of a congruence of geodesics thatintersect the hypersurface orthogonally (so that their tangent vector is equal to n'"on the hypersurface). From this result we conclude that the hypersurface is convexif K > a (the congruence is diverging), or concave if K < a (the congruence isconverging).We see that while hal> is concerned with the purely intrinsic aspects of a hy-persurface's geometry, Kab is concerned with the extrinsic aspects - the way inwhich the hypersurface is embedded in the enveloping spacetime manifold. Takentogether, these tensors provide a virtually complete characterization of the hyper-surface.3.5 Gauss-Codazzi equations3.5.1 GeneralformWe have introduced the induced metric hab and its associated intrinsic covariantderivative. A purely intrinsic curvature tensor can now be defined by the relation(3.37)which of course implies(3.38)The question we now examine is whether this three-dimensional Riemann tensorcan be expressed in terms of - the four-dimensional version - evaluatedon I;.3.5 Gauss-Codazzi equationsTo answer this we start with the identity77which follows immediately from Eq. (3.33). We first develop the left-hand side:Next we turn to the right-hand side:We nowequate the two sides and solve for e{. Subtracting a similar expres-sion for gives - the quantity in which we are interested.After some algebra we findProjecting along edJL gives(3.39)and this is the desired relation between R"bcd and the full Riemann tensor. Pro-jecting instead along nfJ- gives(3.40)Equations (3.39) and (3.40) are known as the Gauss-Codazzi equations. They re-veal that some components of the spacetime curvature tensor can be expressed interms of the intrinsic and extrinsic curvalures of a hypersurface. The missing com-ponents are nV and these cannot be expressed solely in terms of h"b,K"b, and related quantities.78 Hypersurfaces3.5.2 ContractedformThe Gauss-Codazzi equations can also be written in contracted form, in terms ofthe Einstein tensor GafJ = RafJ - RgafJ' The spacetime Ricci tensor is given byRafJ = gfJ-VRfJ-avfJ= (EnfJ- nV+ = ERfJ-avfJnfJ-nV +hl1lnRfJ-avfJef,;e;;,and the Ricci scalar isA little algebra then reveals the relations-2EGafJnanfJ = 3R +E(KabKab - K 2)andG a fJ - K b KafJ ean - alb - ,a'(3.41)(3.42)Here, 3R = hab is the three-dimensional Ricci scalar. The importance ofEqs. (3.41) and (3.42) lies with the fact that they form part of the Einstein fieldequations on a hypersurface 1::; this observation will be elaborated in the next sec-tion. We note that GafJ the remaining components of the Einstein tensor,cannot be expressed solely in terms of hub, Kab, and related quantities.3.5.3 Ricci scalarWe now complete the computation of the four-dimensional Ricci scalar. Our start-ing point is the relationR 2 I abR Ii a v fJ I abhllll1 R fJ- a v fJ= 81 fJ-avfJn eun eb + I /lavfJe/lleae"eb'which was derived previously. The first term is simplified by using the com-pleteness relations (3.12) and the fact that RfJ-aVfJnfJ-nanvnfJ = 0; it becomes2E RafJnanfJ. Using the definition of the Riemann tensor, we rewrite this asR a fJ a fJ+ a fJafJn n = -n ;afJn n ;f!an= + + - 3.6 Initial-value problem 79In the second term of this last expression we recognize K 2, where K = nC\x isthe trace of the extrinsic curvature. The fourth term, on the other hand, cM beexpressed asa f3 f3fJ- avn ;f3n ;a = g g na;f3nfJ-;V= (snf3nfJ- +hf3fJ-)(snanV+haV)ncr ;f3 nfJ-;U= (snf3nfJ- + hf3fJ-)hcruncr;f3nfJ-;Uhf3 fJ-hav= ncr ;f3nfJ-;V,bmhan crf3 fJ-U= 1 na;f3eaebnIL,Ve",en= hbmhanKabKmn= Kab Kba= KabKab'In the second line we have inserted the completeness relations (3.12) and recalledthe notation hcrf3= hab In the third and fourth lines we have used the factthat nana;f3 = (ncr na);f3 = O. In the sixth line we have substituted the definition(3.31) for the extrinsic curvature. Finally, in the last line we have used the fact thatKab is a symmetric three-tensor.The previous manipulations take care of the first term in our starting exprescsion for the Ricci scalar. The second telm is simplified by substituting the Gauss-Codazzi equations (3,39),, abJ mnR fJ- a " f3 , ab, mn [R + (K KKK )J1 1 fJ-crvf3emeaelleb = 1 1 lIlanb S 1111, all - tllIl ab= 3R + s(KalJKab - K 2),Putting all this together, we arrive atR = 3R +s(K2- K abKab) + - (3.43)This is the four-dimensional Ricci scalar evaluated on the hypersurface L Thisresult will be put to good use in Chapter 4.3.6 Initial-value problem3.6.1 ConstraintsIn Newtonian mechanics, a complete solution to the equations of motion requiresthe specification of initial values for the position and velocity of each moving body,In field theories, a complete solution to the field equations requires the specifica-tion of the field and its time derivative at one instant of time,80 HypersurfacesA similar statement can be made for general relativity. Because the Einsteinfield equations are second-order partial differential equations, we would expectthat a complete solution requires the specification of gCl/! and gCl/!,r at one instantof time. While this is essentially correct, it is desirable to convert this decidedlynoncovariant statement into something more geometrical.The initial-value problem of general relativity starts with the selection of aspacelike hypersurface which represents an 'instant of time.' This hypersurfacecan be chosen freely. On this hypersUiface we place arbitrary coordinates ya.The spacetime metric gCl/!' when evaluated on components that character-ize displacements away from the hypersurfacc. (For example, gil is such a compo-nent if is a surface of constant t.) These components cannot be given meaning interms of the geometric properties of alone. To provide meaningful initial valuesfor the spacetime metric, we must consider displacements within the hypersurfaceonly. In other words, the initial values for gCl/! can only be the six components ofthe induced metric hab = gCl/! the remaining four components are arbitrary,and this reflects the complete freedom in choosing the spacetime coordinates XCi.Similarly, the initial values for the 'time derivative' of the metric must be de-scribed by a three-tensor that carries infom1ation about the derivative of the metricin the direction normal to the hypersurface. Because Kab = (ngClf!) theextrinsic curvature is clearly an appropriate choice.The initial-value problem of general relativity therefore consists in specifyingtwo symmetric tensor fields, hab and Kab, on a spacelike hypersurface In thecomplete spacetime, hab is recognized as the induced metric on the hypersur-face, while Kab is the extrinsic curvature. These tensors cannot be chosen freely:They must satisfy the constraint equations of general relativity. These are given byEqs. (3.41) and (3.42), together with the Einstein field equations GCI/! = 8TrTCI/!:andKbK - 8 T. CI /! - 8 .alb - ,a - Tr CI/!ean = TrJa'(3.44)(3.45)The remaining components of the Einstein field equations provide evolution equa-tions for hab and Kab; these will be considered in Chapter 4.3.6.2 Cosmological initial valuesAs an example, let us solve the constraint equations for a spatially fiat, isotropic,and homogeneous cosmology. To satisfy these requirements the three-metric musttake the form3.6 Initial-value problem 81where a is the scale factor, which is a constant on the hypersurface. Isotropy andhomogeneity also imply p = constant, ia = 0, andIKab = - Kh"b,3where K is a constant. The second constraint equation is therefore trivially satis-fied. The first one implies2l6np = K2- K abKab = 3" K 2,and this provides the complete solution to the initial-value problem.To recognize the physical meaning of this last equation, we use the fact thatin the complete spacetime, K = n":cr' where ncr is the unit normal to surfaces ofconstant t. The full metric is given by the Friedmann-Robertson-Walker formds2= -dt2+ a2(t)(d.x2+dy2 +dz2),so that ncr = -acrt and K = 311/a, where an overdot indicates differentiation withrespect to t. The first constraint equation is therefore equivalent to3(it/a)2 = 8np,which is one of the Friedmann equations governing the evolution of the scalefactor.3.6.3 Moment of time symmetryWe notice from the previous example that Kab = 0 when it = 0, that is, the extrin-sic curvature vanishes when the scale factor reaches a turning point of its evolution.Because the dynamical history of the scale factor is time-symmetric about the timet = to at which the turning point occurs, we may call this time a moment of timesymmetry in the dynamical evolution of the spacetime. Thus, Kab = 0 at this mo-n;ent of time symmetry.Generalizing, we shall call any hypersurface I; on which Kab = 0 a momentof time symmetry in spacetime. Because Kab is essentially the 'time derivative'of the metric, a moment of time symmetry corresponds to a turning point of themetric's evolution, at which its 'time derivative' vanishes. The dynamical historyof the metric is then 'time-symmetric' about I;. From Eq. (3.45) we see that amoment of time symmetry can occur only if ia = 0 on that hypersurface.82 Hypersurfaces3.6.4 Stationary and static spacetimesA spacetime is said to be stationary if it admits a timelike Killing vector ta. Thismeans that in a coordinate system (t, x") in which ta~ c S ~ , the metric does notdepend on the time coordinate t: gafJ,t ~ a(see Section 1.5). For example, a rotat-ing star gives rise to a stationary spacetime if its mass and angular velocity do notchange with time.A stationary spacetime is also static if the metric does not change under a timereversal, t ---+ -t. For example, the spacetime of a rotating star is not static becausea time reversal changes the direction of rotation. In the specified coordinate system,invariance of the metric under a time reversal implies gta ~ O. This, in turn, impliesthat the Killing vector is proportional to a gradient: ta ~ gttaat. Thus, a spacetimeis static if the timelike Killing vectorfield is hypersurface orthogonal.We may show that if a spacetime is static, then Kab = aon those hypersurfaces~ that are orthogonal to the Killing vector; these hypersurfaces therefore representmoments of time symmetry. If ~ is orthogonal to ta, then its unit normal mustbe given by na = f-Lta, where 1/f-L2 = -tata. This impl"ies that na:fJ = f-Lta;fJ +taf-L,fJ' and n(a;fJ) = t(af-L,fJ) because ta is a Killing vector. That Kab = a followsimmediately from Eq. (3.35) and the fact that ta is orthogonal to e ~ .3.6.5 Spherical space, moment of time symmetryAs a second example, we solve the constraint equations for a spherically symmetricspacetime at a moment of time symmetry. The three-metric can be expressed asfor some function mer); to enforce regularity of the metric at l' = awe must im-pose m(O) = m'(O) = 0, with a prime denoting differentiation with respect to r.For this metric the Ricci scalar is given by 3R = 4m' /rZ Because Kab = a ata moment of time symmetry, Eq. (3.44) implies 16rrp = 3R. Solving for mer)returnsmer) = 11" 4rrr'Zp(r') dr'.This states, loosely speaking, that mer) is the mass-energy contained inside asphere of radius 1', at the selected moment of time symmetry.3.6.6 Spherical space, empty andfiatWe now solve the constraint equations for a spherically symmetric space empty ofmatter (so that p = 0 = ja). We assume that we can endow this space with a flatmetric, so that3.6 Initial-value plVblem 83We also assume that the hypersurface does not represent a moment of time sym-metry. While the flat metric and Kab = 0 make a valid solution to the constraints,this is a trivial configuration - a flat hypersurface in a flat spacetime.Let na = Bar be a unit vector that points radially outward on the hypersurface.The fact that Kab is a spherically symmetric tensor means that it can be decom-posed aswith K1 representing the radial component of the extrinsic curvature, and K2the angular components. In the usual spherical coordinates (r, e, ) we haveK ~ = diag(K1, K2, K2), which is the most general expression admissible underthe assumption of spherical symmetry.Because the space is empty and flat, the first constraint equation reduces toK2- Kab Kab = 0, an algebraic equation for Kj and K2. This gives us the con-dition (2Kt +K2)K2 = O. Choosing K2 = 0 would eventually return the trivials'olution Kab = O. We choose instead K2 = -2Kj and re-express the extrinsic cur-vature aswhere K = -3Kj is the sole remaining function to be determined.To find K(r) we turn to the second constraint equation, K I ~ l b - K.ll = 0, whichbecomes1 (b b) b3" K. ll + Kn Ib +K,bn nll +Knll!bn = O.With K,a = KIna (with a prime denoting differentiation with respect to r), nb1b =21r, and nalbnb = 0 (because the radial curves are geodesics of the hyperSLIrface),we arrive at 2r KI+3K = O. Integration yields( )3/2K(r) = Ko rolr ,with Ko denoting the value of K at the arbitrary radius rooWe have found a nontrivial solution to the constraint equations for a sphericalspace that is both empty and flat. The physical meaning of this configuration willbe revealed in Section 3.13, Problem 1.84 HypersUifaces3.6.7 Conformally-fiat spaceA powerful technique for generating solutions to the constraint equations consistsof writing the three-metric aswhere 1/J(yQ) is a scalar field on the hypersurface. Such a metric is said to beconformally related to the flat metric, and the space is said to be confofmally flat.For this metric the Ricci scalar is 3R = -81/J-5 V21/J, and Eq. (3.44) takes the formof Poisson's equation,whereis an effective mass density on the hypersurface. At a moment of time symmetrythis simplifies to Peff = 1/J5 P, and one possible strategy for solving the constraintis to specify Peff, solve for 1/J; and then see what this produces for the actual massdensity p. If P = 0 at the moment of time symmetry, then the constraint becomesLaplace's equation V21/J = 0, and this admits many interesting solutions. A well-known example is Misner's (1960) solution, which describes two black holes aboutto undergo a head-on collision. This initial data set has been vigourously studiedby numerical relativists.3.7 Junction conditions and thin shellsThe following situation sometimes presents itself: A hypersurface ~ partitionsspacetime into two regions 1/ + and or ~ (Fig. 3.5). In 1/ + the metric is g;;fl' andit is expressed in a system of coordinates x+. In 1/ - the metric is g;;fl' and it isexpressed in coordinates x ~ . We ask: What conditions must be put on the metrics toensure that 1/ + and 1/ - are joined smoothly at ~ , so that the union of g:fl and g;;flforms a valid solution to the Einstein field equations? To answer this question is notentirely straightforward because in practical situations, the coordinate systems x%will often be different, and it may not be possible to compare the metrics directly.To circumvent this difficulty we will endeavour to formulate junction conditionsthat involve only three-tensors on L In this section we will assume that ~ iseither timelike or spacelike; we will return to the case of a null hypersurface inSection 3.11.3.7 Junction conditions and thin shells 853.7.1 Notation and assumptionsWe assume that the same coordinates ya can be installed on both sides of thehypersurface, and we choose n", the unit normal to ~ , to point from 1/ - to1/ +. We suppose that a continuous coordinate system x", distinct from x ~ f , canbe introduced on both sides of the hypersurface. These coordinates overlap withx ~ in an open region of 1/ + that contains ~ , and they also overlap with x.': inan open region of 1/ - that contains L (We introduce these coordinates for ourshort-term convenience only; the final fonnulation of the junction conditions willnot involve them.)We imagine ~ to be pierced by a cOilgruence of geodesics that intersect itorthogonally. We take f. to denote proper distance (or proper time) along thegeodesics, and we adjust the parameterization so that e= 0 when the geodesicscross the hypersurface; our convention is that f. is negative in 1/ - and positive in1/ +. We can think of eas a scalar field: The point P identified by the coordinatesx" is linked to ~ by a member of the congruence, and f.(x") is the proper distance(or proper time) from ~ to P along this geodesic. Our construction implies that adisplacement away from the hypersurface along one of the geodesics is describedby dx" = net de, and that(3.46)we also have n"n" = s.We will use the language of distributions. We introduce the Heaviside distribu-tion G (n, equal to +1 if f. > 0, 0 if f. < 0, and indetem1inate if f. = O. We notethe following properties:G(f.)8(-f.) = 0, ddf. G(f.) = 8(),where 8(f.) is the Dirac distribution. We also note that the product G(f.)8(f.) is notdefined as a distribution.The following notation will be useful:netFigure 3.5 Two regions of spacetime joined at a common boundary.86 Hypersuifaceswhere A is any tensorial quantity defined on both sides of the hypersurface; [A] istherefore the jump of A across We note the relations(3.47)where = axCl jaya. The first follows from the relation dxCl= nCidf and the con-tinuity of both f and XCi across the second follows from the fact that the coor-dinates ya are the same on both sides of the hypersurface.3.7.2 First junction conditionWe begin by expressing the metric gClfj, in the coordinates XCi, as a distribution-valued tensor:(3.48)where l;fj is the metric in 1/ expressed in the coordinates XCi. We want to knowif the metric of Eq. (3.48) makes a valid distributional solution to the Einstein fieldequations. To decide we must verify that geometrical quantities constructed fromgClfj, such as the Riemann tensor, are properly defined as distributions. We mustthen try to eliminate, or at least give an interpretation to, singular terms that mightarise in these geometric quantities.Differentiating Eq. (3.48) yieldsgClfj.y = S(f) gtfj.y +S(-f) g;;fj,y + c8()[gClfj]ny ,where Eq. (3.46) was used. The last term is singular and it causes problems whenwe compute the Christoffel symbols, because it generates terms proportional toS(f)8(). If the last term were allowed to survive, the connection would not bedefined as a distribution and our program would fail. To eliminate this term weimpose continuity of the metric across the hypersurface: [gClfj] = O. This statementholds in the coordinate system XCi only. However, we can easily turn this into acoordinate-invariant statement: 0 = eg = eg]; this last step followsby virtue of Eq. (3.47). We have obtained(3.49)the statement that the induced metric must be the same on both sides of Thisis clearly required if the hypersurface is to have a well-defined geometry. Equa-tion (3.49) will be our first junction condition, and it is expressed independentlyof the coordinates XCi or x. Coordinate independence explains why Eq. (3.49)produces only six conditions while the original statement [gClfj] = 0 contained ten:The mismatch corresponds to the four coordinate conditions [XCi] = O.3.7 Junction conditions and thin shells 873.7.3 Riemann tensorTo find the second junction condition requires more work: we must calculate thedistribution-valued Riemann tensor. Using the results obtained thus far, we havethat the Christoffel symbols arerafJy = 8m +B(-) where are the Christoffel symbols constructed from A straightforwardcalculation then revealsrafJy." = 8(n +8(-) +c8 () [rafJy ]n",and from this follows the Riemann tensor:(3.50)where(3.51 )We see that the Riemann tensor is properly defined as a distribution, but the 8-function term represents a curvature singularity at :E. Our second junction con-dition will seek to eliminate this term. Failing this, we will see that a physicalinterpretation can nevertheless be given to the singularity. This is our next topic.3.7.4 Surface stress-energy tensorAlthough they are constructed from Christoffel symbols, the quantities AafJy" forma tensor because the difference between two sets of Christoffel symbols is a tenso-rial quantity (see Section 1.2). We would like to find an explicit expression for thistenSOr.The fact that the metric is continuous across :E in the coordinates xaimpliesthat its tangential derivatives also must be continuous. This means that if gafJ.y isto be discontinuous, the discontinuity must be directed along the normal vector na.There must therefore exist a tensor field KafJ such thatthis tensor is given explicitly byKafJ = s[gafJ,y ]nY.Equation (3.52) implies[rafJy] = (K'/Jn y + - KfJyna) ,(3.52)(3.53)88and we obtainHypersuifacesa E:( a a a a )A fJy8 = 2: K 81lfJn y - K ynfJn8 - KfJ81l ny + KfJyn fl8 .This is the 8-function part of the Riemann tensor.Contracting over the first and third indices gives the 8-function palt of the Riccitensor:A _AI' _ E: ({L fL )afJ = afLfJ - 2: KfLan nfJ + KfLfJn na - KnanfJ - E:KafJ 'where K== K';,. After an additional contraction we obtain the 8-function part of theRicci scalar,A - Aa( I' v )= a = E: KfLVn n - E:K .With this we fonn the 8-function part of the Einstein tensor, and after using theEinstein field equations we obtain an expression for the stress-energy tensor:(3.54)where 81T SufJ == AafJ - !AgafJ' On the right-hand side of Eq. (3.54) the first andsecond terms represent the stress-energy tensors of regions )/ + and )/ -, respec-tively. The 8-function term, on the other hand, comes with a clear interpretation:It is associated with the presence of a thin distribution of matter - a surface layer,or a thin shell - at 1;; this thin shell has a surface stress-energy tensor equalto SafJ.3.7.5 Secondjunction conditionExplicitly, the surface stress-energy tensor is given by161TE:SafJ = KfLanfLnfJ + KflfJnfLna - KnanfJ - E:KafJ - (KfLVnfLnV- E:K)gafJFrom this we notice that SafJ is tangent to the hypersurface: SafJnfJ = O. It thereforeadmits the decomposition(3.55)where Sab = S a f J e ~ ef, is a symmetric three-tensor. This is evaluated as follows:I 61TSab = - K a f J e ~ e ~ - E:(KfLVnfLnV- E:K)hab_ _ a fJ _ (fLV _ hmll fL V)l I- KafJelieh KfLvg emen 711b+ K7ab_ a fJ + htrln fL V h- -KafJeaeb KfLvemen abo3.7 Junction conditions and thin shellsOn the other hand we have[na;fJ] = -[r:fJ]ny1= -2: (KyanfJ + KyfJna - KafJny)nYI= 2: (E:KafJ - KyanfJnY- KYf!nanY),which allows us to write[Kab] = = Collecting these results we obtain89(3.56)which relates the surface stress-energy tensor to the jump in extrinsic curvaturefrom one side of :E to the other. The complete stress-energy tensor of the sUifacelayer is(3.57)We conclude that a smooth transition across :E requires [KabJ =0 - the extrinsiccurvature must be the same on both sides of the hypersurface. This requirementdoes more than just remove the o-function term from the Einstein tensor: In Sec-tion 3.13, Problem 4 you will be asked to prove that [KabJ = 0 implies AafJy8 = 0,which means that the full Riemann tensor is then nonsingular at :E.The condition [KabJ = 0 is our second junction condition, and it is expressedindependently of the coordinates xaand If this condition is violated, thenthe spacetime is singular at :E, but the singularity comes with a sound physicalinterpretation: a surface layer with stress-energy tensor is present at the hy-persurface.3.7.6 SummaryThe junction conditions for a smooth joining of two metrics at a hypersurface :E(assumed not to be null) areIf the extrinsic curvature is not the same on both sides of :E, then a thin shell withsurface stress-energy tensor90 Hypersurfacesis present at :E. The complete stress-energy tensor of the surface layer is givenby Eq. (3.57) in the continuous coordinates xO'. In the coordinate system x ~ usedoriginally in 1/ , it isTO'fJ = Sab (3X) ( 3 X ~ ) S().~ 3ya 3ybThis follows from Eq. (3.57) by a simple coordinate transformation from XU to x;such a transformation leaves both f. and sah unchanged.This formulation of the junction conditions is due to Darmois (1927) and Israel(1966). The thin-shell formalism is due to Lanczos (1922 and 1924) and Israel(1966). An extension to null hypersurfaces will be presented in Section 3.1 I.3.8 Oppenheimer-Snyder collapseIn 1939, J. Robert Oppenheimer and his student Hartland Snyder published the firstsolution to the Einstein field equations that describes the process of gravitationalcollapse to a black hole. For simplicity they modelled the collapsing star as a spher-ical ball of pressureless matter with a uniform density. (A perfect fluid with neg-ligible pressure is usually called dust.) The metric inside the dust is a Friedmann-Robertson-Walker (FRW) solution, while the metric outside is the Schwarzschildsolution (Fig. 3.6). The question considered here is whether these metrics can bejoined smoothly at their common boundary, the sUlface of the collapsing star.The metIic inside the collapsing dust (which occupies the region 1/ -) is givenby(3.58)1/+ : Schwarzschild1/-: FRW:E1_--------_1Figure 3.6 The Oppenheimer-Snyder spacetime.3.8 Oppenheimer-Snyder collapse 91where T is proper time on comoving world lines (along which x, e, and are allconstant), and aCT) is the scale factor. By virtue of the Einstein field equations,this satisfies.2 8n 2a + I = 3pa , (3.59)(3.60)where an overdot denotes differentiation with respect to T'. By virtue of energy-momentum conservation in the absence of pressure, the dust's mass density Psatisfies3 3pa = constant == - a max ,8nwhere amax is the maximum value of the scale factor. The solution to Eqs. (3.59)and (3.60) has the parametric forma(1) = !amax(l + cos 1), T(1) = !amax(1) + sin 1);the collapse begins at 1) = 0 when a =amax, and it ends at 1) = n when a = O. Thehypersurface :E coincides with the surface of the collapsing star, which is locatedat X = XO in our comoving coordinates.The metric olltside the dust (in the region 1" +) is given byf = I - 2M/,., (3.61)where M is the gravitational mass of the collapsing star. As seen from the out-side, :E is described by the parametric equations r = R(T), t = T(T), where T isproper time for observers comoving with the surface. Clearly, this is the same Tthat appears in the metric of Eq. (3.58).It is convenient to choose yG = (T, e, ) as coordinates on :E. It follows thate ~ = u(1, where u(1 is the four-velocity of an observer comoving with the surfaceof the collapsing star.We now calculate the induced metric. As seen from 1" - the metric on :E isAs Seen from 1" +, on the other hand,where F = I - 2M/ R. Because the induced metric must be the same on bothsides of the hypersurface, we haveR(T)=a(T)sinxo, (3.62)92 HypersuifacesThe first equation determines R(T) and the second equation can be solved for T:FT = JR2 + F== f3(R, R). (3.63)This equation can be integrated for T (r) and the motion of the boundary in 1/ + iscompletely determined.The unit normal to :E can be obtained from the relations naua= 0, nana=1. As seen from 1/ -, u'=- aa = aT and n;; dxa= a dX; we have chosen n X> 0so that nais directed toward 1/ +. As seen from 1/ +, u+ aa = Tal + Rar andnt dxa= - Rdt +Tdr, with a consistent choice for the sign.The extrinsic curvature is defined on either side of:E by K ab = Thenonvanishing components are Krr = na;/luau/l = = _aana (whereaa is the acceleration of an observer comoving with the surface), KI}& = nl};l}, andK = Il:. A straightforward calculation reveals that as seen from 1/ -,KI} = K = a-I cot XO'-I} - , (3.64)the first result follows immediately from the fact that the comoving world lines ofa FRW spacetime are geodesics. As seen from flj/' +,(3.65)where f3(R, R) is defined by Eq. (3.63).To have a smooth transition at the surface of the collapsing star, we demand thatKab be the same on both sides of the hypersurface. It is therefore necessary foru't to satisfy the geodesic equation (a't. = 0) in 1/ +. It is easy to check that thegeodesic equation produces R2+ F = 2, where = -Uf is the (conserved) en-ergy parameter of the comoving observer. This relation implies f3 =, and the factthat f3 is a constant enforces K+T = 0, as required. On the other hand, [KI}II] = 0gives cot xola = fJ IR = 1(a sin xo), orf3 = = cosxo (3.66)We have found that the requirement for a smooth transition at :E is that the hy-persurface be generated by geodesics of both 1/ - and 1/ +, and that the parameters and XO be related by Eq. (3.66). With the help of Eqs. (3.59), (3.62), and (3.63)we may tum Eq. (3.66) into41T 3M=-pR3 ' (3.67)which equates the gravitational mass of the collapsing star to the product of itsdensity and volume. This relation has an immediate intuitive meaning, and it neatlysummarizes the complete solution to the Oppenheimer-Snyder problem.3.9 Thin-shell collapse 933.9 Thin-shell collapseAs an application of the thin-shell formalism, we consider the gravitational col-lapse of a thin spherical shell. We assume that spacetime is fiat inside the shell(in 1/ -). Outside (in 1/ +), the metric is necessatily a Schwarzschild solution (byvirtue of the spherical symmetry of the matter distribution). We assume also thatthe Shell is made of pressureless matter, in the sense that its surface stress-energytensor is constrained to have the form(3.68)in which a is the surface density and uathe shell's velocity field. Our goal is toderive the shell's equations of motion under the stated conditions.Using the results derived in the preceding section, we haveKT = fJ'1i?,K ~ 1 1 = Kt" = fJIR,fJ+ = Ji?2 + 1 - 2MI R,fJ- = Ji?2 + 1,where R(r) is the shell's radius, and M its gravitational mass. As we did before,we use (r, e, ) as coordinates on:E; in thcse coordinates ua= ayalar. Equation(3.56) allows us to calculate the components of the surface stress-energy tensor,and we find-a = ST = fJ+ - fJ- 0 = SO = fJ+ - fJ- + ~ + - ~ - .T 4][ R ' 11 8][ R 8][ RThe second equation can be integrated immediately, giving (fJ+ - fJ-)R =constant. Substituting this into the first equation yields 4][ R2a = -constant.Wc have obtained4][ R2a == m = constant (3.69)(3.70)and fL - fJ+ = ml R. The first equation states that m, the shell's rest mass, staysconstant during the evolution. Squaring the second equation converts it tov'1+R2 m2M=m 1 +R2_-2R'which comes with a nice physical interpretation. The first term on the right-handside is the shell's relativistic kinetic energy, including rest mass. The second termis the shell's binding energy, the work required to asscmble the shell from its dis-persed constituents. The sum of these is the total (conserved) energy, and this is94 Hypersuljacesequal to the shell's gravitational mass M. Equation (3.70) provides a vivid illus-tration of the general statement that all forms of energy contribute to the totalgravitational mass of an isolated body.Equations (3.69) and (3.70) are the shell's equations of motion. It is interestingto note that when M < m, the motion exhibits a turning point at R = Rmax ==m2/[2(m - M)]: An expanding shell with M < m cannot escape from its owngravitational pull.3.10 Slowly rotating shellOur next application of the thin-shell formalism is concerned with the spacetimeof a slowly rotating, spherical shell. We take the exterior metric to be the slow-rotation limit of the Kerr solution,(3.71)Here, f = 1 - 2M/ r with M denoting the shell's gravitational mass, and a =J / M M, where J is the shell's angular momentum. Throughout this sectionwe will work consistently to first order in a.The metric ofEq. (3.71) is cut off at r = R, which is where the shell is located.As viewed from the exterior, the shell's induced metric isIt is possible to remove the off-diagonal term by going to a rotating frame of ref-erence. We therefore introduce a new angular coordinate 'IjJ related to by'IjJ = - Qt, (3.72)where Q is the angular velocity of the new frame with respect to the inertial frameofEq. (3.71). We anticipate that Q will be proportional to a, and this allows us toapproximate d2 by d'IjJ2 +2Q dt d'IjJ. Substituting this into d s ~ returns a diagonalmetric if Q is chosen to beThe induced metric then becomes2MaQ=--R3 (3.73)(3.74)It is now clear that the shell has a spherical geometry. As Eq. (3.74) indicates, wewill use the coordinates ya = (t, e, 'IjJ) on the shell.3.10 Slowly rotating shell 95We take spacetime to be fiat inside the shell, and we write the Minkowski metricin the form(3.75)where p is a radial coordinate. This metric must be cut off at p = R and joinedto the exterior metric of Eq. (3.71). The shell's intrinsic metric, as computed fromthe interior, agrees with Eq. (3.74). Continuity of the induced metric is thereforeestablished, and we must now tum to the extrinsic curvature.We first compute the extrinsic curvature as seen from the shell's exterior. Inthe metric ofEq. (3.71), the shell's unit normal is nrt = j-I/2artr. The parametricequations of the hypersurface are t = t, e= e, and = 1jJ + Qt, and they havethe generic form xrt= xrt(yO). These allow us to compute the tangent vectors = BxrtlByO = al = = B. Fromall this we find that the nonvanishing components of the extrinsic curvature areKI- t - R2.Jl- 2MIR'I 3Ma sin2eK 1jJ - R2.J1-2MIR''If 3Ma /K I = Ji.4 V 1- 2MIR,II I / 1jJK II = RV I - 2MI R = K 1jJ'As now seen from the shell's interior, the unit normal is n", = o"'P and the tangentvectors are art = at, a", = a(j, and a", = a1jJ. From this we find that K IIII =II R = are the only two nonvanishing components of the extrinsic curvature.This could have been obtained directly by setting M = 0 in our previous results.We have a discontinuity in the extrinsic curvature, and Eq. (3.56) allows us tocalculate sob, the shell's surface stress-energy tensor. After a few lines of algebrawe obtainSll = - (1- /1- 2MIR),I 3Ma sin2eS - 1jJ - 8rr R2.J I - 2MI R '1jJ 3Ma ,.----S I = - 8rrR4 /1 - 2MIR,o I-MIR-.J1-2MIR 1jJSII= 8rrR.JI-2MIR =S1jJ'96 HypersuljacesThese results give us a complete description of the surface stress-energy tensor,but they are not terribly illuminating. Can we make sense of this mess?We will attempt to cast sab in a perfect-fluid form,(3.76)in terms of a velocity field ua, a surface density u, and a surface pressure p. Howdo we find these quantities? First we notice that Eq. (3.76) implies S'"ub= -uua,which shows that uais a normalized eigenvector of the surface stress-energy ten-sor, with eigenvalue -u. This gives us three equations for three unknowns, thedensity and the two independent components of the velocity field. Once those havebeen obtained, the pressure is found by projecting sab in the directions orthogonalto ua. The rest is just a matter of algebra.We can save ourselves some work if we recognize that the shell must moverigidly in the 1jJ direction, with a uniform angular velocity w. Its velocity vectorcan then be expressed as(3.77)where ta= ayalat and 1jJa = ayala1jJ are Killing vectors of the induced metrichah. In Eq. (3.77), w = d1jJ/dt is the shell's angular velocity in the rotating frameofEq. (3.72), and y is determined by the normalization condition habUaUb =-1.We can simplify things further if we anticipate that w will be proportional to a. Forexample, neglecting O(w2) terms when normalizing uagivesy = .Jl - 2MI R' (3.78)(3.79)With these assumptions, we find that the eigenvalue equation produces w = I( -Sll + and u = -SII' After simplification the first equation becomes6Ma 1-2MIRw = -R-3 -'--+-::3'.J;=1=_ and the second isu = _1_ (1- .11- 2MIR).4n R (3.80)We now have the surface density and the velocity field. The surface pressurecan easily be obtained by projecting sab in the directions orthogonal to ua:p = +UaUb)sab = !(S +u), where S = habSab. This gives p = See' andl-MIR-.J1-2MIRp = (3.81)3./0 Slowly rotating shell 97The shell's material is therefore a perfect fluid of density u, pressure p, and angularvelocity w. When R is much larger than 2M, Eqs. (3.79)-(3.81) reduce to w ~3aj(2R2), u ~ Mj(41T R2), and p ~ M2j(161T R3), respectively.The spacetime of a slowly rotating shell offers us a unique opportunity to ex-plore the rather strange relativistic effects associated with rotation. We concludethis section with a brief description of these effects.The metric of Eq. (3.71) is the metric outside the shell, and it is expressed ina coordinate system that goes easily into a Cartesian frame at infinity. This is theframe of the 'fixed stars; and it is this frame which sets the standard of no rotation.The metric of Eq. (3.75), on the other hand, is the metric inside the shelL andit is expressed in a coordinate system that is rotating with respect to the frameof the fixed stars. The transfonnation is given by Eq. (3.72), and it shows thatan observer at constant 1jJ moves with an angular velocity d jdt = Q. Inertialobservers inside the shell are therefore rotating with respect to the fixed stars, withan angular velocity Qin == Q. According to Eq. (3.73), this is(3.82)This angular motion is induced by the rotation of the shell, and the effect is knownas the dragging of inertial frames. It was first discovered in 1918 byThirring andLense.The shell's angular velocity w, as computed in Eq. (3.79), is measured in therotating frame. As measured in the Ilonrotating frame, the shell's angular velocityis Qshell = djdt = d-ifijdt + Q = (V + Qin. According to Eqs. (3.79) and (3.82),this is2Ma I + 2"j I - 2Mj RQshell = IF (I -.,.jl - 2M/R)(1 + 3"jl - 2Mj R)' (3.83)When R is much larger than 2M, QinjQsheIl ~ 4Mj(3R), and the internal ob-servers rotate at a small fraction of the shell's angular velocity. As R approaches2M, however, the ratio approaches unity, and the internal observers find them-selves corotating with the shell. This is a rather striking manifestation of framedragging. (The phrase 'Mach's principle' is often attached to this phenomenon.)This spacetime, admittedly, is highly idealized, and you may wonder whether coro-tation could ever occur in a realistic situation. You will sec in Chapter 5 that theanswer is yes: A very similar phenomenon occurs in the vicinity of a rotating blackhole.98 Hypersuifaces3.11 Null shellsWe saw in Sections 3.1 and 3.2 that the description of null hypersurfaces involvesinteresting subtleties, and we should not be surprised to find that the same is trueof the description of null surface layers. Our purpose here, in the last section ofChapter 3, is to face these subtleties and extend the formalism of thin shells, asdeveloped in Section 3.7, to the case of a null hypersurface. The presentation givenhere is adapted from Barrabes and Israel (1991).3.11.1 GeometryAs we did in Section 3.7, we consider a hypersurface :E that partitions spacetimeinto two regions "'1/ in which the metric is g;.B when expressed in coordinates x%.Here we assume that the hypersurface is null, and our convention is such that 1/ -is in the past of :E, and 1/ + in its future. We assume also that the hypersurface issingular, in the sense that the Riemann tensor possesses a 8-function singularity at:E. We will characterize the Ricci part of this singular curvature tensor, and relateit to the surface stress-energy tensor of the shell. (We shall have nothing to sayabout the interesting physical effects associated with the Weyl part of the singularcurvature tensor.)As we did in Section 3.1.2, we install coordinatesy" = (A, ell)on the hypersurface, and we assume that these coordinates are the same on bothsides of :E. We take A to be an arbitrary parameter on the null generators of thehypersurface, and we use 611to label the generators. It is possible to choose A tobe an affine parameter on one side of the hypersurface. But as we shall see below,in general it is not possible to make A an affine parameter on both sides of :E.As seen from 'Y, :E is described by the parametric relations x%(y"), and usingthese we can introduce tangent vectors e" = ax%/ay" on each side of the hyper-sUiface. These are naturally segregated into a null vector k that is tangent to thegenerators, and two spacelike vectors e ~ 1 I that point in the directions transverse tothe generators. Explicitly,e ~ = (ax:).ae J..(3.84)(Here and below, in order to keep the notation simple, we refrain from using the'' label in displayed equations; this should not create any confusion.) By con-struction, these vectors satisfy(3.85)3.11 Null shellsThe remaining inner productsOAB(>", eC) == ga{3 do not vanish, and we assume that they are the same on both sides of :E:We recall from Section 3.1.3 that the two-tensor 0AB acts as a metric on :E, = 0AB deAdeB,99(3.86)(3.87)and the condition (3.87) ensures that the hypersurface possesses a well-definedintrinsic geometry.As we did in Section 3.1.3, we complete the basis by adding an auxiliary nullvector N'l that satisfiesThe completed basis gives us the completeness relationsgU{3 = _kaNf3 _ NUk{3 (3.88)(3.89)for the inverse metric on either side of:E (in the coordinates x'l); (TAB is the inverseof (TAB, and it is the same on both sides.To complete the geometric setup we introduce a congruence bf geodesics thatcross the hypersurface. In Section 3.7, in which :E was either timelike or space-like, the congmence was selected by demanding that the geodesics intersect thehypersurface orthogonally: The vector field u tangent to the congruence was setequal (on :E) to the normal vector n. When the hypersurface is null, however, thisrequirement does not produce a unique congruence, because a vector orthogonalto kacan still possess an arbitrary component along ka.We shall have to give up on the idea of adopting a unique congruence. An im-portant aspect of our description of null shells is therefore that it involves an ar-bitrary congruence of timelike geodesics intersecting :E. This arbitrariness comeswith the lightlike nature of the singular hypersurface, and it cannot be removed.It can, however, be physically motivated: The arbitrary vector field u that entersour description of null shells can be identified with the four-velocity of a familyof observers making measurements on the shell; since many different families ofobservers could be introduced to make such measurements, there is no reason todemand that the vector field be uniquely specified.We therefore introduce a congruence of timelike geodesics y that arbitrarily in-tersect the hypersurface. The geodesics are parameterized by proper time r, whichis adjusted so that r = 0 when a geodesic crosses :E; thus, r < 0 in 1/ - and r > 0100 Hypersurfacesin 1/ +. The vector tangent to the geodesics is u, and a displacement along amcmber of the congruence is described byctx'" = u'" dL (3.90)To ensure that the congruence is smooth at the hypersUlt'ace, we dcmand that u':l: be'the same' on both sides of L. This means that the tangential projections ofthe vector field, must be equal when evaluated on either side of the hypersurface:(3.91)If, for example, u':- is specified in 1/-, then the three conditions (3.91) are suf-ficient (together with the geodesic equation) to determine the three independentcomponents of u,+ in 1/ +. We note that -u", N"', the transverse projection of thefour-velocity, is allowed to be discontinuous at L.The proper-time parameter on the timelike geodesics can be viewed as a scalarfield rex;) defined in a neighbourhood of L: Select a point x; off the hypersurfaceand locate the unique geodesic y that connects this point to L; the value of thescalar field at x; is equal to the proper-time parameter of this geodesic at thatpoint. The hypersurface L can then be described by the statementr(x"') = 0,and its normal vector k; will be proportional to the gradient of rex;) evaluated atL. It is easy to check that the expression(3.92)is compatible with Eq. (3.90). We recall that the factor -k{tu{t in Eq. (3.92) iscontinuous across L.3.11.2 Surface stress-energy tensorAs we did in Section 3.7, we introduce for our short-term convenience a continu-ous coordinate system x"', distinct from x;, in a neighbourhood of the hypersur-face; the final formulation of our null-shell formalism will be independent of thesecoordinates. We express the metric as a distribution-valued tensor:R",fj = 8(r) g;;fj + 8(-r) g;;fj'wherc (x{t) is the metric in 1/ . We assume that in these coordinates, the met-ric is continuous at L: [g"'fj] = 0; Eq. (3.87) is compatible with this requirement.We also have [k"'] = = [N"'] = [u"'] = O. Differentiation of the metric pro-ceeds as in Scctions 3.7.2 and 3.7.3, except that we now write r instead of ., and3.11 Null shells 101we use Eq. (3.92) to relate the gradient of r to the null vector kU We arrive at aRiemann tensor that contains a singular part given by(3.93)where [ru,By] is the jump in the Christoffel symbols across In order to make Eq. (3.93) more explicit we must characterize the discontin-uous behaviour of gu,B.y' The condition [gu,B] = 0 guarantees that the tangentialderivatives of the metric are continuous:The only possible discontinuity is therefore in gu,B.y NY, the transverse derivativeof the metric. In view of Eq. (3.88) we conclude that there exists a tensor field Yu,Bsuch that(3.94)(3.95)This tensor is given explicitly by Yu,B = [gu,B.y]NY, and it is now easy to checkthat[ru,By] = + - Y,BykU).Substituting this into Eq. (3.93) givesRr,u,By8 = - Y,B8kUky - + Y,BykUk8)8(r),(3.96)and we see that kUand Yu,B give a complete characterization of the singular part ofthe Riemann tensor.From Eq. (3.96) it is easy to form the singular part of the Einstein tensor, and theEinstein field equations then give us the singular part of the stress-energy tensor:(3.97)whereSu,B = _1_ (kUy,B kl-' +kf3 y Ukl-' _ yl-' kUk,B _ y kl-' kVgU,B)161T I-' '" I-' I-'Vis the sUlfacc stress-energy tensor of the null shell- up to a factor -k,LUI-' that de-pends on the choice of observers making measurements on the shell. Its expressioncan be simplified if we decompose su,B in the basis (kU, NU). For this purposewe introduce the projections(3.98)102 Hypersuifacesand we use the completeness relations (3.89) to find that the vector y ~ k l " admitsthe decompositiony ~ k l " = ~ ( Y ~ - uABYAB)kct+ (uABYB) eA- (Yl"vkI"P)Nct.Substituting this into our previous expression for sctf! and involving once morethe completeness relations, we arrive at our final expression for the surface stress-energy tensor:(3.99)Here,1 (AB )/-L =. --- u YAB161Tcan be interpreted as the shell's surface density,IjA =. _ (uABYB)161Tas a surface current, andas an isotropic surface pressure.The surface stress-energy tensor ofEq. (3.99) is expressed in the continuous co-ordinates xct. As a matter offact, the derivation ofEq. (3.99) relies heavily on thesecoordinates: The introduction of Yctf! rests on the fact that in these coordinates, gctf!is continuous at ~ , so that an eventual discontinuity in the metric derivative mustbe directed along kct. In the next subsection we will remove the need to involve thecoordinates xct in practical applications of the null-shell formalism. For the timebeing we simply note that while Eq. (3.99) is indeed expressed in the coordinatesxct, it is a tensorial equation involving vectors (kctand eA)and scalars (/-L, j A, andp). This equation can therefore be expressed in any coordinate system; in partic-ular, when viewed from )/ the surface stress-energy tensor can be expressed inthe original coordinates x.3.11.3 Intrinsic formulationIn Section 3.7, the surface stress-energy tensor of a timelike or spacelike shell wasexpressed in terms of intrinsic three-tensors - quantities that can be defined onthe hypersurface only. The most important ingredients in this formulation werehab, the (continuous) induced metric, and [Kab], the discontinuity in the extIinsiccurvature. We would like to achieve something similar here, and remove the need3. II Null shells 103to involve a continuous coordinate system xato calculate the surface quantities f1.',jA,andp.We can expect that the intrinsic description of the surface stress-energy ten-sor of a null shell will involve (JAB, the nonvanishing components of the inducedmetric. We might also expect that it should involve the jump in the extrinsic cur-vature of the null hypersurface, which would be defined by Kab = ka;fJ =!(kgafJ) Not so. The reason is that there is nothing 'transverse' about thisobject: In the case of a timelike or spacelike hypersurface, the normal napointsaway from the surface, and ngafJ truly represents the transverse derivative of themetric; when the hypersurface is null, on the other hand, kais tangent to the sur-face, and kgafJ is a tangential derivative. Thus, the extrinsic curvature is necessar-ily continuous when the hypersurface is null, and it cannot be related to the tensorYafJ defined by Eq. (3.94).There is, fortunately, an easy solution to this problem: We can introduce a trans-verse curvature Cab that properly represents the transverse derivative of the metric.This shall be defined by Cab = !(NgafJ) = !(Na;fJ + or(3.100)To arrive at Eq. (3.100) we have used the fact that N is a constant, and the iden-tity = eb;fJeg, which states that each basis vector is Lie transported alongany other basis vector; this property ensures that Cab, as defined by Eq. (3.100), isa symmetric three-tensor.In the continuous coordinates xa, the jump in the transverse curvature is givenby[Cab] = = I a a= 2: YafJ eaeb,where we have used Eq. (3.95) and the fact that kais orthogonal to We thereforehave [Cu] = !YafJkakfJ, [CAA] = == !YA, and [CAB] = ==!YAB, where we have involved Eq. (3.98). Finally, we find that the surface quanti-ties can be expressed as.A I AB[ ]J = 81T U CAB,Ip = -- [Cu].81T (3.101)We have established that the shell's surface quantities can all be related to theinduced metric U AB and the jump of the transverse curvature Cab This completesthe intrinsic formulation of our null-shell formalism.104 Hypersuifaces3.11.4 SummaryA singular null hypersurface L: possesses a surface stress-energy tensor charac-terized by tangent vectors kl and 2M, whilethe lower sign refers to r < 2M. The metric (5.6) becomes166 Black holesThis expression motivates the introduction of a new set of null coordinates, U andV, defined byU = =t=e-u/4M, (5.7)In tenns of these the metric will be well behaved near r = 2M. Going back to theexact expression (5.5) for r*, we have that er* /2M =e(u-u)/4M ==t=U V, ore'PM - 1) = -UV2M ' (5.8)(5.9)which implicitly gives r as a function of UV. You may check that theSchwarzschild metric is now given by32M3ds2= e-r/2MdUdV + r 2dn2.rThis is manifestly regular at r = 2M. The coordinates U and V are called nullKruskal coordinates. In a Kruskal diagram (a map of the U- V plane; see Fig. 5.2),outgoing light rays move along curves U = constant, while ingoing light raysmove along curves V = constant.In the Kruskal coordinates, a surface of constant r is described by an equa-tion of the form U V = constant, which corresponds to a two-branch hyperbolain the U- V plane. For example, r = 2M becomes U V = 0, while r = 0 becomesU V = 1. There are two copies of each surface r = constant in a Kruskal diagram.For example, r = 2M can be either U = 0 or V = O. The Kruskal coordinatestherefore reveal the existence of a much larger manifold than the portion coveredby the original Schwarzschild coordinates. In a Kruskal diagram, this portion is la-beled 1. The Kruskal coordinates do not only allow the continuation of the metricthrough r = 2M into region II, they also allow continuation into regions III andU V \, ......... - ........ ,,'... ....':._-::."1--- \ -'l' "\ II':r = 3M: III I : r = 3M,A! IV \" Figure 5.2 Kruskal diagram.5.! Schwarzschild black hole 167IV. These additional regions, however, exist only in the maximal extension of theSchwarzschild spacetime. If the black hole is the result of gravitational collapse,then the Kruskal diagram must be cut off at a timelike boundary representing thesurface of the collapsing body. Regions III and IV then effectively disappear be-low the surface of the collapsing star.5.1.3 Eddington-Finkelstein coordinatesBecause of the implicit nature of the relation between rand U V, the Kruskal coor-dinates can be awkward to use in some computations. In fact, it is rarely necessaryto employ coordinates that cover all four regions of the Kruskal diagram, althoughit is often desirable to have coordinates that are well behaved at r = 2M. In suchsituations, choosing v and r as coordinates, or u and r, does the trick. These coordi-nate systems are called ingoing and outgoing Eddington-Finkelstein coordinates,respectively.It is easy to check that in the ingoing coordinates, the Schwarzschild metrictakes the formwhile in the outgoing coordinates,ds2= -(1 - 2M/r) du2- 2du dr + r2dQ2.(5.10)(5.11)It may also be verified that the (v, r) coordinates cover regions I and II of theKruskal diagram, while u and r cOVer regions IV and 1.The Eddington-Finkelstein coordinates can also be used to construct space-time diagrams (Fig. 5.3), but these do not have the property that both ingoingand outgoing null geodesics propagate at 45 degrees. For example, it follows outgoing" ""I ,'" ~ " ' , , , , , , ", ,.- " , > ' ~':~.' . ,r,," ;, . , , , ',' , .", , . , , " , ,','t , , , :' ,'f .. ", , ,ingoing . , ..\ .,' :. ", ,' .... : 'r=O r=2MFigure 5.3 Spacetime diagram based on the Cv, r) coordinates.168 Black holesfrom Eq. (5.10) that ingoing light rays move with dv = 0, that is, along coordi-nate lines that can be oriented at 45 degrees, but the outgoing rays move withdv/dr = 2/0 - 2M/1'), that is, with a varying slope.5.1.4 Painleve-Gullstrand coordinatesAnother useful set of coordinates for the Schwarzschild spacetime are thePainleve-Gullstrand coordinates first considered in Section 3.13, Problem 1. Here,as with the Eddington-Finkelstein coordinates, the spatial coordinates (1', e, ) arethe same as in the original form of the metric, Eq. (5.1), but the time coordinateis different: T is proper time as measured by a free-falling observer starting fromrest at infinity and moving radially inward.The four-velocity of such an observer is given by u'" a", = I-I at - ~ ar ,where 1= 1 - 2M/I'. From this we deduce that u'" = -a",T, where the timefunction T is obtained by integrating dT = dt + I - l ~ dr == dr, where ris proper time. (Integration is elementary, and the result appears in Section 3.13,Problem 1.) After inserting this expression for dt into Eq. (5.1), we obtain thePainleve-Gullstrand form of the Schwarzschild metric:(5.12)The coordinates (T, 1', e, ) give rise to a metric that is regular at I' = 2M, in corre-spondence with the fact that our free-falling observer does not consider this surfaceto be in any way special. Because this observer originates in region I of the space-time (at I' = (0) and ends up in region II (at I' = 0), the new coordinates coveronly these two regions of the Kruskal diagram. By reversing the motion -lettingdr become -dr in Eq. (5.12) - an alternative coordinate system is produced thatcovers regions VI and I instead.From Eq. (5.12) we infer a rather striking prope11y of the Painleve-Gullstrandcoordinates: The hypersurfaces T = constant are all intrinsically fiat. This can beseen directly from the fact that the induced metric on any such hypersurface isgiven by ds2= dr2+1'2 dQ.5.1.5 Penrose-Carter diagramThe double-null Kruskal coordinates make the causal structure of theSchwarzschild spacetime very clear, and this is their main advantage. Another use-ful set of double-null coordinates is obtained by applying the transformationf) = arctan U, if = arctan V. (5.13)5.1 Schwarzschild black hole 169U VFigure 5.4 Compactified coordinatcs for the Schwarzschild spacetime.This rescaling of the null coordinates does not affect the appearance of radial lightrays, which still propagate at 45 degrees in a spacetime diagram based on thenew coordinates (Fig. 5.4). However, willIe the range of the initial coordinateswas infinite (for example, -00 < U < (0), it is finite for the new coordinates (forexample, -n/2 < U< n /2). The entire spacetime is therefore mapped onto afinite domain of the U-Vplane. This compactification of the manifold introducesbad coordinate singularities at the boundaries of the new coordinate system, butthese are of no concern when the purpose is simply to construct a compact map ofthe entire spacetime.In the new coordinates the surfaces r = 2M are located at U= 0 and V=0, and the singularities at r =0, or U V = I, are now at U+ V= n/2. Thespacetime is also bounded by the surfaces U= n/2 and V= n/2. The fourpoints (U, V) = (n/2, n/2) are singularities of the coordinate transformation:In the actual spacetime the surfaces U = 0, U = 00, and U V = 1 never meet.It is useful to assign names to the various boundaries of the compactified space-time (Fig. 5.5). The surfaces U= n /2 and V= n /2 are called future null infinityand are labelled f + (pronounced 'sen plus'). The diagram makes it clear that f +contains the future endpoints of all outgoing n