Theoretical and Mathematical Physics, 176(1): 843–850 (2013)
CONSERVATION LAWS FOR CLASSICAL PARTICLES IN ANTI-DE SITTER–BELTRAMI SPACE T. Angsachon,∗ S. N. Manida,∗ and M. E. Tchaikovskii∗
The behavior of free classical pointlike particles is governed by conservation laws in the anti-de Sitter space. We present the general form of these laws and their realization in the Beltrami coordinates. In these coordinates, we can pass to the nonrelativistic limit resulting in physics in the R space. We construct the initial covariant distribution function for an ideal gas uniformly filling the entire R space.
Keywords: relativity principle, relativistic kinematics, anti-de Sitter space, Beltrami coordinates
1. Introduction The (anti-)de Sitter space (together with the Minkowski space) is a maximally symmetric solution of the Einstein equations. In this space–time, we have ten symmetry generators as in the Minkowski space, and we have two universal constants in contrast to a single universal constant in the Minkowski space. These two constants have different physical dimensionalities: one of them (as in the Minkowski space) is customarily chosen to have the speed dimensionality c, while the other is chosen to have the dimensionality of the squared length ±R2 . Depending on the sign of this constant, we have either the de Sitter or the anti-de Sitter space. The quantity R has the meaning of the radius of positive or negative internal curvature (in the respective de Sitter or anti-de Sitter space). Despite an extensive literature devoted to studying the properties of these spaces [1]–[10], the particle kinematics and dynamics has not yet been described in full detail. The particle dynamics in the de Sitter space case was analyzed in detail in [11], [12]. We present an analogous analysis of conservation laws and classical particle kinematics in the anti-de Sitter space following the ideas in those papers. In this space, we can perform a “nonrelativistic” limit transition c → ∞ in the Beltrami coordinates. Despite being “nonrelativistic,” this limit transition results in a particle kinematics and conservation laws locally coinciding with their counterparts in the Minkowski space [13]–[15]. But here, in contrast to the Minkowski space case, we can explain why additional conservation laws related to the twelve-parameter Schr¨odinger symmetry group [16]–[21] appear when passing to the Galilei–Newton mechanics. We subsequently construct the spherically symmetric initial distribution of classical particles in this space covariant with respect to the subgroup of translations. We obtain its properties and time dependence in explicit form in the case of one spatial dimension.
2. Symmetry generators in the anti-de Sitter space in the Beltrami coordinates We define the four-dimensional anti de Sitter space–time (generalizing to the case of an arbitrary space dimension is trivial) as a hyperbolic hypersurface AdS4 = {X ∈ M (2,3) , X 2 = ηAB X A X B = R2 } ∗
Saint Petersburg State University, St. Petersburg, Russia, e-mail:
[email protected].
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 176, No. 1, pp. 13–21, July, 2013. c 2013 Springer Science+Business Media, Inc. 0040-5779/13/1761-0843
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in the five-dimensional Minkowski space with the metric ds2 = ηAB dX A dX B , where ηAB = diag{1, 1, −1, −1, −1} in inertial reference frames and in the Lorentz coordinates and the indices A and B take the values −1, 0, 1, 2, 3. The symmetry group SO(2, 3) in the anti-de Sitter space is generated by the generators
LAB =
∂ ∂ XA − X . B ∂X B ∂X A
We can define the Beltrami coordinates as the projection of a half of the hyperboloid on the hyperplane with the Lorentz coordinates Xμ xμ = R , μ = 0, 1, 2, 3. (1) X−1 The thus obtained manifold is called the anti-de Sitter–Beltrami space [10]. For constructing the translation generators L(−1)i , we consider an infinitesimal rotation in the plane (X−1 , Xi ), i = 1, 2, 3: X−1 = X−1 cosh α + Xi sinh α X−1 + αXi ,
(2)
Xi = Xi cosh α + X1 sinh α Xi + αX1 ,
(3)
where α is an infinitesimal parameter. Substituting transformations (2) and (3) in formula (1), we obtain the explicit form of the translation generator in the Beltrami coordinates, L(−1)i = −
∂ xi xj ∂ ∂ +R − xi x0 . R ∂xj ∂xi ∂x0
We analogously construct the generator L0(−1) from an infinitesimal rotation in the plane (X0 , X−1 ): X0 = X0 cos α + X−1 sin α X0 + αX−1 , X−1 = −X0 sin α + X−1 cos α −αX0 + X−1 .
(4)
Substituting transformations (4) in formula (1), we obtain the explicit form of the generator L0(−1) in the Beltrami coordinates: 2 ∂ x0 x0 xi ∂ +R L0(−1) = + . R ∂x0 R ∂xi The explicit expressions for the generators of the boosts Li0 and rotations Ji = εijk Ljk in the Beltrami coordinates coincide with the corresponding expressions in the Minkowski space in the Lorentz coordinates, Li0 = x0
∂ ∂ + xi , ∂xi ∂x0
Ji = εijk xj
∂ . ∂xk
We have thus constructed all ten symmetry generators in the anti-de Sitter–Beltrami space. 844
(5)
3. Conserved quantities in the anti-de Sitter space We can write the action of a free massive classical particle on a hyperboloid in the five-dimensional space in the form (6) S = −mc [(V 2 )1/2 + a(X 2 − R2 )]dλ, where X(λ) is a timelike curve endowed with the additional condition X 2 = R2 , which is ensured by the Lagrange multiplier a. Here, V A (λ) = dX A /dλ is the five-dimensional particle velocity (X · V = 0). Action (6) is invariant under the transformations XA → XA + ωAB X B , where ωAB are ten skew-symmetric infinitesimal transformation parameters. The Noether theorem implies the preservation of ten quantities along a timelike geodesic line [12]: LAB =
m √ (XA VB − XB VA ). R V2
(7)
We further describe the conserved quantities in terms of the parameterized timelike geodesic. Because this geodesic is the intersection of a hyperboloid and a two-dimensional plane passing through its geometric center (0, 0, 0, 0, 0), the problem of parameterizing the geodesic reduces to that of parameterizing the two-dimensional plane. In contrast to the de Sitter space case [12], we obtain this parameterization by introducing two timelike vectors ξ and η (ξ 2 > 0, η 2 > 0): ξ = (0, ξ0 , ξ ),
η = (η−1 , 0, η ).
(8)
In these terms, we parameterize the geodesic function as R(λξ + η) X= . 2 2 λ ξ + η 2 + 2λ(ξ · η)
(9)
Substituting this parameterization in (7), we obtain the conserved quantities in a simple form, m[ηA ξB − ηB ξA ] LAB = . η 2 ξ 2 − (ξ · η)2
(10)
4. Conserved quantities in the anti-de Sitter space in the Beltrami coordinates Writing five-dimensional coordinates (9) in the four-dimensional form and substituting them in formula (1), we obtain xμ = λξμ + ημ . Choosing ξ = (0, c, ξ ) and η = (R, 0, η ) as a specific form of the parameters, we obtain xμ = (λc, λξ + η ). The geodesic parameters introduced in (8) acquire a simple sense in the Beltrami coordinates: x0 = λc,
xi = λξi + ηi ,
dxi = x˙i = ξi , dλ
where ηi is the initial coordinate, ξi is the particle velocity, and λ is time. 845
We now obtain explicit expressions for all the ten conserved quantities in the Beltrami coordinates. We first calculate the denominator in (10), η 2 ξ 2 − (ξ · η)2 = R2 c2 − R2 x˙ 2 − c2 ( x − λ x˙ )2 + ( x × x˙ )2 . We can now write the conserved quantities in the form m H = L0(−1) = , 1 − x˙ 2 /c2 − ( x − λ x˙ )2 /R2 + ( x × x˙ )2 /R2 c2
(11)
m(xi − λx˙i ) Ki = RL0i = , 1 − x˙ 2 /c2 − ( x − λ x˙ )2 /R2 + ( x × x˙ )2 /R2 c2
(12)
mεijk x˙j xk Ji = , 1 − x˙ 2 /c2 − ( x − λ x˙ )2 /R2 + ( x × x˙ )2 /R2 c2
(13)
mx˙i . Pi = cLi(−1) = 1 − x˙ 2 /c2 − ( x − λ x˙ )2 /R2 + ( x × x˙ )2 /R2 c2
(14)
The standard limit transition R → ∞ results in the standard conservation laws in special relativity. The much less known limit transition c → ∞ results in physics in the so-called R space [13], which we now describe.
5. Conservation laws for nonrelativistic cosmological particles We consider conservation laws for particles moving with speeds x˙ i c in the anti-de Sitter–Beltrami space. Expressions (11), (12), and (14) then become m H= , 1 − (λ x˙ − x )2 /R2 Ki =
(15)
m(xi − λx˙i ) , R 1 − (λ x˙ − x )2 /R2
(16)
mx˙i Pi = . 1 − (λ x˙ − x )2 /R2
(17)
The mass-shell equation H 2 − Ki2 = m2 now contains only the quantities H and Ki . We now show that the local kinematics in the R space does not differ from the standard relativistic kinematics. We set λ = T + t and R/T ≡ c0 and consider a small space–time neighborhood of the point (T, 0): t T , | x | R. We note that we do not require the relation ( x − λ x˙ )2 R2 to hold. In this case, expressions (15)–(17) become m , H= 1 − x˙ 2 /c20
Ki +
R m(xi − tx˙i ) Pi = , c0 1 − x˙ 2 /c20
mx˙i Pi = . 1 − x˙ 2 /c20
We thus find that a “cosmological” dynamics in a “nonrelativistic” limit locally coincides with the standard relativistic theory with the speed of light c0 . 846
6. The energy in the nonrelativistic noncosmological limit Passing to the approximation r2 ≡ ( x − λ x˙ )2 R2 in (16) and (17), we obtain the standard threedimensional momentum Pi = mx˙ i together with the center-of-mass motion law Ki +(R/c0 )Pi = m(xi −tx˙ i ). But performing such an approximation in (15) results in the relation 4 r m( x − λ x˙ )2 H =m+ + O . (18) 2R2 R4 We thus obtain a more involved expression instead of the standard nonrelativistic kinetic energy. We set λ = T + τ , τ T , and rewrite (18) in the form m x˙ (t x˙ − x ) m(t x˙ − x )2 m x˙ 2 +T + . 2 R 2 If we additionally require that this expression be preserved independently of the choice of the expansion point T , then the quantities HR2 = mR2 + T 2
m x˙ 2 , 2
(19)
W = m x˙ (t x˙ − x ),
(20)
H0 =
I=
m(t x˙ − x )2 2
(21)
must be independently conserved. The conservation of (20) and (21) in addition to the conservation of kinetic energy (19) in nonrelativistic physics follows from the symmetry of the nonrelativistic action for pointlike free particles with respect to the Schr¨ odinger transformation group [16]–[21]. The above procedure is by no means the derivation of these additional conservation laws, but it does demonstrate that studying the particle kinematics in the R space helps in understanding the symmetry nature of nonrelativistic mechanics.
7. Ideal gas in the R space To illustrate the above kinematics, we consider an arbitrary cloud of pointlike free massive particles in the “nonrelativistic” and “noncosmological” limits. If the gas is collisionless or if collisions are absolutely elastic, then the behavior of this cloud is completely governed by twelve conservation laws: H0 =
mn x˙ 2 n
2
n
P =
,
(22)
mn x˙ n ,
(23)
mn ( xn − t x˙ n ) ≡ K(0),
(24)
mn [( xn − t x˙ n ), x˙ n ],
(25)
n
= K
n
J =
n
W =
mn (( xn − t x˙ n ), x˙ n ) ≡ W (0),
(26)
n
I=
n
mn
( xn − t x˙ n )2 ≡ I(0). 2
(27)
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Here, the index n labels different particles. We consider the following time-dependent constructions: K(t) =
mn xn
(28)
n
is the cloud center-of-mass coordinate multiplied by its mass; W (t) =
mn ( xn , x˙ n )
(29)
n
is the correlation function of particle coordinates and velocities (the cloud is contracting when it is negative and is expanding when it is positive); ( xn )2 I(t) = (30) mn 2 n is the trace of the cloud inertia momentum tensor (divided by two); and we propose to call this quantity the “inergy” by analogy with the “energy” term. From (23) and (24), we easily obtain the explicit time dependence for (28): K(t) = K(0) + tP . This relation is the well-known law of motion of the center of mass of the system. Proceeding with (29) analogously, from (22) and (26), we obtain the explicit time dependence of the correlation function: W (t) = W (0) + 2tH0 .
(31)
Finally, using (22) and (31), we obtain the explicit time dependence of the “inergy” from (30): I(t) = I(0) + tW (0) + t2 H0 . The positive definiteness of the quantity I(t) implies the condition W 2 (0) ≤ 4H0 I(0) for the initial absolute value of the correlation function.
8. A covariant distribution of an ideal gas in the R space The generator of translations in the anti-de Sitter space in the Beltrami coordinates obtained in (3) also preserves its form in the R space, and finite translations must therefore have the form r
r − a = , 1 − a r/R2
r⊥
r⊥ 1 − a2 /R2 = , 1 − a r/R2
1 − a2 /R2 t = . 1 − a r/R2
t
(32)
These translations obviously send the point a to the origin and the sphere of radius R to itself We assume that we have an ideal gas, i.e., a set of pointlike particles with equal masses, in the space under consideration. Formulas (15)–(17) imply that the gas is concentrated inside a sphere of radius R at the initial instant. We assume that the gas distribution is homogeneous and isotropic. We easily obtain the spherical symmetry and obtain homogeneity by imposing the condition ρ(r) d3 r = ρ(r ) d3 r . Using this relation and formulas (32), we find the density at the point r: ρ(r) =
ρ0 , (1 − r2 /R2 )2
ρ0 ≡ ρ(0).
The integral of this density obviously diverges when integrating over the entire volume of the ball. 848
From formulas (32), we obtain transformations of particle velocities under translations. To obtain a covariant distribution, it suffices to relate the velocity at an arbitrary point to the velocity at the origin: u (r) = u (0) 1 −
r2 , R2
u⊥ = u⊥ (0).
Assuming that the angular distribution of particles with respect to velocities is symmetric at the origin, we find that the function for the particle distribution with respect to the velocities has the angular dependence ρ(x, θ) =
ρ(0) 1 1 2 2 4π 1 − x2 1 − x sin θ
at the point x ≡ r/R. Analytically studying the time dependence of this function is rather complicated, and we therefore analyze the simplest one-dimensional model. In the interval [−1, 1], we define the distributions of the density ρ(x) = ρ0 /(1 − x2 ) and particle √ velocities u(x) = u0 1 − x2 at the initial instant. We now find the time dependence of the particle density. We let y = x(t) denote the particle position at the time t and x = x(0) denote the initial position of the √ particle. The velocity of this particle is constant and is equal to u = u0 1 − x2 , y = x + u0 t 1 − x2 ,
u0 tx dy = dx 1 − √ . 1 − x2
The particle density changes during the evolution process, √ 1 − x2 dx = ρ(x, 0) √ . ρ(y, t) = ρ(x, 0) dy 1 − x2 − u0 tx
(33)
We consider the density at the origin, i.e., at y = 0. At this point, we have x = −u0 t
1 − x2 ,
ρ(0, 0) ρ(x, 0) = √ . 1 − x2
Substituting these quantities in (33), we obtain an original and nontrivial result: ρ(0, t) = ρ(0, 0).
(34)
That the density at the origin is constant implies that the density is constant in the whole interval. At the same time, the matter flow outside this interval is nonzero. And particle velocities change with time leaving the density constant. For example, the time dependence of the velocity at the origin is u0 . u(t) = 1 + u20 t2 We note that requiring that the initial density distribution be covariant inside the interval is what leads to such a nontrivial result.
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