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Testing Hofava-Lifshitz gravity using thin accretion disk properties
Tiberiu HarkcB and Zoltan Kovacs
Department of Physics and Center for Theoretical and Computational Physics,
The University of Hong Kong, Pok Fu Lam Road, Hong Kong
Francisco S. N. LobcQ
Centra de Fisica Teorica e Computacional, Faculdade de Ciencias da Universidade de Lisboa,
Avenida Professor Gama Pinto 2, P-1649-003 Lisboa, Portugal
(Dated: August 10, 2009)
Recently, a renormalizable gravity theory with higher spatial derivatives in four dimensions was
proposed by Hof ava. The theory reduces to Einstein gravity with a non- vanishing cosmological con-
stant in IR, but it has improved UV behaviors. The spherically symmetric black hole solutions for an
arbitrary cosmological constant, which represent the generalization of the standard Schwarzschild-
(A)dS solution, has also been obtained for the Hofava-Lifshitz theory. The exact asymptotically
flat Schwarzschild type solution of the gravitational field equations in Hofava gravity contains a
quadratic increasing term, as well as the square root of a fourth order polynomial in the radial
coordinate, and it depends on one arbitrary integration constant. The IR modified Hofava gravity
seems to be consistent with the current observational data, but in order to test its viability more
observational constraints are necessary. In the present paper we consider the possibility of observa-
tionally testing Hofava gravity by using the accretion disk properties around black holes. The energy
flux, temperature distribution, the emission spectrum as well as the energy conversion efficiency are
obtained, and compared to the standard general relativistic case. Particular signatures can appear
in the electromagnetic spectrum, thus leading to the possibility of directly testing Hofava gravity
models by using astrophysical observations of the emission spectra from accretion disks.
PACS numbers: 04.50.Kd, 04.70.Bw, 97.10.Gz
I. INTRODUCTION
Recently, Hofava proposed a renormalizable gravity
theory in four dimensions which reduces to Einstein grav-
ity with a non- vanishing cosmological constant in IR but
with improved UV behaviors p], The latter theory
admits a Lifshitz scale-invariance in time and space, ex-
hibiting a broken Lorentz symmetry at short scales, while
at large distances higher derivative terms do not con-
tribute, and the theory reduces to standard general rela-
tivity (GR) . Since then various properties and character-
istics of the Hofava gravities have been extensively ana-
lyzed, ranging from formal developments [||, cosmology
[j], dark energy [j| and dark matter Q, and spherically
symmetric solutions @, H, H, H3| • Although a generic vac-
uum of the theory is anti-de Sitter one, particular limits
of the theory allow for the Minkowski vacuum. In this
limit post-Newtonian coefficients coincide with those of
the pure GR. Thus, the deviations from the conventional
GR can be tested only beyond the post-Newtonian cor-
rections, that is for a system with strong gravity at as-
trophysical scales.
In this context, IR- modified Hofava gravity seems to
be consistent with the current observational data, but in
order to test its viability more observational constraints
are necessary. In Ref. [11], potentially observable prop-
*Electronic address: harko@hkucc.hku.hk
^Electronic address: flobo@cii.fis.ul.pt
erties of black holes in the Hofava-Lifshitz gravity with
Minkowski vacuum were considered, namely, the grav-
itational lensing and quasinormal modes. It was shown
that the bending angle is seemingly smaller in the consid-
ered Hofava-Lifshitz gravity than in GR, and the quasi-
normal modes of black holes are longer lived, and have
larger real oscillation frequency in the Hofava-Lifshitz
gravity than in GR. In Ref. [12], by adopting the strong
field limit approach, the properties of strong field gravita-
tional lensing in the deformed Horava-Lifshitz black hole
were considered, and the angular position and magnifica-
tion of the relativistic images were obtained. Compared
with the Reissner-Norstrom black hole, a significant dif-
ference in the parameters was found. Thus, it was argued
this may offer a way to distinguish a deformed Horava-
Lifshitz black hole from a Reissner-Norstrom black hole.
In Ref. [13], the behavior of the effective potential was an-
alyzed, and the timelike geodesic motion in the Hofava-
Lifshitz spacetime was also explored. In this paper, we
further explore the possibility of testing the viability of
Hofava-Lifshitz gravity using thin accretion disk proper-
ties.
Recent observations suggest that around almost all of
the active galactic nuclei (AGN's), or black hole candi-
dates, there exist gas clouds surrounding the central com-
pact object, together with an associated accretion disc,
on a variety of scales from a tenth of a parsec to a few
hundred parsecs [ijj]. These gas clouds are assumed to
form a geometrically and optically thick torus (or warped
disc) , which absorbs most of the ultraviolet radiation and
the soft X-rays. The gas exists in either the molecular
2
or the atomic phase. The most powerful evidence for
the existence of super massive black holes comes from
the VLBI imaging of molecular H2O masers in the ac-
tive galaxy NGC 4258 [15j. This imaging, produced by
Dopplcr shift measurements assuming Keplerian motion
of the masering source, has allowed a quite accurate es-
timation of the central mass, which has been found to
be a 3.6 x 10 7 M o super massive dark object, within 0.13
parsecs. Hence, important astrophysical information can
be obtained from the observation of the motion of the
gas streams in the gravitational field of compact objects.
The mass accretion around rotating black holes was
studied in general relativity for the first time in [l6| • By
using an equatorial approximation to the stationary and
axisymmetric space-time of rotating black holes, steady-
state thin disk models were constructed, extending the
theory of non-relativistic accretion [13]. In these mod-
els hydrodynamical equilibrium is maintained by efficient
cooling mechanisms via radiation transport, and the ac-
creting matter has a Keplerian rotation. The radiation
emitted by the disk surface was also studied under the
assumption that black body radiation would emerge from
the disk in thermodynamical equilibrium. The radiation
properties of the thin accretion disks were further ana-
lyzed in [l8| and in [l9| , where the effects of the photon
capture by the hole on the spin evolution were presented
as well. In these works the efficiency with which black
holes convert rest mass into outgoing radiation in the
accretion process was also computed.
More recently, the emissivity properties of the accre-
tion disks were investigated for exotic central objects,
such as wormholes [2(| El} , and non-rotating or rotating
quark, boson or fermion stars, brane-world black holes
or gravastars [H [H, H, H, [2I H3, H| . The radiation
power per unit area, the temperature of the disk and the
spectrum of the emitted radiation were given, and com-
pared with the case of a Schwarzschild black hole of an
equal mass. The physical properties of matter forming
a thin accretion disk in the static and spherically sym-
metric spacetime metric of vacuum f(R) modified gravity
models were also analyzed [2t| . Particular signatures can
appear in the electromagnetic spectrum, thus leading to
the possibility of directly testing modified gravity mod-
els by using astrophysical observations of the emission
spectra from accretion disks.
It is the purpose of the present paper to study the thin
accretion disk models applied for black holes in Hofava-
Lifshitz gravity models, and carry out an analysis of the
properties of the radiation emerging from the surface of
the disk. As compared to the standard general relativistic
case, significant differences appear in the energy flux and
electromagnetic spectrum for Hofava black holes, thus
leading to the possibility of directly testing the Hofava-
Lifshitz theory by using astrophysical observations of the
emission spectra from accretion disks.
The present paper is organized as follows. In Sec. [Til
we present the action and specific solutions of static and
spherically symmetric spacetimes. In Sec. IIII1 we review
the formalism and the physical properties of the thin disk
accretion onto compact objects. In Sec. IIV1 we analyze
the basic properties of matter forming a thin accretion
disk around vacuum black holes in Hofava gravity, and
compare the results with the Schwarzschild solution. We
discuss and conclude our results in Sec. [Vj
II. BLACK HOLES IN HORAVA GRAVITY
In this section, we briefly review the Hofava-Lifshitz
theory, where differential geometry of foliations repre-
sents the proper mathematical setting for the class of
gravity theories studied by Hofava [2j. As foliations can
be equipped with a Riemannian structure, the dynamical
variables in Hofava-Lifshitz gravity is the lapse function,
N, the shift vector N l , and the 3-dimensional spatial
metric, g^. Thus, it is useful to use the ADM formalism,
where the four-dimensional metric is parameterized by
the following
ds 2 = -N' 2 c 2 dt 2 + gij (dx i + N' 1 dt) (dx j + N j dt) . (1)
In this context, the Einstein- Hilbert action is given by
S = Jd 4 x^N (KijK* -K 2 + R.^ - 2 A
(2)
where G is Newton's constant, R^ is the three-
dimensional curvature scalar for g^. The extrinsic cur-
vature, Kij, is defined as
(3)
where the dot denotes a derivative with respect to t, and
V,; is the covariant derivative with respect to the spatial
metric g^ .
The IR-modified Hofava action is given by
S =
J dtd 3 x y/gN
-2 ( K v Ki
+^R^R^ k
2 2
K t 1 !><••' i>(3Jtj
K 2 fi 2 /4A-1
8(3A- 1) V 4
-R^
(R^) 2 -A W R^+3A 2 W ^
2 2
fir // w
8(3A- 1)
(4)
where k, A, v, fi, lu and A\y are constant parameters. C l °
is the Cotton tensor, defined as
Akl \
(5)
Note that the last term in Eq. represents a 'soft' vio-
lation of the 'detailed balance' condition, which modifies
3
the IR behavior. This IR modification term, fi 4 R( 3 ', gen-
eralizes the original Hof ava model (we have used the no-
tation of Ref. [3]). Note that now these solutions with an
arbitrary cosmological constant represent the analogs of
the standard Schwarzschild-(A)dS solutions, which were
absent in the original Hof ava model [|| .
The fundamental constants of the speed of light c,
Newton's constant G, and the cosmological constant A
are defined as
K 2 fl 2 \X W \
8(3A-1) 2
G =
16tt(3A- 1)
A= -A w c 2 .
(6)
Throughout this work, we consider the static and
spherically symmetric metric given by
ds 2
-N 2 (r)dt 2 +
dr 2
7M
r 2 (de 2 +sm 2 0d(/) 2 ), (7)
where N(r) and f(r) are arbitrary functions of the radial
coordinate, r.
Imposing the specific case of A = 1, which reduces to
the Einstein-Hilbert action in the IR limit, one obtains
the following solution of the vacuum field equations in
Hof ava gravity,
N 2 = f = \ + {uj-A w )r 2 - y/r[v(u - 2A w )r 3 + % (8)
where (3 is an integration constant [![.
By considering (3 — —a 2 /A\y and oj = the solution
given by Eq. ([8]) reduces to the Lu, Mei and Pope (LMP)
solution given by
/ = 1 - A w r 2
(9)
Alternatively, considering now (3 — AujM and A^ = 0,
one obtains the Kehagias and Sfetsos's (KS) asymptoti-
cally flat solution [lOj], given by
/ = 1 + ujr 2 - ^r(uj 2 r 3 + 4wM) .
(10)
which is the only asymptotically flat solution in the fam-
ily of solutions ([5]). We shall use the Kehagias-Sfetsos
solution for analyzing the accretion disk properties. Note
that there is an outer (event) horizon, and an inner
(Cauchy) horizon at
r± = M 1 ± a/1 - l/(2wM 2 )
(11)
To avoid a naked singularity at the origin, one also needs
to impose the condition
ojM 2 > -.
~ 2
(12)
Note that in the GR regime, i.e., loM 2 3> 1, the outer
horizon approaches the Schwarzschild horizon, r + ~ 2M,
and the inner horizon approaches the central singularity,
r_ ~ 0.
III. ELECTROMAGNETIC RADIATION
PROPERTIES OF THIN ACCRETION DISKS
To set the stage, we present the general formalism
of electromagnetic radiation properties of thin accretion
disks in a general static, spherically-symmetric space-
time.
A. Spacetime metric and geodesic equations
In this work we analyze the physical properties and
characteristics of particles moving in circular orbits
around general relativistic compact spheres in a static
and spherically symmetric geometry given by the follow-
ing metric
ds 2 = g tt dt 2 + g rr dr 2 + g ee d9 2 + d<\> 2
(13)
Here the metric components gtt, g r r, gee and g^ depend
only on the radial coordinate r. In a static and spher-
ically symmetric spacetime two constants of motion for
particles do exist, the specific energy E and of the spe-
cific angular momentum L, respectively. The geodesic
equations of motion in the equatorial plane (6 — n/2)
can be written in terms of these constants of motion as
9tti = —E , (14)
g^<\> = L, (15)
-gttg rr r 2 + V eff (r) = E 2 . (16)
where the effective potential term is defined as
V eff (r) = -g tt (l + — ) . (17)
For stable circular orbits in the equatorial plane
the following conditions must hold: V e ff(r) = and
V e ff t r {r) — 0, respectively. These conditions provide the
specific energy, the specific angular momentum and the
angular velocity Q of particles moving in circular orbits
for the case of static general relativistic compact spheres,
given by
E
L
gtt
\J-gtt - g^tt 2
g<t><t>&
\J-gu - g^tt 2 '
fi = =f =
dt
~9tt,r
(18)
(19)
(20)
The marginally stable orbit around the central object can
be determined from the condition V e //, rr (r) = 0. This
condition provides the following relation
E
L 2 gtt,rr
{gttg^),
0.
(21)
4
By inserting Eqs. (fl8 ]) -(|20" )) into Eq. (|21"]l. and solving
this equation for r, we obtain the marginally stable orbit
for the explicitly given metric coefficients gu, gt<j> and
g^cf,. For a Schwarzschild black hole we have gu = — (1 —
2M/r), g rr — —g^ 1 and g^ — r 2 , and the geodesic
equation (fT6|) for the radial coordinate r becomes
r 2 + V eff (r)=E 2
with the effective potential given by
Eqs. fI3] ) -(|2"0 ]) leads to the form
r(r - 2M)
£ =
L =
n =
s/r - 3M '
R 5 n
(22)
(23)
(24)
(25)
(26)
for the specific energy, the specific angular momentum,
and the angular velocity for the Schwarzschild metric.
Since for the KS solution, given by Eq. (jTUJ) , g u = —f(r),
g rr = —g^ 1 and g^ = r 2 , the effective potential in
Hof ava-Lifshitz theory can be written as
V eff (r) = [l + ur 2 -f M Ar)] 1 +
L
(27)
with /A/. w (r) = yjruj^r 3, + AM), whereas the specific
energy, the specific angular momentum and the angular
velocity are given by
E =
L
n
^l + r 2 (u-W)- f M ,J
rHl
y/\+1*(u>-n 2 )-fM,J
/ rf M ,u - M - ur 3
(28)
(29)
(30)
M.L
The effective potentials of the Schwarzschild black hole
and of the KS solution are compared for the same ge-
ometrical mass in Fig. [T] As previously shown in
V e f f (r) for the KS solution approaches the Schwarzschild
potential for increasing values of lu.
B. Properties of thin accretion disks
For a thin accretion disk we assume that its vertical
size is negligible, as compared to its horizontal exten-
sion, i.e, the disk height H, defined by the maximum
7.05 -
Schwarzschild BH
U)=0.5M' 2
m=1.0 M 2
co=5.0 NT
0.95
2 3 4 5 6 78 JO 20
r/M
50 100
FIG. 1: The effective potential V e //(r) of the orbit-
ing particles for the Kehagias-Sfetsos solution and for the
Schwarzschild black hole with the same total mass M for the
specific angular momentum L = AM. The parameter ui of the
Kehagias-Sfetsos solution is set to 0.5M -2 , 1M~ 2 and 5M -2 ,
respectively.
half thickness of the disk, is always much smaller than
the characteristic radius r of the disk, H <C r. The thin
disk is in hydrodynamical equilibrium, and the pressure
gradient and a vertical entropy gradient in the accret-
ing matter are negligible. The efficient cooling via the
radiation over the disk surface prevents the disk from cu-
mulating the heat generated by stresses and dynamical
friction. In turn, this equilibrium causes the disk to sta-
bilize its thin vertical size. The thin disk has an inner
edge at the marginally stable orbit of the compact ob-
ject potential, and the accreting plasma has a Keplerian
motion in higher orbits.
In steady state accretion disk models, the mass accre-
tion rate Mq is assumed to be a constant that does not
change with time. The physical quantities describing the
orbiting plasma are averaged over a characteristic time
scale, e.g. At, over the azimuthal angle A(f> = 2ir for a to-
tal period of the orbits, and over the height H [iH [ttI. [l8| .
In the standard accretion disk theory the integration of
the total divergence of the energy-momentum tensor of
the plasma forming the disk provides the disk structure
equations. The radiation flux F emitted by the surface of
the accretion disk can be derived from the conservation
equations for the mass, energy and angular momentum,
respectively, and it is expressed in terms of the specific
energy, angular momentum and of the angular velocity
of the particles orbiting in the disk as [H, [TH ,
F(r)
M
n
-g (e - my
(E - QL)L tr dr, (31)
where Mo is the mass accretion rate, measuring the rate
at which the rest mass of the particles flows inward
through the disk with respect to the coordinate time
t, and r ms is the marginally stable orbit obtained from
Eq. (HJ), respectively.
5
Another important characteristics of the mass accre-
tion process is the efficiency with which the central object
converts rest mass into outgoing radiation. This quan-
tity is defined as the ratio of the rate of the radiation of
energy of photons escaping from the disk surface to infin-
ity, and the rate at which mass-energy is transported to
the central compact gene ral relativistic object, both mea-
sured at infinity [161 1 1 81 ] . If all the emitted photons can
escape to infinity, the efficiency is given in terms of the
specific energy measured at the marginally stable orbit
lmn.n i
1-Er,
(32)
For Schwarzschild black holes the efficiency is about
6%, whether the photon capture by the black hole is con-
sidered, or not. Ignoring the capture of radiation by the
hole, e is found to be 42% for extremely rotating Kerr
black holes (a* = 1) , whereas with photon capture the
efficiency is 40% pj].
The accreting matter in the steady-state thin disk
model is supposed to be in thermodynamical equilib-
rium. Therefore the radiation emitted by the disk sur-
face can be considered as a perfect black body radiation,
where the energy flux is given by F(r) = crT 4 (r) (a is
the Stefan-Boltzmann constant), and the observed lumi-
nosity L (y) has a redshifted black body spectrum [24| :
L i v ) — 47rer / (v) = — - cos 7 / / f -.
nc Jr t Jo exp(hv e /T)-l
(33)
Here d is the distance to the source, I{v) is the thermal
energy flux radiated by the disk, 7 is the disk inclination
angle, and r, and rt indicate the position of the inner
and outer edge of the disk, respectively. We take fj =
r ms and ry — * 00, since we expect the flux over the disk
surface vanishes at r — ► oofor any kind of asymptotically
flat geometry. The emitted frequency is given by v e —
v(l + z), where the redshift factor can be written as
1 + z
1 + f2r sin (j) sin 7
\/-9tt - 20^0 - f2 2 #00
where we have neglected the light bending [3(1 HH .
(34)
IV. ELECTROMAGNETIC SIGNATURES OF
ACCRETION DISKS AROUND
KEHAGIAS-SFESTOS BLACK HOLES
As a first step in the study of the accretion disk prop-
erties, we obtain Eqs. ([2"5)) - ([3T))) for the specific energy E,
the specific angular momentum L and the angular veloc-
ity f2 of any particle orbiting around a KS black holes. By
inserting Eqs. l[28 ]) -([30" l) into the flux integral Eq. ([HI]), we
can derive the radial profile of the emitted photon energy
flux over the whole surface the disk in the KS potential.
Eq. (|3ip is derived by integrating the conservation laws
for the mass, energy and angular momentum, which are
invariant for Hofava gravity, since the extra terms in the
action Eq. do not give any contribution to the total
divergence of the stress energy tensor.
The profiles for the energy flux are presented, for dif-
ferent values of u>, in Fig. [5] For the sake of comparison
we also present the flux distribution over a disk rotating
around a Schwarzschild black hole.
Schwarzschild BH
(0=0.5 M~ 2
m=1.0 M.
03=5.0 NT 2
100
FIG. 2: The energy radiated by a disk around the Kehagias-
Sfetsos and Schwarzschild black holes with the same total
mass M. The parameter ui of the Kehagias-Sfetsos solution
is set to 0.5M -2 , 1M~ 2 and 5M -2 , respectively, and the flux
values are normalized by F max = 1-37 x W~ 5 Mo/M 2 , the
maximal flux value for the Schwarzschild black hole.
Similarly to the case of the effective potential, the de-
viation of F(r) for the KS geometry from the standard
Schwarzschild flux increases as ui tends to 0.5M -2 . The
left edge of the flux profiles, shifting from r/M = 6 to
lower radii, shows that the distance of the inner edge of
the accretion disk and the event horizon of the KS black
hole remains almost the same as for the Schwarzschild
geometry (see Table J]). For u — 0.5M -2 the degener-
ate event horizon of the KS black hole is at r — M, and
the marginally stable orbit approaches r/M = 5. The
maximal flux value also increases for smaller values of ui.
When ljM 2 reaches its lower limit at 0.5, the maximum
value of the flux is already a factor of 1.4 higher than the
maximum value F max — 1.37 x 10~ 5 A/o/M 2 correspond-
ing to the Schwarzschild solution. Similarly to the inner
edge of the disk, the flux maximum is shifted to lower
and lower radii by decreasing oj.
These features can also be observed in the temperature
profiles presented in Fig. [31 However, the differences in
the temperature amplitudes are not so big as they are in
the case of the flux distribution.
In Fig. [31 the spectral energy distribution, calculated
with the use of Eqs. ([3"3")l and ([3~4")l , respectively, shows
a more interesting difference between the disk spectra of
the KS black hole and of the Schwarzschild black hole,
respectively. The disk spectra are very similar for both
the KS and the Schwarzschild black holes in the region
with v < 10 16 Hz. The cut-off frequencies of the spectra
6
r/M
FIG. 3: The disk temperature for Kehagias-Sfetsos and
Schwarzschild black holes with the same total mass M. The
parameter u> of the Kehagias-Sfetsos solution is set to 0.5M -2 ,
1M~ 2 and 5M~ 2 , respectively.
are also of the order of « 10 16 Hz for all cases, but they
are somewhat higher for the KS black holes than for the
Schwarzschild case, which separates the two classes. For
the KS solution the spectral properties do not exhibit any
significant differences with the variation of w: the spectra
are essentially the same for any value of u>. Although
the amplitude and the cut-off frequency of the spectra
are maximal in the limit lo = 0.5M -2 , the differences in
these quantities are negligible even for u> = 1000M -2 .
v[Hz]
FIG. 4: Disk spectra for Kehagias-Sfetsos and Schwarzschild
black holes with the same total mass M. The parameter ui
of the Kehagias-Sfetsos solution is set to 0.5M -2 , 1M -2 and
5Af~ 2 , respectively. Here M = 1M© and M = 10~ 12 M©/yr.
Table Q] shows the conversion efficiency e of the accreted
mass into radiation for both KS and Schwarzschild black
holes. For a given configuration with a fixed value of oj, e
is somewhat higher in the accretion process driven by KS
black holes, as compared to the Schwarzschild geometry.
This means that KS black holes always convert more ef-
ficiently mass into radiation than a standard general rel-
ativistic, static black hole do. The most efficient mecha-
uj [M 2 ] r ms [M] e
0.5 5.2441 0.0630
1.0 5.6644 0.0597
5.0 5.9536 0.0576
6.0000 0.0572
TABLE I: The marginally stable orbit and the efficiency for
Kehagias-Sfetsos and Schwarzschild black hole geometries.
The last line corresponds to the standard general relativis-
tic Schwarzschild black hole.
nism is provided by the KS black holes for the minimal
value of w, where efficiency is 6.3%. For uiM 2 >> 1, the
values of e and r ms approach those of the Schwarzschild
black hole, as expected.
V. DISCUSSIONS AND FINAL REMARKS
In the present paper we have considered the basic phys-
ical properties of matter forming a thin accretion disc
in the vacuum spacetime metric of the Hofava-Lifshitz
gravity models. The physical parameters of the disc -
effective potential, flux and emission spectrum profiles
- have been explicitly obtained for several values of the
parameter u> characterizing the vacuum solution of the
generalized field equations. All the astrophysical quanti-
ties, related to the observable properties of the accretion
disc, can be obtained from the black hole metric. Due to
the differences in the space-time structure, the Hofava-
Lifshitz theory black holes present some very important
differences with respect to the disc properties, as com-
pared to the standard general relativistic Schwarzschild
case.
The determination of the accretion rate for an astro-
physical object can give a strong evidence for the exis-
tence of a surface of the object. A model in which Sgr A*,
the 3.7 x 10 6 A/q super massive black hole candidate at
the Galactic center, may be a compact object with a ther-
mally emitting surface was considered in [32]. For very
compact surfaces within the photon orbit, the thermal
assumption is likely to be a good approximation because
of the large number of rays that are strongly gravitation-
ally lensed back onto the surface. Given the very low
quiescent luminosity of Sgr A* in the near-infrared, the
existence of a hard surface, even in the limit in which
the radius approaches the horizon, places a severe con-
straint on the steady mass accretion rate onto the source,
M < l(T 12 Af© yr^ 1 . This limit is well below the min-
imum accretion rate needed to power the observed sub-
millimeter luminosity of Sgr A*, M > 10~ 10 Af Q yr^ 1 .
Thus, from the determination of the accretion rate it fol-
lows that Sgr A* does not have a surface, that is, it must
have an event horizon.
Therefore, the study of the accretion processes by com-
pact objects is a powerful indicator of their physical na-
7
ture. Since the energy flux, the temperature distribution
of the disk, the spectrum of the emitted black body ra-
diation, as well as the conversion efficiency show, in the
case of the Hof ava-Lifshitz theory vacuum solutions, sig-
nificant differences as compared to the general relativistic
case, the determination of these observational quantities
could discriminate, at least in principle, between stan-
dard general relativity and Hofava-Lifshitz theory, and
constrain the parameter of the model.
Acknowledgments
The work of T. H. was supported by the General Re-
search Fund grant number HKU 701 808P of the govern-
ment of the Hong Kong Special Administrative Region.
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