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REVIEW 4 major objections 6 minor 56 references

Relativistic Fluid Dynamics in Curved Spacetime: a Novel Effective Hamiltonian Approach

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proposes a relativistic Hamiltonian formulation of fluid dynamics in curved spacetime, with dynamics cast in coordinate time, yielding generalized continuity and Euler equations and, in Schwarzschild spacetime, exactly solvable…

desk verdict The flat-space limit already breaks the central claim; the Hamiltonian idea is worth a look, but the mapping is wrong and the stability analysis is circular. read the letter →

arxiv 2507.16444 v1 pith:OZDVFALZ submitted 2025-07-22 gr-qc hep-th

Arpan Krishna Mitra , Subir Ghosh This is my paper · ORCID
classification gr-qchep-th MSC 83C5576Y0583C57
keywords relativisticfluiddynamicsHamiltonianformulationEuleriandescriptioncoordinatetimeSchwarzschildspacetimegeneralizedvorticityperturbationstabilityblackholeaccretion
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 tries to establish that the familiar Hamiltonian structure of non-relativistic fluid mechanics—where density and velocity fields evolve under Poisson brackets and a Hamiltonian—can be lifted to relativistic fluids in curved spacetimes without abandoning coordinate time. The authors construct a relativistic Eulerian Poisson algebra and a fluid Hamiltonian, and derive generalized continuity and Euler equations that hold for generic static metrics. Specializing to Schwarzschild spacetime, they obtain exact stationary radial inflow and outflow solutions whose coordinate velocity freezes at the horizon, and they show that first-order density and velocity perturbations decay exponentially in time. If correct, this offers a first-principles Hamiltonian route to studying fluid behavior near black holes and other strong-field astrophysical settings, with the observable dynamics expressed from the viewpoint of a static observer at fixed position.

What carries the argument

The key machinery is the Eulerian map from Lagrangian particle variables to fields, $\rho(\vec r,t) = \rho_0\int d^3x\, \delta^3(\vec X(\vec x,t)-\vec r)$ and $j^i = \rho U^i$, together with the relativistic Poisson bracket algebra derived from $\{X^\mu, U^\nu\} = (\Phi/m)\,g^{\mu\nu}$ and $\{U^\alpha, U^\beta\}$ containing metric derivatives. The Hamiltonian $H = \int d^3r\, \rho\left(g_{0i}g^{i\alpha}U^\alpha/g_{00} + \sqrt{\Psi}\right)$ generates the equations of motion, and the generalized vorticity $\Omega^{ij} = g^{ik}\partial_k U^j - g^{jk}\partial_k U^i$ organizes the coupling terms in the Euler equation, playing no role in the spherically symmetric background but emerging at first order in perturbations to drive angular dynamics.

What would settle it

Compute the standard covariant conservation law for dust in Schwarzschild coordinates, $\partial_t(\sqrt{-g}\,\rho U^t) + \partial_r(\sqrt{-g}\,\rho U^r) = 0$ with $\sqrt{-g} = r^2\sin\theta$, for a stationary radial flow and compare the resulting density profile with the paper's equation $\partial_r(f\rho_0 U_0)=0$ and the derived velocity profile; any discrepancy would show that the proposed continuity equation does not reproduce relativistic particle conservation.

Watch

Extended reading notes

Core claim

The central discovery is a relativistic extension of the Eulerian Hamiltonian formalism: a Poisson bracket algebra for the density field $\rho(\vec r,t)$ and velocity field $U^i(\vec r,t)$, together with a Hamiltonian, such that Hamilton's equations reproduce relativistic continuity and Euler equations in coordinate time. The algebra is obtained by mapping canonical brackets of Lagrangian variables $(X^\mu, U^\nu)$ with metric-dependent brackets $\{X^\mu, U^\nu\} = (\Phi/m)\,g^{\mu\nu}$, where $\Phi\approx\sqrt{-g_{00}}$, into field brackets. The proposed Hamiltonian is $H = \int d^3r\, \rho\left[g_{0i}g^{i\alpha}U^\alpha/g_{00} + \sqrt{\Psi}\right]$ with $\Psi$ built from the spatial metric and velocities, which reduces to the non-relativistic kinetic-plus-potential form in the flat limit. For a diagonal static metric the continuity equation becomes $\dot\rho + \partial_i\left((-g_{00})\rho U^i/\sqrt{1+Z}\right)=0$ with $Z = g_{ij}U^iU^j$, and the Euler equation acquires a generalized vorticity term $\Omega^{ij} = g^{ik}\partial_k U^j - g^{jk}\partial_k U^i$. Specializing to Schwarzschild, stationary radial background flows satisfy $U_0(r) = \pm\sqrt{(1-B/r)(-1+e^{B/r})}$ and $f\rho_0 U_0 = \text{constant}$, and first-order perturbations about this background decay exponentially in coordinate time, indicating linear stability.

Load-bearing premise

The derivation assumes that the Eulerian density $\rho(\vec r,t) = \rho_0\int d^3x\,\delta^3(\vec X - \vec r)$, with no metric volume element or $\gamma$ factor, together with the particle Hamiltonian $H = -P_0$ evaluated under $\Phi\approx\sqrt{-g_{00}}$, correctly encodes relativistic matter so that the resulting continuity and Euler equations are the true relativistic equations in curved spacetime.

Editorial extensions

If this is right

  • If the framework is correct, the derived bracket algebra and Hamiltonian constitute a new, first-principles Hamiltonian description of relativistic ideal fluids in curved spacetimes, valid for any static metric and reducing to the standard non-relativistic equations in the flat limit.
  • The coordinate-time continuity equation $\dot\rho + \partial_i\left((-g_{00})\rho U^i/\sqrt{1+Z}\right)=0$ provides a concrete prescription for evolving fluid density and velocity from the viewpoint of a fixed Eulerian observer, which the authors argue is better aligned with astrophysical observations than proper-time covariant formulations.
  • The exact stationary radial flow solutions in Schwarzschild, with $U_0$ vanishing at the horizon and $f\rho_0 U_0 = \text{constant}$, give explicit background profiles that can serve as starting points for studying black-hole accretion in this Hamiltonian language.
  • The linear stability result—exponential decay of first-order radial and angular perturbations—implies that the spherically symmetric radial background flow is stable under small perturbations, a property that would support the use of these backgrounds in astrophysical modeling.
  • The generation of vorticity at linear order over an irrotational background indicates that angular motion can be induced purely by perturbations, a feature that could matter for understanding the onset of non-radial structure in accretion flows.

Reading between the lines

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

  • The paper does not check whether its continuity equation matches the covariant conservation law $\nabla_\mu(\rho U^\mu)=0$ when $\rho$ is interpreted as rest-mass density; a direct comparison with the standard result would either validate the mapping or show that the density field needs a $\sqrt{-g}$ factor, and this check is a natural next step.
  • The vanishing of the coordinate velocity at the horizon is an expected coordinate-time artifact (matter takes infinite coordinate time to reach the horizon), so the framework's predictions about the horizon region should be interpreted with proper-time physics in mind.
  • A testable extension is to verify whether the full Poisson algebra with $\Phi = \sqrt{-g_{00}(1+Z)}$ and the complete Hamiltonian $\sqrt{\Psi}$ satisfy the Jacobi identities; if they do, the low-velocity truncation used in the paper could be relaxed to obtain higher-order relativistic corrections.
  • The formalism's explicit dependence on the metric components suggests a direct route toward rotating (Kerr) spacetimes through the $g_{0i}$ terms in the Hamiltonian, as the authors note in their outlook, and such an extension could be tested against known Kerr accretion solutions.
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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

4 major / 6 minor

Summary. The paper proposes an Eulerian Hamiltonian formulation of relativistic fluid dynamics in curved spacetime, in which the fluid is described by a density ρ and a coordinate three-velocity U^i evolving in coordinate (Eulerian) time. The authors posit a relativistic Poisson algebra, Eqs. (16)-(18), and a Hamiltonian, Eq. (20), and claim that these yield generalized continuity and Euler equations, Eqs. (21)-(23), valid for generic stationary metrics. They then specialize to the Schwarzschild metric, derive stationary radial background flows, Eqs. (35)-(37), and study first-order perturbations, concluding that the background flow is linearly stable and that vorticity is generated at linear order. The paper presents explicit plots of the background profiles and perturbation eigenfunctions and argues that the coordinate-time description is advantageous for astrophysical applications near horizons.

Significance. If correct, the framework would provide a Hamiltonian route to relativistic fluid equations in strong-field gravity using only coordinate-time variables, potentially useful for accretion modeling and numerical relativity. The paper does contain a concrete construction: an explicit Poisson algebra, an explicit Hamiltonian, an exact stationary radial background solution in Schwarzschild spacetime, and a perturbation analysis with plotted solutions. These are testable and clearly presented. However, the central claim of validity in generic curved spacetimes is not supported: the density mapping is carried over from the non-relativistic theory without metric volume or Lorentz factors, the derived continuity equation disagrees with standard special-relativistic conservation already in flat space, and the derivation of the central equations is not shown. These are load-bearing issues that affect every subsequent result, including the background solution and the stability conclusion. As it stands, the paper cannot be accepted as a reliable derivation of relativistic fluid dynamics.

major comments (4)
  1. [§2, Eq. (6), and §3, Eqs. (21)-(24)] The Eulerian mapping ρ(r,t)=ρ0∫d^3x δ(X(x,t)-r) and j^i=ρ0∫d^3x U^i δ(...) is used unchanged in curved spacetime, without a √{-g} volume element or a Lorentz factor. This cannot represent the time component of a conserved particle current, since covariant conservation requires ∂_μ(√{-g} n U^μ)=0. The inconsistency is already visible in the flat-space limit: from the bracket (18) and Hamiltonian (20), Hamilton's equation gives ∂_t ρ = -∂_i(ρ U^i/√(1+U^2)), whereas Eq. (24) states a different flux structure. The paper never defines which physical quantity ρ denotes or verifies the equations against standard special-relativistic continuity, so the claim that Eqs. (24)-(27) are the relativistic fluid equations is unsupported.
  2. [§3, Eqs. (21)-(23)] The central equations of motion are introduced after the phrase "after a long and tedious computation" with no derivation shown. These equations are the foundation for all later results, but the paper provides neither the intermediate steps nor an independent consistency check, such as verifying Jacobi identities for the proposed algebra or demonstrating that the equations reduce to a known relativistic limit beyond the non-relativistic case. Without a verifiable derivation, Eqs. (21)-(23) cannot be considered established.
  3. [§3, Eq. (36)] The steady-state continuity equation for the spherically symmetric background, Eq. (33), contains a factor r² inside the radial derivative: ∂_r(r² f ρ U0)=0. However, Eq. (36) drops this factor and asserts ∂_r(f ρ0U0)=0, leading to ρ0=χ/(fU0). The correct stationary solution of the paper's own continuity equation would satisfy r² f ρ0 U0 = constant, not fρ0U0=χ. Consequently, the background density profiles in Figure 3 and all subsequent perturbation results are not solutions of the equations the paper itself writes down.
  4. [§4, Eqs. (46)-(48) and stability claim (I)] The perturbations are written as u(r,t)=h(r)e^{-κt} and ω(r,t)=g(r)e^{-λt}, with κ and λ appearing as free constants. The paper then concludes from the exponential time dependence that the perturbations decay and the background is stable. This is circular: a positive κ would give growth, and the analysis never determines the admissible signs or ranges of κ and λ from the dynamical equations. A genuine stability statement requires a dispersion relation or an eigenvalue problem that fixes these constants; the present treatment assumes the decay it claims to establish.
minor comments (6)
  1. [§2, Eqs. (16) and (18)] There are mismatched braces in the displayed brackets, e.g., {j^i(r), ρ(r′} and {U^i(r), ρ(r′}, and the text refers to "the PB (41)" although no Eq. (41) exists; the reference should be to Eq. (3) or to the corresponding equation in the text.
  2. [Introduction] The phrase "spaceL time broken-up scenario" appears to be a typographical error for "spacetime broken-up scenario."
  3. [§4] The symbol ω is used both for the generalized vorticity components in Eq. (45) and for the angular perturbation satisfying Eq. (44), which makes the stability subsection unnecessarily confusing.
  4. [§2, Eq. (20)] The notation √Ψ is used without a clear definition of Ψ; it appears to be defined implicitly by the second line of Eq. (20), but the same symbol is also used in Eq. (19) with a different meaning.
  5. [References] Reference [15] contains stray bracketed items, including "[12] C.Eckart" and "[13]", which appear to be remnants of an earlier draft; these should be cleaned up.
  6. [Figures] Figure captions are incomplete, and the figures are not described with enough quantitative detail; for example, Figure 3's caption ends with "for" and gives no axis labels or parameter values.

Circularity Check

1 steps flagged · score 6.0 of 10

The linear-stability result is assumed rather than derived: perturbations are written as e^{-κt} with arbitrary κ, and the same exponential decay is then reported as the stability conclusion.

  1. self definitional [Section 4, 'Radial perturbation u(r,t)' and 'Angular perturbation ω(r,t)', item (I), Eqs. (42)-(48).]
    "Invoking separation of t and r variables, u(r, t) is expressed as u(r, t) = h(r) exp(−κt) with h(r) satisfying ... It is worthwhile to mention two important aspects of our analysis: (I) The perturbations are exponentially decaying in time, indicating that the stationary, spherically symmetric background flow considered by us is stable."

    The stability claim is exactly the assumed time dependence. The separation constant κ is not determined by any dispersion relation or boundary-value problem: Eq. (42) is a first-order advection PDE, so for any chosen κ (positive, negative, or complex) one can find an h(r) satisfying Eq. (46). Choosing κ with positive real part, or simply writing e^{−κt} with κ positive, and then reporting 'exponentially decaying' restates the ansatz as the conclusion. No independent mode analysis establishes Re κ > 0 for all perturbations, and no restriction on κ is derived from the equations. The same applies to the angular perturbation ω = g(r)e^{−λt}. Thus the advertised linear stability is an input of the calculation, not a derived result.

full rationale

Most of the paper's Hamiltonian construction is not circular: the Poisson algebra and Hamiltonian are posited, and the continuity and Euler equations are computed from them; this is a constructive model-building step rather than a fit. The stationary background solution is obtained by solving an ODE with a boundary condition, which is legitimate. However, the paper's stability claim — a headline result — reduces to its own ansatz: perturbations are written as e^{−κt}, and the same exponential decay is then reported as 'stability,' with κ unconstrained and no mode analysis. This is a genuine circular step in a central advertised result, so the overall score is 6. The separate concern that the density mapping in Eq. (6) omits metric volume and γ factors, and that the resulting equations are not checked against covariant conservation, is a correctness and validation issue rather than a circularity, and is not counted here.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

No data are fitted, but the framework rests on an unvalidated map from Lagrangian to Eulerian variables in curved spacetime, a low-velocity truncation, and a posited Hamiltonian. The perturbation decay rates are free parameters, which is the main reason the stability claim is unsupported.

free parameters (2)
  • perturbation decay rates kappa and lambda = undetermined; any value solves the separated ODEs
    In Eqs. (46) and (48), the ODEs fix the spatial profiles h(r) and g(r) for arbitrary kappa and lambda. No eigenvalue condition or boundary condition determines their values, so the exponential decay used to claim stability is an input, not a prediction.
  • background density normalization chi = arbitrary integration constant from Eq. (36)
    The stationary density profile rho0 = chi / (f U0) carries an arbitrary integration constant that sets the overall density scale. It does not affect velocities or stability but is a chosen parameter.
assumptions (3)
  • ad hoc to paper The Eulerian map rho(r,t) = rho0 integral d^3x delta(X(x,t)-r) and j^i = rho0 integral d^3x U^i delta(...) remains valid in curved spacetime without metric determinant factors.
    Used in Section 2 to define the fields. If sqrt(-g) or gamma factors are required for a covariant fluid density, the subsequent continuity equation is not the relativistic one.
  • domain assumption The low-velocity approximation Phi approximately sqrt(-g00) and the truncation to O(U^2) preserve the validity of the derived equations and the stationary solutions.
    Introduced in Section 2 and used in the working equations (27) of Section 3. The paper still calls the stationary solutions exact, even though they solve a truncated system.
  • ad hoc to paper The canonical particle Hamiltonian H = -P0 with P_mu = (m/Phi) g_mu nu U^nu and U0 = 1 correctly represents the relativistic fluid Hamiltonian after field mapping.
    Section 2, Eqs. (19) and (20). This identification is posited and not validated against known relativistic fluid action principles or conservation laws.

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Cite this review

Pith. "Pith review of Relativistic Fluid Dynamics in Curved Spacetime: a Novel Effective Hamiltonian Approach." pith.science (2026). https://pith.science/paper/OZDVFALZ

@misc{pith2026250716444,
  author       = {Pith},
  title        = {Pith review of: Relativistic Fluid Dynamics in Curved Spacetime: a Novel Effective Hamiltonian Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZDVFALZ}},
  note         = {Machine review of arXiv:2507.16444}
}
abstract

We present a comprehensive Eulerian (Hamiltonian) framework for relativistic fluid dynamics in curved spacetimes, with emphasis on Schwarzschild geometry. The key innovation lies in the consistent use of density and three-velocity fields, all defined in coordinate (Newtonian) time, while fully incorporating relativistic and curvature effects. {\it We stress that the entire dynamics is developed in coordinate (Eulerian) time, from the viewpoint of a static Eulerian observer.} In the non-relativistic regime, it is well known that Poisson brackets between the density $\rho(\vec{x}, t)$ and velocity fields $v^i(\vec{x}, t)$, together with an appropriate Hamiltonian, yield the continuity and Euler equations as Hamiltonian equations of motion. These field-theoretic Poisson brackets can be derived from canonical phase-space brackets defined on Lagrangian variables $(x_i, p_j)$, mapped to Eulerian fields $(\rho, v^i)$. We extend this framework to curved spacetime by constructing a relativistic version of the Eulerian Poisson algebra and proposing a corresponding Hamiltonian. This yields relativistic continuity and Euler equations valid in generic spacetimes, with evolution in coordinate time. Applying this to the Schwarzschild metric, we obtain exact stationary background solutions for radial flows and perform a perturbative stability analysis. We also introduce a generalised vorticity and examine its effect on perturbations over irrotational backgrounds, revealing its role in flow stability. This work offers a unified, first-principles Hamiltonian formulation of relativistic fluid dynamics in curved geometries, forming a foundation for further study of astrophysical and cosmological fluid phenomena.

Figures

Figures reproduced from arXiv: 2507.16444 by the authors.

Figure 1
Figure 1. Outward stationary background flow velocity [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Inward stationary background flow velocity [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Inward/Outward stationary background flow density [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Spatial part of radial perturbation u = h(r) exp(−κt) vs r or more explicitly h(r) = C exp(Z 1 U0(r) " k f(r) − 1 2U0(r)  B r 2 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Spatial part of angular perturbation ω(r) = R(r)exp(−λt) [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Spatial part of perturbation radial and angular velocity [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Spatial part of total velocity perturbation [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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Works this paper leans on

56 extracted references · 51 canonical work pages

  1. [1]

    Landau and E.M

    L.D. Landau and E.M. Lifshitz, Fluid Mechanics, 2nd edition, Course of Theoretical Physics, Vol. 6 (Pergamon Press, Oxford, 1987)

    work page 1987

  2. [2]

    Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (John Wiley & Sons, 1972)

    S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (John Wiley & Sons, 1972)

    work page 1972

  3. [3]

    The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid,

    C. Eckart, “The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid,” Phys. Rev. 58, 919 (1940)

    work page 1940

  4. [4]

    Action functionals for relativistic perfect fluids,

    J.D. Brown, “Action functionals for relativistic perfect fluids,” Class. Quantum Grav. 10, 1579 (1993). 17

    work page 1993

  5. [5]

    Lamb, Hydrodynamics (Cambridge University Press, Cambridge UK 1932), p

    H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge UK 1932), p. 248; P.J. Morrison, ’Hamiltonian description of the ideal fluid’, Rev. Mod. Phys., 70: 467, 1998

    work page 1932

  6. [6]

    Hamiltonian formulation for perfect fluid equations with the ℓ-conformal Galilei symmetry,

    Timofei Snegirev, “Hamiltonian formulation for perfect fluid equations with the ℓ-conformal Galilei symmetry,” arXiv:2302.01565 (2023)

    arXiv 2023

  7. [7]

    Relativistic fluctuations in stochastic fluid dynamics,

    X. An, G. Basar, M. Stephanov, and Ho-Ung Yee, “Relativistic fluctuations in stochastic fluid dynamics,” EPJ Web of Conferences 259, 13011 (2024)

    work page 2024

  8. [8]

    Hamiltonian description of the ideal fluid,

    P.J. Morrison, “Hamiltonian description of the ideal fluid,” Rev. Mod. Phys. 70, 467 (1998)

    work page 1998

Show all 56 references
  1. [9]

    Hamiltonian Formulation of Fluid and Plasma Dynamics,

    P. J. Morrison, “Hamiltonian Formulation of Fluid and Plasma Dynamics,” AIP Conf. Proc., 88, 13–46 (1982)

    1982

  2. [10]

    Sur la g´ eometrie diff´ erentielle des groupes de Lie de dimension infinie et ses applications ` a l’hydrodynamique des fluides parfaits,

    V.I. Arnold, “Sur la g´ eometrie diff´ erentielle des groupes de Lie de dimension infinie et ses applications ` a l’hydrodynamique des fluides parfaits,” Ann. Inst. Fourier16, 319–361 (1966)

    1966

  3. [11]

    Hamiltonian Theory of a Relativistic Perfect Fluid,

    B.F. Schutz, “Hamiltonian Theory of a Relativistic Perfect Fluid,” Phys. Rev. D 2, 2762 (1970)

    1970

  4. [12]

    Relativistic fluid dynamics: Physics for many different scales,

    N. Andersson and G.L. Comer, “Relativistic fluid dynamics: Physics for many different scales,” Living Reviews in Relativity 10, 1 (2007). https://doi.org/10.12942/lrr-2007-1

    2007 doi

  5. [13]

    Landau, Zh

    These brackets were first posited (in somewhat imprecise form) by L. Landau, Zh. Eksper. Teoret. Fiz. 11, 592 (1941) [English translation: J. Phys. USSR 5, 71 (1941)]; see also P. Morrison and J. Greene, Phys. Rev. Lett. 45, 790 (1980), (E) 48 569 (1982)

    1941

  6. [14]

    A note on the canonical approach to hydrodynamics and Hamiltonian structures,

    Luca Martinoia, Rajeev Singh, “A note on the canonical approach to hydrodynamics and Hamiltonian structures,” arXiv:2408.10698 (2024)

    2024 arXiv

  7. [15]

    A. K. Mitra, R. Banerjee, S. Ghosh, JCAP 10 (2018) 057; P. Das, S. Ghosh, Phys. Rev. D. Phys. Rev. D 98, 084047 (2018). [12] C.Eckart, Phys. Rev. 54, 920 (1938). [13] A constructive discussion of the Darboux theorem can be found it the second work of Ref. [7]. [14]

    2018

  8. [16]

    A canonical Hamiltonian formulation of the Navier-Stokes problem,

    J. W. Sanders, L. Rizzi, W. T. Vetterling, and M. B. Short, “A canonical Hamiltonian formulation of the Navier-Stokes problem,” arXiv:2310.07085 [physics.flu-dyn] (2024)

    2024 arXiv

  9. [17]

    Nonlinear Stability of Fluid and Plasma Equilibria,

    D. D. Holm, J. E. Marsden, T. Ratiu, and A. Weinstein, “Nonlinear Stability of Fluid and Plasma Equilibria,” Physics Reports, vol. 123, no. 1–2, pp. 1–116, 1985

    1985

  10. [18]

    Dynamics of Extended Bodies in General Relativity—I. Momentum and An- gular Momentum,

    W. G. Dixon, “Dynamics of Extended Bodies in General Relativity—I. Momentum and An- gular Momentum,” Proc. Royal Society A, vol. 314, 499–527, 1969

    1969

  11. [19]

    Jackiw, V.P

    R. Jackiw, V.P. Nair, S.-Y. Pi, A.P. Polychronakos; J.Phys. A37 (2004) R327-R432; hep-ph arXiv:hep-ph/0407101 18

    2004 arXiv

  12. [20]

    Clebsch, J

    A. Clebsch, J. Reine Angew. Math. 56, 1 (1859); C.C. Lin, International School of Physics E. Fermi (XXI)

  13. [21]

    Poisson brackets and Clebsch representations for mag- netohydrodynamics, multifluid plasmas, and elasticity,

    D. D. Holm and B. A. Kupershmidt, “Poisson brackets and Clebsch representations for mag- netohydrodynamics, multifluid plasmas, and elasticity,” Physica D: Nonlinear Phenomena , vol. 6, no. 3, pp. 347–363, 1983

    1983

  14. [22]

    F. C. Michel, ”Accretion of Matter by Condensed Objects”, Astrophysics and Space Science, vol. 15, pp. 153–160, 1972

    1972

  15. [23]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman (1973)

    1973

  16. [24]

    R. M. Wald, General Relativity, University of Chicago Press (1984)

    1984

  17. [25]

    Padmanabhan, Gravitation: Foundations and Frontiers , Cambridge University Press (2010)

    T. Padmanabhan, Gravitation: Foundations and Frontiers , Cambridge University Press (2010)

    2010

  18. [26]

    Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, 1983)

    S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, 1983)

    1983

  19. [27]

    Evolution of binary black-hole spacetimes,

    F. Pretorius, “Evolution of binary black-hole spacetimes,” Phys. Rev. Lett. 95, 121101 (2005)

    2005

  20. [28]

    Bondi; ’On spherically symmetrical accretion’, 1952MNRAS.112..195B

    H. Bondi; ’On spherically symmetrical accretion’, 1952MNRAS.112..195B

  21. [29]

    Pseudo-Schwarzschild description of transonic spherical accretion onto compact objects,

    T. K. Das and A. Sarkar, “Pseudo-Schwarzschild description of transonic spherical accretion onto compact objects,” Astronomy & Astrophysics 374, 1150 (2001). https://doi.org/10. 1051/0004-6361:20010736

    2001

  22. [30]

    Accretion powered spherical wind in general relativity,

    T. K. Das, “Accretion powered spherical wind in general relativity,” arXiv:astro-ph/0212089 (2002)

    2002 arXiv

  23. [31]

    Pseudo-Schwarzschild Description of Accretion-Powered Spherical Outflow,

    T. K. Das, “Pseudo-Schwarzschild Description of Accretion-Powered Spherical Outflow,” arXiv:astro-ph/0212217 (2002)

    2002 arXiv

  24. [32]

    Critical properties of spherically symmetric black hole accretion in Schwarzschild geometry,

    I. Mandal, A. Ray, and T. K. Das, “Critical properties of spherically symmetric black hole accretion in Schwarzschild geometry,” Monthly Notices of the Royal Astronomical Society 378, 1400–1416 (2007). https://doi.org/10.1111/j.1365-2966.2007.11876.x

    2007

  25. [33]

    Revealing noncanonical Hamiltonian structures in relativistic fluid dynamics,

    Keiichiro Takeda, Naoki Sato “Revealing noncanonical Hamiltonian structures in relativistic fluid dynamics,” Progress in Physics (2025)

    2025

  26. [34]

    Mathematical Problems in Relativistic Fluid Dynamics,

    NSF Project, “Mathematical Problems in Relativistic Fluid Dynamics,” NSF Award, July 2024

    2024

  27. [35]

    Extending the Hamiltonian formulation of particle motion in gravitational fields,

    R. da Silva, et al., “Extending the Hamiltonian formulation of particle motion in gravitational fields,” Phys. Rev. D 110, 124031 (2024). 19

    2024

  28. [36]

    Relativistic Fluid Dynamics,

    J. Olsthoorn, “Relativistic Fluid Dynamics,” Waterloo Mathematics Review, Vol. 1, 44–59 (2012). Available at: https://mathreview.uwaterloo.ca/archive/voli/2/olsthoorn. pdf

    2012

  29. [37]

    Relativistic fluid dynamics: physics for many dif- ferent scales,

    N. Andersson and G. L. Comer, “Relativistic fluid dynamics: physics for many dif- ferent scales,” Living Reviews in Relativity 24, 3 (2021). https://doi.org/10.1007/ s41114-021-00031-6

    2021

  30. [38]

    Relativistic fluids in cosmological spacetimes,

    D. Fajman, M. Ofner, and Z. Wyatt, “Relativistic fluids in cosmological spacetimes,” Class. Quantum Grav. 41, 233001 (2024). https://doi.org/10.1088/1361-6382/ad84ad

    2024 doi

  31. [39]

    Magnetofluid dynamics in curved spacetime,

    C. Bhattacharjee, R. Das, S. M. Mahajan, “Magnetofluid dynamics in curved spacetime,” Phys. Rev. D 91, 064055 (2015). https://doi.org/10.1103/PhysRevD.91.064055

    2015 doi

  32. [40]

    Hamiltonian formalism for general-relativistic adiabatic fluids,

    D. D. Holm, “Hamiltonian formalism for general-relativistic adiabatic fluids,” Journal of Mathematical Physics 26, 2378 (1985)

    1985

  33. [41]

    Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics, (Benjamin, 1967)

    A. Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics, (Benjamin, 1967)

    1967

  34. [42]

    Relativistic Eulerian and Lagrangian perturbation theory,

    E. Villa and C. Rampf “Relativistic Eulerian and Lagrangian perturbation theory,” JCAP, 2016(01):030, arXiv:1505.04782

    2016 arXiv

  35. [43]

    Perturbation theory of relativistic stars and fluids,

    J. L. Friedman, “Perturbation theory of relativistic stars and fluids,” in Gen. Rel. Grav.: A Centennial Perspective, Cambridge Univ. Press, 2015

    2015

  36. [44]

    Gauge-invariant perturbations of nonperfect fluids with dissipation,

    C. D ´ ıaz-Guerra et al., “Gauge-invariant perturbations of nonperfect fluids with dissipation,” Phys. Rev. D , 110, 124012 (2024), arXiv:2309.08628

    2024 arXiv

  37. [45]

    Relativistic fluid dynamics: Physics for many different scales

    N. Andersson, “Relativistic fluid dynamics: Physics for many different scales” arXiv:2101.07869 [gr-qc], 2021

    2021 arXiv

  38. [46]

    V. F. Mukhanov, Physical Foundations of Cosmology , CUP, 2005

    2005

  39. [47]

    Hamilton principle for chiral anomalies in hydrodynamics,

    P. B. Wiegmann, “Hamilton principle for chiral anomalies in hydrodynamics,” Phys. Rev. D 106, 096013 (2022)

    2022

  40. [48]

    Reference Frames in Astronomy,

    K. J. Johnston and Chr. de Vegt, “Reference Frames in Astronomy,” Reports on Progress in Physics, vol. 66, no. 6, pp. 809–859, 2003

    2003

  41. [49]

    Relativistic Reference Frames for Astrometry and Navigation in the Solar System,

    S. M. Kopeikin, et al., “Relativistic Reference Frames for Astrometry and Navigation in the Solar System,” in Proceedings of the ASTROCON 2006 Conference*, edited by D. Folkner, AIP Conference Proceedings 886, pp. 268–283, 2007, arXiv:astro-ph/0610022

    2006 arXiv

  42. [50]

    Gaia Early Data Release 3 – The celestial reference frame,

    Gaia Collaboration, “Gaia Early Data Release 3 – The celestial reference frame,” Astronomy & Astrophysics, vol. 649, A2, 2021. 20

    2021

  43. [51]

    Topology of rotating stratified fluids with and without background shear flow,

    Ziyan Zhu,, Christopher Li, and J. B. Marston, “Topology of rotating stratified fluids with and without background shear flow,” Phys. Rev. Research 5, 033191 (2023)

    2023

  44. [52]

    E. W. Bertschinger, https://dspace.mit.edu, Physics CourseHome, 2002

    2002

  45. [53]

    The linear stability of the Schwarzschild solution to gravitational perturbations,

    M. Dafermos, G. Holzegel, I. Rodnianski, “The linear stability of the Schwarzschild solution to gravitational perturbations,” (see Dafermos’ website and associated review articles)

  46. [54]

    A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited,

    G. Benomio, “A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited,” arXiv:2211.00616 (2022)

    2022 arXiv

  47. [55]

    Black-hole eddy currents,

    T. Damour, “Black-hole eddy currents,” Phys. Rev. D 18, 3598 (1978)

    1978

  48. [56]

    Analogue Gravity,

    C. Barcel´ o, S. Liberati, and M. Visser, “Analogue Gravity,” Living Rev. Relativity 8, 12 (2005). https://doi.org/10.12942/lrr-2005-12 21 6 Appendices ˙U r = √−g00 2Φ grr (∂rg00)(1 + Z) + g00 (∂rgrr)(U r)2 + (∂rgθθ)(U θ)2 + (∂rgϕϕ)(U ϕ)2. + 2 grrU r∂rU r + 2gθθU θ∂rU θ + 2gϕϕ...

    2005 doi

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