REVIEW 3 major objections 3 minor 116 references
Incidence Relations for Self-Dual Black Holes
T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper constructs explicit closed-form deformed incidence relations for the three self-dual black hole spacetimes—Eguchi-Hanson, self-dual Taub-NUT, and self-dual Plebański-Demiański—using Kerr-Schild coordinates and DM recursion in…
desk verdict Solid closed-form incidence relations for EH and SDTN, with a genuinely nice SDTN linearization; the SDPD section carries a load-bearing gap where the key calculation is skipped. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the deformed incidence relation $\mu^{\dot\alpha}=F^{\dot\alpha}(x,\lambda)$, the curved-space replacement of the flat twistor incidence relation $\mu^{\dot\alpha}=x^{\dot\alpha\alpha}\lambda_\alpha$; it parametrizes α-surfaces and holomorphic twistor lines. The engine is the DM recursion operator $R$, defined by $E_{0\dot\alpha}R[\psi]=E_{1\dot\alpha}[\psi]$, which turns harmonic functions on spacetime into functions on twistor space as a Laurent series in $\lambda_0/\lambda_1$. Applied to the Plebański coordinates $q^{\dot\alpha}$, it generates the corrections to $F^{\dot\alpha}$. The input is the second Plebański scalar $\Phi$, satisfying the second heavenly equation; the paper finds explicit $\Phi$ for EH, SDTN, and SDPD and resums the recursion to all orders.
What would settle it
Substitute the SDPD scalar (6.6) directly into the second heavenly equation (2.3), and run the DM recursion one order beyond the displayed terms to check whether Eq. (6.15) is reproduced; a failure would invalidate the SDPD incidence relation (6.22). A simpler check is to take the $b\to\infty$ limit of (6.22) and verify that, after the stated twistor translation, it reproduces the SDTN formula (5.19).
Extended reading notes
Core claim
The paper's central claim is that the deformed incidence relation $\mu^{\dot\alpha}=F^{\dot\alpha}(x,\lambda)$ is known in closed form for every self-dual black hole, summarized in Eq. (1.5). For Eguchi-Hanson, $F^{\dot\alpha}$ is built from quadratic holomorphic coordinates and given by Eq. (4.19), reproducing known results. For self-dual Taub-NUT, the exact relation is linear in the gravitational coupling $\kappa$ and contains a logarithm whose branch point tracks the Misner string, Eq. (5.19). For self-dual Plebański-Demiański, the relation involves fractional powers with $\Delta=\sqrt{1-8\kappa/b}$ and reduces to the Taub-NUT answer when the acceleration $1/b$ is sent to zero, Eq. (6.22). All three reduce to the flat incidence relation as $\kappa\to 0$, and the paper uses the Taub-NUT formula to write explicit plane-wave zero-rest-mass fields on that background.
Load-bearing premise
The load-bearing premise is that the proposed scalar for the self-dual Plebański-Demiański solution really satisfies the second heavenly equation, and that the calculation omitted in Section 6.3 leading to Eq. (6.15) is correct.
Editorial extensions
If this is right
- For each self-dual black hole, the α-surfaces and twistor lines can be written explicitly in Kerr-Schild coordinates, not just order by order in perturbation theory.
- For self-dual Taub-NUT, the exact linearity in $\kappa$ means the all-order incidence relation is no more complicated than its first correction; the logarithm and its periodicity encode the Misner string in twistor space.
- For self-dual Plebański-Demiański, the closed form with $\Delta=\sqrt{1-8\kappa/b}$ gives a concrete twistor-space description of an accelerating black hole pair, with the non-accelerating limit recovering SDTN.
- The explicit $F^{\dot\alpha}$ provide eikonal phases, so plane-wave zero-rest-mass fields on these backgrounds can be written down; on SDTN this gives Eq. (7.4), with amplitude modulation tied to the Dirac quantization condition for NUT charge.
- Penrose transforms using holomorphic twistor quadrics on the deformed lines yield ASD fields and, in principle, linearized metric perturbations corresponding to inserting 'mini' ASD black holes on SD black hole backgrounds.
Reading between the lines
- The exact linearity of the SDTN incidence relation in $\kappa$ suggests the SDTN deformed twistor space may be an affine object with no higher-order contact terms; a natural test is whether the associated deformed complex structure is likewise linear in the NUT parameter.
- The contrast between the logarithm (SDTN) and the fractional powers (SDPD) points to a twistor-space branch structure tied to the Misner string and the acceleration horizon; this could be probed by computing the monodromy of twistor lines around those loci.
- The EH result's quadratic holomorphic coordinates raise the possibility that other self-dual metrics admit twistor-space coordinates that are low-degree polynomials, which would give a direct dictionary for celestial holography on these backgrounds.
- Because the closed forms give exact null-separation functions and eikonal phases, they are a ready-made input for exact Green's functions and scattering amplitudes on black hole backgrounds; comparing a two-point function built from (5.19) with known instanton Green's functions would be a concrete check.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs local deformed incidence relations μ^α̇=F^α̇(x,λ) for the three self-dual black hole metrics in Kerr-Schild coordinates—Eguchi-Hanson, self-dual Taub-NUT, and self-dual Plebański-Demiański—by implementing Dunajski-Mason recursion in Plebański second heavenly coordinates. For each solution it proposes a Plebański scalar, runs the recursion to all orders, and reports closed-form incidence relations: Eqs. (4.19), (5.19), and (6.22), summarized in Eq. (1.5). The SDTN result is claimed to be exactly linear in the gravitational coupling κ, and the SDPD result is claimed to reduce to SDTN in the non-accelerating limit. The paper also sketches applications to zero-rest-mass fields and to twistor-quadric perturbations. The EH and SDTN sections are explicit and largely checkable; the SDPD section contains several steps that are delegated to omitted or abbreviated calculations.
Significance. Conditional on the SDPD gaps being filled, this is a useful and significant contribution: explicit closed-form incidence relations for SDTN and SDPD in KS coordinates have not appeared in the prior literature, and the exact linearity of the SDTN relation in κ is a striking structural result. The EH section cleanly reproduces known quadratic holomorphic coordinates, and the Penrose-transform applications in Sec. 7 are natural and potentially valuable for scattering and perturbation theory. The paper does not supply machine-checkable algebra, so the omitted SDPD computations are the key correctness risk; if they are supplied and verified, the paper would merit publication.
major comments (3)
- [Sec. 6.2, Eq. (6.6)] The claim that Φ in Eq. (6.6) is a second Plebański scalar for the SDPD metric is not demonstrated. The text computes Φ_α̇β̇ and the Weyl tensor, but it never verifies the second heavenly equation □Φ = κ Φ_α̇β̇ Φ^{α̇β̇}, Eq. (2.3), for this Φ, nor does it show that the KS coordinates (p,q) are Plebański coordinates in the sense of Sec. 2.1. Since the DM recursion in Eqs. (6.10)–(6.11) and hence the final formula (6.22) presuppose (2.3), this is load-bearing. Please include the verification explicitly, or cite and state the theorem from which it follows.
- [Sec. 6.3, Eqs. (6.12)–(6.19)] The derivation of the recursion relation for G is delegated to 'a tedious calculation' in Eq. (6.15), and the subsequent integration to Eq. (6.19) fixes the integration constant only by the κ→0 limit. The final closed form for F^ḋ1 in Eqs. (6.19) and (6.22) depends on this recursion to all orders, so the omitted algebra is not a presentation detail. Please supply the calculation leading to Eq. (6.15), including the PDE/ODE step, and justify the choice of primitive in Eqs. (6.16)–(6.19) with sufficient regularity conditions.
- [Sec. 6.3, Eq. (6.20)] The construction of F^ḋ0 is not established. The text states that 'the details will be omitted for reasons of space' and that it is easy to verify L_γ̇[F^ḋ0 F^ḋ1] = 0 by direct computation, but the incidence relation requires L_γ̇[F^ḋ0] = 0 individually, as in Eq. (2.23); annihilation of the product does not imply this. The displayed F^ḋ0 in Eq. (6.22) is therefore unsupported as it stands. Please provide the direct verification of L_γ̇[F^ḋ0] = 0 and the derivation of the normalization condition stated in Eq. (6.20).
minor comments (3)
- [Sec. 6.1, before Eq. (6.4)] The phrase 'In the spinor notation, Eq. (5.3) boils down to' should refer to Eq. (6.1), since Eq. (6.4) is the spinor form of the SDPD metric, not of the SDTN metric in Eq. (5.3).
- [Sec. A.2.4] The word 'descried' appears twice in the reproduction of the KS metrics; it should read 'described'.
- [Sec. 7.1] The claim that the plane-wave expressions in Eqs. (7.2a) and (7.2b) satisfy the zero-rest-mass equations is asserted rather than shown; a short derivation using L_γ̇[F^α̇] = 0 would make the application self-contained.
Circularity Check
No significant circularity: the incidence relations are derived from Plebański scalars via the Dunajski–Mason recursion, with no fitted target inputs; the SDPD section has omitted verifications, but those are gaps in exposition, not circular steps.
full rationale
The paper's derivation chain is: (i) take known or newly proposed Plebański second heavenly scalars for EH, SDTN, and SDPD; (ii) implement the Dunajski–Mason recursion operator to construct functions F^dotα(x,λ) satisfying L_dotγ[F^dotα]=0; (iii) resum the recursion to closed form; (iv) verify the flat-space limit κ→0. The incidence relations are not assumed in the input scalars. For EH, the scalar (4.11) and quadratic holomorphic coordinates X^dotαdotβ (4.17) are standard inputs, and the closed form (4.19) is a concrete combination of X with a constant reference spinor, derived rather than fitted. For SDTN, the scalar (5.6) is quoted from the author's own Ref. [22], but the headline result—linearity of the incidence relation in κ, Eq. (5.19)—is not present in that input; it emerges from the recursion chain (5.17) and the harmonicity of ζ^n. For SDPD, the scalar (6.6) is new, and the recursion (6.12) plus the first-order equation (6.15) determine F^dot1; the integration constant is fixed by the flat limit, not by the target formula. The product relation (6.21) is a simplification of the derived expression. The self-citations [22,23] are used as input structures (KS/Plebański coordinates, SDTN as KS metric), but the incidence relations are not contained in those cited results, so the self-citation is not circular. The genuine weaknesses are in Sec. 6.3: the second heavenly equation for Φ (6.6) is not explicitly verified, the step to Eq. (6.15) is called 'a tedious calculation', and the check of L_dotγ[F^dot0F^dot1]=0 is delegated as 'easy to verify'. These are omitted or compressed verifications—correctness risks, not circular reductions. No equation in the paper is defined in terms of the target incidence relation, and no parameter is fitted to the claimed closed forms. Accordingly, the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- standard math A self-dual Ricci-flat complex four-manifold locally admits Plebanski coordinates (p,q) and a scalar Φ satisfying the second heavenly equation (2.3).
- domain assumption The Dunajski-Mason recursion operator R exists on ker(□Φ) and the Laurent series (2.36) descends to twistor space, up to primitive choices.
- domain assumption The Kerr-Schild coordinates in each example are also Plebanski's second heavenly coordinates.
- ad hoc to paper For SDPD the posited Plebanski scalar (6.6) satisfies the heavenly equation and the recursion ODE (6.15) holds.
Cite this review
Pith. "Pith review of Incidence Relations for Self-Dual Black Holes." pith.science (2026). https://pith.science/paper/EQOIDTMZ
@misc{pith2026260807774,
author = {Pith},
title = {Pith review of: Incidence Relations for Self-Dual Black Holes},
year = {2026},
howpublished = {\url{https://pith.science/paper/EQOIDTMZ}},
note = {Machine review of arXiv:2608.07774}
}
read the original abstract
In Penrose's nonlinear graviton construction, a self-dual spacetime arises as the moduli space of holomorphic twistor lines in curved twistor space, whose explicit parametrization is concretely given by the incidence relation. We demonstrate a concrete local construction of the incidence relations for self-dual black hole spacetimes in Kerr-Schild coordinates: Eguchi-Hanson, self-dual Taub-NUT, and self-dual Pleba\'nski-Demia\'nski. This is facilitated by implementing the Dunajski-Mason recursion in the framework of Pleba\'nski's second heavenly equation, for which we explicitly find the Pleba\'nski scalar. Our results describe closed-form formulae. Notably, for the self-dual Taub-NUT solution, the exact incidence relation exhibits linear dependence on the gravitational coupling. The construction of zero-rest-mass fields on self-dual black hole backgrounds is briefly sketched as an application.
Reference graph
Works this paper leans on
-
[1]
Twistor theory: An approach to the quantisation of fields and space-time,
R. Penrose and M. A. H. MacCallum, “Twistor theory: An approach to the quantisation of fields and space-time,”Physics Reports6no. 4, (1973) 241–315
1973
-
[2]
Twistor theory at fifty: from contour integrals to twistor strings,
M. Atiyah, M. Dunajski, and L. Mason, “Twistor theory at fifty: from contour integrals to twistor strings,”Proc. Roy. Soc. Lond. A473no. 2206, (2017) 20170530, arXiv:1704.07464 [hep-th]
arXiv 2017
-
[3]
Nonlinear gravitons and curved twistor theory,
R. Penrose, “Nonlinear gravitons and curved twistor theory,”Gen. Rel. Grav.7(1976) 31–52
1976
-
[4]
Gravity, twistors and the MHV formalism,
L. J. Mason and D. Skinner, “Gravity, twistors and the MHV formalism,”Commun. Math. Phys.294(2010) 827–862,arXiv:0808.3907 [hep-th]
arXiv 2010
-
[5]
A connection between the Einstein and Yang-Mills equations,
L. J. Mason and E. T. Newman, “A connection between the Einstein and Yang-Mills equations,”Commun. Math. Phys.121(1989) 659–668
1989
-
[6]
Some solutions of complex Einstein equations,
J. F. Plebanski, “Some solutions of complex Einstein equations,”Journal of Mathematical Physics16no. 12, (1975) 2395–2402
1975
-
[7]
On the separation of Einsteinian substructures,
J. F. Pleba´ nski, “On the separation of Einsteinian substructures,”J. Math. Phys.18(1977) 2511–2520
1977
-
[8]
Self-dual two-forms and gravity,
R. Capovilla, J. Dell, T. Jacobson, and L. Mason, “Self-dual two-forms and gravity,”Class. Quant. Grav.8(1991) 41–57
1991
Show all 116 references
-
[9]
Sur le probl` eme de Pfaff,
G. Darboux, “Sur le probl` eme de Pfaff,”Bulletin des sciences math´ ematiques et astronomiques6no. 1, (1882) 14–36
-
[10]
Twistor sigma models for quaternionic geometry and graviton scattering,
T. Adamo, L. Mason, and A. Sharma, “Twistor sigma models for quaternionic geometry and graviton scattering,”Adv. Theor. Math. Phys.27no. 3, (2023) 623–681,arXiv:2103.16984 [hep-th]
2023 arXiv
-
[11]
Hyper-K¨ ahler hierarchies and their twistor theory,
M. Dunajski and L. J. Mason, “Hyper-K¨ ahler hierarchies and their twistor theory,” Communications in Mathematical Physics213no. 3, (2000) 641–672,arXiv:math/0001008 [math.DG]
2000 arXiv
-
[12]
Heavenly hierarchies and curved twistor spaces,
M. Dunajski and L. Mason, “Heavenly hierarchies and curved twistor spaces,”Twistor Newsletter41(1996) 26–34
1996
-
[13]
A recursion operator for ASD vacuums and z.r.m fields,
M. Dunajski and L. Mason, “A recursion operator for ASD vacuums and z.r.m fields,” Twistor Newsletter43(1997) 24–29
1997
-
[14]
Scattering on self-dual Taub-NUT,
T. Adamo, G. Bogna, L. Mason, and A. Sharma, “Scattering on self-dual Taub-NUT,” Class. Quant. Grav.41no. 1, (9, 2024) 015030,arXiv:2309.03834 [hep-th]
2024 arXiv
-
[15]
Graviton scattering on self-dual black holes,
T. Adamo, G. Bogna, L. Mason, and A. Sharma, “Graviton scattering on self-dual black holes,”Class. Quant. Grav.43no. 5, (2026) 055005,arXiv:2507.18605 [hep-th]
2026 arXiv
-
[16]
Black holes in Klein space,
E. Crawley, A. Guevara, N. Miller, and A. Strominger, “Black holes in Klein space,”JHEP 10(2022) 135,arXiv:2112.03954 [hep-th]
2022 arXiv
-
[17]
Self-dual black holes in celestial holography,
E. Crawley, A. Guevara, E. Himwich, and A. Strominger, “Self-dual black holes in celestial holography,”JHEP09(2023) 109,arXiv:2302.06661 [hep-th]
2023 arXiv
-
[18]
Self dual black holes as the hydrogen atom,
A. Guevara and U. Kol, “Self dual black holes as the hydrogen atom,”arXiv:2311.07933 [hep-th]. – 44 –
-
[19]
An exact black hole scattering amplitude,
A. Guevara, U. Kol, and H. Tran, “An exact black hole scattering amplitude,” arXiv:2412.19627 [hep-th]
-
[20]
New near extremal black holes and love symmetry,
A. Guevara and U. Kol, “New near extremal black holes and love symmetry,” arXiv:2511.18637 [hep-th]
-
[21]
Note on the Kerr spinning-particle equations of motion,
J.-H. Kim, “Note on the Kerr spinning-particle equations of motion,”arXiv:2512.23697 [gr-qc]
-
[22]
Single Kerr-Schild metric for Taub-NUT instanton,
J.-H. Kim, “Single Kerr-Schild metric for Taub-NUT instanton,”Phys. Rev. D111no. 2, (2025) L021703,arXiv:2405.09518 [hep-th]
2025 arXiv
-
[23]
Newman-Janis algorithm from Taub-NUT instantons,
J.-H. Kim, “Newman-Janis algorithm from Taub-NUT instantons,”Phys. Rev. Lett.136 no. 23, (6, 2026) 231401,arXiv:2412.19611 [gr-qc]
2026 arXiv
-
[24]
The dual twistor theory of self-dual black holes,
T. Adamo, B. Araneda, and S. Seet, “The dual twistor theory of self-dual black holes,” arXiv:2601.05037 [hep-th]
-
[25]
Higher-spins on Taub-NUT and higher-spin Taub-NUT,
E. Skvortsov and Y. Yin, “Higher-spins on Taub-NUT and higher-spin Taub-NUT,”JHEP 12(2025) 099,arXiv:2508.18804 [hep-th]
2025 arXiv
-
[26]
Memory effect from the scattering of Taub-NUT black holes,
G. Doran, R. Monteiro, and N. Moynihan, “Memory effect from the scattering of Taub-NUT black holes,”arXiv:2603.24365 [hep-th]
-
[27]
Rotating, charged, and uniformly accelerating mass in general relativity,
J. F. Plebanski and M. Demianski, “Rotating, charged, and uniformly accelerating mass in general relativity,”Annals Phys.98(1976) 98–127
1976
-
[28]
Asymptotically flat selfdual solutions to Euclidean gravity,
T. Eguchi and A. J. Hanson, “Asymptotically flat selfdual solutions to Euclidean gravity,” Phys. Lett. B74(1978) 249–251
1978
-
[29]
Selfdual solutions to Euclidean gravity,
T. Eguchi and A. J. Hanson, “Selfdual solutions to Euclidean gravity,”Annals Phys.120 (1979) 82
1979
-
[30]
Gravitational instantons,
T. Eguchi and A. J. Hanson, “Gravitational instantons,”Gen. Rel. Grav.11(1979) 315–320
1979
-
[31]
Gravitational instantons,
S. W. Hawking, “Gravitational instantons,”Phys. Lett. A60no. 2, (1977) 81
1977
-
[32]
Gravitational multi-instantons,
G. W. Gibbons and S. W. Hawking, “Gravitational multi-instantons,”Phys. Lett. B78 (1978) 430
1978
-
[33]
U(1) x U(1) quaternionic metrics from harmonic superspace,
P.-Y. Casteill, E. Ivanov, and G. Valent, “U(1) x U(1) quaternionic metrics from harmonic superspace,”Nucl. Phys. B627(2002) 403–444,arXiv:hep-th/0110280
2002 arXiv
-
[34]
Killing-Yano tensor and supersymmetry of the self-dual Pleba´ nski-Demia´ nski solution,
M. Nozawa and T. Houri, “Killing-Yano tensor and supersymmetry of the self-dual Pleba´ nski-Demia´ nski solution,”Class. Quant. Grav.33no. 12, (2016) 125008, arXiv:1510.07470 [hep-th]
2016 arXiv
-
[35]
Hidden symmetries of generalised gravitational instantons,
B. Araneda, “Hidden symmetries of generalised gravitational instantons,”Annales Henri Poincare26no. 11, (2025) 4021–4049,arXiv:2309.05617 [gr-qc]
2025 arXiv
-
[36]
Self-dual Kerr-Schild metrics and null Maxwell fields,
K. P. Tod, “Self-dual Kerr-Schild metrics and null Maxwell fields,”J. Math. Phys.23no. 6, (1982) 1147–1148
1982
-
[37]
Sparling-Tod metric = Eguchi-Hanson,
G. Burnett-Stuart, “Sparling-Tod metric = Eguchi-Hanson,”Twistor Newsletter9(Nov.,
-
[38]
The self-dual classical double copy, and the Eguchi-Hanson instanton,
D. S. Berman, E. Chac´ on, A. Luna, and C. D. White, “The self-dual classical double copy, and the Eguchi-Hanson instanton,”JHEP01(2019) 107,arXiv:1809.04063 [hep-th]
2019 arXiv
-
[39]
The non-linear graviton representing the analogue of schwarzschild or kerr black holes,
G. A. J. Sparling, “The non-linear graviton representing the analogue of schwarzschild or kerr black holes,”Twistor Newsletter1(1976) 14–17. – 45 –
1976
-
[40]
Polygons and gravitons,
N. J. Hitchin, “Polygons and gravitons,”Mathematical Proceedings of the Cambridge Philosophical Society85no. 3, (1979) 465–476
1979
-
[41]
Complete ricci-flat k¨ ahler metrics onC n need not be flat,
C. LeBrun, “Complete ricci-flat k¨ ahler metrics onC n need not be flat,” inSeveral Complex Variables and Complex Geometry, Part 2, vol. 52 ofProceedings of Symposia in Pure Mathematics, pp. 297–304. American Mathematical Society, Providence, RI, 1991
1991
-
[42]
Schwarzschild black holes from twistor space,
T. Adamo, B. Araneda, S. Seet, and A. Sharma, “Schwarzschild black holes from twistor space,”arXiv:2607.06236 [hep-th]
-
[43]
The celestial chiral algebra of self-dual gravity on Eguchi-Hanson space,
R. Bittleston, S. Heuveline, and D. Skinner, “The celestial chiral algebra of self-dual gravity on Eguchi-Hanson space,”JHEP09(2023) 008,arXiv:2305.09451 [hep-th]
2023 arXiv
-
[44]
The geometry of complex self-dual Einstein spaces,
C. P. Boyer, “The geometry of complex self-dual Einstein spaces,” inNonlinear Phenomena, pp. 25–46. Springer, 1983
1983
-
[45]
The geometry of W-gravity,
C. Hull, “The geometry of W-gravity,”Physics Letters B269no. 3, (1991) 257–263
1991
-
[46]
Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group,
S. Chakravarty, L. Mason, and E. T. Newman, “Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group,”Journal of mathematical physics32no. 6, (1991) 1458–1464
1991
-
[47]
Self-dual gravity via Hitchin’s equations,
E. Chac´ on and H. Garcia-Compean, “Self-dual gravity via Hitchin’s equations,”Journal of Mathematical Physics60no. 5, (2019) 052502
2019
-
[48]
A construction of hyper-K¨ ahler metrics,
S. G. Gindikin, “A construction of hyper-K¨ ahler metrics,”Functional Analysis and Its Applications20no. 3, (1986) 238–240
1986
-
[49]
The kinematic algebra from the self-dual sector,
R. Monteiro and D. O’Connell, “The kinematic algebra from the self-dual sector,”JHEP07 no. 7, (2011) 007,arXiv:1105.2565 [hep-th]
2011 arXiv
-
[50]
The Moyal deformation of the 2 nd heavenly equation,
J. Plebanski, M. Przanowski, B. Rajca, and J. Tosiek, “The Moyal deformation of the 2 nd heavenly equation,”Acta Physica Polonica B26no. 5, (1995) 889–902
1995
-
[51]
From principal chiral model to self-dual gravity,
J. F. Pleba´ nski, M. Przanowski, and H. Garcia-Compe´ an, “From principal chiral model to self-dual gravity,”Modern Physics Letters A11no. 08, (1996) 663–673
1996
-
[52]
The Lagrangian of a self-dual gravitational field as a limit of the SDYM Lagrangian,
J. F. Pleba´ nski and M. Przanowski, “The Lagrangian of a self-dual gravitational field as a limit of the SDYM Lagrangian,”Physics Letters A212no. 1-2, (1996) 22–28
1996
-
[53]
Nonlinear graviton as a limit of sl(n,C) chiral fields asn→inf ty,
M. Przanowski, S. Formanski, and F. J. Turrubiates, “Nonlinear graviton as a limit of sl(n,C) chiral fields asn→inf ty,”arXiv:gr-qc/9905078
-
[54]
Self-dual gravity as a large-nlimit of the 2d non-linear sigma model,
Q. H. Park, “Self-dual gravity as a large-nlimit of the 2d non-linear sigma model,”Physics Letters B238no. 2-4, (1990) 287–290
1990
-
[55]
SU(inf ty) (super) gauge theories and self-dual (super) gravity,
C. Castro, “SU(inf ty) (super) gauge theories and self-dual (super) gravity,”Journal of mathematical physics34no. 2, (1993) 681–689
1993
-
[56]
Hidden symmetry of the galileon,
K. Hinterbichler and A. Joyce, “Hidden symmetry of the galileon,”Physical Review D92 no. 2, (2015) 023503
2015
-
[57]
On integrability properties of SU(2) Yang–Mills fields. I. Infinitesimal part,
H. Urbantke, “On integrability properties of SU(2) Yang–Mills fields. I. Infinitesimal part,” J. Math. Phys.25no. 7, (1984) 2321–2324
1984
-
[58]
A new characterization of half-flat solutions to einstein’s equation,
A. Ashtekar, T. Jacobson, and L. Smolin, “A new characterization of half-flat solutions to einstein’s equation,”Commun. Math. Phys.115(1988) 631
1988
-
[59]
Selfduality in four-dimensional Riemannian geometry,
M. F. Atiyah, N. J. Hitchin, and I. M. Singer, “Selfduality in four-dimensional Riemannian geometry,”Proc. Roy. Soc. Lond. A362(1978) 425–461. – 46 –
1978
-
[60]
An example of anH-space,
G. A. J. Sparling and K. P. Tod, “An example of anH-space,”J. Math. Phys.22(1981) 331–332
1981
-
[61]
The non-linear graviton representing the analogue of Schwarzschild or Kerr black holes,
G. A. J. Sparling, “The non-linear graviton representing the analogue of Schwarzschild or Kerr black holes,”Twistor Newsletter1(1976) 14–17
1976
-
[62]
An asymptotically flatH-space,
K. P. Tod, “An asymptotically flatH-space,”General Relativity and Gravitation13no. 2, (1981) 109–122
1981
-
[63]
Equivalence of Eguchi-Hanson metric to two-center Gibbons-Hawking metric,
M. K. Prasad, “Equivalence of Eguchi-Hanson metric to two-center Gibbons-Hawking metric,”Phys. Lett. B83(1979) 310–310
1979
-
[64]
The disjointed thermodynamics of rotating black holes with a NUT twist,
A. M. Ghezelbash, R. B. Mann, and R. D. Sorkin, “The disjointed thermodynamics of rotating black holes with a NUT twist,”Nucl. Phys. B775(2007) 95–119, arXiv:hep-th/0703030
2007 arXiv
-
[65]
The flatter regions of Newman, Unti, and tamburino’s generalized Schwarzschild space,
C. W. Misner, “The flatter regions of Newman, Unti, and tamburino’s generalized Schwarzschild space,”J. Math. Phys.4(1963) 924–938
1963
-
[66]
A new interpretation of the NUT metric in general relativity,
W. B. Bonnor, “A new interpretation of the NUT metric in general relativity,”Math. Proc. Cambridge Phil. Soc.66no. 1, (1969) 145–151
1969
-
[67]
Physical interpretation of NUT metric,
A. Sackfield, “Physical interpretation of NUT metric,” inMathematical Proceedings of the Cambridge Philosophical Society, vol. 70, pp. 89–94, Cambridge University Press. 1971
1971
-
[68]
Spinning cosmic strings and quantization of energy,
P. O. Mazur, “Spinning cosmic strings and quantization of energy,”Phys. Rev. Lett.57 (1986) 929–932
1986
-
[69]
J. B. Griffiths and J. Podolsk´ y,3.4.1. A spinning cosmic string, pp. 36–37. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2009
2009
-
[70]
A combined Kerr-NUT solution of the Einstein field equations,
M. Demianski and E. T. Newman, “A combined Kerr-NUT solution of the Einstein field equations,”Bulletin de l’Academie Polonaise des Sciences, Serie des Sciences Mathematiques, Astronomiques et Physiques14(12, 1966) 653–657
1966
-
[71]
The NUT solution as a gravitational dyon,
J. Dowker, “The NUT solution as a gravitational dyon,”General Relativity and Gravitation 5(1974) 603–613
1974
-
[72]
The gravitational analogues of magnetic monopoles,
J. S. Dowker and J. A. Roche, “The gravitational analogues of magnetic monopoles,”Proc. Phys. Soc.92no. 1, (1967) 1–8
1967
-
[73]
Dual-mass in general relativity,
S. Ramaswamy and A. Sen, “Dual-mass in general relativity,”Journal of Mathematical Physics22no. 11, (1981) 2612–2615
1981
-
[74]
A gravitational analog of the Dirac monopole,
J. Samuel and B. Iyer, “A gravitational analog of the Dirac monopole,”Current Science (1986) 818–824
1986
-
[75]
J. B. Griffiths and J. Podolsk´ y,Minkowski space-time, pp. 36–37. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2009. Section 3.4.1: A spinning cosmic string
2009
-
[76]
L’ether lumineux demontre par l’effet du vent relatif d’ether dans un interferometre en rotation uniforme,
G. Sagnac, “L’ether lumineux demontre par l’effet du vent relatif d’ether dans un interferometre en rotation uniforme,”Comptes Rendus de l’Academie des Sciences157 (1913) 708–710
1913
-
[77]
Sur la preuve de la realite de l’ether lumineux par l’experience de l’interferographe tournant,
G. Sagnac, “Sur la preuve de la realite de l’ether lumineux par l’experience de l’interferographe tournant,”Comptes Rendus de l’Academie des Sciences157(1913) 1410–1413. – 47 –
1913
-
[78]
The Sagnac effect in general relativity,
A. Ashtekar and A. Magnon, “The Sagnac effect in general relativity,”Journal of Mathematical Physics16no. 2, (1975) 341–344
1975
-
[79]
Comments on quantum-mechanical interference due to the earth’s rotation,
J. Sakurai, “Comments on quantum-mechanical interference due to the earth’s rotation,” Physical Review D21no. 10, (1980) 2993
1980
-
[80]
Sagnac effect in relativistic and nonrelativistic physics,
J. Anandan, “Sagnac effect in relativistic and nonrelativistic physics,”Physical Review D24 (1981) 338–346
1981
-
[81]
Complex conformal transformations and zero-rest-mass fields,
B. Araneda, “Complex conformal transformations and zero-rest-mass fields,”Phys. Rev. D 108no. 2, (2023) 024032,arXiv:2305.08756 [gr-qc]
2023 arXiv
-
[82]
Taylor’s theorem in the tensor calculus,
H. Ruse, “Taylor’s theorem in the tensor calculus,”Proceedings of the London Mathematical Society2no. 1, (1931) 87–92
1931
-
[83]
A characteristic function in Riemannian space and its application to the solution of geodesic triangles,
J. L. Synge, “A characteristic function in Riemannian space and its application to the solution of geodesic triangles,”Proceedings of the London Mathematical Society2no. 1, (1931) 241–258
1931
-
[84]
The motion of point particles in curved spacetime,
E. Poisson, A. Pound, and I. Vega, “The motion of point particles in curved spacetime,” Living Rev. Rel.14no. 1, (2011) 7,arXiv:1102.0529 [gr-qc]
2011 arXiv
-
[85]
Green’s Functions for Gravitational Multi-Instantons,
D. N. Page, “Green’s Functions for Gravitational Multi-Instantons,”Phys. Lett. B85(1979) 369–372
1979
-
[86]
Gluon scattering on the self-dual dyon,
T. Adamo, G. Bogna, L. Mason, and A. Sharma, “Gluon scattering on the self-dual dyon,” Lett. Math. Phys.115no. 1, (2025) 18,arXiv:2406.09165 [hep-th]
2025 arXiv
-
[87]
S-algebra in gauge theory: twistor, spacetime and holographic perspectives,
A. Kmec, L. Mason, R. Ruzziconi, and A. Sharma, “S-algebra in gauge theory: twistor, spacetime and holographic perspectives,”Class. Quant. Grav.42no. 19, (2025) 195008, arXiv:2506.01888 [hep-th]
2025 arXiv
-
[88]
Solutions of the zero-rest-mass equations,
R. Penrose, “Solutions of the zero-rest-mass equations,”J. Math. Phys.10(1969) 38–39
1969
-
[89]
Note on the Kerr spinning particle metric,
E. T. Newman and A. I. Janis, “Note on the Kerr spinning particle metric,”J. Math. Phys. 6(1965) 915–917
1965
-
[90]
The Weyl double copy from twistor space,
E. Chac´ on, S. Nagy, and C. D. White, “The Weyl double copy from twistor space,”JHEP 05(2021) 2239,arXiv:2103.16441 [hep-th]
2021 arXiv
-
[91]
Twistorial Foundation for the Classical Double Copy,
C. D. White, “Twistorial Foundation for the Classical Double Copy,”Phys. Rev. Lett.126 no. 6, (2021) 061602,arXiv:2012.02479 [hep-th]
2021 arXiv
-
[92]
Reconstructing classical spacetimes from the S-Matrix in twistor space,
A. Guevara, “Reconstructing classical spacetimes from the S-Matrix in twistor space,” arXiv:2112.05111 [hep-th]
-
[93]
Twistor quadrics and black holes,
B. Araneda, “Twistor quadrics and black holes,”Phys. Rev. D108no. 4, (2023) 044040, arXiv:2212.10491 [gr-qc]
2023 arXiv
-
[94]
The theory of gravitation,
H. Weyl, “The theory of gravitation,”Annalen Phys.54(1917) 117–145
1917
-
[95]
Uniformly accelerating charged mass in general relativity,
W. Kinnersley and M. Walker, “Uniformly accelerating charged mass in general relativity,” Phys. Rev. D2(1970) 1359–1370
1970
-
[96]
Linear superposition of two type-N nonlinear gravitons,
J. F. Pleba´ nski, M. Przanowski, and S. Forma´ nski, “Linear superposition of two type-N nonlinear gravitons,”Physics Letters A246no. 1-2, (1998) 25–31
1998
-
[97]
Extending half-flat metrics,
D. Robinson, “Extending half-flat metrics,”Twistor Newsletter44(1998) 10–11
1998
-
[98]
Some real and complex solutions of Einstein’s equations,
D. Robinson, “Some real and complex solutions of Einstein’s equations,”General relativity and gravitation19(1987) 693–698. – 48 –
1987
-
[99]
Magnetic monopoles in Kaluza-Klein theories,
D. J. Gross and M. J. Perry, “Magnetic monopoles in Kaluza-Klein theories,”Nucl. Phys. B 226(1983) 29–48
1983
-
[100]
Penrose and W
R. Penrose and W. Rindler,Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 4, 1988
1988
-
[101]
Twistor algebra,
R. Penrose, “Twistor algebra,”Journal of Mathematical physics8no. 2, (1967) 345–366
1967
-
[102]
Exact vacuum solutions of Einstein’s equation from linearized solutions,
B. C. Xanthopoulos, “Exact vacuum solutions of Einstein’s equation from linearized solutions,”Journal of Mathematical Physics19no. 7, (1978) 1607–1609
1978
-
[103]
Generating exact solutions to Einstein’s equation using linearized approximations,
A. I. Harte and J. Vines, “Generating exact solutions to Einstein’s equation using linearized approximations,”Phys. Rev. D94no. 8, (2016) 084009,arXiv:1608.04359 [gr-qc]
2016 arXiv
-
[104]
Conformal killing forms on riemannian manifolds,
U. Semmelmann, “Conformal killing forms on riemannian manifolds,”Mathematische Zeitschrift245no. 3, (2003) 503–527
2003
-
[105]
Conformal Yano-Killing tensor for the Kerr metric and conserved quantities,
J. Jezierski and M. Lukasik, “Conformal Yano-Killing tensor for the Kerr metric and conserved quantities,”Class. Quant. Grav.23(2006) 2895–2918,arXiv:gr-qc/0510058
2006 arXiv
-
[106]
Black holes, hidden symmetries, and complete integrability,
V. P. Frolov, P. Krtous, and D. Kubiznak, “Black holes, hidden symmetries, and complete integrability,”Living Rev. Rel.20no. 1, (2017) 6,arXiv:1705.05482 [gr-qc]
2017 arXiv
-
[107]
Isometries and the double copy,
D. A. Easson, G. Herczeg, T. Manton, and M. Pezzelle, “Isometries and the double copy,” JHEP09(2023) 162,arXiv:2306.13687 [gr-qc]
2023 arXiv
-
[108]
Black holes and the double copy,
R. Monteiro, D. O’Connell, and C. D. White, “Black holes and the double copy,”JHEP12 no. 12, (2014) 056,arXiv:1410.0239 [hep-th]
2014 arXiv
-
[109]
Type D spacetimes and the Weyl double copy,
A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, “Type D spacetimes and the Weyl double copy,”Class. Quant. Grav.36(2019) 065003,arXiv:1810.08183 [hep-th]
2019 arXiv
-
[110]
Walker and R
M. Walker and R. Penrose, “On quadratic first integrals of the geodesic equations for type
-
[111]
On a quadratic first integral for the charged particle orbits in the charged Kerr solution,
L. P. Hughston, R. Penrose, P. Sommers, and M. Walker, “On a quadratic first integral for the charged particle orbits in the charged Kerr solution,”Commun. Math. Phys.27(1972) 303–308
1972
-
[112]
spacetimes,”Commun. Math. Phys.18(1970) 265–274
1970
-
[113]
A relation between tree amplitudes of closed and open strings,
H. Kawai, D. C. Lewellen, and S. H. H. Tye, “A relation between tree amplitudes of closed and open strings,”Nucl. Phys. B269(1986) 1–23
1986
-
[114]
A cubic action for selfdual Yang-Mills,
A. Parkes, “A cubic action for selfdual Yang-Mills,”Phys. Lett. B286(1992) 265–270, arXiv:hep-th/9203074
1992 arXiv
-
[116]
Coherent states, background fields, and double copy,
A. Ilderton and W. Lindved, “Coherent states, background fields, and double copy,”JHEP 09(2025) 156,arXiv:2505.16852 [hep-th]. – 49 –
2025
-
[1979]
Mathematical Institute, Oxford; 16 November 1979
6–8. Mathematical Institute, Oxford; 16 November 1979
1979
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.