CoChat
Sign Up
Cited by 40Black Holes and Theoretical PhysicsNoncommutative and Quantum Gravity TheoriesRelativity and Gravitational Theory

The DOI matches its Crossref record; 34 of the 34 references it cites were found in a registry or an index; none of the 38 sources its claims cite is checked for support yet.

Geometrodynamical formulation of two-dimensional dilaton gravity

Open access
Cavaglià, Marco·1999·Physical Review D
DOI 10.1103/physrevd.59.084011
Open this paper in CoChat — free accountOpen PDF

Abstract

Two-dimensional matterless dilaton gravity with an arbitrary dilatonic potential can be discussed in a unitary way, both in the Lagrangian and canonical frameworks, by introducing suitable field redefinitions. The new fields are directly related to the original spacetime geometry and in the canonical picture they generalize the well-known geometrodynamical variables used in the discussion of the Schwarzschild black hole. So the model can be quantized using the techniques developed for the latter case. The resulting quantum theory exhibits the Birkhoff theorem at the quantum level.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

The paper constructs a geometrodynamical formulation of two-dimensional matterless dilaton gravity with arbitrary dilatonic potential by a Lagrangian-level field redefinition to the dilaton and mass, generalizing Kuchař/Varadarajan variables. It quantizes the model, showing the quantum theory reduces to quantum mechanics and exhibits a quantum Birkhoff theorem.

Key findings

  • The general matterless dilaton gravity action, Eq. (2), can be recast via a Bäcklund transformation and a “point transformation” into a new Lagrangian, Eq. (22), with fields \(M\) and \(\phi\); the resulting canonical variables generalize the geometrodynamical variables of Kuchař and Varadarajan to arbitrary \(V(\phi)\).
  • The classical general solution is static and \((0+1)\)-dimensional, depending on a single variable; horizons are determined by \(N(\phi)-M=0\). Examples include CGHS \(V=\text{const}\), Jackiw-Teitelboim \(V=\phi\), and dimensionally reduced 4D spherically symmetric Einstein gravity \(V=1/(2\sqrt{\phi})\); Schwarzschild is recovered with \(4\phi=R^2\).
  • Quantization gives constraints \(\phi'=0\), \(M'=0\); the effective action reduces to \(S_{\rm eff}=\int d\tau\,(m\,dp_m/d\tau-m)\), with Schrödinger equation \(i\partial\Psi/\partial\tau=H_{\rm eff}\Psi\), \(H_{\rm eff}=m\). The Hilbert space coincides with that of the \(0+1\) quantization, which the paper defines as the quantum Birkhoff theorem.
  • In the \(0+1\) approach, the Hermitian mass operator \(m\) is not self-adjoint, while \(m^2\) is self-adjoint with positive eigenvalues; \(m\) or \(m^2\) is the only gauge-invariant observable apart from its conjugate variable.

Methods

  • Theoretical study of two-dimensional matterless dilaton gravity with arbitrary dilatonic potential, using Lagrangian and canonical ADM frameworks; starts from Eq. (2), varies to field equations, and uses the Bäcklund transformation, Eqs. (7)–(8), and Filippov’s local integral of motion.
  • Derives the new Lagrangian, Eq. (22), canonical transformation, Eqs. (33)–(34), Poisson brackets, Eq. (35), falloff conditions, Eqs. (38)–(41), boundary term, Eq. (45), and constraints, Eq. (46).
  • Quantizes via Kuchař’s geometrodynamical method, reduces the system to quantum mechanics, and compares with the \(0+1\) quantization; no empirical data or sample are used.

Limitations

  • Stated: the analysis is matterless; coupling to matter and applications such as black hole thermodynamics and gravitational collapse are left for future work.
  • Stated: the exact falloff behavior depends on the particular potential; local asymptotic structure and singularities depend on the choice of \(V(\phi)\), and continuity of fields and derivatives across horizons is assumed.
  • Stated: previous free-field canonical transformations are not directly related to spacetime geometry, may be pathological or ill-defined, and cannot be inverted without extra boundary variables; the paper argues it is hard to believe more complicated potentials can be handled with free fields. Evident from the text: the work is purely formal, with no experimental or observational tests.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevd.59.084011

Cites

34 of 34 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

0 of 38 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The Bäcklund transformation can be used to find a transformation from the original fields (g,phi) to new fields (X_i), i=1,...,4, where one of the new fields coincides with M.

    “The Ba ¨cklund transformation introduced in the previous section can be used to find a transformation from the original fields, (g ,), to new fields (X i ), iϭ1, . . .,4, where one of the new fields X i coincides with M.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The general solution depends on the single variable phi.

    “Thus the general solution depends on the single variable .”
    Quote in the paperSubstantial claim
  • Findingreasoning

    Since transformation (7),(8) is defined when Eq. (6) holds, Eqs. (9),(10) are equivalent to the original field equations (4),(5) except at the horizon(s).

    “Since the transformation ͑7͒,͑8͒ is defined when Eq. ͑6͒ holds, Eqs.͑9͒,͑10͒ are equivalent to the original field equations ͑4͒,͑5͒ except at the horizon͑s͒.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The fields and their derivatives are continuous across horizons (or at any spacetime point).
  • Findingreasoning

    From ∇M=0, M is a locally conserved quantity.

    “From the second equation we find that M is a locally conserved quantity.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The equation ∇∇phi=0 implies phi is a free D'Alembert field.

    “The first equation implies that is a free ͑D'Alembert͒ field.”
    Quote in the paperSubstantial claim
  • It may state two separate claims as one. 33
  • The quote only partly states this claim. 6
  • Probably not a substantial claim. 12

  • It may state two separate claims as one.

    Findingcitation

    Equations (12)-(14) imply that the general solution of the model is (0+1)-dimensional, i.e., any solution possesses a Killing vector.

    “Equations ͑12͒-͑14͒ imply that the general solution of the model is actually (0ϩ1) dimensional, i.e., that any solution possesses a Killing vector ͓29͔.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [29]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Findingown data

    The transformation (33),(34) defines a canonical map, as the only nonvanishing equal-time Poisson brackets are Eq. (35).

    “After some tedious calculations one can check that the only nonvanishing Poisson brackets at equal time t are ͓ M ͑ t,x ͒, M ͑ t,xЈ͔͒ϭ␦͑xϪxЈ͒, ͓ ¯͑t,x ͒, ¯͑ t,xЈ͔͒ϭ␦͑xϪxЈ͒; ͑35͒ so Eqs.͑33͒,͑34͒ define a canonical map.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one. The quote only partly states this claim.

    Findingreasoning

    Using Eq. (16), the line element (19) reads Eq. (20), which reduces to the standard Schwarzschild solution with the substitution 4phi=R^2.

    “Clearly Eq. ͑20͒ reduces to the standard Schwarzschild solution with the substitution 4ϭR 2 .”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingnone given

    Using the ansatz (19), the dimensional reduction of four-dimensional vacuum Einstein gravity can be obtained.

    “The dimensional reduction of the fourdimensional vacuum Einstein gravity,”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    If the asymptotic region is defined by phi→∞ and V(phi)≈phi^k, then the two-dimensional spacetime is asymptotically flat for phi→∞ if k<1, and has constant curvature for phi→∞ if k=1.

    “Thus the two-dimensional spacetime is asymptotically flat for →ϱ if kϽ1, and has constant curvature for →ϱ if kϭ1.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingreasoning

    By differentiation of Eq. (7), Eqs. (26),(27) are satisfied if Eq. (4) is satisfied because Eq. (4) implies ∇M=0.

    “Further, by differentiation of Eq. ͑7͒ one finds that Eqs.͑26͒,͑27͒ are satisfied if Eq. ͑4͒ is satisfied because Eq. ͑4͒ implies ٌ M ϭ0.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    Using conformal lightcone coordinates, Eqs. (9),(10) integrate to phi=U(u)+V(v), M=M0.

    “Using conformal lightcone coordinates Eqs.͑9͒,͑10͒ can be explicitly integrated.The general solution is ϭU͑u͒ϩV͑v ͒, M ϭM 0 .”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one. The quote only partly states this claim.

    Findingreasoning

    The difference of the Liouville forms reads Eq. (36) with F given by Eq. (37).

    “Finally, the difference of the Liouville forms reads”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The local asymptotic structure of the solution (16) and the existence of singularities depend on the choice of the dilatonic potential.

    “The local asymptotic structure of the solution ͑16͒ and the existence of singularities depend on the choice of the dilatonic potential.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Findingcitation

    The above falloff conditions coincide with those used by Kuchar in Ref. [23].

    “The above falloff conditions coincide with those used by Kuchar ˇin Ref. ͓23͔.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [23]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Findingreasoning

    In this case the super-Hamiltonian and supermomentum fall off as Eq. (42), and the Liouville form Eq. (43) is well defined.

    “In this case the super-Hamiltonian and supermomentum fall off as”
    • Also stated in: § ͑37͒ · plain
      “is well defined.Finally, using Eqs.͑41͒ the second term on the right-hand side of Eq. ͑37͒ vanishes at spatial infinities and the difference of the Liouville forms is an exact form.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    By integration by parts, the action can be written as Eq. (22) with surface term Eq. (23).

    “Finally, by an integration per parts we find”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The converse is true provided Eq. (6) is satisfied; when this condition holds Eq. (26) implies ∇M=0.

    “The converse latter statement is also true provided that Eq. ͑6͒ is satisfied.When this condition holds Eq. ͑26͒ implies ٌ M ϭ0.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one. The quote only partly states this claim.

    Findingreasoning

    Using the new fields, Eq. (4) reads ∇∇phi=0 and ∇M=0.

    “Using the new fields Eq. ͑4͒ reads”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    In two dimensions the Ricci scalar R is a total divergence and can be locally written as Eq. (21).

    “The key of the construction is the observation that in two dimensions the Ricci scalar R is a total divergence and can be locally written as”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingown data

    The two charts are related by transformation Eq. (33), which is invertible with inverse Eq. (34).

    “The two charts (, ,, ) and ( ¯, ¯, M , M ) are related by the transformation”
    • Also stated in: § ͑32͒ · plain
      “The transformation given above is easily invertible.The result is”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Findingreasoning

    Varying Eq. (22) yields Eqs. (25)-(27).

    “Varying Eq. ͑22͒ we find”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingreasoning

    In the Schwarzschild black hole case N(phi)=√phi, the spatial boundaries are at ±∞ and the falloff conditions (38) become Eq. (41).

    “For instance, in the Schwarzschild black hole case N()ϭͱ the spatial boundaries are located at x a ϭϪϱ and x b ϭϩϱ and with a little algebra one can check that the falloff conditions ͑38͒ become”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Findingnone given

    Equation (9) is a direct consequence of the canonical field equations.

    “Equation ͑9͒ is a direct consequence of the canonical field equations.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The stationary states are the eigenfunctions of m and the Hilbert space coincides with the Hilbert space obtained in the 0+1 approach.

    “The stationary states are the eigenfunctions of m and the Hilbert space coincides with the Hilbert space obtained in the 0ϩ1 approach.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The Hermitian operator m in the gauge-fixed, positive-norm Hilbert space is not self-adjoint, while its square is a self-adjoint operator with positive eigenvalues.

    “It turns out that the Hermitian operator m in the gauge-fixed, positive-norm, Hilbert space is not selfadjoint, while its square is a self-adjoint operator with positive eigenvalues.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Findingnone given

    M weakly commutes with the constraints, as expected for a local integral of motion.

    “͑Note that M weakly commutes with the constraints, as expected for a local integral of motion.͒”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The quantization becomes trivial and the Hilbert space is spanned by the eigenvectors of the sole gauge-invariant operator m corresponding to the mass.

    “The quantization becomes trivial and the Hilbert space is spanned by the eigenvectors of the sole-apart from its conjugate momentum-gauge-invariant operator m corresponding to the mass of the system.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The effective Hamiltonian is given by the boundary term (45) and the reduced action reads Eq. (47).

    “The effective Hamiltonian is simply given by the boundary term ͑45͒ and the reduced action reads”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The general dilaton gravity action Eq. (2) can be cast into form (22) and the system can be described both in canonical and Lagrangian frameworks using the dilaton and mass as new variables.

    “We have seen that the general dilaton gravity action, Eq. ͑2͒, can be cast into the form ͑22͒ and the system can be described both in the canonical and Lagrangian frameworks using the dilaton and the mass as new variables.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingcitation

    A careful analysis shows the transformation cannot be inverted and, to make it invertible, one has to supplement the new field variables by an extra pair of conjugate variables related to the value of the fields at the boundary.

    “Indeed, a careful analysis shows that the transformation cannot be inverted and, in order to make it invertible, one has to supplement the new field variables by an extra pair of conjugate variables related to the value of the fields at the boundary.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [18]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [6]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Findingnone given

    Equations (46) have the same physical content as Eq. (10).

    “Equations ͑46͒ have the same physical content of Eq. ͑10͒.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one. The quote only partly states this claim.

    Findingreasoning

    Using the falloff conditions (38), the sole nonvanishing boundary term due to the variation is Eq. (44), so we add boundary term Eq. (45).

    “Using the falloff conditions ͑38͒ we have that the sole nonvanishing boundary term due to the variation of the action with respect to the canonical variables has the form”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingnone given

    The equations of motion are analytically integrable and their solution coincides with the finite gauge transformation generated by the constraint H=0.

    “The equations of motion are analytically integrable and their solution coincides with the finite gauge transformation generated by the constraint H ϭ0.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingnone given

    The mass m-or its square-is the only gauge-invariant observable of the system (apart from the conjugate variable), and the Hilbert space of the 0+1 approach coincides with the Hilbert space of the full quantum 1+1 theory obtained through the geometrodynamical formalism.

    “However, the relevant point is that the mass m-or its square-is the only gauge-invariant observable of the system ͑apart from the conjugate variable, of course͒ and the Hilbert space of the 0ϩ1 approach coincides with the Hilbert space of the full quantum 1ϩ1 theory obtained through the geometrodynamical formalism.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Findingreasoning

    The Schrödinger equation is Eq. (48).

    “The Schro ¨dinger equation is”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    Only the CGHS model has been quantized using the geometrodynamical formalism.

    “Again, only the CGHS model has been quantized using this formalism ͓21͔.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [21]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    Classically one may always choose U(phi)=phi and locally set W(phi)=0 by a Weyl rescaling of the metric.

    “Indeed, classically we may always choose U()ϭ and locally set W()ϭ0 by a Weyl rescaling of the metric ͓2͔.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    In the free-field canonical transformation approach, the new canonical variables are not directly related to the original spacetime geometry and important physical quantities cannot be expressed in terms of the new fields.

    “The main drawback of this approach is that the new canonical variables are not directly related to the original spacetime geometry and important physical quantities cannot be expressed in terms of the new fields ͓18͔.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [18]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Study design and conductreasoning

    By requiring the epsilon functions to go to zero rapidly enough, the Liouville form and the super-Hamiltonian and supermomentum are well defined and F reduces to an exact form.

    “By requiring that ⑀ (a,b) go to zero rapidly enough, both the Liouville form and the super-Hamiltonian and supermomentum are well defined and the difference of the Liouville forms F( ¯, ¯, M , M ) reduces to an exact form.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. It may state two separate claims as one.

    Interpretationnone given

    The new fields are the dilaton and the mass of the system, thus avoiding problems related to their interpretation in terms of geometrical variables.

    “The new fields have clear physical meaning-they are the dilaton and the ''mass'' of the system-thus avoiding problems related to their interpretation in terms of the geometrical variables.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Interpretationcitation

    In the canonical picture the new fields generalize the well-known geometrodynamical variables used in the discussion of the Schwarzschild black hole to a generic dilatonic potential.

    “In the canonical framework the new fields generalize the geometrodynamical variables of Kuchar ˇ͓23͔ and Varadarajan ͓21͔ to a generic dilatonic potential.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [23]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [21]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationcitation

    The quantum theory reduces to quantum mechanics and the Hilbert space coincides with the Hilbert space obtained by quantizing the theory first reducing it to a 0+1 dynamical system and then imposing the quantization algorithm.

    “The quantum theory reduces to quantum mechanics and the Hilbert space coincides with the Hilbert space obtained by quantizing the theory first reducing it to a 0ϩ1 dynamical system with a finite number of degrees of freedom and then imposing the quantization algorithm †see Ref. ͓27͔ for the case Vϭ1/(2ͱ) ‡.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Premisenone given

    It is not clear how to generalize the canonical transformation to free fields for an arbitrary dilatonic potential.

    “Further, it is not clear how to generalize the canonical transformation for an arbitrary dilatonic potential.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. It may be too vague to check against evidence.

    Interpretationreasoning

    This is the essence of the quantum Birkhoff theorem.

    “This is the essence of the quantum Birkhoff theorem.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. It may be too vague to check against evidence.

    Interpretationreasoning

    Thus a quantum theory in which quantum operators have a clear physical meaning is easily achieved.

    “Thus a quantum theory in which quantum operators have a clear physical meaning is easily achieved.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

    Premisenone given

    Our definition of the quantum Birkhoff theorem differs from the definition in previous literature; in previous literature a precise definition is missing.

    “Note that our definition of the quantum Birkhoff theorem differs from the definition that can be found in the previous literature ͓2,19͔.͑Actually, in the previous literature a precise definition of the quantum Birkhoff theorem is missing.͒”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [2]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [19]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    The theory reduces formally to quantum mechanics and the quantization can be carried on as usual.

    “The theory reduces formally to quantum mechanics and the quantization can be carried on as usual.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    The quantization of the general dilaton gravity model becomes straightforward and the resulting quantum theory exhibits at the quantum level the Birkhoff theorem.

    “The quantization of the general dilaton gravity model becomes straightforward and the resulting quantum theory exhibits at the quantum level the Birkhoff theorem.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Premisecitation

    In spite of these difficulties, the CGHS model can still be managed and different approaches to quantization can be carried on, leading to a consistent quantum theory.

    “In spite of these difficulties, the CGHS model can still be managed and different approaches to the quantization can be carried on, leading to a consistent quantum theory ͓5,6,18͔.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [5]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [6]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [18]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

45 of 45

  • The authors state this as an assumption.

    The fields and their derivatives are continuous across horizons (or at any spacetime point).

    “This condition can be lifted if one requires the continuity of the fields and of their derivatives at any spacetime point.”
    “This justifies a posteriori the assumption of continuity made below Eq. ͑6͒.”
1 of 1

Citations

Cites

Flagged first, then by citations

Research works printed but not linked: 6

  1. R. Jackiw, 1997, Nucl. Phys. B (Proc. Suppl.)
  2. M. Cavaglià, 1998, AIP Conf. Proc.
  3. T. Klösch, 1997, Nucl. Phys. B (Proc. Suppl.)
  4. H. Verlinde, 1992, Sixth Marcel Grossmann Meeting on General Relativity
  5. M. Cavaglià, 1967, Atti Acad. Sci. Torino Sci. Fis. Mat. Natur., 131, 65
  6. T. Regge, 1974, Ann. Phys. (N.Y.), 174, 463

Cited by

Sorted by citations

Related

Metadata

Authors
Cavaglià, M.
Affiliations
Max Planck Society (DE)
Corresponding author
Cavaglià, Marco
ORCID
Cavaglià, Marco
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
1999-03-12
Volume
59
Issue
8
ISSN
  • 0556-2821 (ISSN-L)
  • 1089-4918
DOI
10.1103/physrevd.59.084011
OpenAlex
W2013932170
Cited by
40
References
34
Citations by year
2026: 1 · 2025: 4 · 2023: 3 · 2022: 5 · 2021: 2 · 2020: 3 · 2019: 1 · 2018: 2 · 2014: 2
FWCI
3.04
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • MPG.PuRe (Max Planck Society) (submittedVersion, other-oa)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • Max Planck Institute for Plasma Physics (submittedVersion)
  • Max Planck Digital Library (submittedVersion, other-oa)
  • arXiv (Cornell University)
Funders
European Commission
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
Keywords
dilaton gravity, two-dimensional gravity, geometrodynamics, canonical quantization, field redefinitions, Birkhoff's theorem, quantum gravity, Schwarzschild black hole

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

A language model drew the claims from the paper on 2026-11-03, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).

A second model labelled each claim on 2026-11-03 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review