Geodesic motion in a swirling universe: The complete set of solutions

Rogério Capobianco, Betti Hartmann, and Jutta Kunz

Phys. Rev. D 109, 064042 – Published 14 March 2024

Abstract

We study the geodesic motion in a space-time describing a swirling universe. We show that the geodesic equations can be fully decoupled in the Hamilton-Jacobi formalism leading to an additional constant of motion. The analytical solutions to the geodesic equations can be given in terms of elementary and elliptic functions. We also consider a space-time describing a static black hole immersed in a swirling universe. In this case, full separation of variables is not possible, and the geodesic equations have to be solved numerically.

Received 9 January 2024
Accepted 21 February 2024

DOI:https://doi.org/10.1103/PhysRevD.109.064042

Physics Subject Headings (PhySH)

Fluids & classical fields in curved spacetime General relativity General relativity equations & solutions

Particles & FieldsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Rogério Capobianco ¹, Betti Hartmann ², and Jutta Kunz ³

¹Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, São Paulo 13560-970, Brazil
²Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
³Institut für Physik, Carl-von-Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany

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Issue

Vol. 109, Iss. 6 — 15 March 2024

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Images

Figure 1
The ergoregion of the swirling universe solution for different values of the parameter $j$ in the $z − ρ$ plane (left) and rotated around the symmetry axis (right).
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Figure 2
We show examples of orbits for massless particles. The left and middle figures are for $j = 2$ , $L = 0.4$ and $k = 5$ , and $E = 5$ and hence the radial motion has two turning points (see the discussion in the text). The left figure shows the projection of the orbit onto the $x − y$ -, $x − z$ and the $y − z$ plane, respectively, and the middle figure shows the motion in three dimensions. The right figure is for $j = \sqrt{\frac{27}{256}}$ , $L = 1$ and $k = 1$ , and $E = 2$ , such that the only possible motion is when choosing the initial condition as $ρ_{0} = \frac{4}{3}$ . Note that both orbits have $\dot{z} (0) < 0$ , and hence the particle starting at $z (0) = 2$ moves downward toward the turning point $z_{+}$ and then escapes to infinity.
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Figure 3
Examples of orbits for massive particles. The figure on the left shows an orbit from Region 1 with $j = \sqrt{1 / 20}$ , $L = 1$ , $k = 2.2$ and $E = 3$ . The figure in the middle shows an orbit from Region 2 with $j = 0.5$ , $L = 1.2$ , $k = 5$ and $E = 5$ . For these two cases the radial motion is restricted to take place between the turning points. The figure on the right shows an orbit from Region 3 with $j = 1 / 16$ , $L = 2$ , $k = 2$ and $E = 3$ for which we have to choose $ρ (0) = 4 \sqrt{\sqrt{2} - 1}$ such that $ρ$ stays constant throughout the motion.
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Figure 4
We show the ergoregions for the space-time describing a Schwarzschild black hole of mass $M = 1$ immersed in a swirling universe for three different values of $j$ . On the left, we give the projection of the ergoregions onto the $r − θ$ plane, while the right figure shows the ergoregions rotated around the symmetry axis.
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Figure 5
Two timelike orbits sharing the same constants of motion $E = \sqrt{0.93}$ and $L = \sqrt{\frac{1}{0.072}}$ . The initial conditions were chosen such that the positions are the same. Moreover, the initial polar velocity and the initial radial velocity are defined to satisfy the normalization condition for each case. The top left panel (a) shows a bound orbit for $M = 1$ , $j = 4 \times 10^{- 5}$ in the $x − y$ plane (solid orange). For comparison the orbit in the Schwarzschild space-time ( $M = 1$ , $j = 0$ ) is also shown (black dashed). Panel (b) shows the orbit of panel (a) in three dimensions. Panel (c) shows an escape orbit with $M = 1$ and $j = 4 \times 10^{- 4}$ . The colors in (b) and (c), respectively, indicate the location above or below the equatorial plane ( $z = 0$ ), while the surface of the black sphere indicates the horizon of the black hole.
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