Generic features of Einstein-Aether black holes
aa r X i v : . [ g r- q c ] N ov Generic features of Einstein-Aether black holes
Takashi Tamaki ∗ and Umpei Miyamoto † Department of Physics, Waseda University, Okubo 3-4-1, Tokyo 169-8555, Japan
We reconsider spherically symmetric black hole solutions in Einstein-Aether theory with the con-dition that this theory has identical PPN parameters as those for general relativity, which is themain difference from the previous research. In contrast with previous study, we allow superluminalpropagation of a spin-0 Aether-gravity wave mode. As a result, we obtain black holes having aspin-0 “horizon” inside an event horizon. We allow a singularity at a spin-0 “horizon” since it isconcealed by the event horizon. If we allow such a configuration, the kinetic term of the Aetherfield can be large enough for black holes to be significantly different from Schwarzschild black holeswith respect to ADM mass, innermost stable circular orbit, Hawking temperature, and so on. Wealso discuss whether or not the above features can be seen in more generic vector-tensor theories.
PACS numbers: 04.40.-b, 04.70.-s, 95.30.Tg. 97.60.Lf.
I. INTRODUCTION
Identifying the contents of dark energy and dark mat-ter (DE/DM) is one of the most important subjects incosmology. It is frequently argued that gravitational the-ories are an alternative to DE/DM. Recently, tensor-vector-scalar (TeVeS) theories have attracted much at-tention since they do not only explain galaxy rotationcurves but also satisfy many constraints from solar ex-periments [1]. Although a deficency in explaining themismatch between luminous and dynamical masses inclusters of galaxies by TeVeS has been pointed out [2],resolution of this problem by considering a generalizedvector-tensor theory has also been reported [3]. More-over, these vector fields might explain an accelerated ex-pansion of the universe [3, 4] and might be important ininflationary scenarios [5, 6]. The origin of such a vectorfield is argued in [7].However, it is nontrivial whether or not these theoriessatisfy the constraints by strong gravity tests. Notice theresult for scalar-tensor theories where compact objectshave strong deviations from those in general relativity(GR) even in the cases that satisfy weak field tests [8].To study vector fields in a general form is difficult. Forexample, results in TeVeS are still limited to special casessuch as [9]. Thus, as a first step, it is important to inves-tigate a simplified model which is tractable and instruc-tive for general cases. One such useful model would beEinstein-Aether (EA) theory [10], where all parameter-ized post-Newtonian (PPN) parameters [11] can be thesame as those in GR [12]. EA theory is a vector-tensortheory, and TeVeS can be written as a vector-tensor the-ory which is the extension of EA theory [13]. In EAtheory, strong gravitational cases including black holeshave been analyzed to some extent [14, 15, 16, 17, 18].Nevertheless, the analysis of black holes has been lim-ited to the case in which the event horizon coincides with ∗ Electronic address: [email protected] † Electronic address: [email protected] the spin-0 horizon [15], and this case does not neces-sarily satisfy weak fields tests. Thus, it is interestingto ask whether or not significant differences from theSchwarzschild black hole appear when weak fields testsare satisfied. For this reason, we argue black holes withthe case in which the EA theory has identical PPN pa-rameters as in GR.This paper is organized as follows. In Sec. II, we ex-plain EA theory and summarize constraints located byprevious research. In Sec. III, we mention our methodof analyzing black holes. In Sec. IV, we show the resultsand compare them with the Schwarzschild black hole. InSec. V, consequences and future subjects are discussed.In Appendix A, we summarize basic equations. We useunits in which c = 1 and follow the sign conventions ofMisner, Thorne, and Wheeler [19], e.g., ( − , + , + , +) formetrics. II. EINSTEIN-AETHER THEORYA. The action and basic equations
We consider the following action [17]: I = 116 πG Z d x √− g L , (2.1) L = R − K abcd ∇ a u c ∇ b u d + λ ( u + 1) , (2.2) K abcd := c g ab g cd + c δ ac δ bd + c δ ad δ bc − c u a u b g cd , (2.3)where u a is a vector field and u := u a u a . c i ( i =1 , , ,
4) are theoretical parameters in EA theory. λ isa Lagrange multiplier ensuring the vector field u a to beunit timelike vector everywhere.Varying this action with respect to λ and u a , we have u + 1 = 0 , (2.4) c ˙ u m ∇ a u m + ∇ m J ma + λu a = 0 , (2.5)where J am := K abmn ∇ b u n , (2.6)˙ u b := u a ∇ a u b . (2.7)Multiplying Eq. (2.5) by u a , we have λ = c ˙ u + u a ∇ m J ma . (2.8)Varying the action with respect to the metric, we have G ab = ∇ m h J m ( a u b ) − J m ( a u b ) + J ( ab ) u m i + c ( ∇ a u m ∇ b u m − ∇ m u a ∇ m u b )+ c ˙ u a ˙ u b + λu a u b − g ab L u , (2.9)where L u := K abcd ∇ a u c ∇ b u d . (2.10) B. Present Constraints in EA Theory
If we assume the weak field and slow-motion limits inEA theory [12], we have to take Newton’s gravitationalconstant as G N = (cid:18) − c + c (cid:19) − G , (2.11)to reproduce Newtonian gravity correctly. For all thePPN parameters to coincide with those in GR, we have c = − c − c c + c c , c = − c c . (2.12)If we assume Friedmann-Robertson-Walker (FRW)space-time and the Aether is aligned with a cosmolog-ical rest frame, the cosmological gravitational constant isgiven by [20] G cosmo = G (cid:18) c + + 3 c (cid:19) − , (2.13)where c + := c + c . Using primordial He abundance,we have | G cosmo /G N − | < / . (2.14)From the maximum mass of neutron stars ∼ M ⊙ [21,22], we have c + c ≤ . ∼ .
6, depending on EOS [18].In [23], the sound modes are analyzed by expandingthe metric and the Aether around the Minkowski met-ric. As in the case in GR, we have two spin-2 modes.As peculiar to EA theory, there are three wave modes.Two correspond to a transverse spin-1 mode, and onecorresponds to a longitudinal spin-0 mode. The squaredspeeds of them are summarized as( s ) = c c − c )(1 − c ) , (2.15)( s ) = c (2 c − c + c )2(1 − c ) c ( c − c ) , (2.16)( s ) = 11 − c , (2.17) where we eliminate c and c with Eq. (2.12).For these sound velocities to be equal to or larger thanthe photon velocity, or, to ensure stability against linearperturbation in Minkowski (or FRW) background andlinearized energy positivity, we have [5, 23, 24, 25]0 < c + < , < c − := c − c < c + − c + ) . (2.18)Radiation damping was also analyzed in [26, 27], whichalmost restricts c + as a function of c − based on the ob-servation of, say, B1913+16 [28]. III. ANALYSIS IN A SINGLE-NULLCOORDINATE SYSTEM
Our purpose in investigating black holes in EA the-ory is not to give a further restriction but to understandgeneric features of vector-tensor theories under the con-dition that weak gravity tests are satisfied. This is themain difference from the previous research [15], whichinvestigates black holes with the parameters [29] c = − ( c + c ) ( c − c ) − c + c )( c − c )(3 c + 4 c − c ) + 2 (3.1)and c = − c , or c = − c + c , or c = − c . In theseparameters, qualitative differences from Schwarzschildblack holes have been shown. It is nontrivial whetheror not this is true even for the case which satisfies weakgravity tests.From this point of view, we take the following strategy.(i) We assume (2.12) since the constraints by the solar ex-periments are severe. (ii) We assume (2.18). Otherwise,a naked singularity appears outside the event horizon ingeneral. As for other constraints, notice that (2.14) issatisfied if (2.12) is satisfied. Constraints from neutronstars and from radiation damping are related to stronggravity tests at least partially. For the above reasons,we do not impose these constraints. Thus, we have twotheoretical parameters ( c + , c − ) with the condition (2.18).We write a static and spherically symmetric line ele-ment in a single null coordinate system as, ds = − N ( r ) dv + 2 B ( r ) dvdr + r d Ω . (3.2)In this coordinate, the vector field takes the form of u = a ( r ) ∂ v + b ( r ) ∂ r . (3.3) b ( r ) = 0 means that the Aether is not aligned with thetimelike Killing field, which is inevitable because of theevent horizon. From Eq. (2.4), − N a + 2 Bab = − . (3.4)We can eliminate λ with Eq. (2.8). Then, from the Ein-stein and Aether equations, we obtain basic equations,which can be written schematically as N ′ = f ( B, N, a, a ′ ) , (3.5) B ′ = f ( B, N, a, a ′ ) , (3.6) a ′′ = f ( B, N, a, a ′ ) , (3.7)where the prime denotes the derivative with respect to r .Here, we have eliminated b with Eq. (3.4). The explicitform is summarized in Appendix A.The boundary condition at the horizon r h is N ( r h ) = 0.We set B ( r h ) = 1. We can also set r h = 1 since there isno scale in the present theory. In this sense, it is assumedthat the area coordinate r is normalized by the horizonradius below.If we use a rescaling freedom of v as dv ′ = B ( ∞ ) dv ,the asymptotic form of the metric is written as ds = − N ( ∞ ) B ( ∞ ) dv ′ + 2 dv ′ dr + r d Ω . (3.8)Thus, the boundary condition at spatial infinity for theasymptotic flatness is N ( ∞ ) = B ( ∞ ) . (3.9)We should require b ( ∞ ) = 0 , (3.10)for the Aether to be aligned with the timelike Killingfield. Then, by Eq. (3.4), we have a ( ∞ ) = 1 B ( ∞ ) . (3.11)We can determine the pair of a h := a ( r h ) and a ′ h := a ′ ( r h )as shooting parameters, one of which is fixed by (3.11).Thus, there remains one freedom. Fixing this freedom isdone as follows.Even in the spherically symmetric case, there is a spin-0 mode. Then, we can define the effective metric for aspin-0 mode as g (0) ab = g ab − [( s ) − u a u b . (3.12)We call the horizon associated with this metric as thespin-0 horizon. The freedom mentioned above is fixed bythe requirement that the regularity at the spin-0 horizonwhich is inside the event horizon.However, since the asymptotic observer is insensitiveto the regularity at the spin-0 horizon, we permit the sin-gularity at the spin-0 horizon. For this reason, we leaveone freedom. In concrete terms, we obtain a h iterativelyfor some a ′ h , which is regarded as a free parameter. Weuse the Bulirsch-Stoer method in our numerics [30]. IV. PROPERTIES OF SOLUTIONSA. Mass and Hawking temperature of EA blackhole
We show several asymptotically flat solutions in Figs. 1(a)-(c) for c + = 0 . c − = 0 .
1. In the figures, we have -1.5-1.4-1.3-1.2-1.1-1-0.9 1 10 100 (a) B * a r a' (cid:1) =-1 -0.500.5 (b) B r a' (cid:0) =-1 -0.500.51 (c) G m r a' (cid:2) =-1 -0.5 0 0.5 1 FIG. 1: Field configurations for c + = 0 . c − = 0 .
1. De-noted numbers in each figure, ranging from − a ′ h . We normalize the quantities Gm and r bythe horizon radius r h . The solution with the smallest a ′ h haslargest deviation from a Schwarzschild black hole. (a) G M A D M a' h (c + ,c - )=(0.4,0.1)(c + ,c - )=(0.5,0.1)(c + ,c - )=(0.6,0.1)(c + ,c - )=(0.6,0.15)(c + ,c - )=(0.6,0.18) (b) a' h G M T O T (c + ,c - )=(0.4,0.1)(c + ,c - )=(0.5,0.1) (c + ,c - )=(0.6,0.1)(c + ,c - )=(0.6,0.15)(c + ,c - )=(0.6,0.18) FIG. 2: (a) a ′ h v.s. GM ADM and (b) a ′ h v.s. GM TOT forseveral sets of c + and c − . Physical quantities are normalizedby the horizon radius r h . Notice that there is a lower limit a ′ h , crit below which there is no regular solution. Near a ′ h , crit , GM ADM and GM TOT depend on a ′ h remarkably. selected five solutions. The differences of these solutionsare the changing boundary value a ′ h , ranging from − a h that satisfies the asymptotic condi-tion (3.11) for various values of a ′ h . We also show B ( r ) inFig. 1 (b). Since B ( r ) = const . = 1 for a Schwarzschildblack hole, it indicates that there are differences in phys-ical quantities from those for Schwarzschild black holes.Figure 1 (c) shows a “mass” function. In AE theory, itis important to distinguish different notions of mass. If we define the mass function m ( r ) by m ( r ) := r G (cid:18) − NB (cid:19) , (4.1)we can interpret m ( ∞ ) as ADM mass M ADM . As wecan see, m ( r ) monotonically decreases. Our calculationsuggests that this is generic. This is not surprising sinceenergy conditions are not necessarily satisfied in EA the-ory [25].Since Figs. 1 show that the deviation from theSchwarzschild black hole is largest for the smallest valueof a ′ h , it is natural to ask whether or not there is a lowerlimit a ′ h , crit below which there is no regular solution. Weshow the relation a ′ h and M ADM for various values of c + and c − in Fig. 2 (a). Typically, M ADM is smallerthan that of a Schwarzschild black hole by about 10%,which is consistent with the result in [15]. Here, we ob-tain a h iteratively to satisfy Eq. (3.9) for each a ′ h . For a ′ h < a ′ h , crit , we cannot find an appropriate value of a h . a ′ h , crit depends on c + and c − . As a ′ h approaches a ′ h , crit , dM ADM /da ′ h tends to diverge. Since we obtain solutionsnumerically, it is nontrivial whether M ADM is boundedor not from below. In particular, it is important to re-veal the positivity of M ADM . However, since the energyconditions are not guaranteed [25], we cannot prove it atpresent.For M ADM , the difference caused by the change of c − is not clear. We can define total energy M TOT by G N M TOT = GM ADM since the gravitational constant wefeel is different from that in GR as seen in Eq. (2.11).We also exhibit the relation a ′ h - M TOT in Fig. 2 (b). Thisfigure shows the differences caused by the change of c − . M TOT decreases as c − increases as similar to c + .If we contemplate these diagrams from a different view-point, we notice that the horizon radius of black holes inEA theory is larger than that of a Schwarzschild blackhole for fixed GM TOT (or GM ADM ). Therefore, onemight think that black holes in EA theory have largerentropy. However, since we have the Lorentz violatingfield, it is nontrivial to establish black hole thermody-namics [31, 32]. Thus, the comparison of the black holeentropy, which is crucial to discuss the stability of blackholes, belongs among our future tasks.It is also important to reveal what happens at thecritical point, a ′ h = a ′ h , crit . The key point is the factor( − N a ) in the denominator in (A4). For a ′ h = a ′ h , crit ,( − N a ) becomes zero at finite r . Thus, solutionsdisappear. We also show the a ′ h dependence of Hawkingtemperature T H for c + = 0 . c − = 0 . T H diverges forthe solution at a ′ h = a ′ h , crit . Thus, It is intriguing toconsider an evaporation process of such black holes. T H a' h FIG. 3: Hawking temperature T H (normalized by r h ) for c + =0 . c − = 0 . T H diverges for the solutionat a ′ h , crit . B. ISCO of EA black hole
We shall turn to more realistic problems. We considerthe possibility of distinguishing black holes in EA the-ory from Schwarzschild black hole by observation. InRef. [18], the innermost stable circular orbit (ISCO) forneutron stars in EA theory was analyzed. The result isthat the deviation from the Schwarzschild black hole is atmost several percent. But this is not necessarily the casein the present situation, as shown below. The differencesoccur since we have the freedom parameterized by a ′ h andthe Aether is not static. These facts will be importantif we consider observations such as Constellation-X [33],which tracks the motion of individual elements near blackholes.From an equation for timelike geodesics for a unit massparticle, we have effective potential V as V ( r ) = NB (cid:18) L r + 1 (cid:19) , (4.2)where L is the angular momentum normalized by thehorizon radius.We show the typical configurations of V in Fig. 4 forEA theory (with c + = 0 . c − = 0 . a ′ h = 0 . ≃ a ′ h , crit ) and for GR (Schwarzschild black hole), where theangular momentum of the test particle L is fixed as L =1 .
5. We find that a potential minimum exists even for L = 1 . r ISCO (normalized by r h )on a ′ h in Fig. 5 (a). Notice that r ISCO = 3 for theSchwarzschild black hole. Therefore, the difference isnearly 10% for a ′ h ≃ a ′ h , crit . It is also impressive towrite the ISCO normalized by GM TOT (or GM ADM ), which is shown in Fig. 5 (b). In this case, we canfind the difference from the Schwarzschild black hole( r ISCO /GM
ADM = 6) is more than 20%.Finally, let us comment on the parameter region of( c + , c − ) in which black hole solutions exist. We obtainedsolutions even for c + , c − >
1, which seems to conflictwith the previous results [15]. However, since we do notassume regularity at the spin-0 horizon against the casein [15], it is not inconsistent. The qualitative propertiesare same as in other parameter regions, although quanti-tative differences from Schwarzschild black hole becomelarger for large ( c + , c − ) as we expect from Fig. 2. Thesefeatures are same as in [15] where the consistency withthe weak gravity tests are not necessarily imposed. V r EA Sch .
FIG. 4: The potential V for EA theory ( c + = 0 . c − = 0 . a ′ h = 0 . ≃ a ′ h , crit ) and for a Schwarzschild black holewhere the angular momentum of the test particle L (normal-ized by r h ) is fixed by L = 1 .
5. There is a potential minimumin EA theory while there is none for a Schwarzschild blackhole.
V. CONCLUSION AND DISCUSSION
We have reanalyzed black hole solutions in EA theorywhile assuming that all the PPN parameters are the sameas those for GR, resulting in two theoretical parameters c + and c − . This is a main difference from the previousstudy [15]. As another difference, we do not assume reg-ularity at the spin-0 horizon since this is inside the eventhorizon. Interestingly, we find a ′ h , crit below which thereis no regular black hole solution. Near a ′ h , crit , the de-viation of black hole mass M TOT (or M ADM ) and ISCO r ISCO from those for the Schwarzschild black hole becomelarge.These results are instructive for other cases. If we con-sider the case with rotation, freedom of the vector field (a) r I S C O a' h (b) a' h r ISCO /GM
TOT r ISCO /GM
ADM
FIG. 5: The a ′ h dependence of the innermost stable circularorbit (ISCO) for EA theory with c + = 0 . c − = 0 .
1. (a)ISCO normalized by r h . (b) ISCO normalized by GM TOT and GM ADM . is added to (3.3). Then, it also contributes the kineticterm of the vector field, enhancing the differences fromthe vacuum solution. This would also be true in othervector-tensor theories. For this reason, it is important toconsider rotational black holes in vector-tensor theories,if we are to constrain them.Although we have revealed many properties of EAblack holes, some important problems remain to be in- vestigated. One is the positivity of the energy, which isnecessary for the stability of the system. Related to this,to establish the black hole thermodynamics is also im-portant. As a consistency check, we should also performthe linear perturbation for the black holes [34].The other is whether or not regular spin-0 horizon hap-pens as a result of gravitational collapse. In [17], it isshown that regular spin-0 horizon happens if we considera gravitational collapse of a massless scalar field. Thus, itis important to investigate this feature in a general case.It is also interesting to investigate the critical behaviorof such a system [36]. Of course, these are not problemsparticular only to EA theory but also issue confronting inmore generic vector-tensor theories. Thus, it is desirableto investigate them in a unified way. Acknowledgments
We would like to thank Kei-ichi Maeda for continuousencouragement. The numerical calculations were carriedout on the Altix3700 BX2 at YITP, Kyoto University.This work is supported in part by a fund from the 21stCentury COE Program (Holistic Research and EducationCenter for Physics of Self-Organizing Systems) at WasedaUniversity.
APPENDIX A: BASIC EQUATIONS FOREINSTEIN-AETHER SYSTEM
The equation for N is X i =0 H i a i = 0 , (A1)where H = c + [(3 c − − c + ) N + 2(3 c − + c + ) rN N ′ +(3 c − − c + ) r N ′ ,H = 2 c + rN [(3 c − + c + ) N + (3 c − − c + ) rN ′ ] a ′ ,H = − c − + c + ) B ( r ) + 2[ − c − ( − c + ) + c + (6 + c + )] N − c − ( − c + ) + ( − c + ) c + ] rN ′ +(3 c − − c + ) c + r N a ′ ,H = − c + (3 c − + c + ) r N ′ a ′ ,H = − c + a [ − c − + c + + 2(3 c − + c + ) r N a ′ ] ,H = − c + (3 c − + c + ) ra ′ ,H = (3 c − − c + ) c + r a ′ . (A1) is the quadratic equation for N ′ (Notice H ). Ifwe solve (A1) about N ′ , we obtain the equation whichsatisfies asymptotically flatness as N ′ = P i =0 h i a i + 2 √ √ X (3 c − − c + ) c + ra , (A2)where h = − c + (3 c − + c + ) N , h = c + ( c + − c − ) rN a ′ ,h = 3 c − ( c + −
2) + ( c + − c + , h = c + (3 c − + c + ) ra ′ ,X = [ − c − ( c + − − ( c + − c + c − c + (6 − c + + c )] a + c + [(3 c − + 2 c − c + − c ) B + 2 c + ( c − + c + − c − c + ) N ] a + c − c N a + c + [3 c − ( c + −
1) + c − ( c + − c + − c ] raa ′ − c + [3 c − ( c + −
1) + c − ( c + − c + + c ] rN a a ′ + c − c r a ′ . Notice the denominator in (A2). For c + = 3 c − , we shoulduse (A1).The equation for B is B ′ = B P i =0 g i a i Y , (A3)where g = − c + [3 c − ( c + −
1) + c − ( c + − c + + c ] N × [(3 c − − c + ) N + 2(3 c − + c + ) rN N ′ + (3 c − − c + ) r N ′ ] ,g = 2 c + [3 c − ( c + −
1) + c − ( c + − c + + c ] rN × [(3 c − + c + ) N + (3 c − − c + ) rN ′ ] a ′ ,g = −{ c − ( c + −
1) + c − ( c + − c + c + c − c + × (4 c + − B N + [ − c (12 + c + ) + c − c (12 − c + +3 c ) + 9 c − (4 − c + + 5 c ) + 3 c − c + (20 − c + + 8 c )] N +4[ − c + c − c (3 − c + − c ) + 9 c − (1 − c + + 2 c ) +3 c − c + (5 − c + + 4 c )] rN N ′ + c + [27 c − ( c + − − c − c + + c + c − c (7 + 5 c + )] r N ′ + c + [ − c − ( c + −
1) +3 c − c + + c − ( − c + ) c + c ] r N a ′ } ,g = − c + rN { [27 c − ( c + − − c − c − c (13 + 5 c + ) +3 c − c + ( −
13 + 6 c + )] N + 2[18 c − ( c + −
1) + 3 c − ( − c + ) c + + c + c − c (4 + 3 c + )] rN ′ } a ′ ,g = 12[9 c − ( c + − − c + c − c ( −
11 + 3 c + ) + c − c + × ( −
19 + 12 c + )] B + [ c (12 + c + ) + 9 c − (12 − c + +7 c ) + c − c (132 − c + + 9 c ) + 3 c − c + (76 − c + +16 c )] N + 2[ − ( c + − c + c − c (66 − c + − c ) +27 c − (2 − c + + c ) + 3 c − c + (38 − c + + 6 c )] rN ′ − c + [45 c − ( c + −
1) + 3 c + 3 c − c + ( −
13 + 4 c + ) + c − c (9 + 7 c + )] r N a ′ ,g = − c + r { [9 c − ( c + −
1) + c + c − c (1 + c + ) +3 c − c + ( − c + )] N + [ − c − ( c + −
1) + c +3 c − c + (5 + 2 c + ) + c − c ( −
11 + 5 c + )] rN ′ } a ′ ,g = − c + { c − ( c + −
1) + c + c − c ( −
17 + 5 c + ) +3 c − c + ( −
15 + 8 c + ) + [ − c − ( c + −
1) + 3 c +3 c − c + (11 + 4 c + ) + c − c ( −
27 + 11 c + )] r N a ′ } ,g = − c + [ − c − ( c + −
1) + 3 c − (13 − c + ) c + + c + c − c (13 + 5 c + )] ra ′ ,g = − c + [27 c − ( c + − − c − c + + c + c − c (7 + 5 c + )] r a ′ ,Y = 12[ c − ( c + −
1) + c − ( c + − c + − c ] ra ×{ − c − ( c + −
1) + c + ] N a +[3 c − ( c + −
1) + c + ]( N a + 1) } . The equation for a is a ′′ = P i =0 f i a i c + r ( − N a ) Y , (A4)where f = 2 c − c N [(3 c − − c + ) N + 2(3 c − + c + ) rN N ′ +(3 c − − c + ) r N ′ ] ,f = 4 c − c rN [(3 c − + c + ) N + (3 c − − c + ) rN ′ ] a ′ ,f = c N {− c − c + ( c − + c + ) B N + [9 c − ( c + − − c + c − c (17 + 7 c + ) − c − c + ( − c + )] N +2[9 c − ( c + −
1) + c − (13 − c + ) c + 7 c − c − c + (1 +4 c + )] rN N ′ + [9 c − ( c + −
1) + 3 c − (5 − c + ) c + − c + c − c (17 + 3 c + )] r N ′ + 2 c − (3 c − − c + ) c r N a ′ } ,f = c rN { [9 c − ( c + −
1) + 15 c − c − c + (9 + 4 c + ) − c − c (3 + 5 c + )] N − c + [ − c − c + + c + c − ( − c + )] rN N ′ + [ − c − ( c + −
1) + 3 c − c + + c − ( − c + ) c + c ] r N ′ } a ′ ,f = c + N (cid:0) − c − ( c + −
1) + c − (3 − c + ) c + 3 c − c − c + (3 + 4 c + )] B N + 2[(18 − c + ) c + 3 c − c + ( − c + ) + c − c (18 − c + + 3 c ) − c − (2 − c + +3 c )] rN N ′ + 6 c + [ − c − ( c + − − c − c + c + c − c + ( − c + )] r N ′ − c + [3 c − ( c + −
1) + c − ( − c + ) c + 3 c + c − c + ( −
11 + 20 c + )] r N a ′ − N { c − ( c + − − c (3 + 5 c + ) + c − c (3 + 10 c + +2 c ) − c − c + ( − c + + 3 c )] − c + [ − c − ( c + −
1) +3 c − c + + c − ( − c + ) c + c ] r N ′ a ′ } (cid:1) ,f = c + rN a ′ { c − ( c + −
1) + c − ( c + − c + c + c − c + (4 c + − B N − c ( − c + ) + c − c (6 − c + + c ) + 9 c − (2 − c + + 3 c ) + 3 c − c + (10 − c + +12 c )] N − c − ( c + − − c + c − c (1 + 10 c + ) + c − c + (5 + 4 c + − c )] rN N ′ + 4 c − c + [9 c − ( c + − − c − c + + c (3 + c + )] r N ′ + c + [ − c − ( c + −
1) +3 c − c + + c − ( − c + ) c + c ] r N a ′ } ,f = c + (cid:0) c − ( c + −
1) + c − (3 − c + ) c + 3 c − c − c + (3 + c + )] B N + 2 c + [27 c − ( c + −
1) + c ( −
48 +5 c + ) + c − c + ( −
96 + 47 c + + c ) + 3 c − ( −
16 + 5 c + +4 c )] rN N ′ + [ c (12 + c + ) − c − c (12 − c + + c ) +9 c − ( − c + + c ) − c − c + (20 − c + + 8 c )] r N ′ − c + [ − c − ( c + −
1) + 2 c + c − c (29 + c + ) − c − c + ( − c + )] r N a ′ + 2 N {− c (2 + c + ) + c − c ( −
18 + 21 c + + c ) + 9 c − (2 − c + + 3 c ) − c − c + ( − c + + 4 c ) + c + [45 c − ( c + −
1) + c +3 c − c + ( −
11 + 2 c + ) + c − c (13 + 5 c + )] r N ′ a ′ } (cid:1) ,f = c + ra ′ {− c − [3 c − ( c + −
1) + ( c + − c +2 c − c + ( − c + )] B N − c + [ − c − ( c + −
1) +6 c − (10 − c + ) c + + 4 c + c − c (37 + c + )] N +8[ c (3 + 2 c + ) + 9 c − ( − c ) + c − c ( − c + +2 c ) − c − c + (5 − c + + 3 c )] rN N ′ − c + [27 c − ( c + − − c − c + + c + c − c (7 + 5 c + )] r N ′ +2 c + [27 c − ( c + −
1) + c + 3 c − c + ( − c + ) + c − c (7 + 3 c + )] r N a ′ } ,f = − (cid:0) c + − c + c − c + ( −
12 + 7 c + ) − c − c × (12 − c + + c ) + c − ( − c + + c )] B +[ − c ( −
72 + 30 c + + c ) + 9 c − (8 − c + + c + c ) + c − c (216 − c + + 23 c + c ) + 3 c − c + (72 − c + +11 c + 2 c )] rN ′ + c + [(6 − c + ) c + c − c (66 − c + − c ) + 27 c − (2 − c + + c ) + 3 c − c + (38 − c + + 40 c )] r N a ′ + N {− c ( −
36 + 9 c + + c ) +18 c − (4 − c + + c ) − c − c ( −
216 + 126 c + − c + c ) − c − c + ( −
72 + 66 c + − c + c ) + c [63 c − ( c + − − c − c + + c + c − c (19 + c + )] r N ′ a ′ } (cid:1) ,f = − c + ra ′ {− c − ( c + − − c + c − c (3 c + −
11) + c − c + ( −
19 + 12 c + )] B + [(6 − c + ) c + c − c × (66 − c + − c ) + 9 c − (6 − c + + 5 c ) + 3 c − c + × (38 − c + + 22 c )] N + 6[ − c + c − c + ( −
19 + 8 c + ) + c − c ( −
11 + 2 c + + c ) + 3 c − ( − c + + c )] rN ′ +2 c − c + [27 c − ( c + − − c − c + − ( − c + ) c ] r N a ′ } , f = c + {− c − − c − c + + 27 c − c + − c − c +39 c − c + 9 c − c − c + 13 c − c + c − c − c +2[4 c (3 + c + ) + 18 c − (6 − c + + c ) + c − c (132 − c + + 3 c ) + 3 c − c + (76 − c + + 31 c )] r N a ′ − c + [ − c − ( c + −
1) + c + 3 c − c + (5 + 2 c + ) + c − c ( −
11 + 5 c + )] r N ′ a ′ } ,f = c + ra ′ { c + c − c (144 − c + − c ) +9 c − (16 − c + + 3 c ) + 3 c − c + (96 − c + + 12 c ) − c + [ − c − ( c + −
1) + c + 3 c − c + (9 + 2 c + ) + c − c ( −
17 + 3 c + )] r N a ′ } ,f = − c + [( −
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