Introduction of the generalized Lorenz gauge condition into the vector-tensor theory

0.14 MB·English

by  Changjun Gao

Introduction of the generalized Lorenz gauge condition into the vector tensor theory
∗
Changjun Gao
The National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China and
Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China
(Dated: January 31, 2012)
Weintroducethegeneralized Lorentzgauge condition in thetheory of quantumelectrodynamics
into the general vector-tensor theories of gravity. Then we explore the cosmic evolution and the
static,sphericallysymmetricsolutionofthefourdimensionalvectorfieldwiththegeneralizedLorenz
gauge. We find that, if the vector field is minimally coupled to the gravitation, it behaves as the
cosmologicalconstant. Ontheotherhand,ifitisnonminimallycoupledtothegravitation,thevector
field could behave as vast matters in the background of the spatially flat Friedmann-Robertson-
Walker Universe. But it may not be the case. The weak, strong and dominant energy conditions,
thestabilityanalysisofclassicalandquantumaspectswouldputconstraintsontheparametersand
2
so theequation of state of matters would be greatly constrained.
1
0
PACSnumbers: 98.80.Cq,98.65.Dx
2

n
a I. INTRODUCTION ics which has the Lagrangianas follows [12]
J

0 1 1
L = − F Fµν +λ ∇ Aµ− γλ
3 4 µν µ 2
Thevector-tensortheoriesofgravityarefirstproposed (cid:18) (cid:19)
] by Will, Nordtvedt and Hellings [1]. Then they are −jµAµ+ψ¯(iγµ∂µ−m)ψ . (1)
c
q used in cosmology to model inflaton [2], dark matter Here j = eψ¯∂ ψ and F is the field strength tensor of
- [3] and dark energy [4]. However, it is recently found µ µ µν
r Maxwell field. λ is the Lagrange multiplier field which
thattheLorentzinvariantvector-tensortheoriesareusu-
g has the dimension of inverse length, l−1. γ is a dimen-
[ ally plagued by instabilities [5] [6]. On the contrary,
sionless constant. The λ term is the generalized Lorentz
the Lorentz violation vector field models [7] are special
3 gauge. γ = 0, 1, 3 correspond to the Landau gauge
such that some of these models are free of instabilities
v [13],theFeynmangauge[14]andthe Yennie-Friedgauge
[8]. The well-known Einstein-aether theory [9] belongs
2 [15], respectively. We see the generalized Lorentz gauge
4 to these special models. The remarkable difference of
condition could also be understood as a constraint on
3 the Einstein-aether theory from the usual vector-tensor
the divergenceof four-vectorfieldAµ. Motivated by this
6 theory comes from the fact that the vector field is con-
point,weintroducethe generalizedLorentzgaugecondi-
. strained to have constant norm. This constraint elimi-
1
tion into the general vector-tensor theories.
1 nates a wrong-signkinetic term for the length-stretching
The paper is organized as follows. In section II, we
1 mode[10],hence givingthe theoryachancetobe viable.
explore the cosmic evolution and the static, spherically
1
: symmetric solution of the four-vector field which is min-
v The fixed-norm constraint in the Einstein-aether is imally coupled to the gravitation. We find the field be-
i
X achievedbythepresenceofaLagrangemultiplierfieldin haves as a cosmological constant. In section III, we in-
theLagrangiandensity. TheLagrangemultipliermethod vestigate the cosmic evolution of the four-vector which
r
a presents the constraints on the motion of some physi- is nonminimally coupled to the gravitation and find it
cal quantities in nature. So taking into account some could play the role of vast matters for some appropriate
physicalconstraintsonthemotionofphysicalquantities, parameters. Section IV gives the conclusion and discus-
one can always introduce the Lagrange multiplier into sion.
the corresponding Lagrange function. Actually, using We shall use the system of units in which 16πG=c=
this method, Mukhanov, Brandenberger and Sornborger ~ = 4πε = 1 and the metric signature (−, +, +, +)
0
have early proposed a nonsingular universe by limiting throughout the paper.
the spacetime curvature to some finite values [11].
II. COSMOLOGICAL CONSTANT FROM THE
Except for the fixed-norm constraint, are there any
FOUR-VECTOR FIELD WITH THE
other constraints on the four-vector (four dimensional
GENERALIZED LORENZ GAUGE
vector) field? The answer is yes. We remember that
there is a unified formulation of quantum electrodynam-
A. cosmology
ThemostgeneralformfortheLagrangiandensitywith
∗Electronicaddress: [email protected] two derivatives acting on the four-vectorfield Aµ can be
2
written as where
L = −c1∇µAν∇µAν −c2(∇µAµ)2−c3∇µAν∇νAµ H = a˙ , (11)
a
1
+λ ∇ Aµ− γλ . (2)
µ 2 is the Hubble parameter. a(t) is the scale factor of the
(cid:18) (cid:19)
Universe. Dot denotes the derivative with respect to the
Hereci aredimensionlessconstants. Thestability analy- cosmic time t. If
sisofthetheoryandthephenomenologicalinvestigations
could put constraints on the sign and value of ci. c1+c3 =0, (12)
Following the Einstein-aether theory [16], If we define
theequationofmotion,Eq.(10),reducestoaverysimple
Jµ ≡ −c ∇µA −c δµ∇ Aα−c ∇ Aµ , case
ν 1 ν 2 ν α 3 ν
Kµν ≡ λδνµ , (3) φ¨+3Hφ˙ +3H˙φ=0. (13)
we can rewrite the Lagrangiandensity as follows Solving the equation, we obtain
L = Jµ ∇ Aν +Kµ ∇ Aν − 1γλ2 . (4) φ˙ +3Hφ=Λ, (14)
ν µ ν µ 2
where Λ is an integrationconstantwhich has the dimen-
Variation of the Lagrangian density with respect to Aµ sion of l−1. Thus under the condition that Eq. (12) is
leads to the equation of motion for Aµ
satisfied, the Lagrangian,Eq. (2) turns out to be
∇µJµν +∇µKµν + 12∇νλ=0. (5) L = −c21FµνFµν −c2(∇µAµ)2
1
On the other hand, variation of the Lagrangian density +λ ∇ Aµ− γλ , (15)
µ
withrespecttoλgivesthegeneralizedLorenzgaugecon- 2
(cid:18) (cid:19)
dition
where
∇ Aµ−γλ=0, (6)
µ F = ∇ A −∇ A , (16)
µν µ ν ν µ
fromwhichwecanderivetheLagrangemultiplierfieldλ.
is the field strength tensor.
Acting onboth sidesof Eq.(6)by ∇ andusing Eq.(5),
ν In this paper, we are interested in the case of c +
we find 1
c =0 for simplicity. Fromthe generalizedLorenz gauge
3
1 condition, Eq. (6), we obtain
γ∇ Jµ +γ∇ Kµ + ∇ (∇ Aµ)=0. (7)
µ ν µ ν 2 ν µ
φ˙+3Hφ=γλ. (17)
This equationdetermines the dynamics of Aµ subject to
the generalized Lorenz gauge condition. Taking account of Eq. (14), we find the Lagrange multi-
The energy-momentum tensor for the vector field is plier field is actually a constant,
found to be
Λ
λ= . (18)
T = 2c (∇ A ∇αA −∇ A ∇µA ) γ
µν 1 α µ ν µ α α
+2∇ J σA −Jσ A −J Aσ So we can solve for φ from Eq. (17) as follows
σ (µ ν) (µ ν) (µν)
+∇σ Kh (µσAν)−Kσ(µAν)−K(µν)Aσi φ= Λ a3dt. (19)
a3
+g Lh . i (8) Z
µν
Then the energy density and pressure of the vector
InthebackgroundofspatiallyflatFriedmann-Robertson- field derivedfromthe energy-momentumtensor take the
Walker(FRW) Universe,the nonvanishingcomponentof form
the vector is the timelike component. So the vector field
1
Aµ can be written as ρ = −T0 = −c + Λ2 ,
A 0 2 2γ
(cid:18) (cid:19)
Aµ =[φ(t), 0, 0, 0] . (9) 1
p = Ti =− −c + Λ2 . (20)
A i 2 2γ
Then the equation of motion, Eq. (7), becomes (cid:18) (cid:19)
This is exactly for the energy density and pressure of
1
− c +c +c − φ¨+3Hφ˙ +3H˙φ Einstein’s cosmological constant. To interpret the cur-
1 2 3
2γ
(cid:18) (cid:19)(cid:16) (cid:17) rent acceleration of the Universe, we expect Λ to be the
−3H2φ(c +c )=0, (10) order of the inverse of present-day Hubble length and
1 3
3
for U,
1
−c + >0. (21)
2 2γ 2ff′′ −4c fψ2U′f′ −4c ψf2U′ψ′ −2c ψ2f2U′′
2 2 2
Ifc2 =0,weconcludethatγ mustbepositive. TheFeyn- −4c2ψUfψ′f′ +4c2Uψ2f′2−2c2ψUf2ψ′′
man and Yennie-Fried gauge satisfy this requirement, −4c Ufψ2f′′ +ψf2λ′ =0, (28)
whiletheLandaugaugeleadstoavanishingcosmological 2
constant.
for ψ,
B. static and spherically symmetric solution −4c Uψff′′ −4c fψU′f′ −4c f2U′ψ′ −2c ψf2U′′
2 2 2 2
−4c Ufψ′f′ +4c Uψf′2−2c Uf2ψ′′
In order to show the four-vector field theory can pass 2 2 2
the solar system tests, in this subsection, let’s seek for +f2λ′ =0, (29)
the static and spherically symmetric solution of Einstein
equationssourcedbytheLagrangianEq.(15). Thestatic and forf,
and spherically symmetric metric can always be written
as −2c1fφ′2−4c2Ufψ2U′′ −12c2UψfU′ψ′ −4c2U2ψfψ′′
ds2 =−U(r)dt2+ U1(r)dr2+f(r)2dΩ2 . (22) −−12c6cf2Uψψ2U2U′2′+f′γ−fλ162c+2U42Uψ′ψf′′f+′ −4U8fc′2′U+2ψ2f2fU′′′′
2
+2Ufψλ′ −2c fU2ψ′2 , (30)
ComparingtosolvingtheEinsteinequations,wepreferto 2
startfromtheLagrangianEq.(15)forsimplicityincalcu-
respectively. Now we have five independent differential
lations. Because of the static and spherically symmetric
equationsandfiveunknownvariables,U, f, φ, ψ, λ. So
property of the spacetime, the vector field A takes the
µ
the system of equations is closed.
form
From Eq. (26), we obtain
A =[φ(r), ψ(r), 0, 0] , (23)
µ
1
φ=φ +φ dr , (31)
where φ and ψ correspond to the electric and magnetic 1 0 f2
Z
part of the electromagnetic potential. Then we have
where φ , φ are two integration constants. In order to
0 1
′
FµνFµν =−2φ′2 , ∇µAµ =(Uψ)′ +2ff Uψ , (24) fifoxrφla1r,gleeteunsoucognhsird.erWthee eaxsypmecpttototichcaovnedfit2ion∼, nra2m. eSlyo,
φ∼ φ +φ /r. Let the asymptotic value of φ= 0 when
1 0
where prime denotes the derivative with respect to r. r is large enough. Then we conclude that
TakingintoaccounttheRicciscalar,R,wehavethetotal
Lagrangianfrom Eq. (15)
φ =0. (32)
1
L = −U′′ −4U′f′ −4Uf′′ + 2 −2Uf′2 On the other hand, from Eq. (27), we obtain
f f f2 f2
′ 2
+c φ′2−c (Uψ)′ +2f Uψ 1 ′ f′
1 2 f λ= (Uψ) +2 Uψ . (33)
" # γ f
" #
′
′ f 1
+λ (Uψ) +2 Uψ− γλ . (25) KeepingEqs.(31),(32)and(33)inmind,weobtainfrom
f 2
" #
the difference of Eq.(28) and Eq. (29)
Using the Euler-Lagrangeequation, we obtain the equa-
tion of motion for φ, ′′
f =0. (34)
′′ ′ ′ So we have
fφ +2φ f =0, (26)
for λ,
f =f +f r . (35)
0 1
′
(Uψ)′ +2f Uψ−γλ=0, (27) f0, f1 are two integration constants. We can always
f rescale r such that
4
c
f0 =0, f1 =1. (36) L = − 1FµνFµν −c2(∇µAµ)2+b1RµνAµAν
2
Then we obtain from Eq. (29) 1
+b RA Aν +λ ∇ Aµ− γλ . (43)
2 µ µ
2
(cid:18) (cid:19)
ψ r ψ
ψ = 0 + 1 , (37) Here b1, b2 are two dimensionless constants and Rµν, R
U r2U
are the Ricci tensor and the Ricci scalar, respectively.
with ψ , ψ two integration constants. Finally, Eq. (30) In the first place, varying the Lagrangian density with
0 1
turns out to be respect to Aµ, we obtain the equation of motion for Aµ
2γr4U′′ +4γr3U′ +9ψ2r4−2c γφ2
0 1 0
−18c γψ2r4 =0, (38) c1∇νFνµ+c2∇µ(∇νAν)+b1RµνAν
2 0
1
+b RA − ∇ λ=0. (44)
from which we obtain 2 µ µ
2
Secondly,varyingtheLagrangiandensitywithrespectto
U c φ2 3 1
U =U − 1 + 1 0 − −c + ψ2r2 , (39) λ, we obtain the equation of motion for λ
0 r 2r2 2 2 2γ 0
(cid:18) (cid:19)
whereU0andU1aretwointegrationconstants. Compare ∇µAµ−γλ=0. (45)
it with the Reissner-Nordstrom-de Sitter solution
Finally, varying the Lagrangian density with respect to
2M Q2 1
U =1− + − Lr2 , (40) gµν, we obtain the energy momentum
r r2 6
we may put
T = 2∇ J σA −Jσ A −J Aσ
µν σ (µ ν) (µ ν) (µν)
h i
U0 =1, U1 =2M , c1 =2, φ0 =Q, +∇σ K(µσAν)−Kσ(µAν)−K(µν)Aσ
Λ 1
ψ = , L= −c + Λ2 . (41) +b 2h∇ ∇ A Aσ −∇ ∇ (AαAσi)g
0 3 2 2γ 1 σ (µ ν) α σ µν
(cid:18) (cid:19)
(cid:2) (cid:0) (cid:1)
Here L denotes the cosmologicalconstant. −∇2(A A )−4AσR A −2b [R A Aσ
µ ν σ(µ ν) 2 µν σ
Therefore, the static and spherically solution for the
Lagrangian density, Eq. (15), is exactly the Reissner- (cid:3)
Nordstrom-de Sitter solution. The field φ, ψ and the +RAµAν −∇µ∇ν(AσAσ)+∇2(AσAσ)gµν
Lagrange multiplier λ is found to be +2c F Fσ +g L . (46)
1 µσ ν µν (cid:3)
φ= Q , λ= Λ , ψ = Λr + ψ1 . (42) Here Jµν is understood with c1 + c3 = 0. Combining
r γ 3U r2U Eq. (44) and Eq. (45), we can express the equation of
motion for Aµ as follows
We recognize that φ is the electric potential sourced by
the change Q. The cosmological constant is closely re- c ∇ Fν +c ∇ (∇ Aν)+b R Aν
lated to the field strength ∂ (r2Uψ) of the magnetic po- 1 ν µ 2 µ ν 1 µν
r
1
tential ψ. Since the static and spherically symmetric so- +b RA − ∇ (∇ Aν)=0. (47)
2 µ µ ν
lution of the theory is just the Reissner-Nordstrom-de 2γ
Sitter solution,we concludethatitwouldnotviolatethe
solar system tests on gravity theory.
B. cosmic evolution
III. NONMINIMALLY COUPLED TO In the backgroundof spatially flat FRW Universe, the
GRAVITATION equations of motion, Eq. (47) takes the form
1 ·
A. equations of motion and energy momentum c − φ˙ +3Hφ −3(b +2b )H˙φ
2 1 2
tensor 2γ
(cid:18) (cid:19)(cid:16) (cid:17)
−3H2φ(b +4b )=0. (48)
1 2
When the four-vector Aµ is nonminimally coupled to
the gravitation, we have the Lagrangian density as fol- From the energy momentum tensor, we obtain the en-
lows: ergy density and pressure of the four-vector field Aµ
5
1
p = − −c + +2b +4b φ˙2
ρ = c 2φφ¨−φ˙2 +6(b +2b )Hφφ˙ A 2 2γ 1 2
A 2 1 2 (cid:18) (cid:19)
+(cid:16)6(c2−b1(cid:17)−2b2)H˙ −9(c2+2b2)H2 φ2 + 6c2+4b2− γ3 −2b1 Hφφ˙
(cid:18) (cid:19)
−φhλ˙ + 1γλ2 , i (49) − 4γ 2b1 + 5b2 +12b2+12b b
2 2c γ−1 γ γ 2 1 2
2 (cid:18)
+3b2−10c b −4c b H˙φ2
1 2 2 2 1
3γ 3 12b +12c
− H2φ2 − (cid:1)+ 2 2
4c γ−2 γ2 γ
p = 2(c −b −2b ) φφ¨+φ˙2 −c φ˙2 2 (cid:18)
A 2 1 2 2
(cid:16) (cid:17) +64b2−12c2+48b b +8b2−24c b .(54)
+(3c −2b −2b ) 4Hφφ˙ + 2H˙ +3H2 φ2 2 2 1 2 1 2 2
2 1 2
1 h (cid:16) (cid:17) i Then we find (cid:1)
−φλ˙ + γλ2 . (50)
2
1−(2c −4b −8b )γ
p − 2 1 2 ρ
A A
2c γ−1
UsingtheequationofmotionforAµ,Eq.(48),andthe 2
4
generalized Lorentz gauge condition, Eq. (45), we can = φ φH˙ +2Hφ˙ +3H2φ
eliminate φ¨and rewrite the energy density as following 1−2c2γ
(cid:16) (cid:17)
· 3b2+12b2−10c b +12b b −4c b γ
1 2 2 2 1 2 2 1
(cid:2)(cid:0) (cid:1)
+2b +5b ] . (55)
1 1 2
ρ = 6Hφ˙φ −c + +b +2b
A 2 1 2
2γ It is apparent if
(cid:18) (cid:19)
1 2 2
+9H2φ2 −c + + b + b b =b =0, (56)
2 1 2 1 2
2γ 3 3
(cid:18) (cid:19)
1 we have the equation of state for the four-vector Aµ
+ −c + φ˙2 . (51)
2
2γ p
(cid:18) (cid:19) ω = A =−1, (57)
A
ρ
A
We have verified that the energy density, Eq. (51), and which is consistent with the result of Eq. (20).
the pressure, Eq. (50), satisfy the energy conservation On the other hand, if b 6=0 (or b 6=0) and
1 2
equation
3b2+12b2−10c b +12b b −4c b γ
1 2 2 2 1 2 2 1
+2b +5b =0, (58)
(cid:0) 1 2 (cid:1)
dρA namely,
+3H(ρ +p )=0, (52)
A A
dt
2b +5b
γ =− 1 2 , (59)
whichisconsistentwiththeequationofmotion,Eq.(48) 3b2+12b2−10c b +12b b −4c b
1 2 2 2 1 2 2 1
and the energy momentum conservation equation
we have the equation of state
5b +14b
∇νTνµ =0. (53) ωA = 3(b1 +2b2)
1 2
14+5b1
In other words, the equation of motion Eq. (48), the en- = b2
ergy conservation equation Eq. (52) and the energy mo- 3 2+ bb12
mentumconservationequationEq.(53)giveactuallythe 1(cid:16)4+5r (cid:17)
same equation. Similarly, using the equation of motion = , (60)
3(2+r)
for Aµ, Eq.(48), and the generalizedLorentz gaugecon-
dition,Eq.(45),wecaneliminateφ¨andrewritethepres- where we define the ratio r of b and b as r = b /b .
1 2 1 2
sure as following In other words, if the value of γ is given by Eq. (59), we
6
IV. CONCLUSION AND DISCUSSION
20
In conclusion, motivated by the Lagrange multiplier
method in the Einstein-aether theory, we introduce the
10 generalized Lorentz gauge condition in the theory of
quantum electrodynamics into the general vector-tensor
theories of gravity. With the fix-norm constraint, it is
0 found that the Einstein-aether field contributes an en-
ergy density [16]
–10 ρ =−3(c +3c +c )m2H2 , (63)
A 1 2 3
where the constant m represents the norm of the four-
vector Aµ. In order that the energy density be positive,
oneshouldhavec +3c +c <0. However,asshownby
–20 1 2 3
–4 –2 0 2
r Lim,theinvestigationsofquantumaspectonthistheory
require c + 3c + c ≥ 0 [18]. Therefore, the energy
1 2 3
FIG. 1: The equation of state wA of the four-vector field Aµ density of the aether field is nonpositive which violets
with the ratio r of b1 and b2. It is not bounded both below the weak energy condition [17].
and up, namely,−∞<wA <+∞. However, with the generalized Lorentz gauge condi-
tion,wefindthatthevectorfiledcouldcontributeacon-
stant energy density. For the minimally coupled case, in
shall have a constant equation of state for vector field
particular, when c = −c , c = 0, we always have a
Aµ. 1 3 2
non-negativeenergydensityfortheLandaugauge,Feyn-
In Fig.1, we plot the equationof state ω ofthe four-
A man gauge and Yennie-Fried gauge. The corresponding
vector field Aµ with the ratio r of b and b . It is not
1 2 quantum aspects have been very well studied. On the
bounded both below and up, namely, −∞ < w < +∞.
other hand, the static and spherically symmetric solu-
This is different from the quintessence which has the
tion sourced by the vector field is exactly the Reissner-
equation of state −1≤w≤+1.
Nordstrom-de Sitter solution. This reveals that the the-
In particular, when
ory may not be conflict with the solar system tests on
gravity theory. For the non-minimally coupled case, the
5 7 vector may play the role of vast matters in the back-
b2 =−14b1 , γ = 2(7c −4b ) , (61) ground of spatially flat Universe. But it may be not the
2 1
case. The stability analysis of classical and quantum as-
we have the equation of state pects would surely constrain the space of parameters. It
is a very important issue. We plan to leave this analysis
for future publications.
ω =0, (62)
A
which indicates that the four-vector field Aµ behaves as
Acknowledgments
a dust matter. So the four-vector field could behave as
vast matters in the background of spatially flat FRW
Universe. However,this may be notthe case. The weak, We thank A. L. Maroto and J. B. Jimenez for help-
stronganddominantenergyconditions[17],thestability ful discussions. This work is supported by the National
analysisofclassicalandquantumaspectswouldputcon- Science Foundation of China under the Key Project
straints on the parameters and so the equation of state Grant No. 10533010, Grant No. 10575004, Grant No.
would be greatly constrained. 10973014,and the 973 Project (No. 2010CB833004).
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