Horizons As Boundary Conditions in Spherical Symmetry

Horizons as boundary conditions in spherical symmetry

Sharmila Gunasekaran∗ and Ivan Booth† Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, Newfoundland and Labrador, A1C 5S7, Canada

(Dated: March 31, 2021) We initiate the development of a horizon-based initial (or rather final) value formalism to describe the geometry and physics of the near-horizon spacetime: data specified on the horizon and a future ingoing null boundary determine the near-horizon geometry. In this initial paper we restrict our attention to spherically symmetric spacetimes made dynamic by matter fields. We illustrate the formalism by considering a black hole interacting with a) inward-falling, null matter (with no outward flux) and b) a massless scalar field. The inward-falling case can be exactly solved from horizon data. For the more involved case of the scalar field we analytically investigate the near slowly evolving horizon regime and propose a numerical integration for the general case.

I. INTRODUCTION

This paper begins an investigation into what horizon dynamics can tell us about external black hole physics. At first thought this might seem obvious: if one watches a numerical simulation of a black hole merger and sees a post-merger horizon ringdown (see for example [1]) then it is natural to think of that oscillation as a source of emitted gravitational waves. However this cannot be the case. Neither event nor apparent horizons can actually (a)Future and past domains of dependence for send signals to infinity: apparent horizons lie inside event H : standard (3+1) IVP horizons which in turn are the boundary for signals that dynamic can reach infinity[2]. It is not horizons themselves that interact but rather the “near horizon” fields. This idea was (partially) formalized as a “stretched horizon” in the membrane paradigm[3]. Then the best that we can hope for from horizons is that they act as a proxy for the near horizon fields with horizon evolution reflecting some aspects of their dynamics. As explored in [4–8] there should then be a correlation between horizon evolution and external, observable, black hole physics. Robinson-Trautman spacetimes (see for example [9]) (b)Future and past domains of dependence for demonstrate that this correlation cannot be perfect. In Hdynamic ∪ N : characteristic IVP those spacetimes there can be outgoing gravitational (or other) radiation arbitrarily close to an isolated (equilib- FIG. 1. Domains of dependence of initial data arXiv:1905.02748v3 [gr-qc] 30 Mar 2021 rium) horizon[10]. Hence our goal is two-fold: both to understand the conditions under which a correlation will exist and to learn precisely what information it contains. and/or marginally trapped surfaces that seek to localize The idea that horizons should encode physical informa- black holes. These include apparent[2], trapping[11], iso- tion about black hole physics is not new. The classical lated [10, 12–14] and dynamical [15] horizons as well as definition of a black hole as the complement of the causal future holographic screens [16]. These quasilocal defini- past of future null infinity [2] is essentially global and so tions of black holes have successfully localized black hole defines a black hole spacetime rather than a black hole mechanics to the horizon[11–13, 15–17] and been particu- in some spacetime. However there are also a range of ge- larly useful in formalizing what it means for a (localized) ometrically defined black hole boundaries based on outer black hole to evolve or be in equilibrium. They are used in numerical relativity not only as excision surfaces (see, for example the discussions in [18, 19]) but also in interpreting physics (for example [4–8, 20–24]). ∗ [email protected] In this paper we work to quantitatively link horizon dy- † [email protected] namics to observable black hole physics. To establish an 2 initial framework and build intuition we for now restrict the ingoing null hypersurfaces and increases into the fu- our attention to spherically symmetric marginally outer ture). Hence, gρρ = 0 and the curves tangent to the trapped tubes (MOTTs) in similarly symmetric space- future-oriented inward-pointing times. Matter fields are included to drive the dynam- ∂ ics. Our primary approach is to take horizon data as N = (1) a (partial) final boundary condition that is used to de- ∂ρ termine the fields in a region of spacetime in its causal are null. We then scale v so that V = ∂ satisfies past. In particular these boundary conditions constrain ∂v the geometry and physics of the associated “near hori- V· N = −1. (2) zon” spacetime. The main application that we have in mind is interpreting the physics of evolving horizons that One coordinate freedom still remains: the scaling of the have been generated by either numerical simulations or affine parameter on the individual null geodesics theoretical considerations. Normally, data on a MOTT by itself is not sufficient to ρ˜ = f(v)ρ . (3) specify any region of the external spacetime. As shown in FIG. 1(a) even for a spacelike MOTT (a dynamical hori- In subsection II C we will fix this freedom by specifying zon) the region determined by a standard (3+1) initial how N is to be scaled along the ρ = 0 surface Σ (which value formulation would lie entirely within the event hori- we take to be a black hole horizon). zon. More information is needed to determine the near- horizon spacetime and hence in this paper we work with a characteristic initial value formulation [25–29] where extra data is specified on a null surface N that is transverse to the horizon (FIG. 1(b)). Intuitively the horizon records inward-moving information while N records the outward-moving information. Together they are sufficient to reconstruct the spacetime. There is an existing literature that studies spacetime near horizons, however it does not exactly address this problem. Most works focus on isolated horizons. [30] and [31] examine spacetime near an isolated extremal horizon as a Taylor series expansion of the horizon while [32] and [33] study spacetime near more general isolated horizons but in a characteristic initial value formulation FIG. 2. Coordinate system for characteristic evolution. We with the extra information specified on a transverse null work with final boundary conditions so that in the region of surface. [34] studied both the isolated and dynamical interest ρ < 0. case though again as a Taylor series expansion off the horizon. In the case of the Taylor expansions, as one goes to higher and higher orders one needs to know higher Next we define the future-oriented outward-pointing and higher order derivatives of metric quantities at the a null normal to the spherical surfaces S(v,ρ) as ` and scale horizon to continue the expansion. While the current so that paper instead investigates the problem as a final value problem, it otherwise closely follows the notation of and ` · N = −1 . (4) uses many results from [34]. It is organized as follows. We introduce the final value With this choice the four-metric gab and induced two- formulation of spherically symmetric general relativity in metricq ãb on the S(v,ρ) are related by

Sec.II. We illustrate this for infalling null matter in III ab ab a b a b and then the much more interesting massless scalar ﬁeld g =q ˜ − ` N − N ` . (5) in Sec.IV. We conclude with a discussion of results in Further for some function C we can write Sec.V. V = ` − CN. (6)

II. FORMULATION The coordinates and normal vectors are depicted in FIG.2 and give the following form of the metric: A. Coordinates and metric ds2 = 2C(v, ρ)dv2 − 2dv dρ + R(v, ρ)2dΩ2 (7)

We work in a spherically symmetric spacetime (M, g) where R(v, ρ) is the areal radius of the S(v,ρ) surfaces. and a coordinate system whose non-angular coordinates Note the similarity to ingoing Eddington-Finkelstein co- are ρ (an ingoing affine parameter) and v (which labels ordinates for a Schwarzschild black hole. However ∂/∂ρ 3 points inwards as opposed to the outward oriented ∂/∂r where by the choice of the coordinates in those coordinates (hence the negative sign on the dvdρ cross-term). κ = LN C. (20) Typically, as shown in FIG.2 we will be interested in regions of spacetime that are bordered in the future by These equations can be derived from the variations for a surface Σ of indeterminate sign on which ρ = 0 and a the corresponding geometric quantities (see, for example, null N which is one of the v=constant surfaces (and so [35] and [34]) and of course are coupled to the matter ρ < 0 in the region of interest). We will explore how data content of the system through the Einstein equations on those surfaces determines the region of spacetime in their causal past. Gab = 8πTab . (21) Using (8) and (9) we can rewrite the constraint and B. Equations of motion evolution equations in terms of the metric coefficients and coordinates as: In this section we break up the Einstein equations relative to these coordinates, beginning by defining some R,v = R` − CRN , (22) geometric quantities that appear in the equations. C (1 + 4R`RN ) R R = κR + − (G + CG ) , First the null expansions for the à and N a congruences `,v ` 2R 2 `` `N are (23)

ab 2 (1 + 4R`RN ) R θ =q ˜ ∇ ` = L R and (8) RN,v = − κRN − + (G`N + CGNN ). (`) a b R ` 2R 2 2 2 (24) θ =q ãb∇ N = L R = R . (9) (N) a b R N R ,ρ and while the inaffinities of the null vector fields are R a b κ = −N N ∇ ` = 0 and (10) R,ρρ = − GNN , (25) N b a 2 a b 2 κV = κ` − CκN = −` Nb∇a` . (11) 1 R (RR`),ρ = − + G`N , (26) By construction κ = 0 and so we can drop it from our 2 2 N 1 2R R 1 equations and henceforth write C = + ` N − G − G , (27) ,ρρ R2 R2 2 q˜ `N κ ≡ κV = κ` . (12) where Finally the Gaussian curvature of S(v,ρ) is: κ = C,ρ. (28) ˜ 1 K = 2 . (13) R For those who don’t want to work through the deriva- Then these curvature quantities are related by con- tions of [35] and [34], these can also be derived fairly eas- straint equations along the surfaces of constant ρ ily (thanks to the spherical symmetry) from an explicit calculation of the Einstein tensor for (7). LV R =L `R − CL N R (by definition) , (14) 1 LV θ(`) =κ θ(`) + C + θ(N)θ(`) − G`N R2 C. Final Data 1 − G + θ2 , (15) `` 2 (`) We will focus on the case where ρ = 0 is an isolated or 1 dynamical horizon H. Thus LV θ(N) = − κ θ(N) − 2 + θ(N)θ(`) − G`N R H H θ(`) = 0 ⇐⇒ R` = 0 . (29) 1 2 + C GNN + θ(N) , (16) 2 The notation =H indicates that the equality holds on H and “time” derivatives in the ρ direction (but not necessarily anywhere else). Further we can use the coordinate freedom (3) to set 2 θ(N) LN θ(N) = − − GNN , (17) H H 2 RN = R,ρ| = −1 . (30) 1 L θ = − − θ θ + G , (18) N (`) R2 (N) (`) `N On H, the constraints (22)-(24) fix three of 1 1 1 L κ = + θ θ − G − G , (19) N R2 2 (N) (`) 2 q˜ `N {C, κ, R, R`,RN ,G``,G`N ,GNN } (31) 4

H given the other ﬁve quantities. For example if R` = 0 H and RN = −1 then (22) and (23) give 2 H H R G`` R,v = C = 2 (32) 1 − R G`N and (24) gives (a) Isolated horizon : dρ is timelike for all values of ρ 1 R κ = C =H − (G + CG ) . (33) ρ 2R 2 `N NN

Thus if G`` and G`N are specified for vi ≤ v ≤ vf on H H and R(vf ) = Rf then one can solve (32) to find R over the entire range. Equivalently one could take R and one of G`` or G`N as primary and then solve for the other component of the stress-energy. Of course, in general the matter terms will also be (b) Dynamical horizon : dρ is spacelike for small constrained by their own equations; these will be treated values of ρ and eventually becomes timelike for large in later sections. Further data on ρ = 0 will generally values of ρ not be sufficient to fully determine the regions of interest and data will also be needed on an N . Again this will FIG. 3. Spacetime foliation for isolated and dynamical hori- depend on the specific matter model. zons Nevertheless if there is a MOTT at ρ = 0 then the constraints provide significant information about the hori- (no matter flows in the outward `-direction). Further, for zon. If G = 0 (no flux of matter through the horizon) `` simplicity we also assume that it is trace-free then we have an isolated horizon with C = 0, a constant R and a null H. This is independent of other components ab g Tab = 0 ⇔ Tq˜ = 2T`N . (35) of the stress-energy. Alternatively if G`` > 0 (the energy conditions forbid Then we can solve for the metric using only the Bianchi 2 it to be negative) and G`N < 1/R then we have a dy- identities namical horizon with C > 0, increasing R and spacelike 1 ab H . Note that this growth doesn’t depend in any way ∇aG = 0, (36) on GNN : there is no sense in which the growing horizon “catches” outward moving matter and hence grows without any reference to detailed equations of motion for even faster. The behaviour of the coordinates relative to the matter field. Keeping spherical symmetry but tem- isolated and dynamical horizons along with I + is illus- porarily suspending the other simplifying assumptions trated in FIG.3. they may be written as: The evolution equations are more complicated and de- L (R2G )+L (R2G ) + R2(2κ G ) pend on the matter field equations. We examine two such ` NN N `N ` NN 1 cases in the following sections. + R2θ G = 0, (37) 2 (N) q˜ 2 2 2 LN (R G``)+L`(R G`N ) + R (−2κN G``) III. TRACELESS INWARD FLOWING NULL 1 MATTER + R2θ G = 0 . (38) 2 (`) q˜

As our first example consider matter that falls entirely In terms of metric coefficients with κN = 0 plus (34) and in the inward N-direction with no outward `-flux. Then (35) these reduce to: data on the horizon should be sufficient to entirely de- 4 termine the region of spacetime traced by the horizon- (R G`N ),ρ = 0 and (39) crossing inward null geodesics: there are no dynamics 1 (R2G ) + (R4G ) = 0. (40) that don’t involve the horizon. `` ,ρ R2 `N ,v Translating these words into equations, we assume that As we shall see, this class of matter includes interesting a b TabN N = 0 (34) examples like Vaidya-Reissner-Nordström(charged null dust). We now demonstrate that given knowledge of G`` and ¯ 1 2 G`N over a region of horizon H = {H : vi ≤ v ≤ vf } G`N > 1/R signals that another MOTS has formed outside the H original one and so a numerical simulation would see an apparent as well as R(vf ) = Rf we can determine the spacetime horizon “jump” [16, 36]. In the current paper all matter satisfies everywhere out along the horizon-crossing inward null 2 G`N < 1/R and so this situation does not arise. geodesics. 5

p 2 2 A. On the horizon where the apparent horizon rH = m + m − q and r is an affine parameter of the ingoing null geodesics. To First consider the constraints on H¯ . In this case it is put it into the form of (7) where the affine parameter measures distance off the horizon we make the transfor- tidier to take R and G`N as primary. Then we can specify mation Q(v) R =H R (v) and G =H (41) H `N 4 r = r − ρ (51) RH H for some functions RH (v) (dimensions of length) and whence the metric takes the form QH (v) (dimensions of length squared) where the form 2 ! of the latter is chosen for future convenience. Then 2 ρ q − rH (rH − ρ) 2 ds = − 2rH,v − 2 dv (52) H rH (rH − ρ) C = RH,v (42) 2 2 − 2dvdρ + (rH − ρ) dΩ . and by (32) That is H 1 Q G`` = RH,v 2 − 4 (43) 2 RH RH ρ q − rH (rH − ρ) C = rH,v − 2 (53) Finally by (33), 2rH (rH − ρ) R = rH − ρ (54) 1 Q κ =H C =H 1 − . (44) ρ 2 and on the horizon 2RH RH

H H C = rH,v and R = rH (55) B. Off the horizon as expected. To do a complete match we calculate the rest of the Next, integrate away from H¯ . First with G = 0 (25) NN quantities. First appropriate null vectors are can be integrated with initial condition (30) to give 2 ! R(v, ρ) = R (v) − ρ . (45) ∂ ρ q − rH (rH − ρ) ∂ H ` = + r − (56) ∂v H,v 2r (r − ρ)2 ∂ρ Then with (41) we can integrate (39) to find H H ∂ Q N = . (57) G = (46) ∂ρ `N R4 Then direct calculation shows that and use this result and (43) to integrate(40) to get 2 2 ρ q − rH (rH − ρ) R − Q RH,v ρ Q H ,v R` = − 2 (58) G`` = 2 2 + 3 . (47) 2rH (rH − ρ) RH R RH R RN = −1 (59) H With these results in hand and initial condition R` = 0 we integrate (26) to get and 2 2 2 r − q r 2ρqq ρ(Q − R + ρRH ) H H,v ,v R = H (48) G`` = + (60) ` 2 r2 r2 r r3 2R RH H H q2 and finally with initial conditions (32) and (33) we can G`N = 2 (61) integrate (27) to find (rH − ρ) GNN = 0 (62) C = RH,v − R` . (49) 2q2 Gq = 2 . (63) (rH − ρ) C. Comparison with Vaidya-Reissner-Nordström 2 It is clear that with RH = rH and Q = q our general We can now compare this derivation to a known ex- results (41)-(49) give rise to the VRN spacetime (as they ample. The Vaidya-Reissner-Nordström(VRN) metric should). takes the form As expected the data on the horizon is sufficient to determine the spacetime everywhere back out along the 2m(v) q(v)2 ds2 = − 1 − + dv2 + 2dvdr + r2dΩ2 ingoing null geodesics: we simply solve a set of (coupled) r r2 ordinary differential equations along each curve. With (50) the matter providing the only dynamics and that matter 6 only moving inwards along the geodesics the problem is where C is the speed of flow of ψ: if C is constant then quite straightforward. In this case there is no need to this has the exact solution specify extra data on N . We now turn to the more interesting case where the ψ = ψ(x − Ct) (73) dynamics are driven by a scalar field for which there will and so any pulse moves with speed dx = C. Any non- be both inward and outward fluxes of matter. dt homogeneous term corresponds to a source which adds or removes energy from the system. Then (70) tells us that the flux in the N-direction (Φ ) is naturally undimin- IV. MASSLESS SCALAR FIELD ` ished as it flows along a (null) surface of constant v and increasing ρ. However the interaction with the flux in the Spherical spacetimes containing a massless scalar field ` direction can cause it to increase or decrease. Similarly φ(v, ρ) are governed by the stress energy tensor given by, (71) describes the flow of the flux in the `-direction (ΦN ) 1 along a surface of constant ρ and increasing v. Rewriting T = ∇ φ∇ φ − g ∇cφ∇ φ (64) with respect to the affine derivative (see Appendix B) ab a b 2 ab c Dv = ∂v + κ it becomes This system has nonvanishing inward and outward fluxes R Φ which are, D Φ + CΦ = − N ` . (74) v N N,ρ R T = (φ )2 (65) `` ` Then, as might be expected, ΦN naturally flows with 2 ∂ ∂ TNN = (φN ) . (66) coordinate speed C (recall that ` = ∂v + C ∂ρ so this is the speed of outgoing light relative to the coordinate ∂ Here and in the following keep in mind that N = ∂ρ and system) but its strength can be augmented or diminished so φN = φ,ρ. We also observe from (64) that, by interactions with the outward flux.

T`N = 0. (67) A. System of first order PDEs These fluxes are related by the wave equation Together (70) and (71) constitute a first-order system φ := ∇α∇ φ = 0 =⇒ (Rφ ) = −R φ . (68) g α ` ,ρ ` ,ρ of partial differential equations for the scalar field. We For our purposes we are not particularly interested in now restructure the gravitational field equations in the the value of φ itself but rather in the associated net flux same way. of energies in the ingoing and outgoing null direction. First with respect to Φ` and ΦN the constraint equa- Hence we define tions (14)-(16) on constant ρ surfaces become: √ √ R,v = R` − CRN (75) Φ` = 4πRφ` and ΦN = 4πRφN . (69) C (1 + 2R R ) Φ2 R = κR + ` N − ` (76) Respectively these are the square roots of the scalar field `,v ` 2R R energy fluxes in the N and ` directions. That is, over a 2 (1 + 2R`RN ) CΦN sphere of radius R,Φ is the square root of the total inte- RN,v = −κRN − + (77) ` 2R R grated flux in the N-direction and ΦN is the square root of the total integrated flux in the `-direction. Though while the “time”-evolution equations (17)-(19) are: not strictly correct, we will often refer to Φ and Φ ` N 2 themselves as fluxes. ΦN R,ρρ = − (78) Then (68) becomes R 1 R Φ (RR`),ρ = − (79) Φ = − ` N (70) 2 `,ρ R 1 + 2R R 2Φ Φ C = ` N − ` N . (80) ,ρρ R2 R2 or, making use of the fact that φ,vρ = φ,ρv, Two of these equations can be simplified. First, inte- RN Φ` H Φ = −κΦ − CΦ − . (71) grating (79) from ρ = 0 on which R` = 0 we find N,v N N,ρ R ρ R = − . (81) These can usefully be understood as advection equations ` 2R with sources. Recall that a general homogeneous advection equation can be written in the form This can be substituted into (76) to turn it into an alge- braic constraint ∂ψ ∂ψ + C = 0 (72) 2 ∂t ∂x C = 2Φ` − 2R` (κR + R`) . (82) 7

Despite these simplifications, the presence of interact- where the null vectors are scaled in the usual way and, ing outward and inward matter fluxes means that in con- as before, the notation =H indicates that all quantities on trast to the dust examples, this is truly a set of coupled both sides of the equality are evaluated on H. partial differential equations. Hence we can expect that This data can be used to evaluate C and R on H¯ . From the matter and spacetime dynamics will be governed by (82) and (84) off-horizon data in addition to data at ρ = 0. H 2 We reformulate as a system of first order PDEs in the C = 2Φ` and (93) following way. First designate Z v H 2 R = Ro + 2 Φ` dv . (94) {R,RN , κ, Φ`, ΦN } (83) vf ¯ as the primary variables. The secondary variables To find ΦN on H we would need to solve {R`,C} are defined by (81) and (82) in terms of the pri- 1 2 2 H 2 Φ` maries. ΦN,v + 1 − 4Φ` ΦN ΦN = −2Φ` ΦN,ρ + (95) Next on ρ = constant surfaces the primary variables 2R R are constrained by which comes from (71) combined with the above results. However at this stage ΦN,ρ isn’t known and so this can R,v = R` − CRN and (84) H only be solved directly in the isolated Φ` = 0 case. There 1 2 RN,v = −κRN − 1 + 2R`RN − 2CΦ (85) H N iso −(v−vf )/2Ro 2R ΦN = ΦNf e (96) along with scalar flux equation (71) while their time evo- where ΦN = ΦN (0, vf ). Equivalently (see Appendix B) lution is governed by f ΦN is affinely constant on an isolated horizon. With RN = −1, (77) tells us that R,ρ = RN (86) Φ2 H 1 2 R = − N (87) κ = 1 − 2CΦN , (97) N,ρ R 2R 1 ¯ κ,ρ = (1 + 2R`RN − 2Φ`ΦN ) (88) and so without ΦN on H we also can’t determine this R2 (away from isolation). However the corner H¯ ∩ N¯ is an R Φ ` N exception to that rule. There we know Φ`,ΦN and Ro Φ`,ρ = − . (89) R and so

We now consider how all of these equations may be H¯ ∩N¯ 1 2 2 κ = 1 − 4Φ` ΦN . (98) used to integrate final data. The scheme is closely related 2Ro to that used in [27]. The situation is less complicated on N¯ . There with ΦN as known data and final values known for all quantities B. Final data on H¯ and N¯ on H¯ ∩ N¯ all other quantities can be calculated in order i) Solve (86) and (87) for R and R . In line with the depiction in FIG.1(b), we specify final N ¯ ¯ data on H ∪ N or rather on the sections H ∪ N where ii) Calculate R` from (81).

H¯ = {(0, v) ∈ H : vi ≤ v ≤ vf } and (90) iii) Solve (89) for Φ`. ¯ N = {(ρ, vf ) ∈ N : ρi ≤ ρ ≤ 0} . iv) Solve (88) for κ.

Their intersection sphere is H¯ ∩ N¯ = (0, vf ). Here and v) Calculate C from (82). in what follows we suppress the angular coordinates. We then have all data on N¯ . The ﬁnal data is

H¯ :Φ (91) ` C. Integrating from the ﬁnal data N¯ :ΦN and

H¯ ∩ N¯ : R = Ro . We now consider how that data can be integrated into the causal past of H¯ ∪ N¯ . The basic steps in the integra- ¯ ¯ Φ` on H is a function of v while ΦN on N is a function tion scheme are demonstrated in a simple numerical inte- of ρ. Ro is a single number. gration based on Euler approximations. This scheme al- Further on H we have ternates between using steps i)-v) to integrate data down the characteristics of constant v followed by an applica- R =H 0 and R =H −1 (92) ` N tion of (71) to calculate ΦN on the next characteristic. 8

D. Spacetime near a slowly evolving horizon

We now apply the formalism to a concrete example: weak scalar fields near the horizon. Physically the black hole will be close to equilibrium and hence the horizon slowly evolving in the sense of [17, 35]. “Near horizon” means that we expand all quantities as Taylor series in ρ and keep terms up to order ρ2. “Weak scalar field” means that we assume ε Φ , Φ ∼ (100) N ` R FIG. 4. The constraint equations along with initial condi- and then expand the terms of the Taylor series up to or- H H 2 0 tions on the horizon, i.e. R` = 0,RN = −1 determine κ, C der . To order the spacetime will be vacuum (and and R on H¯ Schwarzschild), order 1 will be a test scalar field prop- agating on the Schwarzschild background and order 2 will include the back reaction of the scalar field on the geometry.

1. Expanding the equations

We expand all quantities as Taylor series in ρ. That is for X ∈ {R,RN ,R`, κ, C, Φ`, ΦN }

∞ X ρnX(n)(v) X(v, ρ) = (101) n! ∂ n=0 FIG. 5. Evolving Φ in the − ∂v direction. with

(n) (n+1) (n) (n+1) RN = R and κ = C . (102) In more detail, assume a discretization {vm, ρn} (with (0) ¯ ¯ ¯ m and n at their maxima along the final surfaces) by The free final data is Φ` on H, Ro on H ∩ N and the steps ∆v and ∆ρ. Then if all data is known along a Taylor expanded surface vm+1 and R and Φ` are known everywhere on H¯ : ∞ n X ρ (n) ΦN (ρ) = Φ (103) a) Use the knowledge of ΦN on vm+1 to calculate ΦN,ρ. f n! Nf n=0 b) Use (71) at (v , ρ ) to find Φ . Then m n N,v on N¯ . Following [37] we give names to special cases of

ΦN (vm, ρn) ≈ ΦN (vm+1, ρn) − ΦN,v(vm+1, ρn)∆v (99) this free data: ¯ (0) i) out-modes: no flux through H (Φ` = 0), c) Apply (97) to calculate κ at (vm, 0). ¯ (n) non-zero flux through N (ΦN 6= 0 for some n) d) Use (86)-(89) to integrate the values of RN,ρ, κ,ρ and ii) down-modes: non-zero flux through H¯ (Φ(0) 6= 0), Φ`,ρ out along the v = vm characteristic as for the ` ¯ (n) initial data. zero flux through N (ΦN = 0 for all n) This can then be repeated marching all the way along H¯ From the free data we construct the rest of the final as shown in FIG.5. data on H¯ . Equations (93) and (97) give

This is how we would proceed for general cases. How- 2 (0) (0) ever those general studies will be left for a future paper. C = 2Φ` (104) Here instead we will focus on spacetime near a slowly 1 C(1) = κ(0) ≈ . (105) evolving horizon. There, as will be seen in the next sec- 2R(0) tion, ΦN,ρ is negligible and it becomes possible to integrate along surfaces of constant v. Here and in what follows the ≈ indicates that terms of It may not be immediately obvious how this integra- order 3 or higher have been dropped. Further by our tion scheme obeys causality and what restricts it to de- gauge choice termining points inside the domain of dependence. This (0) (1) is briefly discussed in Appendix A. RN = R = −1 (106) 9 and so from (107) Note that the last two terms will include terms of order 2 once the (107) integration is done to calculate R(0). Z v (0) (0) (n) R = Ro + C dv . (107) From (89) we can rewrite Φ` terms with respect to vf (n) ΦN ones: This is an order 2 correction as long as the interval of Φ(1) = 0 (118) integration is small relative to 1/. ` The last piece of final data on H¯ is Φ(0) and comes Φ(0) N Φ(2) ≈ N . (119) from the first order differential equation (95) ` 2 2Ro dΦ(0) Φ(0) Φ(0) The vanishing linear-order term reflects the fact that N + N ≈ ` (108) close to the horizon (where R` = 0) the inward flux de- dv 2Ro Ro couples from the outward (89) and so freely propagates which has the solution into the black hole. Physically this means that (to first Z v order in ρ near the horizon) the horizon flux is approxi- (0) (0) (v −v) v v˜ (0) Φ = Φ e f /2Ro + e− /2Ro e /2Ro Φ d˜v (109) mately equal to the “near-horizon” flux. N Nf ` vf Next, from (88) in which the free data Φ(0) came in as a boundary con- (0) (0) Nf (1) (2) 1 2Φ` ΦN dition. Note that scalar fields that start small on the κ = C ≈ 2 − 2 and (120) R(0) Ro boundaries remain small in the interior, again as long as (0) (0) (1) the integration time is short compared to 1/. We assume 2Φ 2Φ + RoΦ (2) 3 ` N N that this is the case. κ ≈ 2 − 2 . (121) R(0) Ro From the final data, the black hole is close to equilib- 2 rium and the horizon is slowly evolving to order . That Again keep in mind that the R(0) terms will be corrected is, the expansion parameter [17, 35]: to order 2 from (107). Finally these quantities may be substituted into (71) 1 4Φ2 42 C θ2 + G N aN b ≈ ` ∼ . (110) to get differential equations for the Φ(n): 2 (N) ab R2 R2 N (1) (1) (0) (0) Further we already have the first order expansion of C: dΦN ΦN Φ` ΦN + ≈ 2 − 2 (122) dv Ro Ro Ro (0)2 ρ (2) (2) (0) (0) (1) C ≈ 2Φ` + . (111) dΦ 3Φ 2Φ 5Φ 3Φ 2Ro N N ` N N + ≈ 3 − 3 − 2 . (123) dv 2Ro Ro 2Ro Ro That is (to first order) there is a null surface at Like (109) these are easily solved with an integrating fac- (0)2 (1) (2) ρEHC ≈ −4R Φ . (112) tor and respectively have Φ and Φ as boundary con- o ` Nf Nf ditions. This null surface is the event horizon candidate discussed Note the important simplification in this regime that in [34]: if the horizon remains slowly evolving throughout enables these straightforward solutions. The fact that its future evolution and ultimately transitions to isolation R` ∼ ρ has raised the ρ-order of the ΦN,ρ terms. As a then the event horizon candidate is the event horizon. result we can integrate directly across the ρ = constant Moving off the horizon to calculate up to second order surfaces rather than having to pause at each step to first in ρ2, from (86) and (87) we find calculate the ρ-derivative. The Φ(n) are final data for Nf 2 these equations. They can be solved order-by-order and Φ(0) (1) (2) N then substituted back into the other expressions to re- RN = R ≈ − (113) Ro construct the near-horizon spacetime. (0) (0) (1) It is also important that the matter and geometry ΦN ΦN + 2RoΦN (2) equations decompose cleanly in orders of : we can solve RN ≈ − 2 (114) Ro the matter equations at order relative to a fixed back- and so from (81) ground geometry and then use those results to solve for the corrections to the geometry at order 2. (0) R` = 0 (115) 1 R(1) = − (116) 2. Constant inward flux ` 2R(0) (2) 1 We now consider the concrete example of an affinely R` = − . (117) R(0)2 constant flux through H¯ along with an analytic flux 10

¯ v−v through N . Then by Appendix B for V = f/2Ro along with similarly quadratic affine data on N¯ : Φ(0) = Φ(0)eV , (124) ` `f ρ2 (0) (0) v−vf (0) (1) (2) where Φ is the value of Φ at vf and V = while ΦNf = ΦN + ρΦN + ΦN . (133) `f ` 2Ro f f 2 f ΦNf retains its form from (103). We solve the equations for this data up to second order A priori these are uncorrelated but let us restrict the (n) (n) in ρ and . First for ΦN equations we find: initial configuration so that ΦN (vi) = 0. That is, there is no ΦN flux through v = vi. Φ(0) ≈ eV − e−V Φ(0) + e−V Φ(0) (125) N `f Nf Then the process to apply these conditions is, given (0) (0) ¯ 2Φ 2Φ the free final data on H: (1) `f −2V Nf −2V −V ΦN ≈ 1 − e + e − e (126) Ro Ro (n) i) Solve for the ΦN from (108), (122) and (123). + Φ(1) e−2V Nf (n) (n) (0) ii) Solve Φ (vi) = 0 to find the Φ in terms of the Φ N Nf (2) `f V −V −2V −3V a . These are linear equations and so the solution is ΦN ≈ − 2 e + 14e − 48e + 33e (127) n 4Ro straightforward. (0) ΦN + f 7e−V − 24e−2V + 17e−3V (n) 2 iii) Substitute the resulting expressions for Φ into re- 2Ro N sults from the previous sections to find all other 6Φ(1) Nf (2) quantities. + e−3V − e−2V + Φ e−3V (128) Nf Ro and so These calculations are straightforward but quite messy. Here we only present the final results for ΦNf : Φ(0) = eV Φ(0) (129) ` `f (1) 2a (1− e3Vi ) a (1− e4Vi ) (0) 2Vi 1 2 Φ` = 0 (130) Φ ≈(1− e )a0 + + Nf 3 2 Φ(0) Φ(0) (2) `f V −V Nf −V (134) Φ` ≈ 2 e − e + 2 e . (131) 2Ro 2Ro 2a (e2Vi − e3Vi ) a (1 + 8e3Vi − 9e4Vi ) Φ(1) ≈ 0 + 1 (135) Nf The scalar field equations are linear and so it is not sur- Ro 6Ro prising that to this order in each solution can be thought 4Vi 5Vi a2(1 + 5e − 6e ) of as a linear combination of down and out modes. + 5R However for the geometry at order 2, down and out o a (1 + 14e2Vi − 48e3Vi + 33e4Vi ) modes no longer combine in a linear way. These quan- Φ(2) ≈ − 0 (136) (n) Nf 2 4Ro tities can be found simply by substituting the Φ` and (n) (n) (n) 3Vi 4Vi 5Vi (n) (n) a1(1 + 35e − 135e + 99e ) ΦN into the expression for R , RN , R` , C and − (n) 2 κ given in the last section. They are corrected at or- 15Ro 2 4Vi 5Vi 6Vi der by flux terms that are quadratic in combinations a2(1 − 35e + 144e − 110e ) (m) (n) + of Φ and Φ . The terms are somewhat messy and 2 `f Nf 20Ro the details not especially enlightening. Hence we do not write them out explicitly here. where Vi = V (vi). If Vi is sufficiently negative that we can neglect the exponential terms:

¯ ¯ 3. H − N correlations (0) 2a1 a2 Φ ≈ a0 + + (137) Nf 3 2 a a From the preceding sections it is clear that there does Φ(1) ≈ 1 + 2 Nf not need to be any correlation between the scalar field 6Ro 5Ro ¯ ¯ a a a flux crossing H and that crossing N . These fluxes are Φ(2) ≈ − 0 + − 1 + 2 . Nf 2 2 2 actually free data. Any correlations will result from ap- 4Ro 15Ro 20Ro propriate initial configurations of the fields. In this final example we consider a physically interesting case where In either case the flux through H¯ fully determines the flux such a correlation exists. through N¯ . The constraint at vi is sufficient to determine Consider quadratic affine final data (Appendix B) on the Taylor expansion of the flux through N¯ relative to H¯ = {(v, 0) : vi < v < vf }: the expansion of the flux through H¯ . Though we only did

(0) V 2V 3V this to second order in ρ/v we expect the same process Φ` = a0e + a1e + a2e (132) to ﬁx the expansions to arbitrary order. 11

V. DISCUSSION ACKNOWLEDGMENTS

This work was supported by NSERC Grants 2013- In this paper we have begun building a formalism that 261429 and 2018-04873. We are thankful to Jeff Wini- constructs spacetime in the causal past of a horizon H¯ cour for discussions on characteristic evolution during the and an intersecting ingoing null surface N¯ using final 2017 Atlantic General Relativity Workshop and Confer- data on those surfaces. It can be thought of as a special- ence at Memorial University. IB would like to thank Ab- ized characteristic initial value formulation and is par- hay Ashtekar, José-Luis Jaramillo and Badri Krishnan ticularly closely related to that developed in [28]. Our for discussions during the 2018 “Focus Session on Dy- main interest has been to use the formalism to better un- namical Horizons, Binary Coalescences, Simulations and derstand the relationship between horizon dynamics and Waveforms” at Penn State. Bradley Howell pointed out off-horizon fluxes. So far we have restricted our attention a correction to our general integration scheme which is to spherical symmetry and so included matter fields to now incorporated in Sections IV B and IV C. drive the dynamics.

One of the features of characteristic initial value prob- Appendix A: Causal past of H¯ ∪ N¯ lems is that they isolate free data that may be specified on each of the initial surfaces. Hence it is no surprise that In this appendix we consider how the general integra- the corresponding data in our formalism is also free and tion scheme for the scalar field spacetimes of Section IV uncorrelated. We considered two types of data: inward “knows” how to stay within the past domain of depen- flowing null matter and massless scalar fields. dence of H¯ ∪ N¯ . For the inward-flowing null matter, data on the horizon actually determines the entire spacetime running back- wards along the ingoing null geodesics that cross H¯ . Physically this makes sense. This is the only flow of matter and so there is nothing else to contribute to the dynamics.

More interesting are the massless scalar field spacetimes. In that case, matter can flow both inwards and outwards and further inward moving radiation can scat- ter outwards and vice versa. For the weak field near- horizon regime that we studied most closely, the free final data is the scalar field flux through H¯ and N¯ along with the value of R at their intersection. Hence, as noted, these fluxes are uncorrelated. However we also considered the case where there was no initial flux of scalar FIG. 6. Causality restrictions on ∆v: the CFL condition re- field travelling “up” the horizon. In this case the coeffi- stricts the choice of ∆v to ensure that attempted numerical cients of the Taylor expansion of the inward flux on H¯ evolutions respect causality. In this figure the ρ and v coordi- fully determined those on N¯ (though in a fairly compli- nates are drawn to be perpendicular to clarify the connection with the usual advection equation: to compare to other di- cated way). This constraint is physically reasonable: one agrams rotate about 45◦ clockwise and skew so coordinate would expect the dominant matter fields close to a black curves are no longer perpendicular. The dashed lines are null hole horizon to be infalling as opposed to travelling (al- and have slope C in this coordinate system. If data at points most) parallel to the horizon. It is hard to imagine a A, B and C are used to determine ΦN,ρ then the size of the mechanism for generating strong parallel fluxes. discrete v-evolution is limited to lie inside the null line from point C. The largest ∆v allowed by the restriction evolves to While we have so far worked in spherical symmetry D. the current work still suggests ways to think about the horizon-I + correlation problem for general spacetimes. For a dynamic non-spherical vacuum spacetime, gravita- First, it is clear how the process develops spacetime tional wave fluxes will be the analogue of the scalar field up to the bottom left-hand null boundary (v = vi) of fluxes of this paper and almost certainly they will also the past domain of dependence. The bottom right-hand be free data. Then any correlations will necessarily re- boundary is a little more complicated but follows from sult from special initial configurations. However as in our the advection form of the ΦN,v equation (74). Details will example these may not need to be very exotic. It may depend on the exact numerical scheme but the general be sufficient to eliminate strong outward-travelling near picture is as follows. horizon fluxes. In future works we will examine these Assume that we have discretized the problem so that more general cases in detail. we are working at points (vj, ρk). Then in using (74) to 12 move from a surface vi to vi−1, the Courant-Friedrichs- Appendix B: Affine derivatives and final data Lewy (CFL) condition (common to many hyperbolic equations) tells us that the maximum allowed ∆v is The off-horizon ρ-coordinate in our coordinate system is affine while v is not. However, as seen in the main text, ∆ρ when considering the final data on H¯ it is more natural ∆v < , (A1) C to work relative to an affine parameter. This is somewhat complicated because Φ` and ΦN are respectively linearly dependent on ` and N and the scaling of those vectors is where ∆ρ is the coordinate separation of the points that also tied to coordinates via (1), (2) and (6). In this ap- we are using to calculate the right-hand side of (74). pendix we will discuss the affine parameterization of the Then, as shown in FIG. 6, the discretization progres- horizon and the associated affine derivatives for various sively loses points of the bottom right of the diagram: quantities. they are outside of the domain of dependence of the indi- Restricting our attention to an isolated horizon H¯ with vidual points being used to determine them. For example κ = 1 , consider a reparameterization if we are using a centred derivative so that 2Ro v˜ =v ˜(v) . (B1) Φ (v , ρ ) − Φ (v , ρ ) Φ ≈ N j k+1 N j k−1 (A2) Then N,ρ 2∆ρ ∂ d˜v ∂ = (B2) ∂v dv ∂v˜ then we need adjacent points as shown in FIG. 6. and so ` = eV `˜ and N = e−V N˜ (B3) d˜v where we have defined V so that eV = . Hence dv dV κ˜ = −N˜ `ã∇ `˜b = e−V κ − (B4) b a dv

and so for an aﬃne parameterization (κ = ∂vV ): v − v eV = exp f (B5) 2Ro

for some vf and

V v˜ − v˜o = 2Roe (B6)

for somev õ. The vf freedom corresponds to the freedom to rescale an affine parameterization by a constant multiple while thev õ is the freedom to set the zero ofv ˜ wherever you like. FIG. 7. A cartoon showing the CFL-limited past domain of Now consider derivatives with respect to this affine pa- dependence of H¯ ∪ N¯ . Null lines are now drawn at 45◦ so the rameter. For a regular scalar field analytic past domain of dependence is bound by the heavy df df dashed null lines running back from the ends of H¯ and N¯ .A = e−V . (B7) (very coarse) discretization is depicted by the gray lines and d˜v dv the region that cannot be determined with dashed lines. The However in this paper we are often interested in scalar boundary points of that region are heavy dots. quantities that are defined with respect to the null vectors:

(0) V (0) (0) −V (0) Φ` = e Φ˜ and ΦN = e Φ ˜ . (B8) The lower-right causal boundary of FIG. 1 is then en- ` N forced by a combination of the endpoints of N¯ and the Then CFL condition as shown in FiG. 7. Points are progres- (0) ! dΦ d dΦ(0) sively lost as they require greater than the maximum `˜ =e−V e−V Φ(0) =e−2V ` − κΦ(0) (B9) allowed ∆v. The numerical past-domain of dependence d˜v dv ` dv ` necessarily lies inside the analytic domain. The coarse- (0) dΦ d dΦ(0) ness of the discretization in the ﬁgure dramatizes the ef- N˜ = e−V eV Φ(0) = N + κΦ(0) . (B10) fect: a ﬁner discretization would keep the domains closer. d˜v dv N dv N 13

That is these quantities are aﬃnely constant if quadratic”. By this we mean that:

V (0) −V (0) Φ` = e Φ` and ΦN = e ΦN (B11) 2 f f Φ`˜ = Ao + A1v˜ + A2v˜ m for some constants Φ(0) and Φ(0) . `f Nf Φ = a eV + a e2V + a e3V , (B12) In the main text we write this affine derivative on H¯ ` o 1 2 as Dv with its exact form depending on the ` or N dependence of the quantity being differentiated. where for simplicity we have setv õ to zero (so that v = 0 Finally at (132) we consider a Φ` that is “affinely is V˜ = 2Ro) and absorbed the extra 2Ros into the an.

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