Charged Black Holes in GR and Beyond

LIGO-G1801141-v1

Charged black holes in GR and beyond

N. K. Johnson-McDaniel Benasque meeting: NR beyond GR 04.06.2018 Here is the title

Nathan K. Johnson-McDaniel,1 Author name,2 and Other author name3 1Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK 2Affiliation 3Other affiliation (Dated: June 1, 2018) Abstract

I. INTRODUCTION

First section of text.

II. HERE WE DO SOMETHING

Next section of text.

A. Further details

A nice spin-weighted spherical harmonic expression:

h`m(t):= 2Y¯`m(✓, ')h(t, ✓, ')d⌦. (1) 2 ZS

µ (j):= S⇤ (M!j; ✓, ) Y (✓, )d⌦ m``0n0 s `0mn0 s `m (2) Z

q4⇤ + q⇤ T =3 A B (3) 2 (1 + q)5

XA = MA/M (4) Overview of charged BH solutions: Standard electrically chargedq = MA/MB solutions (5)

T T 2 • a 1+n1 + n2  Reissner-Nordström: MNonspinning,!mrg = 0 discovered2 2in 1916, the same year as the (6) 22 pq T T 2 Schwarzschild solution. Simple generalization1+d12 + d2 of 2Schwarzschild:

Electric potential 1 2M Q2 2M Q2 Q ds2 = 1 + dt2 + 1 + dr2 + r2(d✓2 +sin2 ✓d2), = (7) r r2 r r2 r ✓ ◆ ✓ ◆ However, introducing charge gives some radically different properties, similar to the introduction of angular momentum,1. Explanation e.g., a Cauchy horizon and the possibility of extremal black holes with zero temperature. Note that Eq. (1) and the results from [1]...

• Kerr-Newman: Spinning, obtained by an application of the Newman-Janis procedure to Reissner-Nordström by Newman et al. in 1965, two years after the Kerr solution was discovered. By the uniqueness theorems, the most general black hole in Einstein-Maxwell theory.

• [Note that one can obtain magnetically charged versions of these solutions with a duality transformation.] Overview of charged BH solutions: Beyond Einstein-Maxwell

• String theory-inspired charged black hole with rotation (and both electric charge and magnetic dipole) constructed by Sen, PRL, 1992. Satisfies most general low-energy string theory action with at most two derivatives, i.e., Einstein-Maxwell + dilaton and (vector) Chern-Simons terms.

• No stationary Proca black holes with stationary Proca field, due to theorem by Bekenstein, PRD 1972, though there are stationary Proca black hole solutions with a Proca field with a harmonic time dependence, e.g., Herdeiro, Radu and Rúnarsson, CQG 2016.

• Yang-Mills charged black holes: Can be static and not spherically symmetric (e.g., Kleihaus and Kunz, PRL 1997), generally have to be constructed numerically, and are also generally unstable. There are also results in the rotating case, e.g., Kleihaus and Kunz, PRL 2001.

• Higher-dimensional charged black holes (and rings, Saturns): Can be constructed analytically, even with rotation, as in Caldarelli, Emparan, and Van Pol, JHEP 2011; for rings and Saturns, see Grunau, PRD 2014 for solutions in Einstein-Maxwell-dilaton theory. 3 1 6 Black Holes

As shown by Boyer and Lindquist (1967) and Carter (1968a), the s ingularities in the metric components at r = r + and (for a # 0 or e * 0) at r = r- are coordinate singularities of the same nature as the singularity at r = 2M in the Schwarzschild spacetime. Thus, we may extend the spacetime through these coordinate singularities much as in the Schwarzschild case . When these extensions are patched together, a remarkable global structure of the extended charged Kerr spacetimes is obtained . A conformal diagram of the extended Schwarzschild spacetime is shown in Figure 12 .3, and a conformal- diagram: of the extended charged Kerr spacetime with a . U is shown in Figure 12.4 for the "non-extreme" ease a ' + e 2 < M 2 . Region I of Figure 12.4 is the asymptotically flat region covered in a nonsingular fashion by the original coordinates (12.3 .1) with r > r + . By extending through the coordinate singularity at r = r + , we obtain region II - representing a black hole, region Iii representing a white hole, and region IV representing another asymptotically flat region, just as in the Schwarzschild case, Figures 6.9 and 12.3 . However, unl i ke the Schwarzschild case, instead of encountering a true singularity at the "top boundary" of region H and "bottom boundary>, .of region III, we encounter merely another coordinate singularity at _r , = r- . Thus, we can extend region II through r = r- to obtain regions V and VI. These regions contain the ring s clarity at I = 0 and, as described above, one can pass through the, ring singularity to obtain another asymptotically flat region with r --> -flo. (In this asymptotically flat reg ion the ring singularity is a : naked singularity of negative mass ,:.. With respect to the. original asymptotically flat region 1, the ring singularity,, of course, lies-within a black hole . ) One may then continue. toComparisonextend the cha,rged Kerr spaced of "causalupward" ad in finistructuretum (eternal BHs) to obtain a region VII, identical in structure to region Ili, and obtain regions VIII and IX, identical in , structure to regions IV and, 1, stc., ; Similarly, .: one may extend the charged Kerr solutions "downward" ad infinitum. The structureReissner-Nordströmof the extended Reissner-Nordstrom apaeetime {a = 0, e * Q) : is very similar excegt ; that the true Kerr(-Newman) singularity at I =Ono longert a ring structure, and one cannot extend to-negative 158 (nonzeroEXAOT SOLUTIONS q) [5.5 12 .3 The Charged Kerr Black Holes 31 7 values of r.1'he global structure of the "extreme" charged Kerr- ease e z + as =M 2 nonzero a

/ (where r + = r_ =.M) differs from Figure 12 . 4 but has a similar structure consisting/ of "blocks" with r > M and, r . < . M patched together in an infinite chain . II Thus, a n- observer starting in region I of the extended charged Kerr spacetime of Figs . 12.4 may cross the event horizon at r T r+ .and enter the black hole region U . However, instead of inevitablySchwarzschild falling into a singularity within a finite proper time as occurs in the SchwarzschiId spacetime, the observer may pass through the "inner II

/ " ,r - r_ / r-O / ,'=0 (.IIt!!lIll1tll.y) ,, / ,.=,.- (.Iltl!ul,u·lty) , / i o III ). III Cauchy hor-izon ~ ~ = 0 ion I~ I / '. for 9' (Ring Singu larity )

M i+

Fig . 12 . 3. A conformal diagram of the extended Schwarzschild spacetime (see Fig . S 6.9), represented in the sameFiguremanner asfromused iWald:n Figure 12.2 . Note that since the extended Schwarzschi ld space has two distinct asymptotically flat regions, two distinct confom at boundariesGeneralare shown . Relativity

r= 0 Orthogonal surfaces Fig . 12 . 4. A conformal diagram of the extended charged Kerr spacetime in the {t = constant} case n 0 0, a' + e2 < 1FigureKZ . from Wald: FIGURE 26. Penrose diagram for the maximally extended Reissner-Nordstri:im solutionFigure(el < ml). fromAn infinite Hawkingchain of asymptotically and Ellis:flat regions I General Relativity (00 > r > r+) are connected by regions IIhorizon"(r+ > r > r_) andr III= (r_r-> r(>which0); is a Cauchy horizon for the hypersur# "ace S shown in dig . each region III is bounded by a timelike singularity at r = O. Large-scale structure12 .4),of spacetimethereby entering region V or VI. From , there, he may end his existence in the but unlike in the SchwarzBchild solution,ring issingulatimelike andrityso butcan behe also may pass through the ring singularity to a new asymp- avoided by a future-directed timeliketocurveticallyfrom flata regionregion,I which or he may enter the "white hole region" VII and from there enter crosses r = r+. Such a curve can pass through regions II, III and II and re-emerge into another asymptoticallythe newflat regionasymptoticallyI. This raises flat region VIII or IX. From there he may enter the new black the intriguing possibility that one mightholebe assocable toitravelatedto withother theme : asymptotic regions, and continue his journey . How much of this . exte Wed charged Kerr ,spacetime should be taken seriously? What portion of this spacetime would be produced by a phys ically realistic grav- itatiunal collapse, starting from "won-ero tic" initial data, i .e . , from an asymp- totically flat initial data surface S with topology R3? Unlike the Schwarzsetrild case, we have no reason to believe that the exterior gravitational field of:any physically re able"cv s ng body will be described by the Kerr metric, since, as mentioned above, in the nonsghericalcase :we would, in general, expect a complicated dynam- ical evolution which only "setdes down" to a stationary geometry at late times in J-(,O+). Thus, we are not in a position to follow the dynamical evolution of the gravitational collapse of a body which forms a Kerr black hole and thereby determine the detailed spacetime geometry inside the black hole . However, in the case of spherical collapse of a charged body (e * 0), the spacetime geometry exterior to the matter is described by the Reissner-Nordstrom solution since Birkhoff's theorem can be generalized to show that the Reissner-Nordstrom spacetime is the _unique spherical electrovac solution . The dynamical evolution of a simple system like a charged, spherical shell of dust can b e obtained explicitly . In this case, spacetime is flat inside the shell, and the fiat interior of the shell entirely "covers up„ regions II .i and IV of Figure 12.4. Part or all of regions H and V (including, in all cases, the singularity at r = 0 in region V) also are covered up. The behavior of the shell for the various choices of total mass M, total charge e, and total rest mass .M , are chronicled in detail Comparison of ISCO velocities

• While charge mimics some of the effects of spin, in at least one sense the effects of spin are more extreme.

Considering the PN velocity parameter (from the angular velocity at infinity), v = (Mω)1/3 for a neutral particle at the ISCO, we have:

• Schwarzschild: ~0.41

• extreme Reissner-Nordström: ~0.48

• extreme Kerr (prograde orbit): ~0.79 Applications: Astrophysics?

• For standard electric charge, likely no astrophysical applications: There is enough plasma in the universe that even if one manages to form a charged black hole, it will discharge to a dimensionless charge of ~10-21 or ~10-18 (the inverse of the electron or proton’s charge-to-mass ratio) on a timescale of ~1 μs or ~1 ms. [See, e.g., Cardoso et al. JCAP 2016 for a nice review.]

• People have tried various ways of circumventing these results over the years, to allow for the black holes to retain significant charge for a nonnegligible amount of time, but none of them are particularly compelling.

• If one wants to consider magnetic monopoles [as in Liebling and Palenzuela, PRD 2016], then magnetically charged black holes could be a possibility. Applications: Astrophysics?

• If black holes are charged under something other Figure from than standard electromagnetism (e.g., a “dark Cardoso et al. (JCAP 2016) electromagnetic field”), then there is still the e possibility that this could be astrophysically 0 relevant. =0

plasma absorption -5 excluded by galactic dynamics • One still has to worry that there would be ] h associated “dark electrons” that would neutralize [ -10 the black holes similarly quickly as in the standard Log discharge limit -3 10 M> electromagnetic field case. Q/ -15 1 M= Q/ JCAP05(2016)054 • However, as pointed out by Cardoso et al. (JCAP -20 2016), if the particles that are charged under the -6 -4 -2 0 2 4 6 Log[m/GeV] dark EM field have much smaller charge-to-mass Log charge in units of Figure 1. The parameter space of a minicharged DM fermion with mass m and charge q = e ✏2 + ✏2 ratios than the electron, as in the millicharged dark Log mass in GeVh (see main text for details). The left and right panels respectively show the planes ✏ = 0 and h p matter scenario, then✏ =one 0 of can the three avoid dimensional the standard parameter space (m, ✏h, ✏). As a reference, an electron-like particle arguments about discharging(m 0.5 MeV ,black✏ = 1) is holes denoted quickly. by a black marker. In each panel, the red and blue areas below the ⇠ 3 two threshold lines denote the regions where charged BHs with a charge-to-mass ratio Q/M > 10 and Q/M = 1 can exist [cf. eq. (2.23)]. The region above the black dashed line is excluded because in this region extremal BHs would discharge by plasma accretion within less than the Hubble time [cf. eq. (2.20)]. Left panel: the hatched region is excluded by the e↵ects of the magnetic fields of galaxy clusters [21] and it is the most stringent observational constraint on the model (we also show the region excluded by the direct-detection experiment LUX [24], cf. ref. [21] for details and other constraints). Right panel: when ✏ = 0 our model reduces to that of DM with dark radiation [23] and the region above the solid black line is excluded by soft-scattering e↵ects on the galaxy dynamics [23]. In the region above the dark red dot-dashed line hidden photons emitted during the ringdown of a 3 M 60M would be absorbed by hidden plasma of density ⇢DM 0.4 GeV/cm [cf. eq. (3.42)]. ⇠ ⇠

dark fermions do not possess (fractional) electric charge but interact among each other only through the exchange of dark photons, the latter being the mediators of a long-range gauge interaction with no coupling to Standard-Model particles [23]. Although rather di↵erent, both these DM models introduce fermions with a small (elec- tric and/or dark) charge. It is therefore natural to expect that minicharged DM can circum- vent the stringent constraints that are in place for charged BHs in Einstein-Maxwell theory. In this work we will show that this is indeed the case and that even extremal Kerr-Newman BHs are astrophysically allowed in the presence of minicharged DM. Charged BHs are remarkably sensitive to the presence of even tiny hidden charges but are otherwise insensitive to the details of their interaction. This is a consequence of the equivalence principle of general relativity. We shall take advantage of this universality and discuss BHs charged under a fractional electric charge or under a hidden dark interaction on the same footing. Figure 1 summarizes the main results of section 2, showing the parameter 2 2 space of a minicharged fermion with mass m and charge q = e ✏h + ✏ in which astrophysical charged BHs can exist (here and in the following ✏h and ✏ are theq fractional hidden and electric charges of the dark fermion, respectively). Interestingly, such region does not overlap with the region excluded by direct-detection experiments and by cosmological observations. Having established that charged astrophysical BHs can exist in theories of minicharged

–2– Applications: Mathematical/theoretical physics

• Another way of approaching extremality, and thus to test cosmic censorship [e.g., the study of self-force effects on the original overcharging scenario from Hubeny PRD 1999 by Zimmerman et al. PRD 2013].

• Lack of separability of perturbations for Kerr-Newman black holes.

• Existence of hair from non-EM charges.

• Additional tests of isolated and dynamical horizon frameworks, and calculations of black hole entropy. Applications: Toy models

• Charged black holes are excellent toy models for certain effects that it is difficult to calculate otherwise.

• In particular, since Reissner-Nordström BHs exhibit several of the effects of rotation (e.g., extremal black holes and a Cauchy horizon), while retaining the technical simplicity of spherical symmetry, they have been used as a toy model for Kerr black holes

• My primary interest in charged black holes is in using the charge as a proxy for possible beyond-GR effects in binary black holes.

• In particular, generic charged black hole binaries will emit dipole radiation. And even in the case of equal charge-to-mass ratio, where there is no dipole radiation, the binary will have PN phase and amplitude coefficients that differ from those of vacuum BBH systems, in addition to changes to the quasinormal mode spectrum of the final black hole, except in the special case where they form an uncharged final black hole.

• Thus, one can use charged BBH waveforms to check how sensitive current LIGO tests of GR are to completely consistent parameterized deviations from GR (and whether the parameterized test can recover the correct changes to the post-Newtonian coefficients). Studies of charged black holes: Perturbation theory

• Post-Newtonian (actually charged point particles, but useful for studying binaries):

-2 • Conservative dynamics: 1PN [O(c )] N-body Lagrangian and Hamiltonian calculated [Bażański Acta Phys. Pol. 1956 and 1957 and Barker and O’Connell, JMP 1977]. [2PN 2-body Lagrangian EM-gravity terms have been calculated in Gaida and Tretyak, Sov. Phys. J., 1991, but using a method that seems a bit suspect.]

• Dissipative dynamics: Fluxes have just been computed to Newtonian order, as far as I can tell.

Newtonian-order gravitational waveforms are calculated in Cardoso et al. JCAP 2016.

• Quasinormal modes: Difficult to compute for Kerr-Newman, as the equations are not separable, but finally computed in Dias, Godazgar, and Santos, PRL 2015 (plus calculations in the slow-rotation [Pani, Berti, and Gualtieri, PRL and PRD 2013] or weakly charged [Mark et al., PRD 2015] limits).

• Self-force: The electromagnetic self-force is a standard toy model, and there are now self-force calculations for Reissner-Nordström black holes, both of the scalar self-force for circular orbits [Castillo, Vega, and Wardell, arXiv 2018] and the electromagnetic self-force for radial trajectories [Zimmerman et al., PRD 2013]. There are also calculations of inspirals of uncharged point particles around a RN BH in the adiabatic approximation [Zhu and Osburn, PRD 2018]. BLACK HOLE DYNAMICS IN EINSTEIN-MAXWELL- … PHYS. REV. D 97, 064032 (2018)

−3 orders of magnitude smaller, only ϕ1 ≈ 10 , and 2 one expects dynamical differences to scale with ϕ1. And so perhaps it is natural that the differences we 8 see for these parameter choices are as small as we report. Scalarization levels comparable to those neutron star mergers would require BH charges 2 −1 −1 α0q M ≈ 10 (qM ≃ 10 ) for small (very large) values of α0. Because EMD allows for scalar radiation, we can gain additional understanding by extracting it in addition to the Studies of charged black holes: gravitational wave signal. As discussed in Appendix B, the computation of the Newman-Penrose scalar Φ22 indicates that the scalar radiation is expected to scale as Φ22 ≈ α0ϕ;tt Numerical relativity (evaluated asymptotically). One can thus estimate that FIG. 2. The electricthis (left panel) radiation and Hamiltonian in the (right early panel) constraints inspiral along phasethe collision scales axis at time ast =384M for model q+-090_d32. The solid (black) curves display the result obtained for lower resolution hf = M/160 and the dashed (red) curves 2 4 that obtained forΦ higher22 ≈ resolutionα0ϕ1Ωhf =.M/ This196 and scaling amplified isby 1 assumed.2 for the expected in Fig. fourth-order7, which convergence. FIG. 8. The (real part of the) l m 1 and l m 2 modes ¼ ¼ ¼ ¼ EMshows waveformΦ22 as a function of time forGW both waveform the equal and of the scalar gravitational radiation Φ22 of a binary black hole unequal mass cases. In particular, because the orbital with an electric charge qe 0.001 for different values of α0 • Zilhão et al., PRD 2012 and 2014: Evolutions of head-on corresponding to the unequal¼ mass binary case. Here, we have frequency differs only slightly with changes to α0, the scaled up the case α 1000 by a factor of 3 in accordance collisions of nonspinning charged black holes from rest, rescaling depends only on the coupling and scalar charge. 0 ¼ The coupling value is straightforward, but the black hole with expected scaling if the dilaton charge is ignored. Top: The l m 1 mode. Bottom: The l m 2 mode. using either analytic equal charge-to-mass ratio initial data, scalar charge is chosen as the scalar charge of individual ¼ ¼ ¼ ¼ black holes in isolation. Thus, the scalar charges for equal or a simple numerical initial data construction to obtain mass binaries are chosen as ϕ −4.8 × 10−7; −4 × than otherwise expected from simple superposition argu- 1 ¼ f opposite charge-to-mass ratios. 10−4; −6.9 × 10−4 while for unequal mass binaries, we ments and is consistent with the observations made in the choose masses andg scalar charges to be (m , m ): ϕ isolated black hole cases where the scalar charge shows a 1 2 1 ¼ −3;−2 ×10−7; −2.4;−1.6 ×10−4; −4.2;−2.7 ×10−4 trend toward saturation at high coupling values. fð Þ 3 ð 3 Þ ð Þ g This saturation is evident in Fig. 8, which shows the • for α0 1; 10 ; 3 × 10 respectively [which are well Zilhão et al., PRD 2014: Evolutions of perturbed Kerr- FIG. 3. Real partapproximated of the electromagnetic¼ f by (l =1, them analytical= 0 mode)g and expression gravitational (l =2, Eq.m =(38) 0 mode)]. waveforms.l Thesem have 1 and l m 2 modes for the scalar radiation been convenientlyOppositely rescaled and shifted in timecharged so that their peaks head-on coincide. collision ¼ ¼ ¼ ¼ As shown in Fig. 7, reasonably good agreement with the corresponding to the unequal mass binary for α0 1000 Newman black holes to study stability. and 3000. In contrast with Fig. 7, however, both cases¼ here radiated energiesexpected thuswaveforms obtained scaling are plotted is obtainedin Fig. from 4 as duringjust Zilhão presented, the inspiral for aet range phase, al. of cut-o ↵ buts for the former. For functions of thethe charge-to-mass scaling ratio overestimates and quantitatively the magnitudeinstance, setting ofzc =1 the.5M radiation, we make the followingare obser-scaled linearly by their respective value of α0, ignoring illustrate the scaling discussed in the previous paragraph. vations: i) For the configuration Q /M =0.99, the ratio | | the dilaton charge. Focusing only on the merger, this simple These resultsduring contrast with the the merger. corresponding The equal failure! of of energy the in scaling electromagnetic during to gravitational the radiation • charge collisions of Paper I, where the emitted gravita- obtained in our numerical simulations is 5.8scaling (cf. Table in α works quite well, supporting our assertion that Liebling and Palenzuela, PRD 2016: Evolutions of merger indicates that the nonlinear behavior is less radiative ⇠ 0 tional radiation decreases with increasing charge because I), whereas that obtained from our simple analytic ap- of its deceleratingScalar e↵ect and radation the correspondingly lowfrom col- proximationHirschmann is 5.2; cf. Eq. (3.15). et ii) Equalal. amountsthe scalar of charges saturate at large coupling. (weakly) electric and magnetically charged orbiting black lision velocities, and the emitted electromagnetic radia- electromagnetic and gravitational radiated energiesAn are interesting aspect of gravitational radiation in EMD tion peaks at around Q/M =0.6. In the case of oppo- obtained for Q /M 0.37 in the numerical simulations | | ⇠ site charges, in contrast, both gravitational and electro- and Q /M =0.31 in the analytic model. Theis analyti-that it could contain a dipolar component in contrast to holes, including force-free plasma, using metric initial data | | magnetic radiation increase with Q/M, and the electro- cal results thus reproduce the numerical valuesGR, with which an disallows dipolar radiation. Although one magnetic radiation becomes the dominating channel for error between 10% and 20%. A comparison of the ener- Q /M 0.37. gies emitted in gravitational and electromagneticgenerally radia- expects the dipolar component, when allowed for uncharged black holes + boosted charges for the EM | | & As already mentioned at the end of the last section, we tion for the entire range Q/M is shown in Fig.by 4. the Even theory, to dominate over higher multipoles, here the observe a good agreement between our simple analytic though a discrepancy at a level of about 10 % is visible, initial data. model of Sec. III and the numerical simulations we have the analytic prediction captures the main featuresstrength of the of the dipolar component depends on the differ- ence in the scalar charges of the black holes. As a result, the equal mass case produces no dipolar radiation. For the • Hirschmann et al., PRD 2018: Evolutions of orbiting unequal mass binary with m1=m2 3=2 considered here, the scalar charges are different but¼ nevertheless are suffi- binaries of weakly charged black holes in Einstein-Maxwell- ciently close to each other that the resulting l m 1 mode is weaker than the l m 2 mode. dilaton theory, with the same initial data construction as ¼We¼ comment on two further conclusions¼ ¼ that can be drawn from our studies as well as leave open a question above (and a constant dilaton). deserving of investigation. First, we find the ringdown of the merger remnant appears largely insensitive to the value FIG. 7. The (real part of the) l m 2 mode of the scalar of the coupling. As mentioned in the previous section ¼ ¼ concerning the ringdown of the head-on remnant, small gravitational radiation Φ22 of a binary black hole with an electric charge qe 0.001 for different values of α0. Top: The equal mass values of the electric charge produce correspondingly small case. Bottom:¼ The unequal mass case. differences in ringdown versus GR.

064032-9 8

(4–6) and Eq. (21 –22) in to express x and ⇢ in terms of C , and therefore in terms of (B,) and B: ± 8 b sin [(B,)] x = , (62) cos [B⇡] cosh [(B,)] (4–6) and Eq. (21 –22) in to express xand ⇢ in terms of C , and therefore in terms of (B,) and B: b sin [(B,)] ± ⇢ = . b sin [(B,)] (63) cos [B⇡] cosh [(B,x =)] , (62) cos [B⇡] cosh [(B,)] (64) 8 b sin [(B,)] We plug these expressions into the expression for the horizon surface, written⇢ = in coordinate-free form: . (63) cos(4–6) [B⇡ and] Eq.cosh (21 [ –22)(B, in to)] express x and ⇢ in terms of C , and therefore in terms of (B,) and B: ± ( x aˆ x )2 (aˆ x)2 b sin [(B,)] (64) x = , (62) |2 2· | 2 + · 2 =1, (65)cos [B⇡] cosh [(B,)] r + a /M r We plug these expressions into the expression for the horizon surface, written in coordinate-free form:b sin [(B,)] ⇢ = . (63) where x =(x b, ⇢ cos , ⇢ sin ), depending on the location of the black hole, and then use a root-finder to numericallycos solve [B⇡] cosh [(B,)] ( x aˆ x )2 (aˆ x)2 ± 2 2 (64) for (B,) at each (B,). Note that = const for Schwarzschild for| not2 x =2· x| 2 b+, but ·x2= x=1b, 1+R /b . This is (65) probably± not the most elegant way of doinb± this, but we haven’t found a fullyr + analyticala /M± solutionr yet. ± We plug these expressions into thep expression for the horizon surface, written in coordinate-free form: where x =(x b, ⇢ cos , ⇢ sin ), depending on the location of the black hole, and then use( x a root-finderaˆ x )2 (aˆ tox numerically)2 solve ± | · | + · =1, (65) for (B,) at each (B,). Note that = const for Schwarzschild for not x = x b, butr2 +x a=2/Mx 2 b r1+2 R2/b2. This is ± B. Linearized XCTS± equations ± ± probably not the most elegant way of doinb this, but wewhere haven’tx =(x foundb, ⇢ acos fully, ⇢ sin analytical), depending solution on the yet. location of the black hole, and then use a root-finder to numerically solve ± p for (B,) at each (B,). Note that = const for Schwarzschild for not x = x b, but x = x b 1+R2/b2. This is The standard XTCS equations: ± ± probably not the most elegant way of doinb this, but we haven’t found a fully analytical± solution yet. ± p 2 1 1 2 5 B.1 Linearized7 ij XCTS equations5 ˜ R˜ K + A˜ A˜ij = 2⇡ ⇢, (66) r 8 12 8 B. Linearized XCTS equations 7 ij 7 The standard XTCS equations:2 6 i ij 10 i ˜ j L˜ ˜ K ˜ j u˜ =8⇡ J , (67) r 2(↵ ) 3 r r 2(↵1 )The standard1 XTCS equations:1  ⇣ ⌘ ˜ 2✓ ˜ ◆ 2 5 7 ˜ij ˜ 5 R K + A Aij =˜ 2 2⇡ 1 ˜⇢, 1 2 5 1 7 ˜ij ˜ (66) 5 R˜ 5 7 R K + A Aij = 2⇡ ⇢, (66) ˜ 2 2 4 8 ˜ij ˜ 5 r 8k 12 84 r 8 12 8 (↵ ) (↵ ) + K + A Aij +7 (@tK @kK)=(↵ ) 2⇡ (⇢ 7+2S) . (68) 8 12 8 ij 2 7 ij 7 r " ˜ # ˜ 6 ˜ i ˜ ˜ ij ˜ 10 i2 6 ˜ i ˜ ij 10 i j L K j j u˜ =8L⇡ J , K j u˜ (67)=8⇡ J , (67) r 2(↵ ) 3 r r⇥ 2(↵r ) 2(↵⇤ ) 3 r r 2(↵ )  ⇣ ⌘ ✓ ◆ After assuming u˜ij = @tK =0, and including an additional equation⇣ to solve⌘ for , we have (note✓ that for˜ metric◆ quantities in ˜ ˜ 2 R 5 2 4 7 8 ˜ij ˜ 5 k 4 these XCTS equations, we use the conformal2 metric to raiseR and5 lower2 4 indices)7 8 ij (↵ 5) (↵ ) k+ K + A A4ij + (@tK @kK)=(↵ ) 2⇡ (⇢ +2S) . (68) ˜ (↵ ) (↵ )Initial+ dataK + for orbitingA˜ A˜ij r+ charged(@tK "8 @ k12blackK)=(↵ 8 holes) 2⇡ ( #⇢ +2S) . (68) r 8 12 8 ⇥ ⇤ ˜ 2 1 ˜ " 1 2 5 1 7 ˜ij ˜ # 5 R K + A AfterAij = assuming2⇡ u˜ij⇢,= @tK =0, and including an additional(69)⇥ equation to solve⇤ for , we have (note that for metric quantities in r 8 including 12 8 spin theseand XCTS eccentricity equations, we use the conformal reduction metric to raise and lower indices) After assuming u˜ij = @tK =0, and including an additional equation to solve for , we have (note that for metric quantities in 7 ij 2 these XCTS equations, we˜ use the conformal˜ metric6 to˜ i raise and lower10 i indices) ˜ 2 1 ˜ 1 2 5 1 7 ˜ij ˜ 5 j WorkL with SohamK =8 Mukherjee⇡ J , and Wolfgang R (70)TichyK + A Aij = 2⇡ ⇢, (69) r 2(↵ ) 3 r r 8 12 8  • Generalize⇣ ⌘ 2the superposed1 1 Kerr-Schild2 5 1 construction7 ij from Lovelace5 7 et al.ij PRD2 2009 ˜ ˜ R˜ K + A˜ A˜ij = 2⇡ ˜ ⇢, L˜ 6 ˜ iK =8(69)⇡ 10J i, (70) ˜ 2 R 5 2 4 7 to8 include˜ij ˜ r charge,5 8k by superposing 12 Kerr-Newman84 black holesrj (weighted2(↵ ) by 3 r (↵ ) (↵ ) + K + A Aij ( @kK)=(↵ ) 2⇡ (⇢ +2S) , (71) ⇣ ⌘ r " 8 12 8 attenuating# functions) and 7 also solvingij for2 a correction˜ to the superposed electric ˜ 2 6 i R 5 2104 i 7 8 ˜ij ˜ 5 k 4 ˜ L˜ (↵ ) (↵ ˜) K +=8⇡K J+ , A Aij ( @kK)=((70)↵ ) 2⇡ (⇢ +2S) , (71) field to satisfy2 the divergencej j constraint;⇥r the magnetic⇤8 12 divergence8 constraint is = rj(Esp2() ↵, ) 3 r" (72) # satisfiedr automaticallyr  due to⇣ superposing⌘ the vector potentials. 2 j ⇥ ⇤ ˜ = j(Esp) , (72) where ˜ 2 R 5 2 4 7 8 ˜ij ˜ 5 k 4 r r (↵ ) (↵ ) + K + where A Aij ( @kK)=(↵ ) 2⇡ (⇢ +2S) , (71) r • Specifically,8 12 solve the8 XCTS7 equations (with EM sources) for the geometry + 2" k ij #ij 7 (L˜)ij = ˜ ij + ˜ ji g˜ij ˜ k , A˜ = (L˜) , (73)2 k ij ij physical metric 2 (L˜)jij = ˜ ij +⇥˜ ji g˜ij ˜ k ,⇤ A˜ = (L˜) , (73) r r 3 r 2(↵ ) = j(Esp) , r superposedr 3 electricr field 2(↵ )(72) Laplacian r r 1 j j pg F k l 1 j j pg F k l where ⇢ = S = (B B +andE computeE ),J electric= and✏ EmagneticB , fields using: ⇢ = S = (BjB(74)+ EjE ),Ji = ✏iklE B , (74) 8⇡ j j i 4⇡ ikl 8⇡ 4⇡ 7 superposed 1 2 determinant of i 1 ikl i ikl ˜ ˜ ˜ ˜ k ˜ij B = ✏˜F @kij(Asp)l,Evectorj =(E potentialsp)j + j, (75) B = ✏F @k(Asp()L l,E)ij = j =(ijE+sp)jj+i j,g˜ij kphysical, metricA = pg(L) ,(75) r (73) pg r r r 3 r (computed at 2(↵ ) 4 1 where pg is theeach square iteration) root of theg determinant of the physical metric (gij = g˜ij). We also solve for the correction to the scalarj4 potentialj (see Eq. 72), wherep theF covariantk l derivatives are compatible with g in contrast to the covariant derivatives above where pg is the square root of the determinant of the physical⇢ = metricS = (gij(B=jB g˜+ijE).jE We) also,J solvei = for the✏ikl correctionE B , to the ij (74) • 8⇡ which are compatible with g˜ .4 Expressed⇡ in terms of the conformal metric, it can be written as: scalar potential (see Eq. 72), where the covariant derivativesHave are implemented compatible with thisgij methodin contrast in Wolfgang’s to the covariantij SGRID derivatives spectral above code, and are 1 which are compatible with g˜ij. Expressed in terms of thecurrently conformali improving metric,ikl it canboundary be written conditions as: and implementing computations1 of ADM B = ✏F @k(Asp)l,Ej =(Esp)j + ˜j2, +2@k(ln )@ = @ (˜gkm(E ) 2 (75)g˜). (76) quantities.pg rr k 2pg˜ k sp m ˜ 2 k 1 km 2 p +2@ (ln )@k = @k(˜g (EspAlternatively,)m g˜) one. can write this in4 a slightly simpler form(76) as where pg ris the square root of the determinant2pg˜ of the physical metric (gij = g˜ij). We also solve for the correction to the ij scalar potential (see Eq. 72), where the covariant derivatives arep compatible with gij in contrastg˜ [ ˜ i +2 to the@i(ln covariant )][@j +( derivativesEsp)j]=0. above (77) Alternatively, one can write this in a slightly simpler form as r which are compatible with g˜ij. Expressed in terms of the conformal metric, it can be written as: ij ˜ g˜ [ i +2@i(ln )][@j +(Esp)j]=0. 1 (77) r ˜ 2 +2@k(ln )@ = @ (˜gkm(E ) 2 g˜). (76) r k 2pg˜ k sp m p Alternatively, one can write this in a slightly simpler form as g˜ij[ ˜ +2@ (ln )][@ +(E ) ]=0. (77) ri i j sp j 6 6

[see Eq. (2.2) in [25]].7 This is the Ricci scalar[see term Eq. in (2.2) the in Raychaudhuri [25]].7 This is the equation Ricci scalar [see, term e.g., in the Eq. Raychaudhuri (20) in [13], equation which [see, denotes e.g., Eq. our (20) in [13], which denotes our `µ by kµ], so that term vanishes if we are considering`µ by kµ], a so non-expanding that term vanishes horizon. if we are considering a non-expanding horizon. However, it is not clear how to implement this condition on Rµ⌫ as a boundary condition for (and/or as a boundary condition However, it is not clear how to implement this condition on Rµ⌫ as a boundary condition for (and/or as a boundary condition for a possible correction to the magnetic field). Specifically, as discussed in Sec. II C of [25] [see Eqs. (2.16) and (2.17) in for a possible correction to the magnetic field). Specifically, as discussed in Sec.R `µ IIp⌫ C of0 [25] [seep⌫ Eqs. (2.16) and (2.17) in µparticular],⌫ the stronger⌫ condition that µ⌫ , for any tangent to the horizon [from below Eq. (2.3) in [25]] gives particular], the stronger condition that Rµ⌫ ` p µ ⌫0 for any p µ tangent⌫ to the horizon8 [from below Eq. (2.3) in [25]] gives Fµ⌫ `,p , 0 , Fµ⇤⌫ ` p on the horizon. Written in terms of the electric and magnetic fields [using Eqs. (2.28) and (2.29) µ ⌫ µ ⌫ 8 F ` p 0 F ⇤ ` p on the horizon. Written in terms of the electric and magnetic fields [using Eqs. (2.28) and (2.29) µ⌫ , , µ⌫ in [1]], these imply that Eak , (s B)a, as well as Bak , (s E)a, but this follows from the first equation. Here the superscripts denote projection perpendicular⇥ to sa, the unit normal ⇥ to (the 2d section of) the horizon. Note that one can replacek in [1]], these imply that Eak , (s B)a, as well as Bak , (s E)a, but this follows from the first equation. Here the ⇥ k ⇥k k superscripts denote projection perpendicular toBsaaand, theEa unitwith normalBa and E toa on (the the2 right-handd section sides of) the of these horizon. equations Note if one that so one desires. can Thus, replace these boundary conditions are the same as those obtained for electric and magnetic fields on Kerr in the membrane paradigm viewpoint—see Eq. (4.10) in [26]. Ba and Ea with Bak and Eak onBoundary the right-hand sides conditionsThese of boundary these equations conditions if do one not so involve desires. the normal Thus, derivative these boundary of and thus conditions cannot be are used the to obtain a standard (Neumann same as those obtained for electric and magneticor Robin) fields on boundary Kerr in condition the membrane involving the paradigm normal derivative viewpoint—see at the boundary. Eq. (4.10) They in thus [26 also]. do not say anything about the These boundary conditions do not involve thecharge normal on the derivative black hole, of which and can thus be computed cannot be by used a surface to obtain integral a of standard the normal (Neumann component of Ea (see the discussion in • Currently just using superposed fields to set boundary conditions for initial tests. or Robin) boundary condition involving the normalSec. IV Cderivative). We thus can at the thus boundary. fix the charge They on the thus black also hole do with not a Neumann say anything boundary about condition the on . Of course, just fixing the total charge does not specify the normal derivative at every point,E as needed for the boundary condition, just its integral over charge on the black hole, which• Will can use be proper computed theisolated surface. by ahorizon surface A simple (really integral way non-expanding to fix of the the horizon) normal normal boundary derivative component conditions at every point of for isa the to(see just geometry the scale discussion the superposed in field by a constant such that Sec. IV C). We thus can thus fix(as the in chargeLovelace on etthe the al.), total black and charge holeuse is a the with Neumann desired a Neumann one. boundary Specifically, boundary condition we take condition for the scalar on .potential Of course, to fix just fixing the total charge does not specify the normal derivative at every point, as needed for the boundary condition, just its integral over the charge on the black holes. The current proposal is just tosa @scale the(⇣ superposed1)sa(E ) , electric (53) a , sp a the surface. A simple way to fixfield the to normal obtain derivative the desired at charge. every point is toCharge just scale of black the hole superposed field by a constant such that where the total charge is the desired one. Specifically,Normal to we take used in superposition 1 horizon a a ⇣ := 4⇡Q (E )adS 6 (54) (= excision s @a , (⇣ 1)s (Esp)a, H sp a (53)  Sexc surface) 7 I where and [seeQ Eq.is the (2.2) charge in [25]]. on theThis hole. is the This Ricci construction scalar term in obviouslythe Raychaudhuri will not equation work [see, if the e.g., desired Eq. (20) charge in [13 is], which nonzero denotes and ourthe superposed `µ Hby kµ], so that term vanishes if we are considering a non-expanding horizon. However, we’refield trying leads to tothink zero of charge better (or ways a much (where smaller1 the charge boundary than the condition desired charge), involves but the this sort of situation seems unlikely to occur in However, it is not clear how to implement this condition on Rµ⌫ as a boundary condition for (and/or as a boundary condition horizon’s geometry).practice. a ⇣ :=for 4 a⇡ possibleQH correction(Esp to) thedS magnetica field). Specifically, as discussed in Sec. II C of [25] [see(54) Eqs. (2.16) and (2.17) in However, while this is simple, it seemsR a`µ bitp⌫ kludgy0 and insufficientlyp⌫ flexible to provide the very best initial data. One would particular], the strongerSexc condition that µ⌫ , for any tangent to the horizon [from below Eq. (2.3) in [25]] gives µ ⌫ I µ ⌫ a 8 • [N.B.: The isolatedrather Fhorizonµ⌫ have` p the, boundary0 boundary, Fµ⇤⌫ ` conditionsp conditionon the horizon. on implys @aWritten thatrespond in terms to the of geometry the electric of and the magnetic horizon. fields In fact, [using for Eqs. an (2.28) unperturbed and (2.29) Kerr-Newman Q a a and H is the charge on the hole. This constructionblackin hole [ obviously1]], in these a tetrad imply will appropriate that notEak work(s for if theB the) isolated, as desired well horizon as B chargeak formalism,(s is nonzeroE) , the but derivatives this and follows the of fromsuperposeds E thea and firsts equation.Ba are related Here the to the intrinsic , ⇥ a , ⇥ a k field leads to zero charge (or a muchIt doesn’t smaller seem chargeRicci simplesuperscripts scalar than to enforceR theH denoteof desired (a this projection2d section with charge), perpendiculara of) boundary the but horizon. this to condition,s sorta, the Specifically, of unit situation normal since from to it (the seemsinvolves Eq.2d section(56b) unlikely in of) [27 the] to horizon. [useful occur Notedefinitions in that one are can in replace Eqs. (2) and (8)] and Eq. (23) in Smarr [28] (noting that the Gaussian curvature of a 2d surface is half its Ricci scalar), one obtains practice. transverse derivativesBa andof �Ea. Wewith Bareak and waitingEak on theto right-handsee how sides well of this these is equations satisfied if onein our so desires. data Thus,with these boundary conditions are the same as those obtained for2 electric2 and2 magnetic fields on Kerr in the membrane paradigm viewpoint—see2 2 Eq.2 (4.10) in2 [26].2 However, while this is simple,the it seemsabove boundary a bit kludgy conditionsThese anda boundary insufficiently before2Q conditionsH(r+ worryinga do flexible notcos involvetoo✓) to much the provide normala about derivative the how4 veryQH arto of+ bestenforcecosand initial✓ thus) this.] cannot data. be used One2( tor would+ obtain+ a a)( standardr+ 3 (Neumanna cos ✓) a s Ea , 2 2 2 2 ,sBa , 2 2 2 2 ,RH = 2 2 2 3 . (55) rather have the boundary condition on s @a respondor Robin) to the boundary geometry(r condition+ + a ofcos involving the✓) horizon. the normal In derivative fact,(r for+ at+ an thea unperturbed boundary.cos ✓) They thus Kerr-Newman also do not(r+ say+ anythinga cos about✓) the charge on the black hole, which can be computed bya a surface integrala of the normal component of E (see the discussion in black hole in a tetrad appropriate for the isolated horizon formalism, the derivatives of s Ea and s Ba are related to the intrinsica whereSec.r+IVis C). the We radius thus can of thethus outer fix the horizon charge on in the Boyer-Lindquist black hole with a coordinates, Neumann boundary so we have condition on . Of course, just fixing Ricci scalar R of (a 2d section of) the horizon. Specifically, from Eq. (56b) in [27] [useful definitions are in Eqs. (2) and (8)] H the total charge does not specify the@( normalsaE ) derivative2Q atr every point, as@ needed(saB for) the boundary2Q r a condition, just its integral over the surface. A simple way to fix the normala derivative atH every+ point is to just scalea the superposedH + field by a constant such that and Eq. (23) in Smarr [28] (noting that the Gaussian curvature of a 2d surface is half, its2 Ricci2 R scalar),H, one obtains, 2 2 RH. (56) the total charge is the desired one. Specifically,@r+ wer take+ + a @ cos ✓ r+ + a 2 2 2 2 2 2 2 2 a a a 2QH(r+ a cos ✓) (Onea can likely4Q replaceHar+ cosr+ with✓) some radial coordinates2(@ r+ +( defined⇣ a 1))(sr( off+E of) ,3 thea horizoncos ✓) in the first partial derivative,(53) but I have not s E ,sB ,R= a , spa . (55) a , (r2 + a2 cos2 ✓)2 checkeda , this(r explicitly.)2 + a2 cos To2 ✓ obtain)2 the expressionsH for( ther2 normal+ a2 cos components2 ✓)3 of the electric and magnetic fields, we have + where NP+ 1 a n a b 1 a +(3) a c 1 a a a NP noted that 1 := 2 Fab(n ` + m m¯ )= 2 (s Ea ✏abcm m¯ bB1 )= 2 (s Ea + is Ba), so s Ea =2Re1 and a NP (3) a a where r+ is the radius of the outer horizon in Boyer-Lindquists Ba =2Im1 . NKJ-M: coordinates, I need to so check we have the⇣ := sign 4⇡ ofQH ✏abcm(Espm¯) bdSina the tetrad used in [27]. However, it’s not(54) clear if this sort Sexc a of relation extends to perturbeda Kerr-Newman, though it’sI possible that one can obtain something useful in the general case from @(s Ea) the2 isolatedandQHQrH+is horizon the charge formalism on@ the(s hole.B using,a This) e.g.,construction2 (possiblyQHr+ obviouslya just some will not of) work Eqs. if (86) the desired and (89) charge in [29 is]. nonzero But even and the if one superposed obtains something , 2field leads2 R toH zero, charge (or a much, smaller charge2 than2 R theH. desired charge), but this sort of situation seems(56) unlikely to occur in @r+ usefulr+ + in general,a if it just@ constrainscos ✓ the normalr+ + derivativesa of the fields, then it’s unclear how to turn it into a boundary condition for thepractice. normal derivative of . However, while this is simple, it seems a bit kludgy and insufficiently flexible to provide the very best initial data. One would (One can likely replace r+ with some radial coordinate defined off of the horizona in the first partial derivative, but I have not k Turningrather have now the to boundary the non-expanding condition on s horizon@a respond condition to the geometry on the parallel of the horizon. components In fact, for of an the unperturbed fields, Ea Kerr-Newman, (s B)a, this can a a ⇥ checked this explicitly.) To obtain the expressionspresumablyblack for hole be the in satisfied a tetrad normal appropriate by adding components for the the curl isolated of of a horizonthe to-be-determined electric formalism, and the vector magneticderivatives field to of fields,thes E electrica and wes B and/ora haveare related magnetic to the field, intrinsic since this does NP 1 a n a b 1 a R(3) 2 a c 1 a a a NP a noted that := Fab(n ` + m m¯ )=not affectRicci(s E scalar thea valueH of✏ ofabc (a themd section divergence.m¯ bB of))= the If horizon. we( justs Specifically,E considera + is adding fromBa) Eq., the so (56b) curls inE to [a27 the]=2Re [useful electric definitions field,and and are call in Eqs. the (2) vector and (8)] field Z , it has 1 2 2 2 2 1 a NP to satisfyand Eq.(3) (23) in Smarra [28] (noting that the Gaussian curvature of a d surface is half its Ricci scalar), one obtains s Ba =2Im1 . NKJ-M: I need to check the sign of ✏abcm m¯ b in2 the2 tetrad2 used in [27]. However, it’s not clear2 if this2 2 sort 2 2 a 2QH(r+ a cos ✓) a b4QHar+ cos ✓) 2(r+ + a )(r+ 3a cos ✓) s Ea (D,sZ)aBa sas ( Z)b , (,Rs B)Ha= Eak =: a . (55) (57) of relation extends to perturbed Kerr-Newman, though it’s possible, (r2 that+ a2 onecos2 ✓ can)2 obtain⇥ something, (r2 r⇥+ a2 usefulcos2 ✓)2 in⇥ the general (r case2S+ a from2 cos2 ✓)3 the isolated horizon formalism using, e.g., (possibly just some of)+ Eqs. (86) and (89) in [29].+ But even if one obtains something+ where r is the radius of the outer horizon in Boyer-Lindquist coordinates, so we have useful in general, if it just constrains the normal derivatives+ of the fields, then it’s unclear how to turn it into a boundary condition 7 Specifically, `µ =(nµ + sµ)/p2, where@(sanEµ is) the unit2Q normalr to the spatial@(s hypersurfaceaB ) 2 andQ rsµ ais the unit normal to (the 2d section of) the horizon. a H + R , a H + R . (56) for the normal derivative of . 8 In the Newman-Penrose formalism, this can be written, as2 NP 2 0H, where the Newman-Penrose, 2 scalar2 HNP := F `amb, and ma is a complex null vector @r+ r+ +0 a, @ cos ✓ r+ + a 0 ab Turning now to the non-expanding horizon conditionthat here is tangent on to the the horizon.parallel components of the fields, Eak , (s B)a, this can (One can likely replace r+ with some radial coordinate defined off of the horizon in⇥ the first partial derivative, but I have not presumably be satisfied by adding the curl of a to-be-determinedchecked this explicitly.) vector To obtain field the to the expressions electric for and/or the normal magnetic components field, of the since electric this and does magnetic fields, we have NP := 1 F (na`n + mam¯ b)= 1 (saE (3)✏ mam¯ Bc)= 1 (saE + isaBa ) saE =2ReNP not affect the value of the divergence. If we just considernoted that adding1 the2 ab curl to the electric2 field,a andabc callb the vector2 fielda Z a, it, so has a 1 and a NP (3) a s Ba =2Im1 . NKJ-M: I need to check the sign of ✏abcm m¯ b in the tetrad used in [27]. However, it’s not clear if this sort to satisfy of relation extends to perturbed Kerr-Newman, though it’s possible that one can obtain something useful in the general case from the isolatedb horizon formalism using, e.g., (possibly just some of) Eqs. (86) and (89) in [29]. But even if one obtains something (D Z)a usefulsas in( general,Z if)b it just, ( constrainss B) thea normalEak derivatives=: a of the fields, then it’s unclear how to turn it(57) into a boundary condition ⇥ for the normalr⇥ derivative of ⇥. S Turning now to the non-expanding horizon condition on the parallel components of the fields, Eak , (s B)a, this can presumably be satisfied by adding the curl of a to-be-determined vector field to the electric and/or magnetic field,⇥ since this does 7 Specifically, `µ =(nµ + sµ)/p2, where nµ is the unit normalnot affect to the the spatial value of hypersurface the divergence. and Ifs weµ is just the consider unit normal adding to the (the curl2 tod section the electric of) field, the horizon. and call the vector field Za, it has 8 toNP satisfy NP a b a In the Newman-Penrose formalism, this can be written as 0 , 0, where the Newman-Penrose scalar 0 := Fab` m , and m is a complex null vector b that here is tangent to the horizon. (D Z) s s ( Z) (s B) Ek =: (57) ⇥ a a r⇥ b , ⇥ a a Sa

7 Specifically, `µ =(nµ + sµ)/p2, where nµ is the unit normal to the spatial hypersurface and sµ is the unit normal to (the 2d section of) the horizon. 8 NP NP a b a In the Newman-Penrose formalism, this can be written as 0 , 0, where the Newman-Penrose scalar 0 := Fab` m , and m is a complex null vector that here is tangent to the horizon. Conclusions

• There are a wide variety of charged binary black hole solutions involving GR coupled to other fields (which could also be thought of as modifications to GR itself).

• Even concentrating on just the Kerr-Newman family of Einstein-Maxwell theory, there is a possibility that these could still be astrophysically relevant if they are charged under some dark EM field or magnetically charged.

• Regardless, charged black holes make excellent toy models for various effects, including spin and modified gravity effects.

• There have been evolutions of charged black hole binaries with two codes (Lean and HAD), but only with very simple approximate initial data in the orbiting case.

• We have implemented a generalization of the Lovelace et al. superposed Kerr-Schild construction to Kerr-Newman black holes in Wolfgang Tichy’s SGRID code and are currently testing the data with evolutions in the vacuum case and improving the boundary conditions. Extra slides Technical details about SGRID implementation

• Use Ansorg’s ABφ coordinates in their black hole excision version (to give a compactified grid with more resolution close to the black holes) with a Chebyshev-Chebyshev-Fourier spectral expansion.

Figure 5.1: TheIllustrationz = 0 slice produced of by the SGRID AB codeφ for solving the • constraint equations. Lines of constant A and B are shown for b = 5, and the Solve nonlinear equations using Newton’s black holescoordinates are located along the x-axis from and excised George by a spherical region at A = 0. The grid is separated into two subdomains that meet along the yz-plane method, and linear equations using (Seen here onlyReifenberger’s as the line x = 0). thesis dctemplates (+ BLAS & LAPACK) GMRES with block Jacobi preconditioner + UMFPACK (or just UMFPACK if you have enough memory).

• OpenMP parallelized (for, e.g., setting the elements of the matrix to solve).

Figure 5.2: The = 0 slice of the subdomain for the +x-axis black hole produced by the SGRID code for solving the constraint equations. Lines of constant A and B are shown and the excision boundary is located at A = 0.

59