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Mixed Solutions to the Liouville Equation

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Pith's one-line read The paper constructs explicit classical solutions to the Liouville equation with three singularities whose monodromies are mixed hyperbolic and elliptic, and uses them to give explicit metrics for black-hole/particle two-body systems in…

desk verdict A useful, mostly sound extension of classical Liouville solutions to mixed monodromies; the gravity punchline is plausible but rests on two unproven steps. read the letter →

arxiv 2506.02196 v1 pith:QHFP5WAW submitted 2025-06-02 hep-th gr-qc

Sujay K. Ashok , Jan Troost This is my paper
classification hep-thgr-qc MSC 34M3533C0583C80
keywords Liouvilleequationhyperbolicmonodromyelliptichypergeometricfunctionsthree-dimensionalgravityblackholehorizonssaddlepointapproximationtwo-bodyproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper constructs explicit classical solutions to the Liouville equation with three point singularities when the singularities have mixed hyperbolic and elliptic monodromies, extending the previously known all-hyperbolic and all-elliptic families. The solutions are written as ratios of hypergeometric functions whose phase factors are fixed by requiring the monodromies to lie in SL(2,R) or SU(1,1), so that the Liouville field is single-valued. When these two-dimensional metrics are read as time-symmetric slices of three-dimensional Einstein gravity, a hyperbolic singularity corresponds to a black hole and an elliptic singularity to a point particle; the farthest closed geodesic around each hyperbolic singularity becomes a snapshot of a horizon at its turning point. The paper also derives large-hyperbolic-monodromy limits using uniform saddle point approximations, producing simpler expressions for the metrics and for the horizon curves. A proof of the location of the outermost geodesic is left as an outstanding problem.

What carries the argument

The central objects are auxiliary SL(2,R) doublets $\psi_\pm$ (or SU(1,1) doublets $\xi_\pm$) solving the Fuchsian equation $\partial_z^2\psi + \frac{1}{2} T_\phi\psi = 0$, whose monodromies around the three singularities encode the source parameters. The Liouville field is recovered from the single-valued combination $e^{-\phi/2} = \pm \frac{i}{2}(\psi_-\psi_+ - \psi_+\psi_-)$, or equivalently $e^\phi = 4|\partial f|^2/(1-|f|^2)^2$ with $f = \xi_+/\xi_-$. Because the two independent solutions are hypergeometric functions, every solution is fixed by a ratio of ${}_2F_1$'s together with a phase; the phase is determined by hypergeometric connection formulae that impose SL(2,R) or SU(1,1) valued monodromy. The same machinery carries the asymptotics: uniform saddle point approximations to the hypergeometric integrals produce Airy and Bessel-K representations that simplify the large-$\lambda$ metrics and the farthest geodesics.

What would settle it

Numerically locate the length-minimizing geodesic that loops one hyperbolic singularity in a mixed one-elliptic/two-hyperbolic configuration at moderate $\lambda$ and read off the corresponding level; if it is not $l_g^0=-1$, the claimed alternating pattern and horizon interpretation fail. Alternatively, substitute the time-dependent metric built from $Z = f(\zeta(z,t))$ into Einstein's equations with two particle sources to test directly whether the moving two-body metric is exact.

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Extended reading notes

Core claim

The paper's central claim is that the four classes of three-singularity Liouville solutions — three hyperbolic, one elliptic plus two hyperbolic, two elliptic plus one hyperbolic, and three elliptic — can all be written explicitly as ratios of hypergeometric functions, with the undetermined phase factors fixed by demanding that the monodromies around the singularities belong to SL(2,R) or SU(1,1). In the mixed cases, the monodromy consistency conditions take the form of relations such as $\cosh \pi\lambda_3 = \cot(\eta)\sinh(\pi\lambda_1)\sin(4\pi G m_2) + \cosh(\pi\lambda_1)\cos(4\pi G m_2)$, linking the hyperbolic and elliptic parameters. These ratios provide explicit metrics $ds^2 = e^\phi dz d\bar z$; near a hyperbolic singularity the metric has alternating geodesics and divergent boundaries, and the outermost geodesic in the homotopy class around that singularity is the level $|\rho| = e^{-\pi/2\lambda}$. In the gravitational application, these outermost geodesics are reinterpreted as snapshots of black-hole horizons, so the mixed solutions describe two bodies in which a black hole and a particle coexist, or two particles whose combined monodromy forms a black hole. In the large hyperbolic monodromy limit, uniform saddle point integration expresses the metrics in terms of Airy functions and Bessel functions, making the heavy-black-hole geometries and their horizon curves tractable.

Load-bearing premise

The outermost closed geodesic around a hyperbolic singularity is assumed to lie at the particular level $l_g^0=-1$, that is at $|\rho| = e^{-\pi/2\lambda}$, a fact the paper verifies numerically but does not prove; if that level were different, the horizon snapshots in Section 3 would shift.

Editorial extensions

If this is right

  • The four classes of three-singularity Liouville solutions with mixed monodromies become explicit, so quantities derived from the Liouville field — the metric, geodesics, and actions — can be computed directly for these configurations.
  • In three-dimensional gravity, the mixed hyperbolic-elliptic solutions give explicit metrics for a point particle interacting with a black hole, and for two particles whose center-of-mass monodromy is hyperbolic, describing black-hole formation by collision.
  • The farthest geodesic around each hyperbolic singularity is a horizon snapshot, and the alternating sequence of geodesics and divergent boundaries is the spatial structure of a time-symmetric slice containing black holes.
  • In the large hyperbolic monodromy limit, the metrics reduce to Airy- and Bessel-type special functions, making the horizon curves numerically accessible and providing an analytic handle on very heavy black hole geometries.
  • The monodromy consistency relations give the classical mass and velocity relations that must hold for static configurations of particles and black holes in three dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same monodromy-consistency technique should extend to configurations with four or more singularities, at the cost of higher-order differential equations; mixed hyperbolic and elliptic cases would then describe multi-black-hole, multi-particle spatial slices.
  • The uniform saddle point expansions near coalescing saddle points suggest a refined scaling of horizon positions of order $\lambda^{-2/3}$ near branch points; this scaling could be tested numerically at larger $\lambda$ than those plotted in the paper.
  • The explicit metrics invite a direct extension to spacetimes with angular momentum by allowing iso(2,1) valued monodromies rather than only SO(2,1) valued ones, a direction the paper explicitly leaves open.
  • In the semiclassical limit of Liouville conformal field theory, the mixed-monodromy correlation functions may have saddles described by the same ratios of hypergeometric functions, potentially simplifying the computation of classical actions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper constructs classical solutions to the Liouville equation with three punctures whose monodromies are arbitrary mixtures of hyperbolic and elliptic, expressed as ratios of hypergeometric functions with phases fixed by SL(2,R)/SU(1,1) connection formulae. It then studies the large-hyperbolic-monodromy limit using uniform saddle-point, Airy, and Bessel-function approximations, analyzes the outermost geodesics around hyperbolic singularities, and interprets those geodesics as snapshots of black-hole horizons in 2+1 gravity. The final sections propose explicit metrics for moving two-particle systems in flat spacetime and AdS3 by composing a time-dependent coordinate reparametrization with the static Liouville solution.

Significance. If the construction is valid, the paper genuinely enlarges the set of explicit three-singularity Liouville solutions and provides concrete metrics for mixed particle/black-hole geometries that were previously described only qualitatively in [9,25,27]. The paper is careful and transparent in presenting hypergeometric connection formulae, and the uniform saddle-point expressions for the large-monodromy limits are useful and well illustrated by the figures. The authors also honestly flag the two inputs that are not proved: the numerical value l_g^0 = -1 for the outermost geodesic level and the exactness of the moving two-particle metric. The central claim, however, rests on monodromy consistency checks that are asserted rather than demonstrated for the mixed cases, and the gravitational application contains a load-bearing unverified step; both are fixable within the scope of the manuscript.

major comments (3)
  1. [Appendix A.2, Eqs. (A.12), (A.16) and (2.42)-(2.49)] The central existence claim for mixed-monodromy solutions rests on the chosen phases e^{iv_i} making all monodromies simultaneously SL(2,R) or SU(1,1) valued and satisfying the product identity (A.7). For the all-hyperbolic case, the monodromy matrices and the consistency condition are displayed in Appendix A.1. For the hyperbolic-elliptic-hyperbolic case, however, the text only quotes the resulting matrices (A.12) and (A.16) and states that they 'again satisfy' (A.7); for the elliptic-elliptic-hyperbolic case no monodromy matrices are given at all, and the appendix ends with 'a similar analysis can be carried out.' Since a sign error or a wrong Gamma-function argument would change the monodromy class and invalidate the claimed solution, please include explicit verification (or a concise derivation) of the product identity and of the Wronskian normalizations for both mixed classes.
  2. [Sec. 2.3, Eq. (2.51)-(2.52); Sec. 3.1.3 and Sec. 4.2.2] The farthest-geodesic analysis uses the numerically observed value l_g^0 = -1, as the paper itself stresses in Sec. 2.3 ('Numerically, we find that the outermost geodesic is always at l_g^0 = -1') and again in the Conclusions ('A proof of this claim remains an outstanding problem'). This value determines the limiting curves |rho_i| = e^{-pi/(2 lambda)} used in Sec. 3.1.3 (Eq. (3.27)) and hence the horizon snapshots discussed in Sec. 4.2.2-4.2.3. If the true outermost level were different, those curves would shift and the horizon interpretation would need modification. Because this is load-bearing for the gravitational interpretation, the paper should either supply a proof or perturbative argument for l_g^0 = -1, or explicitly mark the limiting-curve and horizon results as conditional on this numerical input.
  3. [Sec. 4.1.2, Eqs. (4.8)-(4.11); Sec. 4.1.3] The moving two-particle metric is claimed to be exact: after substituting Z = f(zeta(z,t)) into (4.3), the paper states that 'we obtain an entirely explicit metric solution by differentiation and simple algebra.' However, because zeta depends on t, dZ contains a dt term, so the resulting three-dimensional metric has dt dz and dt^2 corrections that are not displayed. The paper also does not verify that the prescribed trajectories xi_{1,2}(t) are geodesics of the full time-dependent metric, which is needed for the energy-momentum tensor to be covariantly conserved. The same issue applies to the AdS3 construction in Sec. 4.1.3. Please provide the full time-dependent metric (or a explicit gauge choice that eliminates the cross terms) and verify the geodesic worldline condition before the two-moving-particle claim can be considered established.
minor comments (5)
  1. [Eq. (2.27)] There is a typographical parenthesis in the denominator: 'Gamma((1 + i lambda_1)' should read 'Gamma(1 + i lambda_1)'.
  2. [Appendix A.2, Eqs. (A.9)-(A.14)] The symbol m_2 appears in several Gamma-function arguments but is never defined in this appendix; it should presumably be alpha_2 or else an explicit complementary mass parameter should be introduced.
  3. [Eq. (3.28)] The phrase 'uni-valued metric function' in Sec. 3.2 should be 'single-valued metric function'.
  4. [Eqs. (2.44) and (2.49)] The square-root phases e^{i epsilon} and e^{i eta} are defined only through their squares; for numerical evaluation and analytic continuation, an explicit branch choice should be stated.
  5. [Figure 12 caption] The caption contains a typographical spacing error ('lambda -> infinityis') and the legend text should be checked for consistency with the plotted curves.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the mixed-monodromy solutions are constrained constructions, and the unproven steps are assumptions or verification gaps, not inputs recycled as outputs.

full rationale

The derivation chain is self-contained. The paper constructs Liouville fields from normalized doublets of hypergeometric solutions, fixing the overall phases e^{iv_i} by the requirement that monodromies around all punctures lie in SL(2,R) or SU(1,1) (e.g., equations (2.27), (2.42), (2.47), and (2.49)). These are constraints on the prefactors, not definitions of the claimed predictions: the resulting ratios (2.43) and (2.48) are then exact solutions if the monodromy consistency holds, and the large-λ metrics in Section 3 are asymptotic evaluations of those same hypergeometric expressions via saddle point methods, not fits to the outcomes they describe. The geodesic level l_g^0=-1 is numerically observed and explicitly flagged as preliminary with a proof listed as an outstanding problem; importing it is an assumption that affects the horizon interpretation, but it is not a circular reduction of the Liouville construction. The bibliography contains no self-citations by the present authors, and the cited results [5,7,8,9,25] are external. The unverified product-identity statement for the mixed monodromy matrices in Appendix A.2 is a genuine verification gap and a correctness risk, but a missing proof is not circularity. No step was found in which a quantity defined in terms of another is then claimed to predict that same quantity, nor any fitted input renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; the monodromy parameters encode known masses and deficit angles. The main unproven inputs are the numerical geodesic level and the moving-particle metric ansatz.

free parameters (1)
  • Outermost geodesic level l_g^0 = -1
    Chosen from numerical observation (Sec. 2.3); not proven. The large-λ geodesic curves and the horizon interpretation depend on this choice.
assumptions (5)
  • domain assumption The auxiliary doublet ψ± or ξ± has monodromy in SL(2,R) or SU(1,1) around each singularity.
    Sec. 2.1 establishes the doublet formalism; this restricts the class of solutions considered.
  • domain assumption The phases e^{iv} fixed by monodromy consistency yield a real, single-valued Liouville field.
    Sec. 2.2 uses connection formulae to fix the phases; the reality of e^{-φ/2} is asserted, not proven in detail.
  • ad hoc to paper The outermost geodesic around a hyperbolic singularity is at level l_g^0 = -1.
    Sec. 2.3 reports this as a numerical observation and calls the pictures preliminary; used in Sec. 3.1.3 to define the farthest geodesics.
  • domain assumption A hyperbolic monodromy corresponds to a black hole singularity and an elliptic monodromy to a point particle in 2+1 gravity.
    Sec. 4.2 imports this dictionary from [8,9,25].
  • ad hoc to paper The time-dependent coordinate transformation Z=f(ζ(z,t)) yields an exact solution of Einstein's equations for moving particles.
    Sec. 4.1.2 asserts this without computing the full metric or checking the Einstein equations including cross terms.

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Cite this review

Pith. "Pith review of Mixed Solutions to the Liouville Equation." pith.science (2026). https://pith.science/paper/QHFP5WAW

@misc{pith2026250602196,
  author       = {Pith},
  title        = {Pith review of: Mixed Solutions to the Liouville Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QHFP5WAW}},
  note         = {Machine review of arXiv:2506.02196}
}
read the original abstract

We enlarge the set of explicit classical solutions to the Liouville equation with three singularities to the cases with mixed hyperbolic and elliptic monodromies. We analyze the large hyperbolic monodromy limit of the solutions and the farthest geodesics looping one hyperbolic singularity. These two-dimensional geometries describe a time-symmetric spatial slice of a solution to three-dimensional general relativity. The geodesics are reinterpreted as snapshots of horizons of evolving black holes. We study the spatial slice with three horizons of very heavy black holes in some detail. We use uniform saddle point integration to present the Liouville and heavy black hole geometries in terms of simpler special functions. These make a detailed analysis of mixed particle and black hole geometries possible.

Figures

Figures reproduced from arXiv: 2506.02196 by the authors.

Figure 1
Figure 1. Alternating geodesics and contours around the singularity at [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Alternating geodesics and contours around the singularities at [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. A sketch of the two-dimensional geometry with three hyperbolic monodromies. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: A sketch of the geometry with one elliptic and two hyperbolic singularities. The [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: We schematically represented a geometry with an elliptic singularity that cuts out [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: A sketch of the geometry with one hyperbolic singularity on the left and two [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: There are two elliptic singularities that cut out a wedge from space. The lines bor [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: The numerical outermost geodesics in the [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: The limit curves are shown on the left. The numerical outermost geodesics at a [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: For the value of λ = 8 and α2 = 0.2, we plot the farthest geodesics around z = 0 and z = ∞ in blue and the limit curve in red. We note that the outermost geodesics deviate away from the elliptic singularity. 22 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: On the left, we have the outermost geodesics drawn for the same value of [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: The farthest geodesics are plotted for the case [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]

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Works this paper leans on

27 extracted references · 18 canonical work pages

  1. [1]

    Sur l’´ equation aux diff´ erences partielles 2log d2λ dudv ± λ 2a2 = 0,

    J. Liouville,“Sur l’´ equation aux diff´ erences partielles 2log d2λ dudv ± λ 2a2 = 0,” Journal de Math´ ematiques Pures et Appliqu´ ees18 (1853), 71

    work page

  2. [2]

    Exact Quantum Three Point Function of Liouville Highest Weight States,

    A. Bilal and J. L. Gervais, “Exact Quantum Three Point Function of Liouville Highest Weight States,” Nucl. Phys. B 305 (1988), 33-68 doi:10.1016/0550-3213(88)90685-2

    work page doi:10.1016/0550-3213(88)90685-2 1988

  3. [3]

    Notes on quantum Liouville theory and quantum gravity,

    N. Seiberg, “Notes on quantum Liouville theory and quantum gravity,” Prog. Theor. Phys. Suppl. 102 (1990), 319-349 doi:10.1143/PTPS.102.319

    work page doi:10.1143/ptps.102.319 1990

  4. [4]

    Structure constants and conformal bootstrap in Liouville field theory,

    A. B. Zamolodchikov and A. B. Zamolodchikov, “Structure constants and conformal bootstrap in Liouville field theory,” Nucl. Phys. B 477 (1996), 577-605 doi:10.1016/0550- 3213(96)00351-3 [arXiv:hep-th/9506136 [hep-th]]

    arXiv 1996

  5. [5]

    Classical Liouville action on the sphere with three hyperbolic singularities

    L. Hadasz and Z. Jaskolski, “Classical Liouville action on the sphere with three hyperbolic singularities,” Nucl. Phys. B 694 (2004), 493-508 doi:10.1016/j.nuclphysb.2004.03.012 [arXiv:hep-th/0309267 [hep-th]]

    work page Pith review arXiv 2004

  6. [6]

    Polyakov conjecture for hyperbolic singularities,

    L. Hadasz and Z. Jaskolski, “Polyakov conjecture for hyperbolic singularities,” Phys. Lett. B 574 (2003), 129-135 doi:10.1016/j.physletb.2003.08.075 [arXiv:hep-th/0308131 [hep-th]]

    arXiv 2003

  7. [7]

    Hyperbolic three-string vertex,

    A. H. Fırat, “Hyperbolic three-string vertex,” Journal of High Energy Physics 08 (2021), 035 doi:10.1007/JHEP08(2021)035 [arXiv:2102.03936 [hep-th]]

    arXiv 2021

  8. [8]

    Explicit solutions for Point Particles and Black Holes in spaces of constant curvature in 2+1-D Gravity

    M. Welling, “Explicit solutions for point particles and black holes in spaces of con- stant curvature in (2+1)-Dimensional gravity,” Nucl. Phys. B 515 (1998), 436-452 doi:10.1016/S0550-3213(98)00008-X [arXiv:hep-th/9706021 [hep-th]]

    work page Pith review arXiv 1998

Show all 27 references
  1. [9]

    Time symmetric initial data for multibody solutions in three-dimensions,

    A. R. Steif, “Time symmetric initial data for multibody solutions in three-dimensions,” Phys. Rev. D 53 (1996), 5527-5532 doi:10.1103/PhysRevD.53.5527 [arXiv:gr-qc/9511053 [gr-qc]]

    1996 arXiv

  2. [10]

    Hyperbolic string vertices,

    K. Costello and B. Zwiebach, “Hyperbolic string vertices,” JHEP 02 (2022), 002 doi:10.1007/JHEP02(2022)002 [arXiv:1909.00033 [hep-th]]. 35

    2022 arXiv

  3. [11]

    Sullo svolgimento del quoziente di due serie ipergeometriche in frazione continua infinita,

    B. Riemann, “Sullo svolgimento del quoziente di due serie ipergeometriche in frazione continua infinita,” Collected Works of Bernhard Riemann (1863), 145-153

  4. [12]

    Asymptotic Expansions of Hypergeometric Functions,

    G. Watson, “Asymptotic Expansions of Hypergeometric Functions,” Transactions of Cambridge Philosophical Society 22 (1918), 277–308

    1918

  5. [13]

    An extension of the method of steepest descents,

    C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descents,” Mathematical Proceedings of the Cambridge Philosophical Society 53 No. 3. Cambridge University Press, 1957

    1957

  6. [14]

    Asymptotics of the hypergeometric function,

    D. Jones, “Asymptotics of the hypergeometric function,” Mathematical methods in the applied sciences 24.6 (2001), 369-389

    2001

  7. [15]

    Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited,

    S. F. Khwaja and A. B. Olde Daalhuis, “Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited,” . Proc R Soc A 469: 20130008. http://dx.doi.org/10.1098/rspa.2013.0008

    2013

  8. [16]

    Asymptotics of the Gauss hypergeometric function with large parameters, II,

    R. Paris, “Asymptotics of the Gauss hypergeometric function with large parameters, II,” Journal of Classical Analysis 3.1 (2013), 1-15

    2013

  9. [17]

    F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010

    2010

  10. [18]

    Three-Dimensional Einstein Gravity: Dynamics of Flat Space,

    S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional Einstein Gravity: Dynamics of Flat Space,” Annals Phys. 152 (1984), 220 doi:10.1016/0003-4916(84)90085-X

    1984 doi

  11. [19]

    Three-Dimensional Cosmological Gravity: Dynamics of Con- stant Curvature,

    S. Deser and R. Jackiw, “Three-Dimensional Cosmological Gravity: Dynamics of Con- stant Curvature,” Annals Phys. 153 (1984), 405-416 doi:10.1016/0003-4916(84)90025-3

    1984 doi

  12. [20]

    Exact quasistatic particle scattering in (2+1) gravity,

    A. Bellini and P. Valtancoli, “Exact quasistatic particle scattering in (2+1) gravity,” Phys. Lett. B 348 (1995), 44-50 doi:10.1016/0370-2693(95)00106-U

    1995 doi

  13. [21]

    (2+1) gravity with moving particles in an in- stantaneous gauge,

    A. Bellini, M. Ciafaloni and P. Valtancoli, “(2+1) gravity with moving particles in an in- stantaneous gauge,” Nucl. Phys. B 454 (1995), 449-464 doi:10.1016/0550-3213(95)00442- U [arXiv:hep-th/9507077 [hep-th]]

    1995 arXiv

  14. [22]

    Nonperturbative particle dynamics in (2+1) gravity,

    A. Bellini, M. Ciafaloni and P. Valtancoli, “Nonperturbative particle dynamics in (2+1) gravity,” Phys. Lett. B 357 (1995), 532-538 doi:10.1016/0370-2693(95)00983-R [arXiv:hep-th/9507076 [hep-th]]

    1995 arXiv

  15. [23]

    Perturbative analysis of the two-body problem in (2+1) - AdS gravity,

    P. Valtancoli, “Perturbative analysis of the two-body problem in (2+1) - AdS gravity,” Int. J. Mod. Phys. A 16 (2001), 2945-2964 doi:10.1142/S0217751X01004232 [arXiv:hep- th/9907173 [hep-th]]

    2001

  16. [24]

    Nonperturbative Two Particle Scattering Amplitudes in (2+1)-Dimensional Quantum Gravity,

    G. ’t Hooft, “Nonperturbative Two Particle Scattering Amplitudes in (2+1)-Dimensional Quantum Gravity,” Commun. Math. Phys. 117 (1988), 685 doi:10.1007/BF01218392

    1988 doi

  17. [25]

    Multi - black hole geometries in (2+1)-dimensional gravity,

    D. R. Brill, “Multi - black hole geometries in (2+1)-dimensional gravity,” Phys. Rev. D 53 (1996), 4133-4176 doi:10.1103/PhysRevD.53.R4133 [arXiv:gr-qc/9511022 [gr-qc]]. 36

    1996 arXiv

  18. [26]

    Geometry of the (2+1) black hole,

    M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the (2+1) black hole,” Phys. Rev. D 48 (1993), 1506-1525 [erratum: Phys. Rev. D 88 (2013), 069902] doi:10.1103/PhysRevD.48.1506 [arXiv:gr-qc/9302012 [gr-qc]]

    1993 arXiv

  19. [27]

    Black hole creation in (2+1)-dimensions,

    H. J. Matschull, “Black hole creation in (2+1)-dimensions,” Class. Quant. Grav. 16 (1999), 1069-1095 doi:10.1088/0264-9381/16/3/032 [arXiv:gr-qc/9809087 [gr-qc]]. 37

    1999 arXiv

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