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( x ρ p x steenergy- the is ) ρ n 4-velocity and , ρ ), ), ∂ T U ,ν ,... . . ρ, ν, µ, µ µν ∂ µ ν ( x and 0 = g αβ ρ R ,and ), µν the ; c g Ji ˇ r ´ ˇ ıBi c´ ak are (1) = = = d and 1 ooeeu n stoi ttcslto f()(with (2) of (8), solution metric static isotropic and homogeneous A h imn esrcnb eopsdas decomposed be can tensor Riemann The classification Algebraic 2.1 and characterization Invariant 2 models. of asymptotic classes as general act more to of out states turned which Cosmology: solutions examples “old” entry other many of [[see are There epoch Aspects]]. followed expansion Mathematical era, standard inflationary Einstein a an enters by initial later an and state from static asymptotically our which starts cos- to tuned” universe according “fine suggested a of was 2004 expansion in scenario 1920s, mological the late of the in observa- discovery Universe its the the lost after and relevance unstable tional is it modern Although of cosmology. birth the galaxies”)— universe—marked static distributed Einstein (“uniformly the dust incoherent of Λ of effect” where se .. 1];the [13]); e.g., (see, ed ic,a ie on,i antb eemndin determined tensor be energy-momentum cannot matter it the point, of given terms a at gravitationa since, pure field” the of “characteristic the as considered lsicto fteWy esr hsi etformu- best is This spinors tensor. 2-component Weyl using lated the of classification a and efactorized: be fwihayWy pnrΨ spinor Weyl any which of emnsapicplnl ieto,say, direction, null de- principal spinors a the of termines each symmetrization; denote brackets 1 implies (1) oia emΛ term logical n11,Enti eeaie 1 yadn cosmo- a adding by (1) generalized Einstein 1917, In G lsicto ftesolutions the of classification a sn E.Agbaccasfiaini ae on based is classification Algebraic EE). using can E and R R R µν k > αβγδ µν +1, = opnae h rvttoa attraction gravitational the compensates 0 g − G µν Ψ 0. = e zc Republic Czech ue, 2 1 r osrce from constructed are ABCD = Λ=const): = (Λ g µν elcnomltensor conformal Weyl C a R αβγδ os) nwihte“repulsive the which in const), = Λ + = α + ABCD g ( A µν E β αβγδ B 8 = α γ determining C A πT + δ ( D A G µν ) , αβγδ R 1 = . αβ k C , α R , , ) nterms in 2), αβγδ C T = αβ αβγδ and α a be can (as A α g can ¯ (2) (3) (4) αβ A E ′ l a a α [[see entry Spinors and Spin-coefficients]]. The Petrov- where the time-independent 1-forms ω = Eαdx sat- α 1 a b c Penrose classification is based on coincidences among isfy the relations dω = 2 Cbcω ω , d is the exterior a − ∧ these directions. A solution is of type I (general case), derivative and Cbc are the structure constants [[see entry II, III, and N (“null”) if all null directions are different, Cosmology: Mathematical Aspects for more details]]. or two, three, and all four coincide, respectively. It is of The standard Friedmann-Lemaitre-Robertson-Walker type D (“degenerate”) if there are two double null direc- (FLRW) models admit in addition an isotropy group tions. The equivalent tensor equations are simplest for SO(3) at each point. They can be represented by the type N : metric δ αβγδ ∗αβγδ Cαβγδk =0, CαβγδC =0, CαβγδC =0, (5) dr2 g = dt2+[a(t)]2 + r2(dθ2 + sin2 θdϕ2) , (8) ∗ 1 ρσ − 1 kr2 where Cαβγδ = 2 ǫαβρσC γδ, ǫ is the Levi-Civita pseu- − dotensor. in which a(t), the “expansion factor”, is determined by matter via EE, the curvature index k = 1, 0, +1, the 2.2 Classification according to symme- 3-dimensional spaces t = const have a constant− curva- tries ture K = k/a2; r [0, 1] for closed (k = +1) universe, r [0, ) in open∈ (k = 0, 1) universes (for another Most of the available solutions have some exact contin- ∈ ∞ − uous symmetries which preserve the metric. The corre- description, [[see entry Cosmology: Mathematical As- sponding group of motions is characterized by the num- pects]]). ber and properties of its Killing vectors ξα satisfying the There are 4-dimensional spacetimes of constant cur- vature solving EE (2) with Tµν = 0: the Minkowski, Killing equation (£ξg)αβ = ξα;β + ξβ;α = 0 (£ is the Lie derivative) and by the nature (spacelike, timelike or de Sitter, and anti de Sitter spacetimes. They admit the null) of the group orbits. For example, axisymmetric, same number (10) of independent Killing vectors, but in- stationary fields possess two commuting Killing vectors, terpretations of the corresponding symmetries differ for of which one is timelike. Orbits of the axial Killing vec- each spacetime. α tor are closed spacelike curves of finite length, which van- If ξ satisfies £ξgαβ = 2Φgαβ, Φ =const, it is called a ishes at the axis of symmetry. In cylindrical symmetry, homothetic (Killing) vector. Solutions with proper ho- mothetic motions Φ = 0 are “self-similar”. They cannot there exist two spacelike commuting Killing vectors. In 6 both cases, the vectors generate a 2-dimensional Abelian in general be asymptotically flat or spatially compact group. The 2-dimensional group orbits are timelike in but can represent asymptotic states of more general so- the stationary case and spacelike in the cylindrical sym- lutions. In [13], a summary of solutions with proper ho- metry. motheties is given; their role in cosmology is analyzed by α J. Wainwright and G.F.R. Ellis (eds) 1996, and by A.A. If a timelike ξ is hypersurface-orthogonal, ξα = λΦ,α for some scalar functions λ, Φ, the spacetime is static. In Coley, 2003; for mathematical aspects of symmetries in general relativity, see G.S. Hall, 2004. coordinates with ξ = ∂t, the metric is There are other schemes for invariant classification of g = e2U dt2 + e−2U γ dxidxk, (6) − ik exact solutions (reviewed in [13]): the algebraic classifi- cation of the Ricci tensor and energy-momentum tensor where U,γik do not depend on t. In vacuum, U satisfies the potential equation U :a = 0, the covariant derivatives of matter; the existence and properties of preferred vec- :a tor fields and corresponding congruences; local isometric (denoted by :) are w.r.t. the 3-dimensional metric γik. A classical result of Lichnerowicz states that if the vacuum embeddings into flat pseudo-Euclidian spaces, etc. metric is smooth everywhere and U 0 at infinity, the spacetime is flat (for refinements, see→ M.T. Anderson, 2000). 3 Minkowski (M), de Sitter (dS), In cosmology, we are interested in groups whose re- and anti de Sitter (AdS) space- gions of transitivity (points can be carried into one an- other by symmetry operations) are 3-dimensional space- times like hypersurfaces (homogeneous but anisotropic models These metrics of constant (zero, positive, negative) cur- of the Universe). The 3-dimensional simply transitive vature are the simplest solutions of (2) with T = 0 groups G were classified by Bianchi in 1897 according µν 3 and Λ = 0, Λ > 0, Λ < 0, respectively. The standard to the possible distinct sets of structure constants but topology of M is R4. The dS has the topology R1 S3 their importance in cosmology was discovered only in and is best represented as a 4-dimensional hyperboloid× the 1950’s. There are nine types: Bianchi I to Bianchi v2 + w2 + x2 + y2 + z2 = (3/Λ) in a 5-dimensional flat IX models. The line element of the Bianchi universes can −space with metric g = dv2 + dw2 + dx2 + dy2 + dz2. be expressed in the form The AdS has the topology− S1 R3; it is a 4-dimensional g = dt2 + g (t)ωaωb, (7) hyperboloid v2 w2 + x2 + y×2 + z2 = (3/Λ), Λ < 0, in − ab − − −
2 flat 5-d space with signature ( , , +, +, +). By unwrap- spherically symmetric body (spherically symmetric grav- ping the circle S1 and considering− − the universal covering itational waves do not exist). It is the most influential space, one gets rid of closed timelike lines. solution of EE. The essential tests of general relativity— These spacetimes are all conformally flat and can be perihelion advance of Mercury, deflection of both optical conformally mapped into portions of the Einstein uni- and radio waves by the Sun, and signal retardation—are verse [[see entry Asymptotic Structure and Conformal based on (10) or rather on its expansion in M/r. Space Infinity]]. However, their conformal structure is glob- missions have been proposed that could lead to measure- ally different. In M one can go to infinity along time- ments of “post-post-Newtonian” effects [[see entry Gen- like/null/spacelike geodesics and reach five qualitatively eral Relativity: Experimental Tests and [10]]]. The full different sets of points: future/past timelike infinity i ±, Schwarzschild metric is of importance in astrophysical future/past null infinity ±, and spacelike infinity i 0. processes involving compact stars and black holes. In dS, there are only pastI and future conformal infini- Metric (10) describes the spacetime outside a spher- ties −, +, both being spacelike (on the Einstein cylin- ical body collapsing through r = 2M into a spheri- der,I theI de Sitter spacetime is a “horizontal strip” with cal black hole. In Fig. 1, the formation of an event +/ − as the “upper/lower circle”). The conformal in- horizon and trapped surfaces is indicated in ingoing finityI I in AdS is timelike. Eddington-Finkelstein coordinates (v,r,θ,ϕ) where v = As a consequence of spacelike ± in dS, there ex- t + r +2M log(r/2M 1) so that (v,θ,ϕ) = const are in- − ist both particle (cosmological) andI event horizons for going radial null geodesics. The interior of the star is de- geodesic observers [7]. dS plays a (doubly) fundamental scribed by another metric (e.g., the Oppenheimer-Snyder role in the present-day cosmology: it is an approximate collapsing dust solution—see below). The Kruskal exten- model for inflationary paradigm near the Big Bang and sion of the Schwarzschild solution, its compactification, it is also the asymptotic state (at t ) of cosmologi- the concept of the bifurcate Killing horizon, etc., are an- cal models with a positive cosmological→ ∞ constant. Since alyzed in [[entry Stationary Black Holes]] and in [10], [7], recent observations indicate that Λ > 0, it appears to [2]. describe the future state of our Universe. AdS has come The Reissner-Nordstr¨om solution describes the exte- recently to the fore due to the “holographic” conjecture rior gravitational and electromagnetic fields of a spher- [[see entry AdS/CFT Correspondence]]. ical body with mass M and charge Q. The energy- Christodoulou and Klainermann, and Friedrich proved momentum tensor on the right-hand side of EE is that that M, dS, and AdS are stable with respect to general, nonlinear (though “weak”) vacuum perturbations— result not known for any other solution of EE [[see entry SINGULARITY r = 0 EVENT HORIZON Stability of Minkowski Space]]. r = 2 M
TRAPPED SURFACES OUTGOING PHOTON 4 Schwarzschild and Reissner- Nordstr¨om metrics INFALLING Q PHOTON These are spherically symmetric spacetimes—the SO3 ro- v = const. tation group acts on them as an isometry group with spacelike, 2-dimensional orbits. The metric can be brought into the form P
2ν 2 2λ 2 2 2 2 2 g = e dt + e dr + r (dθ + sin θdϕ ), (9) SURFACE OF STAR − ν(t, r), λ(t, r) must be determined from EE. In vacuum, we are led uniquely to the Schwarzschild metric O
−1 2M 2 2M 2 2 2 2 2 g = 1 dt + 1 dr +r (dθ + sin θdϕ ), Figure 1 Gravitational collapse of a spherical star (the in- − − r − r terior of the star is shaded). The light cones of three events, (10) O, P , Q, at the center of the star, and of three events out- where M = const has to be interpreted as mass, as test side the star are illustrated. The event horizon, the trapped particle orbits show. The spacetime is static at r> 2M, surfaces, and the singularity formed during the collapse are i.e., outside the Schwarzschild radius at r = 2M, and also shown. Although the singularity appears to lie along the asymptotically (r ) flat. direction of time, from the character of the light cone outside → ∞ Metric (10) describes the exterior gravitational field of the star but inside the event horizon we can see that it has a an arbitrary (static, oscillating, collapsing or expanding) spacelike character.
3 of the electromagnetic field produced by the charge; the U, k, and A are functions of ρ,z. field satisfies the curved-space Maxwell equations. The The most celebrated vacuum solution of the form (12) metric reads is the Kerr metric for which U,k,A are ratios of simple 2M Q2 polynomials in spheroidal coordinates (simply related to g = 1 + dt2 + (11) r r2 (ρ,z)). The Kerr solution is characterized by mass M − − 2 2 − and specific angular momentum a. For a > M , it de- 2M Q2 1 + 1 + dr2 + r2(dθ2 + sin2 θdϕ2). scribes an asymptotically flat spacetime with a naked − r r2 singularity. For a2 M 2, it represents a rotating black ≤ The analytic extension of the electrovacuum metric hole that has two horizons which coalesce into a degen- 2 2 (11) is qualitatively different from the Kruskal exten- erate horizon for a = M —an extreme Kerr black hole. 2 2 1 2 2 The two horizons are located at r± = M (M a ) 2 (r sion of the Schwarzschild metric. In the case Q > M ± − there is a naked singularity (visible from r ) at being the Boyer-Lindquist coordinate—[[see entry Sta- r = 0 where curvature invariants diverge. If Q→2 < ∞ M 2, tionary Black Holes]]). As with the Reissner-Nordstr¨om the metric describes a (generic) static charged black hole black hole, the singularity inside is timelike and the inner 2 2 1/2 with two event horizons at r = r± = M (M Q ) . horizon is an (unstable) Cauchy horizon. The analytic The Killing vector ∂/∂t is null at the horizons,± − timelike extension of the Kerr metric resembles Fig. 2 (see [5], at r > r+ and r < r−, but spacelike between the hori- [7], [10]-[14] for details). zons. The character of the extended spacetime is best Thanks to the black-hole uniqueness theorems [[see seen in the compactified form, Fig. 2, in which world- entry Stationary Black Holes]], the Kerr metric is the lines of radial light rays are 45-degree lines. Again, two unique solution describing all rotating black holes in vac- infinities (right and left, in regions I and III) arise (as uum. If the cosmic censorship conjecture holds, Kerr in the Kruskal-Schwarzschild diagram—see [[entry Sta- black holes represent the end states of gravitational col- tionary Black Holes]]), however, the maximally extended lapse of astronomical objects with supercritical masses. geometry consists of an infinite chain of asymptotically According to prevalent views, they reside in the nuclei flat regions connected by “wormholes” between the sin- of most galaxies. Unlike with a spherical collapse, there gularities at r = 0. In contrast to the Schwarzschild are no exact solutions available which would represent singularity, the singularities are timelike—they do not the formation of a Kerr black hole. However, starting block the way to the future. The inner horizon r = r− from metric (12) and identifying, e.g., z = b = const and Cauchy horizon z = b (with the region b
4 - (r = ) r = r - III' + I' + (r = ) r = r Worldline of a shell
i- IV' i-
r = r - r = r - r = 0 r = 0 (singularity) (singularity) V V r = r - r = r - Cauchy horizon for
i+ II i+
(r = ) + + r = r + r = r (r = ) + spacelike hypersurface t=0 i 0 III I i 0
(r = ) r = r +
r = r + - (r = ) -
- i- IV i
Figure 2 The compactified Reissner-Nordstr¨om spacetime representing a non-extreme black hole consists of an indefinite chain of asymptotic regions (“universes”) connected by “wormholes” between timelike singularities. The worldline of a shell ′ collapsing from “universe” I and re-emerging in “universe” I is indicated. The inner horizon at r = r− is the Cauchy horizon for a spacelike hypersurface Σ. It is unstable and thus it will very likely prevent such a process. and [1] where these techniques using B¨acklund transfor- are “inspired” by galactic Newtonian potentials; disks mations, inverse scattering method, etc. are also ap- around black holes and some other special solutions [13], plied in the nonstationary context of two spacelike Killing [3], [2], [12]. vectors (waves, cosmology). In the stationary case, all There are solutions of the Einstein-Maxwell equations asymptotically flat, stationary, axisymmetric vacuum so- representing external fields of masses endowed with elec- lutions can, in principle, be generated. It is known how tric charges, magnetic dipole moments, etc. [13]. Best to generate fields with given values of multipole mo- known is the Kerr-Newman metric characterized by pa- ments, though the required calculations are staggering. rameters M,a, and charge Q. For M 2 a2 + Q2 it By solving the Riemann-Hilbert problem with appropri- describes a charged, rotating black hole. Owing≥ to the ro- ate boundary data, Neugebauer and Meinel constructed tation, the charged black hole produces also a magnetic the exact solution representing a rigidly rotating thin field of a dipole type. All the black hole solutions can disk of dust (cf. [13], [2]). be generalized to include a non-vanishing Λ (for various A subclass of metrics (12) is formed by static Weyl applications, see [12]). Other generalizations incorpo- solutions with A = b = 0. Eq. (13) then becomes the rate the so-called NUT (Newman-Unti-Tamburino) pa- Laplace equation ∆U = 0. The non-linearity of EE en- rameter (corresponding to a “gravomagnetic monopole”) ters only the equations for k: k = ρ(U 2 U 2 ), k = or an “external” magnetic/electric field or a parameter ,ρ ,ρ − ,z ,z 2ρU,ρU,z. The class contains some explicit solutions of leading to “uniform” acceleration (see [13], [2]). Much interest: the “linear superposition” of collinear particles interest has recently been paid to black-hole (and other) with string-like singularities between them which keep solutions with various types of gauge fields and to multi- the system in static equilibrium; solutions representing dimensional solutions. Refs. [5] and [11] are two exam- external fields of counter-rotating disks, e.g., those which ples of good reviews.
5 6 Radiative solutions Impulsive plane waves can be generated by boosting a “particle” at rest to the velocity of light by an appro- 6.1 Plane waves and their collisions priate limiting procedure. The ultrarelativistic limit of, e.g., the Schwarzschild metric (the so-called Aichelburg- The best known class are “plane-fronted gravitational Sexl solution) can be employed as a “limiting incoming waves with parallel rays” (pp-waves) which are defined state” in black hole encounters (cf. monograph by P.D. by the condition that the spacetime admits a covariantly d’Eath, 1996). Plane-fronted waves have been used in α constant null vector field k : kα;β = 0. In suitable null quantum field theory. For a review of exact impulsive α α coordinates u, v such that kα = u,α, k = (∂/∂v) , and waves, see [12]. complex ζ which spans the wave 2-surfaces u = const, A collision of plane waves represents an exceptional v = const with Euclidean geometry, the metric reads situation of nonlinear wave interactions which can be an- alyzed exactly. Fig. 3 illustrates a typical case in which g =2dζdζ¯ 2dudv 2H(u,ζ, ζ¯)du2, (14) − − the collision produces a spacelike singularity. The initial value problem with data given at v = 0 and u = 0 canbe H(u,ζ, ζ¯) is a real function. The vacuum EE imply formulated in terms of the equivalent matrix Riemann- H ¯ = 0 so that 2H = f(u, ζ)+ f¯(u, ζ¯), f is an ar- ,ζζ Hilbert problem [[see entry Riemann-Hilbert problem]]; bitrary function of u, analytic in ζ. The Weyl tensor it is related to the hyperbolic counterpart of the Ernst satisfies equations (5)—the field is of type N as is the equation (13). For reviews, see [6], [13], and [2]. field of plane electromagnetic waves. In the null tetrad α α α α α k ,l ,m (complex) with l kα = 1,m m¯ α = 1, all other{ products vanishing,} the only nonzero− projection of 6.2 Cylindrical waves α β γ δ the Weyl tensor, Ψ = Cαβγδl m¯ l m = H,ζ¯ζ¯, describes Discovered by G. Beck in 1925 and known today as the transverse component of a wave propagating in the α iΘ the Einstein-Rosen waves (1937), these vacuum solutions k direction. Writing Ψ = e , the real > 0 is the helped to clarify a number of issues, such as energy loss amplitude A polarizationA of the wave, Θ describes . Waves due to the waves, asymptotic structure of radiative space- with Θ = const are called linearly polarized. Consider- times, dispersion of waves, quasi-local mass-energy, cos- ing their effect on test particles one finds plane waves are mic censorship conjecture, or quantum gravity in the con- transverse. text of midisuperspaces (see [2], [1]). The simplest waves are homogeneous in the sense In the metric that Ψ is constant along the wave surfaces. One gets 1 iΘ(u) 2 2(γ−ψ) 2 2 2ψ 2 2 −2ψ 2 f(u, ζ) = 2 (u) e ζ . Instructive are sandwich g = e ( dt + dρ )+ e dz + ρ e dϕ , (15) waves, e.g., wavesA with a “square profile”: = 0 for − − A u < 0 and u>a2, = a 2 = const for 0 u a2. ψ(t,ρ) satisfies the flat-space wave equation and γ(ρ,t) is A ≤ ≤ This example demonstrates, within exact theory, that given in terms of ψ by quadratures. Admitting a “cross the waves travel with the speed of light, produce rela- term” ω(t,ρ) dz dφ, one acquires a second degree tive accelerations of test particles, focus astigmatically of freedom∼ (a second polarization) which makes all field generally propagating parallel rays, etc. The focusing equations nonlinear. effects have a remarkable consequence: there exists no global spacelike hypersurface on which initial data could 6.3 Boost-rotation symmetric space- be specified—plane wave spacetimes contain no global Cauchy hypersurface. times These are the only explicit solutions available which are v u radiative and represent the field of finite sources. Fig. 4 shows two particles uniformly accelerated in opposite I directions. In the space diagram (left), the “string” con- II III necting the particles is the “cause” of the acceleration. u=1 v =1 In “Cartesian-type” coordinates and the z-axis chosen as the symmetry axis, the boost Killing vector has a flat- space form, ζ = z(∂/∂t)+ t(∂/∂z), the same is true for u=0 v =0 IV the axial Killing vector. The metric contains two func- tions of variables ρ2 x2 + y2 and β2 z2 t2. One satisfies the flat-space≡ wave equation, the≡ other− is deter- mined by quadratures. Figure 3 A spacetime diagram indicating a collision of two The unique role of these solutions is exhibited by the plane-fronted gravitational waves which come from regions II theorem which states that in axially symmetric, locally and III, collide in region I, and produce a spacelike singular- asymptotically flat spacetimes in the sense that a null in- ity. Region IV is flat. finity [[see entry Asymptotic structure and conformal in-
6 Schwarzschild interior solution with incompressible fluid
t as its source, represents “a star” of uniform density, ρ = ρ0 = const: z 3 1 2 g = 1 AR2 1 Ar2 dt2 + + − 2 − − 2 − ξ p p 2 dr 2 2 2 2 y + + r (dθ + sin θdϕ ), (16) z 1 Ar2 x − + A =8πρ0/3 = const, R is the radius of the star. The equation of hydrostatic equilibrium yields pres- sure inside the star: -z √1 Ar2 √1 AR2 8πp =2A − − − . (17) 3√1 AR2 √1 Ar2 − − − Figure 4 Two particles uniformly accelerated in opposite Solution (16) can be matched at r = R, where p = 0, directions. Orbits of the boost Killing vector (thinner hyper- to the exterior vacuum Schwarzschild solution (10) if the 2 2 1 3 bolas) are spacelike in the region t >z . Schwarzschild mass M = 2 AR . Although “incompress- ible fluid” implies an infinite speed of sound, the above solution provides an instructive model of relativistic hy- finity]] exists but not necessarily globally, the only addi- drostatics. A Newtonian star of uniform density can 2 tional symmetry that does not exclude gravitational ra- have an arbitrarily large radius R = 3pc/2πρ0 and boost 2 diation is the symmetry. Various radiation charac- mass M = (pc/ρ0) 6pc/π, pc is the centralp pressure. teristics can be expressed explicitly in these spacetimes. However, (17) impliesp that (i) M and R satisfy the in- They have been used as tests in numerical relativity and equality 2M/R 8/9, (ii) equality is reached as pc be- approximation methods. The best known example is the comes infinite and≤ R and M attain their limiting values 1 − 2 C-metric (representing accelerating black holes, in gen- Rlim = (3πρ0) = (9/4)Mlim. For a density typical in 15 −3 . eral charged and rotating, and admitting Λ)—see [4], [2], neutron stars, ρ0 = 10 g cm , we get Mlim =3.96M⊙ [13], [12]. (M⊙ solar mass)—even this simple model shows that in Einstein’s theory neutron stars can only be a few solar 6.4 Robinson-Trautman solutions masses. In addition, one can prove that the “Buchdahl’s inequality” 2M/R 8/9 is valid for an arbitrary equa- ≤ These solutions are algebraically special but in general tion of state p = p(ρ). Only a limited mass can thus be they do not possess any symmetry. They are governed by contained within a given radius in general relativity. The − 1 a function P (u,ζ, ζ¯) (u is the retarded time, ζ a complex gravitational redshift z = (1 2M/R) 2 1 from the − − spatial coordinate) which satisfies a 4th-order nonlinear surface of a static star cannot be higher than 2. parabolic differential equation. Studies by Chru´sciel and Many other explicit static perfect fluid solutions are others have shown that RT solutions of Petrov type II ex- known (we refer to [13] for a list), however, none of them ist globally for all positive “times” u and converge asymp- can be considered as really “physical”. Recently, the dy- totically to a Schwarzschild metric, though the extension namical systems approach to relativistic spherically sym- across the “Schwarzschild-like” horizon can only be made metric static perfect fluid models was developed which with a finite degree of smoothness. Generalization to gives qualititative characteristics of masses and radii (cf. the cases with Λ > 0 gives explicit models supporting work by C. Uggla et al.). the cosmic no-hair conjecture (an exponentially fast ap- The most significant nonstatic spacetime describing a proach to the de Sitter spacetime) under the presence of bounded region of matter and its external field is un- gravitational waves. See [4], [2], [13]. doubtedly the Oppenheimer-Snyder model of gravita- tional collapse of a spherical star of uniform density and zero pressure (a “ball of dust”). The model does not rep- 7 Material sources resent any new (local) solution: the interior of the star is described by a part of a dust-filled FLRW universe (cf. Finding physically sound material sources in an analytic (8)), the external region by the Schwarzschild vacuum form even for some simple vacuum metrics remains an metric (cf. Eq. (10), Fig. 1). open problem. Nevertheless, there are solutions repre- Since Vaidya’s discovery of a “radiating Schwarzschild senting regions of spacetimes filled with matter which metric”, null dust (“pure radiation field”) has been are of considerable interest. widely used as a simple matter source. Its energy- α One of the simplest, the spherically symmetric momentum tensor, Tαβ = ̺kαkβ, where kαk = 0, may
7 be interpreted as an incoherent superposition of waves ity and Astrophysics, in Einstein’s Field Equa- with random phases and polarizations moving in a single tions and Their Physical Implications, ed. B. G. direction, or as “lightlike particles” (photons, neutrinos, Schmidt. Lecture Notes in Physics 540, Springer gravitons) that move along kα. The Vaidya metric de- Verlag, Berlin-Heidelberg-New York, 1-126; see also scribing spherical implosion of null dust implies that in gr-qc/0004016. case of a “gentle” inflow of the dust, a naked singular- ity forms. This is relevant in the context of the cosmic [3] Bonnor WB (1992), Physical Interpretation of Vac- censorship conjecture (cf., e.g., [8]). uum Solutions of Einstein’s Equations. Part I. Time- independent Solutions. General Relativity and Grav- itation 24, 551-574. 8 Cosmological models [4] Bonnor WB, Griffiths JB, and MacCallum MAH There exist important generalizations of the standard (1994), Physical Interpretation of Vacuum Solutions FLRW models other than the above-mentioned Bianchi of Einstein’s Equations. Part II. Time-dependent General Relativity and Gravitation 26 models, in particularly those that maintain spherical Solutions. , symmetry but do not require homogeneity. The best 687-729. known are the Lemaˆıtre-Tolman-Bondi models of inho- [5] Frolov VP and Novikov ID (1998), Black Hole mogeneous universes of pure dust, the density of which Physics. Kluwer Academic Publishers, Dordrecht- may vary [9]. (J. Gair recently generalized these models Boston-London. by including anisotropic pressure and null dust.) Other explicit cosmological models of principal inter- [6] Griffiths JB (1991), Colliding Plane Waves in Gen- est involve, e.g., the G¨odel universe—a homogeneous, eral Relativity. Oxford University Press, Oxford. stationary spacetime with Λ < 0 and incoherent rotat- [7] Hawking SW and Ellis GFR (1973), The Large ing matter in which there exist closed timelike curves Scale Structure of Space-Time through every point; the Kantowski-Sachs solutions— . Cambridge Univer- sity Press, Cambridge. possessing homogeneous spacelike hypersurfaces but (in contrast to the Bianchi models) admitting no simply- [8] Joshi PS (1993), Global Aspects in Gravitation and transitive G3; and vacuum Gowdy models (“generalized Cosmology. Clarendon Press, Oxford. Einstein-Rosen waves”) admitting G2 with compact 2- tori as its group orbits and representing cosmological [9] Krasi´nski A (1997), Inhomogeneous Cosmological models closed by gravitational waves. See [[entry Cos- Models. Cambridge University Press, Cambridge. mology: Mathematical Aspects]] and Refs. [13], [1], [2], [7], [9]. [10] Misner C, Thorne KS, and Wheeler JA (1973), Gravitation. W. H. Freeman and Co., San Francisco. [11] Ort´ın T (2004), Gravity and Strings. Cambridge See also: University Press, Cambridge. General Relativity: Overview. Stationary Black Holes. [12] Semer´ak O, Podolsk´yJ, and Zofkaˇ M (eds.) (2002), Asymptotic Structure and Conformal Infinity. Newto- Gravitation: Following the Prague Inpiration. World nian Limit of General Relativity. Cosmology: Mathe- Scientific, Singapore-London. matical Aspects. Spinors and Spin-Coefficients. Space- time Topology, Causal Structure and Singularities. Lie [13] Stephani H, Kramer D, MacCallum MAH, Hoense- Groups and Lie Algebras. General Relativity: Experi- laers C, and Herlt E (2003), Exact Solutions of Ein- mental Tests. stein’s Field Equations - Second Edition. Cambridge University Press, Cambridge. Suggestions for further reading: [14] Wald RM (1984), General Relativity. The University of Chicago Press, Chicago. [1-14]
References
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[2] Biˇc´ak J (2000), Selected Solutions of Einstein’s Field Equations: Their Role in General Relativ-
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