Particle dynamics in spherically symmetric electro-vacuum instantons

In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann...

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Published inThe European physical journal. C, Particles and fields Vol. 84; no. 4; pp. 374 - 26
Main Author Garnier, Arthur
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024
Springer
Springer Nature B.V
Springer Verlag (Germany)
SpringerOpen
Subjects
Algorithms
Astronomy
Astrophysics and Cosmology
Black holes
Deflection
Einstein equations
Elementary Particles
Elliptic functions
Equations of motion
Event horizon
Exact solutions
Hadrons
Heavy Ions
Instantons
Mathematics
Measurement Science and Instrumentation
Nuclear Energy
Nuclear Physics
Orbits
Physics
Physics and Astronomy
Quantum Field Theories
Quantum Field Theory
Regular Article - Theoretical Physics
Singularity (mathematics)
Spacetime
String Theory
Trigonometric functions
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ISSN1434-6052
1434-6044
1434-6052
DOI10.1140/epjc/s10052-024-12719-4

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Abstract In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity v > 1 and we give approximate values for their size when v ≳ 1 . For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass’ elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw “shadows” of the first ones, as it was already done before for classical black holes.
AbstractList In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner-Nordström (ERN) metrics, and the Bertotti-Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista-Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity [Formula omitted] and we give approximate values for their size when [Formula omitted] For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass' elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw "shadows" of the first ones, as it was already done before for classical black holes.
Abstract In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity $$v>1$$ v > 1 and we give approximate values for their size when $$v > rsim 1.$$ v ≳ 1 . For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass’ elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw “shadows” of the first ones, as it was already done before for classical black holes.
In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity v>1 and we give approximate values for their size when v≳1. For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass’ elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw “shadows” of the first ones, as it was already done before for classical black holes.
In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity $$v>1$$ v > 1 and we give approximate values for their size when $$v > rsim 1.$$ v ≳ 1 . For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass’ elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw “shadows” of the first ones, as it was already done before for classical black holes.
In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity v > 1 and we give approximate values for their size when v ≳ 1 . For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass’ elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw “shadows” of the first ones, as it was already done before for classical black holes.
In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions: the Euclidean Reissner–Nordström (ERN) metrics, and the Bertotti–Robinson-like (BR) metrics, the latter having constant Kretschmann scalar. First, we derive the motion equations for the ERN spacetime and we generalize the results of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded if and only if it has an event horizon. We also obtain the Weierstrass form of the polar radial motion, providing an efficient tool for numerical computations. We then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime which, in contrast to the Lorentzian background, can be either positive or negative. We observe the presence of a null and a maximal deflection rings for particles with velocity at infinity $$v>1$$ v > 1 and we give approximate values for their size when $$v > rsim 1.$$ v ≳ 1 . For BR spacetimes, we obtain analytic solutions for the radial motion in proper length, involving (hyperbolic) trigonometric functions and we deduce that orbits either exponentially go to the singularity or are periodic. Finally, we apply the previous results and use algorithms related to Weierstrass’ elliptic functions to produce a Python code to plot orbits of the spacetimes ERN and BR, and draw “shadows” of the first ones, as it was already done before for classical black holes.
ArticleNumber 374
Audience Academic
Author Garnier, Arthur
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  organization: Université de Picardie, LAMFA (UMR 7352 du CNRS)
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CitedBy_id crossref_primary_10_1103_PhysRevD_109_104060
crossref_primary_10_1140_epjc_s10052_025_13957_w
crossref_primary_10_1142_S0219887824400346
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  doi: 10.1103/PhysRevD.77.103005
– volume: 59
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  publication-title: Phys. Lett. B
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– year: 2012
  ident: 12719_CR27
  publication-title: Class. Quantum Gravity
  doi: 10.1088/0264-9381/29/6/065016
– volume: 17
  start-page: 213
  year: 2006
  ident: 12719_CR23
  publication-title: Publ. Astron. Dep. Eotvos Univ.
  doi: 10.48550/arXiv.astro-ph/0602427
– year: 2022
  ident: 12719_CR53
  publication-title: Eur. Phys. J. C
  doi: 10.1140/epjc/s10052-022-10054-0
– ident: 12719_CR51
  doi: 10.1007/978-94-007-5410-2
– year: 2009
  ident: 12719_CR17
  publication-title: ApJ
  doi: 10.1088/0004-637X/696/2/1616
– volume: 468
  start-page: 1252
  year: 2012
  ident: 12719_CR5
  publication-title: Proc. R. Soc. A
  doi: 10.1098/rspa.2011.0616
– volume: 31
  start-page: 265
  issue: 1
  year: 1990
  ident: 12719_CR3
  publication-title: J. Differ. Geom.
  doi: 10.4310/jdg/1214444097
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Snippet In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds of such...
Abstract In this paper, we study the geodesic motion in spherically symmetric electro-vacuum Euclidean solutions of the Einstein equation. There are two kinds...
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SubjectTerms Algorithms
Astronomy
Astrophysics and Cosmology
Black holes
Deflection
Einstein equations
Elementary Particles
Elliptic functions
Equations of motion
Event horizon
Exact solutions
Hadrons
Heavy Ions
Instantons
Mathematics
Measurement Science and Instrumentation
Nuclear Energy
Nuclear Physics
Orbits
Physics
Physics and Astronomy
Quantum Field Theories
Quantum Field Theory
Regular Article - Theoretical Physics
Singularity (mathematics)
Spacetime
String Theory
Trigonometric functions
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Title Particle dynamics in spherically symmetric electro-vacuum instantons
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