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The field equations for Scalar-Tensor-Vector-Gravity (STVG) or modified gravity (MOG) have a static, spherically symmetric black hole solution determined by the mass M with either two horizons or no horizon depending on the strength of the gravitational constant G = GN (1+α) where α is a parameter. A regular singularity-free MOG black hole solution is derived using a nonlinear, repulsive gravitational field dynamics and a reasonable physical energy-momentum tensor. The Kruskal-Szekeres completions of the MOG black hole solutions are obtained. The Kerr-MOG black hole solution is determined by the mass M , the parameter α and the spin angular momentum J = M a. The equations of motion and the stability condition of a test particle orbiting the MOG black hole are derived, and the radius of the black hole photosphere and its shadow cast by the Kerr-MOG black hole are calculated. A traversable wormhole solution is constructed with a throat stabilized by the repulsive gravitational field.
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arXiv:1412.5424v2 [gr-qc] 4 Feb 2015
Black Holes in Modified Gravity (MOG)
J. W. Moffat
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
February 5, 2015
The field equations for Scalar-Tensor-Vector-Gravity (STVG) or modified gravity (MOG) have a
static, spherically symmetric black hole solution determined by the mass Mwith either two horizons or
no horizon depending on the strength of the gravitational constant G=GN(1+α) where αis a parameter.
A regular singularity-free MOG black hole solution is derived using a nonlinear, repulsive gravitational
field dynamics and a reasonable physical energy-momentum tensor. The Kruskal-Szekeres completions
of the MOG black hole solutions are obtained. The Kerr-MOG black hole solution is determined by
the mass M, the parameter αand the spin angular momentum J=M a. The equations of motion and
the stability condition of a test particle orbiting the MOG black hole are derived, and the radius of the
black hole photosphere and its shadow cast by the Kerr-MOG black hole are calculated. A traversable
wormhole solution is constructed with a throat stabilized by the repulsive gravitational field.
1 Introduction
The Scalar-Tensor-Vector (STVG) modified gravitational (MOG) theory [1] has successfully explained the
solar system observations [2], the rotation curves of galaxies [3, 4] and the dynamics of galactic clusters [5,
6], as well as describing the growth of structure, the matter power spectrum and the cosmic microwave
background (CMB) acoustical power spectrum data [7]. In the following, we investigate the nature of a
black hole solution to the field equations. The static spherically symmetric vacuum solution describes the
final stage of the collapse of a body in terms of an enhanced gravitational constant Gand a gravitational
repulsive force with a charge Q=αGNM, where αis a parameter defined by G=GN(1 + α), where GN
is Newton’s constant and Mis the total mass of the black hole. The black hole has two horizons if α < 1
and no horizon if α > 1 with a naked singularity. The vector field φµproduces a repulsive gravitational force
in the theory, which repels a test particle falling through the inner horizon preventing it from reaching the
singularity at r= 0.
A MOG static spherically symmetric vacuum solution regular at r= 0 is obtained which has two horizons
when α < αcrit and the metric is de Sitter (anti-de Sitter) as r0. For α > αcrit the spacetime has no
horizons and is regular at r= 0. The collapsed object described by this spacetime has the topological
structure S3compared with the singular MOG solution which has the topology S2×R. The regular solution
describes a grey hole.
A derivation of the Kruskal-Szekeres [8, 9] analytic completion of the black hole solutions is obtained.
The photosphere produced by photons orbiting around the black hole is calculated in terms of the charge
Q=αGNM. The Kerr-MOG black hole solution is determined by the spin angular momentum parameter
a, the mass Mand the parameter αand the motion and stability condition of a test particle orbiting the
black hole are derived.
A traversable MOG wormhole solution is obtained with a wormhole throat stabilized by the balance
of attractive and repulsive gravitational forces. The wormhole allows observers to causally connect distant
parts of asymptotically flat regions of the universe or connect to different universes.
2 Field Equations and MOG Black Hole Solution
In the following, we will be investigate the STVG field equation for the metric tensor gµν [1]:
Rµν =8πGTφµν ,(1)
where we have set c= 1 and assumed that the measure of gravitational coupling Gis constant, tG= 0.
We have Bµν =µφννφµand the energy-momentum tensor for the φµvector field is given by 1
Tφµν =1
4π(BµαBνα 1
4gµν Bαβ Bαβ ).(2)
The dimensionless constant ωincluded in the STVG action in [1] as a scalar field is now treated as a
constant and in the following ω= 1. The source charge Qof the vector field φµis the only constant needed
to determine solutions in STVG, and when Qvanishes STVG reduces to GR. We neglect the mass of the
φµfield, for in the determination of galaxy rotation curves and galactic cluster dynamics µ= 0.04 (kpc)1,
which corresponds to the vector field φµmass mφ= 2.6×1028 eV [2]. The smallness of the φµfield
mass in the present universe justifies our ignoring it when solving the field equations. However, the scale
µ= 0.04 kpc1plays an important role in fitting galaxy rotation curves. We also need the vacuum field
νBµν =1
gν(gBµν ) = 0,(3)
σBµν +µBνσ +νBσµ = 0,(4)
where νis the covariant derivative with respect to the metric tensor gµν .
We adopt the static spherically symmetric metric:
ds2= exp(ν)dt2exp(λ)dr2r2d2,(5)
where d2=2+ sin2θdφ2. We have for the static solution φ06= 0 and φ1=φ2=φ3= 0, and from (3)
we get
r(gB0r) = sin θr(exp[(λ+ν)/2]r2φ
0) = 0,(6)
where φ
0=rφ0. Integrating this equation, we get
0= exp[(λ+ν)/2] Q
where Qis the source charge of the Bµν field. Moreover, we have
Setting γ= exp(ν) and solving the field equations (1), we obtain λ=νand
γ+= 1 GQ2
r2GM, (10)
where 2GM is a constant of integration. The gravitational field metric is given by
1We have assumed that the potential V(φ) in the definition in [1] of the energy-momentum tensor Tφµν is zero.
where G=GN(1 + α). This has the form of the static, point particle Reissner-Nordstr¨om solution for an
electrically charged body [10, 11].
We now postulate that the charge Qis proportional to the mass of the source particle, Q=κM. We
where we define α= (GGN)/GN[3]. This yields the physical value for the charge Q:
Q=±pαGNM. (13)
To maintain a repulsive, gravitational Yukawa force when the mass parameter µis non-zero, necessary to
describe physically stable stars, galaxies, galaxy clusters and agreement with solar system observational data,
we choose the positive value for the square root in (13) giving Q > 0. We now obtain for the metric (11):
where as before G=GN(1 +α). The metric reduces to the Schwarzschild solution when α= 0. To determine
the solar system tests predicted by the metric, we must use the weak field and slow velocity test particle
equation of motion in which the repulsive Yukawa gravitational force cancels the attractive force at the scale
of the solar system and results in agreement of the theory with solar system tests (see Eq.(30) in Section 4
and [2]).
As for the Reissner-Nordstr¨om and and the Kerr solution [12] the MOG black hole has two horizons:
r±=GN(1 + α)M1±1α
1 + α1/2.(15)
We obtain for α= 0 the Schwarzschild radius event horizon r+=rs= 2GNM. The inner horizon ris
a Cauchy horizon and is expected to be unstable as in the case of the Cauchy horizon for the Reissner-
Nordstr¨om metric [13]. the singularity at r= 0 in the black hole can be removed by quantum gravity [14,
15, 16].
We can also consider singularity-free classical solutions of our modified field equations. A black hole
solution regular at r= 0 has been derived by Ay´on-Beato and Garcia [17, 18, 19]. It is an exact solution
of the Einstein vacuum field equations with mass and electric charge as two parameters in the solution, in
which the source is a nonlinear electrodynamic field satisfying the weak energy condition Tµν VµVν>0,
where Vµis a timelike vector. The metric has the form of a Bardeen-type black hole [20, 21, 22, 23] with a
physical source and energy-momentum tensor regular at r= 0.
A metric solution regular at r= 0 is obtained from the MOG field equations (1), (3) and (4) for a
reasonably physical energy-momentum tensor Tφµν and a nonlinear Bµν field dynamics:
(r2+αGNGM2)3/2+αGNGM 2r2
(r2+αGNGM2)3/2+αGNGM 2r2
The associated gravitational B0rfield is given by
B0r= (pαGN)r4r25αGNGM2
The solution (16) behaves asymptotically like the solution (14):
g00 = 1 2GM
The B0rfield (17) is regular at r= 0. The nonlinear Bµν field dynamics behaves for increasing rlike
the MOG gravitational weak field dynamics and the spacetime geometry is determined by the parameter
αand the total mass M. The metric solution has finite values for the curvature scalar R=Rµµand the
Kretchmann invariant Rµνλσ Rµν λσ at r= 0. It is asymptotically flat as r→ ∞ and is the Schwarzschild
metric for large r. For small rthe metric behaves as
ds2= (1 1
3Λr2)dt2(1 1
where the cosmological constant Λ is given by
Λ = 3
α3/2(1 + α).(21)
This means that the interior material of the regular MOG black hole satisfies the vacuum equation of state
p=ρ, where pand ρ=ρvac denote the pressure and the vacuum density, respectively. The cosmological
constant Λ can be positive or negative depending on the magnitude of α, so that the interior of the black
hole is described by either a de Sitter or anti-de Sitter spacetime.
The horizons determined by g00 = 0 for the regular MOG black hole are given by the real solutions of
the equation:
r8+2(3αGNGM 22G2M2)r6+αGNGM 2(11αGNGM24G2M2)r4+6(αGNGM 2)3r2+ (αGNGM2)4= 0.
It is sufficient to consider the positive roots of this equation and it has two or no positive zeros. For α < αcrit
there are two horizons, an outer horizon r+and an inner Cauchy horizon r. For α > αcrit there are no
horizons and there is no naked singularity at r= 0. The topology of the spacelike sections for the MOG
spacetime described by the metric (14) is S3×Rthroughout. In the regular MOG spacetime described by
the metric (16) the topology switches from S3×Rto S3[21, 22].
3 The Kruskal-Szekeres Coordinates
We write the metric in the form:
γ= 1 2GM
We determine a simultaneous transformation of rand tto new coordinates u(r, t) and v(r, t) for which the
light cones are lines with slope ±1 [8, 9, 24, 25, 26]. In such coordinates the metric (23) takes the form:
ds2=f2(u, v)(dv2du2)r2(u, v )d2.(25)
The solution for the uand vcoordinates in which the metric is nonsingular at the horizon yields coordinate
patches, corresponding to the two real roots of (15). The Penrose diagram showing the singular MOG black
hole geometry is displayed in Fig. 1. The metric written in isotropic coordinates has the form:
ds2=1 + 2GM
r=ρ1 + 2GM
The metric can be imbedded in 4-dimensional space, giving a surface of revolution obtained by rotating
a parabola about a line perpendicular to its axis. The curve to be rotated in the θand φdirections is given,
in the ¯rrplane by
¯r(r) = Zdr(2GMr αGNGM 2)
Figure 1: The Penrose diagram for the singular MOG black hole solution (image by Andrew Hamilton).
where r1and r2denote the minimum values of rin two coordinate patches of the MOG Penrose diagram.
The solution for f2takes the form:
f2(r) = 12GM
where dr=γ1(r)dr,bis a constant, Kis an arbitrary scale factor and f2(r) is regular and positive
throughout the coordinate patch.
The maximal analytic extension of the regular MOG black hole solution with the metric (16) can be de-
rived using the form of the standard metric (23) and the same method used to obtain the Kruskal-Szekeres
extension of the singular MOG solution.
4 Test Particle Motion
Consider the motion of test particles of mass mand charge q=κm =αGNm. The equations of motion
are given by
ds2+ Γµαβ
ds =κBσµdxσ
ds .(30)
Because the test particle mass mhas canceled, the motion of a test particle satisfies the weak equivalence
principle. Photons with mass 0 travel along null geodesics with the equation of motion
ds2+ Γµαβ
= 0,(31)
where λis an affine parameter.
If we allow a negative charge Qfor our MOG solution, then if Q > 0 the force always points in the
positive direction on the initial surface, so that on the sheet u < 0, it points towards the wormhole throat,
while on the sheet u > 0 it points away from the throat. The “charged wormhole” looks like a negative
charge viewed from one sheet, and like a positive charge when viewed from the other sheet. The Bµν field
flux flows continuously from one sheet to the other through the wormhole.
From the equations of motion of a test particle (30), we get after integration in polar coordinates with
ds ,dt
ds =C
where Jand Care constants of integration and Jis the angular momentum. Differentiating the metric (23)
with respect to s, we obtain
ds 2
ds 2
= 1,(33)
which has the form of an energy integral. We can write this equation as
ds 2
12GM/r +αGNGM 2/r2=r4(C21)
The term αG2M2/r2dominates as r0 and we see that the test particle can never reach r= 0. In
contrast to the analytic continuation of the Schwarzschild metric for a point mass, the throat of a wormhole
describing the particle pulsates in time and does not pinch off. No test particle ever hits the singularity
at r= 0. The result that the modified gravity metric is described by a pulsating throat, is in accord with
the dynamic balance between the gravitational attraction and the repulsive gravity pressure (anti-gravity)
produced by the Bµν field.
When an observer falls freely through the MOG horizon and then subsequently falls through the inner
Cauchy horizon, the repulsive gravity pressure propels the observer into a white hole in another universe
as described in Fig. 1. However, in a realistic MOG black hole the spacetime inside the inner horizon is
unstable due to mass inflation and would prevent the formation of a wormhole inside the MOG black hole.
5 Kerr-MOG Black Hole
In addition to mass M, we shall include in our solution the spin angular momentum J=M a:
ρ2(dt asin2θdφ)2sin2θ
∆ = r22GM r +a2+αGNGM 2, ρ2=r2+a2cos2θ. (36)
The spacetime geometry is axially symmetric around the zaxis [27]. Horizons are determined by the roots
of ∆ = 0:
r±=GN(1 + α)M1±s1a2
N(1 + α)2M2α
1 + α.(37)
An ergosphere horizon is determined by g00 = 0:
rE=GN(1 + α)M1 + s1a2cos2θ
N(1 + α)2M2α
1 + α.(38)
Our Kerr-MOG vacuum solution of the MOG field equations is fully characterized by the mass M, spin
parameter aand the parameter α. According to the no-hair theorem [28, 29, 30, 31], no closed time-like
curves exist outside of the horizon. Multipole moments of the Kerr-MOG spacetime Mland Jlexist. The
mass multipole moments Mlvanish if lis odd, and the set of current multipole moments Slvanish if lis
even. The no-hair theorem can then be stated for α > 0 as [32, 33]:
The radial motion of test particles with mass min the equatorial plane θ=π/2 is governed by [25]:
A(r)E22B(r)E+D(r)r4(pr)2= 0, E =B(r) + pB(r)2A(r)D(r) + A(r)r4(pr)2
where Eis the energy, pris the radial momentum pr=mdr/ds and A, B, D are given by
A(r) = (r2+a2)2a2,(41)
B(r) = (Ja +αGNmM r)(r2+a2)Ja,(42)
D(r) = (Ja +αGNmM r)2J2m2r2.(43)
The effective potential V(r) is the minimum allowed value of the energy Eat radius r:
V(r) = B(r) + pB2(r)A(r)D(r)
The permitted regions for a particle of energy Eare the regions with VE, and the turning points for
pr=mdr/ds = 0 occur when V(r) = E. Stable circular test particle orbits occur at the minima of V(r).
The irreducible mass of the Kerr-MOG black hole is given by
Mir =1
2[(GN(1 + α)M+qG2
N(1 + α)2M2αG2
N(1 + α)M2a2)2+a2]1/2.(45)
We have δMir >0, so that no injection of small lumps of matter can ever reduce the irreducible mass of a
Kerr-MOG black hole. This is consistent with the second law of black hole dynamics, for the surface area of
the black hole is A= 16πG2
N(1 + α)2M2
The radius of the photosphere determined by our metric (14) is given by
rps =3
2GN(1 + α)M1 + s18α
9(1 + α).(46)
It is necessary to take into account the screening mechanism for photon trajectories in MOG [2]. However,
this screening mechanism holds for weak gravitational fields and may not hold for the strong gravitational
fields near black holes. When the impact parameter r0takes values near rps the angle of deflection δθdef >2π,
whereby the light ray can turn around the black hole several times before reaching the observer. We have
two infinite sets of images, one produced by clockwise winding around the black hole for δα > 0, and the
other one by counter-clockwise winding for δ
thetadef <0. These images are located at the same side and at the opposite side of the source.
6 Blackhole Shadow Image and Lensing
A remarkable prediction of strong gravity is the existence of a region outside a black hole horizon (or no
horizon), in which there are no closed photon orbits. The photons that cross this region are removed from
the observable universe. This phenomenon results in a shadow (silhouette) imprinted by a black hole on the
bright emission that exists in its vicinity and the apparent shape of a black hole is defined by the boundary
of the black hole [34, 35, 36, 37, 38, 39, 40]. The Hamilton-Jacobi equation determines the geodesics for a
given geometry [36, 41, 42, 43]:
∂λ =1
2gµν ∂S
where λis an affine parameter and Sis the Jacobi action. The action Scan be written as
2m2λEt +Jzφ+Sr(r) + Sθ(θ),(48)
where mthe test particle mass, Eis the energy, Jzis the angular momentum in the direction of the axis of
symmetry. For a photon null geodesic, we have m= 0 and from the Hamilton-Jacobi equations of motion,
we obtain
=a(JzaE sin2θ) + r2+a2
sin2θaE +a
Here, Σ = r2+a2cos2θand Rand Θ are given by
R= [(r2+a2)EaJz]2∆[K+ (JzaE)2],Θ = K+ cos2θa2E2J2
where Kis a constant. The spacetime is asymptotically flat, so that photon paths are straight lines at
infinity. The photon trajectories are determined in terms of two impact parameters ξand η, which are
defined by ξ=Jz/E and η=K/E2. The photon orbits with constant rare derived from (51) and yield the
boundary of the black hole shadow. The circular photon orbits are determined by the conditions R(r) = 0
and dR/dr = 0 and we get
ξ(r) = GN(1 + α)M(r2a2)rαG2
N(1 + α)M2r
a(rGN(1 + α)M),(54)
η(r) = r2
a2(rGN(1 + α)M)2[4GN(1 + α)Mr(a2+αG2
N(1 + α)M2)4αG2
N(1 + α)M2
r2(r3GN(1 + α)M)2].(55)
These equations determine parametrically the critical locus (ξ, η) and give the constants of the motion for
the photon orbits of constant radius.
We use celestial coordinates xand yto determine the black hole shadow. We consider an observer far
away from the black hole and θis the angular coordinate of the observer corresponding to the angle between
the axis of rotation of the black hole and the line of sight of the observer as described in Fig. 2. The celestial
coordinates xand yof the shadow image are given by
x=ξcsc θ, y =±qη+a2cos2θξ2cot2θ. (56)
The photon orbit is parameterized by the conserved quantities (ξ, η) with the observer’s polar angle
θ. Letting ξand ηtake all possible values with a fixed θ, the photon capture region is determined by
Figure 2: The apparent position of a light ray with respect to the observer’s projection plane in the x,y
coordinates containing the center of the spacetime: x denotes the apparent distance from the rotation axis,
and y the projection of the rotation axis itself (dashed line). The angle θdenotes the angle of latitude,
reaching from the north pole at θ= 0 to the south pole at θ=π(image by Math@IT).
the impact parameter space (bx, by) for the given θ. The photon capture region can be regarded as the ap-
parent shadow shape of the black hole. The shape of the black hole shadow is obtained by plotting yversus x.
The lensing equation can be written as [44]:
tan β= tan θDls
(tan θ+ tan(δα θ)),(57)
where βand θare the angular source and image positions, respectively, δα is the deflection angle and Dls
and Dos are the distances from the lens to the source and from the observer to the source, respectively. The
deflection angle is [45]:
δα(r0) = 2 Z
drrr/r0212GN(1 + α)M/r0+αG2
N(1 + α)M2/r2
12GN(1 + α)M/r +αG2
N(1 + α)M2/r21/2
π, (58)
where r0is the distance of closest approach. The impact parameter is given by
J(r0) = r0
p12GN(1 + α)M/r0+αG2
N(1 + α)M2/r2
where J(r0) = Dol sin θwith Dol the distance from the observer to the lens.
7 Traversable MOG Wormhole
The idea of a worm hole in spacetime was realized in 1916 by Flamm [46] who recognized that the hy-
persurfaces of constant Killing time through a Schwarzschild spacetime, embedded in an Euclidean space,
demonstrated that these hypersurfaces consist of two asymptotically flat sheets. These sheets are connected
by an Einstein-Rosen bridge [47], or throat, called a “wormhole” by Wheeler [48]. The possibility that a
wormhole can connect two different universes or two separated spacetime regions in our universe was con-
sidered by Morris and Thorne [49, 50], Hawking and Ellis [26] and Visser [51]. The problem of finding a
wormhole solution connecting to the same asymptotic region can at best be done in an axially symmetric
An explicit construction of a charged wormhole which does connect to the same asymptotic region
was demonstrated by Schein and Aichelburg [52, 53]. The wormhole does not violate the positive energy
conditions and it does not require negative energy to stabilize the wormhole throat. In the following, we
will adapt their wormhole solution to our modified gravity in which the charge Qis proportional to mass:
Q=αGNM. This means that the interior black hole solution is described by the MOG metric with only
the mass Mas source, and the exterior repulsive force that stabilizes two connected spheres Si(i= 1,2) is
due to the gravitational repulsive potential φµ= (φ0,0,0,0). If the stable traversable wormhole only allows
one-way travel through it, a time-reversed stable wormhole can be constructed in the same spacetime region
that would allow a hapless space adventurer to return to his original destination.
The Majumdar-Papapetrou solution of the MOG field equations (1), (3) and (4) allows for a system of
bodies to be held in equilibrium by a balance between the gravitational Bµν field repulsion and gravitational
attraction [54, 55]. In Cartesian coordinates the solution is given by
ds2=V2dT 2V2(dx2+dy2+dz2),(60)
where V(x, y, z ) satisfies Laplace’s equation:
V(x, y, z) = 2
∂z2V(x, y, z ) = 0.(61)
Two interior non-intersecting spheres Siare cut out of the Majumdar-Papapetrou spacetime and the
potential function Vis constant on the surfaces of the spheres. This problem is analogous to determining
the electric potential outside two charged metal spheres. The two-body problem is constructed by choosing
the z-axis to point along the line of symmetry joining the two spheres Siwith radius R. The two spheres
are centered at z=±d1and the function Vis given by
V=V0= 1 + m1
For large distance separation the two spheres appear as two particles with Q=M=m1(in relativistic units
GN=c= 1 and α(1 + α) = 4πor α= 3.08). The metric potential then has the form [52, 53]:
V(x) = 1 + Σ
New coordinates are defined by
sinh µ0
, d1= ˜rcoth µ0,(64)
and the polar coordinates:
coth µ=(x2+y2+z2+ ˜r2)
(2˜rz),cot z=(x2+y2+z2˜r2)
(2˜rpx2+y2),cot φ=x
The metric now takes the form:
ds2=V2dT 2
(cosh µcos η)2(2+2+ sin2η2),(66)
and the potential Vis given by [52, 53]:
V(µ, η, φ) = 1 + m1
R1 + pcosh µcos ηΣ+
n=−∞(1)n+1 1
pcosh(µ+ 20)cos η.(67)
We can now construct a wormhole by attaching different asymptotic regions of our MOG spacetime to the
surfaces µ=±µ0. This is accomplished by placing two infinitely thin shells of Q=αGNMgravitationally
charged matter at the transition surfaces of the spheres Si. The interior metric of the shells is given by
r2dT 2dr2
12GM/r +αGNGM 2/r2r2d2.(68)
Figure 3: A MOG wormhole with a bridge connecting two asymptotically flat spacetimes (from Wikipedia).
We have two asymptotic regions Iand II lying outside the MOG spacetimes, and we cut off the asymp-
totically flat parts. We introduce spherical polar coordinates (T , r, θ, φ) centered at z=d1so that the
radius of the sphere S+
1is given by r+=R. Then, the metric (68) is
ds2=V2dT 2
In terms of the proper time τthe induced metric on the shell S1is
By introducing a second shell S
2in the asymptotic region II, lying in the causal future of S
1, a traversable
wormhole can be constructed [52, 53]. Because of the symmetry of the over all system, the shell energy
densities and pressures are the same for both shells.
The energy density and pressure obtained for the inner MOG region are given in the limit of M0 by
4πV 2
∂r+r+=R=(cosh µ0cos η)
4π¯rV 2
∂µ µ=µ0
, p = 0.(71)
The sign of V/∂µ|µ=µ0can be shown to be negative on S1for arbitrary values of the positive parameter
m1[52, 53], so that the energy density σ > 0, thereby, avoiding any violations of positive energy theorems.
Any spacelike slice which avoids the singularities in the MOG metric cuts S1and S2and connects two
separated asymptotic regions. An observer can enter the wormhole through S1and causally emerge at S2
arbitrarily far in the past or future. In Fig. 3 a MOG wormhole is displayed. A second S3and S4construc-
tion the same as the one for S1and S2in the external Majumdar-Papapetrou region can be constructed
some distance away. Then, the second wormhole can be used by the observer to causally get back to his/her
original starting place.
8 Conclusions
A solution is derived to the MOG vacuum field equations for a point particle with mass M, gravitational
charge Q=αGNM, and the gravitational field equations for a spread out distribution of Bµν energy
density Tφµν. As in the case of the Reissner-Nordstr¨om electric charge solution and the Kerr solution,
the metric has two horizons. However, if a mechanism such as quantum gravity is invoked to remove the
singularity at r= 0, then this solution could be physically viable as a description of the black hole. A
classical Bardeen-type MOG solution is derived which is regular at r= 0 and has two horizons.
A Kruskal-Szekeres maximal analytic set of coordinates is determined for both the MOG singular and
non-singular (at r= 0) solutions. The Kerr-MOG solution for a point particle with mass Mand angular
momentum parameter acan produce a photon sphere which can differ observationally from the photon
sphere produced by a pure Kerr metric solution. The lensing of the black hole solutions and the images
of a black hole shadow can provide observational signatures which can distinguish between the standard
GR and MOG black hole solutions. Future observational data obtained by the Event Horizon Telescope
(EHT) [37, 38, 39, 40] can potentially accomplish this task.
A traversable wormhole is constructed by adapting the Schein-Aichelburg solution [52, 53] to the STVG
field equations. The field external to two connected spheres S1and S2is described by the Majumdar-
Papapetrou solution [54, 55]. The MOG gravitational repulsion balances the attractive gravity resulting in
a stable wormhole throat, allowing an observer to traverse from one asymptotically flat region of spacetime
to another distant one. This could have the consequence of permitting closed timelike curves and the
construction of a time-machine violating causality. However, a quantum description of time travel which
avoids closed time-like curves can potentially circumvent a violation of causality [56].
I thank Avery Broderick, Martin Green and Viktor Toth for helpful discussions. This research was generously
supported by the John Templeton Foundation. Research at the Perimeter Institute for Theoretical Physics is
supported by the Government of Canada through industry Canada and by the Province of Ontario through
the Ministry of Research and Innovation (MRI).
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... the scalar field G can be treated as a constant on the scales of compact objects, ∂ ν G = 0 [13,14]. This means that the aforementioned parameter α can be regarded as a positive dimensionless constant, whose value depends on the mass of the gravitational source [1]: ...
... Under these assumptions (and by setting the speed of light in vacuum to c = 1), one obtains [13] the following line element: ...
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We have used publicly available kinematic data for the S2 star to constrain the parameter space of MOdified Gravity. Integrating geodesics and using a Markov Chain Monte Carlo algorithm, we have provided the first constraint on the scales of the Galactic Centre for the parameter α of the theory, which represents the fractional increment of the gravitational constant G with respect to its Newtonian value. Namely, α≲0.662 at 99.7% confidence level (where α=0 reduces the theory to General Relativity).
... It can well explain the solar observations, the rotation curves of galaxies [31] and the dynamics of galactic clusters [32]. Based on the theory of gravity, a static spherically symmetric modified gravity Schwarzschild black hole metric was first given in [33]. The metric describes the final stage of the collapse of a body by introducing α as a coupling parameter of modified gravity, which enhances the gravitational constant and provides a charge yielding a gravitational repulsive force. ...
... In terms of the scalar-tensor-vector modified gravitational theory, a static spherically symmetric nonrotating black hole [33] is written in Boyer-Lindquist coordinates x µ = (t, r, θ , ϕ) as ...
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Based on the scalar-tensor-vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincare sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can do chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos.
... Another modified theory where some work has been done is Moffat's tensor-vector-scalar theory (Moffat, 2015). The theory admits a well-posed initial value problem, but for sufficiently small scales, it becomes mathematically equivalent to Einstein-Maxwell theory in vacuum, where BHs carry a "gravitational" (not electric) charge, that is determined by the theory's coupling constant. ...
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The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of GWs can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas.
The lack of rotating black holes, typically found in nature, hinders testing modified gravity from astrophysical observations. We present the axially symmetric counterpart of an existing spherical hairy black hole in Horndeski gravity having additional deviation parameter Q, which encompasses the Kerr black hole as a particular case (Q=0). We investigate the effect of Horndeski parameter Q on the rotating black holes geometry and analytically deduce the gravitational deflection angle of light in the weak-field limit. For the S2 source star, the deflection angle for the Sgr A* model of rotating Horndeski gravity black hole for both prograde and retrograde photons is larger than the Kerr black hole values. We show how parameter Q could be constrained by the astrophysical implications of the lensing by this object. The thermodynamic quantities, Komar mass, and Komar angular momentum gets corrected by the parameter Q, but the Smarr relation Meff=2ST+2ΩJeff still holds at the event horizon.
We use the causal horizon entropy to study the asymptotic behaviors of regular AdS black holes. In some literature, the causal horizon entropy is regarded as a generalized holographic c -function. In this paper, we apply this idea to the case of regular AdS black holes. We show that the causal horizon entropy decreases to zero at the center of regular AdS black holes and in particular it is stationary because its derivative with respect to the affine parameter approaches zero asymptotically. Meanwhile, the asymptotic behavior of the metric of regular AdS black holes implies that the black hole center corresponds to an IR fixed point. Therefore, we conclude that the causal horizon entropy is a valid candidate for the holographic c -function of these regular AdS black holes.
We are fitting dynamics of electrically neutral hot-spot orbiting around Sagittarius A* (Sgr A*) source in Galactic center, represented by various modifications of the standard Kerr black hole (BH), to the three flares observed by the GRAVITY instrument on May 27, July 22, July 28, 2018. We consider stationary, axisymmetric, and asymptotically flat spacetimes describing charged BHs in general relativity (GR) combined with nonlinear electrodynamics, or reflecting the influence of dark matter (DM), or in so called parameterized dirty Kerr spacetimes, and test them using the hot-spot data. We show that the orbital frequencies as well as positions of the hot-spots orbiting the considered BHs fit the observed positions and periods of the flare orbits, and give relevant constraints on the parameters of the considered BH spacetimes and the gravity or other theories behind such modified spacetimes.
The null geodesics of the regular and rotating magnetically charged black hole in a non-minimally coupled Einstein-Yang-Mills theory surrounded by a plasma medium is studied. The effect of magnetic charge and Yang-Mills parameter on the effective potential and radius of photon orbits has investigated. We then study the shadow of a regular and rotating magnetically charged black hole along with the observables in the presence of the plasma medium. The presence of plasma medium affects the apparent size of the shadow of a regular rotating black hole in comparison with vacuum case. Variation of shadow radius and deformation parameter with Yang-Mills and plasma parameter has examined. Furthermore, the deflection angle of the massless test particles in weak field approximation around this black hole spacetime in the presence of homogeneous plasma medium is also investigated. Finally, we have compared the obtained results with Kerr-Newman and Schwarzschild black hole solutions in general relativity (GR).
Scalar tensor vector gravity (STVG) is a fully covariant Lorentz invariant alternative theory of gravity also known as MOdified Gravity (MOG) which modifies General Relativity by inclusion of dynamical massive vector field and scalar fields. In STVG the mass μ of the vector field ϕμ and the gravitational constant G acquire the status of dynamical fields. We use the reconstructed total cluster mass of the X-COP sample obtained from X-ray observations by XMM-Newton telescope in combination with Sunyaev–Zel’dovich (SZ) effect observed within Planck all-sky survey to estimate the α and μ parameters of MOG theory. The obtained values are consistent with previous fits by other authors. Hence the MOG is passing another test and proves its consistency strengthening thereby its stance of being a promising alternative to General Relativity.
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In this paper, we investigate the effects of the nonlocality on the shadow cast by two types of rotating black holes corresponding to two specific functionals of the covariant D'Alembertian operator, which introduce one cut-off parameter. Then we calculate the weak deflection angle of light by these two rotating black holes with the Gauss-Bonnet theorem. For both types of black holes, we show that the size of the black hole shadow decreases with the cut-off parameter since the nonlocality weakens the gravitational constant, and the shape of the shadow gets more deformed with the increase of the cut-off parameter. However, if the rotation parameter is small, the shape of the shadow is almost a circle even if the cut-off parameter approaches to its maximum. The energy emission rate of both models is also investigated and it is shown that there is a peak for each curve and the peak decreases and shifts to the low frequency with the increase of the cut-off parameter. In addition, we also explore the shadow of both types of black holes surrounded by a nonmagnetized pressureless plasma which satisfies the separability condition. It is found that the plasma has a frequency-dependent dispersive effect on the size and shape of the black hole shadow. Finally, we give a brief analysis about the weak deflection angle by these two rotating black holes and find that the cut-off parameter of model A makes a positive contribution to the deflection angle, which can be compared with the contribution of the rotation parameter, while the cut-off parameter of model B makes a negative contribution which can be ignored.
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Higher-dimensional theories admit that astrophysical objects like supermassive black holes are somewhat different from standard ones and that their optical properties deviate from general relativity. A black hole casts a shadow as an optical appearance because of its strong gravitational field, and a black hole shadow could be used to determine which theory of gravity is consistent with observations. Measurements of the shadow sizes around the black holes can help evaluate various parameters of the black hole metric and may provide an opportunity to find signatures of extra dimensions. We present an exact five-dimensional (5D) rotating regular black hole metric, with a deviation parameter k≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 0$$\end{document}, that interpolates between the 5D Kerr black hole (k=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document}) and 5D Kerr–Newman (r≫k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \gg k$$\end{document}). This 5D rotating regular black hole is an exact general relativity solution coupled to nonlinear electrodynamics. Interestingly, for a given value of parameter k, there exits a critical angular momentum a=aE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=a_E$$\end{document}, which corresponds to an extremal rotating regular black hole with degenerate horizons. At the same time, for a<aE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a<a_E$$\end{document}, one has a non-extremal rotating regular black hole with outer and inner horizons. Owing to the correction factor (e-k/r2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-k/r^2}$$\end{document}), which is motivated by the quantum arguments, the ergoregion and shadow of 5D black holes are modified.
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The shadows cast by non-rotating and rotating modified gravity (MOG) black holes are determined by the two parameters mass $M$ and angular momentum $J=Ma$. The sizes of the shadows cast by the spherically symmetric static Schwarzschild-MOG and Kerr-MOG rotating black holes increase significantly as the free parameter $\alpha$ is increased from zero. The Event Horizon Telescope (EHT) shadow image measurements can determine whether Einstein's general relativity is correct or whether it should be modified in the presence of strong gravitational fields.
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Galaxy rotation curves determined observationally out to a radius well beyond the galaxy cores can provide a critical test of modified gravity models without dark matter. The predicted rotational velocity curve obtained from Scalar-Vector-Tensor Gravity (STVG or MOG) is in excellent agreement with data for the Milky Way without a dark matter halo, with a mass of $5\times 10^{10}\,M_{\odot}$. The velocity rotation curve predicted by modified Newtonian Dynamics (MOND) does not agree with the data.
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A conformal coupling of the metric in the Jordan frame to the energy-momentum tensor, screens the scalar field gravitational coupling strength $G$ in modified gravity (MOG). The scalar field acquires a mass which depends on the local matter density: the scalar field particle is massive for the Sun and earth, where the density is high compared to low density environments in cosmology and astrophysics. Together with the screening of the vector field $\phi_\mu$, this guarantees that solar system tests of gravity are satisfied. The conformal metric is coupled to the electromagnetic matter field and energy-momentum tensor, screening $G$ for the Sun and the deflection of light by the Sun and the Shapiro time delay in MOG are in agreement with general relativity. For galaxies and galactic clusters the enhanced gravitational coupling constant $G$ leads to agreement with gravitational lensing without dark matter. For compact binary pulsars the screening of $G$ removes the monopole and dipole gravitational radiation modes in agreement with the binary pulsar timing data.
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The image of the emission surrounding the black hole in the center of the Milky Way is predicted to exhibit the imprint of general relativistic (GR) effects, including the existence of a shadow feature and a photon ring of diameter ~50 microarcseconds. Structure on these scales can be resolved by millimeter-wavelength very long baseline interferometry (VLBI). However, strong-field GR features of interest will be blurred at lambda >= 1.3 mm due to scattering by interstellar electrons. The scattering properties are well understood over most of the relevant range of baseline lengths, suggesting that the scattering may be (mostly) invertible. We simulate observations of a model image of Sgr A* and demonstrate that the effects of scattering can indeed be mitigated by correcting the visibilities before reconstructing the image. This technique is also applicable to Sgr A* at longer wavelengths.
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An important piece of evidence for dark matter is the need to explain the growth of structure from the time of horizon entry and radiation-matter equality to the formation of stars and galaxies. This cannot be explained by using general relativity without dark matter. So far, dark matter particles have not been detected in laboratory measurements or at the LHC. We demonstrate that enhanced structure growth can happen in a modified gravity theory (MOG). The vector field and particle introduced in the theory to explain galaxy and cluster dynamics plays an important role in generating the required structure growth. The particle called the phion (a light hidden photon) is neutral and is a dominant, pressureless component in the MOG Friedmann equations, before the time of decoupling. The dominant energy density of the phion particle in the early universe, generates an explanation for the growth of density perturbations. The angular acoustical power spectrum due to baryon-photon pressure waves is in agreement with the Planck 2013 data. As the universe expands and large scale structures are formed, the density of baryons dominates and the rotation curves of galaxies and the dynamics of clusters are explained in MOG, when the phion particle in the present universe is ultra-light. The matter power spectrum determined by the theory is in agreement with current galaxy redshift surveys.
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We apply the weak field approximation limit of the covariant scalar–tensor–vector gravity theory, so-called MOdified gravity (MOG), to the dynamics of clusters of galaxies by using only baryonic matter. The MOG effective gravitational potential in the weak field approximation is composed of an attractive Newtonian term and a repulsive Yukawa term with two parameters α and μ. The numerical values of these parameters have been obtained by fitting the predicted rotation curves of galaxies to observational data, yielding the best-fitting result: α =8.89±0.34 and μ = 0.042 ± 0.004 kpc−1. We extend the observational test of this theory to clusters of galaxies, using data for the ionized gas and the temperature profile of nearby clusters obtained by the Chandra X-ray telescope. Using the MOG virial theorem for clusters, we compare the mass profiles of clusters from observation and theory for 11 clusters. The theoretical mass profiles for the inner parts of clusters exceed the observational data. However, the observational data for the inner parts of clusters (i.e. r < 0.1r500) is scattered, but at distances larger than ∼300 kpc, the observed and predicted mass profiles converge. Our results indicate that MOG as a theory of modified gravity is compatible with the observational data from the Solar system to megaparsec scales without invoking dark matter.
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Gravity is treated as a stochastic phenomenon based on fluctuations of the metric tensor of general relativity. By using a (3+1) slicing of spacetime, a Langevin equation for the dynamical conjugate momentum and a Fokker-Planck equation for its probability distribution are derived. The Raychaud-huri equation for a congruence of timelike or null geodesics leads to a stochastic differential equation for the expansion parameter θ in terms of the proper time s. For sufficiently strong metric fluctuations, it is shown that caustic singular-ities in spacetime can be avoided for converging geodesics. The formalism is applied to the gravitational collapse of a star and the Friedmann-Robertson-Walker cosmological model. It is found that owing to the stochastic behavior of the geometry, the singularity in gravitational collapse and the big-bang have a zero probability of occurring. Moreover, as a star collapses the prob-ability of a distant observer seeing an infinite red shift at the Schwarzschild radius of the star is zero. Therefore, there is a vanishing probability of a Schwarzschild black hole event horizon forming during gravitational collapse.
(Abridged) In General Relativity, the shadow cast by a black hole has a size that depends very weakly on its spin or the orientation of the observer. The half opening angle of the shadow is always equal to 5+-0.2 GM/Dc^2, where M is the mass of the black hole and D is its distance from the Earth. Therefore, measuring the size of the shadow of a black hole of known mass-to-distance ratio and verifying whether it is within the 4% predicted range constitutes a null hypothesis test of GR. We show that Sgr A* is the optimal target for performing this test with the Event Horizon Telescope. We use the results of monitoring of stellar orbits to show that the ratio M/D for Sgr A* is already known to an accuracy of ~6%. We investigate our prior knowledge of the scattering screen towards Sgr A, the effects of which will need to be corrected for in order for the black-hole shadow to appear sharp against the background emission. We argue that, even though the properties of the scattering ellipse at longer wavelengths are known to within ~3-20%, extrapolating them down to the 1.3 mm EHT requires further theoretical work and a model for the wavelength dependence of the axis ratio and position angle in the presence of anisotropic turbulence. Finally, we employ recent GRMHD simulations of the accretion flow around Sgr A* to investigate the visibility and sharpness of the black-hole shadow, which will determine the accuracy at which the GR test can be performed. We explore an edge detection scheme and a pattern matching algorithm based on the Hough/Radon transform and demonstrate that the shadow of the black hole at 1.3 mm can be localized, in principle, to within ~9%. All these results suggest that our prior knowledge of the properties of the black hole, of the scattering broadening in the interstellar medium, and of the accretion flow can only limit this GR test with EHT observations of Sgr A* to <10%.
We consider the Pleba\'nski class of electrovacuum solutions to the Einstein equations with a cosmological constant. These space-times, which are also known as the Kerr-Newman-NUT-(anti-)de Sitter space-times, are characterized by a mass $m$, a spin $a$, a parameter $\beta$ that comprises electric and magnetic charge, a NUT parameter $\ell$ and a cosmological constant $\Lambda$. Based on a detailed discussion of the photon regions in these space-times (i.e., of the regions in which spherical lightlike geodesics exist), we derive an analytical formula for the shadow of a Kerr-Newman-NUT-(anti-)de Sitter black hole, for an observer at given Boyer-Lindquist coordinates $(r_O, \vartheta_O)$ in the domain of outer communication. We visualize the photon regions and the shadows for various values of the parameters.