Black hole accretion in scalar-tensor-vector gravity
BBlack hole accretion in scalar–tensor–vector gravity
Anslyn J. John ∗ National Institute for Theoretical Physics (NITheP),Stellenbosch 7600, South Africa andInstitute of Theoretical Physics, Stellenbosch University, Stellenbosch 7600, South Africa
Abstract
We examine the accretion of matter onto a black hole in scalar–tensor–vector gravity (STVG).The gravitational constant is G = G N (1 + α ) where α is a parameter taken to be constant forstatic black holes in the theory. The STVG black hole is spherically symmetric and characterisedby two event horizons. The matter falling into the black hole obeys the polytrope equation of stateand passes through two critical points before entering the outer horizon. We obtain analyticalexpressions for the mass accretion rate as well as for the outer critical point, critical velocity andcritical sound speed. Our results complement existing strong field tests like lensing and orbitalmotion and could be used in conjunction to determine observational constraints on STVG. ∗ Electronic address: [email protected] a r X i v : . [ g r- q c ] M a r . INTRODUCTION The concordance model of cosmology (ΛCDM) is a remarkably successful paradigm forthe origin and development of large scale structure [1], [2]. In order to account for observedgalaxy rotation curves, weak lensing and the formation of galaxy clusters the model requiresmost of the matter in the universe to only interact gravitationally. Cold dark matter neces-sitates an extension to the standard model of particle physics. The late time acceleration ofthe universe can be explained by introducing an energy component with a negative equationof state parameter viz. dark energy. The cosmological constant is a leading candidate fordark energy whilst other proposals include dynamical scalar fields e.g. quintessence.An alternative school of thought to ΛCDM is to modify general relativity without in-troducing dark matter and dark energy. Hypothesised theories include TeVeS [3] and f ( R )gravity [4].A relativistic theory of gravitation, scalar–tensor–vector gravity (STVG), was introducedin [5]. In this theory the gravitational ‘constant’, G , as well as a vector field coupling, ω , andthe vector field mass, µ , are treated as dynamical scalar fields. STVG is similar to earlierproposed modifications of general relativity viz. non–symmetric gravitational theory (NGT)[6] and metric–skew–tensor gravity (MSTG) [7] in that they all introduce an extra degreeof freedom due to skew–symmetric fields coupling to matter. In MSTG, which is the weakfield approximation to NGT, the skew field is a rank three tensor, whilst in STVG the fieldis a rank two tensor.The trajectory of test particles in STVG obeys a modified acceleration law that providesa good fit to galaxy rotation curves [8] and cluster data [9] without invoking non–baryonicdark matter. The modified acceleration law adds a repulsive Yukawa force to the Newtonianlaw. This corresponds to the exchange of a massive spin 1 boson whose effective massand coupling to matter can vary with distance scale. Adding a scalar component to theNewtonian force law corresponds to exchanging a spin 0 particle and an attractive Yukawaforce. Consequently a purely scalar correction cannot provide an acceleration law satisfyinggalaxy rotation curves and cluster data. Note that the STVG, MSTG, and NGT theoriesall satisfy cosmological tests. The degeneracy between these modified theories could bebroken by testing their predictions in strong gravitational fields. This motivates our studyof accretion onto STVG black holes. 2he final stage of gravitational collapse of a compact object in STVG is a static, spher-ically symmetric black hole [10]. This object has an enhanced gravitational constant, G = G N (1 + α ), and a repulsive gravitational force with charge Q = √ αG N M where G N is Newton’s constant, α is a dimensionless parameter and M is the black hole’s mass.This black hole spacetime admits two event horizons and its Kruskal–Szekeres completionwas found in [10]. In the same article Moffat obtained a rotating black hole characterised byits mass, M , angular momentum, a , and the STVG parameter, α . He also determined themotion and stability of a test particle in orbit about the black hole, the radius of the photo-sphere and constructed a traversable wormhole solution. An earlier paper [11] determinedthe sizes and shapes of shadows cast by STVG black holes. The dynamics of neutral andcharged particles around STVG black holes immersed in magnetic fields was investigated in[12].A substantial body of literature in astrophysics is devoted to the problem of matter ac-creting onto stars and black holes [13–16]. In the context of general relativity black holeaccretion was studied by Michel [17]. Shapiro determined the luminosity and frequency spec-trum of gas accreting onto a black hole [18] as well as the effects of an interstellar magneticfield [19]. He also solved the accretion problem on a rotating black hole [20]. The significanceof the gas backreaction on the accretion rate was explored in [21] and [22]. Charged blackhole accretion was investigated by Michel [17] and Ficek [23] who included the effects of thecosmological constant. Accretion onto a broad class of static, spherically symmetric space-times was analysed by Chaverra and Sarbach [24]. Studies of higher dimensional accretionwere undertaken in [25] and [26] while quantum gravity corrections were included in [27].In section II we briefly describe the action for scalar–tensor–vector gravity, outline thederivation of its static black hole solution and highlight its key features. In section III wedetermine the accretion rate for matter falling into the black hole. In section IV we analysethe accretion rate and critical radius for various values of the modified gravity parameter, α , and adiabatic index, γ . We state our conclusions in the final section. II. BLACK HOLES IN MODIFIED GRAVITY
The scalar–tensor–vector theory of gravity [5] belongs to the class of modified gravitytheories with varying fundamental constants. In addition to a modified Einstein–Hilbert3ction the theory introduces three scalar fields and a vector field. The action governing thetheory is given by S = S grav + S φ + S S + S matter (1)where S grav = 116 π (cid:90) d x √− g (cid:20) G ( R + 2Λ) (cid:21) (2a) S φ = (cid:90) d x √− g (cid:20) ω (cid:18) B ab B ab + V ( φ ) (cid:19)(cid:21) (2b) S S = (cid:90) d x √− g (cid:20) G (cid:18) g ab ∇ a G ∇ b G − V ( G ) (cid:19) + 1 G (cid:18) g ab ∇ a ω ∇ b ω − V ( ω ) (cid:19) + 1 µ G (cid:18) g ab ∇ a µ ∇ b µ − V ( µ ) (cid:19)(cid:21) (2c)and S matter represents the action for the matter component. S grav is the standard Einstein–Hilbert action where Newton’s constant G N has been promoted to a dynamical scalar field, G ( x a ). In S φ we have a Maxwell–like contribution to the action from the vector field φ a which is defined via B ab = ∂ a φ b − ∂ b φ a . (3)Each of the three scalar fields viz. G , ω and µ has an associated potential, V .The static, spherically symmetric black hole solution for STVG was obtained in [10]by solving the vacuum field equations derived from the action (1). The matter energy–momentum tensor vanishes ( T matter = 0) and we neglect the influence of the cosmologicalconstant (Λ = 0). The enhanced gravitational coupling, G = G N (1 + α ), is taken to beconstant i.e. ∂ a G = 0. The field coupling the vector field, φ a , to the action is also taken tobe constant viz. ω = 1. The energy–momentum tensor due to the vector field is given by T ( φ ) ab = − π (cid:0) B cb B ac − B ab B ab (cid:1) . In order to successfully reproduce galaxy rotation curvesand cluster dynamics the vector field mass has to be m φ = 2 . × − eV, which correspondsto a scale of 0 . − . The field mass is negligible on the scale of compact objects andcan be safely ignored for black holes in the theory. The vacuum field equations for the vectorfield are given by ∇ b B ab = 0 (4a) ∇ c B ab + ∇ b B ca + ∇ a B bc = 0 . (4b)4he spherically symmetric spacetime due to a black hole of mass M in STVG is describedby the line element ds = (cid:18) − GMr + αGG N M r (cid:19) dt − (cid:18) − GMr + αGG N M r (cid:19) − dr − r dθ − r sin θdφ (5)where G = G N (1 + α ), G N is Newton’s gravitational constant and α is a dimensionless con-stant. In the complete STVG theory the fundamental ‘constants’ vary with time. The blackhole solution (5) however is static; hence the modified gravitational constant is implicitlyfixed. The speed of light is normalised. The Schwarzschild solution of general relativity isrecovered in the limit where α → r ± = G N M (cid:104) α ± (1 + α ) / (cid:105) and is formally similar to the Reissner-Nordstr¨om line element describing a charged black hole. As in that case the inner horizonis a Cauchy horizon which we expect to also be unstable.The similarity of the STVG black hole to the Reissner-Nordstr¨om solution is not un-surprising given the presence of a Maxwell–like vector field. The STVG vector field is a4–potential sourced by the gravitational ‘charge’ viz. mass. In Einstein–Maxwell theorythe analogous potential is sourced by the electric charge. Electrically charged black holeshave no physical significance as any hypothetical compact object that acquires a charge willaccrete charges of the opposite sign and rapidly neutralize. III. ACCRETION IN STVGA. Conservation laws
In spherically symmetric accretion the gas surrounding a non–rotating black hole is ini-tially at rest. Under the influence of the black hole’s gravitational attraction the gas acceler-ates inwards. The gas velocity reaches its local sound speed and then continues to acceleratetowards the black hole at supersonic velocities.The gas accreting onto the black hole is modelled as a perfect fluid with energy–momentum tensor T ab = ( ρ + p ) u a u b − pg ab (6)where ρ , p and u a are the fluid’s energy density, pressure and 4–velocity respectively. Since5he gas flow is stationary and spherically symmetric, its only non–vanishing velocity com-ponents are u ( r ) and u ≡ v ( r ). Under the normalisation condition u a u a = 1 the temporalcomponent of the 4–velocity is u = (cid:113) − GMr + αG N GM r + v − GMr + αG N GM r . (7)If particle number is conserved during the flow then ∇ a ( nu a ) = 0 (8)where n is the fluid’s number density and ∇ a is the covariant derivative with respect to thecoordinate x a . Conservation of energy–momentum is governed by ∇ a T ab = 0 . (9)For a perfect fluid accreting onto the black hole (5), the continuity equation (8) is1 r ddr (cid:0) r nv (cid:1) = 0 (10)while equation (9) can be re–written as1 r ddr (cid:32) r ( ρ + p ) v (cid:18) − GMr + αG N GM r + v (cid:19) / (cid:33) = 0 (11) v dvdr = − dpdr (cid:16) − GMr + αG N GM r + v (cid:17) ρ + p − GMr (cid:18) − αG N Mr (cid:19) . (12)We restrict our attention to adiabatic flows so the first law of thermodynamics for the fluidis given by T ds = 0 = d (cid:16) ρn (cid:17) + pd (cid:18) n (cid:19) (13)which, upon integration, yields dρdn = ρ + pn . (14)Using the fluid’s adiabatic sound speed, a ≡ dpdρ = dpdn nρ + p , (15)we express the continuity (10) and momentum (12) equations as v (cid:48) v + n (cid:48) n = − r (16) vv (cid:48) + (cid:18) − GMr + αGG N M r + v (cid:19) a n (cid:48) n = − GMr (cid:18) − αG N Mr (cid:19) (17)6here primes denote spatial derivatives. The number density and velocity derivatives canbe written as n (cid:48) = D D (18) v (cid:48) = D D (19)where we have defined D = v − (cid:16) − GMr + αGG N M r + v (cid:17) a nv (20) D = − v (cid:18) v r − GMr (cid:18) − α G N Mr (cid:19)(cid:19) (21) D = 1 n (cid:20)(cid:18) − GMr + αGG N M r + v (cid:19) a r − GMr (cid:18) − α G N Mr (cid:19)(cid:21) . (22)Introducing m , the mass of an individual gas particle, we obtain the mass accretion rate byintegrating the continuity equation (10) over a unit volume˙ M = 4 πmnr v. (23)The accretion rate, ˙ M has dimensions of mass . time − and is independent of r . Equations(10) and (11) can be combined to yield (cid:18) ρ + pn (cid:19) (cid:18) − GMr + αGG N M r + v (cid:19) = E (24)which is the relativistic version of the Bernoulli equation. If the gas is at rest at largedistances from the black hole i.e. v ∞ = 0 the integration constant is E = (cid:16) ρ ∞ + p ∞ n ∞ (cid:17) andhas dimensions of enthalpy squared. B. Critical points
The gas is at rest very far from the black hole. Under the influence of the black hole’sgravitational field it accelerates inwards, eventually falling into the outermost event horizon.In the original Bondi problem (accretion driven by a Newtonian potential) the gas acceleratesfrom rest and passes through a critical point, where its velocity matches its local sound speed.The gas then flows towards the central mass at supersonic velocities in a manner analogousto flow through a de Laval nozzle [16]. A similar velocity profile occurs for accretion onto7 Schwarzschild black hole. In this case, however, the gas velocity at the critical point doesnot equal its sound speed. Here we establish the existence of two critical points in accretiononto a STVG black hole.A critical point occurs whenever the quantity D in (18) - (19) vanishes. In order to avoidinfinite acceleration the expressions D and D must simultaneously vanish. The criticalpoint conditions are thus D = D = D = 0 at some particular values of r .At large distances the gas velocity is subsonic i.e. v < a . Moreover for conventionalmatter the sound speed is always subluminal i.e. a ≤
1. Thus we have D ≈ v − a nv forlarge values of r . Since the gas flows inwards we have v < D >
0. At theoutermost horizon we have D = vn (1 − a ) hence D <
0. Thus D = 0 at some distance r s between the outer event horizon and infinity i.e. there is at least one critical point satisfying r + H < r s < ∞ .The critical point conditions viz. D = D = D = 0 at r s are v s − (cid:18) − GMr s + αGG N M r s + v s (cid:19) a s = 0 (25) (cid:18) − GMr s + αGG N M r s + v s (cid:19) a s r s − GMr s (cid:18) − α G N Mr s (cid:19) = 0 (26)2 v s r s − GMr s (cid:18) − α G N Mr s (cid:19) = 0 (27)where v s ≡ v ( r s ) etc. Introducing the dimensionless variable y ≡ GMr the critical points arelocated at y ± s = 3 a s + 12 αα +1 a s + 1 (cid:104) ± √ ∆ (cid:105) (28)where ∆ ≡ − αα +1 ( a s + 1) a s (3 a s + 1) . (29)We confine our attention to the outermost critical point which we label y s = 3 a s + 12 αα +1 a s + 1 (cid:104) √ ∆ (cid:105) . (30)At this point the critical velocity is given by v s = 12 y s (cid:18) − αα + 1 y s (cid:19) (31)= (3 a s + 1)( a s + 1)(1 − √ ∆)4 αα +1 ( a s + 1) . (32)8 . The accretion rate We now evaluate the Bernoulli equation (24) at the critical point to determine the criticalsound speed, a s , in terms of a ∞ . We employ a polytropic equation of state for the gas viz. p = Kn γ (33)where 1 < γ < /
3. For this polytrope the energy equation (13) can be integrated to obtain ρ = Kγ − n γ + mn (34)where mn is the rest–energy density. Using (15) the Bernoulli equation (24) is rewritten as (cid:18) − γ − a (cid:19) (cid:18) − GMr + αGG N M r + v (cid:19) − = (cid:18) − γ − a ∞ (cid:19) . (35)We use the fact that a ≤ a s (cid:28) y s , and obtain(1 + 3 a s ) (cid:18) − γ − a s (cid:19) ≈ − γ − a ∞ (36)The critical sound speed, to leading order, is thus a s = 25 − γ a ∞ . (37)We can determine the critical number density, n s , by combining (15), (33) and (34) to obtain γKn γ − = ma − a / ( γ −
1) (38) ≈ ma (39)where we have exploited the relation a / ( γ − (cid:28)
1. Since n ∼ a / ( γ − we have (cid:18) n s n ∞ (cid:19) ≈ (cid:18) a s a ∞ (cid:19) γ − . (40)The mass accretion rate, ˙ M , is independent of r . In particular, equation (23) must hold atthe outer critical point, r s , hence ˙ M = 4 πmn s r s v s . (41)The rate at which polytropic matter accretes adiabatically onto a STVG black hole is˙ M = π ( GM ) mn ∞ a ∞ (cid:18) − γ (cid:19) γ − γ − (cid:20) a ∞ (5 − γ ) (cid:21) (cid:34)(cid:18) − γa ∞ (cid:19) + 6 − αα + 1 (cid:35) . (42)9 V. ANALYSIS
In the limit α → M = 4 π (cid:18) GMa ∞ (cid:19) mn ∞ a ∞ (cid:18) (cid:19) γ +12( γ − (cid:18) − γ (cid:19) γ − γ − . (43)Note that as with spherical accretion in general relativity, the accretion rate onto a STVGblack hole is proportional to the square of the black hole’s mass i.e. ˙ M ∝ M . We will onlyconsider solar mass black holes i.e. M = M (cid:12) = 1 . × g. The accreting gas is taken tobe ionized hydrogen with molecular mass m H = 1 . × − g at temperature T = 10 Kand number density n ∞ = 1cm − . For an ideal gas the sound speed is a = γk B T /µm . Themean molecular weight for ionized hydrogen is µ = 1 /
2. We fix the adiabatic index, γ = 5 / γ as the gas falls towards the black hole. Theboundary condition for the gas sound speed is thus a ∞ = 2 . × cm s − . For ouranalysis we utilize geometric units where G = c = 1.In Fig. 1 we plot the mass accretion rate, ˙ M , as a function of the gravitational parameter, α . We don’t consider very large values of α as this would imply significant deviations awayfrom the standard value of Newton’s constant. The values for ˙ M in general relativity,where a polytrope accretes onto a Schwarzschild black hole are recovered at α = 0. Here theaccretion rate increases as the adiabatic index increases. A gas that is close to the isothermallimit, γ = 1, accretes at a lower rate than relativistic gases, γ = 4 /
3. Non–relativistic gases, γ = 5 /
3, accrete at the fastest rate.This behaviour persists when one looks at STVG accretion. A non–relativistic gas ac-cretes at a greater rate than a relativistic gas or an isothermal gas. The accretion rate foreach class of gas rises gently then slowly decreases as the STVG parameter, α is increased.The most pronounced increase occurs as we approach the non–relativistic limit, γ = 5 / IG. 1: (color online) The accretion rate, ˙ M , as a function of the STVG parameter, α , for variousadiabatic indices, γ . Values are expressed in geometric units. responsible for the energy emitted by active galactic nuclei [14]. In this case the efficiencyfor energy conversion is significantly higher than for spherical accretion. V. CONCLUSION
The scalar–tensor–vector theory of gravity (STVG) is a modified theory of gravity thatsatisfies a number of cosmological tests. The black hole solutions of the theory possess twoevent horizons and depend on the black hole’s mass, M , angular momentum, a , and theSTVG parameter, α , which characterises deviations from the gravitational constant, G .We studied the accretion of a polytrope onto a non–rotating black hole in STVG. Thegas is at rest far from the black hole, then accelerates towards its outer event horizon. Weestablished the existence of a critical point, r s in the flow and calculated its location aswell as the gas velocity, v s and sound speed, a s , at the critical point. We determined ananalytical expression for the rate at which gas accretes onto the black hole. The accretionrate, ˙ M is parametrised by α and the adiabatic index, γ . In the limit that α → α increases, the accretion rate for the gas increases then decreases slightly. Thegas properties, characterised by γ , have a greater effect on the accretion rate than the changein gravitational theory, parametrised by α . Uncertainty in gas dynamics thus dominatesover uncertainty in the gravitational theory. Since changes in the accretion rate in thisidealised adiabatic, spherical problem are quite subtle it would appear to be quite difficultto distinguish between general relativity and STVG using accreting systems alone. Strongfield tests of STVG should incorporate accretion, lensing and test particle motion.In order to determine whether STVG dynamics could power active galactic nuclei it isnecessary to formulate the accretion problem for a rotating black hole. This study will bethe subject of a forthcoming paper. It would also be instructive to compare our results tothose for matter accreting onto black holes in similar modified theories of gravity such asnon–symmetric gravitational theory (NGT) and metric–skew–tensor gravity (MSTG). Acknowledgements
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