# de Sitter relativity in static charts

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## Abstract

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 \(\varLambda \). These are the Minkowski flat spacetime (with \(\varLambda =0\)), and the hyperbolic spacetimes, de Sitter (dS) with \(\varLambda >0\) and Anti-de Sitter (AdS) having \(\varLambda <0\) [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, 4, 5, 6] or for describing the classical relative geodesic motion [7, 8, 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 \(\varLambda <0\) 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 \(\varLambda >0\) and positive constant curvature. This is a hyperboloid of radius \(R=\frac{1}{{{\omega }}} = \sqrt{\frac{3}{\varLambda }}\) embedded in the \((1\,{+}\,4)\)-dimensional pseudo-Euclidean spacetime \((M^5,\eta ^5)\) of Cartesian coordinates \(z^A\) (labeled by the indices \(A,\,B,\ldots = 0,1,2,3,4\)) and metric \(\eta ^5=\mathrm{diag}(1,-1,-1,-1,-1)\). These coordinates are global, corresponding to the pseudo-orthonormal basis \(\{\nu _A\}\) of the frame into consideration, whose unit vectors satisfy \(\nu _A\cdot \nu _B=\eta ^5_{AB}\). Any point \(z\in M^5\) is represented by the five-dimensional vector \(z=\nu _A z^A=(z^0,z^1,z^2,z^3,z^4)^T\), which transforms linearly under the gauge group *SO*(1, 4) which leave the metric \(\eta ^5\) invariant.

*M*,

*g*) giving the set of functions \(z^A(x)\) which solve the hyperboloid equation,

## 3 Conserved quantities

*SO*(1, 4) whose transformations leave invariant the metric \(\eta ^5\) of the embedding manifold \(M^5\) and implicitly Eq. (1). For this group we adopt the canonical parametrization,

*so*(1, 4) algebra carried by \(M^5\) having the matrix elements,

The *so*(1, 4) basis-generators with an obvious physical meaning [6, 7, 8] are the energy \({\mathfrak {H}}={{\omega }}{\mathfrak {S}}_{04}\), angular momentum \({\mathfrak {J}}_k=\frac{1}{2}\varepsilon _{kij}{\mathfrak {S}}_{ij}\), Lorentz boosts \({\mathfrak {K}}_i={\mathfrak {S}}_{0i}\), and the Runge–Lenz-type vector \({\mathfrak {R}}_i={\mathfrak {S}}_{i4}\). In addition, we consider the momentum \({\mathfrak {P}}_i=-{{\omega }}({\mathfrak {R}}_i+{\mathfrak {K}}_i)\) and its dual \({\mathfrak {Q}}_i={{\omega }}({\mathfrak {K}}_i-{\mathfrak {R}}_i)\), which are nilpotent matrices of two Abelian three-dimensional subalgebras [6].

*SO*(1, 4) isometries which are defined (up to a multiplicative constant) as [4],

*m*and momentum \(\mathbf {P}\) have the general form

*SO*(3) vectors having the components,

Notice that all the conserved quantities carrying space indices (*i*, *j*, ...) transform alike under rotations as *SO*(3) vectors or tensors. Moreover, the condition \(z^i\propto x^i\) fixes the same (common) three-dimensional basis \(\{\mathbf {\nu }_1,\mathbf {\nu }_2,\mathbf {\nu }_3\}\) in both the Cartesian charts, of \(M^5\) and *M*. This means that the *SO*(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.

*so*(1, 4) algebra. In the flat limit, \({{\omega }}\rightarrow 0\), when \(\mathbf{Q} \rightarrow \mathbf{P}\), this identity becomes just the usual mass-shell condition \(p^2=m^2\) 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].

## 4 Timelike geodesics

*E*and

*L*as well as on \(t_0\) and \(\phi _0\). In the simpler case of \(\phi _0=0\) their non-vanishing components read

*m*at \(t=0\) (Fig. 1) such that for \(t_0=0\) this vector lays over the semi major axis being orthogonal on \(\frac{\mathbf {R}}{E}\) (Fig. 2). It is remarkable that for any \(t_0\) the vector \(\mathbf {P}\) is oriented along the lower asymptote, while \(\mathbf {Q}\) gives the direction of the upper one (as in Figs. 1 and 2). Thus the vector \(\mathbf {Q}\), whose role in comoving charts was rather unclear [10], now gets a precise physical meaning.

All these properties are independent on the value of \(\phi _0\) which gives only the rotation of the major axis in the plane \((\mathbf {\nu }_1,\mathbf {\nu }_2)\). Nevertheless, in the appendix we give the general form of all these conserved quantities calculated for an arbitrary \(\phi _0\not =0\).

## 5 Relativity

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 \(\partial _t\) [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 \(\mathbf {L}=0\)) in a given direction determined by its conserved momentum \(\mathbf {P}\) 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 *SO*(1, 4) group. Therefore, we set the synchronization condition at \(t=t'=0\) when \(\mathbf {x}(0)={\mathbf {x}}'(0)=0\) such that the origins of both frames, *O* and \(O'\), overlap the point \(z_o=(0,0,0,0,{{{\omega }}}^{-1})^T \in M^5\), which was the fixed point in constructing the dS manifold as the space of left cosets \(SO(1,4)/ L^{\uparrow }_{+}\) where the Lorentz group \(L^{\uparrow }_{+}\) is the stable group of \(z_o\) [10].

*SO*(1, 4) transformation generating the Lorentzian isometry between the frames \(O'\) and

*O*. This has the form [10]

*E*and \({P}^i\) are the components of the energy-momentum four-vector of the carrier particle when this is passing through the origin of the fixed frame.

*m*of the carrier particle. This can be done by changing the parametrization of \({\mathfrak {g}}(\mathbf {P})\), setting

### 5.2 Simple relativistic effects

*O*and guarantees that after this transformation we obtain well-defined Cartesian coordinates that satisfy the condition \(|\mathbf {x}(t',\mathbf {x}')|\le \frac{1}{{{\omega }}}\) imposed by the existence of the cosmological horizon. For the inverse Lorentzian isometry, we obtain a similar condition defining the domain \({\mathscr {D}}\) of this transformation.

*A*of arbitrary position vector \(\mathbf{a}\), fixed rigidly to the mobile frame \(O'\). Then we may write the general relations

*O*and \(O'\).

*A*indicating \(\delta t'\) without changing its position such that \(\delta x^{\prime \, i}=0\). Then, after a little calculation, we obtain the time dilation observed by

*O*, \(\delta t(t)=\delta t'\, \tilde{\gamma }(t)\), given by the function

*O*increases to infinity as

*O*sees how the clock in \(O'\) lats more and more such that

*t*tends to infinity when \(t'\) is approaching to \(t_m\). 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 \(\tilde{\gamma }(t)\) for different norms \(a=|\mathbf{a}|\) of the position vector \(\mathbf{a}\) oriented parallel with \(\mathbf{V}\). Other interesting and attractive conjectures may be studied numerically starting with the above presented approach.

## 6 Remark on the dS–AdS symmetry

Concluding we can say that the dS relativity in the conformal 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 \({{\omega }}\rightarrow 0\)) 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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