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Accretion disks around black holes in Scalar-Tensor-Vector Gravity

Daniela Perez,∗ Federico G. Lopez Armengol,† and Gustavo E. Romero‡

Instituto Argentino de Radioastronomıa(CCT - La Plata, CONICET; CICPBA),Camino Gral Belgrano Km 40 C.C.5,

1894 Villa Elisa, Buenos Aires, Argentina

(Dated: May 9, 2017)

Scalar Tensor Vector Gravity (STVG) is an alternative theory of gravitation that has successfullyexplained the rotation curves of nearby galaxies, the dynamics of galactic clusters, and cosmologicaldata without dark matter, but has hardly been tested in the strong gravity regime. In this work,we aim at building radiative models of thin accretion disks for both Schwarzschild and Kerr blackholes in STVG theory. In particular, we study stable circular equatorial orbits around stellarand supermassive black holes in Schwarzschild and Kerr STVG spacetimes. We also calculate thetemperature and luminosity distributions of accretion disks around these objects. We find thataccretion disks in STVG around stellar and supermassive black holes are colder and less luminousthan in GR. The spectral energy distributions obtained do not contradict current astronomicalobservations.

I. INTRODUCTION

General Relativity (GR) is a theory about the interac-tion of spacetime and other material objects. One hun-dred years after the discovery of the field equations [1],GR continues proving its extraordinary predictive power[2]. Still, GR is a defective theory as pointed out byEinstein himself with regard to the problem of spacetimesingularities [3]. GR also fails at reproducing rotationcurves of nearby galaxies, mass profiles of galaxy clus-ters, some gravitational lensing effects, and cosmologicaldata. In order to accommodate GR to astronomical ob-servations, the theory is modified adding a term witha cosmological constant to the field equations, and theexistence of dark matter is postulated. However, everyexperiment aimed at directly measuring the properties ofsuch matter has failed in its quest so far [4–6].A different approach to explain the astronomical data

is to introduce a more drastic modification to the the-ory of gravitation. Following this path, Moffat[7] postu-lated the Scalar-Tensor-Vector Gravity theory (STVG),also called MOdified Gravity theory (MOG). In STVG,in addtion to the tensor metric field, a vector field isintroduced, and the universal gravitational constant G,along with the mass µ of the vector field are consideredas dynamical scalar fields. In particular, for weak grav-itation, STVG leads to a modified acceleration law thathas two key features: first, an enhanced Newtonian ac-celeration law, quantified by G = GN(1 + α) that hassuccessfully explained the rotation curves of many galax-ies [8–10], the dynamics of galactic clusters [11–13], and

∗ danielaperez@iar.unlp.edu.ar† flopezar@iar.unlp.edu.ar‡ romero@iar.unlp.edu.ar; Also at Facultad de Ciencias As-tronomicas y Geofısicas, UNLP, Paseo del Bosque s/n CP, 1900La Plata, Buenos Aires, Argentina

cosmological observation [14], without dark matter. Sec-ond, a repulsive Yukawa force term that counteracts theaugmented Newtonian acceleration law at certain scales,in such a way that the Solar System results of GR arerecovered.

The strong gravity regime of STVG has just started tobe explored [15]. Moffat [16] found solutions of STVGfield equations that represent a static, spherically sym-metric black hole and also a stationary, axially symmet-ric black hole, hereafter called Schwarzschild STVG andKerr-STVG black holes, respectively. The equations tocalculate the black hole shadow and lensing effects forSchwarzschild and Kerr STVG black holes were derivedin [16, 17]. Hussain and Jamil [18] analysed the dynamicsof neutral and charged particles around a SchwarzschildSTVG black hole inmersed in a weak magnetic field, andstudied the stability of the circular orbits. Recently, John[19] studied the spherically symmetric accretion of a gaswith a polytropic equation of state into a SchwarzschildSTVG black hole. Analytical expressions were obtainedfor the mass accretion rate, critical velocity, and criticalsound speed.

In the present work, we construct thin relativistic ac-cretion disks around Schwarzschild and Kerr black holesin STVG theory. In particular, we calculate tempera-ture and spectral energy distributions of accretion disksaround stellar and supermassive black holes, and com-pare the results with those obtained in GR. We analysethe viability of the theory in the strong gravity regimeby contrasting our predictions with current astronomicaldata.

The paper is organised as follows. In Sect. II we pro-vide a brief review of STVG theory. We study circularorbits in both Schwarzschild and Kerr STVG spacetimesin Sect. III. Section IV is devoted to the calculation ofthe properties of accretion disks in these spacetimes, andin Sect. V we analyse the results previously obtained.We offer the conclusions of the work in the last section.

2

II. STVG GRAVITY

A. STVG action and field equations

The action [20] in STVG theory is [7]:

S = SGR + Sφ + SS + SM, (1)

where

SGR =1

16π

∫

d4x√−g

1

GR, (2)

Sφ = −∫

d4x√−g

(

1

4BµνBµν − 1

2µ2φµφµ

)

, (3)

SS =

∫

d4x√−g

1

G3

(

1

2gµν∇µG∇νG− V (G)

)

(4)

+

∫

d4x1

µ2G

(

1

2gµν∇µµ∇ν µ− V (µ)

)

. (5)

Here, gµν is the spacetime metric, R denotes the Ricciscalar, and ∇µ is the covariant derivative; φµ standsfor a Proca-type massive vector field, µ is its mass, andBµν = ∂µφν − ∂νφµ; G(x), and µ(x) are scalar fieldsthat vary in space and time, and V (G), and V (µ) are thecorresponding potentials. We adopt the metric signatureηµν = diag(−1,+1,+1,+1). The term SM in the actionrefers to possible matter sources.The full energy-momentum tensor for the gravitational

sources is:

Tµν = TMµν + T φ

µν + T Sµν , (6)

where

TMµν = −

2√−g

δSM

δgµν, (7)

T φµν = − 2√

−g

δSφ

δgµν, (8)

T Sµν = − 2√

−g

δSS

δgµν. (9)

Following the notation introduced above, TMµν denotes the

ordinary matter energy-momentum tensor and T φµν the

energy-momentum tensor of the field φµ; T Sµν denotes the

scalar contributions to the energy-momentum tensor.The Schwarzschild and Kerr STVG black hole solutions

were found under the following assumptions [16]:

• The mass of the vector field φµ is neglected; sincethe effects of mφ manifest at kiloparsec scales fromthe source[21], it can be dismissed when solvingthe field equations for compact objects such asblack holes.

• G is a constant that depends on the parameter α[7]:

G = GN (1 + α) , (10)

where GN denotes Newton’s gravitational constant,and α is a free adimensional parameter.

Given these hypotheses, the action (1) takes the form,

S =

∫

d4x√−g

(

R

16πG− 1

4BµνBµν

)

. (11)

Variation of the latter expression with respect to gµνyields the STVG field equations:

Gµν = 8πGT φµν , (12)

where Gµν is the Einstein tensor, and the energy-momentum tensor for the vector field φµ is given by[22]

T φµν = −1

4

(

Bµ33αBνα − 1

4gµνB

αβBαβ

)

. (13)

If we vary the action (11) with respect to the vector fieldφµ, we obtain the dynamical equation for such field:

∇νBµν = 0. (14)

The equation of motion for a test particle in coordi-nates xµ is given by

(

d2xµ

dτ2+ Γµ

αβ

dxα

dτ

dxβ

dτ

)

=q

mBµ

νdxν

dτ, (15)

where τ denotes the particle proper time, and q is thecoupling constant with the vector field.Moffat [16] postulates that the gravitational source

charge q of the vector field φµ is proportional to the massof the source particle,

q = ±√

αGNm. (16)

The positive value for the root is chosen (q > 0) to main-tain a repulsive, gravitational Yukawa-like force whenthe mass parameter µ is non-zero (see the Appendix forfurhter details). Then, we see that in STVG theory thenature of the gravitational field has been modified withrespect to GR in two ways: an enhanced gravitationalconstant G = GN (1 + α), and a vector field φµ that ex-erts a gravitational Lorentz-type force on any materialobject through Eq. (15).In what follows, we present the Schwarzschild and Kerr

STVG black holes and analyse their main properties.

B. STVG black holes

1. Schwarzschild STVG black hole

The line element of the spacetime metric for aSchwarzschild STVG black hole is given by [16]:

ds2 = −(

1− 2GN(1 + α)M

c2r+

G2NM2(1 + α)α

c4r2

)

dt2

+

(

1− 2GN (1 + α)M

c2r+

G2NM2(1 + α)α

c4r2

)−1

dr2

+ r2(

dθ2 + sin θ2dφ2)

, (17)

3

where M stands for the mass of the black hole, and αdetermines the strength of the gravitational vector forces.For α → 0, the Schwarzschild metric in GR is recovered.The metric (17) has an outer (r+) and an inner (r−)

event horizons:

r± =GNM

c2

[

1 + α± (1 + α)1/2]

. (18)

If we set α = 0, then r+ = rs = 2GM/c2 is theSchwarzschild event horizon in GR. From Eq. (18) wesee that if α > 0 the radius of the Schwarzschild STVGblack hole outer horizon is larger than the correspondinghorizon in GR.Next, we introduce the spacetime metric for the Kerr

STVG black hole.

2. Kerr STVG black hole

The line element of the spacetime metric of a black holeof mass M and angular momentum J = aM in STVGtheory is [16]:

ds2 = −c2(

∆− a2 sin2 θ) dt2

ρ2+

ρ2

∆dr2 + ρ2dθ2

+2ac sin2 θ

ρ2[

(r2 + a2)−∆]

dtdφ

+[

(r2 + a2)2 −∆a2 sin2 θ] sin2 θ

ρ2dφ2, (19)

∆ = r2 − 2GMr

c2+ a2 +

αGNGM2

c4(20)

= r2 − 2GN (1 + α)Mr

c2+ a2 +

α(1 + α)G2NM2

c4,

ρ2 = r2 + a2 cos2 θ. (21)

The metric above reduces to the Kerr metric in GR whenα = 0. By setting a = 0 in Eq. (19), we recover themetric of a Schwarzschild STVG black hole.The radius of the inner r− and outer r+ event horizons

are determined by the roots of ∆ = 0:

r± =GN (1 + α)M

c2

1±

√

1− a2c4

G2N (1 + α)M2

− α

(1 + α)

.

(22)If we set α = 0 in the latter expression, we obtain theformula for the inner and outer horizons of a Kerr blackhole. Inspection of Eq. (22) also reveals that for α > 0the outer horizon of a Kerr black hole in STVG is largerthat the corresponding one in GR.The radial coordinate of the ergosphere is determined

by the roots of gtt = 0:

rE =GNM (1 + α)

c2

[

1±

√

1− a2 cos2 θ c4

GN2 (1 + α)

2M2

− α

1 + α

]

.

(23)

We see that the ergosphere grows in size as the parameterα increases.

In the next section, we calculate and analyse the equa-torial circular motion of test particles around these blackholes. We develop the formalism for the calculation ofthe circular orbits in Kerr STVG spacetime. The corre-sponding expressions in Schwarzschild STVG spacetimecan be easily obtained by setting a = 0.

III. CIRCULAR ORBITS AROUND STVG

BLACK HOLES

The radial motion of a test particle of mass m in theequatorial plane θ = π/2 of a Kerr STVG black holesatisfies the energy equation[23]:

χ(r, α)E2−2β(r, L, α)E+γ0(r, L, α)+r4 (pr)2 = 0, (24)

where E is the energy of the particle, pr = m dr/ds theradial momentum, and:

χ(r, α) =(

r2 + a2)2 −∆a2, (25)

β(r, L, α) =

(

L a+αGN mM r

c

)

(

r2 + a2)

− L a∆, (26)

γ0(r, L, α) =

(

L a+αGN mM r

c

)2

−∆ L2

− c2r2m2∆. (27)

In the above formulas, L denotes the angular momentumof the test particle. The solution of Eq. (24) is:

E =β(r, L, α) +

√

β(r, L, α)2 − χ(r, α)γ0(r, L, α)− r4pr2

χ(r, l, α).

(28)Since we aim at studying circular orbits, we set pr = 0 inthe latter equation, and we derive an expression for theeffective potential V (r, L, α):

V (r, L, α) =β(r, L, α) +

√

β(r, L, α)2 − χ(r, α)γ0(r, L, α)

χ(r, L, α).

(29)The allowed regions for the motion of a test particle withenergy E at infinity are the regions with V (r) ≤ E, andthe turning points (pr = dr/ds = 0) occur for V (r) = E.

We compute the radius of the circular orbits by solvingthe equation:

dV (r, L, α)

dr= 0, (30)

for a given value of L, and α.

We express the effective potential in terms of the fol-

4

lowing dimensionless quantities:

x =r

rg, (31)

a =a

rg, (32)

L =L

mcrg(33)

where rg = GNM/c2 is the gravitational radius.Equations (20), (25), (26), and (27) are now written

as:

∆(x, α) = x2 − 2x(1 + α) + a2 + α(1 + α), (34)

β(x, L, α) =(

L a+ αx)

(x2 + a2)

− L a(1 + α)2∆(x, α), (35)

γ0(x, L, α) =(

L a+ αx)2

− (x2 + L2)(1 + α)2∆(x, α), (36)

χ(x, α) = (x2 + a2)2 − (1 + α)2∆(x, α)a2. (37)

We have derived all the necessary formulae to calculatethe value of the radial coordinate of the last stable circu-lar orbit around a Kerr STVG black hole of mass M , andangular momentum J = a M . We now need to estimatethe range of values of the parameter α that correspondsto stellar and supermassive black holes.The value of the parameter α depends on the mass of

the gravitational central source. Since we aim at model-ing accretion disks around stellar and supermassive blackholes, we first need to set the range of the permissible val-ues of the parameter α.For stellar mass sources, Lopez and Romero [24] found

that:

α < 0.1. (38)

For supermassive black holes (107M⊙ ≤ M ≤ 109M⊙),we estimate the lower and upper limit for α as follows:

• Lower limit for α: Moffat et al. [25] showed thatfor globular clusters (104M⊙ ≤ M ≤ 106M⊙) thepredicted value of the parameter α is α = 0.03.

• Upper limit for α: Brownstein et al. [11] placedrestrictions on the parameter α by fitting rotationcurves of dwarf galaxies (1.9×109M⊙ ≤ M ≤ 3.4×1010M⊙). They obtained:

2.47 ≤ α ≤ 11.24. (39)

Since the values of the masses of supermassive blackholes lay in between the masses of globular clusters anddwarf galaxies, we assume that for such black holes:

0.03 < α < 2.47. (40)

FIG. 1. Plot of the effective potential as a function of theradial coordinate for different values of the parameter α inthe case of a stellar mass black hole in Schwarzschild STVGspacetime. Here, xisco = risco/rg stands for the location ofthe innermost stable circular orbit.

Given the conditions (38) and (40), we now proceed tocalculate the location of the innermost stable circular or-bit (ISCO) in Schwarzschild and Kerr STVG spacetimes,respectively. The stellar and supermassive black hole arecharacterised by the parameters displayed in Table I. Forthe stellar mass black hole, we adopt the best estimatesavailable for the well-known galactic black hole Cygnus-X1 [26, 27].

In Figs. 1 and 2, we show the plot of the effec-tive potential as a function of the radial coordinate inSchwarzschild STVG spacetime for a stellar and super-massive black hole. In both cases, and for all values of theparameter α, the ISCO is larger than for a Schwarzschildblack hole in GR. This diference, however, is minor forstellar black holes: at most 0.45 percent. On the con-trary, the last stable circular orbit for a supermassiveblack hole in STVG spacetime can lay as far as 16.44 rg,a departure of 174 percent with respect to GR.

The last stable circular orbit for Kerr black holes inSTVG is also larger that for a Kerr black hole of thesame spin in GR (see Figs. 3, and 4). As the value of theparameter α rises, the radius of the ISCO increases. Forstellar mass black holes, the radius of the ISCO growthsup to 49 percent with respect to the value of the ISCOin GR; for supermassive black holes, the ISCO increasesup to 716 percent for α = 2.45.

The analysis of the circular orbits presented in this sec-tion will be applied next to the construction of accretiondisks around STVG black holes.

5

TABLE I. Values of the relevant parameters. The angular momentum a has been normalised as a = a/rg; here, a has unit oflength.

Parameters, symbol [units] Stellar mass BH Supermassive BH

Mass, M [M⊙] 14.8 108

Angular momentum a 0.99 0.99

Mass accretion rate, M [MEdd] 2.53 0.5

FIG. 2. Plot of the effective potential as a function of theradial coordinate for different values of the parameter α inthe case of a supermassive black hole in Schwarzschild STVGspacetime. Here, xisco = risco/rg stands for the location ofthe innermost stable circular orbit.

IV. ACCRETION DISKS IN STVG THEORY

A. Analysis of the structure of the disk

Novikov and Thorne [28] and Page and Thorne [29]made the first relativistic analysis of the structure of anaccretion disk around a black hole in General Relativity.They supposed the disk self-gravity is negligible, and thatthe background spacetime geometry is stationary, axiallysymmetric, asymptotically flat, and reflection-symmetricwith respect to the equatorial plane. They also assumedthat the central plane of the disk is located at the equa-torial plane of the spacetime geometry; since the disk issupposed to be thin (∆z = 2h << r, where z is a co-ordinate that measures the height above the equatorialplane, and h is the thickness of the disk at radius r) themetric coefficients only depend on the radial coordinater.

The three fundamental equations for the time-averagedradial structure of the disk were derived in [29]. In partic-

FIG. 3. Plot of the effective potential as a function of theradial coordinate for different values of the parameter α inthe case of a stellar mass black hole in Kerr STVG spacetime;the spin of the black hole is a = 0.99. Here, xisco = risco/rgstands for the location of the innermost stable circular orbit.

ular, an expression for the heat emitted by the accretiondisk was obtained that depends on the specific energyE†,the specific angular momentum L†, and the angular ve-locity Ω of the particles that move on equatorial circulargeodesic orbits around the black hole:

Q(r) =M

4π√−g

Ω,r

(E† − ΩL†)2

∫ r

risco

(

E† − ΩL†)

L†,r dr,

(41)

where M stands for the mass accretion rate, and g isthe metric determinant. The lower limit of this integralcorresponds to the radius of the innermost stable circularorbit.

Equation (41) is derived from the laws of conservationof rest mass, angular momentum, and energy. The spe-

6

FIG. 4. Plot of the effective potential as a function of theradial coordinate for different values of the parameter α in thecase of a supermassive black hole in Kerr STVG spacetime;the spin of the black hole is a = 0.99. Here, xisco = risco/rgstands for the location of the innermost stable circular orbit.

cific quantities are:

E†(r) ≡ −ut(r), (42)

L† ≡ uφ(r), (43)

Ω(r) ≡ uφ

ut, (44)

where ut and uφ are the t and φ components of the four-velocity of the particle, respectively. Expressions (42)and (43) no longer hold in STVG. Because of the presenceof the gravitational vector field φµ, a neutral test particlein STVG is subjected to a gravitational Lorentz force,whose vector potential in Boyer-Lindquist coordinates is:

φ =−Qr

ρ2(

dt− a sin2 θ dφ)

. (45)

Following [30] we redefine expressions (42) and (43) as:

E = −ptm

+q

mφt, (46)

L =pφm

+q

mφφ. (47)

The conservation laws (Equations 32a, and 32b in [29])take the form:

(

L− w)

,r+

q

mφφ,r = f

(

L+q

mφφ

)

, (48)

(

E − Ωw)

,r+

q

mφt,r = f

(

E +q

mφt

)

, (49)

where

f = 4πeν+Φ+µF/M0, (50)

w = 2πeν+Φ+µWφr/M0. (51)

F stands for the emitted flux, Wφr denotes the torque per

unit circunference, eν+Φ+µ = (−g)1/2 is the square root

of the metric determinant, and M0 the mass accretionrate.

If we multiply Eq. (48) by Ω, and substract from Eq.(49), we obtain the relation:

w =

[(

E − ΩL)

+ q/m (φt − Ωφφ)− q/m(

φt,r − Ωφφ,r

)]

−Ω,r.

(52)The latter expression reduces to Eq. (33) in [29] if φ = 0.

If we insert Eq. (52) into Eq. (48), after some algebra,we get a first order differential equation for f :

a0(x) = a1(x)f + a2(x)df

dx, (53)

where,

a0(x) =(

E − ΩL) d

dx

(

E − ΩL)

,x

−Ω,x

, (54)

a1(x) =

(

E − ΩL)2

− Ω,x

,x

−

(

E − ΩL)(

E,x − ΩL,x

)

−Ω,x

, (55)

a2(x) =

(

E − ΩL)2

−Ω,x

. (56)

Below we write the expressions for the specific energyE, angular momentum L[31], and angular velocity for amassive particle in a circular orbit around a STVG Kerrblack hole:

E = −e1

(

e2 +k1√β

)

, (57)

L = −E(

Ω ˜gφφ + ˜gtφ

)

(

gtt + Ω ˜gtφ

) −α

[

−(

x2 + a2)

Ω + a]

x(

gtt + Ω ˜gtφ

) , (58)

7

where,

β = −gtt − Ω(

2 ˜gtφ + ˜gφφΩ)

, (59)

k1 = gtt + Ωgtφ, (60)

e1 = −

√

1 +(e11 + e12)α2

−x2(

˜gtφ2 − gtt ˜gφφ

) , (61)

e11 = a2(

a2gtt + 2agtφ + ˜gφφ)

, (62)

e12 = x2(

2a2gtt + 2agtφ + ˜gφφ2 − gtt ˜gφφ

)

+ gttx4,(63)

e2 = −(

a2gtt + agtφ + gttx2)

α

e1x(

˜gtφ2 − gtt ˜gφφ

) . (64)

If we replace Eq. (57) into Eq. (58), and the latterinto Eq. (30), after considerable algebra, we obtain an

equation for the angular velocity Ω:

B0 +B1

√

β +B2β = 0, (65)

where,

B0 = e21k21x

2[

k2 ( ˜gtt,xk2 − 2 ˜gtφ,xk1) + ˜gφφ,xk21

]

, (66)

B1 = 2k1e1αk1(

a (ak1 + k2)− k1x2)

+ 2k1e1k3x2 ( ˜gtφ,xk1 − ˜gtt,xk2)

+ 2e21e2k1x2[

k2 ( ˜gtt,xk2 − 2 ˜gtφ,xk1) + ˜gφφ,xk21

]

,(67)

B2 = e21e22x

2[

k2 ( ˜gtt,xk2 − 2 ˜gtφ,xk1) + ˜gφφ,xk21

]

+ x2[

k21 (gtt ˜gφφ,x + ˜gtt,x ˜gφφ) + ˜gtt,xk23

]

− 2gtφ ˜gtφ,xk21x

2 − 2ak1k3α

+ 2e1e2k3x2 ( ˜gtφ,xk1 − ˜gtt,xk2)

+ 2e1e2k3x2αk1

[

a (ak1 + k2)− k1x2]

. (68)

and

k2 = ˜gtφ + Ω ˜gφφ, (69)

k3 = −α

[

−(

x2 + a2)

Ω + a]

x. (70)

Finally,

gtt = −

(

∆− a2)

x2, (71)

gtφ =

[

−(

x2 + a2)

+ ∆]

a

x2, (72)

˜gφφ =

[

(

x2 + a2)2 − ∆a2

]

x2, (73)

˜gtt,x = −2(x− α) (1 + α)

x3, (74)

˜gtφ,x = 2 a(x− α) (1 + α)

x3, (75)

˜gφφ,x = 2x4 − a2 (x− α) (1 + α)

x3. (76)

Here, ∆ is given by Eq. (34).

The heat Q is related to the dimensionless function fas [29]:

Q(x) =c3M

4π√−g

f(x) =c3M

4πcr2gx2f(x) (77)

=c6M

4πGN2M2x2

f(x), [Q(x)] = erg cm−2 s−1.

If we assume that the disk is optically thick in the z-direction, every element of area on its surface radiatesas a blackbody at the local effective surface temperatureT (x). By means of the Stefan Bolzmann’s law, we obtainthe temperature profile:

T (x) = z(x)

(

Q(x)

σ

)1/4

, [T (x)] = K. (78)

As usual, σ denotes the Stefan-Boltzmann constant; thefunction z(x) stands for the correction due to gravita-tional redshift and takes the form:

z(x) =

√

x2 − 2(1 + α)x + a2 + α(1 + α)

x. (79)

Given the blackbody assumption, the emissivity perunit frecuency Iν of each element of area on the disk isdescribed by the Planck function:

Iν(ν, x) = Bν(ν, x) =2hν3

c2[

e(hν/kT (x)) − 1] . (80)

Here h and k are the Planck and Boltzmann constants,respectively. The total luminosity at frecuency ν is:

Lν(ν) =

∫ xout

xisco

2πr2gIνx dx, (81)

=4πhGN

2M2 ν3

c6

∫ xout

xisco

x dx[

e(hν/kT (x)) − 1] .(82)

We adopt for the radius of the outer edge of the disk[32] rout = 11risco.We also investigate the structure of the accretion disk

in the z-(vertical) direction in STVG theory. In thesteady thin disk model developed by [33], the typicalscale-height of the disk in the z-direction is given by:

H ∼= cs

(

r

GNM

)1/2

r, (83)

where cs is the sound speed. The latter expression wasderived under the assumption of hydrostatic equilibriumin the z-direction.For an accretion disk in STVG, expression (83) takes

the form:

HSTVG∼= cs

(

r

GN (1 + α)M

)1/2

r. (84)

8

We estimate the deviation in scale-height H of an ac-cretion disk in STVG with respect to an accretion diskin Shakura & Sunyaev model as:

HSTVG

H≈ (1 + α)

−1/2. (85)

Hence, an accretion disk in STVG is thinner than in GR;this difference increases as the parameter α growths.

B. Results

1. Primary cinematic parameters

We have computed numerically the specific angularmomentum L, the specific energy E, and angular velocityΩ for a test particle in a circular equatorial orbit aroundSchwarzschild and Kerr STVG black holes, respectively;the spin of the Kerr-STVG black hole is a = 0.99. Themost relevant results are display in Figs. 5, 6, and 7 forthe case of a supermassive Kerr black hole with α = 1,and α = 2.45, respectively. We also show for comparisonthe corresponding plots for a Kerr black hole of the samespin in GR.We see that both L, and E are larger in STVG than

in GR. As the value of the parameter α increases thedeviation with respect to GR is more prominent. On thecontrary, there are no significant differences between theangular velocity in STVG and GR.Notice that the slopes of the functions L and E in

STVG (Figs. 5 and 6) are smaller than in GR. Such pe-

culiarity in the behaviour of L and E becomes relevant inthe calculation of the heat Q produced by the accretiondisk. From Eqs. (54), (55), and (56), we see that the co-efficients of the differential equation (53) for the function

f depend on the derivatives of L and E. As we will showin the following subsections, the amount of heat radiatedby an accretion disk in STVG is smaller than in GR. As aresult, the temperature and luminosity distributions alsodecrease.

2. Temperature and luminosity of a Schwarzschild-STVGblack hole

We show in Fig. 8 the plot of the temperature as afunction of the radial coordinate (above), and the cor-responding temperature residual with GR (below). InFig. 9 we plot the luminosity as a function of the en-ergy (above) and its corresponding residual (below) for astellar-Schwarzschild-STVG black hole. We observe thatthe disk is colder as the parameter α increases. This fea-ture is noticeable at the inner parts of the disk, whilefor large values of the radial coordinate, the value of thetemperature tends to the corresponding value in GR. Thedeviations with repect to GR, however, are small: forα = 10−1 the largest difference in temperature is only of0.2 percent and 0.27 percent in luminosity.

1

2

3

4

5

6

7

8

0 10 20 30 40 50

Lz

r/rg

α=0 (GR)α=1

α=2.45

FIG. 5. Plot of the specific angular momentum as a functionof the radial coordinate for a test particle in a circular equa-torial orbit around a supermassive Kerr STVG black hole ofspin a = 0.99.

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 10 20 30 40 50

E

r/rg

α=0 (GR)α=1

α=2.45

FIG. 6. Plot of the specific energy as a function of the radialcoordinate for a test particle in a circular equatorial orbitaround a supermassive Kerr STVG black hole of spin a =0.99.

For a supermassive black hole in Schwarzschild-STVGspacetime, the temperature of the inner parts of the diskis almost 3.5 percent lower than in GR (α = 2.45) (seeFig. 10). Accordingly, the peak of the emission de-creases up to 2.18 percent, and the corresponding energyis shifted toward lower energies (Fig. 11).

9

0

0.02

0.04

0.06

0.08

0.1

0 10 20 30 40 50

Ω

r/rg

α=0 (GR)α=1

α=2.45

FIG. 7. Plot of the angular velocity as a function of theradial coordinate for a test particle in a circular equatorialorbit around a super massive Kerr STVG black hole of spina = 0.99.

3. Temperature and luminosity for Kerr-STVG black hole

We display in Figs. 12 and 13 the plot of the tem-perature as a function of the radial coordinate, and theplot of the luminosity as a function of the energy for astellar Kerr-STVG black hole, respectively. The corre-sponding residual plots are shown at the botton of eachfigure. The accretion disk has a lower temperature thanin GR; the largest difference in temperature is of 0.8 per-cent for α = 10−1. The maximum luminosity of the diskalso decreases in 0.67 percent with respect to GR and thecorresponding energy is shifted toward lower energies.The most significant deviations from GR arise for su-

permassive Kerr-STVG black holes, as shown in Figs. 14and 15. For α = 2.45, the maximum temperature re-duces up to 12.30 percent, and the peak of the emissionis almost 4 percent lower than in GR.

V. DISCUSSION

Accretion disks in STVG around stellar and supermas-sive black holes are colder and underluminous than inGR. We can attribute two main reasons to these phenom-ena. First, the last stable circular orbit in Schwarzschildand Kerr-STVG black holes is farther from the hole thanin GR; the higher the value of the parameter α, the largerthe radius of the ISCO. Since accretion disks get more lu-minous the closer they are to the hole, the contributionto the luminosity at high energies reduces for larger val-ues of the parameter α. Second, as already mention inSection IVB1, the calculation of the heat Q of the diskdepends on the derivatives of L and E which are smaller

−2

−1

0

6 8 10 12 14 16 18 20

[%]

r/rg

5

5.2

5.4

5.6

5.8

6

6.2

6.4

log

(T [

K])

α=0 (GR)α=10

−4

α=5x10−4

α=10−3

α=5x10−3

α=10−2

α=5x10−2

α=10−1

FIG. 8. Top: Temperature as a function of the radial coor-dinate for a stellar Schwarzschild-STVG black hole. Bottom:Residual plot of the temperature as a function of the radialcoordinate for a stellar Schwarzschild-STVG black hole.

−1

0

2.2 2.4 2.6 2.8 3 3.2

[%]

log10(E [eV])

35

35.1

35.2

35.3

35.4

35.5

35.6

35.7

35.8

log

10(ν

Lν [

erg s

−1])

α=0 (GR)α=10

−4

α=5x10−4

α=10−3

α=5x10−3

α=10−2

α=5x10−2

α=10−1

FIG. 9. Top: Luminosity as a a function of the energy fora stellar Schwarzschild-STVG black hole. Bottom: Residualplot of the luminosity as a function of the energy for a stellarSchwarzschild-STVG black hole.

10

−8

−6

−4

−2

0

5 10 15 20 25 30 35 40

[%]

r/rg

3.2

3.4

3.6

3.8

4

4.2

4.4lo

g(T

[K

])

α=0 (GR)α=5x10

−2

α=10−1

α=0.25α=0.5

α=1α=1.5

α=2α=2.45

FIG. 10. Top: Temperature as a function of the radial co-ordinate for a supermassive Schwarzschild-STVG black hole.Bottom: Residual plot of the temperature as a function ofthe radial coordinate for a supermassive Schwarzschild-STVGblack hole.

−2

−1

0

−0.5 0 0.5 1 1.5

[%]

log10(E [eV])

40

40.5

41

41.5

42

log

10(ν

Lν [

erg s

−1])

α=0 (GR)α=5x10

−2

α=10−1

α=0.25α=0.5

α=1α=1.5

α=2α=2.45

FIG. 11. Top: Luminosity as a function of the energy for a su-permassive Schwarzschild-STVG black hole. Bottom: Resid-ual plot of the luminosity as a function of the energy for asupermassive Schwarzschild-STVG black hole.

−2

−1

0

0 5 10 15 20

[%]

r/rg

6

6.2

6.4

6.6

6.8

log

(T [

K])

α=0 (GR)α=10

−4

α=5x10−4

α=10−3

α=5x10−3

α=10−2

α=5x10−2

α=10−1

FIG. 12. Top: Temperature as a function of the radial coor-dinate for a stellar Kerr-STVG black hole of spin a = 0.99.Bottom: Residual plot of the temperature as a function of theradial coordinate for a stellar Kerr-STVG black hole of spina = 0.99.

in STVG than in GR. This behavior is opposite to whatis found in other modified theories of relativistic gravita-tion, such as f(R), where disks tend to be sistematicallyhotter and more luminous than in GR [34].We can understand why the ISCO in STVG is far-

ther from the black hole than GR as follows: a) In thestrong gravity regime, the gravitational repulsion due tothe Yukawa-type force cannot counteract the enhancedgravitational attraction caused by the modified gravita-tional constant G = GN(1 + α). As a result, the gravi-tational field is stronger than in GR, and the region ofinstability around the black hole increases displacing theposition of the ISCO at larger radii. b) Because of thepresence of the gravitational vector field φµ, and for ro-tating sources such as Kerr black holes, the vector po-tential in Boyer-Lindquist coordinates is:

φ =−Qr

ρ2(

dt− a sin2 θ dφ)

. (86)

The components of the four-velocity of a test particle ina circular equatorial orbit in STVG are:

uα =(

ut, 0, 0, uφ)

. (87)

Inserting Eqs. (86) and (87) into (15), where Bµν =

∂µAν − ∂νAµ, a simple calculation shows that such aparticle is subjected to gravitomagnetic Lorenz force in

11

−2

−1

0

2.6 2.8 3 3.2 3.4 3.6

[%]

log10(E [eV])

36

36.2

36.4

36.6

36.8

37

log

10(ν

Lν [

erg s

−1])

α=0 (GR)α=10

−4

α=5x10−4

α=10−3

α=5x10−3

α=10−2

α=5x10−2

α=10−1

FIG. 13. Top: Luminosity as a function of the energy fora stellar Kerr-STVG black hole of spin a = 0.99. Bottom:Residual plot of the luminosity as a function of the energy fora stellar Kerr-STVG black hole of spin a = 0.99.

the radial direction, thus increasing the strenght of thegravitational field. Notice (see Section III) that the mostsignificant deviations in the location of the ISCO with re-spect to GR are for supermassive Kerr black holes withα = 2.45. The bigger the black hole, the stronger the de-viation from GR. This provides a tool to observationallydifferenciate between both theories if reliable estimatesof the black hole masses and luminosities are available

The spectral energy distributions obtained do not seemto contradict astronomical observations (see for example[27]). Thus, in the present work, no further limits can beimposed to the range of values of the parameter α thatcorresponds to stellar and supermassive black holes.

The observation of low luminosity Galactic nuclei, suchas Sg A⋆, led to propose another accretion regime modeldifferent from the geometrically thin, optically thick ac-cretion disk model, called Advection-Dominated Accre-tion Flows (ADAF). As STVG theory predicts thin rel-ativistic accretion disks that are less luminous than inGR, it could be possible that a modified theory of gravi-tation could lead to the same observational predictions asADAF in GR, making individual cases difficult to com-pare in the absence of independent information about theaccretion regime.

−10

−8

−6

−4

−2

0

0 5 10 15 20 25 30 35 40

[%]

r/rg

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

log

(T [

K])

α=0 (GR)α=5x10

−2

α=10−1

α=0.25α=0.5

α=1α=1.5

α=2α=2.45

FIG. 14. Top: Temperature as a function of the radial co-ordinate for a supermassive Kerr-STVG black hole of spina = 0.99. Bottom: Residual plot of the temperature as a func-tion of the radial coordinate for a supermassive Kerr-STVGblack hole of spin a = 0.99.

VI. CONCLUSIONS

We have studied stable circular orbits and relativis-tic accretion disks around Schwarzschild and Kerr blackholes in the strong regime in STVG theory, for both stel-lar (0 < α < 0.1) and supermassive sources (0.03 < α <2.47). We have found that the last stable circular or-bit is larger than in GR. As the value of the parameterα increases, the ISCO is located farther from the blackhole. This is due to the enhanced gravitational constantG = GN(1+α), and the presence of a gravitational radialLorentz force that manifests in the strong regime.

In order to calculate the heat radiated by the accretiondisk, we generalised the relativistic formula given by [29],for the case in which external vector fields are present.We, then, computed the temperature and luminosity dis-tributions for Schwarzschild and Kerr-STVG black holes,respectively. The disks are colder and underluminuous incomparison with thin relativistic accretion disks in GR.Accretion disks around supermassive Kerr STVG blackholes present the largest deviations in temperature andluminosity. The spectral energy distributions predictedby STVG theory are not in contradiction with currentastronomical observations given our ignorance of the de-tails of the accretion regime.

Recently, Romero and collaborators [35] provided a cri-

12

−4

−2

0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

[%]

log10(E [eV])

36

37

38

39

40

41

42

43

log

10(ν

Lν [

erg

s−

1])

α=0 (GR)α=5x10

−2

α=10−1

α=0.25α=0.5

α=1α=1.5

α=2α=2.45

FIG. 15. Top: Luminosity as a function of the energy for a su-permassive Kerr-STVG black hole of spin a = 0.99. Bottom:Residual plot of the luminosity as a function of the energy fora supermassive Kerr-STVG black hole of spin a = 0.99.

terion for the identification of some supermassive blackhole binaries in which the less massive black hole carvesan annular gap in the circumbinary disk. If the mostmassive of the two black holes lauches a relativistic jet,the spectral energy distribution (SED) at gamma ray en-ergies has a unique signature that can be used to identifysuch systems. The distance between the black holes andhence, the location of the gap in the disk, will be dif-ferent in STVG theory. This in turn implies that theSEDs at very high energies will differ from those ob-tained by [35]. Future observations of supermassive bi-nary black hole systems with Cherenkov telescopes, suchas the Cherenkov Telescope Array (CTA; [36]), couldbe used to further constrain STVG theory in the stronggravity regime.

In the present paper, we have focused our investiga-tions on the dynamical aspects of STVG as well as onthe structure and emission in the continuous of thin rel-ativistic accretion disks. More stringent constraints inthe parameters of STVG can be achieved by studyingquasi-periodic oscillations (QPOs), and the profile andequivalent width of the X-ray iron emission lines in blackhole systems. We will adress such issues in a future work.

ACKNOWLEDGMENTS

This work was supported by grants AYA2016-76012-C3-1-P (Ministerio de Educacion, Cultura y Deporte,Espana) and PIP 0338 (CONICET, Argentina).

Appendix A: Modified acceleration law in the weak

field regime

The equation of motion for material test particles, as-suming 2GM/r ≪ 1 and the slow motion approximationdr/ds ≈ dr/dt ≪ 1 is [7]:

d2r

dt2− J2

N

r3+

GM

r2=

q

m

dφ0

dr, (A1)

where JN is the Newtonian orbital angular momentum,and φ0 is the t-component of the gravitational vector fieldφµ.For weak gravitational fields to first order, the static

equations for φ0 for the source-free case are (see Eq.(14)):

~∇2φ0 − µ2φ0 = 0, (A2)

where ~∇2φ0 is the Laplacian operator, and the contri-bution from the self-interaction potential W (φ) has beenneglected. For a spherically symmetric static field φ0,Eq. A2 takes the form:

φ′′0 +

2

rφ′0 − µ2φ0 = 0, (A3)

which has the Yukawa solution:

φ0(r) = −GNMαexp(−µ r)

µ r, (A4)

If we replace the latter equation in Eq. (A1), we obtain:

d2r

dt2− J2

N

r3+

GM

r2= GNMα

exp(−µr)

µr2(1 + µ r) . (A5)

Then, the radial acceleration takes the form:

a(r) = −GN (1 + α) M

r2+GNMα

exp(−µ r)

r2(1 + µr) ,

(A6)The first term of Eq. (A6) represents an enhanced gravi-tational attraction, and serves to explain galaxy rotationcurves, light bending phenomena, and cosmological data,without dark matter. The second term represents grav-itational repulsion and becomes relevant when µ r ≪ 1.This Yukawa-type force counteracts the enhanced attrac-tion, and from the interplay, the Newtonian accelerationlaw is recovered at µ r ≪ 1 scales.

13

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14

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