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Accelerated Motion Under the Influence of Expanding Space CSIRO PUBLISHING www.publish.csiro.au/journals/pasa Publications of the Astronomical Society of Australia, 2010, 27, 15–22 The Adventures of the Rocketeer: Accelerated Motion Under the Influence of Expanding Space∗ Juliana KwanA,D, Geraint F. LewisA, and J. Berian JamesB,C A Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW 2006 B Institute for Astronomy, Royal Observatory, Edinburgh EH9 3HJ, UK C Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark D Corresponding author. Email: [email protected] Received 2009 July 31, accepted 2009 September 8 Abstract: It is well known that interstellar travel is bounded by the finite speed of light, but on very large scales any rocketeer would also need to consider the influence of cosmological expansion on their journey. This paper examines accelerated journeys within the framework of Friedmann–Lemaître–Robertson–Walker universes, illustrating how the duration of a fixed acceleration sharply divides exploration over interstellar and inter- galactic distances. Furthermore, we show how the universal expansion increases the difficulty of intergalactic navigation, with small uncertainties in cosmological parameters resulting in significantly large deviations. This paper also shows that, contrary to simplistic ideas, the motion of any rocketeer is indistinguishable from Newtonian gravity if the acceleration is kept small. Keyword: cosmology: theory 1 Introduction upon the evolution of the energy densities. Furthermore, The discovery of the accelerated expansion of the universe it is shown that although far more realistic than constant (Riess et al. 1998; Perlmutter et al. 1999) poses an inter- acceleration, an initial burst of acceleration followed by esting challenge to future space travellers. In the currently a ‘coasting’ period is of little use if we wish to traverse favoured cosmological model, the future of intergalactic intergalactic distances. We also contribute to the current travel appears grim as the universe becomes dominated debate of whether space really expands (see Lewis et al. by dark energy and interesting structures, such as galaxies (2008) and Abramowicz (2008) for recent developments) and quasars, will have receded beyond the maximum dis- in the context of accelerated motion. tance that the rocketeer can travel to, even after an infinite Several authors have explored unaccelerated paths proper time. The ‘end of cosmology’ will be realised as through a FLRW universe. See, for example, Whiting the rocketeer infers the existence of an ‘island universe’, (2004) and Grøn & Elgarøy (2007) for the general solu- being able to reach fewer and fewer destinations as these tion for free particle motion in FLRW cosmologies as well are accelerated beyond their event horizon (Loeb 2002; as Barnes et al. (2006) who clarify the interpretation of Krauss & Scherrer 2007). these solutions in the context of joining the Hubble flow. In this paper, we consider the path of a rocketeer, However, the treatment of accelerated motion is uncom- who intends to travel as far as possible in a Friedmann– mon, despite its potential for illuminating the nature of Lemaître–Robertson–Walker (FLRW) universe with dif- expanding space as a physical phenomenon or a trick of ferent cosmologies. The nature of the path depends on coordinates. Including the effect of acceleration increases several important features of the cosmology and this the complexity of the problem and indeed have only been is treated in terms of the limiting horizons alluded to discussed at length in the context of cosmology by Rindler by Krauss & Scherrer (2007) and Loeb (2002), and also (1960) and Heyl (2005). Rindler (1960) presented a novel in the use of phase portraits to describe the motion of method for calculating accelerated paths by reducing the the rocketeer. We also address some of the problematic order of the differential equations albeit by increasing the aspects of travelling in an expanding universe, includ- number of coupled equations to solve. Of more relevence, ing the difficulty of achieving a return trip due to the Heyl (2005) considered the problem of how far a rocketeer complicated dependence of the motion of the rocketeer can travel in a human lifespan. The layout of this paper is as follows: Section 2 ∗ presents the necessary background including the equations Research undertaken as part of the Commonwealth Cosmology Ini- tiative (CCI: www.thecci.org), an international collaboration sup- of motion of the rocketeer, while Section 3 addresses the ported by the Australian Research Council. return journey of a rocketeer in a FLRW universe and its © Astronomical Society of Australia 2010 10.1071/AS09050 1323-3580/10/01015 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.58, on 25 Sep 2021 at 06:11:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1071/AS09050 16 J. Kwan et al. implications for the concept of expanding space; Section 4 coordinates, in which the spatial coordinates must be mod- discusses the journey of an intergalactic explorer who ini- ified as well (see Infeld & Schild (1945) and Lewis et al. tially accelerates but then ‘coasts’ at a constant velocity (2007) for a complete derivation). close to the speed of light. Furthermore we also discuss Simply solving Equation 1 is not sufficient to derive the the importance of accelerated motion in the context of motion of the rocketeer; she must also satisfy the following future measurements in extragalactic astronomy. In the constraints from the normalisation of uµ, the orthogonality Appendix, we provide a first-order perturbative solution of uµ and µ and the magnitude of µ (Equations 4, 5 and for the motion of the rocketeer. 6) that are inserted into their initial conditions. These are expressed in the FLRW metric, as follows: 2 Background µ ν = 2 r2 − η2 =− Accelerated motion in general relativity is most conve- gµνu u a(η) (u u ) 1, (4) niently approached by solving the geodesic equation with µ ν 2 r2 t2 2 gµν = a(η) ( − ) = , (5) a four-acceleration term. The equation of motion of the g uµν = urr − utt = 0, (6) rocketeer is given by: µν Duµ duµ dxν dxσ where the rocketeer is restricted to travel in the radial direc- µ ≡ = + µ , (1) tion only. Since this scenario is purely intended to address dτ dτ νσ dτ dτ the issue of inertial observers in expanding space, we have where µ is the four acceleration, uµ is the four-velocity ignored the details on how such a journey might be tech- and τ is the proper time of the rocketeer. The magnitude of nically feasible and the rocketeer is represented by a point the acceleration, , corresponds to the force experienced particle of unspecified mass that is capable of sustaining by the rocketeer. Equation 1 constitutes a set of differ- a constant acceleration for an indefinite period. ential equations that are generally coupled for non-trivial systems once an appropriate geometry has been inserted. 3 Accelerated Motion Solving these is the first step in determining the path of the To begin, we would like to consider the simplest case rocketeer, but for many cases this is analytically intractable of accelerated motion in an expanding universe: constant and we have had to recourse to numerical techniques acceleration in a single direction. For a universe described extensively. We present a pertubative solution (whose purely by special relativity, it is trivial to show that the path accuracy is limited to specific regimes of proper time) in of a rocketeer is a hyperbola of the form: the Appendix, but this is unwieldy and cumbersome. It is also necessary to choose an appropriate means of slicing 1 1 t = sinh(τ); r = cosh(τ). (7) the time coordinate for the FLRW model; in this case it is most expedient to work in conformal coordinates.The path In this instance, constant acceleration means that the three- of the rocketeer asymptotes to a null geodesic, as shown velocity of the rocketeer rapidly approaches the speed of in Heyl (2005), which determines the maximum comov- light and her path is well approximated by that of a photon ing distance that can be travelled in an infinite proper as η = r + 1/, where r is the comoving radial coordinate. time. This is easily visualised when all photon paths trace ◦ A similar analysis may be performed in conformal coor- out straight lines oriented at ±45 , as in special relativity dinates to determine how far the rocketeer can travel after and although this is not true in general for curved space- an infinite time has elasped because her path is bounded times, we can enforce this behaviour through a conformal by a light cone projected from her point of departure. For representation. Note that such a transformation preserves universes with a non-zero cosmological constant, the con- angles at a point and as such paths that are timelike in formal time in the future is finite after an infinite cosmic FLRW coordinates remain so in the conformal geometry time has elapsed1. Heyl (2005) showed that the furthest the and vice versa. Since we are only considering motion in a rocketeer can travel is a comoving distance equal to the radial direction, we can express the FLRW metric: remaining conformal time for that cosmological model, ds2 = a2(η)(dr2 − dη2), (2) which corresponds to travelling for an infinite cosmic time; the transformation is purely for convenience.As cos- where η is the conformal time and a(η) is the scale factor mic time approaches infinity, a universe with a non-zero that governs the expansion of the universe. The conformal cosmological constant will have a scale factor that is well = time is simply a rescaling of the coordinate time (the time approximated by an exponential (and is exact for ,0 1 = measured by a stationary observer in a FLRW metric), and m,0 0).
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