Observational Limits on Gauss-Bonnet and Randall-Sundrum Gravities

0.16 MB·English

by  S. O. Alexeyev

Observational Limits on Gauss-Bonnet and Randall-Sundrum Gravities
Stanislav O. Alexeyev1, Kristina A. Rannu1, Polina I. Dyadina2, Boris N. Latosh3,4, and Slava G. Turyshev5
1Sternberg Astronomical Institute, Lomonosov Moscow State University,
Universitetsky Prospekt, 13, Moscow 119991, Russia
2Physics Department, Lomonosov Moscow State University, Vorobievy Gory, 1/2, Moscow 119991, Russia
3Faculty of Natural and Engineering Science, Dubna International University,
Universitetsaya Str., 19, Dubna, Moscow Region, Russia
4Physics Department, Institute for Natural Sciences,
Ural Federal University, Kuibyshev Str., 48, Yekaterinburg, Russia and
5Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, CA 91109-0899, USA
(Dated: December 22, 2014)
5
1 WediscussthepossibilitiesofexperimentalsearchfornewphysicspredictedbytheGauss-Bonnet
0 andtheRandall-Sundrumtheoriesofgravity. Theeffectivefour-dimensionalspherically-symmetrical
2 solutions of these theories are analyzed. We consider these solutions in the weak-field limit and in
n the process of the primordial black holes evaporation. We show that the predictions of discussed
a models are the same as of General Relativity. So, current experiments are not applicable for such
J search therefore different methods of observation and higher accuracy are required.

7
PACSnumbers: 04.50.Kd,04.70.-s,04.80.Cc,98.80.Es
1

]
c I. INTRODUCTION ometry. All the matter and the three fundamental inter-
q actions are localized on this brane except gravity which
-
r A set of multidimensional gravity models beginning isallowedtopropagateintothebulkalongtheextranon-
g from the Kaluza-Klein one [1] result from the attempts compact dimension. Thus the Randall-Sundrum model
[
to construct an unified field theory. As we live in a contains the description of the four-dimensional space-

1 space-time with four noncompact dimensions any mul- time from the very beginning and therefore does not
v tidimensionaltheoryneeds anappropriateeffective four- need any special theory serving as its effective four-
7 dimensional limit consistent with the predictions of gen- dimensional limit. The Randall-Sundrum I (RSI) model
1 eral relativity (GR) and the results of observations and includes two branes with different properties helping to
2
4 experiments. solve the hierarchy problem [12]. Moving the second
0 String theory [2] along with the loop quantum grav- brane to the infinity led to the Randall-Sundrum II
. ity [3] is a promising candidate for a quantum theory of (RSII) model with one brane [13]. In this paper we deal
1
gravity nowadays. Lovelock gravity [4] appeared to be a with RSII only.
0
5 ghost-free four-dimensional low-energy effective limit of Black hole solution is a basic one for any theory of
1 the string theory [5, 6]: gravity. First ofall, it describes the compactobject that
:
v a very massive star at the end of its life cycle collapses
L=√ g R+α S2+α S3+... , (1)
i − 2 3 into. It also features the curvature of the space-time
X (cid:0) (cid:1) produced by the presence of matter specific for the con-
where Sn is Euler characteristic of the n-th order. The
r sidered gravity model. Any extended theory of gravity
a leading and the most studied among them is the sec-
should be consistent with the predictions of GR and the
ond order curvature correctiongiven by a Gauss-Bonnet
observationalresultsthereforetheexistenceofblackholes
term S2 = S = R Rijkl 4R Rij +R2. The ef-
GB ijkl − ij and their properties are important indicators of the the-
fective four-dimensional limit of the string theory also
ory’s viability.
includes the scalar field that is the projection of the
g component of the ten-dimensional string metric The Gauss-Bonnet solution has been studied explic-
10 10
to the four-dimensional manifold. The Gauss-Bonnet itly during the recent years [7, 9–11]. On the contrary
term coupled to the scalar (dilatonic) field [7–11] de- thereareseveraldifferentsolutionsforRandall-Sundrum
scribes the influence of the compactextra dimensions on model [14–18]. It was argued that static black holes
the four-dimensional space-time. Therefore the Gauss- cannot exist in RSII with a radius much greater than
Bonnet theory with the dilaton scalar field serves as the the AdS length ℓ [19–21] and that even very small RSII
effective four-dimensional limit of the string theory. static black holes do not exist [22, 23]. Therefore RSII
Unlike the string theory the Randall-Sundrum model solutionsforlargeblackholesthathavebeenfoundinde-
allows the only extra dimension to be large and even pendently by FiguerasandWiseman[24, 25]andAbdol-
infinite [12, 13]. This model considers four-dimensional rahimi,Catto¨en,PageandYaghoobpour-Tari[26]arean
branes with tension embedded into a five-dimensional important improvement of the Randall-Sundrum model
space-time (bulk) that is assumed to have an AdS ge- interesting for further consideration. In their work [26]
5
2
Abdolrahimi,Catto¨enetal. comparetheobtainedblack- values we jump to the usual ones)
hole solution to the one Figueras and Wiseman [25] and
show that these solutions agree closely. In this paper we 1
S = d4x√ g R+2∂ φ∂µφ+
use the ACPY solution [26] as it contains the necessary 16πZ − h− µ (4)
details. The Figueras-Wiseman solution [24] is consid-
+λe−2φS +... ,
ered separately. GB
i
The outline of this paper is as follows: in Sec-
where φ is the potential of the dilatonic field, S is the
tion “Weak field limit” we discuss the weak field and GB
Gauss-Bonnettermandλisthestringcouplingconstant.
slow motion approximation of the Gauss-Bonnet and
For this purpose we construct a post-Newtonian param-
Randall-Sundrum theories. Section “Thermodynamics
eterization of the static asymptotically flat spherically-
and PBHs” is devoted to the analysis of thermodynami-
symmetric Gauss-Bonnet solution
cal properties of these models and their influence on the
primordialblack holes mass spectra. In Section“Discus-
σ2
sion and Conclusions” we discuss the results obtained, ds2 =∆dt2 dr2 r2 dθ2+sin2θdϕ2 , (5)
offer conclusions, and outline the next steps. − ∆ − (cid:16) (cid:17)
2M
∆=1 + r−2 , σ =1+ r−2 ,
− r O O
(cid:0) (cid:1) (cid:0) (cid:1) (6)
II. WEAK FIELD LIMIT D
φ=φ∞+ + r−2 ,
r O
(cid:0) (cid:1)
As a weak-field limit we consider the dynamical con-
where(t,r,θ,ϕ)areusualsphericalcoordinatesandfunc-
ditions in the solar system i.e. post-Newtonian approxi-
tions∆andσ dependontheradialcoordinater only, M
mation. The metrictensorg canbe representedasthe
µν
is the Arnowitt-Deser-Misner (ADM) mass, D is dila-
perturbationh aroundMinkowskispace-time η [27]:
µν µν
tonic charge i. e. the effective charge of the scalar field
source and φ∞ is the asymptotic value of dilatonic po-
tential [7, 8]. As it was argued in [10] D 1/M.
gµν =ηµν +hµν. (2) ∝
We put metric (5) with the expansions (6) to the field
equationswrittendowninthemostcomputationallycon-
In this paper we consider only spherically-symmetric
venient form [29]:
solutions and therefore the static gravity field at the
distance r from its source. In the first post-Newtonian
G =8π Tm +Tφ +TGB , (7)
(PPN)order the correctionto the gravitationalfield h µν µν µν µν
µν
(cid:0) (cid:1)
can be expressed by the series with respect to the nega-
whereTm isamatterstress-energytensorwhileTφ and
tivepowersofthe radialcoordinater upto thefollowing µν µν
TGB describethescalarfieldandtheGauss-Bonnetterm
order terms: µν
presence:
h (r−3), h (r−4), h (r−2) (3)
00 0j ij
∼O ∼O ∼O 1 1
Tφ = ∂ φ ∂ φ g ∂ρφ ∂ φ ,
and use geometric units ~ = c = G = 1 with non- µν 8π(cid:18) µ ν − 2 µν ρ (cid:19)
dimensional masses expressed in units of Plank mass. 1
The PPN limit is well tested by experiments [27, 28]. TµGνB = 8π h(∇µ∇ν −gµν(cid:3))(e−2φR)
The better the experimental accuracy becomes [28] the
moreopportunitiestotestsmallgravitationaleffectspre- + 2 (cid:16)(cid:3)δµσδνσ+ gµν∇ρ∇σ−∇ρ∇(µδνσ)(cid:17)(e−2φ Rρσ)
dictedbycurrentlyviabletheoriesshouldappear. Weuse
2 ρ σ(e−2φ R ) .
the expansion(3)to comparethe magnitudes ofthe pre- µρνσ
− ∇ ∇ i
dicted effects in order to see if the specific effects of the
considered solutions can be tested. As the PPN requires Using the standard computational techniques [27] the
the weak-field limit approximation we apply our results leading order for the nontrivial correction to the Gauss-
to solar system where the post-Newtonian parameters Bonnet metric tensor is
(PPN-parameters)aremeasuredwithhighprecision[32].
DM
Our results are inapplicable to the strong field limit. δhGB =8 + (r−5). (8)
00 r4 O
Comparing this result with (3) we see that the correc-
A. Gauss-Bonnet gravity tion term (8) lies beyond the PPN order that should be
proportionalto 1/r2. Thustheparametersofthe Gauss-
We begin our consideration by exploring the weak Bonnetmodelcannotbeconstrainedbythe solarsystem
field limit of the Gauss-Bonnettheory (Here and further tests. Thisresultisconsistentwithconclusionsfrom[29]
during solving the equations we use the dimensionless where the cosmologicallimit of the discussed model was
Planckianunits,onlyatthestepofnumericalestimation studied.
3
B. Randall-Sundrum gravity where M is the mass of the massive central object. In
the considered case it equals the solar mass. It also is
TheblackholesolutionoftheRandall-Sundrummodel expressedin Plank units of mass and therefore is dimen-
was constructed by Figueras and Wiseman [24] using an sionless.
associated five-dimensional anti-de Sitter space (AdS ) TheconstraintonthePPNparameterβ obtainedfrom
5
and dS -CFT correspondence [30]. The Figueras- analysis of the lunar laser ranging data [31] is β 1
Wisema5n solut4ion describes a static black hole with ra- 1.1 10−4[32]. TheadmittedregionoftheAdS|len−gth| ≤is
dius up to 20ℓ and reproduces four-dimensional GR limi×tedbytheresultsoftheNewton’slawtestℓ<10−5m
on the bran∼e in the low curvature and the low energy [33]. Therefore the upper limit on the value of ǫ is:
limit. We intend to use the fact that Schwarchildmetric ǫ 5.7 10−47cm l . (13)
Pl
can be used not only as a black hole one but also as a ≤ × ≪
description of gravitational (stellar) system far from the Originally the parameter ǫ was assumed to be negli-
centralbody. For example, for Solar system (taking into gibly small and the vanishing value found in (13) im-
account all the limitations and corrections). plies that in fact ǫ = 0. Thus the Figueras-Wiseman
The five-dimensional metric can be written near the four-dimensional black hole solution is not only is self-
conformal boundary z =0 as consistent but well consistent with the solar system con-
straints as well. Therefore this solution is indistinguish-
l2 able form GR in the PPN limit after all.
ds2 = dz2+g˜ (z,x) dxµdxν , (9)
z2 µν The other recent Randall-Sundrum solution obtained
(cid:2) (cid:3) by Abdolrahimi, Catto¨en, Page and Yaghoobpour-
where z is a coordinate of the brane along the extra di- Tari (ACPY) [26] is asymptotically conformal to the
mension and g˜µν(z,x) is the metric on the brane deter- Schwarzschild metric and includes a negative five-
mined by the Fefferman-Graham expansion [30]. The dimensional cosmologicalconstant Λ :
5
corresponding effective four-dimensional field equations
v(r) F(r)
[24] are: ds2 = u(r)dt2+ dr2+ r2+ dΩ2,
− u(r) (cid:20) Λ (cid:21)
5
−
G =8πG Tbrane+ǫ2 16πG TCFT[g] + u(r)=1 2M/r,
µν 4 µν n 4h µν i (10) − ′
r 2M F(r)
+aµν[g]+logǫ bµν[g] +O(ǫ4logǫ), v(r)=1+ − ,
o (cid:18)r (3M/2)(cid:19)(cid:20) Λ5r(cid:21) (14)
− −
whereG4 istheusualfour-dimensionalgravitationalcon- 2M 2M 2
stant, Tbrane isthe stress-energytensorofthe matterlo- F(r)=1 1.1241 +1.956
µν − (cid:18) r (cid:19) (cid:18) r (cid:19) −
calizedonthebrane,tensors TCFT[g] ,a [g]andb [g]
h µν i µν µν 2M 3 2M 11
resultfromtheextradimensionanddependonthemetric
9.961 + ... +2.900 ,
tensor components and ǫ is a small perturbation param- − (cid:18) r (cid:19) (cid:18) r (cid:19)
eter indicating the deviation of the brane position from ′
where d/dr. The function F(r) describes the per-
the equilibrium state z =0. ≡
turbation caused by the bulk. The best fit for it was
The additional term in the post-Newtonian expansion
obtained in [26].
oftheFigueras-Wisemansolutioncalculatedinthispaper
The field equations induced onthe brane were derived
is
by Sasaki, Shiromizu and Maeda [34]:
δhF00W = 12271 ℓǫ22 Mr22. (11) Gµν =− Λ4gµν + M8π2 Tµν + M8π3 Sµν −Eµν, (15)
Pl4 Pl5
The obtained value (11) lies within the PPN limit (3) where Λ is usual four-dimensional cosmological con-
4
andpointsatapotentiallyobservableeffect. InRandall- stant, g is the metric on the brane, T is the stress-
µν µν
Sundrum model gravity is allowed propagate into the energy tensor of the matter localized on the brane, S
µν
bulk along the extra dimension therefore the effect de- is the localquadraticstress-energycorrection,E is the
µν
scribed by (11) most likely leads to the negative non- four-dimensionalprojectionofthe five-dimensionalWeyl
linearity in gravitational superposition. In other words tensor. M is usual four-dimensional Planck mass and
Pl4
theresultinggravitationalfieldproducedbytwoormore M is the fundamental five-dimensional Planck mass.
Pl5
massiveobjects canbe lessthanthe directvectorsumof The induced metric on the brane is flat, the bulk is
their contributions. The parameterized post-Newtonian an anti-de-Sitter Space-time as in the original Randall-
(PPN) parameter β is responsible for such an effect Sundrum scenario [13], then E = 0 [16]. Therefore
µν
[27, 28]. Therefore the result (11) should be expressed the correction term due to the contribution from ACPY
as follows: topology (14) that follows from (15) has the form
ǫ2 121 ℓ2M2 1
β =1 M2, (12) δhAP = + (r−5). (16)
− ℓ2 108 00 96 r4 O
4
Accordingto(3)the expansiontermofthe PPN-order for GRB progenitors and therefore the limit estimation
should be proportional to r−2. The correction (16) con- for black hole evaporation rate can be obtained in such
tainsthenextperturbationorderwhichliesbeyondPPN a way.
similarly to the Gauss-Bonnet case (8). Therefore the Differenttheoriesofgravitypredictdifferentblackhole
obtainedcontribution(16) cannotbe observedinthe so- evaporation rate and therefore different initial masses of
lar system experiments as well. This conclusion on the the PBHs that fully evaporate for the Universe lifetime.
Randall-Sundrum model predictions confirms the result In this paper we compare the evaporation rates for the
for the Figueras-Wisemanandcoincideswith the Gauss- Gauss-BonnetandtheRSIIblackholesolutions. Accord-
Bonnet case. ingtothe GRB dataandthe precisionoftheFermiLAT
telescope the closest distance d at which the telescope
willbeabletodetecttheevaporationofprimordialblack
III. THERMODYNAMICS AND PBHS holes is [36]
−0.5 0.7 0.8
It is conjectured that density fluctuations in the early Ω E T
d 0.04 pc, (20)
Universe could have created black holes with arbitrarily ≃ (cid:18)sr(cid:19) (cid:18)GeV(cid:19) (cid:18)TeV(cid:19)
small masses even to the Planck scale [35]. These black
where Ω is the angular resolution of the telescope, E –
holes are referred to as primordial black holes (PBHs)
energy range of the telescope, T – temperature of the
[36] and can be used to consider viable theories in cos-
blackhole. Spending thesamereverseprocedureandus-
mological conditions.
ing a telescope detected gamma-ray bursts lead to the
Hawkingevaporation[37,38]is one ofthe mostsignif-
observable difference of the PBH initial mass on its fi-
icant properties of a black hole and can be described by
nal evaporation stage can vary from the GR predictions
the mass-loss rate equation [39]:
within the following limits:
dM 1 k
B
= , (17) M
− dt 256 π3M2 investigated theory >105. (21)
M
GR
whereM isthemassofablackholeandk istheStefan-
B
Boltzmannconstant. Hawkingevaporationisaquantum Therefore we use this limit as the mass cutoff threshold
process forbidden in classicalphysic. An outgoing radia- in our calculations.
tionhasto crossapotentialbarrierofblackhole horizon Using the method by Shankaranarayanan,Padmanab-
[40] so a radiation surrounding a black hole is in a ther- han and Srinivasan [43] it is possible to rewrite the ex-
mal equilibrium and can be described as a black body pression for the Gauss-Bonnet black hole temperature
radiation. Therefore the black hole evaporation obeys and then use (7). In the astrophysical case the dilatonic
the following law: charge D 1/M [10]. Therefore the right part of (18)
≃
can be expanded in series as
dM
− dt =kBST4, (18) dM 1 kB + 1 kB + M−10 . (22)
− dt ≃ 256 π3M2 512 π3M6 O
where S is its surface area. We use this formula to esti- (cid:0) (cid:1)
mate the lifetime of the black holes in the Gauss-Bonnet Theinitialmassofthe PBHthatfully evaporatesduring
and the Randall-Sundrum models. the lifetime of the universe in this case is
According to (17) the black holes with stellar masses
M =8 1014g. (23)
evaporate very slowly and do not lose mass through this GB ×
processnoticeably. Onthe otherhand PBHswith initial
Thedifferencebetweentheobtainedvalueandthesimilar
masses smaller than
GR quantity (19) is smaller than the cutoff threshold
set by (21). Thus the specific features of the Gauss-
M 5.0 1014 g (19)
0 ≈ × Bonnetevaporationratearenegligibleatthecurrentlevel
have already evaporated and can contribute to the ex- of accuracy and the predictions of Gauss-Bonnetgravity
tragalactic background radiation [38]. PBHs with ini- for the Hawking evaporation are indistinguishable from
tialmassgreaterthanM (19)shouldbeevaporatingtill those of the GR.
0
now [41]. According to some models of black hole evap- One of the first and most studied black hole solu-
oration [7, 8, 41] the last stages of this process can be tions of Randall-Sundrum model was found by Dadhich,
accompaniedbyburstsofhigh-energyparticles[8]includ- Maartens et al. [14, 16]. They obtained an exact local-
ing gamma radiationwith energyof the MeV-TeV range izedblackholesolution,whichremarkablyhasthemath-
that occur at the distances about z 9.4 [42]. Such ematical form of the Reissner-N¨ordstrom solution, but
events should be rather rare and on th≤e other hand the without electric charge being present [14]:
set of more simple explanations for most part of gamma
2M q 1
ray bursts (GRB) exists. Nevertheless PBHs at the last g =g =1 + , (24)
stage of evaporation can serve as additional candidates − tt rr − r (cid:18)MP2l5(cid:19)r2
5
The Reissner-N¨ordstrom-type correction to the not contribute into the requiredorder of the spherically-
Schwarzschild potential in (24) can be thought of symmetric solution’s expansion. This result agrees with
as a dimensionless “tidal charge” parameter q, arising the previous conclusions by Sotiriou and Barausse [29]
from the projection onto the brane of free gravitational who considered the cosmological solution of the action
field effects in the bulk transmitted via the bulk Weyl (4) and showed that the influence of the Gauss-Bonnet
tensor [14]. The projectedWeyltensor, transmitting the termisnegligibleatsolarsystemscales. Combiningthese
tidal charge stresses from the bulk to the brane, is [14]: two results we can state that the leading term of Love-
lockexpansion(1)describingsecond-ordercurvaturecor-
E = q 1 (u u 2r r +h ), rection does not provide any visible deviation from GR
µν −(cid:18)M2 (cid:19)r4 µ ν − µ ν µν predictions in the weak-field limit therefore such theory
Pl5
of gravity fully agrees with GR.
whereh =g +u u projectsorthogonalto4-velocity
µν µν µ ν Thisconclusionisalsovalidforanymodelwithhigher-
field uµ and r is a unit radial vector.
µ order curvature corrections having a proper Newtonian
The mass loss rate obtained similarly to the Gauss-
limit. As the Gauss-Bonnet term is the leading curva-
Bonnet case equals to
ture correction of the Lovelock gravity its contribution
into the post-Newtonian expansion of the metric also is
dM 1 k
= B + M−6 . (25) the largest one. Taking into account the others Euler
− dt 216 π3M2 O
(cid:0) (cid:1) characteristics i.e. the next orders of curvature correc-
The leading term in (25) cannot produce the needed 5- tions isn’t able to change the picture as their influence
orderdifferencedefinedbythethresholdparameter(21). is even less and obviously lies far beyond the PPN limit.
The initial mass of the Dadhich-Rezania black hole that ThustheconclusionsfortheGauss-Bonnetmodelcanbe
evaporatescompletelyduringthelifetimeoftheUniverse generalized to the Lovelock gravity.
proves this fact: The theories with curvature power series are not the
only method of geometrical extending of GR. In the
MDR =5.3 1014g. (26) generic case the Lanrangian can contain an arbitrary
×
function of the Ricci scalar R. Such theories set up
As the obtained difference is much less than the cutoff
f(R)-gravity [44, 45] and Lovelock gravity is its partic-
threshold (21) the “tidal charge” influence is vanishing
ular case. Many f(R)-gravity models such as ln(R) or
and cannot have experimentally verifiable consequences.
1/R [44, 46] were originally introduced as attempts to
TheblackholeevaporationfortheACPYsolutiondis-
explain dark energy or dark matter. They don’t have a
cussed in the previous section also was considered in a
proper PPN limit [44] and are inapplicable to solar sys-
similar manner. The evaporation rate of this solution
tem scale. Therefore our conclusions for Gauss-Bonnet
has completely the same form as the original Hawking
theory inthe weak-fieldlimit areapplicable for Lovelock
formula (17) up to M−10 terms thus the value of the
gravity and f(R)-gravity of the Lovelock type.
initial mass equals to that given by the GR:
The thermodynamicalproperties of the Gauss-Bonnet
M =5.0 1014g. (27) black hole solution were considered in details earlier
AP
× [7, 8, 10] however only the black holes of Planck scales
The results for Figueras-Wiseman solution are the same were investigated. For the black holes with larger mass
because of the form of the solution (11). the influence of the Gauss-Bonnet term and the scalar
The obtained results (22, 23, 25-27) lead to a conclu- field becomes negligibly small therefore the evaporation
sionthattheprecisionofthecurrentlyexistingGRBdata is predictably the same as in the GR case.
isnotsufficienttodistinguishtheGR,the Gauss-Bonnet SinceRandallandSundrumproposedatheoryofgrav-
and the Randall-Sundrum gravity from each other via ity with noncompact extra dimension [12, 13] several
the PBH consideration. black hole solutions have been found [14, 17, 24–26].
Consideration of the post-Newtonian expansion of the
Figueras-Wiseman solution [24] reveals such possible ef-
IV. DISCUSSION AND CONCLUSIONS fect as a negative nonlinearity of gravitational superpo-
sition (12). It naturally results from the theory itself
In this paper we discussed the possibilities to test the because gravity is allowed to propagate to the extra di-
theories extending GR in different ways by the example mensioninRandall-Sundrummodel. Howeverthebreak-
of the Gauss-Bonnet and the Randall-Sundrum models ingofgravitationalsuperpositionturnsouttodependon
both in the weak field and the cosmological limits. For a negligibly small parameter thus the predictions of the
thispurposethepost-Newtonianexpansionandtheblack Figueras-WisemansolutionfullyagreeswithGRandthe
hole evaporationofthese theories’solutionswereconsid- presentobservations. Thiseffectmayinfluencethestrong
ered. field regime (close binary systems, black holes) as a con-
AstheGauss-Bonnettermcoupledwiththescalarfield sequence of curvature growth. So the next step could be
does not influence the post-Newtonian limit (8) there- thesearchofsuchfeaturesoftheRandall-Sundrummodel
forethe nontrivialscalarhairgeneratedbyit[7,11]does in the strong field limit. Fortunately this investigation
6
is admissible as the large stable black hole solutions for modelsshouldbelimitedviathesecondorthethirdpost-
RSII black holes have been found [24, 26]. Newtonian orders. The corresponding formalisms of the
The consideration of the other black hole solution by 2PN and 3PN really do exist [47, 48]. These formalisms
Abdolrahimi, Page et al. [26] shows that the terms de- considerthe gravitationalradiationand its subtle effects
scribing the bulk influence (16) greatly exceed the limits on pulsar timing and orbit parameters. However many
of the post-Newtonian approximation. As a result both calculations there are based on GR and do not suit for
large Randall-Sundrum black holes solutions cannot be comparingarbitraryextendedtheoriesofgravitylikethe
distinguished from the Schwarzschild metric at the solar PPN formalism [27] does.
system scales.
We have examined the evaporation rate for the There are also the other ways to test astrophysical
Randall-Sundrum black holes as well. The results for predictions of the extended theories of gravity such as
oneofthefirstsolutionsobtainedbyDadhich,Maartens, accretion onto massive objects and microlensing. After
PapadopoulosandRezania[14]andthelatestonebyAb- computingtheaccretionrateforsomesolutiontheresult
dolrahimi, Page et al. [26] are presented (22, 23, 25-27). can be compared with GR predictions and some other
ThedifferencebetweentheDadhich-Rezaniasolutionand extended gravity cases. The investigation of the data
theGRisnegligiblysmallandthePagesolutioncoincides of gravity lensing events also is a perspective method as
with GR completely. these data become more and more complete. Verifica-
Asiseasytosee,manyextendedgravitymodelscannot tion of the extended gravity models via studying binary
bedistinguishedfromGRandfromeachotherneitherat systems and particularly the pulsar data needs special
the solar system scales nor by the black holes thermo- methods and approaches. Their construction is the sub-
dynamic properties. Therefore the coincidence of these ject of further considerations.
extended theories with the GR serves a good argument Acknowledgments
in favor of their validity. However it does not mean that
no difference can be found by other verification meth-
ods. Except the weak field and the cosmological tests a The work was supported by Federal Agency on Sci-
strong field approximation is widely used. It has such a ence and Innovations of Russian Federation, state con-
verificationlaboratoryas close binary systems firstof all tract 02.740.11.0575. S.A. and B.L. also were supported
containing pulsars as one or even both its components. byindividualgrantsfromDmitryZiminFoundation“Dy-
Agreatamountofdatahasbeenobtainedfromtheseob- nasty”. AuthorswouldliketothankProf. S.Capozziello,
servationsanditobviouslyshouldbeusedfortestingthe Dr. M. Smolyakov and Dr. D. Levkov for useful discus-
extendedgravitymodelsthoughthis methodhasitsown sions on the subject of this work. This work was per-
shortcomings. Iftheordersofthepost-Newtoniancorrec- formed at the Jet Propulsion Laboratory, California In-
tionsofthe extendedgravitymodelsliebeyondthe PPN stituteofTechnology,underacontractwiththeNational
order it is natural to suggest that the parameters of the Aeronautics and Space Administration.
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