Strong and weak gravitational field in $R+μ^4/R$ gravity

0.17 MB·English

by  Kh. Saaidi

AstrophysicsandSpaceScience
DOI10.1007/s•••••-•••-••••-•
Strong and weak gravitational field in R + µ4/R gravity
aKh. Saaidi1 • aA. Vajdi • aS. W. Rabiei •
bA. Aghamohammadi • aH. Sheikhahmadi
2
1
0
2
(cid:13)c Springer-Verlag••••
n
a
Not to appear in Nonlearned J., 45. part of Hilbert-Einstein lagrangian (Capozziello 2002;
J
Carrollet al. 2004, 2005; Clifton and Barrow 2005;
3 Abstract We introduce a new approach for investi-
gating the weak field limit of vacuum field equations Nojiri and Odintsov 2003; Sawicki and Hu 2007b;
] Evans et al 2008; Aghmohammadi et al. 2009). One
c in f(R) gravity and we find the weak field limit of
q f(R) = R+µ4/R gravity. Furthermore, we study the of the initiative f(R) models supposed to explain the
r- stronggravityregimeinR+µ4/Rmodeloff(R)grav- positive acceleration of expanding universe has f(R)
g ity. We show the existence of strong gravitationalfield action as f(R) = R µ4/R (Carroll et al. 2004). Af-
−
2 [ iµn va0cu,utmhewfoeraksuficehldmliomdietl.anWdtehefinstdroonugtgirnavtihtaetiloimnaitl taeprpeparroepdostihnigs tmhoedfel(Rsu)ff=er sRev−eraµl4/pRrobmleomdse.l, Iitnwthaes
v fie→ldcanberegardedasaperturbedSchwarzschildmet- metric formalism, initially Dolgov and Kawasaki dis-
7 ric. covered the violent instability in the matter sector
5
(Dolgov and Kawasaki 2003). The analysis of this in-
1
4 Keywords Sphericallysymmetricsolution. f(R) grav- stabilitygeneralizedtoarbitraryf(R)models(Faraoni
. ity. General relativity 2006; Sawicki and Hu 2007a) and it was shown than
1
0 an f(R) model is stable if d2f/dR2 > 0 and unsta-
0 ble if d2f/dR2 < 0. Thus we can deduce R µ4/R
1 Introductions −
1 suffer the Dolgov-Kawasaki instability but this insta-
:
v bility removes in the R+µ4/R model, where µ4 > 0.
Observations on supernova type Ia (Riess et al. 1998;
i Furthermore, one can see in the R µ4/R model the
X Perlmutter et al. 1999),cosmicmicrowavebackground −
cosmology is inconsistent with observation when non-
r (Spergel et al. 2003)andlargescalestructure(Tegmark et al.
a relativisticmatter ispresent. Infactthereis nomatter
2004), all indicate that the expansion of the universe
dominant era (Amendola et al. 2007a,b; Evans et al
is not proceeding as predicted by general relativity, if
the universe is homogeneous, spatially flat, and filled 2008). However, the recent study shows the standard
withrelativisticmatter. Aninterestingapproachtoex- epoch of matter domination can be obtained in the
plain the positive acceleration of the universe is f(R) R+µ4/R model (Evans et al 2008).
theories of gravity which generalize the geometrical It is obvious that a viable theory of gravity must
have the correct newtonian limit. Indeed a viable the-
oryoff(R)gravitymustpass solarsystemtests. After
aKh. Saaidi
the R µ4/R was suggested as the solution of cosmic-
aA.Vajdi −
accelerationpuzzle, ithasbeenarguedthatthis theory
aS.W.Rabiei
is inconsistent with solar system tests (Chiba 2003).
bA.Aghamohammadi
Thisclaimwasbasedonthefactthatmetricf(R)grav-
aH.Sheikhahmadi
ity is equivalent to ω = 0 Brans-Dicke theory, while
aDepartment of Physics, Faculty of Science, University of Kur-
the observational constraint is ω > 40000. But this
distan,Sanandaj,Iran.
is not quite the case and it is possible to investigate
[email protected].
thesphericalsymmetricsolutionsoff(R)gravitywith-
bDepartmentofPhysics,SanandajBranch,IslamicAzadUniver-
outinvokingthe equivalenceoff(R)gravityandscalar
sity,Sanandaj,Iran
tensor theory (Clifton and Barrow 2005; Cembranos
2
2006; Sawicki and Hu 2007b; Multamaki and Vilja The analysis of spherically symmetric solution can be
2006; Capozziello et al. 2008, 2009; Saaidi et al 2010; carried out using schwarzschildcoordinate
Capozziello et al. 2010). It has been shown that
some f(R) models accept the Schwarzschild-de Sitter ds2 = A(r)dt2+B(r)−1dr2+r2dΩ2. (5)
−
space-time as a spherical symmetric solutions of field
In the weak field limit approximation the metric de-
equation(Multamaki and Vilja 2006). HenceR µ4/R
− viates slightly from the Minkowski metric, so we can
model has a Schwarzschild-de Sitter solution with con-
write
stantcurvatureasR= 3µ4 wherethisisnotthecase
in R+µ4/R model.
p A(r)=1+a(r),
In this paper we study the R+µ4/R model of f(R)
B(r)=1+b(r),
gravity. We find the static spherically symmetric solu-
tion of vacuum field equation in both weak field limit a , b 1. (6)
| | | |≪
and strong gravity regime, moreover, the weak field
Whensolvingthe fieldequations(3,4)wewillkeeponly
analysis can be expanded on f(R) models of the form
terms linear in the perturbations a(r), b(r). Hence
f(R)=R+ǫh(R).
equations (3,4) leads to
′
a b 2dϕ(R)
2 Weak field limit + = ǫ
r r2 − r dr
′
b b
In this section we investigate the weak field solution of + = ǫ 2ϕ(R), (7)
r r2 − ∇
vacuum field equation in f(R) theories of gravity. We
are interested in model of the form f(R)=R+ǫh(R), and
with ǫ an adjustable small parameter. The motivation
R=3ǫ 2ϕ(R). (8)
for discussing these models is that the nonlinear cur- ∇
vature terms that grow at low curvature can lead to
where () indicates a derivation with respect to r.
thelatetimepositiveacceleration,butduringthestan- ′
dard matter dominated epoch, where the curvature is
2.1 f(R)=R1+ǫ
assumed to be relatively high, could have a negligible
effect.
Thismodelisconsideredin(Clifton and Barrow 2005).
The field equations for these models are
It is shown that this model has an exact spherically
g symmetric vacuum solution and regarding the general
G = ǫ G +g ✷ + µν
µν − µν µν −∇µ∇ν 2 line-element in Eq.(5), it may be written as
h h(R)
R ϕ(R)+kTµν, (1) A(r) = r2ǫ(1+2ǫ)/(1−ǫ)+c r−(1−4ǫ)/(1−ǫ),
× − ϕ(R)
(cid:18) (cid:19)(cid:21)
(1 ǫ)2
B(r) = −
where ϕ(R) = dh(R)/dR. Contracting the field equa- (1 2ǫ+4ǫ2)(1 2ǫ 2ǫ2)
tion we obtain − − −
1+c r−(1−2ǫ+4ǫ2)/(1−ǫ) ,
2h(R) ×
R=ǫ R +3✷ ϕ(R) kT. (2) (cid:16) (cid:17)
− ϕ(R) − wherecisaconstant. Inthelimitǫ 0,thesesolutions
(cid:20) (cid:21) →
become
Where for the vacuum T ,T = 0. If ǫ = 0 the above
µν
equations reduce to Einstein equation. Hence we sup- ds2 = 1+2ǫlnr+ c dt2+ 1+2ǫ+ c −1dr2
pose G and R in the r.h.s of Eqs.(1,2) can be ne- − r r
µν
glected for small values of ǫ. Furthermore if the con- +r(cid:16)2dΩ2. (cid:17) (cid:16) (cid:17) (9)
dition lim [h(R)/ϕ(R)] = 0 is satisfied we can neglect
R→0 because we seek the weakfield limit, inaboveequation
this term too. Neglecting these terms leads to the fol-
we assume c/r 1.
lowing equations ≪
Since we are interested in the limit ǫ 0, we may
→
G = ǫ[g ✷ ]ϕ(R), (3) expand f(R)=R1+ǫ around ǫ=0. Then we have
µν µν µ ν
− −∇ ∇
f(R) = R+ǫRlnR,
and
h(R) = RlnR,
R=ǫ3✷ϕ(R). (4)
ϕ(R) = 1+lnR.
Stronggravitationalfieldin... 3
It is clear that h(R) satisfies the condition Indeed by this approach the Post-Newtonian parame-
ter is found as γ = 1/2 while the measurements
PPN
lim [h(R)/ϕ(R)]=0. indicateγ =1+(2.1 2.3) 10−5 (Bertotti et al.
R→0 PPN ± ×
2003). Also we must note that using equivalence be-
Insertingϕ(R)inthetraceequation(8),theRicciscalar tween f(R) gravity and scalar tensor gravity one can
is obtained as find models which are consistent with the solar sys-
tem tests. This consistency can be made by giving the
6ǫ
R= . (10) scalarahighmassorexploitingtheso-calledchameleon
−r2
effect(Mota and Barrow 2004; Khoury and Weltman
Then we arrive at the solutions of Eq.(7) 2004;Capozziello and Tsujikawa 2008;Faulkner et al.
2007). However,whenoneisusingequivalencebetween
c c
a= +2ǫlnr, b= +2ǫ, (11) f(R) gravity and scalar tensor gravity, the continuity
r r
of scalar field or its equivalent, the Ricci scalar, at the
where c is a constant. We can see our solutions are in matter boundary is crucial condition which is not the
agreement with the exact solutions (9). Also one can case in Einstein gravity. But in this work we don’t
check neglecting R, Gµν and h(R)/ϕ(R) in Eq.(1, 2) is adoptthe continuityofRicciscalarforsolvingthe field
reasonable. equations. Instead, we suppose that when µ tends to
zero we arrive at the Einstein gravity. Thus we find a
2.2 f(R)=R+ǫlnR solutionfor1/Rmodelwhichisradicallydifferentfrom
other solutions in (Erickcek et al. 2006; Chiba et al.
Forthis model ϕ(R)=1/R. Solvingtraceequation(8) 2007).
and field equations (7) we obtain For this model we have
√6ǫ h(R) = 1/R,
R= , (12) ±
r ϕ(R) = 1/R2, (16)
∓
and
where h(R) fulfills the condition lim [h(R)/ϕ(R)]=0.
R→0
2M ǫ Solving Eqs.(7,8) we obtain
a=b= r. (13)
− r − 6
r 4 −2
R= 7αµ3r 3,
whereM isaconstant. Thereforethespacetimemetric µ4 ∓ 1 4 4
for empty space in this model is = µ3r3,
R2 49α2
ds2 = 1 2M ǫr dt2 a=−2Mr ± 43αµ34r43,
− − r − 6
(cid:18) r (cid:19) 2M 4 4
−1 b= αµ3r3. (17)
+ 1 2M ǫr dr2+r2dΩ2. (14) − r ±
− r − 6
(cid:18) r (cid:19) where α3 = 4/147 and M is a constant. Therefore the
We can see, the generalized Newtonian potential is metric for space time is
M 1 ǫ ds2 = 1 2M 3αµ34r43 dt2
Φ = r. (15)
G − − r ± 4
− r − 2 6 (cid:18) (cid:19)
r −1
This generalizedgravitationalpotential has two terms. + 1 2M αµ34r43 dr2+r2dΩ2. (18)
− r ±
The first term is the standard Newtonian potential (cid:18) (cid:19)
and the second term make a constant acceleration, We can use the isotropic form, by introducing
+ ǫ/24, which is independent of the mass of star. In a new radius, ρ, which defined as
(Saffari and Rahvar 2008) this metric is used to ad-
p
dress the Pioneer’s anomalous. 2M 3 4 4
r=ρ 1+ αµ3ρ3,
s ρ ± 4
2.3 f(R)=R µ4/R
± and therefore the equivalent metric can be read
Basedonequivalencebetweenf(R)gravityandBrans- as
Disicinkceotnhseiostreynwtiwthitωh=sol0a,ritsywsatsemargtueestdst(hCathitbhaist2h0e0o3r)y. ds2 = (1 2M 3αµ43ρ34)dt2
− − ρ ± 4
4
+ (1+ 2M 3αµ43ρ34)(dρ2+ρ2dΩ2). where in the above equation we used the empty space
ρ ± 4 solution Eq.(18). From the above equation we may
determine the parameter M. It is seen that in the
Fromabovemetriconecanseeintheasymptotic
µ 0 limit this constant reduces to the Schwarzschild
behavior,µ 0, γPPN 1 can be obtained. →
→ ≃ radius. Furthermore,accordingto cosmologicalstudies
From Eq.(17) it is obvious that in the limit µ 0,
→ µ2 = 10−52m−2 (Carroll et al. 2004) so, regarding a
µ4/R2 tends to zero, so there is not singularity in the
typical solar system, in Eq.(26) we may neglect terms
field equations. Also one can check neglecting R, G ,
and h(R)/ϕ(R) in Eq.(1, 2) is reasonable. µν at order O k2µT42 .
From eq(cid:16)uation(cid:17)(26) it is obvious that the physical
2.4 Interior solution in the f(R)=R+µ4/R model interpretation of the parameter M differ from that of
general relativity. Also from the above equation it is
Inthissectionwediscusstheinteriorgravitationalfield clear that in the 1/R gravity the external solution de-
in the spherically symmetric case of static mass distri- pends on the shape of matter distribution.
bution in the f(R) = R+µ4/R model where µ 0.
→
So we seek a sphericallysymmetric, static solutionand
we adopt the metric(5). In this model we may rewrite 3 Strong Gravity Regime in R+µ4/R Model
field equation (1) and trace equation (2) as
In this section we investigate the existence of strong
µ4
Gν = δνR+Gν +δν✷ ν +kTν, (19) gravitationalfield for f(R)=R+µ4/R model of f(R)
µ µ µ µ −∇µ∇ R2 µ
gravity. We can rewrite the field equation (1) as
(cid:0) (cid:1)
µ4
R = 3(R ✷) kT. (20) µ4 1 µ4
− R2 − Gνµ 1− R2 =−3δµνR−∇µ∇ν R2 , (27)
(cid:18) (cid:19) (cid:18) (cid:19)
From Eq.(2) it is obvious that as µ 0, R kT, so
assumingµ4 kT,inther.h.sofE→q.(19)w→e−mayne- where we have used the trace equation
≪−
glect those terms that containµ4/R2. Thus field equa-
R= 3[R+✷] µ4/R2 . (28)
tions (19) reduce to Einstein equations hence we may −
write (cid:0) (cid:1)
In the above equation we have neglected the energy-
momentumtensorofmatterbecauseweinvestigatethe
Gν kTν. (21)
µ ≃ µ stronggravitationalfieldarounda sphericallysymmet-
furthermore the conservation equation, Tµν = 0, ric distribution of matter. Adopting the general spher-
;ν
leads to ically symmetric metric (5), we can rewrite the trace
equation (28) and (rr),(tt) components of field equa-
′
A
p′ = (p+ρ), (22) tion (27) as
−2A
d2 2 d 1 BA′ d
wherep,ρarepressureanddensityofmatter. Toobtain B + + B′+ +R
− dr2 rdr 2 A dr
metric components (A,B), we use Eq.(22) and rr and (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
tt components of Eq.(21) µ4/R2 = R, (29a)
× 3
′
Gr = A B + B−1 kp, (23) (cid:0) ′ (cid:1)
r A r r2 ≃ BA + B−1 1 µ4/R2
rA r2 −
′ (cid:18) (cid:19)
Grr = Br + Br−2 1 ≃−kρc2. (24) + B d2 + B′ d (cid:0)µ4/R2 =(cid:1) R, (29b)
dr2 2 dr −3
(cid:18) (cid:19)
Solving Eq.(23) we may write (cid:0) (cid:1)
′
B B 1
+ − 1 µ4/R2
1 r µ4 r r2 −
B =1 kc2 ρ(x)x2dx+ . (25) (cid:18) (cid:19)
− r O k2T2 BA′ d (cid:0) R (cid:1)
Z0 (cid:18) (cid:19) + µ4/R2 = , (29c)
2A dr −3
From continuity of the metric component B(r), on
(cid:0) (cid:1)
the boundary surface r =r we find where() denotes derivation with respect to the (r).
0 ′
In the previous section we showed, (R+µ4/R) model
krc2 r0ρ(x)x2dx+αµ43r034 +O kµ2T42 = 2rM,(26)
0 Z0 (cid:18) (cid:19) 0
Stronggravitationalfieldin... 5
0
0
-2
-4
-5
a a b -6 b
-30 -30 -8 0.
1 1
0.02
-10 -10
0.01
-12
0. -0.01
0. 0.01 0.02 x -14 0. 0.01 0.02 x
-15
0 1 2 3 4 0 1 2 3
10-3x 10-3x
Fig. 1 aagainst x. Thered-solid lineshowsnumericalre- Fig. 2 b against x. Thered-solid lineshows numericalre-
sults of Eqs.(32). The green-dashed line representsapprox- sultsof Eqs.(32). Thegreen-dashed line represents approx-
imate solution for x ≪ 1 (Eq.(34a)) and the black-dotted imate solution for x ≪ 1 (Eq.(34b)) and the black-dotted
line is the approximate solution for x ≫ 1 (Eq.(36a)). A line is the approximate solution for x ≫ 1 (Eq.(36b)). A
close up on theorigin of main figure is presented . close up on theorigin of main figureis presented .
has the week field solution as x d2 + 1 d +γ b d2 + b′ d φ2
x+1dx2 2(x+1)2dx dx2 2 dx
ds2 = 1 2M + 3α(µr)43 dt2 (cid:18) 1 b a′ (cid:18) b a (cid:19)(cid:19)
− − r 4 = + + + −
(cid:20) (cid:21) 3φ (x+1)2 x+1 x+γa(x+1)
+ 1− 2Mr +α(µr)43 −1dr2+r2dΩ2, (30) 1(cid:18) +γa′ (1+γφ2) (32b)
(cid:20) (cid:21) × (x+1)2
(cid:18) (cid:19)(cid:19)
where α = (4/147)1/3 . It is obvious this metric re-
duces to Schwarzschild metric in the limit µ 0. 1 (x+1)(b a) 1 dφ2
→ 1+γ − +γa′
Now we seek the solution of field equation in the limit 2 x+γa(x+1) (x+1)2 dx
(cid:18) (cid:19)(cid:18) (cid:19)
(r 2M). Without loss of generality we can assume ′
2M→= 1. In order to solve equations (29) we use some = 1 + b + b 1+γφ2 , (32c)
3φ (x+1)2 x+1
definitions as (cid:18) (cid:19)
(cid:0) (cid:1)
where () denotes derivation with respect to the (x).
φ=γ/R, ′
Forthelimitµ 0,inthe aboveequationswesuppose
γ = µ4/3, that we can ne→glect terms containing γ . After solv-
−
1 ing equations we check this assumption. By neglecting
A=1 +γa(r),
− r these terms, equations 32 can be rewritten as
1
B =1 +γb(r). (31)
− r 1 = x d2 + 2x+1 d φ2 (33a)
3φ x+1dx2 (x+1)2dx
Because we seek the solution in the limit r 1, we (cid:18) (cid:19)
→
may define a new variable as x = r 1. Using these
− ′
definitions we can rewrite Eq.(29) as b a b a 1
+ + − =
(x+1)2 x+1 x(x+1)2 −3φ
′ ′
d 2b b +a (x+1)(b a)
γ b + + + − x d2 1 d
dx x+1 2 2(x+γa(x+1)) + + φ2 (33b)
(cid:18) x+1dx2 2(x+1)2dx
1 dφ2 1 1 (cid:18) (cid:19)
+γa′ = γφ2
(x+1)2 dx 3 − φ
(cid:18) (cid:19)(cid:19) (cid:18) (cid:19) 1 1 dφ2 1 b b′
x d2 + 2x+1 d φ2 (32a) 2(x+1)2 dx = 3φ + (x+1)2 + x+1. (33c)
− x+1dx2 (x+1)2dx
(cid:18) (cid:19)
6
2.0 conformal transformation and changing coordinate we
can see the strong field solution (35) is
1.5 ds2 =
Φ 1/3
2M 3 4 r
Φ 2.0 1 (2Mµ)4/3( 1)2/3 dt2
-20 1.0 1.5 − − r − 8(cid:18)3(cid:19) 2M − !
1
1.0 2M 1 4 1/3 r
0.5 + 1 + (2Mµ)4/3( 1)2/3 dr2
0.5 − r 8 3 2M − !
0.0 (cid:18) (cid:19)
0 1 2 3 4 5 6 x +r2dΩ2,
0.0
whichisvalidinthe rangeof(2Mµ)4 r/2M 1 1
0 2 4 6 8 10
≪ − ≪
and farther where r 2M, the metric of space time
10-3x canbe approximated≫by the metric (30). Furthermore,
wehavesolvedfieldequations(32)numericallyandpre-
Fig. 3 ϕ versus x. The red-solid line shows numerical re-
sentedtheresultsinfigures1,2,and3. Theplotsshow
sultsofEq.(32a). Thegreen-dashedlinerepresentsapprox-
imate solution for x ≪ 1 (Eq.(34c)) and the black-dotted that the numerical results are in agreement with the
line is the approximate solution for x ≫ 1 (Eq.(36c)). A analytical solutions (34,36) in their region of validity.
close up on theorigin of main figure is presented .
4 Discussion
In the limit x 1, solutions of Eq. (33) are
≪
We studied spherically symmetric solution of f(R)
3 4 1/3
a = x2/3, (34a) gravity. At first a new approach for investigating the
0
8 3
(cid:18) (cid:19) weakfieldlimitofvacuumfieldequationsinf(R)grav-
1 4 1/3 ity was introduced. Our results for the weak field
b = x2/3, (34b)
0 −8 3 limit of some studied f(R) models are in agreement
(cid:18) (cid:19)
with their known solutions. We solved the field equa-
3 1/3
φ = x1/3. (34c) tions for f(R) = R+µ4/R gravity at weak field limit
0
4
(cid:18) (cid:19) and obtained a solution which differs slightly from the
schwarzschild metric. Our results are against the ar-
Thus we obtain the metric for x 1 as
≪ guments that f(R) models are ill defined because of
1 3 4 1/3 the equivalence of f(R) gravity and Brans-Dicke grav-
ds2 = 1 µ4/3(r 1)2/3 dt2
− − r − 8 3 − ! ity with ωBD = 0 which leads to γPPN = 1/2. In fact
(cid:18) (cid:19)
our results are in agreement with the recent article of
+ 1 1 + 1 4 1/3µ4/3(r 1)2/3 dr2 Capozziello et al. (Capozziello et al. 2010), in which
− r 8(cid:18)3(cid:19) − ! they have studied Newtonian limit of the f(R) gravity
byconsideringthatfourthordergravitymodelsaredy-
+ r2dΩ2. (35)
namically equivalent to the O’Hanlon lagrangian and
Furthermore, for x 1, we can obtain the solutions of they have shown fourth order gravity models can not
equations (33) as ≫ be ruled out only on the base of analogy with Brans-
Dicke gravity with ω = 0. Moreover, regarding the
BD
a∞ = 3αx4/3, (36a) results for the weak field limit, we investigated the
−4 strong field regime for this model and showed that if
b∞ = αx4/3, (36b) (r 2M)/(2M)5 µ4, where r and 2M are radius
− − ≫
1 and Schwarzschild radius in the Schwarzschild coordi-
φ∞ = 7αx2/3, (36c) nate respectively, the gravitational field is a perturbed
Schwarzschildmetric even in strong gravity regime. fi-
whichareinagreementwithweekfieldlimit (30). Now
nally we solved the master equations numerically by
we can check the validity of our assumption. Consid-
setting the initial value conditions using the analytical
ering the solutions (36), shows that neglecting terms
answersof the strong gravityregion. In figures (1) and
containing γ in Eqs. (32) is valid only for x γ3
(2) we plotted the analytical and numerical solutions
≫| |
or x µ4. Hence the metric (34) is solution of field
of the components of the metric, a and b, versus ra-
≫
equationsintherangeofµ4 x 1. Byperforminga
dius in two weak and strong gravity region. It is seen
≪ ≪
Stronggravitationalfieldin... 7
that in the strong region (the close up part) the rele-
vant analytical answer and the numerical solution are
agree together while the analytical weak field approx-
imation solution deviates from the numerical solution.
The close up part of figures show that with increas-
ing the radius and going to the weak filed region, the
analytical solutions of strong filed approximation and
numerical answers get separated from each other, and
at last in the weak field region, i.e. x 1, the ana-
≫
lytical weak field answers coincide with the numerical
solution,whiletheanswersforthestronggravityregion
has agratedeviationfromthe numericalresults inthis
region.
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