Eur. Phys. J.

C (2019) 79:942
https://doi.org/10.1140/epjc/s10052-019-7455-3

Regular Article - Theoretical Physics

NAT black holes

Metin Gürses1,a , Yaghoub Heydarzade1,b , Çetin Şentürk2,c
1 Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara, Turkey
2 Department of Aeronautical Engineering, University of Turkish Aeronautical Association, 06790 Ankara, Turkey

Received: 25 July 2019 / Accepted: 4 November 2019

© The Author(s) 2019

Abstract We study some physical properties of black holes The event horizon of a black hole is a globally-defined
in Null Aether Theory (NAT) – a vector-tensor theory of grav- causal boundary which separates the inside of the black hole
ity. We first review the black hole solutions in NAT and then from the outside. More formally, it is a null surface separat-
derive the first law of black hole thermodynamics. The tem- ing those light rays reaching infinity from those falling to the
perature of the black holes depends on both the mass and the singularity inside. Since it is defined globally, the determina-
NAT “charge” of the black holes. The extreme cases where tion of the location of the event horizon requires in general
the temperature vanishes resemble the extreme Reissner– the knowledge of the global structure of the spacetime. How-
Nordström black holes. We also discuss the contribution of ever, in the case of static spherically symmetric spacetimes,
the NAT charge to the geodesics of massive and massless one can introduce convenient coordinate systems in which
particles around the NAT black holes. the determination is made by looking for places where the
local light cones tilt over. This implies that the existence of
event horizons (and of black holes) has to do with the local
1 Introduction Lorentz invariance of the spacetime. Therefore, it is of great
importance to explore the properties of black holes in gravity
Black holes are of fundamental importance today. This is theories that exhibit violations of local Lorentz invariance.
because of the fact that studies of their properties from Lorentz symmetry is built in GR which describes grav-
both theoretical and observational points of view are being itation well at low energy scales by assuming the space-
expected to shed much light on the nature of the gravity at time structure as continuous and smooth, excluding singu-
strong gravity regimes and at very high energy scales where larities. But this symmetry might be broken at very high
the gravitational force becomes dominant over the other inter- energy scales, especially at the Planck or quantum gravity
actions. For this reason, they have always been at the heart scales, where quantum gravitational effects must be taken
of the theoretical investigations involving gravitational phe- into account. In fact, there are theories, such as string theory
nomena, especially since the discovery of the four laws of and loop quantum gravity, contemplating that the quantum
black hole mechanics [1] and Hawking radiation [2] in the fluctuations at or beyond the Planck scale might be so vio-
context of general theory of relativity (GR). More impor- lent that the spacetime ceases to be continuous and has a
tantly, the thermodynamic interpretation of the four laws [3] discrete structure, and thereby the Lorentz symmetry is not
and the attributions of temperature and entropy to black hole valid [15,16]. This way of reasoning immediately leads to
horizon have provided useful information about the nature the contemplation of gravity theories in which Lorentz sym-
of quantum gravity through holography [4,5] and its spe- metry is broken explicitly.
cific realization AdS/CFT correspondence [6–8]. Observa- One way to construct a Lorentz-violating gravity theory
tionally, recent GW events [9–13] and the image taken by is to assume the existence of a vector field of constant norm
Event Horizon Telescope Collaboration [14] have proven the which dynamically couples to the metric tensor at each point
existence of black holes by direct observations, which has of spacetime. In other words, the spacetime curvature is deter-
also justified the theoretical studies conducted so far. mined together by the metric tensor and the coupled vector
field in spacetime. Such a vector field is referred to as the
a e-mail: “aether” because that generally defines a preferred direction
gurses@fen.bilkent.edu.tr
b e-mail: in spacetime and breaks the local Lorentz invariance. Eintein-
yheydarzade@bilkent.edu.tr
c e-mail:
aether theory [17] is such a theory in which the vector field is
csenturk@thk.edu.tr

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 942 Page 2 of 14 Eur. Phys. J. C (2019) 79:942

timelike everywhere and explicitly breaks the boost sector of metric solutions of NAT, we will first discuss the possible
the Lorentz symmetry. The internal structure and dynamics effect of the null aether field on the solar system dynam-
of this theory have been studied extensively in the literature ics by extracting the so-called Eddington-Robertson-Schiff
[18–42]. parameters β and γ for our solutions, which explicitly appear
Recently, a new vector-tensor theory of gravity called the in the perihelion precession and the light deflection expres-
Null Aether Theory (NAT) [43] has been introduced into the sions. We will see that, at the post-Newtonian order, there
realm of modified gravities. This theory assumes the dynam- is no contribution from the aether field to the deflection of
ical vector field (the aether) inherent in the theory to be null at light rays passing near a massive body; that is the same as
evey point in spacetime. In the paper [43], we first studied the in GR! However, there is an explicit contribution, at the
Newtonian approximation of the theory and showed that it post-Newtonian order, from the aether field to the perihe-
reproduces the Poisson equation at the perturbation order by, lion precession of planetary orbits. This fact can be used
in some cases, rescaling the Newton’s constant G N . Then we to constrain the parameters of the theory from solar sys-
obtained exact spherically symmetric solutions in this theory tem observations. Then we shall present the details of the
by properly choosing the null vector field and we showed that black hole spacetimes by discussing the singularity struc-
there is a large class of solutions depending on the parameters ture, the ADM mass of the asymptotically flat solutions, and
of the theory. Among these, there are Vaidya-type nonstation- the thermodynamics in order. In the thermodynamics of NAT
ary solutions because of the null aether behaving as a null black holes especially, it is interesting to note that an appro-
matter source, and for some special values of the parameters, priate definition of the NAT “charge” reduces the horizon
stationary Schwarzschild–(A)dS and Reissner–Nordström– thermodynamics to that of the Reissner–Nordström–(A)dS
(A)dS type solutions with some effective cosmological con- black hole in GR and the first law takes the standard form
stant and some “charge” sourced by the aether, respectively. if the theory’s parameters c2 and c3 satisfy a strict condi-
We also discussed the existence of stationary black holes tion. Lastly, we will also discuss the circular geodesics of
among these exact solutions for arbitrary values of the param- massive and massless particles around the NAT black holes
eters of the theory. (See [43] for details and explicit structures to see the effect of the null aether on the particle trajec-
of these solutions.) To see the effect of the null aether in cos- tories in the spacetime. We will show that the null aether
mology, we studied the flat FLRW metric and, taking the substantially changes the behavior of the circular orbits of
spatial component of null aether lying along the x axis, we massive and massless particles. We will also calculate the
found all possible perfect fluid solutions of NAT. We also perihelion precession of planets and the deflection of light
discussed the existence of the Big-bang singularity and the rays explicitly in the case of a nonzero cosmological con-
accelerated expansion of the universe in NAT. In addition stant.
to these, to better understand the internal dynamics of the The organization of the paper is as follows. In Sect. 2, we
theory, we constructed exact wave solutions by specifically give the Null Aether Theory in detail. In Sect. 3 we review
considering the Kerr–Schild–Kundt (KSK) class of metrics the Newtonian approximation of the theory and observe that
[44,45] with maximally symmetric backgrounds. After giv- the results we obtained in this section are consistent with the
ing the exact AdS-plane wave solutions of NAT in D ≥ 3 exact solutions in the next section. In Sects. 4 and 5, we dis-
dimensions, we also obtained all possible pp-wave solutions cuss exact spherically symmetric solutions and black hole
of the theory propagating in the flat background spacetime. spacetimes in NAT, respectively. In Sect. 6, we obtain the
These exact wave solutions are consistent with the linearized ADM mass of the asymptotically flat NAT black holes. In
waves of the theory [46]. In Einstein-aether theory, spheri- Sect. 7, we study the first law of black hole thermodynamics.
cally symmetric black hole solutions exist for special values In Sect. 8, we obtain the circular orbits of massive and mass-
of parameters of the theory (see, for example, [23] and [26]). less particles around the NAT black holes. Finally, in Sect. 9,
So the solution class is restricted. On the other hand, as we we conclude by summarizing our work and indicating some
presented in our paper, in NAT there is a large class of black possible future directions.
hole solutions for any values of the theory’s parameters (see We use the metric signature (−, +, +, +, . . .) throughout
also [43]). Similarly, gravitational plane wave solutions in the paper.
Einstein-aether theory exist under certain conditions on the
parameters of the theory (see, for instance, [41]). However, in
NAT we have exact plane wave and pp-wave solutions valid 2 Null Aether theory
for any values of the parameters of the theory (see [43]).
In this paper, we will continue our explorations in the Aether theory is a generally covariant theory of gravity in
implications of the exact spherically symmetric solutions and which the metric tensor (gμν ) of the spacetime dynamically
black hole spacetimes found in [43]. After giving a brief couples, through covariant derivatives, to a vector field (v μ )
review of the Newtonian limit and static spherically sym- – referred to as the “ aether.” In the absence of matter fields,

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the action of the theory can be written as [43] n μ are real null vectors with lμ n μ = −1, and m μ is a com-
 plex null vector orthogonal to lμ and n μ , and then assume the
1 √
I = d 4 x −g [R − 2 − K μν αβ ∇μ v α ∇ν v β null aether vμ is proportional to the one null leg of this tetrad,
16π G
say lμ ; i.e. vμ = φ(x)lμ . Thus this geometric construction
+ λ(vμ v μ + ε)], (1)
enables us to naturally introduce a scalar function φ(x) – the
where R is the Ricci scalar,  is the bare cosmological con- spin-0 part of the aether field – which generally contains the
stant, and physical meaning of the aether by carrying a nonzero “ aether
μ
charge.”
K μν αβ = c1 g μν gαβ + c2 δαμ δβν + c3 δβ δαν − c4 v μ v ν gαβ ,
(2)
3 Newtonian limit of Null Aether theory
with the dimensionless constant parameters c1 , c2 , c3 , c4 .
From now on, throughout the text, we shall use the shorthand
The Newtonian limit of the theory can be achieved by assum-
notation ci j = ci + c j for combinations of these constants.
ing the gravitational field is weak and static and produced by
When ε = −1, the aether field is timelike and this case
a nonrelativistic matter field. Also the cosmological constant
corresponds to the Einstein-Aether theory of [17]. In our
plays no role in this context so that it can be set equal to zero.
case, however, ε = 0 and the aether becomes a null vector
Therefore in taking the Newtonian limit, we can write the
field. The Lagrange multiplier λ in (1) is introduced into the
metric in x μ = (t, x, y, z) as
theory to explicitly enforce the nullity of the vector field; that
is, to have ds 2 = −[1+2 (
x )]dt 2 +[1−2 (
x )](d x 2 +dy 2 +dz 2 ),
vμ v μ = 0 (3) (9)

at each point of the spacetime. Therefore the independent where ( x ) is the gravitational potential on the order of G,
variables in the theory are g μν , v μ , and λ. The field equations and take the matter energy-momentum tensor as
are then obtained by varying the action (1) with respect to
these fields: Varying with respect to λ immediately leads to
matter
Tμν = (ρm + pm )u μ u ν + pm gμν + tμν , (10)
the null constraint (3) and, making use of it, varying with √
where u μ = 1 + 2 δμ0 is the four-velocity of the matter
respect to g μν and v μ respectively yields
field, ρm and pm are the mass density and pressure, and tμν
 
G μν + gμν = ∇α J α (μ vν) − J(μ α vν) + J(μν) v α is the stress tensor with u μ tμν = 0. Then perturbing also
  the aether field appropriately, we consider only the zeroth
+ c1 ∇μ vα ∇ν v α − ∇α vμ ∇ α vν
and first order (linear) terms in vμ and gμν in the Eistein–
1
+ c4 v̇μ v̇ν + λvμ vν − Lgμν , (4) Aether Eqs. (4) and (5). At this point, however, there appear
2 three distinct cases in perturbing the aether field, with the
c4 v̇ α ∇μ vα + ∇α J α μ + λvμ = 0, (5) associated Newtonian limits:

where we used the identifications Case 1: Let us decompose the null aether field as

v̇ μ ≡ v α ∇α v μ , (6) vμ = aμ + kμ , (11)
J μ α ≡ K μν αβ ∇ν v β , (7) where aμ = (a0 , a1 , a2 , a3 ) is a constant null vector repre-
μ α senting the background aether field and kμ = (k0 , k1 , k2 , k3 )
L≡J α ∇μ v . (8)
is the perturbation which need not necessarily be a null vec-
Obviously, the Minkowski metric (ημν ) together with a con- tor. The null constraint (3) then implies that
stant null vector (vμ = const.) and λ = 0 constitute a solu-
tion to NAT. Since being null, the zero ather field (i.e. vμ = 0) a02 = a · a , (12)
with an arbitrary λ reduces the theory to the usual general rel- 1
ativity; however, this trivial case can be distinguished from k0 = [a · k + 2a02 ], (13)
a0
the nontrivial aether case by imposing certain initial and
boundary conditions on the solutions of the Einstein-Aether at the perturbation order. Since the metric is symmetric under
equations (4) and (5). (See the discussion in [43].) rotations, we can take, without loosing any generality, a1 =
Since the aether field in NAT is null by construction, one a2 = 0 and for simplicity we will assume that k1 = k2 = 0.
can always introduce a scalar degree of freedom into the the- Then one can show that
ory. The reasoning is as follows: First set up, at each point in 2a33 c4
a = (l , n , m , m̄ ), where l and
spacetime, a null tetrad eμ c3 = −c1 = −c2 , k3 = − , (14)
μ μ μ μ μ c1

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4π G In the case of spherical symmetry, outside the mass dis-

∇2 = ρm = 4π G N ρm . (15)
1 − c1 a32 tribution of mass M, the Poisson Eq. (21) gives
The last Eq. (15) is in the form of the Poisson equation and GM
(r ) = − , (23)
implies that Newton’s gravitation constant G N is an effective r
one defined by the scaling and the condition (22) gives
G a1 a2
GN = . (16) φ(r ) = (1+q)/2 + (1−q)/2 , (24)
1 − c1 a32 r r
where a1 and a2 are arbitrary constants on the order of G and
Similar scaling also appears in the context of Einstein-Aether
we have defined the parameter
theory [30,31]. The constraint c3 +c1 = 0 can be removed by

taking the stress part tμν into account in the energy momen- c1
q ≡ 9+8 , (25)
tum tensor, then there remains only the constraint c2 = c1 . c23
Case 2: Take the null aether field as vμ = φ( x )lμ where lμ which is always positive by definition. Therefore, we can
is a null vector defined by the geometry (9) as immediately see that the three of the parameters of NAT must
satisfy the constraint
xi i
lμ = δμ0 + (1 − 2 ) δ , (17) c1 9
r μ ≥− . (26)
 c23 8
with r = x 2 + y 2 + z 2 and i = 1, 2, 3. Note that any
Specifically, when q = 0 (c1 = −9c23 /8), we have
multiplicative function of x can be absorbed into the scalar
function φ(x ). Now assuming the perturbation φ( x ) = φ0 + a1 + a2
φ(r ) = √ ; (27)
φ1 (
x ) where φ0 = const. = 0 and φ1 is at the same order as r
G, we obtain when q = 3 (c1 = 0), we have
c1 + c3 = 0, c2 = 0, c4 = 0, φ1 = 2φ0 , (18) a1
φ(r ) = 2 + a2 r ; (28)
4π G r
∇2 = ρm = 4π G N ρm . (19)
1 − c1 φ02 or when q = 1 (c1 = −c23 ), we have
a1
Again, the effective value of Newton’s constant can be seen φ(r ) = + a2 . (29)
from (19) r

G
GN = . (20) 4 Spherically symmetric static solutions in Null Aether
1 − c1 φ02
Theory
This is, however, a very restricted aether theory because there
is only one independent parameter c1 left in the theory. In this section, we shall review the spherically symmetric
Case 3: Take the zeroth order scalar aether field in Case 2 as static solutions in NAT found previously in the original work
zero; i.e., φ0 = 0. This means that φ( x ) = φ1 (
x ) and is at [43]. The metric written in the Eddington–Finkelstein coor-
the same order as G. Therefore, there is no contribution to dinates x μ = (u, r, θ, ϕ) is
the Eq. (4) from the aether field at the linear order in G, and  2
from the 00 component of (4), we get ds 2 = − 1 − r − 2 f (r ) du 2
3
∇2 = 4π Gρm , (21) + 2dudr + r 2 dθ 2 + r 2 sin2 θ dϕ 2 , (30)

which is the Poisson equation unaffected by the null aether where u is the null coordinate, then taking the null aether field
field at the perturbation order. On the other hand, from the – assumed to be present at each spacetime point in the theory
ith component of the aether Eq. (5) we obtain, at the linear – is aligned with this coordinate, we obtain the solution
order in G, vμ = φ(r )δμu , (31)
(c2 + c3 )r x ∂ j ∂i φ − (2c1 + c2 + c3 )x x ∂ j φ
2 j i j a1 a2
φ(r ) = (1+q)/2 + (1−q)/2 , (32)
r
⎧ r
+[2c1 + 3(c2 + c3 )]r 2 ∂i φ − 2(c1 + c2 + c3 )x i φ = 0,
⎪ a 2b 2
⎪ 1 + 2 b2 + m̃ , for q = 0,
1 a
(22) ⎪
⎨ r 1+q r 1−q r
f (r ) = (33)
after eliminating the Lagrange multiplier λ by using the ⎪m
⎪
⎪
⎩ ,
zeroth order equation. for q = 0,
r

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where a1 , a2 , m̃, and m are just integration constants and Schwarzschild-(A)dS type solution if A = 0. Solutions
 involving terms like A/r 4 can be found in, e.g., [23,48].
c1 1
q ≡ 9 + 8 , b1 = [c3 − 3c2 + c23 q],
c23 8
Before concluding this section, one last remark must be
1
b2 = [c3 − 3c2 − c23 q]. (34) made on the possible effects of the null aether field on the
8
solar system observations. For this purpose, we will consider
As we will show later, the constants m̃ and m are the mass the post-Newtonian parameters in the case of a static, spher-
parameters of the solutions. At this point, it is also important ically symmetric mass distribution like the Sun. Since the
to note that the exact solution (32) is the same as the linearized cosmological constant is totally negligible in this setting, the
one (24) obtained in the previous section. This means that the metric produced by such a body can be expanded to post-
null aether contribution to the metric [see Eq. (33)] comes in Newtonian order as [49,50]
at the order of G 2 .  
Now performing the coordinate transformation 2G M G2 M 2
ds 2 = − 1 − + 2(β − γ ) 2 + · · · dt 2
r r
dr  
du = dt +  2
, (35) GM
1− 3r − 2 f (r ) + 1 + 2γ + · · · dr 2
r
one can bring the metric (30) into the Schwarzschild coordi-
+ r 2 dθ 2 + r 2 sin2 θ dϕ 2 , (39)
nates
dr 2 where M is the mass of the body and β and γ are the so-called
ds 2 = −h(r )dt 2 + + r 2 dθ 2 + r 2 sin2 θ dϕ 2 , (36) Eddington-Robertson-Schiff parameters. These two param-
h(r )
eters explicitly appear in the expressions for the perihelion
where precession of a planetary orbit and the deflection of light rays
 passing near the body which are respectively given by
h(r ) ≡ 1 − r 2 − 2 f
⎧
3  
2 − β + 2γ 6π G M
⎪
⎪  2 2a12 b1 2a22 b2 2m̃ ϕ = , (40)
⎨ 1 − 3 r − r 1+q − r 1−q − r (for q = 0),
⎪ 3 a(1 − e2 )
 
= 1 + γ 4G M
⎪
⎪ ψ= , (41)
⎩ 1 −  r 2 − 2m
⎪
(for q = 0). 2 b
3 r
(37) where a is the semi-major axis and e is the eccentricity of
the orbit and b is the impact parameter. In general relativity,
together with from the Schwarzschild metric, it can immediately be seen
 
1 r that β = γ = 1.
vμ = φ(r ) δμ + δμ .
t
(38) In NAT, we have the solutions given by (36) and (37).
h
So taking  = 0, for the case q = 0, since we recover
The metric (36) describes the spherically symmetric static the usual Schwarzschild solution, we can immediately have
solutions in NAT, and interestingly we have lots of them due β = γ = 1 just as in GR, but when q is a positive integer,
to the free parameters q, b1 , and b2 in the theory. The solution the expanded metric is
for q = 0 is the usual Schwarzschild-(A)dS spacetime but  
there are also solutions corresponding to some other specific 2m̃ 2a12 b1
values of the parameter q which are of special importance; ds = − 1 −
2
− + · · · dt 2
r r2
for instance,  
2m̃
+ 1+ + · · · dr 2
• When q = 1 (c1 = −c23 ), h(r ) ≡ 1 − A − r 2 /3 − r
B/r 2 − 2m̃/r , where A ≡ 2a22 b2 and B ≡ 2a12 b1 : This + r 2 dθ 2 + r 2 sin2 θ dϕ 2 , (42)
is a Reissner–Nordström–(A)dS type solution if A = 0.
• When q = 2 (c1 = −5c23 /8), h(r ) ≡ 1 − r 2 /3 − where we have assumed a2 = 0 just for simplicity. It should
A/r 3 − Br − 2m̃/r , where A ≡ 2a12 b1 and B ≡ 2a22 b2 : be noted that the terms with q > 1 do not contribute to the
this solution with A = 0 has been obtained by Mannheim post-Newtonian order. In other words, only the term with
and Kazanas [47] in conformal gravity who also argue q = 1 has contribution to the post-Newtonian order. Now,
that the linear term Br can explain the flatness of the knowing that m̃ ∼ G and a1 ∼ G and comparing (42) with
galaxy rotation curves. (39), we can read off the post-Newtonian parameters as
• When q = 3 (c1 = 0), h(r ) ≡ 1 − A/r 4 − Br 2 − 2m̃/r , a12 b1
where A ≡ 2a12 b1 and B ≡ /3 + 2a22 b2 : This is a β =1− , γ = 1. (43)
m̃ 2

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Therefore, we can see from (41) that the null aether does not corresponding to the event horizon. When there is only one
affect the light deflection at the post-Newtonian order; it is event horizon, the metric function h(r ) can be written as
the same as in GR. However, it is obvious from (40) that it
h(r ) = (r − r0 )g(r ), (45)
does affect the perihelion precessions of planets as
  where g(r ) is a continuous function for r ≥ r0 and g(r ) > 0
a12 b1 6π m̃
ϕ = 1 + , (44) because h(r ) must be positive for r > r0 . This means that
3m̃ 2 a(1 − e2 )
h (r0 ) = g(r0 ) > 0 (46)
This result tells us that, if b1 > 0, the perihelion advance is
greater than that of GR, and if b1 < 0, it is less than that of due to the continuity of g(r ). When there are multiple event
GR. horizons, say the number is m, the metric function h(r )
should be in the form
h(r ) = (r − r1 )(r − r2 ) · · · (r − rm )g(r ), (47)
5 Black hole solutions in Null Aether Theory
where g(r ) > 0 for r greater than the largest root, say r0 .
The metric (36) also describes spherically symmetric static Again, due to the continuity of g(r ) for r ≥ r0 ,
black hole solutions in NAT. The event horizons of these
h (r0 ) = (r0 − r1 )(r0 − r2 ) · · · (r0 − rm )g(r0 ) > 0, (48)
solutions are in principle determined by the positive real
roots of the equation h(r ) = 0 [see Eq. (37)]. In general, where we assume that all the roots are distinct and the event
the existence of these roots crucially depends on the signs horizon is at r0 , the largest root of (47); that is, r0 > r1 >
and/or values of and the relation between the parameters · · · > rm . When some or all of the roots are coincident, we
(q, , a1 , a2 , b1 , b2 , m̃, m) appearing in (37). For example, have the extreme case. For example, for two coincident roots,
in the case q = 0, there are two distinct positive real roots,
which are those of the usual Schwarzschild-dS black hole, h(r ) = (r − r0 )2 g(r ), (49)
if  > 0 and 0 < 9m 2 < 1, and there is only one posi- where g(r ) > 0 for r > r0 . Then
tive root, which is that of the usual Schwarzschild-AdS black
hole, if  < 0. On the other hand, the determination of the h (r0 ) = 0. (50)
positive real roots of the equation h(r ) = 0 in the other case From now on, we shall admit this condition as the indicator
q = 0 is not that easy. However, we can generally make of an extreme black hole.
the following points. If q is an integer, h(r ) = 0 becomes To understand the singularity structure of our solutions
a polynomial equation which may have at least one positive given in (36) and (37), we shall calculate the two of the cur-
real root representing the event horizon of the corresponding vature scalars; namely, the Ricci and Kretschmann scalars.
black hole. And, if q is not an integer, the limits limr →0+ h(r ) For q = 0, they are
and limr →∞ h(r ) may be used to just determine the existence
of the real roots; more explicitly, since h(r ) is a continuous A1 (q − 1) A2 (q + 1)
R = 4 + 2q 3+q
+ , (51)
function of r , when the signs of the limits are opposite, it is r r 3−q
certain that there is at least one real root of h(r ). For example, K = Rμναβ R μναβ
in Table 1, we classified the cases in which there is at least
48m̃ 2 82 8q A1 (q − 1) A2 (q + 1)
one real root of the equation h(r ) = 0. There might be other = 6
+ + 3+q
+
r 3 3 r r 3−q
possibilities, of course, but by giving these examples, we are
trying to point out that there are black hole solutions in the A1 (q + 2)(q + 3) A2 (q − 2)(q − 3)
+ 16m̃ 6+q
+
general case q = 0 as well. r r 6−q
Black hole solutions may have one or multiple horizons. A21 (12 + 20q + 17q 2 + 6q 3 + q 4 )
+4
We call r = r0 the largest root of h(r ) and hence the one r 2(3+q)
2 A1 A2 (12 − 9q 2 + q 4 )
+
Table 1 Some cases in which black holes certainly exist in NAT r6
 limr →0+ h(r ) limr →∞ h(r )
A (12 − 20q + 17q 2 − 6q 3 + q 4 )
2
q b1 b2 + 2 , (52)
r 2(3−q)
(0, 3) + ± − − +
(0, 3) − ± + + − where we made the definitions A1 ≡ a12 b1 and A2 ≡ a22 b2 .
(3, ∞) + − ± − + It can be seen that the only singularity is at r = 0. From
(3, ∞) − + ± + − these, we can also recover the standard Schwarzschild-(A)dS
expressions by setting A1 = 0 and A2 = 0 simultaneously.

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6 ADM mass of asymptotically flat solutions  

1 a12 b1 a22 b2
M AD M = m̃ + (1 + q) q + (1 − q) −q |r →∞ .
G r r
To obtain asymptotically flat solutions, we should immedi-
ately take  = 0, and the metric (36) becomes (60)

dr 2 Then one realizes that, for having an asymptotically well

ds 2 = −h(r )dt 2 + + r 2 dθ 2 + r 2 sin2 θ dϕ 2 , (53)
h(r ) defined ADM mass for NAT black holes,
where ⎧
for q = 1 (if a1  = 0 and a2  = 0) or (if a1 =0 or b1 =0),
⎧ m̃ ⎨
⎪ 2a12 b1 2a22 b2 2m̃ M AD M =
⎪ G⎩
⎨ 1 − r 1+q − r 1−q − r (for q = 0),
⎪ for 0 < q (if a2 = 0 or b2 = 0).
h(r ) = (54) (61)
⎪
⎪
⎪
⎩ 1 − 2m (for q = 0).
r It should be noted that having an asymptotically flat space-
As is obvious, in the q = 0 case, the metric is just the usual time with a well defined ADM mass is guaranteed only
Schwarzschild spacetime which is explicitly asymptotically by the second case in (61). In all these cases, the ADM
flat. However, in the q = 0 case, to achieve asymptotically mass is rescaled through the definition of G in the the-
flat boundary conditions, one should consider the following ory; for example, in the Newtonian limit Case 1 of Sect.
cases separately: Since q > 0 by definition (see Eq. (34)), 3, G = G N (1 − c1 a32 ) and

h(r ) |r →∞ = 1 m̃
⎧ M AD M = . (62)
⎨ for 0<q<1 (if a1  = 0 and a2  = 0) or (if a1 =0 or b1 =0), G N (1 − c1 a32 )
×
⎩ Although both cases a2 = 0 and b2 = 0 give the same
for 0 < q (if a2 = 0 or b2 = 0).
(55) ADM mass (61) for q > 0 for an observer at infinity, they
differ if one considers the aether field φ by putting different
For stationary spacetimes with the time translation Killing constraints on the parameter q. That is,
⎧
vector χ μ , the ADM and Komar masses are identical. So, the ⎪ a1
⎪
⎪ If a2 = 0 ⇒ φ = (1+q)/2 , 0 < q,
ADM mass can be calculated from ⎨ r
 ⎪
1 ⎪
⎪ a1 a2 c3 −3c2
M AD M = − ∇ μ χ ν dμν , (56) ⎩ If b2 = 0 ⇒ φ=
(1+q)/2
+ (1−q)/2 , 0 < q= <1.
4π G B∞ r r c23
(63)
where dμν = −u [μ sν ] d A, with d A = r 2 sin θ dθ dϕ, is
the differential surface element on a two-sphere B living in For both of these cases, the constraints on q parameter
a√ √  of the spacetime. Here, u μ =
spacelike hypersurface guarantees that the aether field is also well behaved at asymp-
− hδμt and sμ = δμr / h are the unit timelike and spacelike totic region.
normals to B, respectively, and B∞ is a two-sphere at spatial We define the NAT charge in the following way. Let
infinity. Regarding the stationary nature of our spacetime
μ
(36), the corresponding Killing vector field is χ μ = δt and f μν = ∇μ vν − ∇ν vμ (64)

h be an antisymmetric tensor constructed from the null aether

∇ μ χ ν dμν = − d A, (57)
2 vector field vμ and let the conserved current J μ be defined
by
where h(r ) is given by (37) with  = 0 and the prime denotes
differentiation with respect to r . Then, the ADM mass in (56)
∇ν f μν = 4π G J μ . (65)
reduces to
r2 It should be noted that the Einstein-Aether field equations (4)
M AD M = h |r →∞ . (58) and (5) can indeed be written in a form including the tensor
2G
f μν . Then, from the conservation equation ∇μ J μ = 0, we
For the case q = 0, the ADM mass reads as can define the conserved charge as
m 
M AD M = , (59)
G QN = J μ dσμ , (66)

but for the case q = 0, we obtain

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where dσμ ≡ −u μ d V with d V being the volume element of ⎧  1/2q

a spacelike hypersurface  in the spacetime. Now with the ⎪
⎪ a12 b1
⎪
⎪r > (for b1 > 0 and b2 < 0),
help of the Stokes’s theorem, we can write ⎪
⎪
⎪
⎪ a22 |b2 |
⎪
⎪
  ⎪
⎨
1  1/2q
QN = J μ dσμ = ∇ν f μν dσμ , (71)
4π G  ⎪ a12 |b1 |

  ⎪
⎪ r0 < r < (for b1 < 0 and b2 > 0),
1 1 ⎪
⎪ 2
a2 b2
= μν
f dμν − f μν dμν , (67) ⎪
⎪
⎪
⎪
4π G B∞ 4π G B H ⎪
⎪
⎩
√ r0 < r (for b1 < 0 and b2 < 0).
where dμν = −u [μ sν ] d A with u μ = − hδμt , sμ =
√
δμr / h, and d A = r 2 sin θ dθ dϕ, as before. Here, B∞ is
the boundary of  at spatial infinity and B H is the bound- 8 Thermodynamics of NAT Black holes
ary on the horizon. For the asymptotically flat black hole
solutions (36) the contribution of the B∞ integral becomes Now we shall study the thermodynamics of NAT black holes
zero. Thus, integrating the angular part and inserting the null that we reviewed in Sect. 5. Here we first consider the case
aether vector field (38), we obtain a2 = 0. Then the metric function h(r ) and the scalar aether
field φ(r ) take the forms
r2
QN = − φ |r →r0 . (68)
G  2 2a12 b1 2m̃
h(r ) = 1 − r − 1+q − , (72)
This is the conserved NAT charge for the black hole solutions 3 r r
a1
(36). As an example, when a2 = 0, the conserved charge Q N φ(r ) = (1+q)/2
. (73)
r
is proportional to the parameter a1 in the solution as
The location of the event horizon r0 is given by h(r0 ) = 0
1+q a1 and the area of the event horizon is A = 4πr02 . Now let a1 =
QN = . (69)
2 Gr (q−1)/2 G Qr0
(q−1)/2
, where Q is related to the conserved NAT charge
0
Q N in (69) as Q = 2 Q N /(1 + q). With this identification,
(72) and (73) become

 2 2G 2 Q 2 b1  r0 q−1 2m̃
7 Energy conditions
h(r ) = 1 − r − − , (74)
3 r2 r r
G Q  r0 (q−1)/2
At this point, it would be interesting to see if there is any
condition on the parameter q in the solution (37) to have φ(r ) = . (75)
a physical aether source in the field Eq. (4). The energy- r r
momentum tensor, tμν , of the aether field can be read from At the event horizon location r0 , we then have
the left hand side of (4), and so the weak energy condition
states that tμν u μ u ν ≥ 0 for an arbitrary timelike vector u μ .  2 2G 2 Q 2 b1 2m̃
h(r0 ) = 1 − r0 − 2
− = 0, (76)
Assuming  = 0, considering the metric (36), and taking 3 r0 r0
μ
u μ = δt / h, we obtain GQ
φ(r0 ) = . (77)
r0
1
ρ = tμν u μ u ν = 2 [1 − (r h) ]
It is interesting that the horizon condition (76) is independent
⎧  r 
⎪ 2a12 b1 2a22 b2 of the parameter q and, when b1 ≡ 18 [c3 − 3c2 + c23 q] =
⎪
⎪
q
⎨− 3 + −q (for q = 0), −1/2, it becomes that of the Reissner–Nordstrom–(A)dS
r rq r
= (70) black hole in GR. In addition, the scalar aether field φ(r )
⎪
⎪
⎪
⎩ resembles the electric potential at r = r0 .
0 (for q = 0).
Now assuming the entropy as S = k A, where k is a posi-
Here we know that r > r0 and q > 0 by definition. Then, for tive constant which takes the value 1/4 [2], and varying that,
q = 0, to satisfy the weak energy condition, ρ ≥ 0, we have we obtain
the following cases.  
δS = 8π kr0 r0m̃ δ m̃ + r0Q δ Q + r0 δ , (78)

• If a1 = 0 or b1 = 0, then b2 ≤ 0. where r0m̃ = ∂r ∂r0 ∂r0

∂m , r0Q = ∂ Q , and r0 = ∂ . This relation
0

• If a2 = 0 or b2 = 0, then b1 ≤ 0. can be translated into the form of the first law of thermody-
• If a1 = 0 and a2 = 0, then namics as

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Eur. Phys. J. C (2019) 79:942 Page 9 of 14 942

δ m̃ GQ
= T δS + Vφ δ Q + V δ P, (79) φ(r0 ) = . (91)
G r0
where the temperature T , the NAT charge potential Vφ , the The rest goes on like in the case of a2 = 0; the only difference
event horizon volume V , and the pressure P are given by is that b1 must be replaced by b2 in all the Eqs. (78)–(85).
1 1
T = = h (r0 ), (80)
8π Gkr0 r0m̃ 16π Gk 9 Null and timelike geodesics
1 r0Q GQ
Vφ = − = −2b1 = −2b1 φ(r0 ), (81)
G r0m̃ r0 9.1 Circular orbits
r0 4
V = 8π = πr03 , (82)
r0m̃ 3 Here, we study the circular orbits at the equatorial plane, i.e
 θ = π2 , for the metric function (72) with a2 = 0. Accord-
P=− , (83) μ μ
8π G
μ
 μ vector fields K = (∂t ) =
ingly, we have two Killing
where b1 takes −1/2 to get the standard expression for the (1, 0, 0, 0) and R = ∂ϕ = (0, 0, 0, 1) corresponding to
xμ
fist law. By using the discussions in Sect. 5, we can now the conserved energy E = −K μ ddσ and conserved angular
μ
explicitly see from (80) that T > 0 for the non-extreme momentum L = Rμ dσ , respectively, where σ is an affine
dx

cases and T = 0 for all the extreme cases. parameter along the geodesics. Then, regarding the metric,
As a remark, it is worth mentioning the following point. the energy and angular momentum magnitude of the orbiting
The extremal event horizon r0 is a radius where h(r0 ) = 0 body are given by
and h (r0 ) = 0, and so, when  = 0, the extremal event    
dt dϕ
horizon for (76) can be obtained as E =h , L = r2 . (92)
dσ dσ
r0 = m̃, (84) x dx
On the other hand, using the geodesics equation gμν ddσ
μ ν
dσ
which can equivalently be written in terms of mass and aether = , where  = 0 and −1 denote the null and timelike
charge as geodesics, respectively, we obtain
 2  2    
2
m̃ 2 = −2b1 G 2 Q 2 . dt dr 2 dϕ
(85) −h 2
+ +h r −  = 0. (93)
dσ dσ dσ
This relation tells us that b1 must always be less than zero
and particularly for b1 = − 21 , one can obtain the relation Using the energy and angular momentum (92), we arrive at
m̃ 2 = G 2 Q 2 similar to the one in the case of the Reissner–  
1 dr 2
Nordstrom black hole in Einstein gravity, which is also obvi- + V = E, (94)
2 dσ
ous from (76).
2
The thermodynamics of the other case a1 = 0 is similar where E = E2 and the potential V reads as
to the case above in which a2 = 0. In this case, the metric  2 
function h(r ) and the scalar aether field φ(r ) become 1 L
V= h −  . (95)
2 r2
 2 2a22 b2 2m̃
h(r ) = 1 − r − 1−q − , (86) Substituting the metric function h in (72), we find the poten-
3 r r tial as
a2
φ(r ) = (1−q)/2
. (87)   m̃ L2 m̃ L 2 1
r V=− + + 2 − 3 − L 2
−(q+1)/2 2 r 2r r 6
This time, defining a2 = G Qr0 , where Q is the NAT a 2b a 2b L 2
1 1 1
“charge” again, we can write (86) and (87) as + r 2 + 1+q 1
− 1 3+q , (96)
6 r r
 2 2G 2 Q 2 b2  r0 −(q+1) 2m̃ where the first four terms are the standard terms as in GR
h(r ) = 1 − r − − , (88)
3 r2 r r [51], and the last four terms are the new correction terms by
 
G Q r0 −(q+1)/2 the cosmological constant and aether field, respectively. In
φ(r ) = . (89)
r r Fig. 1, we have plotted the potential function V versus r for
At the event horizon location r0 , however, we obtain the same some sets of q, L and a12 b1 parameters for the massive and
Eqs. (76) and (77) massless particles, respectively. For each set of parameters,
one can see that in general the deviation of the potential V
 2 2G 2 Q 2 b2 2m̃ from GR potential for the massive particles is more than for
h(r0 ) = 1 − r − − = 0, (90)
3 0 r02 r0 the massless particles. For both the massive and massless

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Fig. 1 The upper and lower plots are denoting the potential V for some typical values of the parameters for the massive and massless particles,
respectively

cases, by increasing q, the potential tends to GR. However, where one can see that cosmological constant does not con-
by increasing L, the potential increases and deviates more tribute for the null geodesics but the aether field does as the
from GR. For b1 > 0, the potential decreases by increasing last term. Here, one may consider the particular case q = 1.
a12 b1 values and vice versa. This case has two solutions as
The circular orbits can be obtained as the radii where the 
potential is flat, i.e ddrV |r =rc = 0. Here rc denotes the circular 3m̃ 3m̃ 16a12 b1
r c± = ± 1+ . (101)
orbits. Then, the equation governing the circular orbits can 2 2 9m̃ 2
be obtained as
Considering 9m̃ 2  16a12 b1 , we have
 m̃ L2 3m̃ L 2
− 2 − 3 + ⎧ 2
rc rc rc4 ⎪ 4 a 1 b1
⎨ 3m̃ +
⎪ 3 m̃ ,
1 (1 + q)a12 b1 (3 + q)a12 b1 L 2 r c±  (102)
+ r − + = 0. (97) ⎪
3 2+q
rc
4+q
rc ⎪ 2
⎩ − 4 a1 b1 .
3 m̃
For the GR limit by turning off the cosmological constant
and aether field ( = 0 and a1 = 0), we arrive at Then, the second solution is a physical orbit only for b1 < 0.
Thus, in contrast to GR which has only one null circular orbit
− L 2 rc + 3m̃ L 2 −  m̃rc2 = 0, (98) as in (99), in the presence of aether field for b1 < 0, there
are two null circular orbits in which the radius of the outer
which admits the following solutions for the massless and one is smaller than GR. For b1 > 0, there is only one null
massive particles respectively circular orbit greater than the one in GR.
⎧ For the case of timelike circular orbits, solving Eq. (97)
⎪
⎨  = 0 : rc = 3m̃, for a generic q is impossible. Thus, one may consider the
√ (99) particular case of q = 1 where the resulted equation will be
⎪
⎩  = −1 : r ± = L 2 ± L 4 −12m̃ 2 L 2 .
c 2m̃ a 6th order equation for rc as (97) reduces to
In the presence of the cosmological constant and aether 1
field, the Eq. (97) for the null geodesics reduces to L 2 rc2 − 3m̃ L 2 rc − m̃rc3 + rc6 − 4a12 b1 L 2 −2a12 b1rc2 =0.
3
(103)
(3 + q)a12 b1
rc − 3m̃ − q = 0, (100)
rc Finding the general real and positive solutions to this equation
is not an easy task. However, for realizing the effect of aether

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Eur. Phys. J. C (2019) 79:942 Page 11 of 14 942

field, one may consider  = 0 and rc > 2m̃  2a12 b1 in the where
 
 
Eq. (97) which leads to
1 
Ṽ(x) = 1 − 2m̃x + 2 − 2a12 b1 x 1+q x2 − 2 .
L 2 rc − 3m̃ L 2 − m̃rc2 − 2a12 b1 rc = 0. (104) 2 3x L
(111)
This equation has two solutions as
L 2 − 2a12 b1 | L 2 − 2a12 b1 | Then, Eq. (110) for the timelike geodesics becomes
r c± = ±
 2m̃ 2m̃ d2x m̃
+ x = 2 + 3m̃x 2 +
 1

 12m̃ 2 dϕ 2 L 3L 2 x 3
1 −  2 . (105)
 a 2 b1 (1 + q) q
 2a 2b
+ 1 x + a12 b1 (q + 3)x q+2 .
L 2 1 − L2 1 1
(112)
L2

Following Carroll [51] for large L values, we obtain This equation is the master equation for the perihelion pre-
⎧   cession in the context of NAT for generic q and b1 parame-
⎪
⎪ L2
−
2a12 b1
, ters. Analytically solving this equation for generic q is not an
⎪
⎪ 1
⎪
⎪
m̃ L2 easy task and one may consider specific cases. For the case
⎨
r c±  (106) of q = 1, this equation reduces to
⎪
⎪ 
3m̃
,
⎪
⎪ d2x  1 2a12 b1
⎪
⎪ 2a 2 b m̃
⎩ 1 − L12 1 + x= + 3m̃x 2
+ + 2 x + 4a12 b1 x 3 .
dϕ 2 L2 3L 2 x 3 L
(113)
where we have considered 1 − 2a12 b1 > 0. Then, one can see
that the aether field changes the inner and outer circular orbits Then, in comparison to the Newtonian gravity possessing the
2
of massive particles in GR given by rc− = 3m̃ and rc+ = Lm̃ , d2 x
equation dϕ 2 +x = L 2 , one can realize the GR, cosmological
m̃
respectively. Accordingly, for b1 < 0, the outer and inner constant and aether field corrections, respectively. One can
circular orbits will be larger and smaller, respectively, relative show that this equation admits the following solution [52]
to GR and vice versa. For
 2 m̃
2a12 b1 x(ϕ) = [1 + e cos(ϕ)]
12m̃ = L 1 −
2 2
, L2
(107)  
L2 3m̃ 3 e2 1
+ 4 1 + eϕ sin(ϕ) + 1 − cos(2ϕ)
L 2 3
these orbits coincide at
L 4 3
6m̃ + 1 − eϕ sin(ϕ)
rc = . (108) 3m̃ 3 2
2a12 b1
1− L2 2m̃a12 b1 1
+ 1 + eϕ sin(ϕ)
One can see from (108) that the aether field changes the L4 2
smallest possible circular orbit for the massive particles as 
4m̃ 3 a12 b1 3
rc = 6m̃ in GR. + 6
1 + eϕ sin(ϕ)
L 2

3e2 1
9.2 Perihelion precession + 1 − cos(2ϕ) , (114)
2 3
The perihelion precession represents that non-circular orbits where the first term is the solution for the Newtonian gravity
are not perfect closed ellipses. To derive it, one should obtain with the eccentricity parameter e, and the other terms are the
the evolution of the radial coordinate r as a function of angu- corrections by GR, cosmological constant and aether field.
lar coordinate ϕ, i.e. r = r (ϕ). To do this, using (92) we Neglecting the higher order terms of the small eccentricity
write the Eq. (94) in the following form m̃ 2 2a12 b1
parameter e and using the conditions  1,  1,
    L2 L2
1 dr 2 L 2 one can rewrite the above equation as
+ V = E. (109)
2 dϕ r2 m̃
x(ϕ)  {1 + e cos [(1 − ζ ) ϕ]} , (115)
For more convenience, we introduce a new variable as x = 1 L2
r.
Then, the above equation takes the following form where
 
1 dx 2 E 3m̃ 2 L 6 a 2 b1
+ Ṽ(x) = 2 , (110) ζ = − + 12 . (116)
2 dϕ L L 2 2m̃ 4 L

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Then, during each orbit of the planet, there is a perihelion where the first term represents a straight line in polar coordi-
advance given by nates (x, ϕ), and r0 denotes the distance of closest approach
  of the light from the gravitational center. Then, the second
3m̃ 2 L 6 a12 b1 and third terms denote the GR and aether field contributions
ϕ = 2π ζ = 2π − + 2 . (117)
L2 2m̃ 4 L to the light deflection angle, respectively. The light deflection
angle, ψ, can be obtained using the condition x(π + ψ) = 0
One can rewrite this relation by converting L to the geomet-
in (123) as
ric quantities of each orbit. For this end, using the relation
governing ordinary ellipses as 4m̃ 3π a12 b1
ψ + , (124)
(1 − e2 )a r0 2 r02
r (ϕ) = , (118)
1 + e cos(ϕ) where we have used the approximation relations sin(π +
where a is the semi-major axis, one can obtain the angular ψ)  −ψ and cos(π + ψ)  −1 and dropped higher order
momentum as terms in m̃ and a12 b1 . Here, one realizes that depending on
the sign of the aether field parameter b1 , the light deflection
L 2 ≈ m̃(1 − e2 )a. (119) can be more or less than the GR value given by the above first
Then, by substituting (119) in (117), we obtain term. For b1 < 0, the aether field decreases the light deflec-
  tion angle relative to the Schwarzschild case in GR. This
6π m̃ (1 − e2 )4 a 4 a12 b1 is similar to the effect of charge in the Reissner–Nordström
ϕ = 1− + . (120)
(1 − e2 )a 6m̃ 2 3m̃ 2 solution [52,53].

Here, the correction term by the aether field is exactly same

as the one we previously obtained in (44) by using the post- 10 Conclusion
Newtonian approximation. It is seen that for  > 0 and
b1 < 0, we always have less perihelion precession relative In this work, we investigated the properties of the black hole
to GR. However, for  > 0 and b1 > 0, depending on solutions found in NAT [43] which is a vector-tensor theory
the value of contributions by aether field and cosmological of gravity with the vector field being null and defining the
constant, we may have more or less precession. Also, there aether field at each point of the spacetime. We first reviewed
is an interesting case for b1 > 0 in which the cosmological the Newtonian limit of the theory and showed that the Pois-
constant and aether fields cancel out the effect of each other, son equation is recovered at the linear order in the grav-
i.e for (1−e2 )4 a 4 = 2a12 b1 , leading to the same precession itation constant G of the theory which, depending on the
as in GR. form of the null vector, is related to the Newton’s constant
G N by a scaling factor. We also reviewed the exact spher-
9.3 Light deflection ically symmetric static solutions in NAT and extracted the
post-Newtonian parameters β, γ from these solutions when
To obtain the deflection angle of null geodesics, we set  = 0  = 0. In GR, these parameters are β = γ = 1 and in
in the potential V in (111). Then, the equation governing null NAT, for q = 0, we have the same values because the solu-
geodesics takes the form of tion is the usual Schwarzschild metric in this case. How-
d2x ever, for solutions with q > 0, taking a2 = 0 for simplicity,
+ x = 3m̃x 2 + a12 b1 (q + 3)x q+2 , (121) a2 b
dϕ 2 we found that β = 1 − m̃1 21 and γ = 1, meaning that, at
the post-Newtonian order, the aether does not contribute to
which shows that similar to the closed null geodesics in
the light deflection expression, which is determined only by
Sect. 9.1, the cosmological constant does not contribute to the
γ , while it contributes to the perihelion advance expression,
light deflection angle. However, the aether field contributes.
which is determined by both β and γ . Since the perihelion
Considering the case of q = 1, this equation reduces to a2 b
advance differs from the GR value by the term ( 31m̃ 21 ) where
d2x
+ x = 3m̃x 2 + 4a12 b1 x 3 , (122) b1 ≡ 18 [c3 − 3c2 + c23 q] [see Eq. (44)], the effect of the null
dϕ 2 aether is such that the GR value for the perihelion advance of
which has the following solution [52] planets is increased (for b1 > 0) or decreased (for b1 < 0).
That is to say, solar system observations can be used to put
1 m̃
x(ϕ) = sin(ϕ) + 2 [1 − cos(ϕ)]2 some constraints on the parameters of the theory.
r0 r0 We also studied the exact static black hole solutions in
a12 b1 1 NAT. We observed that, depending on the parameter q,
+ −3ϕ cos(ϕ) + sin(3ϕ) , (123)
2r03 4 there is a large class of black hole solutions in the theory

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and showed, by calculating the curvature scalars Ricci and contributions of the cosmological constant and the null aether
Kretschmann, that all the these solutions are singular only at field and showed that, when  = 0, the aether field contri-
r = 0. These black holes possess in general multiple event bution is the exactly the same as the one obtained in the post-
horizons and the locations of these horizons are dependent Newtonian order. Finally, we investigated the issue of light
on the parameters (q, , a1 , a2 , b1 , b2 , m̃, m) and the rela- deflection angle. We showed that the cosmological constant
tions between them. There are also extreme cases in which does not contribute to the light deflection angle. However,
some or all of the event horizons coincide. To determine the the aether field contributes in which, depending on the sign
mass parameters of these solutions, we calculated the ADM of the b1 parameter, the light deflection can be more or less
mass of the asymptotically flat black holes and showed that, than in GR. Indeed, for b1 < 0, the aether field decreases the
just like the mass parameter m in the case q = 0, the mass light deflection angle relative to the Schwarzschild solution
parameter in the case q > 0 reads m̃ = G M AD M , where G in GR.
is the gravitational constant appearing in the theory. GR has been shown to pass all the experimental tests per-
In the thermodynamics discussion of the NAT black holes, formed so far within the solar system with great precisions
we carried out a generic analysis in which the cosmological [50]. The corrections that we obtained in this work due to the
constant is nonzero. First, defining the NAT “charge” appro- null aether field to the GR predictions are expected to be very
priately, we showed that the horizon condition h(r0 ) = 0 small and may fall into the error bands of the solar system
and the scalar aether field φ(r0 ) at the horizon become simi- experiments. However, one can use these experiments to put
lar to the ones of the Reissner–Nordström–(A)dS black hole observational bounds on the parameters of NAT.
in GR, independently of the values of the parameter q. Then NAT is a new modified theory of gravity recently intro-
we obtained the first law of thermodynamics in which the duced [43]. So far, we have investigated this theory from var-
contribution of the aether field appears as Vφ δ Q, where ious respects: Newtonian limit, spherically symmetric solu-
Vφ = −2b1 φ(r0 ) with φ(r0 ) = Gr0Q and Q is the NAT charge. tions, black holes, thermodynamics, circular geodesics, flat
Therefore, for consistency, it turns out that b1 = −1/2 to cosmological solutions, exact plane waves, etc. But there are
recover the standard form of the first law. some open problems regarding, for example, the stability
Lastly, we studied both the null and timelike geodesics of the theory, PPN parameters, linearized waves, rotating
in the NAT black hole geometries. We explicitly derived the black holes, generic cosmological solutions, and inflationary
general expression for the effective potential governing the cosmologies. Therefore, to gain more understanding on the
motion of the particles in the gravitational field including the internal structure and dynamics of NAT, one needs to further
correction terms due to the cosmological constant and the investigate and pose analytical solutions to the theory.
aether field. As is shown in Fig. 1, depending on the values
of (q, L , a12 b1 ), it turns out that the deviation of the potential Acknowledgements This work is partially supported by the Scientific
and Technological Research Council of Turkey (TUBITAK).
from the GR value is more in the case of massive particles
than in the case of massless particles. In addition, by increas- Data Availability Statement This manuscript has no associated data
ing q, the potential tends to the GR one, while, by increasing or the data will not be deposited. [Authors’ comment: This is a purely
theoretical work, so we have not used any real data.]
L, it deviates more from the GR value for both the massive
and massless particles. We also obtained the general equa- Open Access This article is distributed under the terms of the Creative
tion governing the location of the circular geodesics for both Commons Attribution 4.0 International License (http://creativecomm
massive and massless particles to which there is no contri- ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
bution from the cosmological constant for the null geodesics
to the original author(s) and the source, provide a link to the Creative
as in GR. For specifically q = 1, we showed that, in contrast Commons license, and indicate if changes were made.
to GR possessing only one null circular orbit, there are two Funded by SCOAP3 .
circular orbits in the presence of the aether field for b1 < 0,
and of them, the outer one has a smaller radius than that of
the one in GR. For b1 > 0, on the other hand, there is always
only one circular orbit the radius of which is greater than References
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