Spherically Symmetric Static Black Holes in Einstein-Aether Theory

Spherically symmetric static black holes in Einstein-aether theory

Chao Zhang1,∗ Xiang Zhao1,† Kai Lin2,3,‡ Shaojun Zhang4,5,§ Wen Zhao6,7,¶ and Anzhong Wang1 ∗∗†† 1 GCAP-CASPER, Physics Department, Baylor University, Waco, Texas, 76798-7316, USA 2 Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan, Hubei, 430074, China 3 Escola de Engenharia de Lorena, Universidade de SãoPaulo, 12602-810, Lorena, SP, Brazil 4 Institute for Theoretical Physics & Cosmology, Zhejiang University of Technology, Hangzhou 310032, China 5 United Center for Gravitational Wave Physics, Zhejiang University of Technology, Hangzhou 310032, China 6 CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China and 7 School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China (Dated: September 18, 2020) In this paper, we systematically study spherically symmetric static spacetimes in the framework of Einstein-aether theory, and pay particular attention to the existence of black holes (BHs). In the theory, two additional gravitational modes (one scalar and one vector) appear, due to the presence of a timelike aether field. To avoid the vacuum gravi-Cerenkovˇ radiation, they must all propagate with speeds greater than or at least equal to the speed of light. In the spherical case, only the scalar mode is relevant, so BH horizons are defined by this mode, which are always inside or at most coincide with the metric (Killing) horizons. In the present studies we first clarify several subtle issues. In particular, we find that, out of the five non-trivial field equations, only three are independent, so the problem is well-posed, as now generically there are only three unknown functions, F (r),B(r),A(r), where F and B are metric coefficients, and A describes the aether field. In addition, the two second-order differential equations for A and F are independent of B, and once they are found, B is given simply by an algebraic expression of F,A and their derivatives. To simplify the problem further, we explore the symmetry of field redefinitions, and work first with the redefined metric and aether field, and then obtain the physical ones by the inverse transformations. These clarifications significantly simplify the computational labor, which is important, as the problem is highly involved mathematically. In fact, it is exactly because of these, we find various numerical BH solutions with an accuracy that is at least two orders higher than previous ones. More important, these BH solutions are the only ones that satisfy the self-consistent conditions and meantime are consistent with all the observational constraints obtained so far. The locations of universal horizons are also identified, together with several other observationally interesting quantities, such as the innermost stable circular orbits (ISCO), the ISCO frequency, and the maximum redshift zmax of a photon emitted by a source orbiting the ISCO. All of these quantities are found to be quite close to their relativistic limits.

I. INTRODUCTION ever, the details of these detections have not yet been released [5]. The outbreak of interest on GWs and BHs has further gained momentum after the detection of the The detection of the first gravitational wave (GW) shadow of the M87 BH [6–11]. from the coalescence of two massive black holes (BHs) by the advanced Laser Interferometer Gravitational-Wave One of the remarkable observational results is the dis- Observatory (LIGO) marked the beginning of a new era, covery that the mass of an individual BH in these bi- arXiv:2004.06155v4 [gr-qc] 16 Sep 2020 the GW astronomy [1]. Following this observation, soon nary systems can be much larger than what was pre- more than ten GWs were detected by the LIGO/Virgo viously expected, both theoretically and observationally scientific collaboration [2–4]. More recently, about 50 [12–14], leading to the proposal and refinement of vari- GW candidates have been identified after LIGO/Virgo ous formation scenarios [15, 16]. A consequence of this resumed operations on April 1, 2019, possibly including discovery is that the early inspiral phase may also be the coalescence of a neutron-star (NS)/BH binary. How- detectable by space-based observatories, such as as the Laser Interferometer Space Antenna (LISA) [17], Tian- Qin [18], Taiji [19], and the Deci-Hertz Interferometer Gravitational wave Observatory (DECIGO) [20], for sev- ∗∗Corresponding Author eral years prior to their coalescence [21, 22]. Such space- ∗Electronic address:C [email protected] based detectors may be able to see many such systems, † Electronic address: Xiang [email protected] which will result in a variety of profound scientific conse- ‡Electronic address: [email protected] §Electronic address: [email protected] quences. In particular, multiple observations with differ- ¶Electronic address: [email protected] ent detectors at different frequencies of signals from the ††Electronic address: Anzhong [email protected] same source can provide excellent opportunities to study 2 the evolution of the binary in detail. Since different de- the future. tectors observe at disjoint frequency bands, together they In this paper, we shall continuously work on GWs and cover different evolutionary stages of the same binary BHs in the framework of æ-theory, but move to the ring- system. Each stage of the evolution carries information down phase, which consists of the relaxation of the highly about different physical aspects of the source. perturbed, newly formed merger remnant to its equilib- As a result, multi-band GW detections will provide an rium state through the shedding of any perturbations in unprecedented opportunity to test different theories of GWs as well as in matter waves. Such a remnant will typ- gravity in the strong field regime [23–28]. Massive sys- ically be a Kerr BH, provided that the binary system is tems will be observed by ground-based detectors with massive enough and GR provides the correct description. high signal-to-noise ratios, after being tracked for years This phase can be well described as a sum of damped ex- by space-based detectors in their inspiral phase. The ponentials with unique frequencies and damping times - two portions of signals can be combined to make precise quasi-normal modes (QNMs) [44]. tests for different theories of gravity. In particular, joint observations of binary black holes (BBHs) with a total The information contained in QNMs provide the keys mass larger than about 60 solar masses by LIGO/Virgo in revealing whether BHs are ubiquitous in our Universe, and space-based detectors can potentially improve cur- and more important whether GR is the correct theory rent bounds on dipole emission from BBHs by more than to describe the event even in the strong field regime. In six orders of magnitude [23], which will impose severe fact, in GR according to the no-hair theorem [45], an constraints on various theories of gravity [29]. isolated and stationary BH is completely characterized by only three quantities, mass, spin angular momentum In recent works, some of the present authors gener- and electric charge. Astrophysically, we expect BHs to alized the post-Newtonian (PN) formalism to certain be neutral, so it must be described by the Kerr solution. modified theories of gravity and applied it to the quasi- Then, the quasi-normal frequencies and damping times circular inspiral of compact binaries. In particular, we will depend only on the mass and angular momentum of calculated in detail the waveforms, GW polarizations, re- the final BH. Therefore, to extract the physics from the sponse functions and energy losses due to gravitational ringdown phase, at least two QNMs are needed. This will radiation in Brans-Dicke (BD) theory [30], and screened require the signal-to-noise ratio (SNR) to be of the order modified gravity (SMG) [31–33] to the leading PN or- 100 [46]. Although such high SNRs are not achievable der, with which we then considered projected constraints right now, it was shown that [47] they may be achiev- from the third-generation detectors. Such studies have able once the advanced LIGO and Virgo reach their de- been further generalized to triple systems in Einstein- sign sensitivities. In any case, it is certain that they will aether (æ-) theory [34, 35]. When applying such for- be detected by the ground-based third-generation detec- mulas to the first relativistic triple system discovered in tors, such as Cosmic Explorer [48, 49] or the Einstein 2014 [36], we studied the radiation power, and found that Telescope [50], as well as the space-based detectors, in- quadrupole emission has almost the same amplitude as cluding LISA [17], TianQin [18], Taiji [19], and DECIGO that in general relativity (GR), but the dipole emission [20], as just mentioned above. can be as large as the quadrupole emission. This can provide a promising window to place severe constraints In the framework of æ-theory, BHs with rotations have on æ-theory with multi-band GW observations [23, 26]. not been found yet, while spherically symmetric BHs More recently, we revisited the problem of a binary have been extensively studied in the past couple of years system of non-spinning bodies in a quasi-circular inspi- both analytically [51–62] and numerically [63–69]. It was ral within the framework of æ-theory [37–42], and pro- shown that they can also be formed from gravitational vided the explicit expressions for the time-domain and collapse [70]. Unfortunately, in these studies, the param- frequency-domain waveforms, GW polarizations, and re- eter space has all been ruled out by current observations sponse functions for both ground- and space-based de- [71]. Therefore, as a first step to the study of the ring- tectors in the PN approximation [43]. In particular, down phase of a coalescing massive binary system, in we found that, when going beyond the leading order in this paper we shall focus ourselves mainly on spherically the PN approximation, the non-Einsteinian polarization symmetric static BHs in the parameter space that sat- modes contain terms that depend on both the first and isfies the self-consistent conditions and the current ob- second harmonics of the orbital phase. With this in mind, servations [71]. As shown explicitly in [72], spherically we calculated analytically the corresponding parameter- symmetric BHs in the new physically viable phase space ized post-Einsteinian parameters, generalizing the exist- can be still formed from the gravitational collapse of re- ing framework to allow for different propagation speeds alistic matter. among scalar, vector and tensor modes, without assum- It should be noted that the definition of BHs in æ- ing the magnitude of its coupling parameters, and mean- theory is different from that given in GR. In particu- while allowing the binary system to have relative motions lar, in æ-theory there are three gravitational modes, the with respect to the aether field. Such results will partic- scalar, vector and tensor, which will be referred to as the ularly allow for the easy construction of Einstein-aether spin-0, spin-1 and spin-2 gravitons, respectively. Each templates that could be used in Bayesian tests of GR in of them moves in principle with a different speed, given, 3 respectively, by [73] not be concerned with this question. First, we consider æ-theory as a low-energy effective theory, and second the 2 c123(2 − c14) constraints on the breaking of the Lorentz symmetry in cS = , c14(1 − c13)(2 + c13 + 3c2) the gravitational sector is much weaker than that in the matter sector [76]. So, to avoid this problem, in this pa- 2c1 − c13(2c1 − c13) c2 = , per we simply assume that the matter sector still satisfies V 2c (1 − c ) 14 13 the Lorentz symmetry. Then, all the particles from the 2 1 matter sector will travel with speeds less or equal to the cT = , (1.1) 1 − c13 speed of light. Therefore, for these particles, the Killing (or metric) horizons still serve as the boundaries. Once where c ’s are the four dimensionless coupling constants i inside them, they will be trapped inside the metric hori- of the theory, and c ≡ c + c , c ≡ c + c + c . ij i j ijk i j k zons (MHs) forever, and never be able to escape to spatial The constants c , c and c represent the speeds of the S V T infinities. spin-0, spin-1 and spin-2 gravitons, respectively. In or- With the above in mind, in this paper we shall carry der to avoid the existence of the vacuum gravi-Cerenkovˇ out a systematical study of spherically symmetric space- radiation by matter such as cosmic rays [74], we must times in æ-theory, clarify several subtle points, and then require present numerically new BH solutions that satisfy all the current observational constraints [71]. In particular, we c , c , c ≥ c, (1.2) S V T shall show that, among the five non-trivial field equa- where c denotes the speed of light. Therefore, as far as tions (three evolution equations and two constraints), the gravitational sector is concerned, the horizon of a BH only three of them are independent. As a result, the should be defined by the largest speed of the three dif- system is well defined, since in the current case there are ferent species of gravitons. However, in the spherically only three unknown functions: two describe the space- symmetric spacetimes, the spin-1 and spin-2 gravitons time, denoted by F (r) and B(r) in Eq.(3.1), and one are not excited, and only the spin-0 graviton is relevant. describes the aether field, denoted by A(r) in Eq.(3.2). Thus, the BH horizons in spherically symmetric space- An important result, born out of the above observa- times are defined by the metric [75], tions, is that the three independent equations can be divided into two groups, which decouple one from the g(S) ≡ g − c2 − 1 u u , (1.3) other, that is, the equations for the two functions A(r) µν µν S µ ν and F (r) [cf. Eqs.(3.5) and (3.6)] are independent of the function B(r). Therefore, to solve these three field equa- where uµ denotes the four-velocity of the aether field, µ tions, one can first solve Eqs.(3.5) and (3.6) for A(r) and which is always timelike and unity, uµu = −1. Because of the presence of the aether in the whole F (r). Once they are found, one can obtain B(r) from the third equation. It is even more remarkable, if the third spacetime, it uniquely determines a preferred direction at v each point of the spacetime. As a result, the Lorentz sym- equation is chosen to be the constraint C = 0, given metry is locally violated in æ-theory [76] 1. It must be by Eq.(3.10), from which one finds that B(r) is then di- emphasized that the breaking of Lorentz symmetry can rectly given by the algebraic equation (3.11) without the have significant effects on the low-energy physics through need of any further integration. Considering the fact that the interactions between gravity and matter, no matter the field equations are in general highly involved math- how high the scale of symmetry breaking is [81], unless ematically, as it can be seen from Eqs.(3.5)-(3.10) and supersymmetry is invoked [82]. In this paper, we shall Eqs.(A.1)-(A.4), this is important, as it shall significantly simplify the computational labor, when we try to solve these field equations. Another important step of solving the field equations is 1 It should be noted that the invariance under the Lorentz sym- Foster’s discovery of the symmetry of the action, the so- metry group is a cornerstone of modern physics and strongly called field redefinitions [83]: the action remains invariant supported by experiments and observations [77]. Nevertheless, under the replacements, there are various reasons to construct gravitational theories with µ µ broken Lorentz invariance (LI). For example, if space and/or (gµν , u , ci) → (ˆgµν , uˆ , cî) , (1.4) time at the Planck scale are/is discrete, as currently understood [78], Lorentz symmetry is absent at short distance/time scales µ whereg ˆµν ,u ˆ andc î are given by Eqs.(2.23) and (2.24) and must be an emergent low energy symmetry. A concrete through the introduction of a free parameter σ. Taking example of gravitational theories with broken LI is the Hoˇrava theory of quantum gravity [79], in which the LI is broken via the advantage of the arbitrariness of σ, we can choose 2 2 the anisotropic scaling between time and space in the ultraviolet it as σ = cS, where cS is given by Eq.(1.1). Then, the −z i −1 i (UV), t → b t, x → b x , (i = 1, 2, ..., d), where z denotes spin-0 and metric horizons for the metricg ˆµν coincide the dynamical critical exponent, and d the spatial dimensions. [63, 65, 67]. Thus, instead of solving the field equations Power-counting renormalizability requires z ≥ d at short dis- µ µ tances, while LI demands z = 1. For more details about Hoˇrava for (gµν , u ), we first solve the ones for (ˆgµν , uˆ ), as in gravity, see, for example, the review article [80], and references the latter the corresponding initial value problem can µ therein. be easily imposed at horizons. Once (ˆgµν , uˆ ) is found, 4 using the inverse transformations, we can easily obtain cases that satisfy all the observational constraints and µ (gµν , u ). obtain various new static BH solutions. With the above observations, we are able to solve nu- Then, in Sec. V, we present the physical metric and merically the field equations with very high accuracy, as æ-field for these viable new BH solutions, by using the to be shown below [cf. TableI]. In fact, the accuracy is inverse transformations from the effective fields to the significantly improved and in general at least two orders physical ones. In this section, we also show explicitly µ higher than the previous works. that the physical fields, gµν and u , are also asymptot- µ In theories with breaking Lorentz symmetry, another ically flat, provided that the effective fieldsg ˜µν andu ˜ µ important quantity is the universal horizon (UH) [66, 67], are, which are related tog ˆµν andu ˆ via the coordinate which is the causal boundary even for particles with in- transformations given by Eq.(3.42). Then, we calculate finitely large speeds. The thermodynamics of UHs and explicitly the locations of the metric, spin-0 and univer- relevant physics have been extensively studied since then sal horizons, as well as the locations of the innermost (see, for example, Section III of the review article [80], stable circular orbits (ISCO), the Lorentz gamma fac- and references therein). In particular, it was shown that tor, the gravitational radius, the orbital frequency of the such horizons can be formed from gravitational collapse ISCO, the maximum redshift of a photon emitted by a of a massless scalar field [72]. In this paper, we shall also source orbiting the ISCO (measured at the infinity), the identify the locations of the UHs of our numerical new radii of the circular photon orbit, and the impact param- BH solutions. eter of the circular photon orbit. All of them are given The rest of the paper is organized as follows: Sec. II in TableIV-V. In TableVI we also calculate the differ- provides a brief review to æ-theory, in which the intro- ences of these quantities obtained in æ-theory and GR. duction of the field redefinitions, the current observa- From these results, we find that the differences are very tional constraints on the four dimensionless coupling con- small, and it is very hard to distinguish GR and æ-theory stants ci’s of the theory, and the definition of the spin-0 through these quantities, as far as the cases considered horizons (S0Hs) are given. in this paper are concerned. In Sec. III, we systematically study spherically sym- Finally, in Sec. VI we summarize our main results metric static spacetimes, and show explicitly that among and present some concluding remarks. There is also an the five non-trivial field equations, only three of them appendix, in which the coefficients of the field equations µ µ are independent, so the corresponding problem is well for both (gµν , u ) and (˜gµν , u˜ ) are given. defined: three independent equations for three unknown functions. Then, from these three independent equations we are able to obtain a three-parameter family of ex- II. Æ-THEORY act solutions for the special case c13 = c14 = 0, which depends in general on the coupling constant c2. How- ever, requiring that the solutions be asymptotically flat In æ-theory, the fundamental variables of the gravitational sector are [84], makes the solutions independent of c2, and the metric reduces precisely to the Schwarzschild BH solution with a µ non-trivially coupling aether field [cf. Eq.(3.34)], which (gµν , u , λ) , (2.1) is timelike over the whole spacetime, including the region inside the BH. To further simplify the problem, in with the Greek indices µ, ν = 0, 1, 2, 3, and gµν is the this section we also explore the advantage of the field four-dimensional metric of the spacetime with the signa- redefinitions [83]. In particular, we show step by step ture (−, +, +, +) [37, 70], uµ is the aether four-velocity, how to choose the initial values of the differential equa- as mentioned above, and λ is a Lagrangian multiplier, tions Eqs.(3.63) and (3.64) on S0Hs, and how to re- which guarantees that the aether four-velocity is always duce the phase space from four dimensions, spanned by timelike and unity. In this paper, we also adopt units so ˜ ˜0 ˜ ˜0 (FH , FH , AH , AH ), to one dimension, spanned only by that the speed of light is one (c = 1). Then, the general action of the theory is given by [75], A˜H . So, finally the problem reduces to finding the values of A˜H that lead to asymptotically flat solutions of the form (3.79)[63, 67]. S = Sæ + Sm, (2.2) In Sec. IV, we spell out in detail the steps to carry out our numerical analysis. In particular, as we show where Sm denotes the action of matter, and Sæ the grav- explicitly, Eq.(3.65) is not independent from other three itational action of the æ-theory, given, respectively, by differential equations. Taking this advantage, we use it to Z monitor our numerical errors [cf. Eq.(4.7)]. To check our 1 √ 4 h α Sæ = −g d x Læ (gµν , u , ci) numerical code further, we reproduce the BH solutions 16πGæ obtained in [63, 67], but with an accuracy two orders i + L (g , uα, λ) , higher than those obtained in [67] [cf. Table I]. Unfortu- λ µν Z nately, all these BH solutions have been ruled out by the √ 4 h α i current observations [71]. So, in Sec. IV.B we consider Sm = −g d x Lm (gµν , u ; ψ) . (2.3) 5

Here ψ collectively denotes the matter fields, R and g are, (spin-0), vector (spin-1) and tensor (spin-2) modes [73], respectively, the Ricci scalar and determinant of gµν , and with their squared speeds given by Eq.(1.1). In addition, among the 10 parameterized post- α β Lλ ≡ λ gαβu u + 1 , Newtonian (PPN) parameters [86, 87], in æ-theory the αβ µ ν only two parameters that deviate from GR are α1 and Læ ≡ R(gµν ) − M µν (Dαu )(Dβu ) , (2.4) α2, which measure the preferred frame effects. In terms where Dµ denotes the covariant derivative with respect of the four dimensionless coupling constants ci’s of the αβ æ-theory, they are given by [88], to gµν , and M µν is defined as

αβ αβ α β α β α β 8 (c1c14 − c−c13) M µν ≡ c1g gµν + c2δµ δν + c3δν δµ − c4u u gµν . α = − , 1 2c − c c (2.5) 1 − 13 1 (c14 − 2c13) (3c2 + c13 + c14) α2 = α1 + , (2.13) Note that here we assume that matter fields couple not 2 c123(2 − c14) µ only to gµν but also to the aether field u . However, in order to satisfy the severe observational constraints, such where c− ≡ c1 −c3. In the weak-field regime, using lunar a coupling in general is assumed to be absent [75]. laser ranging and solar alignment with the ecliptic, Solar The four coupling constants ci’s are all dimensionless, System observations constrain these parameters to very and Gæ is related to the Newtonian constant GN via the small values [86], relation [85], −4 −7 |α1| ≤ 10 , |α2| ≤ 10 . (2.14) G G = æ . (2.6) N 1 − 1 c Recently, the combination of the GW event GW170817 2 14 [89], observed by the LIGO/Virgo collaboration, and the

The variations of the total action with respect to gµν , event of the gamma-ray burst GRB 170817A [90] pro- uµ and λ yield, respectively, the field equations, vides a remarkably stringent constraint on the speed of −15 −16 the spin-2 mode, −3×10 < cT −1 < 7×10 , which, 1 Rµν − gµν R − Sµν = 8πG T µν , (2.7) together with Eq.(1.1), implies that 2 æ −15 Æµ = 8πGæTµ, (2.8) |c13| < 10 . (2.15) α β gαβu u = −1, (2.9) Requiring that the theory: (a) be self-consistent, such as free of ghosts and instability; and (b) satisfy all the where Rµν denotes the Ricci tensor, and observational constraints obtained so far, it was found h µ µ µi that the parameter space of the theory is considerably Sαβ ≡ Dµ J uβ) + J(αβ)u − u(βJ (α α) restricted [71]. In particular, c14 and c2 are restricted to h µ µ i +c1 (Dαuµ)(Dβu ) − (Dµuα)(D uβ) −5 0 . c14 . 2.5 × 10 , (2.16) 1 0 c c 0.095. (2.17) +c a a + λu u − g J δ D uσ, . 14 . 2 . 4 α β α β 2 αβ σ δ α α The constraints on other parameters depend on the Æµ ≡ DαJ µ + c4aαDµu + λuµ, √ values of c14. If dividing the above range into three inter- µν 2 δ ( −gLm) −7 −7 −6 T ≡ √ , vals: (i) 0 . c14 ≤ 2×10 ; (ii) 2×10 < c14 . 2×10 ; −g δgµν −6 −5 √ and (iii) 2 × 10 . c14 . 2.5 × 10 , in the first and last 1 δ ( −gL ) intervals, one finds [71], √ m Tµ ≡ − µ , (2.10) −g δu −7 (i) 0 . c14 ≤ 2 × 10 , with c14 . c2 . 0.095, (2.18) −6 −5 α αβ ν µ α µ (iii) 2 × 10 c14 2.5 × 10 , J ≡ M Dβu , a ≡ u Dαu . (2.11) . . µ µν −7 0 . c2 − c14 . 2 × 10 . (2.19) From Eq.(2.8), we find that −7 −6 In the intermediate regime (ii) 2×10 < c14 2×10 , αβ 2 α . λ = uβDαJ + c4a − 8πGæTαu , (2.12) in addition to the ones given by Eqs.(2.16) and (2.17), the following constraints must be also satisfied, 2 λ where a ≡ aλa . It is easy to show that the Minkowski spacetime is a c (c + 2c c − c ) − 10−7 ≤ 14 14 2 14 2 ≤ 10−7. (2.20) solution of æ-theory, in which the aether is aligned along c2 (2 − c14) 0 the time direction,u ¯µ = δµ. Then, the linear perturbations around the Minkowski background show that the Note that in writing Eq.(2.20), we had set c13 = 0, for −15 theory in general possess three types of excitations, scalar which the errors are of the order O (c13) ' 10 , which 6 can be safely neglected for the current and forthcoming and experiments. The results in this intermediate interval of σ h 2i cˆ = 1 + σ−2 c + 1 − σ−2 c − 1 − σ−1 , c14 were shown explicitly by Fig. 1 in [71]. Note that in 1 2 1 3 this figure, the physically valid region is restricted only cˆ = σ c + 1 − σ−1 , to the half plane c ≥ 0, as shown by Eq.(2.16). 2 2 14 σ cˆ = 1 − σ−2 c + 1 + σ−2 c − 1 − σ−2 , Since the theory possesses three different modes, and 3 2 1 3 all of them are moving in different speeds, in general " σ 1 2 1 these different modes define different horizons [75]. These cˆ4 = c4 − 1 − c1 + 1 − c3 horizons are the null surfaces of the effective metrics, 2 σ σ2 # 1 2 (A) 2 − 1 − , (2.24) gαβ ≡ gαβ − cA − 1 uαuβ, (2.21) σ with σ being a positive otherwise arbitrary constant. where A = S, V, T . If a BH is defined to be a region that Then, the following useful relations between ci andc î traps all possible causal influences, it must be bounded by hold, a horizon corresponding to the fastest speed. Assuming that the matter sector always satisfies the Lorentz sym- cˆ2 = σ(c2 + 1) − 1, cˆ14 = c14, metry, we can see that in the matter sector the fastest cˆ13 = σ(c13 − 1) + 1, cˆ123 = σc123, speed will be the speed of light. Then, overall, the fastest cˆ = σ−1(c + σ − 1). (2.25) speed must be one of the three gravitational modes. − − αβ α β However, in the spherically symmetric case, the spin- Note thatg ˆ gˆβγ = δγ andu ˆα ≡ gˆαβuˆ . Then, from 1 and spin-2 modes are not excited, so only the spin-0 Eq.(2.23), we find that gravitons are relevant. Therefore, in the present paper α β the relevant horizons for the gravitational sector are the gˆαβuˆ uˆ = −1, gˆ = σg, (2.26) S0Hs 2. In order to avoid the existence of the vacuum whereg ˆ is the determinant ofg ˆαβ. Thus, replacing Gæ gravi-Cerenkovˇ radiation by matter such as cosmic rays and L by Gˆ and Lˆ in Eq.(2.3), where [74], we assume that c ≥ 1, so that S0Hs are always λ æ λ S √ inside or at most coincide with the metric horizons, the α β Gˆæ ≡ σGæ, Lˆλ ≡ λ gˆαβuˆ uˆ + 1 , (2.27) null surfaces defined by the metric gαβ. The equality happens only when cS = 1. we find that

α α Sæ (gαβ, u , ci,Gæ, λ) = Sˆæ gˆαβ, uˆ , cˆi, Gˆæ, λ . (2.28)

As a result, when the matter field is absent, that is, Lm = A. Field Redefinitions 0, the Einstein-aether vacuum field equations take the α same forms for the fields (ˆgαβ, uˆ , cî, λ),

Due to the specific symmetry of the theory, Foster µν 1 µν µν α Rˆ − gˆ Rˆ = Sˆ , (2.29) found that the action Sæ (gαβ, u , ci) given by Eqs.(2.3)- 2 (2.5) does not change under the following field redefini- Æˆ µ = 0, (2.30) tions [83], α β gˆαβuˆ uˆ = −1, (2.31)

α α ˆµν ˆ (gαβ, u , ci) → (ˆgαβ, uˆ , cî) , (2.22) where R and R are the Ricci tensor and scalar made µν ofg ˆαβ. Sˆ and Æˆ µ are given by Eq.(2.10) simply by µ µ replacing (gµν , u , ci) by (ˆgµν , uˆ , cî). where Therefore, for any given vacuum solution of the µ Einstein-aether field equations (gµν , u , ci, λ), using the 1 gˆ = g − (σ − 1)u u , uˆα = √ uα, above field redefinitions, we can obtain a class of the αβ αβ α β σ vacuum solutions of the Einstein-aether field equations, √ µ 3 αβ αβ −1 α β given by (ˆgµν , uˆ , cî, λ) . Certainly, such obtained solu- gˆ = g − (σ − 1)u u , uˆα = σuα, (2.23) tions may not always satisfy the physical and observational constraints found so far [71].

2 If we consider Hoˇrava gravity [79] as the UV complete theory of the hypersurface-orthogonal æ-theory (the khronometric theory) [91–94], even in the gravitational sector, the relevant boundaries 3 It should be noted that this holds in general only for the vacuum will be the UHs, once such a UV complete theory is taken into case. In particular, when matter presence, the aether ﬁeld will account [80]. be directly coupled with matter through the metric redeﬁnitions. 7

In this paper, we shall take advantage of such field Thus, by properly choosing c0, we can always eliminate redefinitions to simplify the corresponding mathematic one of the three parameters, c1, c3 and c4, or one of problems by assuming that the fields described by their combinations. Therefore, in this case only three µ (gµν , u , ci, λ) are the physical ones, while the ones de- combinations of ci’s appear in the field equations. Since µ scribed by (ˆgµν , uˆ , cî, λ) as the “effective” ones, although both of the two metrics are the vacuum solutions of the c¯13 = c13, c¯14 = c14, c¯2 = c2, (2.39) Einstein-aether field equations, and can be physical, pro- without loss of the generality, we can always choose these vided that the constraints recently given in [71] are sat- three combinations as c , c and c . isfied. 13 14 2 To understand the above further, and also see the phys- The gravitational sector described by (ˆg , uˆµ, cˆ , λ) µν i ical meaning of these combinations, following Jacobson has also three different propagation modes, with their [94], we first decompose Dβuα into the form, speedsc Â given by Eq.(1.1) with the replacement ci by cî. Each of these modes defines a horizon, which is now 1 D u = θh + σ + ω − a u , (2.40) a null surface of the metric, β α 3 αβ αβ αβ α β (A) 2 where θ denotes the expansion of the aether field, hαβ gˆαβ ≡ gˆαβ − cÂ − 1 uˆαuˆβ, (2.32) the spatial projection operator, σαβ the shear, which is where A = S, V, T . It is interesting to note that the symmetric trace-free part of the spatial projection of Dβuα, while ωαβ denotes the antisymmetric part of the c2 spatial projection of D u , defined, respectively, by cˆ2 = A . (2.33) β α A σ λ hαβ ≡ gαβ + uαuβ, θ ≡ Dλu , 2 Thus, choosing σ = cS, we havec ˆS = 1, and from 1 Eq.(2.32) we find that σαβ ≡ D(βuα) + a(αuβ) − θhαβ, 3 (S) 2 ωαβ ≡ D[βuα] + a[αuβ], (2.41) gˆαβ =g ˆαβ, σ = cS , (2.34) with (A, B) ≡ (AB + BA)/2 and [A, B] ≡ (AB − BA)/2. that is, the S0H of the metricg ˆ coincides with its MH. αβ Recall that aµ is the acceleration of the aether field, given Moreover, from Eqs.(2.21) and (2.23) we also find that by Eq.(2.11). In terms of these quantities, Jacobson found (S) 2 gαβ =g ˆαβ, σ = cS . (2.35) Z Z " 4 √ 4 √ 1 2 2 d x −gLæ = d x −g R − cθθ Therefore, with the choice σ = cS, the MH of gˆαβ is also 3 the S0H of the metric g . αβ # Arnowitt-Deser-Misner (ADM) 2 2 2 +caa − cσσ − cωω , (2.42)

B. Hypersurface-Orthogonal Aether Fields where

When the aether field uµ is hypersurface-orthogonal cθ ≡ c13 + 3c2, cσ ≡ c13, (HO), the Einstein-aether field equations depend only on cω ≡ c1 − c3, ca ≡ c14, (2.43) three combinations of the four coupling constants ci’s. To see this clearly, let us first notice that, if the aether and µ µ is HO, the twist ω vanishes [75], where ω is defined as 1 µ µναβ σ2 = − θ2 + (D u )(Dµuν ) + a2. (2.44) ω ≡ uν Dαuβ. Since 3 µ ν µ ν µ µ ν ωµω = (Dµuν )(D u ) − (Dµuν )(D u ) Note that in the above action, there are no crossing terms µ ν α of (θ, σ , ω , a ). This is because the four terms on − (u Dµuα)(u Dν u ) , (2.36) αβ αβ α the right-hand side of Eq.(2.40) are orthogonal to each we can see that the addition of the term other, and when forming quadratic combinations of these quantities, only their “squares” contribute [94]. µ ∆Læ ≡ c0ωµω , (2.37) From Eq.(2.43) we can see clearly that c14 is related to the acceleration of the aether field, c13 to its shear, to Læ will not change the action, where c0 is an arbitrary while its expansion is related to both c2 and c13. More real constant. However, this is equivalent to replacing ci interesting, the coefficient of the twist is proportional to byc ¯i in Læ, where c1 − c3. When uµ is hypersurface-orthogonal, we have ω2 = 0, so the last term in the above action vanishes c¯ ≡ c + c , c¯ ≡ c , 1 1 0 2 2 identically, and only the three free parameters cθ, cσ and c¯3 ≡ c3 − c0, c¯4 ≡ c4 − c0. (2.38) ca remain. 8

It is also interesting to note that the twist vanishes if where dΩ2 ≡ dθ2 + sin2 θdφ2 and xµ = (v, r, θ, φ), while and only if the four-velocity of the aether satisfies the the aether field takes the general form, conditions [95], 2 α 1 − F (r)A (r) u ∂α = A(r)∂v − ∂r, (3.2) u[µDν uα] = 0. (2.45) 2B(r)A(r) which is respect to the spherical symmetry, and satisfies When the aether is HO, it can be shown that Eq.(2.45) is α satisfied. In addition, in the spherically symmetric case, the constraint uαu = −1. Therefore, in the current Eq.(2.45) holds identically. case, we have three unknown functions, F (r),A(r) and B(r). Moreover, it can be also shown [95] that Eq.(2.45) is Then, the vacuum field equations Eµν ≡ Gµν −Sµν = 0 the necessary and sufficient condition to write the four- and Æµ = 0 can be divided into two groups [63, 67]: one velocity u in terms the gradient of a timelike scalar field µ represents the evolution equations, given by φ, Evv = Eθθ = Æv = 0, (3.3) φ u = ,µ . (2.46) µ p ,α −φ,αφ and the other represents the constraint equation, given by

Substituting it into the action (2.42), one obtains the v action of the infrared limit of the healthy extension [91, C = 0, (3.4) 92] of the Hoˇrava theory [79], which is often referred to as where Cα ≡ Erα + urÆα = 0, and 4 the khronometric theory , where φ is called the khronon Gµν [≡ Rµν − Rgµν /2] denotes the Einstein tensor. field. Note that in Eq.(35) of [67] two constraint equations It should be noted that the khronometric theory and Cv = Cr = 0 were considered. However, Cr and Cv are the HO æ-theory are equivalent only in the action level. not independent. Instead, they are related to each other In particular, in addition to the scalar mode, the khrono- by the relation Cr = (F/B)Cv. Thus, Cv = 0 implies metric theory has also an instantaneous mode [66, 96], a Cr = 0, so there is only one independent constraint. On mode that propagates with an infinitely large speed. This the other hand, the three evolution equations can be is mainly due to the fact that the field equations of the cast in the forms 6, khronometric theory are the four-order differential equa- 00 0 0 tions of φ. It is the presence of those high-order terms F = F (A, A ,F,F , r, ci) that lead to the existence of the instantaneous mode 5. 1 = f + f F + f F 2 + f F 3 On the other hand, in æ-theory, including the case with 2r2A4D 0 1 2 3 the HO symmetry, the field equations are of the second 4 + f4F , (3.5) order for both the metric gµν and the aether field uµ. As a result, this instantaneous mode is absent. For more 00 0 0 A = A (A, A ,F,F , r, ci) details, we refer readers to [80] and references therein. 1 = a + a F + a F 2 + a F 3 , (3.6) 2r2A2D 0 1 2 3 0 B 0 0 III. SPHERICALLY SYMMETRIC VACUUM = B (A, A ,F,F , r, ci) SPACETIMES B 1 2 = 2 b0 + b1F + b2F , (3.7) µ 2rA D A. Field Equations for gµν and u where a prime stands for the derivative with respect to As shown in the last section, to be consistent with ob- r, and servations, we must assume c ≥ 1. As a result, S0Hs S D ≡ d J 2 + 1 + 2d J, (3.8) must be inside MHs. Since now S0Hs define the bound- − + aries of spherically symmetric BHs, in order to cover with J ≡ FA2 and spacetimes both inside and outside the MHs, one way 2 is to adopt the Eddington-Finkelstein (EF) coordinates, d± ≡ (cS ± 1)c14(1 − c13)(2 + c13 + 3c2). (3.9)

2 µ ν ds ≡ gµν dx dx = −F (r)dv2 + 2B(r)dvdr + r2dΩ2, (3.1) 6 It should be noted that in [67] the second-order differential equation for F [cf. Eq.(36) given there] also depends on B. But, since from the constraint Cv = 0, given by Eq.(3.10), one can express B in terms of A, F and their derivatives, as shown explicitly 4 In [93, 94], it was also referred to as T-theory. by Eq.(3.11), so there are no essential differences here, and it 5 In the Degenerate Higher-Order Scalar-Tensor (DHOST) theo- should only reflect the facts that different combinations of the ries, this mode is also referred to as the “shadowy” mode [97]. field equations are used. 9

The coeﬃcients fn, an and bn are independent of F (r) To solve Eqs.(3.5) and (3.6), we can consider them as and B(r) but depend on F 0(r), A(r) and A0(r), and are the “initial” value problem at a given “moment”, say, given explicitly by Eqs.(A.1), (A.2) and (A.3) in Ap- r = r0 [63, 67]. Since they are second-order diﬀerential pendix A. The constraint equation (3.4) now can be cast equations, the initial data will consist of the four initial in the form, values,

2 n 0 0 o n0 + n1F + n2F = 0, (3.10) A(r0),A (r0),F (r0),F (r0) . (3.12) where nn’s are given explicitly by Eq.(A.4) in Appendix In principle, r0 can be chosen as any given (finite) mo- A. ment. However, in the following we shall show that the Thus, we have three dynamical equations and one con- most convenient choice will be the locations of the S0Hs. straint for the three unknown functions, F,A and B. As It should be noted that a S0H does not always exist for a result, the system seems over determined. However, any given initial data. However, since in this paper we a closer examination shows that not all of them are in- are mainly interested in the case in which a S0H exists, dependent. For example, Eq.(3.7) can be obtained from so whenever we choose r0 = rS0H , it always means that Eqs.(3.5), (3.6), and (3.10). In fact, from Eq.(3.10), we we only consider the case in which such a S0H is present. find that the function B can be written in the form To determine the location of the S0H for a given ( spherical solution of the metric (3.1), let us first con- 1 2h B(r) = ± √ 2A 4J(1 + 2c2 + c13) sider the out-pointing normal vector, Nµ, of a hyper- 2 2 2A surface r = Constant, say, r0, which is given by Nµ ≡ µ r 2i ∂(r−r0)/∂x = δµ. Then, the metric and spin-0 horizons − (2c2 + c13) J + 1 of gµν are given, respectively, by h 0 0 +4rA 2AJ − 4JA α β gαβN N = 0, (3.13) 0 0 0i (S) α β + c2 J − 1 JA − A − AJ gαβ N N = 0, (3.14) 2h 0 0 02 +r c14 JA + A − AJ µ µν (S) where N ≡ g Nν , and gαβ is defined by Eq.(1.3). For )1/2 the metric and aether given in the form of Eqs.(3.1) and 0 0 02i − (c2 + c13) JA − A − AJ . (3.11) (3.2), they become

F (rMH ) = 0, (3.15) 2 2 2 2 Recall that J = FA . Note that there are two branches cS − 1 J(rS0H ) + 1 + 2 cS + 1 J(rS0H ) = 0, of solutions for B(r) with opposite signs, since Eq.(3.10) (3.16) is a quadratic equation of B. However, only the “+” sign will give us B = 1 at the spatial infinity, while the “-” sign where r = rMH and r = rS0H are the locations of will yield B(r → ∞) = −1. Therefore, in the rest of the the metric and spin-0 horizons, respectively. Note that paper, we shall choose the “+” sign in Eq.(3.11). Then, i Eqs.(3.15) and (3.16) may have multiple roots, say, rMH first taking the derivative of Eq.(3.11) with respective to j and rS0H . In these cases, the location of the metric (spin- r, and then combining the obtained result with Eqs.(3.5) 0) horizon is always taken to be the largest root of ri and (3.6), one can obtain Eq.(3.7) 7. MH (rj ). To solve these equations, in this paper we shall adopt S0H Depending on the value of c , the solutions of Eq.(3.14) the following strategy: choosing Eqs.(3.5), (3.6) and S are given, respectively, by (3.11) as the three independent equations for the three unknown functions, F , A, and B. The advantage of this 1 ∓ c J(r± ) = S ≡ J ±, c 6= 1, (3.17) choice is that Eqs.(3.5) and (3.6) are independent of the S0H 1 ± c S function B. Therefore, we can first solve these two equa- S tions to find F and A, and then obtain the function B and directly from Eq.(3.11). In this approach, we only need to solve two equations, which will significantly save the J(rS0H ) = 0, cS = 1. (3.18) computation labor, although we do use Eq.(3.7) to monitor our numerical errors. It is interesting to note that on S0Hs, we have

D(rS0H ) = 0, (3.19)

as it can be seen from Eqs.(3.8), (3.9) and (3.16). 7 From this proof it can be seen that obtaining Eq.(3.7) from Eq.(3.11) the operation of taking the ﬁrst-order derivatives was As mentioned above, for some choices of ci, Eq.(3.14) involved. Therefore, in principle these two equations are equiv- does not always admit a solution, hence a S0H does not alent modulated an integration constant. exist in this case. A particular choice was considered in 10

[63], in which we have c1 = 0.051, c2 = 0.116, c3 = −c1 Then, we find that Eqs.(3.5)-(3.6) now reduce to and c4 = 0. For this choice, we find that cS ' 1.37404, J + ' −0.157556 and J − ' −6.34696. As shown in Fig. 2 c Fˆ(r) F 00 = − F 0 + 2 , (3.21) 1, the function J(r) is always greater than J ±, so no S0H r 4r2A4 is formed, as first noticed in [63]. Up to the numerical 2 h A00 = r2(A0)2 − rAA0 − A2 errors, Fig.1 is the same as that given in [63], which r2(A + A3F ) provides another way to check our general expressions of 3 0 0 4 i the field equations given above. −rA A (F + rF ) + A F c Fˆ(r) In addition, we also find that the two exact solutions − 2 , (3.22) obtained in [53] satisfy these equations identically, as it 4r2(A + A3F ) is expected. where

2 Fˆ(r) ≡ rA0 − 2A + rA2A0F + A3 (2F + rF 0) . (3.23) 4 Combining Eqs.(3.21) and (3.22), we ﬁnd the following F A B J equation, 3 J+ 2 2 W 00 + W 02 + W 0 − = 0, (3.24) r r2 2 where 1 − FA2 1 W ≡ ln . (3.25) A

0 Eq.(3.24) has the general solution, 3 0 1 2 3 4 5 6 1 + w1r W = ln w2 + ln 2 , (3.26) r/rMH r

FIG. 1: The solution for c1 = 0.051, c2 = 0.116, c3 = −c1 where w1 and w2 are two integration constants. Then, and c4 = 0, ﬁrst considered numerically in [63]. There are an the combination of Eqs.(3.25) and (3.26) yields, outer and inner MHs, at which F vanishes. But, J does not cross the constant line of J +, so that a S0H is absent. This 1 w 1 F (r) = − 2 + w r . (3.27) graph is the same as the one given in [63] (up to the numerical A2 A r2 1 errors). Substituting Eq.(3.27) into Eq.(3.21), we ﬁnd 2 F 00 = − F 0 + F , (3.28) r 0

2 2 where F0 ≡ 9c2w1w2/4. Integrating Eq.(3.28), we ﬁnd 2m F0 2 F (r) = F2 1 − + r , (3.29) B. Exact Solutions with c14 = c13 = 0 r 6

where m and F2 are two other integration constants. On From Eqs.(2.15)-(2.20) we can see that the choice the other hand, from Eq.(3.27), we find that c = c = 0 satisfies these constraints, provided that c 14 13 2 " satisfies the condition 8, w 1 A(r) = − 2 + w r 2F r2 1 s  2 0 . c2 . 0.095. (3.20) 4F 1 ± 2 + 2 + w1r  . (3.30) w2 r

8 Substituting the above expressions for A and F into the When c14 = c13 = 0, the speeds of the spin-0 and spin-1 modes can be inﬁnitely large, as it can be seen from Eq.(1.1). Then, constraint (3.11), we ﬁnd that cautions must be taken, including the calculations of the PPN p parameters [88]. B = F2. (3.31) 11

Note that the above solution is asymptotically ﬂat only where C0 is an arbitrary real constant, and C(r) is a when w1 = 0, for which we have function of r. Then, choosing C(r) so that

2 √ 2m p dC(r) 2C0A B ( σ − 1) F (r) = F2 1 − ,B(r) = F2, = √ √ , (3.37) r dr J ( σ − 1) + ( σ + 1) s ! w2 1 4F 1 we ﬁnd that in the coordinatesx ˜µ = (˜v, r, θ, φ) the line A(r) = − 2 ± 2 + 4 . (3.32) 2F r w2 r element (3.35) takes the form,

0 2 µ ν µ ν Using the gauge residual v = C0v + C1 of the metric dsˆ ≡ gˆµν dx dx =g ˜µν dx˜ dx˜ (3.1), without loss of the generality, we can always set = −F˜(r)dv˜2 + 2B˜(r)dvdr˜ + r2dΩ2, (3.38) F2 = 1, so the corresponding metric takes the precise form of the Schwarzschild solution, where 2 2m 2 2 2 J 2(σ − 1) + 2J(σ + 1) + (σ − 1) ds = − 1 − dv + 2dvdr + r dΩ , (3.33) F˜ = , r 4C2A2 √ 0 while the aether field is given by σB B˜ = . (3.39) C w ± pw2 + 4r3(r − 2m) 0 A(r) = − 2 2 . (3.34) 2r(r − 2m) On the other hand, in terms of the coordinatesx ˜µ, the aether four-velocity is given by It is remarkable to note that now the aether field has no contribution to the spacetime geometry, although it ˜ ˜2 α ∂ α ∂ 1 − F (r)A (r) uˆ =u ˜ = A˜(r)∂v˜ − ∂r, (3.40) does feel the gravitational field, as it can be seen from ∂xα ∂x˜α 2B˜(r)A˜(r) Eq.(3.34). It should be also noted that Eqs.(3.33) and (3.34) were where a particular case of the solutions first found in [53] for the 2C0A case c14 = 0 by further setting c13 = 0. But, the general A˜ = √ √ , (3.41) solutions given by Eqs.(3.29)-(3.31) are new, as far as J ( σ − 1) + ( σ + 1) we know. α β which satisfies the constraintu ˜ u˜ g˜αβ = −1, with

µ ∂xα ∂xβ ∂xα C. Field Equations for g˜µν and u˜ g˜ ≡ gˆ , u˜ ≡ uˆ . (3.42) µν ∂x˜µ ∂x˜ν αβ µ ∂x˜µ α Note that, instead of solving the three independent It should be noted that the metric (3.38) still has the equations directly for A, B and F , we shall first solve gauge residual, the corresponding three equations for A˜, B˜ and F˜, by taking the advantage of the field redefinitions introduced v˜ = C1v˜ + C2, (3.43) in the last section, and then obtain the functions A, B and F by the inverse transformations of Eqs.(3.39) and where C1 and C2 are two arbitrary constants, which will (3.41) to be given below. This will considerably simplify keep the line element in the same form, after the rescal- mathematically the problem of solving such complicated ing, equations. ˜ ˜ To this goal, let us first note that, with the filed redef- ˜ F ˜ B F = 2 , B = . (3.44) initions (2.23), the line element corresponding tog ˆµν in C1 C1 the coordinates (v, r, θ, φ), takes the form, Later we shall use this gauge freedom to fix one of the 2 µ ν dsˆ ≡ gˆµν dx dx initial conditions. " 2 # α (σ − 1) A2F + 1 In the rest of this paper, we always refer (˜gµν , u˜ ) = − F + dv2 as the field obtained by the field redefinitions. The 4A2 α latter is related to (ˆgµν , uˆ ) via the inverse coordinate 1 transformations of Eq.(3.42). Then, the Einstein-aether +2 B + (σ − 1)B A2F + 1 dvdr α 2 field equations for (˜gµν , u˜ ) will take the same forms as 2 2 2 2 2 those given by Eqs.(2.29)-(2.31), but now in terms of −(σ − 1)A B dr + r dΩ . (3.35) α µ (˜gµν , u˜ , c˜i) in the coordinatesx ˜ , wherec ˜i ≡ cî. To bring the above expression into the standard EF form, On the other hand, since the metric (3.38) forg ˜µν takes we first make the coordinate transformation, the same form as the metric (3.1) for gµν , and so does the aether field (3.40) foru ˜µ as the one (3.2) for uµ, it is v˜ = C0v − C(r), (3.36) not difficult to see that the field equations for F˜(r), A˜(r) 12 and B˜(r) will be given precisely by Eqs.(3.5)-(3.10), if Comparing the field equations given in this subsection we simply make the following replacement, with the corresponding ones given in the last subsection, we see that we can get one set from the other simply by (F, A, B, ci) → F,˜ A,˜ B,˜ c˜i . (3.45) the replacement (3.45). In addition, in terms ofg ˆαβ and Nα, Eqs.(3.51) and As a result, we have (3.52) reduce, respectively, to α β 00 0 0 gˆαβN N = 0, (3.57) F˜ = F˜ A,˜ A˜ , F,˜ F˜ , r, c˜i (S) α β gˆαβ N N = 0. (3.58) 1 h ˜ ˜ ˜ 2 ˜ 3 = f0 + f1F˜ + f2F˜ + f3F˜ 2r2A˜4D˜ Sincer ˜ = r, we find that ˜ ˜4i + f4F , (3.46) r˜MH =r ˆMH , r˜S0H =r ˆS0H , (3.59)

00 0 0 wherer ˜ andr ˜ (ˆr , rˆ ) are the locations of A˜ = A˜ A,˜ A˜ , F,˜ F˜ , r, c˜i MH S0H MH S0H the metric and spin-0 horizons of the metricg ˜αβ (ˆgαβ). 1 h 2 3i The above analysis shows that these horizons determined = a˜0 +a ˜1F˜ +a ˜2F˜ +a ˜3F˜ , (3.47) 2 2 2r A˜ D˜ by g˜αβ are precisely equal to those determined by gˆαβ. B˜0 = B˜ A,˜ A˜0, F,˜ F˜0, r, c˜ ˜ i B 2 D. σ = cS 1 h˜ ˜ ˜ 2i = b0 + b1F˜ + b2F˜ , (3.48) 2rA˜2D˜ To solve Eqs.(3.46)-(3.49), we take the advantage of 2 and the choice σ = cS, so that the speed of the spin-0 mode of the metricg ˆµν becomes unity, i.e.,c ˆS = 1. Sincec ˜i =c î, ˜v˜ ˜ ˜2 C ≡ n˜0 +n ˜1F +n ˜2F = 0, (3.49) we also havec ˜S =c ˆS = 1. Then, from Eq.(1.1) we find that this leads to, where 2 2˜c14 − 2˜c13 − c˜13c˜14 ˜ ˜ ˜2 ˜ ˜ c˜2 = . (3.60) D(r) ≡ d− J (r) + 1 + 2d+J(r), 2 − 4˜c14 + 3˜c13c˜14 ˜ ˜ ˜2 ˜ J(r) ≡ F (r)A (r), For such a choice, from Eq.(3.50) we find that d− = 0, ˜ 2 and d± ≡ (˜cS ± 1)˜c14(1 − c˜13)(2 +c ˜13 + 3˜c2).(3.50) ˜ ˜ ˜ ˜ ˜2 ˜ ˜ ˜ D(r) = 2d+J(r) = 2d+A (r)F (r). (3.61) The coefficients fn, añ, bn andn ñ are given by fn, an, bn and nn after the replacement (3.45) is car- Then, Eq.(3.56) yields F˜(˜rS0H ) = 0, since A˜ 6= 0, ried out. which also represents the location of the MH, defined Then, the metric and spin-0 horizons forg ˜ are given, 2 µν by Eq.(3.54). Therefore, for the choice σ = cS the MH respectively, by coincides with the S0H for the effective metricg ˜µν , that α β is, g˜αβN˜ N˜ = 0, (3.51) 2 (S) ˜ α ˜ β r˜MH =r ˜S0H , σ = cS . (3.62) g˜αβ N N = 0, (3.52) As shown below, this will significantly simplify our com- ˜ µ α r r where Nα = (∂x /∂x˜ )Nµ = δα˜ = δα and putational labor. In particular, if we choose this surface as our initial moment, it will reduce the phase space of (S) 2 g˜αβ ≡ g˜αβ − c˜S − 1 u˜αu˜β. (3.53) initial data from 4 dimensions to one dimension only. Forc ˜S = 1, Eqs.(3.46)-(3.49) reduce to, In terms of F˜ and A˜, Eqs.(3.51) and (3.52) becomes, ˜ 00 1 f0 ˜ ˜ ˜ 2 F˜ = + f1 + f2F˜ + f3F˜ ˜ ˜ 2 6 ˜ F (˜rMH ) = 0, (3.54) 4d+r A˜ F 2 ˜ 2 2 ˜ ˜ 3 c˜S − 1 J(˜rS0H ) + 1 + 2 c˜S + 1 J(˜rS0H ) = 0, +f4F˜ , (3.63) (3.55) 00 1 a˜0 2 A˜ = +a ˜1 +a ˜2F˜ +a ˜3F˜ , (3.64) ˜ 2 4 ˜ 4d+r A˜ F where r =r ˜MH and r =r ˜S0H are respectively the locations of the metric and spin-0 horizons for the metric ˜0 ˜ ! B 1 b0 ˜ ˜ = + b1 + b2F˜ , (3.65) g˜µν . Similarly, at r =r ˜S0H we have ˜ ˜ 4 ˜ B 4d+rA˜ F 2 D˜(˜rS0H ) = 0. (3.56) n˜0 +n ˜1F˜ +n ˜2F˜ = 0. (3.66) 13

As shown previously, among these four equations, only (3.73). For each set of (A˜H , r˜S0H ), in general there are three of them are independent, and our strategy in this eight sets of A˜0 , F˜0 . paper is to take Eqs.(3.63), (3.64) and (3.66) as the three H H If we choose r =r ˜ as the initial moment, such independent equations. The advantage of this approach S0H ˜ ˜0 ˜0 ˜ is that Eqs.(3.63), (3.64) are independent of B(r), and obtained AH , FH , together with FH = 0, and a proper Eq.(3.66) is a quadratic polynomial of B˜(r). So, we choice of A˜H , can be considered as the initial conditions can solve Eqs.(3.63), (3.64) as the initial value problem for the differential equations (3.63) and (3.64). ˜ ˜ first to find F (r) and A(r), and then insert them into However, it is unclear which one(s) of these eight sets Eq.(3.66) to obtain directly B˜(r), as explicitly given by of initial conditions will lead to asymptotically flat so- Eq.(3.11), after taking the replacement (3.45) and the ˜0 lutions, except that the one with FH < 0, which can be choice ofc ˜2 of Eq.(3.60) into account. discarded immediately, as it would lead to F˜ = 0 at some From Eqs.(3.63) and (3.64) we can see that they be- radius r > r˜ , which is inconsistent with our assump- ˜ S0H come singular at r =r ˜S0H (Recall F (˜rS0H ) = 0), un- tion that r =r ˜S0H is the location of the S0H [67]. So, in ˜ less f0(˜rS0H ) =a ˜0(˜rS0H ) = 0. As can be seen from general what one needs to do is to try all the possibilities. the expressions of f0(r), a0(r) given in Appendix A, Therefore, if we choose r =r ˜S0H as the initial mo- ˜ ˜ f0(˜rS0H ) =a ˜0(˜rS0H ) = 0 imply b0(˜rS0H ) = 0. There- ment, the four-dimensional phase space of the initial con- fore, to have the field equations regular across the S0H, 0 0 ditions, F˜H , F˜ , A˜H , A˜ , reduces to one-dimensional, we must require ˜b (˜r ) = 0. It is interesting that this is H H 0 S0H ˜ also the condition for Eq.(3.65) to be non-singular across spanned by AH only. the S0H. In addition, using the gauge residual (3.43), we In the following, we shall show further thatr ˜S0H can be chosen arbitrarily. In fact, introducing the dimensionless shall set B˜H = 1, so Eq.(3.66) [which can be written in the form of Eq.(3.11), after the replacement (3.45)] will quantity, ξ ≡ r˜S0H /r, we find that Eqs. (3.63)-(3.65) ˜0 ˜ and (3.49) can be written in the forms, provide a constraint among the initial values of FH , AH 0 0 0 and A˜ , where F˜ ≡ F˜ (˜rS0H ) and so on. In summary, H H d2F˜(ξ) on the S0H we have the following = G (ξ, c˜ ) , (3.74) dξ2 1 i F˜ = 0, (3.67) H d2A˜(ξ) ˜ ˜ ˜0 ˜0 2 = G2 (ξ, c˜i) , (3.75) b0 AH , AH , FH , r˜S0H = 0, (3.68) dξ ˜ B˜ = 1. (3.69) 1 dB(ξ) H = G3 (ξ, c˜i) , (3.76) B˜(ξ) dξ From the expression for ˜b given in Appendix A, we can 0 v˜ 0 0 ˜0 C A˜(ξ), A˜ (ξ), F˜(ξ), F˜ (ξ), B˜(ξ), ξ, c˜i = 0, (3.77) see that Eq.(3.68) is quadratic in AH , and solving it on the S0H, in general we obtain two solutions, v˜ 2 v˜ ± ± where G ’s are all independent ofr ˜ , C ≡ r C˜ , ˜0 ˜0 ˜ ˜0 i S0H S0H A H = A H AH , FH , r˜S0H . (3.70) and the primes in the last equation stand for the derivatives respect to ξ. Therefore, Eqs.(3.74)-(3.77), or equiv- Then, inserting it, together with Eqs.(3.67) and (3.69), alently, Eqs.(3.46)-(3.49), are scaling-invariant and inde- into Eq.(3.10), we get pendent ofr ˜S0H . Thus, without loss of the generality, we ± ˜ ˜0 can always set n˜0 AH , FH , r˜S0H = 0, (3.71) r˜ = 1, (3.78) 0 S0H where the “±” signs correspond to the choices of A˜ H = ± ˜0 A H . In general, Eq.(3.71) is a fourth-order polynomial which does not affect Eqs.(3.74)-(3.77), and also ex- ˜0 of FH , so it normally has four roots, denoted as plains the reason why in [63, 67] the authors setr ˜S0H = 1 directly. At the same time, it should be noted that once (±,n) (±,n) ˜0 ˜0 ˜ F H = F H AH , r˜S0H , (3.72) r˜S0H = 1 is taken, it implies that the unit of length is fixed. For instance, if we have a BH withr ˜S0H = 1 km, (±,n) ˜0 then settingr ˜S0H = 1 means the unit of length is in km. where n = 1, 2, 3, 4. For each given F H , substituting (±,n) Once A˜ is chosen, we can integrate Eqs.(3.74) and it into Eq.(3.70) we find a corresponding A˜0 , given H H (3.75) in both directions to find F˜(ξ) and A˜(ξ), one is by toward the center, ξ =r ˜S0H /r = ∞, in which we have ξ ∈ (±,n) (±,n) [1, ∞), and the other is toward infinity, ξ =r ˜ /r = 0, A˜0 = A˜0 A˜ , r˜ . (3.73) S0H H H H S0H in which we have ξ ∈ (0, 1]. Then, from Eq.(3.11) we can ˜ (±,n) find B(ξ) uniquely, after the replacement of Eq.(3.45). ˜ ˜0 Thus, once AH andr ˜S0H are given, the quantities F H Again, to have a proper asymptotical behavior of B˜(r), (±,n) ˜0 and A H are uniquely determined from Eqs.(3.72) and the “+” sign will be chosen. 14

At the spatial inﬁnity ξ =r ˜S0H /r → 0, we require that However, it is worth emphasizing again that, for the 9 2 the spacetime be asymptotically ﬂat, that is [63, 67] , choice σ = cS we havec ˜S =c ˆS = 1, so the metric and spin-0 horizons of bothg ˆαβ andg ˜αβ all coincide, and are 1 3 3 given by the same rH , as explicitly shown by Eq.(3.81). F˜(ξ) = 1 + F˜1ξ + c˜14F˜ ξ + ··· , 48 1 More importantly, it is also the location of the S0Hs of

1 1 1 the metric gαβ. A˜(ξ) = 1 − F˜ ξ + A˜ ξ2 − c˜ F˜3 2 1 2 2 96 14 1 ! 1 1 IV. NUMERICAL SETUP AND RESULTS − F˜3 + F˜ A˜ ξ3 + ··· , 16 1 2 1 2 A. General Steps ˜ 1 ˜2 2 1 ˜3 3 B(ξ) = 1 + c˜14F1 ξ − c˜14F1 ξ + ··· ,(3.79) 16 12 It is difficult to find analytical solutions to Eqs.(3.74)- 0 00 (3.77). Thus, in this paper we are going to solve them where F˜1 ≡ F˜ (ξ = 0) and A˜2 ≡ A˜ (ξ = 0). It should be noted that the Minkowski spacetime is numerically, using the shooting method, with the asymp- given by totical conditions (3.79). In particular, our strategy is the following: q (i) Choose a set of physical c ’s satisfying the con- ˜ ˜ ˜ 1 ˜ ˜ i F = FM , A = p , B = FM , (3.80) straints (2.16)-(2.20), and then calculate the correspond- F˜M 2 ingc ˜i’s with σ = cS. (ii) Assume that for such chosen c ’s the corresponding where F˜ is a positive otherwise arbitrary constant. i M solution possesses a S0H located at r = r , and then Therefore, in the asymptotical expansions of Eq.(3.79), H follow the analysis given in the last section to impose the we had set F˜M = 1 at the zeroth order of ξ. However, conditions F˜H = 0 and B˜H = 1. the initial conditions imposed at r =r ˜S0H given above (iii) Choose a test value for A˜ , and then solve usually leads to F˜ 6= 1, even for spacetimes that are H M Eq.(3.68) for A˜0 in terms of F˜0 and A˜ , i.e., A˜0 = asymptotically flat. Therefore, we need first to use the H H H H ˜0 ˜0 ˜ gauge residual (3.43) to bring F˜(ξ = 0) = A˜(ξ = 0) = AH (FH , AH ). ˜0 B˜(ξ = 0) = 1, before using Eq.(3.79) to calculate the (iv) Substitute AH into Eq.(3.71) to obtain a quartic equation for F˜0 and then solve it to find F˜0 . constants A˜2 and F˜1. H H ˜ ˜ ˜0 ˜0 From the above analysis we can see that finding spher- (v) With the initial conditions {FH , AH , FH , AH }, in- ically symmetric solutions of the æ-theory now reduces to tegrate Eqs.(3.74) and (3.75) from ξ = 1 to ξ = 0. finding the initial condition A˜H that leads to the asymp- However, since the field equations are singular at ξ = 1, totical behavior (3.79), for a given set of ci’s. we will actually integrate these equations from ξ = 1 − Before proceeding to the next section, we would like to ξ ' 0, where is a very small quantity. To obtain the to recall that when σ = c2 , we have g(S) =g ˆ , as values of z(ξ) at ξ = 1 − , we first Taylor expand them S αβ αβ in the form, shown by Eq.(2.35). That is, the S0H for the metric gαβ now coincides with the MH ofg ˆαβ. With this same very 2 (k) 2 X z |ξ=1 k k 3 choice, σ = cS, the MH forg ˆαβ also coincides with its (1 − ) = (−1) + O , (4.1) z k! S0H. Thus, we have k=0 r =r ˆ =r ˆ =r ˜ =r ˜ ≡ r , σ = c2 . n o S0H S0H MH S0H MH H S where z ≡ A,˜ A˜0, F,˜ F˜0 and z(k) ≡ dkz/dξk. For (3.81) each z, we shall expand it to the second order of , so the errors are of the order 3. Thus, if we choose = 10−14, It must be noted that r defined in the last step denotes H the errors in the initial conditions (1−) is of the order the location of the S0H of g , which is usually different z αβ −42 ˜ ˜ from its MH, defined by 10 . For z = A, F , we already obtained z(1) and z0(1) from the initial conditions. In these cases, to get α β ˜00 ˜00 gαβN N = 0, (3.82) A (1) and F (1), we use the field equations (3.74) and r=rMH (3.75) and L’Hospital’s rule. On the other hand, for z = (S) A˜0, expanding it to the second order of , we have since in general we have cS 6= 1, so gαβ ≡ gαβ − 2 cS − 1 uαuβ 6= gαβ. As a result, we have rMH 6= rS0H 1 A˜0(1 − ) = A˜0(1) − A˜00(1) + A˜(3)(1)2 + O 3 , (4.2) for cS 6= 1. 2

where A˜(3)(1) ≡ d3A˜(ξ)/dξ3 can be obtained by ξ=1 9 Note that in [63, 67] a factor 1/2 is missing in front of A2 in the ﬁrst taking the derivative of Eq.(3.75) and then tak- expression of A(x). ing the limit ξ → 1, as now we have already known 15

A˜(1), A˜0(1), A˜00(1), F˜(1), F˜0(1) and F˜00(1). Similarly, for TABLE I: The cases considered in [63, 67] for variousc ˆ with = F˜0, from Eq.(3.74) we can find F˜(3)(1). 1 z the choice of the parametersc ˆ2,c ˆ3 andc ˆ4 given by Eq.(4.3). (vi) Repeat (iii)-(v) until a numerical solution matched Note that for each physical quantity, we have added two more to Eq.(3.79) is obtained, by choosing different values of digits, due to the improved accuracy of our numerical code. A˜ with a bisectional search. Clearly, once such a value H ˜0 ˜2 cˆ1 r˜g/rH FH AH γ˜ff of A˜H is found, it means that we obtain numerically an asymptotically flat solution of the Einstein-aether field 0.1 0.98948936 2.0961175 1.6028048 equations outside the S0H. Note that, to guarantee that Eq.(3.79) is satisfied, the normalization of {F,˜ A,˜ B˜} need 0.2 0.97802140 2.0716798 1.5769479 to be done according to Eq.(3.80), by using the remaining 0.3 0.96522924 2.0391972 1.5476848 gauge residual of Eq.(3.43). 0.4 0.95054650 1.9965155 1.5140905 (vii) To obtain the solution in the internal region ξ ∈ 0.5 0.93304411 1.9405578 1.4748439 (1, ∞), we simply integrate Eqs.(3.74) and (3.75) from 0.6 0.91106847 1.8666845 1.4279611 ξ = 1 to ξ → ∞ with the same value of A˜H found in the 0.7 0.88131278 1.7673168 1.3702427 last step. As in the region ξ ∈ (0, 1), we can’t really set 0.8 0.83583029 1.6283356 1.2959142 the “initial” conditions precisely at ξ = 1. Instead, we 0.9 0.74751927 1.4155736 1.1921231 will integrate them from ξ = 1 + to ξ = ξ∞ 1. The 0.91 0.73301185 1.3870211 1.1790400 initial values at ξ = 1 + can be obtained by following 0.92 0.71650458 1.3563710 1.1652344 what we did in Step (v), that is, Taylor expand z(ξ) at ξ = 1 + , and then use the field equations to get all the 0.93 0.69745439 1.3232418 1.1506047 quantities up to the third-order of . 0.94 0.67507450 1.2871125 1.1350208 (viii) Matching the results obtained from steps (vi) 0.95 0.64816499 1.2472379 1.1183101 and (vii) together, we finally obtain a solution of 0.96 0.61476429 1.2024805 1.1002331 {F˜(ξ), A˜(ξ)} on the whole spacetime ξ ∈ (0, ∞) (or 0.97 0.57133058 1.1509356 1.0804355 r ∈ (0, ∞)). 0.98 0.51038168 1.0889067 1.0583387 (ix) Once F˜ and A˜ are known, from Eq.(3.11), we can 0.99 0.41063001 1.0068873 1.0328120 calculate B˜, so that an asymptotically flat black hole solution for {A,˜ B,˜ F˜} is finally obtained over the whole space r ∈ (0, ∞). Our results are presented in TableI, where Before proceeding to the next subsection to consider α obs the physically allowed region of the parameter space of γ˜ff ≡ u˜ uα , (4.5) ci’s, let us first reproduce the results presented in Table dF˜(ξ) I of [67], in order to check our numerical code, although r˜g ≡ −rH × lim = 2GæMADM, (4.6) ξ→0 dξ all these choices have been ruled out currently by observations [71]. To see this explicitly, let us first note that obs where uα is the tangent (unit) vector to a radial free- the parameters chosen in [63, 67] correspond to fall trajectory that starts at rest at spatial infinity, and MADM denotes the Komar mass, which is equal to the cˆ3 Arnowitt-Deser-Misner (ADM) mass in the spherically cˆ = − 1 , cˆ = 0 =c ˆ = 0, (4.3) 2 2 3 4 symmetric case for the metricg ˜ [53]. 3ˆc1 − 4ˆc1 + 2 αβ From TableI we can see that our results are exactly the same as those given in [67] up to the same accuracy. so that now onlyc ˆ1 is a free parameter. With this But, due to the improved accuracy of our numerical code, choice ofc î’s, the corresponding ci’s can be obtained from Eqs.(2.25) with σ = c2 , which are given by, for each of the physical quantity, we provided two more S digits. Additionally, in Fig.2 we plotted the functions F˜, B˜, c =c ˆ , 14 1 A˜ and C˜ for four representative cases listed in TableI −2c + 2ˆc + 2c cˆ − 2ˆc2 − c cˆ2 13 1 13 1 1 13 1 (ˆc1 = 0.1, 0.3, 0.6, 0.99). Here, the quantity C˜ is defined c2 = 2 , (4.4) 2 − 4ˆc1 + 3ˆc1 as

d ln B˜ where c13 is arbitrary. This implies that Eqs.(2.25) are ˜ C ≡ − G3 , (4.7) degenerate for the choices of Eqs.(4.3). It can be seen dξ from Eq.(4.4), in all the cases considered in [67], we have −5 c14 > 2.5×10 . Hence all the cases considered in [63, 67] which vanishes identically for the solutions of the ﬁeld do not satisfy the current constraints [71]. equations, as it can be seen from Eq.(3.76). In the rest With the above in mind, we reproduce all the cases of this paper, we shall use it to check the accuracy of our considered in [63, 67], including the ones withc ˜1 > 0.8. numerical code. 16

a 10 1 b 10-13 d a 10-17 0 c 5 10-21 b -1 10-25 0 1 2 10-5 10-2 1 10 104 0

a -11 1 b 10 -15 d 10 -5 0 -19 c c 10 1 10-23 -1 2 10-27 -10 0 1 2 10-5 10-2 1 10 104 0 1 2 3 4 r/rS0H a 1 10-11 b −7 −7 -15 d FIG. 3: The solution for c14 = 2 × 10 , c2 = 9 × 10 , and 10 GR 0 c3 = −c1. Here, A˜, B˜, J˜, F˜ and F˜ are represented by the -19 c 10 red line (labeled by a), green line (labeled by b), orange line -1 10-23 (labeled by c), blue line (labeled by 1) and cyan line (labeled 10-27 by 2) respectively. 0 1 2 10-5 10-2 1 10 104

-4 1 b 10 a -9 d 2 ˜ 10 TABLE II: cS , AH andr ˜g/rH calculated from diﬀerent -14 0 10 {c2, c14} with c13 = 0 and a ﬁxed ratio of c2/c14. 10-19 c 2 ˜ 10-24 c2 c14 cS AH r˜g/rH -1 10-29 0 1 2 10-5 10-2 1 10 104 9 × 10−7 2 × 10−7 4.4999935 2.4558992 1.1450729

FIG. 2: In the above graphs, we use a, b, c and d to represent 9 × 10−8 2 × 10−8 4.4999994 2.4559003 1.1450730 A˜, B˜, F˜ and C˜. In each row,c ˆ1 is chosen, respectively, as −9 −9 cˆ1 = 0.1, 0.3, 0.6, 0.99, as listed in TableI. The horizontal axis 9 × 10 2 × 10 4.4999999 2.4559004 1.1450730 is rH /r.

In our calculations, we stop repeating the bisection search ˜ ˜ ˜ From Fig.2, we note that the properties of {F, A, B} for A˜H , when the value A˜H giving an asymptotically flat −23 depend on the choice ofc ˆ1. The quantity C˜ is approx- solution is determined to within 10 . Technically, these imately zero within the whole integration range, which accuracies could be further improved. However, for our means that our numerical solutions are quite reliable. current purposes, they are already sufficient. As we have already mentioned, theoretically Eq. (3.76) will be automatically satisfied once Eqs. (3.74), (3.75) B. Physically Viable Solutions with S0Hs and (3.77) hold. However, due to numerical errors, in practice, it can never be zero numerically. Thus, to mon- With the above verification of our numerical code, we itor our numerical errors, we always plot out the quantity ˜ turn to the physically viable solutions of the Einstein- C defined by Eq.(4.7), from which we can see clearly the aether field equations, in which a S0H always exists. numerical errors in our calculations. So, in the right-hand panels of Fig.4, we plot out the curves of C˜, denoted by Since c13 is very small, without loss of the generality, in d, in each case. this subsection we only consider the cases with c13 = 0. Clearly, outside the S0H, C˜ 10−17, while inside the As the first example, let us consider the case c14 = . −7 −7 S0H we have C˜ 10−10. Thus, the solutions inside the 2×10 , c2 = 9×10 , and c3 = −c1, which satisfies the . constraints (2.18). Fig.3 shows the functions F,˜ A,˜ B˜, horizon are not as accurate as the ones given outside of in which we also plot J˜ ≡ FÃ˜2 and the GR limit of F˜, the horizon. However, since in this paper we are mainly GR GR concerned with the spacetime outside of the S0H, we shall denoted by F˜ with F˜ ≡ 1 − rH /r. In plotting Fig.3, we chose = 10−14. With the shoot- 10 ing method, A˜H is determined to be A˜H ' 2.4558992 .

behavior (3.79) of the metric coeﬃcients at ξ ≡ rH /r ' 0 sensi- tively depends on the value of A˜H . To make our results reliable, among all the steps in our codes, the precision is chosen to be 10 During the numerical calculations, we ﬁnd that the asymptotical not less than 37. 17

10 10-7 2 8 TABLE III: c , A˜H andr ˜g/rH calculated from diﬀerent S 10-12 {c , c } with c = 0 and changing c /c . 6 a 2 14 13 2 14 10-17 4 10-22 10-10 2 2 d 10-13 ˜ b 10-16 c2 c14 c AH r˜g/rH -19 S 10-27 10 0 10-22 c -32 0.990 0.995 1.000 1.005 1.010 -2 10 2.01 × 10−5 2 × 10−5 1.0049596 1.4562430 1.0005850 0 1 2 3 4 10-9 10-4 10 106 1011

7 × 10−7 5 × 10−7 1.3999982 1.6196457 1.0381205 (a) (b)

10 −7 −8 10-6 9 × 10 2 × 10 44.999939 6.4676346 1.2629671 8 10-11 6 a −5 −7 10-16 9 × 10 2 × 10 449.93921 19.053220 1.3091657 4 10-21 2 d 10-13 b 10-18 10-26 0 10-23 10-28 c -31 0.990 0.995 1.000 1.005 1.010 -2 10 0 1 2 3 4 10-9 10-4 10 106 1011 not consider further improvements of our numerical code (c) (d) 2 inside the horizon. The other quantities, such as cS and 10 r˜g, are all given by the first row of TableII. 10-6 8 10-11 6 a Following the same steps, we also consider other cases, 10-16 4 and some of them are presented in TablesII-III. In par- 10-21 2 10-12 b d 10-16 ticular, in TableII, we fix the ratio of c /c to be 9/2. In -26 -20 2 14 0 10 10 10-24 c 0.990 0.995 1.000 1.005 1.010 addition, the values of {c2, c14} are chosen so that they -31 -2 10 -9 -4 6 11 satisfy the constraints of Eq.(2.18). In TableIII, the ratio 0 1 2 3 4 10 10 10 10 10 c2/c14 is changing and the values of {c2, c14} are chosen (e) (f) so that they are spreading over the whole viable range of c14, given by Eqs.(2.16)-(2.20). FIG. 4: A˜, B˜ and F˜ for different combinations of {c , c } ˜ 2 14 From these tables we can see that quantities like AH listed in TableII and their corresponding C˜’s. Here the hor- andr ˜g are sensitive only to the ratio of c2/c14, instead izontal axis is rH /r. A˜, B˜, F˜ and C˜ are represented by of their individual values. This is understandable, as the red solid line (labeled by a), green dotted line (labeled −5 for c13 = 0 and c14 . 2.5 × 10 , Eq.(1.1) shows that by b), blue dash-dotted line (labeled by c) and orange solid cS ' cS(c2/c14). Therefore, the same ratio of c2/c14 line (labeled by d) respectively. To be specific, (a) and (b) implies the same velocity of the spin-0 graviton. Since are for the case {9 × 10−7, 2 × 10−7}, (c) and (d) are for −8 −8 S0H is defined by the speed of this massless particle, it the case {9 × 10 , 2 × 10 }, (e) and (f) are for the case −9 −9 is quite reasonable to expect that the related quantities {9 × 10 , 2 × 10 }. Note that the small graphs inserted in (b), (d) and (f) show the amplifications of C˜’s near r = rH . are sensitive only to the value of cS. The resulting F˜, A˜, B˜ and C˜ for the cases listed in TablesII andIII are plotted in Figs.4 and5, respectively. tial infinity, Eqs.(3.79), (3.39) and (3.41) lead to

C2 1 F (ξ) = 0 1 + F ξ + c F 3ξ3 +O ξ4, σ 1 48 14 1 C 1 1 B(ξ) = √0 1 + c F 2ξ2 − c F 3ξ3 σ 16 14 1 12 14 1

µ 4 V. PHYSICAL SOLUTIONS (gαβ , u ) +O ξ , √ σ 1 1 2 A(ξ) = 1 − F1ξ + A2ξ C0 2 2 The above steps reveal how we find the solutions of µ 1 1 3 1 3 3 the effective metricg ˜µν and aether fieldu ˜ . To find the − A2F1 − F + c14F ξ µ 1 1 corresponding physical quantities gµν and u , we shall 2 16 96 follow two steps: (a) Reverse Eqs.(3.39) and (3.41) to find +O ξ4 , (5.1) a set of the physical quantities {F (ξ),A(ξ),B(ξ)} (Note that we have ξ =r ˜S0H /r = rS0H /r). (b) Apply the where rescaling v → C v to make the set of {F (ξ),A(ξ),B(ξ)} 0 ˜ take the standard form at spatial infinity r = ∞. F1 = F1, c14 =c ˜14, √ 3 √ A = σA˜ − ( σ − 1)F˜2. (5.2) To these purposes, let us first note that, near the spa- 2 2 4 1 18

3 the four-order of ξ4. 2 a 10-13 1 b -18 0 10

-1 -23 10 10-13 -18 -2 d 10 A. Metric and Spin-0 Horizons c 10-28 10-23 10-28 -3 0.990 0.995 1.000 1.005 1.010 -33 -4 10 0 1 2 3 4 5 10-9 10-4 10 106 1011 −7 −7 Again, we take the case of c14 = 2×10 , c2 = 9×10 , and c = −c as the ﬁrst example. The results for the (a) (b) 3 1 normalized F , A, B and J in this case are plotted in

3 a 10-7 Fig.6. To see the whole picture of these functions on 2 10-12 r ∈ (0, ∞), they are plotted as functions of r/rH inside 1 b 0 10-17 the horizon, while outside the horizon they are plotted as −1 -1 10-22 10-9 functions of (r/r ) . This explains why in the left-hand 10-13 H -2 c d 10-17 -27 10 10-21 panel of Fig.6, the MH ( r = rMH ) stays in the left-hand -3 10-25 0.990 0.995 1.000 1.005 1.010 -32 -4 10 side of the S0H, while in the right-hand panel, they just 0 1 2 3 4 5 10-9 10-4 10 106 1011 reverse the order. In this ﬁgure, we didn’t plot the GR (c) (d) limits for B and F since they are almost overlapped with their counterparts. From the analysis of this case, we 40 ﬁnd the following: 10-5 30 a 10-10 (a) The values of F and B are almost equal to their GR

20 10-15 limits all the time. This is true even when r is approach-

-20 10 10 10-9 ing the center r = 0, at which a spacetime curvature d 10-13 -25 b 10 10-17 singularity is expected to be located. 0 c 0.990 0.995 1.000 1.005 1.010 10-30 0 1 2 3 4 5 10-9 10-4 10 106 1011 (b) Inside the S0H, the oscillations of A and J become visible, which was also noted in [63] 11. Such oscillations (e) (f) continue, and become more violent as the curvature singularity at the center is approaching. 40 10-2 The functions of {F, A, B, J} for the other cases listed 30 a 10-7 in TablesII-III are plotted in Fig.7. In this figure, -12 20 10 the plots are ordered according to the magnitude of c2 . -17 S 10 10-4 10 10-7 -10 Besides, some amplified figures are inserted in (a)-(d) -22 10 10 d 10-13 b 10-16 0 -27 10-19 near the region around the point of F = 0. Similarly, c 10 0.990 0.995 1.000 1.005 1.010 0 1 2 3 4 5 10-9 10-4 10 106 1011 in (e)-(f), some amplified figures are inserted near the region around the point of J = J +. The position of (g) (h) r = rMH , at which we have F (rMH ) = 0, is marked by a full solid circle, while the position of r = rS0H , at which we have J(r ) = J + [cf., Eq. (3.17)], is marked ˜ ˜ ˜ S0H FIG. 5: A, B and F for different combinations of {c2, c14} by a pentagram, and in all these cases we always have listed in TableIII and their corresponding C˜’s. Here the + − ˜ ˜ ˜ ˜ rMH > rS0H . The values of J and J are given by horizontal axis is rH /r. A, B, F and C are represented the brown and purple solid lines, respectively. Note we by the red solid line (labeled by a), green dotted line (la- always have J + > J − for c > 1. By using these two beled by b), blue dash-dotted line (labeled by c) and or- S lines, we can easily find that there is only one rS0H in ange solid line (labeled by d) respectively. To be specific, + −5 −5 (a) and (b) are for the case {2.01 × 10 , 2 × 10 }, (c) and each case, i.e., rS0H in Eq. (3.17). (d) are for the case {7 × 10−7, 5 × 10−7}, (e) and (f) are for From the studies of these representative cases, we find the case {9 × 10−7, 2 × 10−8}, (g) and (h) are for the case the following: (i) As we have already mentioned, in all {9 × 10−5, 2 × 10−7}. Note that the small graphs inserted in these cases the functions B and F are very close to their ˜ 2 (b), (d), (f) and (h) show the amplifications of C near r = rH . GR limits. (ii) Changing cS won’t influence the maximum of A much. In contrast, the maximum of |J| inside 2 the S0H is sensitive to cS. (iii) The oscillation of A(r) gets more violent as c2 is increasing. (iv) The value of The above expressions show clearly that the spacetimes S |r − r | is getting bigger as c2 deviating from 1. described by (g , uµ) are asymptotically flat, provided MH S0H S µν (v) In all these cases, we have only one r , i.e., only that the effective fields (˜g , u˜µ) are. In particular, set- S0H √ µν one intersection between J(r) and J ±, in each case. (vi) ting C = σ, a condition that will be assumed in the 0 Just like what we saw in TablesII-III, in the cases with rest of this section, the functions F,A and B will take their standard asymptotically-flat forms. It is remarkable to note that the asymptotical behavior of the functions F,A and B depends only on c14 up to 11 In [63], the author just considered the oscillational behavior of the third-order of ξ, but c2 will show up starting from A˜. The physical quantities F , A, and B were not considered. 19

1 1

0 ● 0 ● ★ ★ -1

-1 -2

F A B J -3 -2 F A B J J+ J- -4 J+ J- -3 -5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

rS0H/r r/rS0H

−7 −7 FIG. 6: The evolutions of the physical quantities F , A, B and J for the case c13 = 0, c2 = 9×10 and c14 = 2×10 . Here, A, B, J and F are represented by the red solid line, green dotted line, orange dashed line, and blue dash-dotted line, respectively. The positions of r = rMH and r = rS0H are marked by a small full solid circle and a pentagram, respectively. Note that we + − + − have rMH > rS0H . The values J and J are given respectively by the brown and purple solid lines with J > J . The left panel shows the main behaviors of the functions outside the S0H in the range rS0H /r ∈ (0, 1.105), while the right panel shows their main behaviors inside the S0H in the range r/rS0H ∈ (0, 1.2).

the same cS (but different values of c14 and c2), the cor- where Eq.(5.4) was used. For the expansion of F to be responding functions {F, A, B, J} are quite similar. finite, we must assume From TablesII-III and Fig.7, we would like also to note that the value of rMH is always close to the cor- respondingr ˜g. To understand this, let us consider Eq. (5.1), from which we find that

1 3 3 4 F (ξ) = 1 + F1ξ + c14F1 ξ + O ξ , c14, c2 , (5.3) 48 " 3 2# 3 1 r˜g rS0H O ξ , c14, c2 . O c14 . (5.7) after normalization. Recall ξ ≡ rH /r and rH ≡ rS0H . 48 rS0H rMH Then, from Eqs.(4.6), (3.79), (5.2) and (5.3), we also ﬁnd At the same time, recall that we have c 2.5 × 10−5 that 14 . and rS0H 6 rMH . Besides, we also haver ˜g/rS0H 'O(1). r˜g Thus, from Eq. (5.6) we ﬁnd = −F˜1 = −F1. (5.4) rS0H On the other hand, from Eq. (3.15), we have

r 1 r 3 F (ξ)| = 1 + F S0H + c F 3 S0H r=rMH 1 14 1 rMH 48 rMH

+ O ξ4, c , c rMH r˜g 14 2 − . O(c14). (5.8) = 0, (5.5) rS0H rS0H from which we obtain,

2 3 rMH 1 3 rS0H rS0H = −F1 − c14F1 + O rS0H 48 rMH rMH 3 2 r˜g 1 r˜g rS0H = + c14 This result reveals why the values of rMH /rS0H and rS0H 48 rS0H rMH r˜g/rS0H are very close to each other, although not nec- 3 +O ξ , c14, c2 , (5.6) essarily the same exactly.

Finally, let us take a closer look at the diﬀerence between GR and æ-theory, although in the above we al- 20

2 2 c 2 r r c 2 r r A S ≈1.0049596 ( MH/ S0H≈1.0005847) A S ≈1.3999982 ( MH/ S0H≈1.0381204) 1 1 B F B F 0 0

-1 0.0005 -1 0.05 0.0000 ● 0.00 ● -0.0005 -2 -0.0010 -2 -0.05 -0.0015 ★ -0.10 ★ -0.0020 -0.15 -3 0.999 1 1.001 -3 1 1.02 1.05 J J -4 -4 0 2 4 6 8 0 2 4 6 8

r/rS0H r/rS0H

(a) (b)

2 2 c 2 r r c 2 r r A S ≈4.4999994 ( MH/ S0H≈1.1450730) A S ≈4.4999999 ( MH/ S0H≈1.1450730) 1 1 B F B F 0 0

-1 0.2 -1 0.2 0.0 ● 0.0 ● -2 -0.2 -2 -0.2 J -0.4 ★ J -0.4 ★ -0.6 -0.6 -3 0.9 1.0 1.1 1.2 1.3 -3 0.9 1.0 1.1 1.2 1.3

-4 -4 0 2 4 6 8 0 2 4 6 8

r/rS0H r/rS0H

2 2 c 2 r r c 2 r r A S ≈44.999939 ( MH/ S0H≈1.2629669) A S ≈449.93921 ( MH/ S0H≈1.3091654) 1 B 1 B J F J F 0 ● 0 ●

-1 -1 -0.6 -0.85 -0.8 ★ -0.90 ★ -2 -2 -0.95 -1.0 -1.00 -1.2 -1.05 -3 -1.4 -3 -1.10 -1.15 0.2 0.6 1 1.4 0.6 1 1.4 -4 -4 0 2 4 6 8 0 2 4 6 8

r/rS0H r/rS0H

(e) (f)

FIG. 7: Solutions for different combinations of {c2, c14} listed in TablesII-III. Here, A, B, J and F are represented by the red solid line, green dotted line, orange dashed line, and blue dash-dotted line, respectively. These figures are ordered according to 2 + − the magnitude of cS . In each of the figure, the values J and J are given respectively by the brown and purple solid lines with + − J > J , while the positions of r = rMH and r = rS0H are marked by a small full solid circle and a pentagram, respectively. Additionally, the value of rMH /rS0H is also given in each case. 21

0 0.000025 -5.×10 -10

-1.×10 -9 0.000020

-9 Δ F

Δ F -1.5×10 0.000015 -2.×10 -9 0.000010 -2.5×10 -9

-3.×10 -9 5.×10 -6 0.0 0.2 0.4 0.6 0.8 1.0 1×1011 3×1011 5×1011 7×1011 9×1011 ξ ξ

(a) (b)

0.0010 5.×10 -8

4.×10 -8 0.0005

3.×10 -8 Δ B Δ B 0.0000 2.×10 -8

1.×10 -8 -0.0005

0 0.0 0.2 0.4 0.6 0.8 1.0 1×106 3×106 5×106 7×106 9×106 ξ ξ

−7 −7 FIG. 8: ∆F and ∆B for c2 = 9 × 10 , c14 = 2 × 10 and c13 = 0. The panels (a) and (c) show the region outside the S0H, while the panels (b) and (d) show the region inside the S0H. ready mentioned that the results from these two theories B. Universal Horizons are quite similar. To see these more clearly, we first note that the GR counterparts of F and B are given by In theories with the broken LI, the dispersion relation of a massive particle contains generically high-order mo- r mentum terms [80], F GR = 1 − MH ,BGR = 1. (5.9) r  2(z−1) n X k E2 = m2 + c2 k2 1 + a , (5.11) k  n M  Thus, the relative differences can be defined as n=1 ∗ from which we can see that both of the group and phase F − F GR B − BGR velocities become unbounded as k → ∞, where E and k ∆F ≡ , ∆B ≡ . (5.10) F GR BGR are the energy and momentum of the particle considered, and ck and an’s are coefficients, depending on the species of the particle, while M∗ is the suppression energy scale of −7 Again, considering the representative case c2 = 9 × 10 , the higher-dimensional operators. Note that there must −7 c14 = 2 × 10 and c13 = 0, we plot out the differences be no confusion between ck here and the four coupling ∆F and ∆B in Fig.8, from which we find that in the constants ci’s of the theory. As an immediate result, the −12 −9 range ξ ∈ (10 , 1) we have O(∆F ) . 10 . On the causal structure of the spacetimes in such theories is quite 12 other hand, in the range ξ ∈ (1, 10 ), we have O(∆F ) . different from that given in GR, where the light cone at 10−5. Similarly, in the range ξ ∈ (10−12, 1), we have a given point p plays a fundamental role in determining −8 7 O(∆B) . 10 . In addition, in the range ξ ∈ (1, 10 ) we the causal relationship of p to other events [99]. In a UV −3 have O(∆B) . 10 . Thus, we confirm that F and B complete theory, the above relationship is expected even are indeed quite close to their GR limits. in the gravitational sector. One of such examples is the 22

t particles even with inﬁnitely large speeds cannot escape from it, once they are inside it [cf. Fig.9]. So, it acts as an absolute horizon to all particles (with any speed), which is often called the UH [66, 67, 80]. At the horizon, ζ μ as can be seen from Fig.9, we have [102],

dϕ

ϕ=ϕ 1 0 ζ · u| = − (1 + J) = 0, (5.12) r=rUH 2A r=rUH

Black Hole where J ≡ FA2. Therefore, the location of an UH is ex- r r 0 UH MH r actly the crossing point between the curve of J(r) and the horizontal constant line J = −1, as one can see from Figs.6 and7. From these figures we can also see FIG. 9: Illustration of the bending of the φ = constant sur- that they are always located inside S0Hs, as expected. faces, and the existence of the UH in a spherically symmetric In addition, the curve J(r) is oscillating rapidly, and static spacetime, where φ denotes the globally timelike scalar crosses the horizontal line J = −1 back and forth infinite field, and t is the Painlevé-Gullstrand-like coordinates, which times. Therefore, in each case we have infinite number covers the whole spacetime [101]. Particles move always along µ µ of rUH−i (i = 1, 2, ...). In this case, the UH is defined as the increasing direction of φ. The Killing vector ζ = δv always points upward at each point of the plane. The vertical the largest value of rUH−i (i = 1, 2, ...). In TableIV, we dashed line is the location of the metric (Killing) horizon, show the locations of the first eight UHs for each case, r = rMH . The UH, denoted by the vertical solid line, is lo- listed in TablesII andIII. It is interesting to note that cated at r = rUH , which is always inside the MH. the formation of multi-roots of UHs was first noticed in [67], and later observed in gravitational collapse [72]. healthy extension [91, 92] of Hoˇrava gravity [79, 80], a possible UV extension of the khronometric theory (the HO æ-theory [93, 94]). However, once LI is broken, the causal structure will be dramatically changed. For example, in the Newtonian C. Other Observational Quantities theory, time is absolute and the speeds of signals are not limited. Then, the causal structure of a given point p is uniquely determined by the time difference, ∆t ≡ Another observationally interesting quantity is the t − t , between the two events. In particular, if ∆t > 0, p q ISCO, which is the root of the equation, the event q is to the past of p; if ∆t < 0, it is to the future; and if ∆t = 0, the two events are simultaneous. In theories with breaking LI, a similar situation occurs. 2 To provide a proper description of BHs in such theo- 2rF 0(r) − F [3F 0(r) + rF 00(r)] = 0. (5.13) ries, UHs were proposed [66, 67], which represent the absolute causal boundaries. Particles even with infinitely large speeds would just move on these boundaries and Note that in GR we have rISCO/rH = 3 [98]. Due to cannot escape to infinity. The main idea is as follows. the tiny differences between the Schwarzschild solutions In a given spacetime, a globally timelike scalar field φ and the ones considered here, as shown in Fig.8, it is may exist [100]. In the spherically symmetric case, this expected that rISCO’s in these cases are also quite close globally timelike scalar field can be identified to the HO to its GR limit. As a matter of fact, we find that this aether field uµ via the relation (2.46). Then, similar to is indeed the case, and the differences in all the cases the Newtonian theory, this field defines globally an ab- considered above appear only after six digits, that is, solute time, and all particles are assumed to move along GR −6 rISCO − rISCO ≤ 10 , as shown explicitly in Tables the increasing direction of the timelike scalar field, so the V andVI. causality is well defined. In such a spacetime, there may exist a surface at which the HO aether field uµ is orthog- In TableV, we also show several other physical quan- onal to the timelike Killing vector, ζ (≡ ∂v). Given that tities. These include the Lorentz gamma factor γff , the all particles move along the increasing direction of the gravitational radius rg, the orbital frequency of the ISCO HO aether field, it is clear that a particle must cross this ωISCO, the maximum redshift zmax of a photon emitted surface and move inward, once it arrives at it, no matter by a source orbiting the ISCO (measured at the infinity), how large its speed is. This is a one-way membrane, and and the impact parameter bph of the circular photon orbit 23

(CPO), which are deﬁned, respectively, by [67], To avoid the vacuum gravi-Cerenkovˇ radiation, all these modes must propagate with speeds greater than 1 γff = A + , (5.14) or at least equal to the speed of light [74]. However, 4A r=rMH in the spherically symmetric spacetimes, only the spin-0

dF mode is relevant in the gravitational sector [75], and the rg = −rS0H , (5.15) dξ boundaries of BHs are deﬁned by this mode, which are ξ→0 (S) r the null surfaces with respect to the metric gµν deﬁned dF/dr in Eq.(1.3), the so-called S0Hs. Since now cS ≥ c, where ωISCO = , (5.16) 2r cS is the speed of the spin-0 mode, the S0Hs are always r=rISCO inside or at most coincide with the metric (Killing) hori- −1/2 1 + ωISCOrF zons. Then, in order to cover spacetimes both inside and zmax = − 1, (5.17) pF − ω2 r2 outside the MHs, working in the Eddington-Finkelstein ISCO r=rISCO coordinates (3.1) is one of the natural choices. r In the process of gravitational radiations of compact bph = √ , (5.18)

F r=rph objects, all of these three fundamental modes will be emitted, and the GW forms and energy loss rate should where the radius rph of the CPO is defined as be different from that of GR. In particular, to the leading dF order, both monopole and dipole emissions will co-exist 2F − r = 0. (5.19) dr with the quadrupole emission [34, 35, 38–41, 43]. De- r=rph spite of all these, it is remarkable that the theory still re- As pointed previously, these quantities are quite close mains as a viable theory, and satisfies all the constraints, to their relativistic limits, since they depend only on the both theoretical and observational [71], including the re- spacetimes described by F and B. As shown in Fig. cent detection of the GW, GW170817, observed by the 8, the differences of these spacetimes between æ-theory LIGO/Virgo collaboration [89], which imposed the severe and GR are very small. To see this more clearly, let us constraint on the speed of the spin-2 gravitational mode, −15 −16 introduce the quantities, −3 × 10 < cT − 1 < 7 × 10 . Consequently, it is one of few theories that violate Lorentz symmetry and mean- GR rISCO rISCO time is still consistent with all the observations carried ∆rISCO ≡ − , rMH rMH out so far [71, 103]. GR Spherically symmetric static BHs in æ-theory have ∆ωISCO ≡ rgωISCO − (rgωISCO) , GR been extensively studied both analytically [51–61] and ∆zmax ≡ zmax − (zmax) , numerically [63–69], and various solutions have been ob- GR bph bph tained. Unfortunately, all these solutions have been ruled ∆bph ≡ − , (5.20) out by current observations [71]. rg rg Therefore, as a first step, in this paper we have in- where the GR limits of rISCO/rMH , rg√ωISCO, zmax√ and vestigated spherically symmetric static BHs in æ-theory −3/2 bph/rg are, respectively, 3, 2×6 , 3/ 2−1 and 3 3/2. that satisfy all the observational constraints found lately As can be seen from TableVI, all of these quantities are in [71] in detail, and presented various numerical new fairly close to their GR limits. BH solutions. In particular, we have first shown explic- Therefore, we conclude that it is quite difficult to dis- itly that among the five non-trivial field equations, only tinguish GR and æ-theory through the considerations of three of them are independent. More important, the two the physical quantities rISCO, ωISCO, zmax or bph, as far second-order differential equations given by Eqs.(3.5) and as the cases considered in this paper are concerned. Thus, (3.6) for the two functions F (r) and A(r) are independent it would be very interesting to look for other choices of of the function B(r), where F (r) and B(r) are the met- {c2, c13, c14} (if there exist), which could result in distin- ric coefficients of the Eddington-Finkelstein metric (3.1), guishable values in these observational quantities. and A(r) describes the aether field, as shown by Eq.(3.2). Thus, one can first solve Eqs.(3.5) and (3.6) to find F (r) and A(r), and then from the third independent equa- VI. CONCLUSIONS tion to find B(r). Another remarkable feature is that the function B(r) can be obtained from the constraint (3.10), In this paper, we have systematically studied static and is given simply by the algebraic expression of F,A spherically symmetric spacetimes in the framework of and their derivatives, as shown explicitly by Eq.(3.11). Einstein-aether theory, by paying particular attention to This not only saves the computational labor, but also black holes that have regular S0Hs. In æ-theory, a time- makes the calculations more accurate, as pointed out ex- like vector - the aether, exists over the whole spacetime. plicitly in [67], solving the first-order differential equation As a result, in contrast to GR, now there are three grav- (3.7) for B(r) can “potentially be affected by numerical itational modes, referred to as, respectively, the spin-0, inaccuracies when evaluated very close to the horizon”. spin-1 and spin-2 gravitons. Then, now solving the (vacuum) field equations of 24

TABLE IV: rUH−i’s for diﬀerent cases listed in TablesII andIII. Note that here we just show ﬁrst eight UHs of Eq.(5.12) for each case.

2 cS rMH /rUH−1 rMH /rUH−2 rMH /rUH−3 rMH /rUH−4 rMH /rUH−5 rMH /rUH−6 rMH /rUH−7 rMH /rUH−8

1.0049596 1.40913534 9.12519836 68.6766490 524.111256 4006.80012 30638.7274 234291.582 1791613.54

1.3999982 1.39634652 6.27835216 33.1700700 178.825436 967.454326 5237.31538 28355.5481 153524.182

4.4999935 1.36429738 2.74101697 6.42094860 15.6447753 38.6164383 95.7784255 238.000717 591.850964

4.4999994 1.36429738 2.74101595 6.42094387 15.6447581 38.6163818 95.7782501 238.000194 591.849442

4.4999999 1.36429738 2.74101584 6.42094340 15.6447564 38.6163762 95.7782326 238.000141 591.849289

44.999939 1.33939835 1.56980254 1.91857535 2.41278107 3.08953425 4.00154026 5.22142096 6.84725395

449.93921 1.33429146 1.39226716 1.46010811 1.53855402 1.62835485 1.73026559 1.84507183 1.97362062

TABLE V: The quantities rS0H , γff , rISCO, ωISCO, zmax and bph for diﬀerent cases listed in TablesII andIII.

2 cS rMH /rS0H γff rISCO/rMH rgωISCO zmax bph/rg

1.0049596 1.00058469 1.62614814 3.00000083 0.13608278 1.12132046 2.59807604

1.3999982 1.03812045 1.63971715 3.00000002 0.13608276 1.12132035 2.59807621

4.4999935 1.14507287 1.67376648 3.00000000 0.13608276 1.12132034 2.59807621

4.4999994 1.14507298 1.67376647 3.00000000 0.13608276 1.12132034 2.59807621

4.4999999 1.14507299 1.67376647 3.00000000 0.13608276 1.12132034 2.59807621

44.999939 1.26296693 1.69777578 3.00000000 0.13608276 1.12132034 2.59807621

449.93921 1.30916545 1.70149318 3.00000000 0.13608276 1.12132034 2.59807621 spherically symmetric static spacetimes in æ-theory sim- initions is that it allows us to choose the free parameter σ ply reduces to solve the two second-order differential equa- involved in the field redefinitions, so that the S0H of the tions (3.5) and (3.6). This will considerably simplify the redefined metricg ˜µν will coincide with its MH. This will mathematical computations, which is very important, es- reduce the four-dimensional space of the initial condi- ˜ ˜0 ˜ ˜0 pecially considering the fact that the field equations in- tions, spanned by FH , FH , AH , AH , to one-dimension, volved are extremely complicated, as one can see from spanned only by A˜H , if the initial conditions are imposed Eqs.(3.5)-(3.10) and (A.1)-(A.4). Then, in the case on the S0H. In Sec. III.D. we have shown step by step c13 = c14 = 0 we have been able to solve these equa- how one can do it. In addition, in this same subsection tions explicitly, and obtained a three-parameter family we have also shown that the field equations are invari- of exact solutions, which in general depends on the cou- ant under the rescaling r → Cr. In fact, introducing pling constant c2. However, requiring that the solutions the dimensionless coordinate ξ ≡ rS0H /r, the relevant be asymptotically flat, we have found that the solutions four field equations take the scaling-invariant forms of become independent of c2, and the corresponding metric Eqs.(3.74)-(3.77), which are all independent of rS0H . reduces precisely to the Schwarzschild BH solution with Thus, when integrating these equations, without loss of a non-trivially coupling aether field given by Eq.(3.34), generality, one can assign any value to rS0H . which is always timelike even in the region inside the BH. We would like also to note that in Section III.C To simplify the problem further, we have also taken we worked out the relations in detail among the fields µ µ µ the advantage of the field redefinitions that are allowed (gµν , u , ci), (ˆgµν , uˆ , cî) and (˜gµν , u˜ , c˜i), and clarified by the internal symmetry of æ-theory, first discovered by several subtle points. In particular, the redefined met- Foster in [83], and later were used frequently, including ricg ˆµν through Eqs.(2.23) and (2.24) does not take the the works of [63, 65, 67]. The advantage of the field redef- standard form in the Eddington-Finkelstein coordinates, 25

TABLE VI: ∆rISCO, ∆ωISCO, ∆zmax and ∆bph for diﬀerent cases listed in TablesII andIII.

2 cS ∆rISCO ∆ωISCO ∆zmax ∆bph

1.0049596 8.3 × 10−7 1.3 × 10−8 1.2 × 10−7 −1.7 × 10−7

1.3999982 1.8 × 10−8 2.2 × 10−10 2.0 × 10−9 −3.2 × 10−9

4.4999935 4.0 × 10−9 1.5 × 10−12 4.9 × 10−11 −4.2 × 10−10

4.4999994 4.0 × 10−10 1.5 × 10−11 −7.2 × 10−11 −3.2 × 10−10

4.4999999 4.0 × 10−11 2.3 × 10−11 −1.2 × 10−10 −4.5 × 10−10

44.999939 1.5 × 10−10 9.6 × 10−11 −5.6 × 10−10 −1.9 × 10−9

449.93921 1.1 × 10−9 1.1 × 10−11 −4.5 × 10−10 −1.1 × 10−9

−7 −7 as shown explicitly by Eq.(3.35). Instead, only after 9 × 10 , c14 = 2 × 10 and c13 = 0, but similar results a proper coordinate transformation given by Eqs.(3.36) also hold for the other cases, listed in TablesII andIII. and (3.37), the resulting metricg ˜µν takes the standard In this section, we have also identified the locations form, as given by Eq.(3.38). Then, the field equations of the UHs of these solutions and several other observa- µ for (˜gµν , u˜ , c˜i) take the same forms as the ones for tionally interesting quantities, which include the ISCO µ (gµν , u , ci). Therefore, when we solved the field equa- rISCO, the Lorentz gamma factor γff , the gravitational tions in terms of the redefined fields, they are the ones of radius rg, the orbital frequency ωISCO of the ISCO, the µ µ (˜gµν , u˜ ), not the ones for (ˆgµν , uˆ ). maximum redshift zmax of a photon emitted by a source After clarifying all these subtle points, in Sec. IV, we orbiting the ISCO (measured at the infinity), the radii have worked out the detail on how to carry out explicitly rph of the CPO, and the impact parameter bph of the our numerical analysis. In particular, to monitor the nu- CPO. All of them are given in TableIV-V. In TableVI merical errors of our code, we have introduced the quan- we also calculated the differences of these quantities ob- tity C˜ through Eq.(4.7), which is essentially Eq.(3.65). tained in æ-theory and GR. Looking at these results, we Theoretically, it vanishes identically. But, due to numer- conclude that it’s very hard to distinguish GR and æ- ical errors, it is expected that C˜ has non-zero values, and theory through these quantities, as far as the cases con- the amplitude of it will provide a good indication on the sidered in this paper are concerned. We would also like numerical errors that our numerical code could produce. to note that for each BH solution, there are infinite num- To show further the accuracy of our numerical code, ber of UHs, r = rUH−i, (i = 1, 2, 3, ...), which was also we have first reproduced the BH solutions obtained in observed in [67]. In TableIV we have listed the first [63, 67], but with an accuracy that are at least two orders eight of them, and the largest one is usually defined as higher [cf. TableI]. It should be noted that all these BH the UH of the BH. In contrast, there are only one S0H solutions have been ruled out by the current observations and one MH for each solution. These features are also [71]. So, after checking our numerical code, in Sec. IV.B, found in the gravitational collapse of a massless scalar we considered various new BH solutions that satisfy all field in æ-theory [72]. the observational constraints [71], and presented them in An immediate implication of the above results is that TablesII andIII, as well as in Figs.3-5. the QNMs of these BHs for a test field, scalar, vector or Then, in Sec. V, we have presented the physical metric tensor [104], will be quite similar to these given in GR. µ gµν and æ-field u for these viable new BH solutions Our preliminary results on such studies indicate that this obtained in Section IV. Before presenting the results, we is indeed the case. However, we expect that there should µ have first shown that the physical fields, gµν and u , be significant differences from GR, when we consider the are also asymptotically flat, provided that the effective metric perturbations of these BH solutions - the gravi- µ fieldsg ˜µν andu ˜ are [cf. Eqs.(5.1) and (5.2)]. Then, the tational spectra of perturbations [105], as now the BH physical BH solutions were plotted out in Figs.6 and boundaries are the locations of the S0Hs, not the loca- 7. Among several interesting features, we would like to tions of the MHs. This should be specially true for the point out the different locations of the metric and spin-0 cases with large speeds cS of the spin-0 modes, as in horizons for the physical metric gµν , denoted by full solid these cases the S0Hs are significantly different from the circles and pentagrams, respectively. MHs, and located deeply inside them. Thus, imposing Another interesting point is that all these physical BH the non-out-going radiation on the S0Hs will be quite solutions are quite similar to the Schwarzschild one. In different from imposing the non-out-going radiation on Fig.8 we have shown the differences for the case c2 = the corresponding MHs. We wish to report our results 26

along this direction soon in another occasion. Appendix A: The coeﬃcients of fn, an, bn and nn

Acknowledgments In this appendix, we shall provide the explicit expres- We would like very much to express our gratitude to sions of the coefficients of fn, an, bn and nn, encountered Ted Jacobson, Ken Yagi, and Nico Yunes, for their valu- in the Einstein-aether field equations in the spherically able comments and suggestions, which lead us to im- symmetric spacetimes, for which the metric is written prove the paper considerably. This work is partially sup- in the Eddington-Finkelstein coordinates (3.1), with the ported by the National Natural Science Foundation of aether field taking the form of Eq.(3.2). In particular, China (NNSFC) under Grant Nos. 11675145, 11805166, the coefficients of fn, an and bn appearing in Eqs.(3.5)- 11975203, 11773028, 11633001, and 11653002. (3.7) are given by,

0 f0 = −4 (c2 + c13)(c2 + c13 − (c2 + 1) c14) rA(r)A (r) 2 2 2 2 2 0 2 − c14c2 − c2 + c13 + (c2 + 1) c14 − (c2 + 2) c13c14 r A (r) 2 2 4 0 −4 (c14 + 1) c13 + 2 (c2 + 1) c13 + (c2 − 2) c14c13 + c2 (c2 + 2) − 2 c2 + 3c2 + 1 c14 rA(r) F (r) 2 2 2 2 3 0 0 +2 c2 + (2 − 3c2) c14c2 − 9c13c14c2 + (c2 + 1) c14 − c13 (4c14 + 1) r A(r) A (r)F (r) 2 2 2 2 2 6 0 2 − 5c14c2 − c2 + (c2 + 1) c14 + (7c2 − 2) c13c14 + c13 (4c14 + 1) r A(r) F (r) 3 5 0 0 2 3 8 0 3 −2 (c2 + c13) c14 (−c2 + c13 + c14 − 4) r A(r) A (r)F (r) + (c2 + c13) c14 (c2 − c13 + c14) r A(r) F (r) 2 3 0 2 0 − (c2 + c13) A(r) −c14 (c2 − c13 + c14) r A (r) F (r) + 4c2 (c14 − 1) + 2c13c14 , 3 0 f1 = −8 (c2 + c13)(c14c2 + c2 + c13 + c14) rA(r) A (r) 2 2 2 2 2 0 2 +4 c2 − 3c13c14c2 + 2c14c2 + (c2 + 1) c14 − c13 (c14 + 1) r A(r) A (r) 2 2 6 0 −4 (1 − 2c14) c13 + (−c14c2 + 2c2 + c14 + 4) c13 + c2 (c2 + 4) + 5c2 + 9c2 + 4 c14 rA(r) F (r) 2 2 2 2 2 5 0 0 +2 c14c2 + 3c2 − 3 (c2 + 1) c14 + (11c2 + 6) c13c14 + c13 (4c14 − 3) r A(r) A (r)F (r) 2 2 2 2 8 0 2 +2 c2 + (3c2 + 2) c14c2 + 3c13c14c2 + (c2 + 1) c14 + c13 (2c14 − 1) r A(r) F (r) 3 7 0 0 2 +2 (c2 + c13) c14 (c2 − c13 + c14) r A(r) A (r)F (r) 4 3 0 2 0 −2 (c2 + c13) c14A(r) (−c2 + c13 + c14 − 4) r A (r) F (r) − 4 (2c2 + c13 + 1) , 2 2 2 2 2 4 0 2 f2 = 2 c14c2 + 3c2 − 3 (c2 + 1) c14 + (11c2 + 6) c13c14 + c13 (4c14 − 3) r A(r) A (r) 2 2 8 0 +4 − (c14 − 1) c13 + 2 (c2 − 1) c13 + (c2 + 4) c14c13 + (c2 − 2) c2 + 4c2 + 8c2 + 2 c14 rA(r) F (r) 2 2 2 2 7 0 0 +6 (c14 + 1) c2 + c14 (−c13 + c14 + 2) c2 − c13 + c14 r A(r) A (r)F (r) 2 2 2 10 0 2 + − (c14 − 1) c2 + (c13 − c14) c14c2 − (c13 − c14) r A(r) F (r) 6 3 0 2 0 + (−c2 − c13) A(r) −c14 (c2 − c13 + c14) r A (r) F (r) + 8c2 + 4 (6c2 + 3c13 + 4) c14 , 7 0 2 2 8 f3 = 8 (c2 + c13)(c14c2 + c2 + c13 + c14) rA(r) A (r) + 8 2c2 + 3c13c2 + c2 + c13 + c13 c14A(r) 2 2 2 2 6 0 2 +4 c2 − 3c13c14c2 + 2c14c2 + (c2 + 1) c14 − c13 (c14 + 1) r A(r) A (r) 10 0 +4 (c2 + c13)(c2 + c13 − (c2 + 1) c14) rA(r) F (r) 2 9 0 0 +2 (c2 − c13 + c14)(c2 + c13 − (c2 + 1) c14) r A(r) A (r)F (r), 9 0 10 f4 = 4 (c2 + c13)(c2 + c13 − (c2 + 1) c14) rA(r) A (r) − 2 (c2 + c13) (2c2 (c14 − 1) + c13c14) A(r) 2 2 2 2 2 8 0 2 + −c14c2 + c2 − c13 − (c2 + 1) c14 + (c2 + 2) c13c14 r A(r) A (r) , (A.1) 27

2 2 2 0 a0 = 4 − (c14 − 1) c13 + (−c14c2 + 2c2 + 2c14 − 2) c13 + (c2 − 2) c2 + 2 c2 + 3c2 + 1 c14 rA(r) A (r) 2 2 2 2 0 2 + c13 + (5c2c14 + 8) c13 − (c2 + 1) c14 − (c2 − 8) c2 − 5c2 + 18c2 + 8 c14 r A(r)A (r) 3 0 3 5 0 + (c2 + c13) c14 (c2 − c13 + c14) r A (r) + 4 (c2 + c13)(c14c2 + c2 + c13 + c14) rA(r) F (r) 2 2 2 2 4 0 0 −2 (2c14 − 1) c13 + ((3c2 − 2) c14 + 4) c13 − (c2 + 1) c14 + c2 (c2 + 4) + 3c2 + 4c2 + 4 c14 r A(r) A (r)F (r) 2 2 2 2 7 0 2 + −c2 − (c2 + 2) c14c2 + c13c14c2 + c13 − (c2 + 1) c14 r A(r) F (r) 3 6 0 0 2 + (c2 + c13) c14 (c2 − c13 + c14) r A(r) A (r)F (r) 3 3 0 2 0 −2 (c2 + c13) A(r) c14 (−c2 + c13 + c14 − 4) r A (r) F (r) + 2c2 + 2c2c14 + c13c14 + 4 , 2 2 4 0 a1 = 4 (2c14 + 1) c13 + (3c14c2 + 2c2 + c14 − 4) c13 + (c2 − 4) c2 − 3c2 + 7c2 + 4 c14 rA(r) A (r) 2 2 2 2 3 0 2 + − (8c14 − 3) c13 − ((23c2 + 14) c14 − 8) c13 + 3 (c2 + 1) c14 − c2 (3c2 − 8) − c2 − 8c2 − 8 c14 r A(r) A (r) 3 2 0 3 7 0 −2 (c2 + c13) c14 (−c2 + c13 + c14 − 4) r A(r) A (r) − 8 (c2 + 1) (c2 + c13) c14rA(r) F (r) 2 2 2 2 6 0 0 +4 (c14 + 1) c13 + 2 (c2c14 − 1) c13 − (c2 + 1) c14 − c2 (c2 + 2) + c2 + 2c2 + 2 c14 r A(r) A (r)F (r) 2 2 2 2 2 9 0 2 + c14c2 − c2 + c13 + (c2 + 1) c14 − (c2 + 2) c13c14 r A(r) F (r) 5 3 0 2 0 −2 (c2 + c13) A(r) −c14 (c2 − c13 + c14) r A (r) F (r) − 2c2 − (6c2 + 3c13 + 4) c14 , 7 a2 = −2 (c2 + c13) ((6c2 + 3c13 + 4) c14 − 2 (c2 + 2)) A(r) 2 6 0 −4 (c14 + 1) c13 + (c2 (3c14 + 2) + 2) c13 + c2 (c2 + 2) − 2 (2c2 + 1) c14 rA(r) A (r) 2 2 2 2 5 0 2 + −3c2 + (5c2 − 6) c14c2 + 19c13c14c2 − 3 (c2 + 1) c14 + c13 (8c14 + 3) r A(r) A (r) 3 4 0 3 9 0 + (c2 + c13) c14 (c2 − c13 + c14) r A(r) A (r) − 4 (c2 + c13)(c2 + c13 − (c2 + 1) c14) rA(r) F (r) 2 8 0 0 −2 (c2 − c13 + c14)(c2 + c13 − (c2 + 1) c14) r A(r) A (r)F (r), 9 8 0 a3 = 2 (c2 + c13) (2c2 (c14 − 1) + c13c14) A(r) − 4 (c2 + c13)(c2 + c13 − (c2 + 1) c14) rA(r) A (r) 2 2 2 2 2 7 0 2 + c14c2 − c2 + c13 + (c2 + 1) c14 − (c2 + 2) c13c14 r A(r) A (r) , (A.2) and

2 2 0 b0 = 4 (c2 + 1) (c2 + c13) c14A(r) − 4 (c2 + c13) c14rA(r)A (r) 2 0 2 2 4 0 + (c2 + c13) c14 (c2 − c13 + c14) r A (r) − 4 (c2 + c13) c14rA(r) F (r) 2 3 0 0 −2 (c2 + c13) c14 (−c2 + c13 + c14 − 4) r A(r) A (r)F (r) 2 6 0 2 + (c2 + c13) c14 (c2 − c13 + c14) r A(r) F (r) , 2 2 0 2 4 b1 = −2 (c2 + c13) c14 (−c2 + c13 + c14 − 4) r A(r) A (r) − 8 (c2 + 1) (c2 + c13) c14A(r) 2 5 0 0 2 6 0 +2 (c2 + c13) c14 (c2 − c13 + c14) r A(r) A (r)F (r) + 4 (c2 + c13) c14rA(r) F (r), 2 5 0 6 b2 = 4 (c2 + c13) c14rA(r) A (r) + 4 (c2 + 1) (c2 + c13) c14A(r) 2 4 0 2 + (c2 + c13) c14 (c2 − c13 + c14) r A(r) A (r) . (A.3)

On the other hand, the coeﬃcients nn’s appearing in Eq.(3.10) are given by

c A0(r) c c 1 n = 2 − 2 − 13 − 0 2rA(r)3B(r)3 2r2A(r)2B(r)3 4r2A(r)2B(r)3 r2B(r) (c + c + c ) A0(r)F 0(r) (c + c − c ) A0(r)2 (c + 2) F 0(r) − 2 13 14 − 2 13 14 + 2 4A(r)B(r)3 8A(r)4B(r)3 2rB(r)3 (c + c − c ) A(r)2F 0(r)2 − 2 13 14 , 8B(r)3 (−c − c − c ) A0(r)2 c A(r)2F 0(r) 2c + c + 2 n = 2 13 14 − 2 + 2 13 1 4A(r)2B(r)3 2rB(r)3 2r2B(r)3 (−c − c + c ) A(r)A0(r)F 0(r) + 2 13 14 , 4B(r)3 c A(r)A0(r) (−c − c + c ) A0(r)2 (−2c − c ) A(r)2 n = − 2 + 2 13 14 + 2 13 . (A.4) 2 2rB(r)3 8B(r)3 4r2B(r)3 28

2 When cS = 1, i.e. c2 = −2c13 + 2c14 − c13c14 / (2 − 4c14 + 3c13c14), the coeﬃcients f0, a0, b0 and n0 reduce to 2 0 c14c13 + 2c13 − 2c14 f0 = rA(r) F (r) + b0, 2 (c13 − 1) c14 0 2 (c13 − 1) c14rA (r) + (c13 − 2) (c14 − 2) A(r) a0 = 2 0 2 0 f0, c13 (c14 (2rA(r) F (r) + 1) + 2) − 2c14 (rA(r) F (r) + 1) 1 2 2 2 2 2 0 2 b0 = 2 −2 (c13 − 1) c14 4c14c13 + −3c14 − 4c14 + 4 c13 + 4 (c14 − 1) c14 r A (r) ((3c13 − 4) c14 + 2) 2 2 0 2 2 2 2 0 −4 (c13 − 1) c14rA(r)A (r) 4c14c13 + 3c14 − 16c14 + 4 c13 − 4 c14 − 4c14 + 2 rA(r) F (r) 2 2 2 2 2 2 +4 (c13 − 1) c14 − 2 (c13 − 1) c14A(r) 4c14c13 + −3c14 − 4c14 + 4 c13 2 4 0 2 2 2 0 +4 (c14 − 1) c14) r A(r) F (r) + 8 (c13 − 1) c14rA(r) F (r) + 4 (c13 − 1) ((c13 − 2) c14 + 2) , (A.5) and

2 2 0 2 c14 −2c13 + (3c14 + 4) c13 − 4c14 A(r) F (r) n0 = 3 8 ((3c13 − 4) c14 + 2) B(r) 0 2 0 2 +F (r) c14 −2c13 − 3c14c13 + 4c13 + 4c14 − 4 rA (r) − 2 c14c13 + (2 − 6c14) c13 + 6c14 − 4 A(r) 3−1 2 4 3−1 × 4 ((3c13 − 4) c14 + 2) rA(r)B(r) + 8 ((3c13 − 4) c14 + 2) r A(r) B(r) 2 2 0 2 2 0 × c14 −2c13 + (3c14 + 4) c13 − 4c14 r A (r) − 4 c14c13 + 2c13 − 2c14 rA(r)A (r) 4 2 2 2 −8 ((3c13 − 4) c14 + 2) A(r) B(r) + −2c14c13 + (8c14 + 4) c13 − 8c14 A(r) . (A.6)

It should be noted that, due to the complexities of presented in [63, 67]. In the latter, we find that there are the expressions given in Eqs.(A.1)-(A.6), we extract no differences between our numerical solutions and the these coefficients directly from our Mathematica code. In ones presented in [63, 67], within the errors allowed by addition, they are further tested by the exact solutions the numerical codes. presented in [51, 53], as well as by the numerical solutions

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