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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
and
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
February 5, 2015
Abstract
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
1
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
equations:
νBµν =1
gν(gBµν ) = 0,(3)
and
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
φ
0= exp[(λ+ν)/2] Q
r2,(7)
where Qis the source charge of the Bµν field. Moreover, we have
Tφ1
1=Tφ2
2=Tφ3
3=Tφ0
0=1
2exp(λν)(φ
0)2=1
8π
Q2
r4.(8)
Setting γ= exp(ν) and solving the field equations (1), we obtain λ=νand
γ+= 1 GQ2
r2,(9)
=r+GQ2
r2GM, (10)
where 2GM is a constant of integration. The gravitational field metric is given by
ds2=12GM
r+GQ2
r2dt212GM
r+GQ2
r21
dr2r2d2,(11)
1We have assumed that the potential V(φ) in the definition in [1] of the energy-momentum tensor Tφµν is zero.
2
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
have
κ=±pαGN,(12)
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):
ds2=12GM
r+αGNGM2
r2dt212GM
r+αGNGM2
r21
dr2r2d2,(14)
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:
ds2=12GMr2
(r2+αGNGM2)3/2+αGNGM 2r2
(r2+αGNGM2)2dt2
12GMr2
(r2+αGNGM2)3/2+αGNGM 2r2
(r2+αGNGM2)21
dr2r2d2.(16)
The associated gravitational B0rfield is given by
B0r= (pαGN)r4r25αGNGM2
(r2+αGNGM2)4+15
2
GM
(r2+αGNGM2)7/2.(17)
The solution (16) behaves asymptotically like the solution (14):
g00 = 1 2GM
r+αGNGM2
r2+O(1/r3),(18)
and
B0r=αGN
r2+O(1/r3).(19)
3
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
3Λr2)1dr2r2d2,(20)
where the cosmological constant Λ is given by
Λ = 3
G2
NM2α1/22
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.
(22)
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:
ds2=γdt2γ1dr2r2d2,(23)
where
γ= 1 2GM
r+αGNGM2
r2.(24)
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
2ρ2
αGNGM2
(2ρ)22
(2+ρ2d2),(26)
where
r=ρ1 + 2GM
2ρ2
αGNGM2
(2ρ)2.(27)
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)
(rr1)(rr2)1/2
,(28)
4
r
=
r
=
r
=
r
=
r
=−∞
r
=−∞
r
=−∞
r
=−∞
P
arallel
A
ntihorizon
A
ntihorizon
P
arallel
U
niverse
U
niverse
P
arallel
H
orizon
H
orizon
B
lack
H
ole
I
nner
H
orizon
P
arallel
I
nner
H
orizon
W
ormhole
P
arallel
W
ormhole
A
ntiverse
P
arallel
A
ntiverse
W
hite
H
ole
I
nner
A
ntihorizon
P
arallel
I
nner
A
ntihorizon
N
ew
P
arallel
U
niverse
N
ew
U
niverse
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
r+αGNGM2
r2exp(2br)
4K2b2,(29)
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.
5
4 Test Particle Motion
Consider the motion of test particles of mass mand charge q=κm =αGNm. The equations of motion
are given by
d2xµ
ds2+ Γµαβ
dxα
ds
dxβ
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
d2xµ
ds2+ Γµαβ
dxα
dxβ
= 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
θ=π/2:
J=r2
ds ,dt
ds =C
γ,(32)
where Jand Care constants of integration and Jis the angular momentum. Differentiating the metric (23)
with respect to s, we obtain
γdt
ds2
γ1dr
ds 2
r2
ds 2
= 1,(33)
which has the form of an energy integral. We can write this equation as
dr
ds 2
+r2
12GM/r +αGNGM 2/r2=r4(C21)
J2.(34)
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:
ds2=
ρ2(dt asin2θdφ)2sin2θ
ρ2[(r2+a2)adt]2ρ2
dr2ρ22,(35)
where
∆ = 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
G2
N(1 + α)2M2α
1 + α.(37)
6
An ergosphere horizon is determined by g00 = 0:
rE=GN(1 + α)M1 + s1a2cos2θ
G2
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]:
Ml+iSl=M(ia)l.(39)
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
A(r),(40)
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)
A(r).(44)
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
ir.
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.
7
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]:
∂S
∂λ =1
2gµν ∂S
∂xµ
∂S
∂xν,(47)
where λis an affine parameter and Sis the Jacobi action. The action Scan be written as
S=1
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
Σdt
=a(JzaE sin2θ) + r2+a2
[(r2+a2)EaJz],(49)
Σ
=Jz
sin2θaE +a
[(R2+a2)EaJz],(50)
Σdr
=R,(51)
Σ
=Θ.(52)
Here, Σ = r2+a2cos2θand Rand Θ are given by
R= [(r2+a2)EaJz]2∆[K+ (JzaE)2],Θ = K+ cos2θa2E2J2
z
sin2θ,(53)
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
8
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
Dos
(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
r0
drrr/r0212GN(1 + α)M/r0+αG2
N(1 + α)M2/r2
0
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
0
,(59)
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
9
wormhole solution connecting to the same asymptotic region can at best be done in an axially symmetric
solution.
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
∂x2+2
∂y2+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
R.(62)
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
n=1
mn
|x±dn|.(63)
New coordinates are defined by
R=˜r
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
y.(65)
The metric now takes the form:
ds2=V2dT 2
+V2˜r2
(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
ds2=12GM
r+αGNGM2
r2dT 2dr2
12GM/r +αGNGM 2/r2r2d2.(68)
10
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
+V2(dr2
++r2
+d2).(69)
In terms of the proper time τthe induced metric on the shell S1is
ds2=2(RV0)2d2.(70)
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
σ=1
4πV 2
0
∂V
∂r+r+=R=(cosh µ0cos η)
4π¯rV 2
0
∂V
∂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
11
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].
Acknowledgments
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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