CC BY
29
Mathematical
Structures and Modeling 2014. N. 1(29). PP. 38-42
UDC 514.8
ON HYPERBOLIC MOTION IN TWO HOMOGENEOUS SPACE TIMES (RESEARCH ANNOUNCEMENT)
A.V. Levichev1, Doctor of Mathamatics, e-mail: [email protected] O. Simpson2, Student, e-mail: [email protected] B. Vadala-Roth2, Student, e-mail: [email protected]
1Sobolev Institute of Mathematics, Novosibirsk, Russia 2Department of Mathematics and Statistics, Boston University, USA
Abstract. In 1960 W. Rindler generalized the concept of hyperbolic motion to an arbitrary Lorentzian manifold and studied this motion in the case of de Sitter space-time. We specify Rindlers (non-linear) system of differential equations in the case of the Segals compact cosmos D (which is locally isometric to the Einstein static universe), and in the case of the Heraclitian space-time F. This F is the real Lie group U(1,1) with a certain bi-invariant metric on it whereas D is U(2) with a bi-invariant metric on it. In each case, we present a particular solution to the Rindlers system.
Keywords: Lie group, hyperbolic motion, Lorentzian manifold.
1. Motivation and Introduction
This work is partly motivated by publication [7] where W. Rindler says (p.2082) that "in the special theory of relativity the term 'hyperbolic motion' is commonly applied to the motion of a test particle moving with constant proper acceleration along a straight line in a suitable Galilean frame of reference. (Proper acceleration is the acceleration relative to the instantaneous Galilean rest frame.) Hyperbolic motion was first noted by Minkowski [4] and was further studied by Born [2], who also coined its name. This name derives from the fact that the plot of distance against time is a rectangular hyperbola (see equation (9) of [7]). By the same terminology the classical motion with constant acceleration is 'parabolic'."
Let us notice that (as it is obvious from [7, p.2083] calculations) a Galilean frame (from above) is another name for an inertial frame in special relativity theory. W. Rindler generalizes the concept of hyperbolic motion to a general space-time ([7, p.2083]) by which (as it becomes clear from [7, p.2084]) Rindler understands a manifold with Lorentzian metric on it. In his article he only solves the proposed equations (that is, our (1.4) below) for the particular case of de Sitter space-time ([7, p.2085]).
Again, the equations mentioned above are deduced as the ones which describe the motion of a uniformly accelerated (point-like) particle. In order to present these equations let us first describe our notation(s). Let a particle (in a portion of
space-time with coordinates X/ , xx , X/ , ^c 3) have world line
xm = xm(p) (1.1)
where parameter p is the arc length. Velocity U and acceleration A are defined ([7, p.2084]) as follows:
dxnm
Um = dX-, (1.2)
dp
DU m
Am = D^ (1.3)
dp
where the uppercase D is an indication of covariant differentiation. The Rindlers equations ([7, (16) on p.2084]) read as follows:
n Am
DA- = a2Um (1.4)
dp
where a (positive) constant a is the magnitude of the acceleration A given by (1.3).W. Rindler only analyzed these equations for the de Sitter world (which is a well-known example of a homogeneous space-time).
We introduce below two homogeneous space-times, D and F. Each of them can be viewed as a four-dimensional real Lie group equipped with certain bi-invariant metric of Lorentzian signature. In both cases we specify equations (1.4) and present particular solutions.
2. Space-Times D and F
The Lie groups U(2) and U(1,1) can each be defined as the totality of all 2 by 2 matrices Z (with complex entries allowed) which satisfy
Z*sZ = s (2.1)
where s is the unit matrix in case of U(2), and in case of U(1,1) s is the diagonal two by two matrix with entries 1,-1. To make U(2) and U(1,1) space-times, we only need to supply them with metric tensors of Lorentzian signatures: see Section 3.1 (respectively, 3.2) of [3] for these (and more) details on U(2) = D (respectively, U(1,1) = F). At this point in our article, we only need to know that a left-invariant orthonormal basis of vector fields X0,Xi,X2,X3 is chosen on D, and of H0, Hi, H2, H3 on F. In each case, the choice of signature is +, —, —, - . The corresponding metric is bi-invariant (on each of the two groups). The space-time F is known as a tachyonic fluid, [3, Theorem 10] (see more details in that Theorem 10 which justify the Heraclitian world name).
To specify equations (1.4), we will use the following result from [5, Section 8]: in terms of such a basis of vector fields, each Christoffel symbol Gj is nothing but one-half of the structure constant Cj.
Recall that the structure constants Cj are detected from the commutation relations
[Xi,Xj ] = Ck Xk (2.2)
(where summation in k goes from k = 0 to k = 3). The Christoffel symbols Gj are coefficients in the decomposition of the covariant derivative A(Xj) with respect to this very basis of four basic vector fields. Here A(Xj) denotes covariant derivative of vector field Xj w.r.t. vector field X,. Clearly, we have to use vector fields H (rather than Xi) when we deal with space-time F. The commutation tables for the two sets of basic vector fields are given in Section 8 of [3].
We can now think of the vector field U from (1.2) as a vector field on the entire space-time, D or F. Each curve of constant acceleration is thus an integral curve of U. By Uo, Ui, U2, U3 we now understand coordinates of U with respect to the (above chosen) basis on D (or F). By f , f", etc. below we understand the corresponding (ordinary) derivative of a function f(p) on an integral curve.
Proposition 1. In the case of D, the vector field U from (1.4) is a solution of the system U0 = a2Uo,
U' + U2U3 - U2U3 = a2Ui, U' + U1U3 - U1U3 = a2U, U' + U1U2 - U2Ui = a2U. The 'F-system' reads as follows: Uo + U1U2 - U2U1 = a2Uo, U(' + U0U - U2Uo = a2U, U' + U1 Uo - uoU1 = a2U, U' = a2U3.
The proof is omitted: it is a straightforward application of Milnors result [5, Section 8] to Rindlers system (1.4).
In this article we only discuss the following two solutions. From time to time, we use the abbreviations C = cosh(ap), S = sinh(ap).
Proposition 2. The vector field
U = {cosh(ap), sinh(ap), 0, 0} = CXo + SX1 (2.3)
satisfies the D-system, whereas
U = {cosh(ap), 0,0, 0} = CHo (2.4)
satisfies the F-system.
The proof is an easy verification.
Our next goal is to present the corresponding D-curve which (when p = 0) passes through the neutral element of D = U(2). To do so, we use ('excessive') coordinates u-1, uo, u1, u2, u3, u4 on D (see [6, p.92]) rather than deal with matrices defined by (2.1).
Theorem 1. The curve
Z(p) = {cos [a] , sin [a] , sin ] , 0, 0, cos ] } (2.5)
is an integral curve of vector field (2.3). It passes (when p = 0) through the neutral element of D = U(2).
The proof amounts to the direct calculation, and to the careful application of [6, p.92 and p.95] data.
Remark 1. The curve (2.5) is a subset of the 2-dimensional torus T in D: T is defined by equations u_i + u0 = 1, uf + uf = 1, uf = u3 = 0.
To deal with the F-system, let us recall the conformal embedding E of F into D. This E is (indirectly) defined in the proof of [3, Theorem 6]. From that last proof, it follows that an inverse map G is defined on the orbit of the four-dimensional group which is generated by vector fields H0, Hi, Hf, H3 (which are now viewed as vector fields on D). Here we have the orbit of the U(2) neutral element in mind. A directly defined analogue of G is given by formula (3.4) of [1]. Once again, we can use coordinates u_i, uo, ui, uf, u3, u4.
Theorem 2. The image of the curve
Z(p) = {cos [a] , sin [a] , 0, 0, 0,1} (2.6)
under G is an integral curve of vector field (2.4).
The proof is based on how the vector field U = CH0 is expressed in terms of fields X0, Xi, Xf, X3 (see section 3.2 of [3]) and on the conformal embedding E of F into D. Once again, it involves application of [6, p.92 and p.95] data and direct calculation.
More details of the Theorem 2 proof are to be presented elsewhere.
Remark 2. It is of interest to consider more details on the F-system within the space-time F itself. To do so, one can start with an explicit formula for the conformal mapping G from above.
Remark 3. The curve (2.6) is a geodesic in the space-time D but its image under G is not a geodesic in F otherwise it would not be a curve of constant nonzero acceleration in F. In this regard, John Stachel suggested studying how curves of constant acceleration transform when a conformal transformation is applied. Such (and other) questions arise naturally in the scope of his Unimodular Conformal Projective Relativity (UCPR), see [8].
3. Acknowledgments
In regards to the second author, this project was funded (in part) by Boston University UROP. The work of both the second and the third authors was funded (in part) by the BU GUTS 2012-2013 grant.
References
1. Akopyan A.A., Levichev A.V. The Sviderskiy formula and a contribution to Segals chronometry // Mathematical Structures and Modeling. V. 25 (2012). P. 44-51.
2. Born M. Ann. Physik 30, 1 (1909), Sec. 5.
3. Levichev A.V. Pseudo-Hermitian realization of the Minkowski world through the DLF-theory // Physica Scripta. 83 (2011), N. 1. P. 1-9.
4. Minkowski H. Physik. Z. 10, 104 (1909).
5. Milnor J. Curvatures of Left Invariant Metrics on Lie Groups // Advances in math. V. 21 (1976), N. 3. P. 293-329.
6. Paneitz S.M. and Segal I.E. Analysis in space-time bundles I: General considerations and the scalar bundle // Journal of Functional Analysis. V. 47 (1982). P.78-142.
7. Rindler W. Hyperbolic Motion in Curved Space Time // Phys. Rev. 1960. V. 119, Issue 6. P. 2082-2089.
8. Stachel J. Conformal and projective structures in general relativity, Gen. Relativ. &Gravit. (2011) 43: 3399-34-09.
О ГИПЕРБОЛИЧЕСКОМ ДВИЖЕНИИ В ДВУХ ОДНОРОДНЫХ ПРОСТРАНСТВАХ-ВРЕМЕНАХ (АНОНС ИССЛЕДОВАНИЯ)
А.В. Левичев1, с.н.с., д.ф.-м.н., профессор, e-mail: [email protected] О. Симпсон2, студент, e-mail: [email protected] B. Вадала-Рот2, студент, e-mail: [email protected]
1 Институт Математики им. С.Л.Соболева СО РАН, Новосибирск 2Факультет математики и статистики, Бостонский университет, США
Аннотация. В 1960 году В. Риндлер обобщил понятие гиперболического движения для произвольного лоренцева многообразия и изучил это движение в случае пространства-времени де Ситтера. Мы определяем систему (нелинейных) дифференциальных уравнений Риндлера в случае Сигаловского компактного пространства-времени D (которое локально изометрично статической Вселенной Эйнштейна) и в случае Гераклитианского пространства-времени F. При этом пространство-время F является вещественной группой Ли U(1,1) с определённой на ней биинвариантной метрикой, в то время как мир D представляет собой группу Ли U(2) с также определённой на ней биинвариантной метрикой. Для каждого случая представлено частное решение системы Риндлера.
Ключевые слова: группа Ли, гиперболическое движение, лоренцево многообразие.
