JacekJezierski,UniwersytetWarszawski e-mail: Jacek ...jurekfest.fuw.edu.pl/slides/jezierski.pdfJacek Jezierski, Uniwersytet Warszawski Geometry of null hypersurfaces 6/39 Curvatureendomorphism,Raychaudhuriequation

Geometry of null hypersurfaces

Jacek Jezierski, Uniwersytet Warszawskie-mail: [email protected]

Jurekfest, Warszawa

Abstract:We discuss geometry of null surfaces (and its possible applicationsto the horizons, null shells, near horizon geometry, thermodynamicsof black holes)

Jacek Jezierski, Uniwersytet Warszawski Geometry of null hypersurfaces 1/39

Old ideas

In Synge’s festshrift volume [GR, O’Raifeartaigh, Oxford 1972,101-15] Roger Penrose distinguished three basic structures which anull hypersurface N in four-dimensional spacetime M acquires fromthe ambient Lorentzian geometry:

the degenerate metric g |N (see [P. Nurowski, D.C Robinson,CQG 17 (2000) 4065-84] for Cartan’s classification of themand the solution of the local equivalence problem)the concept of an affine parameter along each of the nullgeodesics from the two-parameter family ruling Nthe concept of parallel transport for tangent vectors alongeach of the null geodesics


Geometric structures on screen distribution

Natural geometric structures on TN/K – screen distributiontime-oriented Lorentzian manifold M (−,+,+,+)null hypersurface N – submanifold with codim=1 withdegenerate induced metric g |N (0,+,+), K – time-orientednon-vanishing null vector field such that K⊥p = TpN at eachpoint p ∈ N

1 K is null and tangent to N, g(X ,K )=0 iff X ∈ ΓTN2 integral curves of K – null geodesic generators of N3 K is determined by N up to a scaling factor – positive function


Screen distribution TN/K

TpN/K :={X : X ∈ TpN

}where X = [X ]mod K is an

equivalence class of the relation mod K defined as follows:

X ≡ Y (mod K ) ⇐⇒ X −Y is parallel to K

TN/K := ∪p∈NTpN/K vector bundle over N with2-dimensional fibers (equipped with Riemannian metric h), thestructure does not depend on the choice of K (scaling factor)

h : TpN/K ×TpN/K −→ R , h(X ,Y ) = g(X ,Y )

Remark: If t(K , ·) = 0 then t̄(X̄ , ·) can be correctly defined onTN/K . This implies that g , b, B are well defined on TN/K .


Null Weingarten map and second fundamental form

null Weingarten map b̄K (depending on the choice of scalingfactor, in non-degenerate case one can always take unitnormal to the hypersurface but in null case the vectorfield Kis no longer transversal to N and has always scaling factorfreedom because its length vanishes)

b̄K : TpN/K −→ TpN/K , b̄K (X ) = ∇XK

b̄fK = f b̄K , f ∈ C∞(N) , f > 0

null second fundamental form B̄K (bilinear form associated tob̄K via h)

B̄K : TpN/K ×TpN/K −→ R

B̄K (X ,Y ) = h(b̄K (X ),Y ) = g(∇XK ,Y )

b̄K is self-adjoint with respect to h and B̄K is symmetric


Null mean curvature

N is totally geodesic (i.e. restriction to N of the Levi-Civitaconnection of M is an affine connection on N, any geodesic inM starting tangent to N stays in N) ⇐⇒ B = 0,(non-expanding horizon is a typical example)0 = (∇XK |Y ) ⇒ ∇XK = w(X )Knull mean curvature of N with respect to K

θ := trb =2

∑i=1

B̄K (ei ,ei ) =2

∑i=1

g(∇eiK ,ei )

S – two-dimensional submanifold of N transverse to K ,ei – orthonormal basis for TpS in the induced metric,ei – orthonormal basis for TpN/K


Curvature endomorphism, Raychaudhuri equation

Assume that K is an (affine-)geodesic vector field i.e. ∇KK = 0

We denote by ′ covariant differentiation in the null direction:

Y ′ := ∇KY , b′(Y ) := b(Y )′−b(Y′)

curvature endomorphism

R : TpN/K −→ TpN/K , R(X ) = Riemann(X ,K )K

Ricatti equationb′+b2 +R = 0

Taking the trace we obtain well-known Raychaudhuri equation:

θ′ =−Ricci(K ,K )−B2 , B2 = σ2 + 12θ2 (1)

σ – shear scalar corresponding to the trace free part of BJacek Jezierski, Uniwersytet Warszawski Geometry of null hypersurfaces 7/39

Well-known fact

The following proposition is a standard application of theRaychaudhuri equation.

PropositionLet M be a spacetime which obeys the null energy condition i.e.Ricci(X ,X )≥ 0 for all null vectors X, and let N be a smooth nullhypersurface in M. If the null generators of N are futuregeodesically complete then N has non-negative null meancurvature i.e. θ≥ 0.


Weingarten map – two possibilities

b(X ) = ∇XK , b̄(X̄ ) = ∇XK , π(X ) = X̄

TN π // TN/K

TN

b

OO

π //

TN/K

||

b̄

OO

N


Properties of b and B on TN

Properties of ∇K : TN → TN and g(∇K ) : TN → T ∗N

b and B

b(X ) := ∇XK , B(X ,Y ) := g(∇XK ,Y )

∇X (K |K ) = 0⇒ (∇XK |K ) = 0⇒ b(TN)⊂ TNB is symmetric and bilinear£Kg = 2B (£ – Lie derivative)K is geodesicfor all X (b(X )|K ) = 0 ⇒ B(·,K ) = 0BfK (X ,Y ) = fBK (X ,Y ) (scaling)


Questions:What is the analog of canonical ADM momentum for the nullsurface?What are the ”initial value constraints”?Are they intrinsic objects?

Applications:Dynamics of the light-like matter shell from matter Lagrangianwhich is an invariant scalar density on N [Dynamics of a selfgravitating light-like matter shell: a gauge-invariant Lagrangian andHamiltonian description, Physical Review D 65 (2002), 064036]Dynamics of gravitational field in a finite volume with nullboundary and its application to black holes thermodynamics[Dynamics of gravitational field within a wave front andthermodynamics of black holes, Physical Review D 70 (2004),124010]


Non-degenerate hypersurface – reminderCanonical ADM momentum

Pkl =√

detgmn(gklTrK −K kl )

where K kl is the second fundamental form (external curvature) of theembedding of the hypersurface into the space-time M

Gauss-Codazzi equations for non-degenerate hypersurface

Pi l |l =√

detgmnGiµnµ (= 8π√

detgmnTiµnµ)

(detgmn)R −PklPkl +12 (P

klgkl )2 = 2(detgmn)Gµνnµnν

(= 16π(detgmn)Tµνnµnν)R is the (three–dimensional) scalar curvature of gkl ,nµ is a four–vector normal to the hypersurface,Tµν is an energy–momentum tensor of the matter field,and the calculations have been made with respect to the non-degenerateinduced three–metric gkl ("|" denotes covariant derivative, indices areraised and lowered etc.)


Coordinates convention

A null hypersurface in a Lorentzian spacetime M is athree-dimensional submanifold N ⊂M such that the restriction gabof the spacetime metric gµν to N is degenerate.We shall often use adapted coordinates:

coordinate x3 is constant on N.Space coordinates will be labeled by k, l = 1,2,3;coordinates on N will be labeled by a,b = 0,1,2;coordinates on S will be labeled by A,B = 1,2.Spacetime coordinates will be labeled by Greek charactersα,β,µ,ν.


Repetition in coordinates

The non-degeneracy of the spacetime metric implies that themetric gab induced on N from the spacetime metric gµν hassignature (0,+,+). This means that there is a non-vanishingnull-like vector field K a on N, such that its four-dimensionalembedding K µ to M (in adapted coordinates K 3 = 0) is orthogonalto N. Hence, the covector Kν = K µgµν = K agaν vanishes onvectors tangent to N and, therefore, the following identity holds:

K agab ≡ 0 . (2)

It is easy to prove that integral curves of K a are geodesic curves ofthe spacetime metric gµν. Moreover, any null hypersurface N mayalways be embedded in a one-parameter congruence of nullhypersurfaces.


Volume elementSince our considerations are purely local, we fix the orientation of the R1component (along K ) in N = R1×S2 and assume that null-like vectorsK describing degeneracy of the metric gab of N will be always compatiblewith this orientation. Moreover, we shall always use coordinates suchthat the coordinate x0 increases in the direction of K , i.e., inequalityK (x0) = K 0 > 0 holds. In these coordinates degeneracy fields are of theform K = f (∂0−nA∂A), where f > 0, nA = g0A and we rise indices withthe help of the two-dimensional matrix ˜̃gAB , inverse to gAB .Denote by λ the two-dimensional volume form on each surface x0 = const:

λ :=√

detgAB , (3)then for any degeneracy field K of gab the following object

vK :=λ

K (x0)is a well defined scalar density on N. This means that

vK := vKdx0∧dx1∧dx2

is a coordinate-independent differential three-form on N. However, vKdepends upon the choice of the field K .


Canonical vector density associated with null vectorfield KIt follows immediately from the above definition that the following object:

Λ = vK K

is a well defined (i.e., coordinate-independent) vector density on N.Obviously, it does not depend upon any choice of the field K :

Λ = λ(∂0−nA∂A) (4)

and it is an intrinsic property of the internal geometry gab of N. Thesame is true for the divergence ∂aΛa, which is, therefore, an invariant,K -independent, scalar density on N. Mathematically (in terms ofdifferential forms), the quantity Λ represents the two-form:

L := Λa(∂a cdx0∧dx1∧dx2

),

whereas the divergence represents its exterior derivative (a three-from):dL := (∂aΛa)dx0∧dx1∧dx2.In particular, a null surface with vanishing dL is the non-expandinghorizon.


L and vK without coordinates

Both objects L and vK may be defined geometrically, without any use ofcoordinates. For this purpose we note that at each point p ∈ N, thetangent space TpN may be quotiented with respect to the degeneracysubspace spanned by K . The quotient space TpN/K carries anon-degenerate Riemannian metric h and, therefore, is equipped with avolume form ω (its coordinate expression would be: ω = λ dx1∧dx2).The two-form L is equal to the pull-back of ω from the quotient spaceTpN/K to TpN.

π : TpN −→ TpN/K , L := π∗ω

The three-form vK may be defined as a product:

vK = α∧L ,

where α is any one-form on N, such that < K ,α >≡ 1.We have

dL = θvKwhere θ is a null mean curvature of N.


Connection needs (null) isometry

The degenerate metric gab on N does not allow to define via thecompatibility condition ∇g = 0, any natural connection, whichcould be applied to generic tensor fields on N.Moreover, such connection drastically reduces the degeneratemetric structure g on N.Existence of any symmetric connection ∇̄ on N compatible with gimplies £Kg = 0 and N becomes totally geodesic.

symmetric connection implies trivial second fundamental form

∇̄g = 0 ⇒ £Kg = 0


Divergence of (contravariant-covariant) tensor density

Nevertheless, there is one exception: the degenerate metric definesuniquely a certain covariant, first order differential operator. Theoperator may be applied only to mixed (contravariant-covariant)tensor density fields Hab, satisfying the following algebraicidentities:algebraic properties of H needed for divergence

HabKb = 0 , (5)Hab = Hba , (6)

where Hab := gacHc b. Its definition cannot be extended to othertensorial fields on N. Fortunately, the extrinsic curvature of anull-like surface and the energy-momentum tensor of a null-likeshell are described by tensor densities of this type.


Divergence of tensor density

The operator, which we denote by ∇a, is defined by means of thefour-dimensional metric connection in the ambient spacetime M inthe following way:Given Hab, take any its extension Hµν to a four-dimensional,symmetric tensor density, “orthogonal” to N, i.e. satisfyingH⊥ν = 0 (“⊥” denotes the component transversal to N). Define∇aHab as the restriction to N of the four-dimensional covariantdivergence ∇µHµν.


Divergence of tensor densityThe ambiguities which arise when extending three-dimensional objectHab living on N to the four-dimensional one cancel out and the result isunambiguously defined as a covector density on N. It turns out, however,that this result does not depend upon the spacetime geometry and maybe defined intrinsically on N as follows:

∇aHab = ∂aHab−12H

acgac,b ,

where gac,b := ∂bgac , a tensor density Hab satisfies identities (5) and (6),and moreover, Hac is any symmetric tensor density, which reproducesHab when lowering an index:

Hab = Hacgcb . (7)

It is easily seen, that such a tensor density always exists due to identities(5) and (6), but the reconstruction of Hac from Hab is not unique,because Hac +CK aK c also satisfies (7) if Hac does. Conversely, twosuch symmetric tensors Hac satisfying (7) may differ only by CK aK c .Fortunately, this non-uniqueness does not influence the value of (7).


Intrinsic divergence of tensor density

Hence, the following definition makes sense:

intrinsic divergence on N

∇aHab := ∂aHab−12H

acgac,b . (8)

The right-hand-side does not depend upon any choice ofcoordinates (i.e., transforms like a genuine covector density underchange of coordinates).


2+1 decomposition of Hab

To express the result directly in terms of the original tensor density Hab,we observe that it has five independent components and may be uniquelyreconstructed from H0A (2 independent components) and the symmetrictwo-dimensional matrix HAB (3 independent components). Indeed,identities (5) and (6) may be rewritten as follows:

HAB = ˜̃gAC HCB−nAH0B , (9)

H00 = H0AnA , (10)

HB0 =(

˜̃gBC HCA−nBH0A)nA . (11)

The correspondence between Hab and (H0A,HAB) is one-to-one.


Non-uniqueness in the reconstruction of Hab

To reconstruct Hab from Hab up to an arbitrary additive term CK aKb,take the following, coordinate dependent, symmetric quantity:

FAB := ˜̃gAC HCD ˜̃gDB−nAH0C ˜̃g

CB−nBH0C ˜̃gCA

, (12)

F0A := H0C ˜̃gCA =: FA0 , (13)

F00 := 0 . (14)

It is easy to observe that any Hab satisfying (7) must be of the form:

Hab = Fab + H00K aKb . (15)

The non-uniqueness in the reconstruction of Hab is, therefore, completelydescribed by the arbitrariness in the choice of the value of H00. Usingthese results, we finally obtain:

∇aHab := ∂aHab−12H

acgac,b = ∂aHab−12F

acgac,b

= ∂aHab−12

(2H0A nA,b−HAC ˜̃g

AC,b

). (16)


Divergence of tensor density on M restricted to N

The operator on the right-hand-side of (16) is called the(three-dimensional) covariant derivative of Hab on N with respect to itsdegenerate metric gab. It is well defined (i.e., coordinate-independent) fora tensor density Hab fulfilling conditions (5) and (6). One can also showthat the above definition coincides with the one given in terms of thefour-dimensional metric connection and due to (7), it equals:

divergence on N induced from ambient M

∇µHµb = ∂µHµb−

12H

µλgµλ,b = ∂aHab−12H

acgac,b , (17)

and, whence, coincides with ∇aHab defined intrinsically on N.


Canonical tensor density – analog of ADM momentum

To describe exterior geometry of N we begin with covariant derivativesalong N of the “orthogonal vector K”. Consider the tensor ∇aK µ. Unlikethe non-degenerate case, there is no unique “normalization” of K and,therefore, such an object does depend upon a choice of the field K . Thelength of K vanishes. Hence, the tensor is again orthogonal to N,i.e., the components corresponding to µ = 3 vanish identically in adaptedcoordinates. This means that ∇aKb is a purely three-dimensional tensorliving on N. For our purposes it is useful to use the “ADM-momentum”version of this object, defined in the following way:

null “ADM-momentum”

Qab(K ) :=−s {vK (∇bK a−δab∇cK c) + δab∂cΛc} , (18)

where s := sgng03 =±1. Due to above convention, the object Qab(K )feels only external orientation of N and does not feel any internalorientation of the field K .Remark: If N is a non-expanding horizon, the last term in the abovedefinition vanishes.


Comment to the definition of Q

The last term δab∇cK c in (18) is K -independent. It has beenintroduced in order to correct algebraic properties of the quantity

vK (∇bK a−δab∇cK c) .

One can show that Qab satisfies identities (5)–(6) and, therefore,its covariant divergence with respect to the degenerate metric gabon N is uniquely defined.


Div of Q versus Einstein tensor density

This divergence enters into the Gauss–Codazzi equations, whichrelate the divergence of Q with the transversal component G⊥b ofthe Einstein tensor density

Gµν =√|detg |

(Rµν−

12δ

µνR).

The transversal component of such a tensor density is a welldefined three-dimensional object living on N. In coordinate systemadapted to N, i.e., such that the coordinate x3 is constant on N,we have G⊥b = G3b. Due to the fact that G is a tensor density,components G3b do not change with changes of the coordinate x3,provided it remains constant on N. These components describe,therefore, an intrinsic covector density living on N.


Vector constraint on N

PropositionThe following null-like-surface version of the Gauss–Codazziequation is true:

∇aQab(K ) + svK ∂b(

∂cΛcvK

)≡−G⊥b . (19)

We remind that the ratio between two scalar densities: ∂cΛc andvK , is a scalar function θ. Its gradient is a covector field. Finally,multiplied by the density vK , it produces an intrinsic covectordensity on N. This proves that also the left-hand-side is a welldefined geometric object living on N.The component KbG⊥b of the equation (19) is nothing but adensitized form of Raychaudhuri equation (1) for the congruence ofnull geodesics generated by the vector field K .


2+1 decomposition of constraints

K = ∂0−nA∂A , ∇aQab(K ) + svK ∂b(

∂cΛcvK

)≡−G⊥b

The quantities lAB :=−BAB =−gAcbc B and wa =−b0a represent 2+1decomposition of Weingarten map bab.

K a∂al + (waK a) l−12 l

2− l̄AB l̄AB = 8πTabK aKb ,

where we have decomposed lAB into its trace l (expansion) and itstraceless part l̄AB := lAB−

12gAB l (shear).

waK a corresponds to surface gravity.

∂0wB−wB‖AnA−wAnA‖B− (waKa)‖B−wB l + l̄AB‖A−

12 l‖B =−8πTaBK

a

In case of vacuum spacetimes the right-hand sides of the aboveconstraints vanish.


(Foliation dependent) fourth constraint

if we add slicing of N as an additional structure we can derivefourth constraint:“half”-intrinsic on N fourth constraint

−G(K ,Z ) = (∂0−w0− l)k +12

(2)R +wA||A−wAwA

where(2)R is a scalar curvature of the Riemannian metric structure

gAB and Z is null

g(K ,K ) = g(Z ,Z ) = 0 , g(K ,Z ) = 1 , g(K ,∂A) = g(Z ,∂A) = 0

lab = K µΓµab , kab = Z µΓµab ,

−wa = Z µK νΓµνa = Kbkab , Kb lab = 0


Conclusions/Applications

Crossing null shells – Dray-t’Hooft-Redmount formulaB=0 (totally geodesic null surface) – (non-expanding, Killing)horizons, Near Horizon GeometryConstraints – three of them are intrinsic and correspond todivergence of tensor density, the fourth one needs extrastructure – foliation of NQ and g play a role of ‘initial/boundary data’ on N, they canbe used to define local first law of black hole thermodynamicsfor privileged field KQ and g reduce to covector wA and two-metric gAB in thecase of Near Horizon Geometry and vacuum Einsteinequations lead to basic equation:


Basic equation – non-trivial part of NHG

wA||B +wB ||A +2wAwB = RAB =12Rδ

AB , (20)

where wA is a vector field, wB = gABwB , || denotes covariant derivativewith respect to the metric gAB and RAB is its Ricci tensor.

The above equation appears not only in the context ofKundt’s class, it also arises in the study of vacuum degenerateisolated horizons.Any degenerate Killing horizon also implies this equation.For axial symmetry and spherical topology there is a uniquesolution – extremal Kerr.When one-form wBdxB is closed (e.g. static degeneratehorizon) there are no solutions of (20). However, in general,the space of solutions is not known.


Linearized basic equation

Existence of (general non-symmetric) solutions to linearized basicequation around Kerr has the answer – no solutions

J. Jezierski, B. Kamiński: Towards uniqueness of degenerate axiallysymmetric Killing horizon, Gen Relativ Gravit 45 (2013) 987-1004, DOI10.1007/s10714-013-1506-0, arXiv: 1206.5136 [gr-qc]PT Chruściel, SJ Szybka and P Tod: Towards a classification of vacuumnear-horizons geometries, Class. Quantum Grav. 35 (2018) 015002


Green function on a sphere −→ extremal Kerr

Extremal Kerr has natural representation (in NHG) by generalizedGreen function

J. Jezierski: On the existence of Kundt’s metrics and degenerate (orextremal) Killing horizons, Class. Quantum Grav. 26 (2009) 035011


Solution of the problem with axial symmetry

g = 2m2[1+ cos2 θ

2 dθ2 + 2sin

2 θ1+ cos2 θdϕ

2]

(21)

wθ =− sinθcosθm2(1+ cos2 θ)2 , wϕ = 12m2(1+ cos2 θ) , (22)

represents extremal Kerr with mass m and angular momentum m2.



It is worth to notice that the Kerr solution (22) in terms of

ΦA :=wA

wBwB

has a simple and natural form.Linear equations for ΦA extended through the “poles” are

linear part of basic equation

ΦA||C εAC = 4πm2 (δθ=π−δθ=0) , (23)

ΦA||A = 1−4πm2 (δθ=π + δθ=0) , (24)

where by δp we denote a Dirac delta at point p and 8πm2(=∫

λ)is a total volume of the (Kerr) sphere (21).



Let Gp be a Green function satisfying{4Gp = 1−8πm2δp ,∫

λGp = 0 .(25)

The potentials Φ, Φ̃ for the covector field ΦA defined (up to aconstant) as follows

ΦA = ∂AΦ + εAB∂BΦ̃ (26)

take a simple form

Φ = 12(Gθ=0 +Gθ=π)

Φ̃ = 12(Gθ=0−Gθ=π)



because equations (23), (24) and (26) imply

4Φ = 1−4πm2 (δθ=π + δθ=0) ,4Φ̃ = 4πm2 (δθ=π−δθ=0) .

Green functions for extremal Kerr (21)

closed form of Green function for extremal Kerr

Gθ=0 = 4m2[12 sin

2 θ2 +

18 sin

2 θ− log(sin θ2 ) +13

],

Gθ=π = 4m2[12 cos

2 θ2 +

18 sin

2 θ− log(cos θ2 ) +13

].


Documents

JacekJezierski,UniwersytetWarszawski e-mail: Jacek ...jurekfest.fuw.edu.pl/slides/jezierski.pdfJacek Jezierski, Uniwersytet Warszawski Geometry of null hypersurfaces 6/39 Curvatureendomorphism,Raychaudhuriequation