An Action for Black Hole Membranes

Frank Wilczek; Maulik K. Parikh

REVIEW

An Action for Black Hole Membranes

Maulik K. Parikh, Frank Wilczek

General Relativity and Quantum Cosmology · Submitted 17 Dec 1997 · 23 references

Pith's verdict: A genuinely useful action formulation of the membrane paradigm, with clean derivations; the thermodynamic and 'minimum entropy production' claims are framed more strongly than sections V–VI actually deliver.

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arXiv IDgr-qc/9712077 (version 2)

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CitationFrank Wilczek, Maulik K. Parikh. "An Action for Black Hole Membranes." arXiv:gr-qc/9712077 (1997). https://pith.science/paper/RIQBFCV5

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The thing worth knowing: this gives the membrane paradigm an action principle, and it works. The move is simple — split the action at the stretched horizon, add a surface term that cancels the residual boundary variation, then read that term as surface charge/current and surface stress. You get j_s^a = F^{ab}n_b/4π (3.6) and t_s^{ab} = (Kh^{ab} − K^{ab})/8π (3.24) essentially for free, and the α→0 limits reproduce Ohm's law with ρ = 4π, the Navier-Stokes equation, and the fluid coefficients η = 1/16π, ζ = −1/16π. The axidilaton extension in III.C is new, and the conductivity tensor (3.53) with ρ = 1/(4λ₂) is a nice check against the Kaluza-Klein picture. Section IV — future dynamically closed sets, the horizon normal being tangent to itself — is the part I'd hand to a student; it's the honest answer to why an action can produce dissipation at all, and it's better than the abstract lets on. The observation that S_out + S_surf gives zero external entropy is also sharper than the space it gets. Soft spots, in order of weight. First, the abstract's 'minimum entropy production principle' is oversold: what section VI shows is that −δḢ/δφ = 0 follows from the Hamiltonian constraint being time-independent (6.9), which is automatic. The real content is the quasi-static reduction to D[φ] = (1/4π)∫(αE_∥)² and δD/δφ = 0 — a true statement about stationary dissipative systems, not a new variational principle for black holes. Second, the entropy coefficient η in (3.36) is not independently fixed; section V imports the Gibbons-Hawking normalization (5.3) and recovers A/4. That's a consistency check, not a derivation of the coefficient. Third, the regularity condition (3.12) is acknowledged as not rigorous and the finite-α error is never quantified. The stress-test note is right that this is load-bearing for Ohm's law and Joule heating, but I'd push back on the framing: the paper never claims to derive it from the action. It's the standard membrane-paradigm input, and section IV is explicit that it's a coarse-graining. What's missing is a leading-correction estimate, not a derivation. Minor: I couldn't reconcile the sign of the g term between (3.28) and (3.31) on a quick pass, though the final coefficients match the literature; a line of clarification would help. Who it's for: horizon fluid dynamics, membrane-paradigm users, anyone who wants the action-side reference. Cite it. Send it out.

Referee report

Major objections (3)

[§VI, eq. 6.13; abstract sentence s6] The abstract claims a 'minimum entropy production principle,' but eq. 6.13 is not shown to be minimized. With ζ = −1/16π from eq. 3.31, the integrand ζθ_H² + 2ησ_H² is not positive definite, so extremizing D[v^A] does not by itself establish a minimum; the second variation should be computed, or the claim should be weakened to 'stationary' or 'extremal.' By contrast, eq. 6.11 is manifestly positive, so the electromagnetic case is fine and could be noted as such.
[§III.A, eq. 3.12; §IV] The step from the surface current (eq. 3.6) to Ohm's law (eq. 3.13) and Joule heating (eq. 3.15) requires the regularity condition E_∥ = n̂ × B_∥ (eq. 3.12), which is an independent physical input rather than a consequence of the action. The authors acknowledge this in §IV ('not rigorously true,' 'coarse-graining'), and eq. 4.1 gives the scale λ_max ~ 2Mα, so the concern is not hidden. What remains unresolved is quantitative: no estimate is given for the leading finite-α violation of eq. 3.12, and the abstract's 'shows how an action can produce dissipative equations' would read more accurately if eq. 3.12 were labelled as an additional input at the point where it is used.
[§III.B, eqs. 3.36–3.37; §V] The paper promises that the Euclidean path integral will fix the coefficient η in S = η k_B A/ħ (eq. 3.36), but §V never states the identification explicitly. Eq. 5.10 gives S = A/4, which implies η = 1/4, but the reader is left to infer this, and the symbol η is already used for the shear viscosity (eq. 3.31), where η = 1/16π would give S = A/(16π). Please rename the constant and state the identification explicitly. This is not circularity — eqs. 3.36–3.37 satisfy the first law for any η — but the promised closure should be visible.

Minor objections (8)

[Eq. 3.55] Using eq. 3.53, |j_s|² = 16|λ|²E², so 4λ₂E² = λ₂|j_s|²/(4|λ|²); the factor |λ|⁴ in eq. 3.55 appears to be a typo for |λ|².
[§V, after eq. 5.11] The quoted Φ = Q/r_+ and Ω = J/M hold for Reissner-Nordström; for Kerr-Newman the horizon potential is Φ = Qr_+/(r_+²+a²) and the angular velocity is Ω = a/(r_+²+a²). Please correct or state the restriction.
[§V, charged-hole paragraph] The gauge transformation written as A_a = ∇_a(t−Φ) gives A_t = 1, not zero; presumably the intended statement is A → A − Φ dt with Φ = Q/r_+, so that A_t vanishes at r = r_+.
[Eq. 3.38 and following text] The solution relaxes monotonically to θ = 2g as t increases, and the divergence occurs for t < t_0; the discussion of the 'runaway' and of teleological boundary conditions would be clearer if this direction were stated.
[Eq. 3.39] With λ = i/16π, (i/4)(λF²_+ − λ̄F²_-) = −F²/(32π) using F²_+ + F²_- = 2F², whereas the Maxwell action is −F²/(16π); please state the normalization convention that makes the stated reduction exact.
[Abstract vs §VI] The abstract says 'minimum entropy production' while §VI says 'minimum heat production'; these agree only for a stationary process at temperature T, and the abstract's 'establish the validity' should be qualified by the quasi-stationary, slowly-rotating approximation used in eq. 6.11.
[References] Reference [18] duplicates [5] (Hawking, Phys. Rev. Lett. 26, 1344 (1971)); the focusing-equation citation should be checked.
[Title page] The date line reads November 26, 2024, which is inconsistent with the arXiv v2 date of 9 March 1998; please correct.

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The paper

The paper claims

The black hole membrane picture is rebuilt as a boundary term in the gravitational and electromagnetic actions, so Ohm's law, fluid viscosity, and the area entropy all fall out of one variational principle.

The reading

The membrane paradigm says that an outside observer sees a black hole horizon as an ordinary physical surface: a conducting sheet with friction, obeying Ohm's law, the Joule heating law, and the Navier–Stokes equation. That picture was previously assembled case by case, by massaging the equations of motion. This paper argues that the whole thing follows from a single move: because nothing can classically leave a black hole, physics outside must be stationary under variations of the action restricted to the exterior, and the leftover boundary term at the stretched horizon is cancelled by adding a surface source term. Reading those sources back as a surface charge-and-current sheet and a surface stress tensor reproduces the known membrane results, extends them to dyonic and axidilaton black holes, and — through a Euclidean path integral — fixes the coefficient in the Bekenstein–Hawking entropy formula rather than merely matching it up to a constant.

What carries the argument

The load-bearing device is the surface counterterm at the stretched horizon and its two faces. As a variational object it is fixed by the requirement that the exterior action be stationary; as a physical object it is a two-dimensional source sheet whose stress tensor $t_s^{ab} = frac{1}{8pi}(K h^{ab} - K^{ab})$ and current $j_s^a = frac{1}{4pi}F^{ab}n_b$ are the only things an exterior observer can measure. The identification is made possible by the defining property of a null surface: the null generator $l^a$ is simultaneously normal and tangential, so that $alpha n^a to l^a$ and $alpha U^a to l^a$ as the stretched horizon is pushed onto the true horizon. That single degeneracy is what converts $K_{ab}$ into the intrinsic shear and expansion of the horizon and thereby converts a boundary term into fluid pressure, viscosity, conductivity, and dissipation. The stretched horizon, a timelike surface with a non-degenerate induced metric, exists only so that a conventional action can be written down at all; the regulator $alpha$ is taken to zero at the end of every computation.

Extended reading notes

Core claim

The paper's central claim is that the membrane paradigm is a statement about boundary terms. Splitting the total action as $(S_{out}+S_{surf}) + (S_{in}-S_{surf})$, the authors demand that each half be stationary on its own; the added surface term is exactly what cancels the residual boundary variation that a Dirichlet problem with a horizon, rather than a spatial infinity, leaves behind. For Maxwell theory the required term is $S_{surf}[A] = +int d^3x sqrt{-h}, j_s cdot A$ with $j_s^a = +frac{1}{4pi} F^{ab} n_b$, which reads as a surface charge terminating the normal electric field and a surface current terminating the tangential magnetic field. For Einstein–Hilbert gravity the required variation is $delta S_{surf}[h] = -frac{1}{2}int d^3x sqrt{-h}, t_s^{ab}, delta h_{ab}$ with $t_s^{ab} = +frac{1}{8pi}(K h^{ab} - K^{ab})$, which is precisely an Israel junction-condition stress tensor whose matching requires the extrinsic curvature on the interior side to vanish. Decomposing $K_{ab}$ in the null limit then gives the stress of a two-dimensional Newtonian fluid with pressure $p = g/8pi$, shear viscosity $eta = 1/16pi$, and bulk viscosity $zeta = -1/16pi$, and the contracted Gauss–Codazzi equations become the Navier–Stokes equation and the focusing (heat-transfer) equation. Combined with the regularity condition that tangential fields measured by a fiducial observer are purely ingoing, the surface current becomes $vec{E}_parallel = 4pi vec{j}_s$ — Ohm's law with resistivity $4pi approx 377,Omega$ — and the Poynting flux becomes Joule heating at rate $dM_{irr}/dt = +int alpha^2 j_s^2 rho, dA$. The Euclidean continuation then does real work: the external and membrane actions cancel exactly, giving zero entropy for the outside, while the internal-minus-surface combination gives $+4pi M^2$ and hence $S_{B-H} = A/4$, fixing the undetermined constant $eta$ that the classical equations alone left free. A Hamiltonian treatment closes the argument with a minimum-heat-production principle, $delta D/deltaphi = 0$ for the electromagnetic case and $delta D/delta v^A = 0$ for gravity.

Load-bearing premise

The step that turns a surface current into Ohm's law and Joule heating is the regularity condition that all tangential electromagnetic fields seen by a hovering observer are ingoing; the authors themselves say in Section IV that this is not rigorously true and amounts to discarding the very-high-frequency outgoing modes that such an observer cannot resolve.

Editorial extensions

If this is right

The membrane equations of motion for arbitrary non-equilibrium black holes become consequences of a least-action principle rather than of term-by-term manipulation of Einstein's and Maxwell's equations, so the derivations simplify and generalize mechanically to new fields.
The paradigm extends to the axidilaton sector of the string effective action: the membrane carries a Peccei–Quinn surface charge with $q = -n^a partial_a lambda / lambda_2^2$, the conductivity becomes a tensor built from the dilaton and axion, and the resistivity in the pure dilaton case is $rho = 1/(4lambda_2)$.
For dyonic black holes carrying both electric and magnetic charge, the surface charge and current are $sigma = 4(lambda_2 E_perp + lambda_1 B_perp)$ and $vec{j}_s = 4(lambda_2,hat{n}timesvec{B}_parallel - lambda_1,hat{n}timesvec{E}_parallel)$, so the membrane's resistive response is fixed by the dilaton, not by a universal constant.
Because the path-integral normalization is now fixed by the action, the Bekenstein–Hawking entropy $S = A/4$ is derived rather than fitted, and the outside observer's entropy is shown to be zero — the membrane absorbs everything and hides the hole.
A minimum-entropy-production principle holds for black holes in quasi-stationary situations, with the dissipation functional $D[phi] = frac{1}{4pi}int d^2x sqrt{gamma},(alpha vec{E}_parallel)^2$ extremized by the membrane's own equation of motion.

Reading between the lines

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

If the surface term is genuinely the whole story, then the membrane's transport coefficients are not independent inputs but are fixed by the boundary counterterm's normalization — which suggests that any theory whose horizon boundary term can be written down (higher-derivative gravity, Gauss–Bonnet, Lovelock) inherits a membrane with computable, not conjectured, viscosity-to-entropy ratios.
The paper's remark that the horizon's growth is acausal, requiring final rather than initial conditions, sits oddly with the claim that the same action yields a well-posed Euclidean partition function; reconciling the Lorentzian teleology with the Euclidean periodicity $beta = 2pi/g_H$ looks like the sharpest internal tension a follow-up would need to resolve.
The derivation is silent about the value of $hbar$ in the dissipation relation and about how the coarse-graining scale $lambda_{max} = 2Malpha$ enters loop corrections; a quantum version would have to say whether the membrane's resistivity $4pi$ is protected or runs.
Because the same boundary-term trick works for any field with a kinetic term, one could test the framework by asking whether a scalar or spin-2 field induces a membrane with a consistent sign of bulk viscosity, or whether the negative bulk viscosity $zeta = -1/16pi$ is a generic artifact of the null-normal-tangent degeneracy.

Original abstract

The membrane paradigm is the remarkable view that, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane, obeying such pre-relativistic equations as Ohm's law and the Navier-Stokes equation. It has traditionally been derived by manipulating the equations of motion. Here we provide an action formulation of this picture, clarifying what underlies the paradigm, and simplifying the derivations. Within this framework, we derive previous membrane results, and extend them to dyonic black hole solutions. We discuss how it is that an action can produce dissipative equations. Using a Euclidean path integral, we show that familiar semi-classical thermodynamic properties of black holes also emerge from the membrane action. Finally, in a Hamiltonian description, we establish the validity of a minimum entropy production principle for black holes.

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What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.84 · mentions code or data availability 0.01.

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The membrane paradigm is the remarkable view that, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane, obeying such pre-relativistic equations as Ohm's law and the Navier-Stokes equation. It has traditionally been derived by manipulating the equations of motion. Here we provide an action formulation of this picture, clarifying what underlies the paradigm, and simplifying the derivations. Within this framework, we derive previous membrane results, and extend them to dyonic black hole solutions. We discuss how it is that an action can produce dissipative equations. Using a Euclidean path integral, we show that familiar semi-classical thermodynamic properties of black holes also emerge from the membrane action. Finally, in a Hamiltonian description, we establish the validity of a minimum entropy production principle for black holes.

background contribution method result interpretation main claim

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s0	The membrane paradigm is the remarkable view that, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane, obeying such pre-relativistic equations as Ohm's law and the Navier-Stokes equation.	background 1.00	0.05	0.06	asserted 1.00
s1	It has traditionally been derived by manipulating the equations of motion.	background 0.99	0.05	0.02	asserted 1.00
s2	Here we provide an action formulation of this picture, clarifying what underlies the paradigm, and simplifying the derivations.	contribution 1.00	0.95	0.34	asserted 1.00
s3	Within this framework, we derive previous membrane results, and extend them to dyonic black hole solutions.	contribution 0.56	0.95	0.18	asserted 1.00
s4	We discuss how it is that an action can produce dissipative equations.	contribution 0.85	0.56	0.06	unclear 0.84
s5	Using a Euclidean path integral, we show that familiar semi-classical thermodynamic properties of black holes also emerge from the membrane action.	result 0.78	0.94	0.12	asserted 1.00
s6	Finally, in a Hamiltonian description, we establish the validity of a minimum entropy production principle for black holes.	result 0.78	0.95	0.19	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.84 · evidence from proof or derivation 1.00.

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References

Works cited (23)

[1]

For a review see, e.g., S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Spacetime (Cambridge University Press, Cambridge, England, 1973)

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[2]

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[8]

R. L. Znajek, Mon. Not. R. Astron. Soc. 185, 833 (1978). Znajek’s approach diﬀers from ours in that he considers a membrane with a non-zero thic kness and volume con- ductivities

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[9]

T. Damour, th` ese de doctorat d’´ etat, University of Paris VI, 1979 (unpublished)

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[10]

T. Damour, in Proceedings of the Second Marcel Grossman Meeting on Genera l Rela- tivity, edited by R. Ruﬃni (North-Holland, Amsterdam, 1982), p. 58 7

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[11]

D. A. Macdonald and K. S. Thorne, Mon. Not. R. Astron. Soc . 198, 345 (1982)

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[12]

R. H. Price and K. S. Thorne, Phys. Rev. D 33, 915 (1986)

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[13]

Black Holes: The Membrane Paradigm , edited by K. S. Thorne, R. H. Price, and D. A. Macdonald (Yale University Press, London, 1986). It should be noted that these authors imply by “the membrane paradigm” not only the existence of a s tretched horizon, but also a 3+1 split of spacetime. Their motivation for severing space from time is pragmatic: astrophysi...

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[14]

S. Carlip, Phys. Rev. D 51, 632 (1995)

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[15]

M. K. Parikh and F. Wilczek, in preparation

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

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[17]

See, e.g., L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1987), §49

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S. W. Hawking, Phys. Rev. Lett. 26, 1344 (1971)

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How later papers used it

81 later papers cite it

Evidence: substantial. 80 of the citing papers share no author with it, and 20 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 142 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 1 built on it, 2 used its method or tool and 1 cited it as support.

Builds on it 1 paper

Extended an earlier action principle formulation of the membrane at the horizon given by this paper.

Black hole jet power from impedance matching (2015) · 0.95 · paraphrased: the licence does not allow quoting

Uses its method or tool 2 of 8 papers with this label

Followed this paper in adding a boundary term at the horizon to the one arising from variations of its action.

Conductivity in the gravity dual to massive ABJM and the membrane paradigm (2013) · 0.91 · paraphrased: the licence does not allow quoting
Followed an approach similar to this paper's, approaching a null hypersurface by a timelike hypersurface on which the computation is performed before taking the limit.

Asymptotic symmetries and memories of gauge theories in FLRW spacetimes (2022) · 0.96 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 4 papers with this label

Said its attribution of entropy to sections of its timelike stretched horizon is consistent with the black hole membrane paradigm, in which this paper holds that the timelike stretched horizon can also be thought of as having thermodynamic properties.

Emergence of Spacetime: From Entanglement to Einstein (2020) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

76 arXiv papers cite this in 131 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 74 sentences without a confident label

Builds on it 1 paper

Recently, we have given an action principle formulation of the membrane at infinity [ref] , extending an earlier action principle formulation of the membrane at the horizon [ref] .

Black hole jet power from impedance matching · Introduction · 0.95

Uses its method or tool 8 papers

Let us calculate the membrane transport coefficient of the near extremal black hole induced by $h_ij$ [ref] .

Transport coefficients of D1-D5-P system and the membrane paradigm · Membrane transport coefficients induced by minimal scalars · 0.92
Following [ref] , we need to add a boundary term at the horizon to the one that arises from variations of this action.

Conductivity in the gravity dual to massive ABJM and the membrane paradigm · Scalar field calculation · 0.91
To obtain equations of motion we use the so-called 'membrane paradigm' [ref] .

Direct photons emission rate and electric conductivity in twice anisotropic QGP holographic model with first-order phase transition · Perturbation · 1.00

Compares with it 8 papers

Moreover, the derivation [ref] and ours differs significantly from the well-known membrane derivation [ref] briefly discussed in Section [ref], in which it is essential that viscosity appears as a quantity visible to an infinitely distant observer.

Viscosity, entanglement and acceleration · Unruh effect and KSS bound: resume · 0.98
When $c_1=0$, $$ agrees with the membrane paradigm result [ref] .

The Incompressible Rindler fluid versus the Schwarzschild-AdS fluid · Thermodynamic quantities · 1.00
This is the same stress-energy tensor that appears in the original black hole membrane paradigm [ref] but with an overall minus sign.

Energy extraction from boosted black holes: Penrose process, jets, and the membrane at infinity · Membrane stress-energy tensor · 0.99

Cites it as support 3 papers

There are enough hints [ref] which seems to point to an affirmative answer to this question.

Near-Horizon Analysis of $\eta/s$ · Introduction and Summary · 0.97
These include membrane paradigm [ref] - [ref] , as well as related fluid/gravity correspondence in AdS/CFT [ref] , [ref] , the near horizon conformal symmetry with analogous structure of a single copy of the “near-horizon Virasoro algebra” [ref] - [ref] , as noticed in [ref] , Kerr/CFT correspondence [ref] , Carrolian description of near horizon physics [ref] - [ref] , which may provide a natural holographic framework for non-extremal black holes, even beyond the AdS setting.

Modular quantization and black holes · Discussion · 0.99
This is consistent with the black hole membrane paradigm in which the timelike stretched horizon can also be thought of as having thermodynamic properties [ref] .

Emergence of Spacetime: From Entanglement to Einstein · Stretched Lightcone Thermodynamics · 0.97

Cites it as background 31 papers

According to the membrane paradigm, black hole event horizons behave like 2+1 dimensional fluids [ref] .

BMS invariance and the membrane paradigm · Introduction · 0.96
One of them is the membrane paradigm [ref] .

Shear viscosity of a highly excited string and the black hole membrane paradigm · Introduction · 1.00
The membrane paradigm states that a distant observer sees a fictitious membrane or fluid with some transport coefficients such as viscosities and conductivities on a stretched horizon of a black hole [ref] .

Transport coefficients of D1-D5-P system and the membrane paradigm · Introduction · 1.00

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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For a review see, e.g., S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Spacetime (Cambridge University Press, Cambridge, England, 1973)

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R. L. Znajek, Mon. Not. R. Astron. Soc. 185, 833 (1978). Znajek’s approach diﬀers from ours in that he considers a membrane with a non-zero thic kness and volume con- ductivities

View on Pith 1978

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Black Holes: The Membrane Paradigm , edited by K. S. Thorne, R. H. Price, and D. A. Macdonald (Yale University Press, London, 1986). It should be noted that these authors imply by “the membrane paradigm” not only the existence of a s tretched horizon, but also a 3+1 split of spacetime. Their motivation for severing space from time is pragmatic: astrophysi...

View on Pith 1986

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S. Carlip, Phys. Rev. D 51, 632 (1995)

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M. K. Parikh and F. Wilczek, in preparation

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See, e.g., L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1987), §49

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