#### de Sitter relativity in static charts

Eur. Phys. J. C
de Sitter relativity in static charts
Ion I. Cota˘escu 0
0 West University of Timis ̧oara , V. Pârvan Ave. 4, 300223 Timisoara , Romania
The relative geodesic motion in static (and spherically symmetric) local charts on the (1 + 3)-dimensional de Sitter spacetimes is studied in terms of conserved quantities. The Lorentzian isometries are derived, relating the coordinates of the local chart of a fixed observer with the coordinates of a mobile chart considered as the rest frame of a massive particle freely moving on a timelike geodesic. The time dilation and Lorentz contraction are discussed pointing out some notable features of the de Sitter relativity in static charts.
1 Introduction
The simplest (1 + 3)-dimensional spacetimes of special or
general relativity are vacuum solutions of the Einstein
equations whose geometry is determined only by the value of
the cosmological constant Λ. These are the Minkowski flat
spacetime (with Λ = 0), and the hyperbolic spacetimes,
de Sitter (dS) with Λ > 0 and Anti-de Sitter (AdS) having
Λ < 0 [1]. All these spacetimes have highest possible
isometries [2] representing thus a good framework for studying the
role of the conserved quantities with physical meaning in
quantum theory [3–6] or for describing the classical
relative geodesic motion [7–9]. With their help we constructed
recently the dS relativity [10] in comoving charts [11] and
the AdS relativity [12] in static and spherically symmetric
local charts that complete our image of the special relativity
in spacetimes with maximal symmetry.
Our approach is based on the idea that the inertial (natural)
frames are local charts playing the role of rest frames of
massive particles freely moving along timelike geodesics.
Moreover, we impose a synchronization condition requiring
the origins of the fixed and moving frames to overlap at a
given time. The conserved quantities on these geodesics help
us to mark the different inertial frames whose relative motion
can then be studied by using the Nachtmann boosting method
of introducing coordinates in different dS local charts [3]. In
this manner, we derived the Lorentzian isometries relating
the coordinates of the moving and fixed inertial frames on
dS or AdS backgrounds [10,12].
The (1 + 3)-dimensional AdS spacetime is the only
maximally symmetric spacetime which does not have space
translations [2], since its Λ < 0 produces an attraction of elastic
type such that the geodesic motion is oscillatory around the
origins of the static charts with ellipsoidal closed trajectories.
The AdS relativity relates these charts such that, according to
the synchronization condition, the moving frames may have
only rectilinear geodesics whose oscillatory motions are
centered in the origin of the fixed frame [12]. On the contrary,
in the comoving local charts we used so far (i.e. the
conformal Euclidean and de Sitter–Painlevé ones), the dS
relativity seems to be closer to the Einstein special relativity since
here we have translations and conserved momenta such that at
least in the conformal Euclidean chart all the geodesic
trajectories are rectilinear along the momentum direction [6,10].
However, apart from the comoving charts, the dS
spacetime has, in addition, static charts where the geodesic
trajectories are no longer rectilinear such that the role of the
conserved momentum becomes somewhat obscure. Since the dS
relativity in these charts is not yet formulated, we focus here
on this problem studying the role of the conserved quantities
along geodesics in describing the relative geodesic motion.
In order to preserve the coherence of our dS relativity,
we use here the same definitions, conventions and initial
conditions as in Ref. [10] since then we can take over the
results obtained therein without revisiting the entire
boosting method which allowed us to construct the dS and AdS
relativity. In this manner we obtain a version of the dS
relativity in static charts which is perfectly symmetric with the AdS
one with respect to the change of the hyperbolic functions
into trigonometric ones.
The principal new result we report here concern the role of
the conserved quantities in determining the parametrization
of the Lorentzian isometries relating fixed and moving static
charts. Moreover, we briefly discuss some notable properties
of these isometries and their consequences upon simple
relativistic effects as the time dilation and Lorentz contraction.
We start in the second section with a short review of the
static charts where we consider the dS conserved quantities
presented in the third section. The next section is devoted to
the timelike geodesics in static charts showing how their
integration constants depend on the conserved quantities with
physical meaning and pointing out the kinematic role of
these quantities. In Sect. 5 we solve the relativity problem in
static charts deriving the Lorentzian isometries with different
parametrizations. In the last part of this section we discuss the
above-mentioned simple relativistic effects, giving the
general formulas, allowing the analytical and numerical study of
particular cases. In the last section we present the dS–AdS
symmetry which involves all the conserved quantities of both
these spacetimes.
2 Static charts on dS spacetimes
Let us consider the (1 + 3)-dimensional dS spacetime (M, g)
which is a vacuum solution of the Einstein equations with
Λ > 0 and positive constant curvature. This is a
hyperboloid of radius R = ω1 = Λ3 embedded in the (1 +
4)dimensional pseudo-Euclidean spacetime (M 5, η5) of
Cartesian coordinates z A (labeled by the indices A, B, . . . =
0, 1, 2, 3, 4) and metric η5 = diag(
1, −1, −1, −1, −1
).
These coordinates are global, corresponding to the
pseudoorthonormal basis {νA} of the frame into consideration,
whose unit vectors satisfy νA · νB = η5AB . Any point z ∈ M 5
is represented by the five-dimensional vector z = νAz A =
(z0, z1, z2, z3, z4)T , which transforms linearly under the
gauge group S O(
1, 4
) which leave the metric η5 invariant.
The local static charts {x }, of coordinates x μ (α, ..μ,
ν · · · = 0, 1, 2, 3), can be introduced on (M, g) giving the
set of functions z A(x ) which solve the hyperboloid equation,
where r = |x| ≤ ω1 and χ (r ) = 1 − ωx 2 =
Hereby one obtains the line element,
√
√1 − ω2r 2.
1
z0(x ) = ω χ (r ) sinh(ωt ),
zi (x ) = xi ,
1
z4(x ) = ω χ (r ) cosh(ωt ),
The usual static chart {t, x} with Cartesian spaces coordinates
xi (i, j, k, ... = 1, 2, 3) is defined by
ds2 = η5AB d z A(x )d z B (x )
= χ (r )2dt 2 − δi j + ω
2 xi x j
χ (r )2
d xi d x j .
The associated static chart {t, r, θ , φ} with spherical
coordinates, canonically related to the Cartesian ones, x →
(r, θ , φ), has the line element
dr 2
ds2 = χ (r )2dt 2 − χ (r )2 − r 2(dθ 2 + sin2 θ dφ2).
Apart from the above usual charts, it is useful to consider
the static chart {x˜} = {t, ρ , θ , φ} resulting after the
substitution [13,14],
ρ
r = χ˜ (ρ) , χ˜ (ρ) =
1 + ω2ρ2,
where ρ ∈ [0, ∞). Then the embedding equations become
1
z0(x˜) = ωχ˜ (ρ) sinh(ωt ),
ρ
z1(x˜) = χ˜ (ρ) sin θ cos φ,
ρ
z2(x˜) = χ˜ (ρ) sin θ sin φ,
ρ
z3(x˜) = χ˜ (ρ) cos θ ,
1
z4(x˜) = ωχ˜ (ρ) cosh(ωt ),
while the line element reads
(
1
)
(2)
1
ds2 = χ˜ (ρ)2
dρ2
dt 2 − χ˜ (ρ)2 − ρ2(dθ 2 + sin2 θ dφ2) .
In this chart the components of the four-velocity are denoted
˜ = ddx˜sμ .
uμ
3 Conserved quantities
The dS spacetimes are homogeneous spaces of the gauge
group S O(
1, 4
) whose transformations leave invariant the
metric η5 of the embedding manifold M 5 and implicitly Eq.
(
1
). For this group we adopt the canonical parametrization,
g(ξ ) = exp
i
− 2 ξ AB SAB
∈ S O(
1, 4
),
with skew-symmetric parameters, ξ AB = − ξ B A, and the
covariant generators of the fundamental representation of the
so(
1, 4
) algebra carried by M 5 having the matrix elements,
(SAB )C·D· = i δCA η5B D − δCB η5AD .
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
In any local chart {x }, defined by the functions z = z(x ),
each transformation g ∈ S O(
1, 4
) gives rise to the
associated isometry x → x = φg(x ) derived from the system of
equations z[φg(x )] = g z(x ).
The so(1, 4) basis-generators with an obvious physical
meaning [6–8] are the energy H = ωS04, angular
momentum Jk = 21 εki j Si j , Lorentz boosts Ki = S0i , and the
Runge–Lenz-type vector Ri = Si4. In addition, we
consider the momentum Pi = −ω(Ri + Ki ) and its dual
Qi = ω(Ki − Ri ), which are nilpotent matrices of two
Abelian three-dimensional subalgebras [6].
In general, after integrating the geodesic equations, one
obtains the geodesic trajectories depending on some
integration constants that must get a physical interpretation. This
is possible only by expressing them in terms of conserved
quantities on geodesics. These are given by the Killing
vectors associated to the S O(
1, 4
) isometries which are defined
(up to a multiplicative constant) as [4],
SAB → k(AB) μ = z A∂μz B − z B ∂μz A,
where z A = η5AC zC . The principal conserved quantities
along a timelike geodesic of a pointlike particle of mass m
and momentum P have the general form
K(AB)(x , P) = ωk(AB) μmuμ
fwohuer-rveeuloμcit=y thdaxdtμs(ssa)tiasrfey tuh2e c=omgpμoνnueμnutsν o=f th1e. Tcohvearcioann-t
served quantities with physical meaning [6–8,10] are the
well-known energy and angular momentum,
(11)
(12)
(13)
(14)
and the S O(3) vectors having the components,
related to the conserved momentum, and P and its dual Q
defined as [6],
Pi = −ω(Ri + Ki ),
Qi = ω(Ki − Ri ).
Thus we can construct the five-dimensional matrix,
⎛ 0
⎜ −ωK1
K (x , P) = ⎜⎜ −ωK2
⎜⎝ −ωK3
−E
ωK1
0
−ωL3
ωL2
−ω R1
ωK2
ωL3
0
−ωL1
−ω R2
ωK3
−ωL2
ωL1
0
−ω R3
whose elements transform as a skew-symmetric tensor on
M 5, according to the rule
(
19
)
K(AB)(x , P ) = g·AC· g·BD· K(C D)(x , P),
for all g ∈ S O(
1, 4
). Here g·AB· = ηAC gC·D· η5 B D are the
5
matrix elements of the adjoint matrix g = η5 g η5. Thus,
Eq. (
19
) can be written as K (x , P ) = g K (x , P) gT or
simpler, K = g K gT .
Notice that all the conserved quantities carrying space
indices (i, j, ...) transform alike under rotations as S O(3)
vectors or tensors. Moreover, the condition zi ∝ xi fixes
the same (common) three-dimensional basis {ν1, ν2, ν3} in
both the Cartesian charts, of M 5 and M . This means that
the S O(3) symmetry is global [4] such that we may use the
vector notation for the conserved quantities as well as for the
local Cartesian coordinates on M . However, this basis must
not be confused with that of the local frames on M which are
orthogonal in the sense of the dS geometry.
For studying the conserved quantities on the timelike
geodesics we chose the chart {t, ρ , θ , φ} taking the
angular momentum along the third axis, L = Lν3 = (0, 0, L),
π
for restricting the motion in the equatorial plane, with θ = 2
and u˜θ = 0. Then the non-vanishing conserved quantities
can be written as
E = χm2 u˜t , (20)
˜
L = mχρ22 u˜φ , (21)
˜
m
K1 = ωχ˜ 2 ωρu˜t cosh ωt cos φ
− u˜ρ sinh ωt cos φ + ρu˜φ sinh ωt sin φ ,
m
K2 = ωχ˜ 2 ωρu˜t cosh ωt sin φ
− u˜ρ sinh ωt sin φ − ρu˜φ sinh ωt cos φ ,
m
R1 = ωχ˜ 2 ωρu˜t sinh ωt cos φ
− u˜ρ cosh ωt cos φ + ρu˜φ cosh ωt sin φ ,
m
R2 = ωχ˜ 2 ωρu˜t sinh ωt sin φ
− u˜ρ cosh ωt sin φ − ρu˜φ cosh ωt cos φ ,
while K3 = R3 = 0. Hereby we deduce the following
obvious properties:
E ⎞
ω R1 ⎟
ω R2 ⎟⎟ ,
ω R3 ⎟⎠
0
(18)
K · L = R · L = 0,
and verify the identities
E 2 − ω2 L 2 + R 2 − K 2
E
K ∧ R = − ω L,
= E 2 − ω2L 2 − Q · P = m2u˜2 = m2,
(22)
(23)
(24)
(25)
(26)
(
27
)
defining the principal invariant corresponding to the first
Casimir operator of the so(
1, 4
) algebra. In the flat limit,
ω → 0, when Q → P, this identity becomes just the usual
mass-shell condition p2 = m2 of special relativity [6,10].
We note that in the classical theory the second invariant of
the so(
1, 4
) algebra vanishes since there is no spin [6].
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
κ2 − κ1, ρ+ =
In the case of the timelike geodesics we may exploit the
identity u˜2 = 1 and Eqs. (20) and (21) for obtaining the
radial component
u˜ρ = χ˜ (ρ)2
E 2
m2 χ˜ (ρ)2 −
ω2 L2
m2
L2
− m2ρ2 − 1
which allows us to derive the following prime integrals:
dρ
dt
2
2 2 L2
− ω ρ + E 2ρ2 = 1 −
ω2 L2
E 2
m2
− E 2 ,
dφ L
dt = Eρ2 ,
which give the geodesic equations in the plane (ν1, ν2) as
1
ρ(t ) = [−κ1 + κ2 cosh 2ω(t − t0)] 2 ,
φ (t ) = φ0 + arctan
κ2 + κ1 tanh ω(t − t0) ,
κ2 − κ1
where
κ1 =
E 2 − m2 − ω2 L2
2ω2 E 2
,
satisfying the identity
κ22 − κ12 = ω2 E 2 .
L2
κ2 = 2ω12E 2 (E + m)2 + ω2 L2 21 (E − m)2 + ω2 L2 21 ,
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
where
ρ− =
√
Thus we solve the geodesic equation in terms of conserved
quantities which give a physical meaning to the principal
integration constants. The remaining ones, t0 and φ0,
determine only the initial position of the mobile and implicitly of
its trajectory.
In this manner, we recover the well-known behavior of
the timelike geodesic trajectory which is a hyperbola that for
φ0 = 0 can be written easily in Cartesian space coordinates
(ρ , φ) → (x˜1, x˜2) of the plane (ν1, ν2) as
x˜1(t ) = ρ(t ) cos φ (t ) = ρ− cosh ω(t − t0),
x˜2(t ) = ρ(t ) sin φ (t ) = ρ+ sinh ω(t − t0),
K1 = Eρ− cosh ωt0,
K2 = −Eρ+ sinh ωt0,
R1 = Eρ− sinh ωt0,
R2 = −Eρ+ cosh ωt0,
complying with the specific properties,
1
K = |K| = E [−κ1 + κ2 cosh 2ωt0] 2 ,
1
R = |R| = E [κ1 + κ2 cosh 2ωt0] 2 ,
K · R = E 2κ2 sinh 2ωt0.
P = |P| =
Q = |Q| =
satisfying
P · Q = − 2κ1ω2 E 2.
2κ2 ω E eωt0 ,
2κ2 ω E e−ωt0 ,
Moreover, the conserved momentum and its dual defined by
Eq. (17) have the norms
Hereby we understand that the vector KE indicates the position
of the particle of mass m at t = 0 (Fig. 1) such that for t0 = 0
this vector lays over the semi major axis being orthogonal
on RE (Fig. 2). It is remarkable that for any t0 the vector
P is oriented along the lower asymptote, while Q gives the
direction of the upper one (as in Figs. 1 and 2). Thus the
vector Q, whose role in comoving charts was rather unclear
[10], now gets a precise physical meaning.
All these properties are independent on the value of φ0
which gives only the rotation of the major axis in the plane
(ν1, ν2). Nevertheless, in the appendix we give the general
form of all these conserved quantities calculated for an
arbitrary φ0 = 0.
vFeigct.o1rs adreeS:(it1te)rKEti,m(e2l)ikREe,g(e3o)de−sicωPwEiathndφ0(4=) ω0QEand t0 > 0. The marked
vFeigct.o2rs adreeS:(it1te)rKEti,m(e2l)ikREe,g(e3o)de−sicωPwEiathndφ0(4=) ω0QEand t0 = 0. The marked
An important particular case is when the geodesic is pass
ing through the origin since then the trajectory is rectilinear
with L = 0 and κ− = 0. Consequently, the vectors P and Q
become parallel, having the norms
Recently we have studied the relative geodesic motion on
dS [10] and AdS manifolds [12], applying the Nachtmann
method of boosting coordinates [3]. In the case of the AdS
spacetimes we used static charts, while for the dS
spacetimes we considered only comoving charts (i.e. the conformal
Euclidean and de Sitter–Painlevé ones) where the geodesics
are rectilinear [10]. Here we complete this study constructing
the dS relativity in static charts by taking the results obtained
previously in comoving charts, without revisiting the entire
boosting method.
5.1 Lorentzian isometries
The problem of the relative motion is to find how an arbitrary
geodesic trajectory and the corresponding conserved
quantities can be measured by different observers. The local charts
may play the role of inertial frames related through
isometries. Each observer has its own proper frame {x } in which he
stays at rest in the origin on the world line along the vector
field ∂t [10]. Here we are interested in the inertial frames
defined as proper frames of massive particles freely
moving along geodesics. Then each mobile inertial frame can be
labeled by the conserved quantities determining the geodesic
of the carrier particle which stays at rest in its origin [10].
In what follows we consider two observers assuming that
the first one, O, is fixed in the origin of his proper frame
{x } observing what happens in a mobile frame {x } of the
observer O , which is simultaneously the proper frame of O
and of a carrier particle of mass m moving along a timelike
geodesic with given parameters. This relativity does make
sense only if we can compare the measurements of these
observers imposing the synchronization condition of their
clocks. This means that, at a given common initial time, the
origins of these frames must coincide. However, this
condition is restrictive since this forces the geodesic of the particle
carrying the mobile frame to cross the origin of the fixed
frame O. Consequently, its trajectory is rectilinear (with
L = 0) in a given direction determined by its conserved
momentum P as in Eq. (53).
The choice of the synchronization condition is a delicate
point since the form of the isometry relating the fixed and
mobile frames, called Lorentzian isometry, is strongly
dependent on this condition. For this reason we use the same
condition as in the case of the comoving frames [10] since then
the Lorentzian isometry is generated by the same
transformation of the S O(
1, 4
) group. Therefore, we set the
synchronization condition at t = t = 0 when x(0) = x (0) = 0
such that the origins of both frames, O and O , overlap the
point zo = (0, 0, 0, 0, ω−1)T ∈ M 5, which was the fixed
point in constructing the dS manifold as the space of left
cosets S O(
1, 4
)/L↑ where the Lorentz group L↑+ is the
sta+
ble group of zo [10].
Under such circumstances, the synchronization condition
is the same as in the case of the comoving charts and we
can take over the S O(
1, 4
) transformation generating the
Lorentzian isometry between the frames O and O. This has
the form [10]
g(P) = exp
1
−i Pi Ki P arcsinh
P
m
⎛ E
m
⎜ P1
⎜⎜ m
⎜ P2
= ⎜⎜⎜⎜ Pm3
⎝⎜ m
0
P1
m
1 + n1p2ν
n1 n2 ν
p p
n1 n3 ν
p p
0
P2
m
n1 n2 ν
p p
1 + n2p2ν
n2 n3 ν
p p
0
P3
m
n1 n3 ν
p p
n2 n3 ν
p p
1 + n3p2ν
0
0 ⎞
0 ⎟⎟⎟
0 ⎟⎟ ,
⎟⎟
0 ⎟⎟
⎠
1
(56)
where nP = PP and ν = mE − 1 . The four-dimensional
restriction of this transformation is a genuine Lorentz boost
such that g(P)−1 = g(−P) and g(0) = e. This
transformation generates the Lorentzian isometry and transforms the
conserved quantities according to Eq. (
19
).
The direct Lorentzian isometry, x = φg(P)(x ), between
the coordinates of the mobile and fixed frames, results from
the system of equations z(x ) = g(P)z(x ) as
1
t (t , x ) = ω arctanh
E
m
tanh ωt
ω
+ m
x · P
1 − ω2|x |2
sech ωt
x · P
E + m
while the inverse one has to be obtained by changing x ↔ x
and P → −P. Obviously, in the limit of ω → 0 we recover
the usual Lorentz transformations of special relativity.
We verify first that the geodesic trajectory of the carrier
particle can be recovered from the parametric equations in t
obtained by substituting x = 0 in Eqs. (57) and (58). Then
we obtain the trajectory of the origin O , denoted
x∗i(t ) =
Pi sinh ωt
ω
E 2 + P2 sinh2 ωt
,
which is just Eq. (53) with t0 = 0, corresponding to our initial
condition x∗(0) = 0. The components of the four-velocity
are those given by Eqs. (54) and (55) for t0 = 0 when we
have E = mu0∗(0) and Pi = mui∗(0). This means that E and
Pi are the components of the energy-momentum four-vector
of the carrier particle when this is passing through the origin
of the fixed frame.
This suggests us to consider as principal parameter the
velocity of the carrier particle at t = 0, defined usually
as V = EP . Then we may put the above formulas in forms
closer to those of special relativity, eliminating the mass m
of the carrier particle. This can be done by changing the
parametrization of g(P), setting
E = γ m, P = γ mV, γ = √
such that we can rewrite
g(P) → g(V) = exp
1
1 − V 2
,
1
−i V i Ki V arctanh (V ) ,
obtaining the new expression of the Lorentzian isometry
1
t (t , x ) = ω arctanh
γ tanh ωt
+
γ ω x · V
1 − ω2|x |2
x(t , x ) = x + γ V x · V
sech ωt
1
+ ω
1 − ω2|x |2 sinh ωt
which may be used in applications.
The transformations g(V), generating these isometries,
transform simultaneously all the conserved quantities. If
those of the mobile frame are encapsulated in the matrix
(58)
(59)
(60)
(61)
(62)
(63)
K as in Eq. (18), then the corresponding ones measured in
the fixed frame are the matrix elements of the matrix
K = g(V) K g(V)T .
Thus we obtain the principal tools in studying the relative
motion in static charts on dS spacetimes.
5.2 Simple relativistic effects
The principal feature of the Lorentzian isometries in dS static
charts is that the domains of these mappings do not span the
entire static charts involved in such transformations. Indeed,
the condition | tanh z| ≤ 1, ∀z ∈ R indicates that Eq. (62)
does make sense only in the domain D where the function
V · x V · x
B(x ) = ω χ (x ) = ω √(1 − ω2|x |2)
satisfies
(64)
(65)
(66)
1 1
− γ cosh ωt − sinh ωt ≤ B(x ) ≤ γ cosh ωt − sinh ωt ,
restricting thus the domain of the coordinates (t , x ) of the
mobile frame. This condition determines the field of view of
the observer O and guarantees that after this transformation
we obtain well-defined Cartesian coordinates that satisfy the
condition |x(t , x )| ≤ ω1 imposed by the existence of the
cosmological horizon. For the inverse Lorentzian isometry,
we obtain a similar condition defining the domain D of this
transformation.
It remains to investigate how this restriction works
determining the domain D . Assuming that the observation is
along an arbitrary direction we denote α = angle(x , V) such
that we can write B(x ) = ωVρ(x ) cos α. Here ρ is the radial
coordinate defined by Eq. (5) that is free of any restriction,
taking values in the domain [0, ∞). Then the condition (66)
restricts the observation at the points (t , x ) which satisfy
|ρ(x ) cos α| ≤ ρlim (V , t ) where the function
1
ρlim (V , t ) = ωV
1
γ
cosh ωt − sign(t ) sinh ωt
(67)
is positively defined on the domain [−tm , tm ], vanishing for
t = ± tm where tm = ω1 arctanh γ1 . Hence, we may conclude
that Eq. (66) gives rise to non-trivial restrictions that seem to
be specific for the static charts.
More interesting are the simple relativistic effects as the
time dilation (observed in the twin paradox) and the Lorentz
contraction. In general, these effects are quite complicated
since they are strongly dependent on the position where the
(72)
(73)
(74)
time and length are measured. Let us address this issue,
assuming that the measurement is performed in the point
A of arbitrary position vector a, fixed rigidly to the mobile
frame O . Then we may write the general relations
δt =
δx j =
∂t (t , x )
∂t
∂ x j (t , x )
∂t
x =a
x =a
δt +
δt +
∂t (t , x )
∂ x i
x =a
∂ x j (t , x )
∂ x i
δx i ,
δx i ,
(68)
(69)
x =a
allowing us to relate among themselves the quantities δt, δx j
and δt , δx j measured by the observers O and O .
We consider first a clock in A indicating δt without
changing its position such that δx i = 0. Then, after a
little calculation, we obtain the time dilation observed by O,
δt (t ) = δt γ˜ (t ), given by the function
γ˜ (t ) = γ 1 − B(a) sinh ωt (t )
cosh2 ωt
cosh2 ωt (t )
,
(70)
which depends on the parameter B(a) defined by Eq. (65)
for x = a and the function
1 1
t (t ) = ω arctanh γ (B(a)2 + 1) (tanh ωt
+ B(a) γ 2 B(a)2 + γ 2 − 1 + sech2ωt
,
(71)
resulting after inverting Eq. (62) with x = a. Similarly but
with the supplemental simultaneity condition δt = 0 we
derive the Lorentz contraction of an arbitrary δx , which reads
δx(t ) = γ nV · δx − B(a) a · δx sinh ωt (t )
γ 2 − 1 cosh2 ωt (t )
+ γ B(a) sinh ωt (t ) − 1
(V · a)(a · δx )
× V · δx + χ (a)2 ,
where nV = VV .
Thus we obtained the general formulas of the simple
relativistic effects which hide a large phenomenology that
cannot be exhaustively treated here because of the
difficulties in studying analytically the general case of an
arbitrary a. For this reason we restrict ourselves to the case of
a · V = 0, assuming that δx is parallel with V. Then, by
taking B(a) = 0 in Eqs. (70)–(72) we obtain
δt = δt γ˜ (t ), δx = δx
1
γ˜ (t )
where the function γ˜ (t ) takes the simple form
γ
γ˜ (t ) = cosh2 ωt (t ) − γ 2 sinh2 ωt (t )
1
= γ cosh2 ωt − γ sinh2 ωt,
since now γ tanh ωt (t ) = tanh ωt . Hereby we recover again
the well-known condition of the flat case, δt δx|| = δt δx||,
which also holds in the dS relativity in comoving charts [10].
The function γ˜ (t ) is defined on the domain (−∞, ∞)
taking values in the codomain [γ , ∞). The time dilation
observed by O increases to infinity as
1
γ˜ (t ) ∼ 4
1
γ − γ
e2ωt
when t → ∞, since the observer O sees how the clock in
O lats more and more such that t tends to infinity when t is
approaching to tm . Notice that the observer O measures the
same dilation of the time t of a clock staying at rest in O.
In general, for the clocks situated in arbitrary space points
the problem is much more complicated and cannot be solved
without resorting to numerical method. As an example, we
present in Fig. 3 the functions γ˜ (t ) for different norms
a = |a| of the position vector a oriented parallel with V.
Other interesting and attractive conjectures may be studied
numerically starting with the above presented approach.
(75)
6 Remark on the dS–AdS symmetry
The Lorentzian isometry given by Eqs. (57) and (58) is related
to the corresponding AdS isometry [12] through the
transformation ω → i ω. This is the effect of the well-known dS–AdS
symmetry under this transformation, arising when one uses
the same type of local charts in both these spacetimes.
Moreover, we must specify that this symmetry is general since the
dS conserved quantities transform into AdS ones as
dS[10]
E
L
K
R
P = −ω(K + R)
Q = ω(K − R)
ω −→ i ω AdS[12]
−→ E
−→ L
−→ K
−→ i N
−→ ω(N − i K)
−→ ω(N + i K)
regardless of the local charts we consider. In other respects,
this explains why in AdS spacetimes we do not have a
realvalued conserved momentum.
Concluding we can say that the dS relativity in the confor
mal charts is closer to the Einstein special relativity having
only rectilinear geodesics along the momentum directions,
while in static charts the dS relativity is symmetric with the
AdS one. Obviously, in the flat limit (when ω → 0) the
dS and AdS relativity tend to the usual special relativity in
Minkowski spacetime [6,9].
Finally, we note that this symmetry also holds at the level
of the quantum theory where the quantum observables are
conserved operators corresponding to the conserved
quantities considered above, having the same physical meaning
[4,6]. We remind the reader that in Ref. [4] the conserved
observables of the covariant quantum fields of any spin on
dS and AdS backgrounds are derived explicitly involving the
dS–AdS symmetry. However, now it is premature to discuss
how this symmetry may be extended to the quantum field
theory, since even on dS spacetimes we have already the QED
in Coulomb gauge [15]; on AdS spacetimes a similar theory
has not yet been constructed.
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A Inverse problem
There are situations when we know the integration constants
κ1, κ2, φ0 and t0 and we need to find the physical conserved
quantities. Then from Eqs. (33) and (35) we deduce,
E =
L =
m
(κ1ω2 − 1)2 − κ22ω4
2 2
mω κ2 − κ1
(κ1ω2 − 1)2 − κ22ω4
K1 = Eρ− cos φ0 cosh ωt0 + Eρ+ sin φ0 sinh ωt0,
K2 = Eρ− sin φ0 cosh ωt0 − Eρ+ cos φ0 sinh ωt0
R1 = −Eρ− cos φ0 sinh ωt0 + Eρ+ sin φ0 cosh ωt0,
R2 = −Eρ− sin φ0 sinh ωt0 − Eρ+ cos φ0 cosh ωt0,
(78)
(79)
(80)
(81)
while K3 = N3 = 0. These components satisfy the
properties (46)–(48).
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