Dynamical Horizons: Energy, Angular Momentum, Fluxes and Balance Laws

Abhay Ashtekar; Badri Krishnan

REVIEW

Dynamical Horizons: Energy, Angular Momentum, Fluxes and Balance Laws

Abhay Ashtekar, Badri Krishnan

General Relativity and Quantum Cosmology · Submitted 21 Jul 2002 · 8 references

Pith's verdict: The area-balance identity is exact and the flux formulae are the real contribution; the Bondi–Sachs analogy is asserted rather than computed, which is a minor soft spot, not a flaw.

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arXiv IDgr-qc/0207080 (version 3)

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CitationAbhay Ashtekar, Badri Krishnan. "Dynamical Horizons: Energy, Angular Momentum, Fluxes and Balance Laws." arXiv:gr-qc/0207080 (2002). https://pith.science/paper/JQC6JC7I

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Editor's note

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This is the paper that set up dynamical horizons as a working tool, and it holds up. The core is Eq. (14), specialized to area radius in Eq. (16): the change in horizon area equals matter flux plus a gravitational term built from |σ|² + 2|ζ|². That identity is an exact consequence of the scalar and vector constraints, Gauss–Codazzi, and θ_(ℓ)=0. I checked the chain and found no algebraic slip. The angular momentum balance in Eq. (7) and the generalized first law in Eq. (20) follow cleanly, and the paper is honest that the canonical Kerr-anchored choice in Eqs. (21)–(23) is a choice, not a derivation — the Smarr formula is the Kerr mass formula by construction, and they say so. The genuinely new content, the flux formulae and the area balance, does not depend on that anchor. Credit where due: this is careful, local, coordinate-free work that numerical relativity actually uses, and the positivity of the energy flux is a real result, not a slogan. One thing I'd push back on: the reader's worry that σ and ζ might not exhaust the gravitational degrees of freedom is mis-framed. F_g is defined as the second term in an identity that is derived, so it is complete by construction; if extra modes crossed H, the identity would fail, and it doesn't. The narrower soft spot is the Bondi–Sachs analogy in [abstract:s3]. The paper lists four properties — no coordinates, positivity, locality, vanishing in spherical symmetry — and calls the result 'analogous.' It never takes a null-infinity limit showing F_g reduces to the News flux. That's a hedged analogy, not a claim of equality, so it's a minor gap, not an error. Two smaller presentation points: the step from Eq. (9) to Eq. (12) is compressed to 'simplifying,' and the S² topology claim is asserted rather than shown. Both are Letter-length omissions. Also worth noting: the classifier reports no stated limitations, but §IV does acknowledge that E_t may not be local on a cross-section for general lapse, and that the Kerr-anchored choice needs the right side of (21) positive, i.e., non-extremal. Those are stated, not hidden. Who should read it: anyone doing black-hole initial data or numerical evolution, and anyone working on quasi-local horizons. It deserves a serious referee, and I'd cite it without hesitation.

Referee report

Significance

If the derivation holds, this is a foundational contribution: it supplies local, manifestly positive flux formulae for gravitational radiation crossing a quasi-locally located horizon, without reference to null infinity or to a background. The strengths are concrete and worth naming. The area-balance identity (14)/(16) is an exact consequence of the constraints and is parameter-free; the gravitational flux is a sum of squares, hence positive under the stated energy condition; it vanishes identically in spherical symmetry; and the whole construction reduces correctly to the irreducible mass R/2G in the spherical case and to the Kerr mass formula (23) in the axisymmetric case. The balance laws (19) are finite identities that can be checked directly against numerical simulations, which is exactly the use the authors advertise. The framework has since become standard machinery in numerical relativity, and the paper's careful restriction to zero charge and its explicit statement that the canonical choice (21) requires a non-extremal horizon are appropriate hedges.

Minor objections (11)

[§IV, Eq. (22)] As printed, Ω_o(R) = 2GJR/√(R⁴+4G²J²) is dimensionally inconsistent: it has dimensions of length rather than inverse length, whereas an angular velocity must scale as 1/length. It also fails the paper's own Smarr check (23). Substituting the printed form into 2(k_o a/8πG + Ω_o J) with a = 4πR² gives (R⁴−4G²J²+8G²J²R²)/(2GR√(R⁴+4G²J²)), not √(R⁴+4G²J²)/(2GR). The Kerr angular velocity expressed in the areal radius is Ω = a/R² = 2GJ/(R√(R⁴+4G²J²)); with that replacement the Smarr identity holds identically. Please correct the equation or confirm that the printed form is a transcription error.
[§III, Eq. (13)] The step ∮_S R̃ d²V = 8π used in (13) is Gauss–Bonnet with Euler characteristic χ = 2, and therefore presupposes S ≅ S². The justification for that topology is given only later, in §I, as 'One can show that the topology of S is necessarily S² if the flux of matter or gravitational energy across H is non-zero', with neither proof nor reference. Either supply a one-line argument or a citation before (13), or write (13) with the Euler characteristic and note that the non-zero-flux case forces χ = 2.
[§III, Eq. (15)] The asserted positivity of F_g requires N_r > 0. The paper defines permissible lapses through D_a r = N_r r̂_a but never states that they are positive, although positivity is used in the sentence 'the energy flux is manifestly non-negative'. Please state the restriction explicitly.
[§III, between Eqs. (9) and (12)] The reduction from (9) to (12) is summarized in a single word, 'simplifying'. The intermediate steps — inserting (10), using (11), and cancelling the total derivative 2D_a α^a against the surface terms — are not shown, and sign conventions for K_ab and K̃_ab differ across the literature. A short appendix with one intermediate line would let a reader verify the identity without reconstructing it.
[§IV, Eq. (24)] Because k_o and Ω_o in (21)–(22) are chosen to reproduce the Kerr values, the canonical first law (24) is an identity by construction rather than an independent derivation. The paper is transparent about this ('each cross-section is assigned simply that mass which it would have in the Kerr family'), but the abstract's phrase 'generalizations of the first law' would read more accurately if the derived content were attributed to (19)–(20) and (24) described as a canonical specialization of that family.
[Abstract and §I] The Bondi–Sachs analogy is supported by four structural properties — no coordinates, positivity, locality, and vanishing in spherical symmetry — but no calculation is given showing that F_g reduces to the News flux in an appropriate null limit. Since the abstract states the analogy, one sentence on the precise sense in which the fluxes are 'analogous', or a pointer to where such a limit is worked out, would strengthen the claim.
[§I, first paragraph] The comparison with Hayward's trapping horizons ('our analysis and results differ considerably from those of Hayward's') is asserted rather than substantiated. One concrete point of difference in the flux formula, or in the treatment of angular momentum, would make the comparison useful to a reader familiar with that work.
[Reference [4]] Reference [4] includes 'A. Ashtekar and B. Krishnan (in preparation)' and is cited in support of the claim that the framework 'has been used to extract physics from initial data of widely separated black holes'. Either mark the citation as unpublished in the text or rely on the thesis portion of [4] for that claim.
[§IV, Eq. (19)] The notation ∫_{Ω₁}^{Ω₂} dΩ ∮_S j_φ d²V is ambiguous; presumably it denotes an integral over the one-parameter family of cross-sections weighted by dΩ. Please define it explicitly, since the passage from (19) to (20) depends on how this term is interpreted.
[§I, after the definition] The inference 'Since θ_(ℓ) = 0 and θ_(n) < 0, it follows that K̃ > 0' is correct but compressed; a one-line indication of how K̃ is related to θ_(n) would help the reader who wants to check the monotonicity of the area.
[Production] The title and abstract contain stray spaces ('B alance', 's econd', 'ﬂuxes'), and 'k_R := 1/2R' in §IV should read '1/(2R)'. A production pass is needed before publication.

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The paper

The paper claims

A growing black hole's area change is written as a local sum of matter and gravitational energy fluxes, giving a first law that holds while the hole is still evolving.

The reading

This letter asks what can be said about a black hole's growth using only geometry on the horizon itself, without knowing the whole spacetime. The authors define a dynamical horizon as a space-like three-dimensional surface foliated by two-spheres on which one null expansion vanishes and the other is strictly negative, then feed the standard $3+1$ constraint equations on that surface into a $2+1$ split. The payoff is a surface integral for angular momentum and volume integrals for the fluxes of matter and gravitational energy crossing the horizon, all built from fields defined locally on the horizon. The central identity states that the change in the horizon's area radius equals the matter energy flux plus a positive gravitational flux built from the shear of the horizon's null normal and a transverse acceleration term, $F_{(R)g} = (1/16πG)∫ N_R(|σ|^2 + 2|ζ|^2)d^3V$. If correct, this delivers a physical-process version of the first law of black hole mechanics that holds while the hole is accreting and radiating, and balance laws that mirror the Bondi–Sachs ones at null infinity but can be checked on a finite region of the horizon.

What carries the argument

The engine is the pair of Hamiltonian and momentum constraints satisfied by Cauchy data $(q_{ab}, K_{ab})$ on the space-like horizon, combined with two geometric identities specific to the setting. The first is the Gauss–Codazzi relation expressing the three-dimensional Ricci scalar $R$ in terms of the two-dimensional curvature $R̃$ of the leaves plus extrinsic-curvature squares and a total derivative. The second is the identity $K + K̃ = K_{ab}ŕ^aŕ^b$, which follows from the vanishing of the expansion $θ_{(ℓ)}$ and converts the expansion condition into an algebraic relation between the horizon and cross-section extrinsic curvatures. Because the leaves are two-spheres, Gauss–Bonnet turns the integral of $N_r R̃$ into $8π(r_2 - r_1)$, a boundary term depending only on the two chosen cross-sections. What remains is a volume integral over the horizon whose integrand splits cleanly into a matter part and a piece quadratic in the shear $σ_{ab}$ and the transverse vector $ζ_a$ — the piece the authors identify as gravitational radiation.

Extended reading notes

Core claim

The paper's claim is that on a dynamical horizon — a space-like three-surface foliated by marginally trapped two-spheres — the constraint equations plus a Gauss–Codazzi decomposition force an exact relation between area change and energy flux. With the lapse $N_r$ tied to a radial coordinate by $D_a r = N_r ŕ_a$, equation (14) reads $(r_2 - r_1)/2G = F_{(r)m} + F_{(r)g}$, where the matter term is $∫ T_{ab}τ̂^a ξ^b_{(r)} d^3V$ and the gravitational term is $(1/16πG)∫ N_r(|σ|^2 + 2|ζ|^2)d^3V$. Choosing $r$ to be the area radius $R = √(a/4π)$ makes the statement concrete: the increase in area radius is exactly accounted for by matter energy and by a gravitational flux that is local, manifestly non-negative, and zero in spherical symmetry. Combining this with the angular-momentum balance law yields, for time-evolution fields $t^a = N_r ℓ^a - Ω φ^a$, the generalized first law $dr/2G + Ω dJ = (k_r/8πG)da + Ω dJ = dE_t$, with effective surface gravity $k_R = 1/2R$. When the horizon metric is axisymmetric, a canonical choice of lapse and angular velocity — the ones the Kerr family would have at that area and angular momentum — assigns each cross-section the mass $M = √(R^4 + 4G^2J^2)/(2GR)$ and gives $dM = (k_o/8πG)da + Ω_o dJ$.

Load-bearing premise

The load-bearing premise is that the combination $|σ|^2 + 2|ζ|^2$ captures the complete gravitational energy flux across the horizon; the paper argues this by positivity, locality, and vanishing in spherical symmetry, and calls the identification natural rather than proving that no other radiative modes contribute.

Editorial extensions

If this is right

The second law of black hole mechanics holds on a dynamical horizon as a derived statement: since $θ_{(ℓ)} = 0$ and $θ_{(n)} < 0$ imply $K̃ > 0$, the area of the cross-sections increases monotonically along the outward radial direction.
A physical-process version of the first law, $dM = (k_o/8πG)da + Ω_o dJ$, holds for evolving horizons rather than only for equilibrium ones, with the surface gravity and angular velocity allowed to vary with radius.
Each cross-section of an axisymmetric dynamical horizon can be assigned a mass equal to the mass a Kerr black hole would have with the same area and angular momentum, so the mass is fixed by local fields on that cross-section alone.
The matter and gravitational fluxes satisfy finite balance equations over any segment $ΔH$ of the horizon, giving an infinite family of identities that a numerical evolution should reproduce.
Because the time-evolution vector $t^a$ need not be causal and $φ^a$ need not be a Killing field of $q_{ab}$, the generalized energy $E_t$ can decrease even while area grows, which the authors describe as the analogue of the Penrose process for a dynamical horizon.

Reading between the lines

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

The flux formula gives numerical relativity a local diagnostic: the integral of $|σ|^2 + 2|ζ|^2$ over the apparent horizon can be compared against the Bondi–Sachs energy loss at null infinity in the same simulation, an internal consistency check that does not require evolving to future null infinity.
Because the lapse rescaling freedom is by a positive function of radius rather than a constant, the construction suggests a one-parameter family of quasi-local mass definitions on a single horizon, interpolating between the irreducible mass $R/2G$ and the Kerr-matched mass $M$ depending on how much rotation is attributed to the cross-section.
If future work shows that gravitational radiation crossing a dynamical horizon requires degrees of freedom beyond $σ_{ab}$ and $ζ_a$, the area identity would survive but the reading of the second term as the complete gravitational energy flux would not, and the balance law would need an additional term; a strongly sheared, non-spherical collapse is the natural place to look for the discrepancy.
The Kerr-matched mass assignment suggests a practical definition for the mass of a common apparent horizon in a binary black hole merger, where no global Killing field exists but the horizon is briefly axisymmetric.

Original abstract

Dynamical horizons are considered in full, non-linear general relativity. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local, the energy flux is positive and change in the horizon area is related to these fluxes. The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,546 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.80 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Dynamical horizons are considered in full, non-linear general relativity. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local, the energy flux is positive and change in the horizon area is related to these fluxes. The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Dynamical horizons are considered in full, non-linear general relativity.	background 0.80	0.35	0.04	asserted 0.87
s1	Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained.	result 0.73	0.91	0.31	asserted 1.00
s2	Fluxes are local, the energy flux is positive and change in the horizon area is related to these fluxes.	result 0.98	0.83	0.26	asserted 1.00
s3	The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics.	interpretation 0.80	0.82	0.57	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (8)

[1]

A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krish - nan, J. Lewandowski and J. Wisniewski, Phys. Rev. Lett. 85, 3564-3567 (2000)

View on Pith 2000
[2]

A. Ashtekar, S. Fairhurst, and B. Krishnan, Phys. Rev. D62 104025 (2000)

View on Pith 2000
[3]

A. Ashtekar, C. Beetle, and J. Lewandowski, Phys. Rev. D64, 044016 (2001)

View on Pith 2001
[4]

and from numerical simulations of the ﬁnal phases of black hole collisions [1, 5]. The purpose of this letter is to outline a generalization of these ideas to fully dynami- cal situations in which matter and gravitational radiation can fall into black holes. Our deﬁnition of a dynamical horizon is geared to prac- tical applications in astrophysical situat...

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[5]

B. Krishnan, Thesis dissertation, Penn State (2002); A. Ashtekar and B. Krishnan (in preparation)

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[6]

O. Dreyer, B. Krishnan, E. Schnetter and D. Shoemaker, gr-qc/0206008

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[7]

S. Hayward, Phys. Rev. D49, 6467-6474 (1994)

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[8]

R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press) (1994)

View on Pith 1994

Forward citations

Cited by 22 Pith papers

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Dynamical Horizons and their Properties
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The change in a dynamical horizon's area equals a local, positive flux of matter plus gravitational energy, yielding a finite first law of black hole mechanics in fully non-linear general relativity.
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The paper derives finite, physical-process versions of the first and second laws of black hole thermodynamics for dynamical horizon segments, extending the first law to black holes arbitrarily far from equilibrium.
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A Schwinger-Keldysh EFT for dissipative systems coupled to dynamical gravity requires a dynamical environment sector, modeled here by HydroEFT, and yields dissipative scalar and gravitational wave dynamics plus a gene...
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hep-th 2019-08 conditional novelty 6.0 of 10

A family of horizon boundary conditions labeled by s yields BMS-like algebras with spin-s supertranslations, while another choice gives a non-linear Heisenberg-like algebra whose generators build the BMS-like ones.
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hep-th 2018-06 conditional novelty 6.0 of 10

For marginally trapped surfaces satisfying the paper's 'minimar' conditions, the coarse-grained outer entropy equals one quarter the area, and the entropy-maximizing interior is constructed explicitly.
A New Area Law in General Relativity
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The area of a hypersurface foliated by marginally trapped surfaces increases monotonically along the foliation, even across changes of signature, yielding an area law that applies in cosmology.
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gr-qc 2006-04 conditional novelty 6.0 of 10

Dynamical horizon diagnostics are implemented in three numerical spacetimes, showing how black holes grow, settle to Kerr, and form inner timelike membranes.
Dynamical Entropy Is a Noether Charge
hep-th 2026-07 reject novelty 5.0 of 10

A Noether-charge derivation of the known HWZ dynamical entropy for generic null surfaces, whose local second-law proof is invalid as printed because Eq. (19) has the wrong sign in K.
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How later papers used it

81 later papers cite it

Evidence: substantial. 74 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 167 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

5 later papers by other authors are listed below: 3 built on it and 2 compared their own results with it.

Builds on it 3 papers

Generalised this paper's definition of dynamical horizons by removing the restriction that it be a spacelike surface.

On the paradox of Hawking radiation in a maximally extended Schwarzschild solution (2013) · 0.98 · paraphrased: the licence does not allow quoting
Generalised this paper's definition of dynamical horizons by removing the restriction that it be a spacelike surface.

Astrophysical black holes may radiate, but they do not evaporate (2013) · 0.98 · paraphrased: the licence does not allow quoting
Generalised this paper's definition of dynamical horizons, removing the restriction that it is a spacelike surface, to define MOTS.

Astrophysical Black Hole horizons in a cosmological context: Nature and possible consequences on Hawking Radiation (2014) · 0.98 · paraphrased: the licence does not allow quoting

Compares with it 2 of 4 papers with this label

Compared its own expression with this paper's expression for the total flux.

Gravitational radiation and angular momentum flux from a spinning dynamical black hole (2013) · 1.00 · paraphrased: the licence does not allow quoting
Calculated the energy flux for dynamical horizons and compared it with this paper's results for a particular choice of the displacement vector.

Quasi-Local Energy Flux of Spacetime Perturbation (2008) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

75 arXiv papers cite this in 160 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 2 papers

The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics.

How Black Holes Grow · 1.00
These in turn lead to generalizations of the first and second laws of black hole mechanics.

Dynamical Horizons and their Properties · 0.96

Papers citing it · 99 sentences without a confident label

Builds on it 3 papers

We generalise the definition of dynamical horizons by Ashtekar and Krishnan [ref] by removing the restriction that it be a spacelike surface.

Astrophysical black holes may radiate, but they do not evaporate · Trapping 3-surfaces · 0.98
We generalise the definition of dynamical horizons by Ashtekar and Krishnan [ref] by removing the restriction that it be a spacelike surface.

On the paradox of Hawking radiation in a maximally extended Schwarzschild solution · Trapping 3-surfaces · 0.98
To define MOTS we generalise the definition of dynamical horizons by Ashtekar and Krishnan [ref] by removing the restriction that it is a spacelike surface.

Astrophysical Black Hole horizons in a cosmological context: Nature and possible consequences on Hawking Radiation · Equation for MOTS is LRS-II spacetimes · 0.98

Uses its method or tool 4 papers

[ref] , we compute the energy and angular momentum fluxes emitted in gravitational waves.

Distinguishing scale-dependent Planck stars from renormalization group improved Schwarzschild black holes by Gravitational waves · Introduction · 0.98
The argument is detailed in [ref] and in [ref] , where numerical methods for this purpose are given.

Astrophysical black holes may radiate, but they do not evaporate · The laws of black hole thermodynamics · 0.86
We will next calculate both the mass and the angular momentum of a frozen star using the standard expressions [ref] , which are also valid for an approximate, isolated horizon [ref] , such as the outer surface of a frozen star.

Sourcing the Kerr geometry · Mass and angular momentum · 0.98

Compares with it 4 papers

Recall that the total flux of Ashtekar-Krishnan [ref] , we compare our expression with Ashtekar's expression.

Gravitational radiation and angular momentum flux from a spinning dynamical black hole · Angular momentum and energy-momentum and their flux of a spinning DH · 1.00
The flux formula ([ref]) differs from that given in [ref] .

Hawking radiation from dynamical horizons · Introduction · 0.94
In section V, we calculate the energy flux for dynamical horizons and compare with the results obtained by Ashtekar and Krishnan [ref] for a particular choice of the displacement vector.

Quasi-Local Energy Flux of Spacetime Perturbation · Introduction · 1.00

Cites it as support 2 papers

Continuous accretion of the in-falling matter (satisfying the physically reasonable energy conditions), onto the black hole makes the MOTS spacelike [ref] , and no radiation can escape from the points arbitrarily close to these MOTS.

Accreting Black Holes radiate classical Vaidya radiation to pave way for Hawking radiation · Introduction · 0.97
In this regard black holes can be treated as local objects ( [ref] , [ref] ), and you don't have to worry about the far future fate of the universe in order to know what happens.

Astrophysical black holes may radiate, but they do not evaporate · Astrophysical implications · 0.96

Cites it as background 37 papers

However, due to the global and teleological property of event horizon, the notion of dynamical horizon was developed and its properties were investigated, where, in particular, the first and second laws of black hole mechanics was generalized to the dynamical horizon [ref] .

The black hole dynamical horizon and generalized second law of thermodynamics · Introduction · 0.84
To evaluate the Hawking flux, we recall that there are two well known (and related) definitions of local black hole horizon, the future outward trapping horizon (FOTH) [ref] and the Dynamical Horizon [ref] (or its equilibrium version called the isolated horizon).

Hawking radiation from dynamical horizons · Introduction · 0.96
Two possible candidates for the horizon in the dynamical case are the “future outer trapping horizon” (FOTH) introduced by Hayward [ref] and the “dynamical horizon” (DH) proposed by Ashtekar [ref] .

When null energy condition meets ADM mass · Generalization to dynamical black holes · 0.90

Discussion

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A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krish - nan, J. Lewandowski and J. Wisniewski, Phys. Rev. Lett. 85, 3564-3567 (2000)

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A. Ashtekar, S. Fairhurst, and B. Krishnan, Phys. Rev. D62 104025 (2000)

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A. Ashtekar, C. Beetle, and J. Lewandowski, Phys. Rev. D64, 044016 (2001)

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and from numerical simulations of the ﬁnal phases of black hole collisions [1, 5]. The purpose of this letter is to outline a generalization of these ideas to fully dynami- cal situations in which matter and gravitational radiation can fall into black holes. Our deﬁnition of a dynamical horizon is geared to prac- tical applications in astrophysical situat...

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[5] [5]

B. Krishnan, Thesis dissertation, Penn State (2002); A. Ashtekar and B. Krishnan (in preparation)

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O. Dreyer, B. Krishnan, E. Schnetter and D. Shoemaker, gr-qc/0206008

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S. Hayward, Phys. Rev. D49, 6467-6474 (1994)

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R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press) (1994)

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