Implication of Spatial and Temporal Variations of the Fine-Structure Constant

0.53 MB·English

by  Sze-Shiang Feng

USTC-ICTS-14-15
Implication of Spatial and Temporal Variations of the
Fine-Structure Constant
Sze-Shiang Feng 1, Mu-Lin Yan 2
5
1 Interdisciplinary Center for Theoretical Study,
0
2 Department of Modern Physics,

n University of Science and Technology of China, Hefei, Anhui 230026, China
a
J

9

Abstract
]
c
q
-
r Temporal and spatial variation of fine-structure constant α e2/~c in cosmology has
g ≡
[ been reported in analysis of combination Keck and VLT data. This paper studies

this variation based on consideration of basic spacetime symmetry in physics. Both
1
v laboratoryα anddistantα arededucedfromrelativisticspectrumequationsofatoms
0 z
0
(e.g.,hydrogen atom) defined in inertial reference system. When Einstein’s Λ = 0, the
5
6
0 metric of local inertial reference systems inSMof cosmology is Beltrami metric instead
2
of Minkowski, and the basic spacetime symmetry has to be de Sitter (dS) group. The
0
. corresponding special relativity (SR) is dS-SR. A model based on dS-SR is suggested.
4
0 Comparingthepredictionsonα-varyingwiththedata, theparameters aredetermined.
5
The best-fit dipole mode in α’s spatial varying is reproduced by this dS-SR model. α-
1
: varyings in whole sky is also studied. The results are generally in agreement with the
v
i estimations of observations. The main conclusion is that the phenomenon of α-varying
X
cosmologically with dipole mode dominating is due to the de Sitter (or anti de Sitter)
r
a
spacetime symmetry with aMinkowski point inan extended specialrelativity called de
Sitter invariant special relativity (dS-SR) developed by Dirac-In¨onu¨-Wigner-Gu¨rsey-
Lee-Lu-Zou-Guo.
PACS numbers: 06.20.Jr, 95.30.Sf, 03.65.Pm, 98.62.Ra, 95.36.+x
Key words: Fine-structure constant varying; Spacetime symmetry in Special Relativity;
Dirac equation of Hydrogen atom; Friedmann-Robertson-Walker (FRW) Universe; Local
inertial coordinate systems.
1Email: [email protected]
2Email: [email protected]; Corresponding author.
1 Introduction
Temporal and spatial variation of fundamental constants is a possibility, or even a neces-
sity, in an expanding Universe (see, [1], and a review of [2]). A change in the fine structure
constant α e2/(~c) could be detected via shifts in the frequencies of atomic transitions in
≡
quasar absorption systems. Recent analysis of a combined sample of quasar absorption line
spectra obtainedusing UVES(theUltraviolet andVisual Echelle Spectrograph) onVLT(the
Very Large Telescope ) and HIRES (the High Resolution Echelle Spectrometer) on the Keck
Telescope have provided hints of a spatial variation of the fine structure constant (α), which
is well represented by an angular dipole model [3] [4]. That is, α could be smaller in one
direction in the sky yet larger in the opposite direction at the time of absorption. Prior to
that, measurements of possible time variations of the fine-structure constant was achieved by
the same method [5–9] in the Keck telescope. It has been shown that the time variation of α
exists. Thus, α is a “constant” varying with both red-shift z (or cosmologic time) and direc-
tion in the sky equatorial coordinates. Namely, α = α (Ω) where Ω indicates the direction
z
in the sky. Those direct measurements of possible space-time variations of the fine-structure
constant are of utmost importance for a complete understanding of fundamental physics.
A straightforward conjecture for this phenomenon is that the space-time function of α(x)
may be thought as a scalar field ϕ(x) or a function of ϕ(x) in the spacetime with some
suitable dynamics (see, e.g, [10] [11] [12]). The ϕ(x) is a matter field and fills the Universe
everywhere. Sometime one could call it dilaton-like scalar field. Along this way of thinking,
authors of reference [13] argued that the spatial variation of the fine structure constant α
may be attributable to the domain wall of ϕ(x) in the Universe.
In this present paper, we would like to present a matter-field-free scheme to answer the
challenging questions such as why the fine-structure “constant” α varies over space-time,
and why spatial variation of α is well represented by an angular dipole mode. The scheme
is still in the framework of standard cosmology and of the Special Relativity (SR) theory
except that SR’s spacetime symmetry will be extended. Concretely, we shall apply the de
Sitter invariant Dirac equation to the distant hydrogen atom to explain such variations of
α in cosmology. The calculations are based on the theory of the de Sitter invariant special
relativity (dS-SR) developed by Dirac-In¨onu¨-Wigner-Gu¨rsey-Lee-Lu-Zou-Guo [14–22]. To
study atom physics in dS-SR were firstly called for by P.A.M. Dirac in 1935 [14].
Toshow thefine-structure constant α isunvarying over space-time intheStandardModel
(SM) ofphysics including cosmology, weexamine therelativistic wave equationofanelectron
in hydrogen in SM of physics. First, we consider the laboratory atom. From the viewpoint
of cosmology, the energy level E of a free hydrogen atom in laboratory is determined by the
Dirac equation in a local inertial coordinates system located at the Earth in the Universe
described by Friedmann-Robertson-Walker (FRW) metric. The spacetime metric of the local
inertial system is Minkowski metric:
1 0 0 0
0 1 0 0
η =  − , (1.1)
µν
{ } 0 0 1 0
 − 
 0 0 0 1 
 − 
 
1
which is spacetime independent. E satisfies Dirac spectrum equation:
e2
Eψ = i~cα +m c2β ψ, (1.2)
e
− ·∇− r
(cid:18) (cid:19)
where α α1i+α2j+α3k, β are Diracmatrices, (∂/∂x1)i+(∂/∂x2)j+(∂/∂x3)k and
{ ≡ } ∇ ≡
r = (x1)2 +(x2)2 +(x3)2. This matrix-differential equation is integrable and the solution
of the eigenvalue E is (see, e.g., [23])
p
α2 −1/2
E W = m c2 1+ (1.3)
≡ n,κ e (n κ +s)2
(cid:18) −| | (cid:19)
e2
α , κ = (j +1/2) = 1, 2, 3
≡ ~c | | ···
s = √κ2 α2, n = 1, 2, 3 .
− ···
We keep in mind that the coefficients of operators iα and 1/r in Eq.(1.2) are ~c and
− ·∇ −
e2 respectively, and their ratio is the definition of α (see Eq(1.3)).
Next, we consider a distant atom of hydrogen located on the light-cone of FRW-Universe
(see Fig.1), i.e., the nucleus coordinate is Qµ(z) Q0, Q , and electron’s is Lµ(z)
≡ { } ≡
L0, L . Noting that the metric of the local inertial coordinate system at Qµ in FRW-
{ }
Universe is still η (1.1) because of the spacetime-independency of η and denoting Lµ(z)
µν µν
−
Qµ(z) xµ, the electron wave equation in the distant atom reads
′
≡
e2
Eψ = i~cα +m c2β ψ, (1.4)
′ e
− ·∇ − r
(cid:18) ′ (cid:19)
where (∂/∂x1)i+ (∂/∂x2)j+ (∂/∂x3)k and r = (x1)2 +(x2)2 +(x3)2. Then we
′ ′ ′ ′ ′ ′ ′ ′
∇ ≡
find out that
p
e2
α α = . (1.5)
′ ≡ z ~c
Comparing (1.5) with (1.3), we conclude that
α = α, (1.6)
z
which indicates that the fine-structure constant is unvarying indeed in SM of physics.
The argument above on α-unvarying in SM made by comparing Dirac equation for lab-
oratory hydrogen atom with that for a distant hydrogen atom is deeply related to aspects
of Special Relativity (SR), General Relativity (GR) and Cosmology. In other words, the α-
varying phenomena reported in [3–8] implies some new physics beyond SM. Further remarks
on this issue are follows:
1. The spectrum equations (1.2) (1.4) come from the following inhomogeneous-Lorentz
(or Poincar´e) invariant (i.e., ISO(3,1)) Dirac equation and Maxwell equation in local
inertial systems of FRW Universe:
m c
(iγµDL e )ψ = 0, (1.7)
µ − ~
Fµν = jµ = δµ04πeδ(3)(x), (1.8)
,ν −
whereDL = ∂ ie/(c~)η Aν,andtheelectromagneticpotentialAν φ = e/r, A .
µ ∂Lµ− µν ≡ { }
So, the operator structure of (1.2), (1.4) and a dimensionless combination of universal
constants α = e2/(~c) are rooted in the symmetry assumption of the theory.
2
2. The point that α is unvarying is deduced from the constancy of the adopted metric
of local inertial coordinate system in the FRW Universe (i,e., η =const.). So, the
µν
{ }
fact of the α-varying in real world reported in [3–8] indicates that the metric of local
inertial coordinate system in the real Universe may be spacetime-dependent.
3. Minkowski metric η = diag +, , , is the basic spacetime metric of Einstein’s
µν
{ − − −}
Special Relativity (E-SR) in SM. The most general transformation to preserve metric
η is Poincar´e group (or inhomogeneous Lorentz group ISO(1,3)). It is well known
µν
that the Poincar´e group is the limit of the de Sitter group with pseudo-sphere radius
R . Therefore, E-SR may possibly be extended to a SR theory with de Sitter
| | → ∞
space-time symmetry. Since P.A.M. Dirac’s work in 1935 [14] many discussions (e.g.,
E. In¨onu¨ and E. P. Wigner in 1968 [15]; F. Gu¨rsey and T.D. Lee in 1963 [16], etc )
pointed to such a possible extension of E-SR. In 1970’s, K.H.Look (Qi-Keng Lu) and
his collaboratorsZ.L.Zou, H.Y.Guo suggested the deSitter Invariant Special Relativity
(dS-SR)[17][18](seeAppendix A,andalso[19,20]andAppendixin[24]fortheEnglish
version). IthasbeenprovedthatLu-Zou-Guo’sdS-SRisasatisfying andself-consistent
special relativity theory. In 2005, one of us (MLY) and Xiao, Huang, Li suggested dS-
SR Quantum Mechanics (QM) [20].
4. Beltrami metric (see Appendix A)
B (x) = η /σ(x)+η xλη xρ/(R2σ(x)2), with σ(x) = 1 η xµxν/R2 > 0 (1.9)
µν µν µλ νρ µν
−
isthebasicmetricofdS-SRwithMinkowski pointcoordinatesMµ = 0(i.e., B (x)
µν x=M=0
|
= η ) [20]. Both η and B lead to the inertial motion law for free particles x¨ = 0,
µν µν µν
which is the precondition to define inertial reference systems required by special rel-
ativity theories. However, the B -preserving coordinate transformation group is de
µν
Sitter group SO(4,1) (or SO(3,2)) rather than E-SR’s inhomogeneous Lorentz group
ISO(1,3) [17–20], different from the case of η . In addition, η does not satisfy the
µν µν
Einstein equation with Λ (Einstein cosmology constant) in vacuum, but B (x) does
µν
satisfy it (see below). Generally, when Mµ = 0, the basic metric of dS-SR is modified
6
to be
η η (xλ Mλ)η (xρ Mρ)
B(M)(x) B (x M) = µν + µλ − νρ − , (1.10)
µν ≡ µν − σ(M)(x) R2σ(M)(x)2
where
η (xµ Mµ)(xν Mν)
σ(M)(x) σ(x M) = 1 µν − − . (1.11)
≡ − − R2
which will be called Modified Beltrami metric, or M-Beltrami metric. Based on
B(M)(x), the dS-invariant special relativity with Mµ = 0 can be built. The procedures
µν
6
and formulation are similar to ordinary dS-SR in [17–20], which is actually a slight
(M)
extension of usual dS-SR (see Appendix B). It is essential, however, that B (x) is
µν
spacetime dependent andhasmoreparameters R, Mµ , whichmayprovide apossible
{ }
clue to solve the puzzle of α-varying.
5. Our strategy in this present paper for solving this puzzle is to pursue the following
dS-SR Dirac equation for both electron in laboratory hydrogen and electron in distant
3
hydrogen in the FRW Universe (see Eq.(25) in [21]):
m c
(ie µγa L e )ψ = 0, (1.12)
a Dµ − ~
where L = ∂ iωabσ ie/(c~)B(M)Aν, e µ is the tetrad, ωab is spin-connection,
Dµ ∂Lµ − 4 µ ab− µν a µ
and the electromagnetic potential Aν φ , A . Unlike E-SR Dirac equation (1.7),
B
≡ { }
the spacetime symmetry of (1.12) is de Sitter invariant group SO(4,1) (or SO(3,2))
instead of former ISO(3,1). Specifically, the dS-SR Dirac spectrum equation can
be deduced from (1.12). It is essential that the result will be different form E-SR
equation (1.2). Following the method used in (1.2) (1.3), the coefficients of resulting
( iα )-type and ( 1/r)-type operator terms in the dS-SR Dirac spectrum equation
− ·∇ −
are of ~ (Ω)c and e (Ω)2 respectively. Then their ratio yields prediction of α (Ω)
z z z
≡
e (Ω)2/(~ (Ω)c). The adjustable parameters in this model are R and the position of
z z
Minkowski point Mµ. For simplicity, we take Mµ = M0, M1, 0, 0 . It turns out to
{ }
be a good a choice for solution to the puzzle of α-varying.
6. Different from Quantum Mechanics (QM) wave equation (1.7) deduced from η , the
µν
equation (1.12) is actually a time-dependent Hamiltonian problems in QM. This is
(M)
because B (x) is time-dependent.Therefore the corresponding Lagrangian L (see
µν dS
Eq.(A.150)) and hence Hamiltonian is time-dependent [20]. In this paper, the adia-
batic approach [26] [27] [28] will be used to deal with the time-dependent Hamilto-
nian problems in dS-SR QM. Generally, to a H(x,t), we may express it as H(x,t) =
H (x) + H (x,t). Suppose two eigenstates s and m of H (x) are not degenerate,
0 ′ 0
| i | i
i.e., ∆E ~(ω ω ) ~ω = 0. The validity of for adiabatic approximation relies
m s ms
≡ − ≡ 6
on the fact that the variation of the potential H (x,t) in the the Bohr time-period
′
(∆T(Bohr))H˙ = (2π/ω )H˙ is much less than ~ω , where H m H (x,t) s .
ms m′ s ms m′ s ms m′ s ≡ h | ′ | i
That makes the quantum transition from state s to state m almost impossible.
| i | i
Thus, the non-adiabatic effect corrections are small enough (or tiny) , and the adia-
batic approximations are legitimate . For the wave equation of dS-SR QM of atoms
discussed in this paper, we show that the perturbation Hamiltonian describes the time
evolutions of the system H (x,t) (c2t2/R2) (where t is the cosmic time). Since R is
′
∝
cosmologically large and R >> ct, the factor (c2t2/R2) will make the time-evolution
of the system so slow that the adiabatic approximation works. We shall provide a
calculations to confirm this point in the paper. By this approach, we solve the station-
ary dS-SR Dirac equation for one electron atom, and the spectra of the corresponding
Hamiltonian with time-parameter are obtained. Consequently, we find out that the
electron mass m , the electric charge e, the Planck constant ~ and the fine structure
e
constant α = e2/(~c) vary as cosmic time goes by. These are interesting consequences
since they indicate that the time-variations of fundamental physics constants are due
to solid known quantum evolutions of time-dependent quantum mechanics that has
been widely discussed for a long history (e.g., see [28] and the references within).
7. Finally, we argue that it is reasonable to assume that the Beltrami metric is the
appropriate metric for the spacetime of the local inertial system in real world. If we
express the total energy momentum tensor T as the sum of a possible vacuum term
µν
ρ g and a term TM arising from matter (including radiation), then the complete
− (v) µν µν
4
Einstein equation is [29] [30] [31]:
1
g +Λg = 8πGTM 8πGρ g , (1.13)
Rµν − 2 µνR µν − µν − (v) µν
where ρ is the dark energy density, so Λ = 8πGρ , and Λ is originally
(v) darkenergy (v)
introduced by Einstein in 1917, and serves as a universal constant in physics. We call
it the Einstein (or geometry) cosmological constant. The effective cosmologic constant
Λ = Λ+Λ 1.26 10 56 cm 2 is the observed value determined via effects
eff darkenergy − −
≃ ×
of accelerated expansion of the universe [32] and the recent WMAP data [33]. We have
no any a priori reason to assume the geometry cosmologic constant to be zero, so the
vacuum Einstein equation is:
1
g +Λg = 0, (1.14)
µν µν µν
R − 2 R
(M)
instead of G = 0, and hence the vacuum solution to (1.14) is g = B (x) with
µν µν µν
R = 3/Λ instead of g = η . Therefore, we conclude that the metric of the
µν µν
| |
local inertial coordinate system in real world should be Beltrami metric rather than
p
Minkowski metric. Thus, the dS-SR Dirac equation (1.12) (instead of E-SR Dirac
equation (1.7)) is legitimate to characterize the spectra in the real world, and then the
α-varying over the real world space-time would occur naturally.
This paper provides an understanding of the α-varying in cosmology reported in [3] [4] by
means of extending the basic spacetime symmetry in the local inertial coordinate systems
in the standard cosmologic model. The contents of the paper are organized as follows. In
section 2, light-cone of Friedmann-Robertson-Walker universe is described, and relation of
cosmological time to redshift z is shown. The relation of t z is based on ΛCDM model and
−
the cosmology parameters (H , Ω , Ω 1 Ω ) in real world; In section 3, we describe
0 m0 Λ m0
≃ −
the local inertial coordinate system in light-cone of FRW Universe with Einstein cosmology
constant Λ. The metric of such local inertial systems is M-Beltrami metric that services as
the basic metric of de Sitter invariant special relativity (dS-SR); In section 4, we derive the
electric Coulomb law at light-cone of FRW Universe in terms of dS-SR Maxwell equations.
Asiswell known, Coulombforcedominatesthedynamics oftheatomicspectrums. Insection
5, we discuss the fine-structure constant variation along the best-fit dipole direction shown
in [3,4]. The α-varying ∆α/α (α (Ω) α )/α (where α is α’s value in laboratory)
0 z 0 0 0
≡ −
in this region is derived. Using the data along this best-fit dipole reported by [3,4], the
model’s parameters are determined. The theoretical predictions are consistent with the
observations. In section 6, we examine the α-varying in whole sky. The results are also in
agreement with the estimate from Keck- and VLT data. Finally, we briefly summarize and
discuss our results. In Appendix A we briefly recall the Betrami metric and the de Sitter
invariant special relativity. In Appendix B, a remark on the modified Beltrami metric used
in this paper is provided.
2 Light-Cone of Friedmann-Robertson-Walker Universe
The isotropic and homogeneous cosmology solution of Einstein equation in GR (General
Relativity) is Friedmann-Robertson-Walker (FRW) metric. In this section we discuss the
5
Light-Cone of FRW Universe because all visible quasars in sky must be located on it (see
Figure 1).
Figure 1: Sketch of the light cone of the Friedmann-Robertson-Walker Universe. Only 3 coordinate axes
Q0 =ct, Q1, Q2 are shown in this three dimensional figure. The Q3 axis could be imagined. The Earth
{ }
is located in the origin. The position vector for nucleus of atom between the QSO and the Earth is Q, and
for electron is L. The distance between nucleus and electron is r¯ L Q. The location of the Minkowski
∼| − |
point of Betrami metric is denoted by notation “ ” with M =(M0, M1, 0, 0).
×
The Friedmann-Robertson-Walker (FRW) metric is (see, e.g., [34])
dr2
ds2 = c2dt2 a(t)2 +r2dθ2 +r2sin2θdφ2
− 1 kr2
(cid:26) − (cid:27)
k(QidQi)2
= (dQ0)2 a(t)2 dQidQi +
− 1 kQiQi
(cid:26) − (cid:27)
g (Q)dQµdQν, (2.15)
µν
≡
where a(t) is scale (or expansion) factor and r = QiQi Q, Q1 = Qsinθcosφ, Q2 =
≡
Qsinθsinφ, Q3 = Qcosθ and (Q0)2 = c2t2. Hereafter, for the sake of convenience, we take
p
t to be looking-back cosmologic time, so that t < 0. As is well know FRW metric satisfies
homogeneity and isotropy principle of present day cosmology. For simplicity, we take k = 0
and a(t) = 1/(1 + z(t)) (i.e., a(t ) = 1). And the red shift function z is determined by
0
ΛCDM model [29,35,36](see, e.g., Eq.(64) of [29]):
0 dz
′
t(z) = , (2.16)
H(z )(1+z )
Zz ′ ′
where
H(z ) = H Ω (1+z )3 +Ω (1+z )4 +1 Ω ,
′ 0 m0 ′ R0 ′ m0
−
H0 = 100ph 100 0.705km s−1/Mpc,
≃ × ·
Ω 0.274, Ω 10 5. (2.17)
m0 R0 −
≃ ∼
Figure of t(z) of Eq.(2.16) is shown in Figure 2.
The Light-Cone of FRW Universe is defined by ds2 = 0. From Eq.(2.15), we have the
light-cone equation:
1
(dQ0)2 a(t)2(dQ)2 = 0, or cdt = a(t)dQ = dQ. (2.18)
− − 1+z(t)
6
-tHGyrL
12
10
8
6
4
2
zHredshiftL
1 2 3 4 5
Figure 2: The t z relation in ΛCDM model (eq.(2.16)).
−
Substituting (2.16) into (2.18) gives
z dz
′
Q(z) = c . (2.19)
H(z )
Z0 ′
Figure of Q(z) of Eq.(2.19) is shown in figure 3. Ratio of Q over Q0 is shown in figure 4.
QHzLHGlyrL
25
20
15
10
5
zHredshiftL
1 2 3 4 5
Figure 3: Function Q(z) in ΛCDM model (eq.(2.19)).
Q(cid:144)H-Q0L
2.0
1.5
1.0
0.5
zHredshiftL
1 2 3 4 5
Figure 4: Function of Q(z)/Q0(z). Q(z) and Q0(z)=ct are given in Eqs. (2.19) and (2.16).
3 Local Inertial Coordinate System in Light-Cone of
FRW Universe with Einstein Cosmology Constant
Inprinciple, almost all calculations onquantum spectrums inatomic physics are achieved
in the inertial coordinate systems. From the cosmological point of view, the phenomena
of atomic spectrums should be described in the local inertial coordinate systems of FRW
7
Universe. Therefore, we are interested in how to determine the local inertial coordinate
system in light-cone of FRW Universe when the Einstein cosmology constant Λ is present.
Existence of local inertial coordinate system isrequired by theEquivalence Principle. The
principle states that experiments in a sufficiently small falling laboratory, over a sufficiently
short time, give results that are indistinguishable from those of the same experiments in
an inertial frame in empty space of special relativity [37]. Such a sufficiently small falling
laboratory, over a sufficiently short time represents a local inertial coordinates system. This
principle suggests that the local properties of curved spacetime should be indistinguishable
from those of the spacetime with inertial metric of special relativity. A concrete expression
of this ideal is the requirement that, given a metric g in one system of coordinates xα, at
αβ
each point P of spacetime it is possible to introduce new coordinates xα such that
′
g (x ) = inertial metric of SR at x , (3.20)
α′β ′P ′P
and the connection at x is the Christoffel symbols deduced from g (x ).
′P α′β ′P
In usual Einstein’s general relativity (without Λ), the above expression is
g (x ) = η , and Γλ = 0, (3.21)
α′β ′P αβ αβ
which satisfies the Einstein equation of E-GR in empty space: G = 0.
µν
In dS-GR (GR with a Λ), the local inertial coordinate system at xα is characterized by
′P
g (x ) = B(M)(x ) ηµν + ηµλ(x′λ −Mλ)ηνρ(x′Pρ −Mρ), (3.22)
α′β ′P αβ ′P ≡ σ(M)(x ) R2σ(M)(x )2
′P ′P
with σ(M)(x ) = 1 ηµν(x′Pµ −Mµ)(x′Pν −Mν),
′P − R2
1
Γλ = (B(M))λρ(∂ B(M) +∂ B(M) ∂ B(M))
αβ 2 α ρβ β ρα − ρ αβ
1
= R2σ(M)(x )(δµληνρ +δνληµρ)(x′Pρ −Mρ), (3.23)
′P
(or ∂gρ′β(x′P) = x′Pν −Mν 2η η +η η +η η + 4ηρµηβνηαλ(x′Pµ −Mµ)(x′Pλ −Mλ) )
∂xα R2σ(M)(x )2 ρβ αν ρα βν βα ρν R2σ(M)(x )
′P ′P (cid:20) ′P (cid:21)
(M)
where B (x )were givenin(1.10), whichsatisfies theEinsteinequationofdS-GRinempty
αβ ′P
spacetime: G +Λg = 0 with Λ = 3/R2. (Note η does not satisfy that equation, i.e.,
µν µν µν
G (η) +Λη = 0. So it cannot be the metric of the local inertial system in dS-GR with
µν µν
6
Λ).
To the light cone of FRW Universe with Λ, the coordinate-components Q0(z) = ct(z)
and Q(z) have been shown in Eqs.(2.16) (or Figure 2) and (2.19) (or Figure 3) respec-
tively. Therefore from (1.10), the space-time metric of the local inertial coordinate system
at position Q(z) of the light cone is determined to be
η η (Qλ Mλ)η (Qρ Mρ)
B(M)( ) B ( M) = µν + µλ − νρ − , (3.24)
µν Q ≡ µν Q− σ(M)( ) R2σ(M)( )2
Q Q
where
η (Qµ Mµ)(Qν Mν)
σ(M)( ) σ( M) = 1 µν − − . (3.25)
Q ≡ Q− − R2
8
We see from Figure 1 that the visible atom is embedded into the light cone at Q-point. Since
Q L (i.e., comparing with the Universe, atoms are very very small), we can reasonably
≃
treatthemetricofthespacetimeintheatomicregionasaconstant. Thisisjusttheadiabatic
approximation adopted in [21]. When Qµ Mµ, we have B(M)( ) η . So, Qµ = Mµ is
µν µν
→ Q ⇒
(M)
the Minkowski point of the Beltromi metric B ( ).
µν
Q
Now let’s derive e µ and ωab from (3.24). Setting
a µ
qµ Qµ Mµ, (3.26)
≡ −
then equations (3.24) and (3.25) become
η η qλη qρ
B(M) = µν + µλ νρ , (3.27)
µν σ(M) R2(σ(M))2
where
η qµqν
σ(M) σ(M)(q) = 1 µν . (3.28)
≡ − R2
We introduce notations:
q¯ η qλ, qλ q¯λ = ηµλq¯ , (3.29)
µ µλ µ
≡ ≡
q¯2 η qµqν = q¯ q¯ν, (3.30)
µν ν
≡
and construct two project operators in spacetime q¯µ with metric η :
µν
{ }
q¯ q¯ q¯ q¯
¯ µ ν µ ν
θ η , ω¯ . (3.31)
µν ≡ µν − q¯2 µν ≡ q¯2
It is easy to check the calculation rules for project operators:
θ¯ θ¯λ θ¯ ηλρθ¯ = θ¯ , or in short θ¯ θ¯= θ¯, (3.32)
µλ ν ≡ µλ ρν µν ·
ω¯ ω¯λ ω¯ ηλρω¯ = ω¯ , or in short ω¯ ω¯ = ω¯, (3.33)
µλ ν ≡ µλ ρν µν ·
θ¯ ω¯λ θ¯ ηλρω¯ = 0, or in short θ¯ ω¯ = 0, (3.34)
µλ ν ≡ µλ ρν ·
¯ ¯
θ +ω¯ = η , or in short θ+ω¯ = I. (3.35)
µν µν µν
(M)
B (Q) can be written as
µν
η q¯ q¯
B(M) = µν + µ ν , (3.36)
µν σ(M) R2(σ(M))2
where
η qµqν q¯ q¯ν q¯2
σ(M) = 1 µν = 1 ν 1 . (3.37)
− R2 − R2 ≡ − R2
Since B(M) is a tensor in the spacetime q¯µ, η , it can be written as follows from (3.36):
µν µν
{ }
1 1
B(M) = θ¯ + ω¯ . (3.38)
µν σ(M) µν (σ(M))2 µν
Furthermore, by means of (B(M))µνB(M) = δµ ηµ and the rules (3.31)-(3.35), the above
νλ ν ≡ ν
(M)
expression of B leads to:
µν
(B(M))µν = σ(M)θ¯µν +(σ(M))2ω¯µν. (3.39)
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