Black hole thermodynamics in the presence of nonlinear electromagnetic fields
aa r X i v : . [ g r- q c ] M a r ZTF-EP-21-02RBI-ThPhys-2021-6
Prepared for submission to JHEP
Black hole thermodynamics in the presence ofnonlinear electromagnetic fields
A. Bokuli´c, a T. Juri´c, b and I. Smoli´c a, a Department of Physics, Faculty of Science, University of Zagreb,Bijeniˇcka cesta 32, 10000 Zagreb, Croatia b Rudjer Boˇskovi´c Institute,Bijeniˇcka cesta 54, HR-10002 Zagreb, Croatia
E-mail: [email protected] , [email protected] , [email protected] Abstract:
As the interaction between the black holes and highly energetic infallingcharged matter receives quantum corrections, the basic laws of black hole mechanics haveto be carefully rederived. Using the covariant phase space formalism we generalize the firstlaw of black hole mechanics, both “equilibrium state” and “physical process” version, inthe presence of nonlinear electrodynamics fields, defined by Lagrangians depending on bothquadratic electromagnetic invariants, F ab F ab and F ab ⋆F ab . Derivation of this law demandsa specific treatment of the Lagrangian parameters, similar to embedding of the cosmologicalconstant into thermodynamic context. Furthermore, we discuss the validity of energyconditions, several complementing proofs of the zeroth law of black hole electrodynamicsand some aspects of the recently generalized Smarr formula, its (non-)linearity and relationto the first law. Corresponding author. ontents
Thermodynamics has played a pivotal historical role in our understanding of the inter-nal structure of matter. Establishment of the laws of black hole mechanics [9] and theircorrespondence to the basic laws of thermodynamics [12, 13, 61] provides us with similarinvaluable guiding insight into the elusive microscopic nature of spacetime. Stationaryblack holes have constant surface gravity and gauge scalar potentials (zeroth law), obeyenergy constraints upon perturbations (first law), Hawking’s law of nondecreasing hori-zon area (second law) and Smarr formula (Gibbs–Duhem equation). Augmented by thetheoretical prediction of Hawking’s radiation, there is a strong indication that black holesurface gravity and horizon area correspond, respectively, to the temperature and entropy.Over the course of five decades vast effort has been invested into understanding of var-ious aspects of black hole thermodynamics beyond the original Einstein–Maxwell context.Whereas far greater progress has been made in gravitational sector [121], culminating in– 1 –ald’s entropy formula [120] and its subsequent generalizations [15, 110], the gauge sec-tor still lacks unifying picture, especially with respect to nonlinear generalizations of theclassical Maxwell’s electrodynamics.Nonlinear electrodynamics (NLE) is an umbrella term for a broad class of theories,usually those defined by a Lagrangian constructed from two quadratic electromagnetic in-variants, F ab F ab and F ab ⋆F ab . In order to simplify nomenclature, we may sort the NLEtheories into the F -class , with Lagrangians depending only on invariant F ab F ab , and the FG -class , with Lagrangians depending on both invariants. Earliest NLE theories appearedin 1930s, at the dawn of the quantum field theory. In order to cure the inconsistencies ofthe Maxwell’s electrodynamics associated with the infinite self-energy of the point charges,Max Born proposed a F -class NLE theory [16], which was subsequently expanded in collab-oration with Leopold Infeld to a FG -class NLE theory [17]. Born–Infeld theory reappearedhalf a century later, at the beginning of the first superstring revolution, in low energy limitsof the string theory [47], with the string tension α ′ and the BI parameter b being relatedvia 2 πα ′ = 1 /b [113] (for analysis on lattice see [75]). On the other hand, not long afterthe work of Born and Infeld, Werner K. Heisenberg and Hans H. Euler [62] found a 1-loopQED correction to Maxwell’s Lagrangian.Nonlinearities in the electromagnetic interaction are revealed in the scattering of “lightby light”, that is the γγ → γγ process, and the first direct experimental evidence has beenrecently found by the ATLAS Collaboration [1], leading to strengthening of the constraintson parameters of the NLE Lagrangians [41, 89, 90] (for an overview of earlier experimen-tal constraints on NLE theories see [11, 46]). Also, complementary to the conclusionscoming from experiments performed in terrestrial particle colliders, there are cosmologicalconstraints [20], as well as proposed neutrino astrophysics tests [88].Interest in NLE theories within gravitational physics was ignited with realization thatsome modifications of the Maxwell’s electrodynamics may resolve the black hole singulari-ties, up to constraints given by [22, 24] (see also [23, 26]). Unfortunately, neither electricallycharged Einstein–Born–Infeld black holes [36, 44, 50, 102] nor electrically charged Einstein–Euler–Heisenberg black holes [101, 126] are regular. Early analyses of static sphericallysymmetric solutions of gravitational-NLE Maxwell’s equations appeared back in 1960s[91, 93], with further developments in [34, 37]. A prominent example of a regular blackhole spacetime, proposed by Bardeen [8], was much later interpreted by Ay´on-Beato andGarc´ıa [2, 4] as a solution of Einstein-NLE Maxwell field equations for a particular NLEtheory (and generalized to a rotating solution in [5, 112]). Over the years quest for a reg-ular black hole solutions became intertwined with proliferation of new NLE theories basedon various Lagrangian functions, such as logarithmic [107], hyperbolic tangent [3], power[58, 59], exponential [63] and so on. We note in passing that Wald’s solution [118], describ-ing black hole immersed in a homogeneous magnetic field, has been recently perturbativelygeneralized to NLE theories [14].First systematic approach to thermodynamics of black holes with NLE fields by Rash-eed [98] contains a proof of the zeroth law of black hole electrodynamics (via Einstein’sgravitational field equation), an incomplete attempt to prove first law of black hole ther-– 2 –odynamics (missing the crucial NLE terms) and an ambiguous conclusion that Smarrformula does not hold. Two subsequent decades of research in this subfield brought seriesof papers mostly focused on the simplest, static spherically symmetric black hole solu-tions. Here we have analyses of the black hole thermodynamics for some specific theories(e.g. power-Maxwell in arbitrary number of dimensions [52], Born–Infeld [27, 54, 127] andEuler–Heisenberg [83]) or more general discussions (Smarr formula via assumed first lawand scaling arguments [19]; electrically charged black holes [38] but with highly implicitform of the first law and Smarr formula; Smarr formula for the F -class NLE Lagrangian, us-ing assumed first law and scaling argument [43]; various analyses of phase transitions in thepresence of cosmological constant and NLE fields [6, 54, 64, 83]; thermodynamical stability[21]). Early attempt [85, 86] to generalize the first law using more rigorous, covariant phaseformalism, for static black holes with constant-curvature transversal ( D − F -class NLE theories, suggested absence of NLE corrections. However,first complete generalization of the Smarr formula for rotating black hole with NLE fields[53] has revealed presence of additional NLE terms, inconsistent with unaltered form ofthe first law (see also remarks in [81]). Derivation of the first law for the F -class NLEtheories [128], obtained by variation of the Bardeen–Carter–Hawking mass formula, offersan important step towards the resolution of this problem.The scope of this paper is broad, motivated by the fact that proper understandingof the black hole thermodynamics in the presence of NLE fields is still quite incomplete,with numerous assumptions and technical details being usually swept under the rug. Mostimportantly, we shall offer complete, rigorous derivation of the first law for the rotatingblack holes with electromagnetic fields defined by the Lagrangian which is a member ofthe FG -class NLE theories. Necessity of such generalizations is emphasized by the factthat QED corrections to classical Maxwell’s electrodynamics, defined by Euler–HeisenbergLagrangian, is a FG -class NLE theory. Only when a consistent framework of black holemechanics is reached, we can hope to distillate clear physical points and speculate aboutthe implications of these generalizations.The paper is organized as follows. In section 2 we briefly review the basic elementsof NLE theories, while in section 3 we analyse the conditions leading to energy conditionsand comment on their consequences. In section 4 we revisit and complete several different,complementing approaches to the proof of the zeroth law of black hole nonlinear electro-dynamics. Section 5 is the central part of the paper, where we put covariant phase spaceunder scrutiny in order to prepare it for NLE theories, then derive the first law of blackhole thermodynamics in the presence of NLE fields, both “equilibrium state” and “physicalprocess” versions. In section 6 we discuss several aspects of the NLE Smarr formula, itsconsistency with the first law and conditions under which it can take a linear form. In Ap-pendices we collect important identities, discuss Stokes’ theorem on Lorentzian manifoldsand present a brief list of most important NLE Lagrangians. Notation and conventions . Throughout the paper we use the “mostly plus” metricsignature and the natural system of units, such that G = c = 4 πǫ = 1. Spacetime( M , g ab ) is a four-dimensional, connected, smooth manifold M with a smooth Lorentzian– 3 –etric g ab . We denote differential forms either by “indexless” boldface letters or withabstract index notation, whenever the former becomes cumbersome. Volume 4-form isdenoted by ǫ = ⋆
1. Contraction of a symmetric tensor S ab with a vector X a produces a1-form S ab X b , which we briefly denote by S ( X ). Commutator between two vector fields, X a and Y a is denoted by [ X, Y ] a := X b ∇ b Y a − Y b ∇ b X a . On-shell equalities are denotedby ≈ . Let us briefly review basic elements of the Einstein-NLE field equations. The NLE La-grangian density L ( F , G ) considered here is a smooth function of two electromagneticinvariants F := F ab F ab and G := F ab ⋆F ab . (2.1)For example, classical Maxwell’s Lagrangian density is L (Max) = − F /
4, while an overviewof commonly used NLE Lagrangians is presented in the appendix C. It may seem thatone could construct even more general NLE Lagrangians by inclusion of invariants such as F ab F bc F ca and F ab ⋆F bc F ca . However, it is not too difficult to see, using identities (A.5)and (A.6), that any scalar constructed from F and ⋆ F without any additional derivatives,may be reduced to a function of two basic quadratic invariants F and G [42]. This doesnot hold any more once we include, for example, covariant derivatives of electromagnetic2-form F or nonminimal coupling to gravitation, which we won’t pursue here. In orderto simplify expressions, partial derivatives of the Lagrangian density L are denoted withabbreviations such as L F := ∂ F L , L G := ∂ G L , L FG := ∂ G ∂ F L and so on.We assume that gravitational part of the action is the standard, Einstein–Hilbert one,so that the total Lagrangian 4-form is L = 116 π (cid:0) R + 4 L ( F , G ) (cid:1) ǫ . (2.2)The corresponding Einstein’s gravitational field equation is G ab = 8 πT ab (2.3)with the NLE energy-momentum tensor T ab = − π (cid:0) ( L G G − L ) g ab + 4 L F F ac F cb (cid:1) . (2.4)The NLE Maxwell’s equations ared F = 0 and d ⋆ Z = 0 , (2.5) For a Lagrangian 4-form L = ς (cid:0) R + 4 L (em) (cid:1) ǫ with normalization ς >
0, the electromagnetic energy-momentum tensor is defined as T (em) ab := − πς √− g δS (em) δg ab , with S (em) = 4 ς Z L (em) ǫ . Our choice ς = 1 / (16 π ) is consistent with e.g. [49, 60], whereas ς = 1 normalization is used in [116]. – 4 –here we have introduced auxiliary 2-form Z := − L F F + L G ⋆ F ) . (2.6)We shall refer to the system of equations (2.3)-(2.5) as the gNLE (gravitational-NLE) fieldequations. Alternative, convenient way to write the NLE energy-momentum tensor is toseparate it into “Maxwell part” and the “trace part”, T ab = − L F T (Max) ab + 14 T g ab (2.7)with T (Max) ab := 14 π (cid:18) F ac F cb − g ab F (cid:19) and T := g ab T ab = 1 π ( L − L F F − L G G ) . (2.8)Note that the Maxwell’s energy-momentum tensor T (Max) ab is traceless. Yet another way towrite the NLE energy-momentum tensor, using identity (A.6), is T ab = 14 π (cid:0) Z ac F cb + L g ab (cid:1) . (2.9)Throughout the discussion some special spacetime points will recurringly appear as a tech-nical obstacle. We say that an electromagnetic field is degenerate at point x ∈ M if L F ( x ) = 0. Whereas the Born–Infeld theory is devoid of degenerate points ( L (BI) F doesnot have any real zeros), the Euler–Heisenberg theory formally has a degenerate pointwhenever F = 45 m e / α , but this is inconsistent with the assumption of weak field limit,with which this form of the Lagrangian has been written. Moreover, one might argue thatat least in a weak field limit, that is near the origin of the F - G plane, the derivative L F should take values in a neighbourhood of Maxwellian − /
4, without any zeros.
Measurements of macroscopic physical fields are supporting local positivity of the energydensity and its dominance over the pressure. These observations are captured by various(pointwise) energy conditions [32], among which the four most known are • dominant energy condition (DEC): T ab u a v b ≥ u a and v a or, equivalently, − T ab v b is future directed causal vector for any future directed timelike vector v a ; • weak energy condition (WEC): T ab v a v b ≥ v a ; • null energy condition (NEC): T ab ℓ a ℓ b ≥ ℓ a ;– 5 – strong energy condition (SEC): T ab v a v b ≥ T g ab v a v b for any future directed timelike vector v a .Energy conditions listed above are not independent, but are related by implicationsDEC ⇒ WEC ⇒ NEC ⇐ SEC.Foundational results in general relativity, e.g. singularity theorems [60], are universalon the account of relying on very few details about physical fields, the most prominent beingsome of the energy conditions [32]. As one of the versions of the zeroth law of black holemechanics assumes that the energy-momentum tensor satisfies DEC [116] and Hawking’sblack hole area law [60, 116] assumes that the energy-momentum tensor satisfies NEC, weshall look more closely into these conditions for NLE theories.Analysis of the energy conditions for the electromagnetic energy-momentum tensor iseasiest to perform in spinorial formalism [92, 108]. The electromagnetic 2-form F and itsHodge dual ⋆ F correspond, respectfully, to spinors F ABA ′ B ′ and ⋆F ABA ′ B ′ , which may bedecomposed as F ABA ′ B ′ = ǫ AB φ A ′ B ′ + φ AB ǫ A ′ B ′ , (3.1) ⋆F ABA ′ B ′ = i (cid:0) ǫ AB φ A ′ B ′ − φ AB ǫ A ′ B ′ (cid:1) , (3.2)with symmetric (electromagnetic) spinor φ AB and antisymmetric nondegenerate spinor ǫ AB (symplectic structure on spinor space) . Furthermore, contraction of electromagneticspinors admits decomposition φ AC φ CB = 12 ǫ AB φ DC φ DC . (3.3)One must be cautious about conventions, as spinor formalism is usually done in the “mostlyminus” metric signature. Suppose that η := sgn( η ). Then the spacetime metric g ab corresponds to spinor g ABA ′ B ′ = ηǫ AB ǫ A ′ B ′ and ηF ACA ′ C ′ F C C ′ B B ′ = − φ AB φ A ′ B ′ + 12 ǫ AB ǫ A ′ B ′ (cid:16) φ CD φ CD + φ C ′ D ′ φ C ′ D ′ (cid:17) . (3.4)Electromagnetic invariants are F = 2 (cid:16) φ AB φ AB + φ A ′ B ′ φ A ′ B ′ (cid:17) , G = − i (cid:16) φ AB φ AB − φ A ′ B ′ φ A ′ B ′ (cid:17) . (3.5)Given that we normalize Maxwell’s energy-momentum tensor as T (Max) ab := − η π (cid:18) F ac F cb − g ab F cd F cd (cid:19) , (3.6)the corresponding spinor representation reduces to T (Max) ABA ′ B ′ = 12 π φ AB φ A ′ B ′ (3.7) Here we assume “ l eft to l ower, r ight to r ise” convention of lowering and raising of spinor indices, ǫ AB α A = α B = − ǫ BA α A and ǫ AB α B = α A = − ǫ BA α B . – 6 –ndependently of the metric signature sign η . Finally, electromagnetic spinor may bedecomposed [92, 108] as φ AB = α ( A β B ) . If α A and β A are not proportional, then wesay that φ AB is algebraically general (type I in Petrov classification), whereas in case when α A and β A are proportional, we say that φ AB is algebraically special (type N). Spinor φ AB is algebraically special if and only if the electromagnetic fields is null, that is F = 0 = G .It is well-known that Maxwell’s electromagnetic energy-momentum tensor (3.7) satis-fies both DEC and, since it is traceless, SEC. Namely, for any pair of spinors κ A , λ A andcorresponding pair of future directed null vectors, k AA ′ = κ A κ A ′ and ℓ AA ′ = λ A λ A ′ , wehave T (Max) ABA ′ B ′ k AA ′ ℓ BB ′ = 12 π φ AB φ A ′ B ′ κ A κ A ′ λ B λ B ′ = 12 π (cid:12)(cid:12) φ AB κ A λ B (cid:12)(cid:12) ≥ . (3.8)Since any future directed causal vector is a sum of a pair of future directed null vectors itfollows that T (Max) ab u a v b ≥ u a and v a .Let us now present a simple way to translate energy conditions for NLE theories, whichcomplements some earlier attempts [35, 94]. Theorem 3.1.
The NLE energy-momentum tensor, in η = − signature, satisfies • NEC if and only if L F ≤ ; • DEC if and only if L F ≤ and T ≤ ; • SEC if L F ≤ and T ≥ .Proof . One direction of the claims, the “if” direction, follows immediately from the(2.7) form of the NLE energy-momentum tensor and fact that Maxwell’s electromagneticenergy-momentum tensor T (Max) ab satisfies DEC.For the converse in the NEC case we need to find a future directed null vector ℓ a ,such that T (Max) ab ℓ a ℓ b >
0. Using decomposition φ AB = α ( A β B ) , for the algebraically generalcase we may choose auxiliary spinor λ A = α A + β A , so that λ A α A = 0 = λ A β A , while inthe algebraically special case λ A may be any spinor such that λ A α A = 0. Furthermore, let ℓ AA ′ = ± λ A λ A ′ , with sign choice such that ℓ a is future directed. Then, in both algebraicallygeneral and special case, we have 2 πT (Max) ab ℓ a ℓ b = (cid:12)(cid:12) φ AB λ A λ B (cid:12)(cid:12) >
0. Finally, assuming thatNEC holds, we have 0 ≤ T ab ℓ a ℓ b = − L F T (Max) ab ℓ a ℓ b , so that L F ≤ L F ≤
0. Proof of the remainingclaim, that DEC implies T ≤
0, has already appeared in [94], which we briefly sketch here.If L F = 0, DEC immediately implies T ≤
0, so let as assume that L F <
0. Using theNewman–Penrose null tetrad [108], ( ℓ a = o A o A ′ , n A = ι A ι A ′ , m a = o A ι A ′ , m a = ι A o A ′ ), wemay decompose a timelike vector v a appearing in DEC as v a = aℓ a + bn a + cm a + cm a with some complex numbers ( a, b, c ), normalized for convenience with ab = 1 + | c | . Oneof the forms of DEC, ( T ab v b )( T ac v c ) ≤
0, after a straightforward but tedious calculation,is reduced to an inequality S + (1 + 2 | c | ) L F T ≥
0, with some quantity S independent ofthe parameters ( a, b, c ). Thus, condition T > | c | . – 7 –s we may always choose a NLE Lagrangian such that L (0 ,
0) = 0, then, given that L is differentiable at the origin of the F - G plane, it follows that T = 0 for null electromagneticfields. In other words, at least for null electromagnetic fields, L F ≤ L (BI) F = − W , πT (BI) = 4 b ( W − − F W , W := r F b − G b . (3.9)We immediately see that L (BI) F ≤ √ x − y ≤ √ x ≤ x + 1 for nonnegative x and y ≤ x , we have 2 W ≤ F / b ), so that T (BI) ≤
0. In other words, Born–Infeldtheory obeys DEC and NEC.(b) Euler–Heisenberg: L (EH) F = −
14 + 8 α m e F , πT (EH) = − α m e (cid:0) F + 7 G (cid:1) . (3.10)We see that Euler–Heisenberg theory satisfies DEC and NEC for electromagneticfields with F ≤ m e / α (e.g. weak field, null electromagnetic field).In both of these theories SEC is satisfied for null electromagnetic fields, but this conditionhas to be carefully examined for non-null electromagnetic fields (see e.g. [95]). Constancy of intensive variables over stationary black hole horizons is one of the corner-stones of the black hole thermodynamics. Just as with many other black hole theorems,choice of the assumptions required to establish this result depends on the type of generalitywe strive for, whether we want it to hold for solutions with particular geometric propertiesof the black hole (independent of the field equations) or for solutions of some particularclass of field equations (independent of particular geometric details of the spacetime).The zeroth law of black hole mechanics, constancy of the surface gravity κ over thestationary black hole horizon, can be proved(a) using Einstein’s gravitational field equations, under the assumption that matter sat-isfies dominant energy condition [116],(b) for bifurcate Killing horizons [73], and(c) for horizons generated by Killing vector fields which satisfy some additional geometricproperties [97]. – 8 –he zeroth law of black hole electrodynamics, constancy of the electromagnetic scalarpotentials over the stationary black hole horizon, can be established using similar techniques[104, 105], at least for Maxwell’s electromagnetic fields. Nonlinear electromagnetic fields,on the other hand, demand more careful treatment. As the analyses of the NLE zerothlaw in the literature are incomplete we shall first review various approaches.Suppose that spacetime ( M , g ab ) admits a smooth Killing vector field ξ a and theelectromagnetic field F inherits the symmetry, £ ξ F = 0. One should bear in mind thatthe latter assumption is rather nontrivial, as there are known electrovac spacetimes withsymmetry noninheriting electromagnetic fields [10, 84, 106]. Symmetry inheritance of theelectromagentic fields has been extensively studied within the Maxwell’s theory [31, 33, 84,99, 111, 114, 115, 124, 125] and recently analysed for NLE fields [10]. In general the Liederivative £ ξ F is a linear combination a⋆ F + b F , with b = 0 for Maxwell’s electrodynamics,and there are various sufficient conditions implying a = 0 = b , which we tacitly take to besatisfied.In this context it is convenient to introduce decomposition of F to electric and magneticfields (differential forms) with respect to the Killing vector field ξ a . First of all, we have1-forms E = − i ξ F and H = i ξ ⋆ Z which, as a consequence of the symmetry inheritanceand NLE Maxwell equations (2.5), are closedd E = ( − £ ξ + i ξ d) F = 0 and d H = ( − £ ξ + i ξ d) ⋆ Z = 0 . (4.1)Thus, given that a domain is simply connected, we can define on it associated scalarpotentials, electric Φ and magnetic Ψ, via E = − dΦ and H = − dΨ . (4.2)For completeness, we may introduce two additional 1-forms, B = i ξ ⋆ F and D = − i ξ Z ,with the caveat that in general B and D are not closed. These 1-forms are related by D = − L F E − L G B ) , (4.3) H = − L F B + L G E ) , (4.4)while electromagnetic invariants may be expressed as( ξ a ξ a ) F = 2( E a E a − B a B a ) , (4.5)( ξ a ξ a ) G = − E a B a . (4.6)By construction we immediately know that scalar potentials are constant along the orbitsof the Killing vector field ξ a , namely £ ξ Φ = − i ξ E = 0 and £ ξ Ψ = − i ξ H = 0. The questionis what can be deduced about Φ and Ψ on a Killing horizon H [ ξ ], that is a null hypersurfacegenerated by ξ a . Given that one can prove that ξ ∧ E H = 0 and ξ ∧ H H = 0 , (4.7)contraction with a tangent vector X a ∈ T p H [ ξ ] implies that ( £ X Φ) ξ = 0 and ( £ X Ψ) ξ = 0.Thus, at each point where ξ = 0, we know that £ X Φ = 0 and £ X Ψ = 0, whereas at point– 9 –here ξ a = 0 by definition we immediately have dΦ = 0 and dΨ = 0. In conclusion,(4.7) imply that Φ and Ψ are constant over the Killing horizon H [ ξ ]. Let us review threeapproaches to (4.7) mentioned above.(a) Gravitational field equation approach [98]. Using the identity R ab ξ a ξ b H = 0 and con-traction πT ab ξ a ξ b H = − L F E a E a , Einstein’s field equation implies that the electricfield E a is null at each nondegenerate point of the horizon H [ ξ ]. As ξ a E a = 0, itfollows that ξ ∧ E = 0 at any of these points. Furthermore, (4.5) implies that B a is null as well on H [ ξ ], so that ξ ∧ B H = 0 and, consequently, ξ ∧ H H = 0. The maindrawback here is that it is not quite clear how to generalize the method beyond theEinstein’s gravitational field equation.(b) Bifurcate horizon approach is, arguably, the simplest method. We assume that theKilling horizon H [ ξ ] is of bifurcate type, with vanishing ξ a on bifurcation surface B ⊆ H [ ξ ]. The potentials Φ and Ψ are immediately constant over the bifurcationsurface B and, as they are constant along the orbits of ξ a , they are constant overeach component of H [ ξ ] connected to B . The drawback of this approach is that ahorizon does not have to be of bifurcate type, most notable counterexample beingextremal black hole horizons.(c) Frobenius approach [10, 104, 105], in which we are relying on some additional geomet-ric conditions. Assume that the spacetime is stationary and axially symmetric, withassociated Killing vector fields, respectfully k a and m a , which commute, [ k, m ] a = 0,and satisfy Frobenius condition [79] k ∧ m ∧ d k = 0 = k ∧ m ∧ d m . (4.8)Furthermore, spacetime contains Killing horizon H [ χ ], generated by the Killing vectorfield ξ a = χ a := k a +Ω H m a , where constant Ω H is so-called “horizon angular velocity”.Since k a and m a are tangent to H [ χ ] and χ a is normal to H [ χ ], it follows [66] that k a k a + Ω H k b m b H = 0 , k a m a + Ω H m b m b H = 0 and ( k a k a )( m a m a ) H = ( k a m a ) . (4.9)Finally, we assume that electromagnetic field inherits both symmetries, £ k F = 0 and £ m F = 0. Applying the identity i X £ Y − i Y £ X = i X i Y d − d i X i Y + i [ X,Y ] , (4.10)with X a = k a and Y a = m a on F and ⋆ Z it follows that F ab k a m b and ⋆Z ab k a m b are constant. Thus, on any connected domain of the spacetime containing the pointswhere either k a or m a vanish (an example for the latter is the rotation axis), theseconstants are zero and, consequently, on each nondegenerate point of such a domain ⋆F ab k a m b = 0. These two conditions may be rephrased as k ∧ m ∧ ⋆ F = 0 and k ∧ m ∧ F = 0 . (4.11)– 10 –ontraction with i m i k lead us to (4.7) on each nondegenerate point of the horizonwhere m a m a = 0. Special points on the horizon where m a m a = 0 are usually justmeasure zero sets at which the rotation axis is intersecting the horizon, so that con-stancy of a potential over the whole horizon follows from continuity of the potential.In order to repeat the strategy from (c) to a static, not necessarily axially symmetric,spacetime with associated hypersurface orthogonal Killing vector field k a (satisfying k ∧ d k = 0) and Killing horizon H [ k ], we would need relations of the form k ∧ ⋆ F = 0 and k ∧ Z = 0. These, however, do not necessarily hold under given assumptions, as we may havedyonic solutions. Instead, we may treat some special subcases, defined by the additionalassumptions.(e ) “Purely electric case” in a sense that B = 0. Then (4.5) implies that E is againnull on the horizon H [ k ], which is enough to finalize the proof as in the approach (a)above.(e ) “Purely electric case” in a sense that H = 0, that is k ∧ Z = 0. Here k ∧ H = 0implies L G k ∧ E + L F k ∧ B = 0 and contraction of k ∧ Z = 0 with k a implies L F k ∧ E − L G k ∧ B H = 0. Given that ( L F ) + ( L G ) = 0, we may deduce (4.7).(m ) “Purely magnetic case” in a sense that E = 0. Then (4.5) implies that B is againnull on the horizon H [ k ], which is enough to finalize the proof as in the approach (a)above.(m ) “Purely magnetic case” in a sense that D = 0, that is k ∧ ⋆ Z = 0. Here k ∧ D = 0implies L F k ∧ E − L G k ∧ B = 0 and contraction of k ∧ ⋆ Z = 0 with k a implies L G k ∧ E + L F k ∧ B H = 0. Given that ( L F ) + ( L G ) = 0, we may deduce (4.7).Note that for the test electromagnetic fields, weak in a sense that associated energy-momentum tensor in the gravitational field equation may be neglected, approach (a) isuseless, but any of the other methods may suffice. The first law of black hole mechanics essentially captures energy conservation for slightlyperturbed black holes. Following the nomenclature from [117], approaches to derivation ofthis law may be classified as follows:(1) “equilibrium state” version, in which we are comparing two “nearby” stationary blackhole configurations, with two varieties(1a) original, somewhat cumbersome procedure [9], in which one takes variation ofthe Bardeen–Carter–Hawking mass formula, and We note in passing that on any open set which is devoid of degenerate points and on which k a ishypersurface orthogonal and timelike, the NLE field cannot be null; proof is essntially same as in [111]. – 11 –1b) covariant phase space formalism [69, 80, 96, 120];(2) “physical process” version, in which we are looking at physical, quasistatic processof matter infalling into a black hole [49].Generalization of the first law of black hole mechanics in the F -class NLE theorieswas recently presented in [128], using approach (1a). Our aim is to extend this result forrotating black holes in the FG -class NLE theories, using rigorous approaches (1b) and (2).Basic assumption at the foundation of the first law is that the spacetime is a solutionof gNLE equations with stationary axially symmetric, asymptotically flat metric g ab anda symmetry inheriting electromagnetic field F . Corresponding Killing vector fields are k a = ( ∂/∂t ) a , timelike at infinity, and axial m a = ( ∂/∂ϕ ) a , with compact orbits. As above,we assume that k a and m a commute, [ k, m ] a = 0, and satisfy Frobenius conditions (4.8).Both the “equilibrium state” and the “physical process” version of the first law inspectCauchy surfaces intersecting the black holes. More concretely, in the former case thespacetime contains a bifurcate Killing horizon H [ χ ], a pair of null hypersurfaces generatedby the null Killing vector field χ a = k a + Ω H m a with constant Ω H and surface gravity κ ,which intersect in so-called bifurcation surface B , a smooth, compact, embedded 2-surface.The Killing vector field χ a vanishes on B . Derivation of the “equilibrium state” version ofthe first law is built on a spacelike Cauchy surface Σ ⊆ M , smoothly embedded in M withnowhere vanishing normal, whose boundary ∂ Σ consists of an asymptotically flat end andbifurcation surface B = Σ ∩ H [ χ ]. On the other hand, in the “physical process” version ofthe first law we only need a portion of the Killing horizon (cut by two Cauchy surfaces),which does not need to be of the bifurcate type (accordingly, none of the Cauchy surfacedoes not have to end in bifurcation surface).For any smooth closed 2-surface S we define the Komar mass M S and the Komarangular momentum J S [76] with integrals M S := − π I S ⋆ d k and J S := 116 π I S ⋆ d m . (5.1)More concretely, if S is a “sphere at infinity” S ∞ , that is a limit of these integrals evaluatedon sphere of radius r as r → ∞ , we use simple symbols M := M S ∞ and J := J S ∞ . In ourgeometric setting ADM definitions of mass and angular momentum [66, 116] coincide with M and J . Furthermore, we define the electric charge Q S and the magnetic charge P S withintegrals [66] Q S := 14 π I S ⋆ Z and P S := 14 π I S F . (5.2)Again, as above, we use simple symbols Q := Q S ∞ and P := P S ∞ for charges evaluatedat infinity. It is important to note that, given that gauge 1-form A is globally well-defined, Stokes’ theorem implies P S = 0. Thus, nonvanishing magnetic charge comes withtopologically nontrivial electromagnetic field, treatment of which demands the fibre bundletools. – 12 – .1 Covariant phase space scrutinized Before we outline the general scheme of covariant phase space formalism, we have to addressone of the crucial questions for black hole mechanics with NLE fields, the role of Lagrangian(coupling) parameters. Suppose that NLE Lagrangian is defined with finite number of realparameters, { β , . . . , β n } . Given that we treat these parameters as constants which arenot varied, the result will be the first law which in general is not consistent with thegeneralized Smarr formula. Since the Smarr formula in the presence of NLE fields maybe derived [53] completely independently of the first law, this tension must be resolved.One of the evident options is to extend the phase space with Lagrangian parameters, sothat we consider them constant within fixed spacetime (that is, ∇ a β i = 0), but analysevariations which comprise changes of parameters . Hence, in variational procedure the NLELagrangian is formally treated as function of electromagnetic invariants and parameters, L ( F , G ; { β i } ). Such framework is closely related to treatment of the cosmological constantΛ in black hole thermodynamics, leading to its identification with the pressure in V d p term[68, 70–72, 78].The other possible alternative is to consider even more general framework, in whichLagrangian parameters are spacetime dependent functions [100]. However, note that, using(2.9) with identities (A.12) and (A.13), we have covariant divergence4 π ∇ a T ab = ∇ a ( Z ac F bc + L δ ab )= ( ∇ a Z ac ) F bc + Z ac (d F ) abc + n X i =1 L β i ∇ b β i , (5.3)which for nonconstant parameters β i will not necessarily vanish on-shell. This indicatesthat one needs to complete such theory with additional equations for parameters, but wewill not pursue such generalizations here.We now turn to application of the covariant phase space formalism under the assump-tions given above. In this subsection, for simplicity, we shall denote all dynamical fields(first of all, spacetime metric g ab and gauge field A ) collectively by φ , with all indicessuppressed. Similarly, index of coupling parameters β i will be suppressed in arguments,but we shall keep them in sums involving variations δβ i . Within the variational procedurewe assume that the action of the “variation operator” δ on fields φ and parameters β i isdefined [80, 116] as δφ ( x ) := ∂φ ( x ; λ ) ∂λ (cid:12)(cid:12)(cid:12) λ =0 and δβ i := ∂β i ( λ ) ∂λ (cid:12)(cid:12)(cid:12) λ =0 , (5.4)where φ ( x ; λ ) and β i ( λ ) are smooth 1-parameter configurations of fields and coupling pa-rameters. One must bear in mind that variations of the metric and its inverse are relatedby δg ab = − g ac g bd δg bd , (5.5) The authors in [56] even propose a criterion for distinction between “physical” and “redundant” La-grangian parameters. – 13 –hile the variation of the volume form may be decomposed as δ ǫ = − ǫ g ab δg ab . (5.6)Variation of the Lagrangian 4-form consists of the following terms [69] δ L [ φ ; β ] = E [ φ ; β ] δφ + Λ i [ φ ; β ] δβ i + d Θ [ φ, δφ ; β ] . (5.7)Field equations are contained in the 4-form E , indexed 4-form Λ i is the variation of theLagrangian with respect to coupling parameter β i , while the boundary terms are collectedin the 3-form Θ . Next we introduce the Noether current 3-form J ξ := Θ [ φ, £ ξ φ ; β ] − i ξ L [ φ ; β ] , (5.8)defined with respect to an arbitrary fixed vector field ξ a , which will later be promoted toa Killing vector field. Now, asd J ξ = − E [ φ ; β ] £ ξ φ − Λ i [ φ ; β ] £ ξ β i (5.9)and £ ξ β i = 0, the Noether 3-form is closed on-shell, d J ξ ≈
0, and at least locally exists[119] a 2-form Q ξ , such that J ξ ≈ d Q ξ . In other words, as will be explicitly shown below,we may write J ξ = i ξ C + d Q ξ , (5.10)where C is a 4-form, which vanishes on-shell, C ≈
0. As our focus is mainly on theorieswith the Lagrangian which is a sum of the gravitational and electromagnetic parts, itfollows that the 3-form Θ and the 2-form Q ξ split accordingly, Θ = Θ (g) + Θ (em) and Q ξ = Q (g) ξ + Q (em) ξ . The symplectic current 3-form is defined with respect to two variations δ and δ , ω [ φ, δ φ, δ φ ; β ] := δ Θ [ φ, δ φ ; β ] − δ Θ [ φ, δ φ ; β ] , (5.11)and the presymplectic form is obtained by integrating symplectic current 3-form over aspacelike Cauchy surface ΣΩ Σ [ φ, δ φ, δ φ ; β ] := Z Σ ω [ φ, δ φ, δ φ ; β ] . (5.12)A tacit assumption here is that volume form (orientation) on Σ is given by i ˜ n ǫ , where ˜ n a isa unit, future directed timelike normal on Σ. Taking into account that δξ a = 0, variationof the Noether current (5.10) gives δ J ξ = i ξ δ C + d δ Q ξ , while variation of (5.8) leads to δ J ξ = − i ξ E [ φ ; β ] δφ + ω [ φ, δφ, £ ξ φ ; β ] + d i ξ Θ [ φ, δφ ; β ] − i ξ Λ i [ φ ; β ] δβ i , (5.13)so that ω [ φ, δφ, £ ξ φ ; β ] = i ξ ( E δφ + δ C ) + d( δ Q ξ − i ξ Θ [ φ, δφ ; β ]) + i ξ Λ i [ φ ; β ] δβ i . (5.14)– 14 –mmediately, using Stokes’ theorem (B.1), we haveΩ Σ [ φ, δφ, £ ξ φ ; β ] = Z Σ i ξ ( E δφ + δ C ) + Z ∂ Σ ( δ Q ξ − i ξ Θ [ φ, δφ ; β ]) − K iξ ( β ) δβ i , (5.15)where we have introduced auxiliary functions K iξ , K iξ ( β ) := − Z Σ i ξ Λ i [ φ ; β ] . (5.16)As the top compactly supported de Rham cohomology group for smooth oriented (compactand non-compact) manifolds with nonempty boundary is trivial (see e.g. theorems 8.3.10and 8.4.8 in [123]), we know that pullback of the i ξ Λ i to Σ is globally exact and we canrewrite, via Stokes’ theorem (B.1), K iξ as an integral over ∂ Σ.In order to connect this procedure with Hamiltonian mechanics , encapsulated in rela-tion δH ξ = Ω Σ [ φ, δφ, £ ξ φ ; β ], one has to prove the existence of Hamiltonian H ξ , conjugateto ξ a on Σ. Given that φ is a solution of field equations (thus E = 0) and δφ is a solutionof linearized equations (thus δ C = 0), the first integral on the right hand side of (5.15)will be zero. Thus, the question is whether remaining terms can be written on-shell as avariation of something.In the absence of contribution from parameters, K iξ δβ i , Hamiltonian exists [122] if andonly if Z ∂ Σ i ξ ω [ φ, δ φ, δ φ ] = 0 (5.17)for any two variations δ and δ . More concretely, it is known [69] that Einstein–Hilbertgravitational contribution to i ξ Θ term may be written as a variation, with help of a 3-form b such that Z ∂ Σ i ξ Θ (g) = δ Z ∂ Σ i ξ b . (5.18)As will be demonstrated in the following subsection, electromagnetic contribution to i ξ Θ term will vanish due to boundary conditions and gauge choices. Finally, we have to addressintegrability of the term K iξ δβ i . As local condition ∂ β i K jξ = ∂ β j K iξ is satisfied under mildsmoothness assumptions, we know that I ξ ( β ) exists, such that δI ξ = K iξ δβ i . In the simplestcase, with a single coupling parameter ( n = 1), I ξ is simply a primitive function of K ξ .Now we specialize to a geometric setting described in the introduction of the section5. First we assume that ξ a is a Killing vector field and all dynamical fields inherit cor-responding symmetry, £ ξ φ = 0, so that Ω Σ [ φ, δφ, £ ξ φ ; β ] = 0. Then (5.15) decomposes Let us do a brief recap of Hamiltonian mechanics: building elements consist of a phase space manifoldwith local canonical coordinates s µ = ( q , . . . , p , . . . ), symplectic (closed, nondegenerate) 2-form ω andcorrespondence between a function f and tangent vector X f via d f = − i X f ω , that is X f = ∂f∂p i ∂∂q i − ∂f∂q i ∂∂p i . Dynamics is defined by Hamiltonian scalar H , ˙ f = X H ( f ) and variation δH = ( ∇ µ H ) δs µ = ω µν δs µ ˙ s ν . In more general context this implication demands a careful justification [69], but here it will be imme-diately clear that symplectic current 3-form ω [ φ, δφ, £ ξ φ ; β ], constructed from the Einstein–Hilbert grav-itational 3-form Θ (g) and NLE 3-form Θ (em) , vanish for Killing vector field ξ a and symmetry inheritingelectromagnetic fields. – 15 –n-shell as δ I S ∞ ( Q ξ − i ξ b ) − δ I B ( Q ξ − i ξ b ) − K iξ δβ i ≈ . (5.19)Secondly, we assume that ξ a = χ a = k a + Ω H m a and identify various contributions toboundary integrals.The gravitational part of the Lagrangian 4-form (2.2) is conventional Einstein–Hilbertterm, whose variational properties are well-known [116, 120],116 π δ ( R ǫ ) = 116 π G ab δg ab ǫ + d Θ (g) , Θ (g) := 116 π ⋆ v (5.20)where v is an auxiliary 1-form defined by v a := ∇ b δg ab − g cd ∇ a δg cd , (5.21)and Q (g) ξ = − π ⋆ d ξ . (5.22)Gravitational contributions to (5.19) give us [69] mass and angular momentum of the blackhole spacetime, defined respectfully by M = I S ∞ ( Q (g) k − i k b ) and J = − I S ∞ Q (g) m . (5.23)Absence of the i m b term in the integral for the angular momentum (pullback of i m b to anysphere to which m a is tangent vanishes) is reflected in different normalization of Komarmass and angular momentum [69]. Gravitational contribution at horizon produces theentropy term [120] δ I B Q (g) ξ = κ π δ A , (5.24)where A is the area of the bifurcation surface B . Altogether, the interim form of the firstlaw we have obtained reads δM − Ω H δJ + δ I S ∞ Q (em) χ = κ π δ A + δ I B Q (em) χ + K iχ δβ i . (5.25) Now we turn to the NLE contributions to the first law of black hole mechanics. Variationof the NLE Lagrangian, δ ( L ǫ ) = L F δ F ǫ + L G δ G ǫ + L δ ǫ + n X i =1 L β i δβ i ǫ (5.26)may be conveniently written as δ ( L ǫ ) = L F δ ( F ǫ ) + L G δ ( G ǫ ) + πT δ ǫ + n X i =1 L β i δβ i ǫ . (5.27)– 16 –irst term in (5.27) is, up to factor L F , just the standard Maxwellian contribution L F δ ( F ǫ ) = 8 π L F T (Max) ab δg ab ǫ − L F ∇ a F ab δA b ǫ + 4 L F ∇ a ( F ba δA b ) ǫ . (5.28)Combination of the first term in (5.28) and the third term in (5.27) gives us the NLEenergy-momentum tensor8 π L F T (Max) ab δg ab ǫ + πT δ ǫ = − πT ab δg ab ǫ . (5.29)Also, since − L F ∇ a F ab δA b + L F ∇ a ( F ab δA b ) = −∇ a ( L F F ab ) δA b + ∇ a ( L F F ba δA b ) , (5.30)sum of the first and the third term in (5.27) gives us L F δ ( F ǫ ) + πT δ ǫ = − πT ab δg ab ǫ − ∇ a ( L F F ab ) δA b ǫ + 4 ∇ a ( L F F ba δA b ) ǫ . (5.31)Second term in (5.27) may be written, using (A.8), as L G δ ( G ǫ ) = 4 L G (cid:16) ∇ a (( ⋆F ab ) δA b ) − ( ∇ a ⋆F ab ) δA b (cid:17) ǫ = 4 (cid:16) ∇ a ( L G ( ⋆F ba ) δA b ) − ∇ a ( L G ⋆F ab ) δA b (cid:17) ǫ . (5.32)Altogether, we have obtained a sought form of the variation of the Lagrangian 4-form,14 π δ ( L ǫ ) = 116 π − πT ab δg ab + 4( ∇ a Z ab ) δA b + 4 X i L β i δβ i ! ǫ + d Θ (em) (5.33)with Θ (em) := 116 π ⋆ w , w a = − Z ba δA b . (5.34)Auxiliary 1-form w may be also written as w = − ⋆ ( ⋆ Z ∧ δ A ). Here we can see [96] thatfor the electromagnetic field F of class O ( r − ) and perturbation δ A of class O ( r − ) as r → ∞ , the 3-form Θ (em) does not contribute to the integral at S ∞ .Let us turn to Noether current 3-form16 π J ξ = ⋆ ( v + w ) − ( R + 4 L ) ⋆ ξ . (5.35)Using the identity ∇ b ∇ b ξ a − ∇ a ∇ b ξ b = R ( ξ ) a − ( ⋆ d ⋆ d ξ ) a (5.36)we see that auxiliary 1-form v for δ = £ ξ is equal to ∇ b £ ξ g ab − g cd ∇ a £ ξ g cd = 2 R ( ξ ) a − ( ⋆ d ⋆ d ξ ) a . (5.37)For the NLE 1-form w we have to find objects containing contraction Z ba £ ξ A b . As the Liederivative £ ξ A is contained in the electric 1-form defined with respect to the vector field ξ a , E = − i ξ F = − i ξ d A = − £ ξ A + d i ξ A (5.38)– 17 –ur focus is on the contraction i E Z . Here we need one auxiliary identity, i E ⋆ F = 14 G ξ , (5.39)following directly from (A.6), so that4 i E Z = −
16 ( L F i E F + L G i E ⋆ F ) = 16 π T ( ξ ) − L ξ . (5.40)On the other hand, 4 i E Z = − ⋆ ( ⋆ Z ∧ E )= 4 ⋆ ( ⋆ Z ∧ £ ξ A ) − ⋆ ( ⋆ Z ∧ d i ξ A )= − w − ⋆ d(( i ξ A ) ⋆ Z ) + 4( i ξ A ) ⋆ d ⋆ Z (5.41)which leads to w − L ξ = − π T ( ξ ) − ⋆ d(( i ξ A ) ⋆ Z ) + 4( i ξ A ) ⋆ d ⋆ Z . (5.42)As the variational procedure introduces electromagnetic field via gauge 1-form A , we mustestablish the relation between A and scalar potential. Supposing that the electromagneticfield inherits the symmetry, £ ξ F = 0, and F = d A for some initial gauge choice of gauge 1-form A , we still might have a technical problem as £ ξ A = 0. Then, as d £ ξ A = £ ξ F = 0we know that £ ξ A is a closed form and on a simply connected domain there is a function α , such that £ ξ A = d α . Thus, choosing a gauge function λ defined by £ ξ λ = − α , wehave A = A + d λ , for which £ ξ A = 0. Even after this gauge fixing, we still have aremaining symmetry-consistent gauge freedom, as for any function µ such that £ ξ µ = 0,we have £ ξ ( A + d µ ) = 0. Furthermore,d(Φ + i ξ A ) = − E + ( £ ξ − i ξ d) A = 0 (5.43)proves that Φ and − i ξ A differ by a constant, say Φ = − i ξ A + Φ for some Φ ∈ R . Thisallows us to write J ξ = 18 π ⋆ ( G ( ξ ) − π T ( ξ )) − Φ − Φ π d ⋆ Z + d( Q (g) ξ + Q (em) ξ ) (5.44)with Q (g) ξ = − π ⋆ d ξ and Q (em) ξ = 14 π (Φ − Φ ) ⋆ Z . (5.45)The 4-form C is given by C abcd = 18 π ( G ea − πT ea − A a ∇ r Z re ) ǫ ebcd . (5.46)Again, this confirms that d J ξ ≈ J ξ ≈ d Q ξ .Before we evaluate remaining terms for the first law (5.25), it is convenient to make agauge choice. If we take A such that i ξ A will give nonvanishing contribution at bifurca-tion surface, we are tacitly using gauge field which is divergent there. Take for a simple– 18 –xample non-extremal Reissner–Norstr¨om black hole solution. Using tortoise radial coor-dinate d r ∗ = d r/f ( r ), we can introduce Eddington–Finkelstein coordinates u = t − r ∗ and v = t + r ∗ , and then Kruskal coordinates U = − e − κu and V = e κv . The Killing horizon isgenerated by the Killing vector field k = κ ( V ∂ V − U ∂ U ) and the conventional gauge field(vanishing at infinity) is A = − Qr d t = − Q κr (cid:18) V d V − U d U (cid:19) . (5.47)However, in this gauge A is divergent at the bifurcation surface ( U, V ) = (0 , A ′ = − Q κ (cid:18) r − r + (cid:19) (cid:18) V d V − U d U (cid:19) , (5.48)where r + is the radius of the outer horizon, to obtain regular A on the horizon. Likewise,we shall pursue here an alternative gauge choice, in which A is finite and smooth at H [ χ ]and Φ vanishes at infinity . Thus, i ξ A | B = 0, so that − i ξ A = Φ − Φ H (that is, Φ = Φ H )and i ξ A | ∞ = Φ H . With this choice the Q (em) ξ term drops at the bifurcation surface, butmakes contribution at infinity, δ I S ∞ Q (em) ξ = − Φ H δQ . (5.49)Thus, (5.25) lead to the final form of the first law of black hole mechanics, δM = κ π δ A + Ω H δJ + Φ H δQ + K iχ δβ i (5.50)with K iχ := − π Z Σ L β i ⋆ χ . (5.51)An important lesson here is that K iχ appears as a thermodynamic variable conjugate to β i . In section 6 we shall demonstrate that this form of the first law is consistent with thegeneralized Smarr formula.The first law obtained in (5.50) does not contain the Ψ H δP term, which is occasionallyincluded for generality. Formal reason for its absence is that the gauge field A is tacitlyassumed to be globally well-defined and the whole procedure of the covariant phase spaceformalism should be carefully re-examined to adopt it for solutions with magnetic charge.The only reference, known to us, which has addressed this problem [74], takes into accountcontributions on the edges of the local spacetime patches with well-defined gauge field.These issues are seemingly absent in the approach (1a) to the first law, rendering Ψ H δP term [66, 128], but it is not clear if any of the aforementioned formal issues have beenjust swept under the rug. From another point of view [96], the magnetic charge P is atopological charge and it should not vary under perturbations, nor contribute to the firstlaw. We note in passing that there is also an alternative procedure [48] with a Cauchy surface Σ which doesnot intersects the horizon H [ ξ ] at the bifurcation surface, but we shall not utilize it here. – 19 –ome of the earlier analyses of the first law of black hole thermodynamics in thepresence of NLE fields propose the form of the law with suspicious absence of the K iξ δβ i term. For example, Herdeiro and Radu [65] look at nonrotating, dyonic black holes intheory with the NLE Lagrangian L = L (Max) + α G and derive the first law in the form δM = κδ A / (8 π )+ Φ H δQ + Ψ H δP . However, this result has to be taken with a grain of salt,as the variation “ δ ” used here keeps the product αP fixed. Similarly, one could write therestricted first law ˆ δM = κ ˆ δ A / (8 π ) for perturbations with uncharged, nonrotating matterand the corresponding variation ˆ δ .Following the recent development of the black hole thermodynamics with the cosmo-logical constant [70–72], one is inclined to interpret black hole mass M in the first law (5.50)as a generalized “enthalpy”, related to the internal energy E via Legendre transformation M = E + K iχ β i , so that δ E = κ π δ A + Ω H δJ + Φ H δQ + β i δK iχ . (5.52)It is not quite clear what is the proper, general interpretation of the quantity K iξ . Given thatthe Lagrangian is written such that coupling parameter β i has the same physical dimensionas F / (e.g. β = b in Born–Infeld and β = m e /α in Euler–Heisenberg theory), that isdimension of the electric field, associated K iξ may be interpreted, based on dimensionalargument, as a “NLE vacuum polarization” (this was remarked in [54] for the Born–Infeldtheory).Let us now turn to different approach to the first law of black hole mechanics, thephysical process version. Instead of looking at stationary black hole configurations which are “nearby” in some ab-stract phase space, here we want to describe physical process in which a (possibly charged)matter is thrown into a black hole. Geometric setting is the same as above, except that theKilling horizon H [ ξ ] no longer needs to be of the bifurcate type. Suppose that Σ and Σ are, respectfully, initial and final smooth, spacelike, asymptotically flat Cauchy surfaces,both of which terminate on the horizon H [ ξ ], as sketched in the figure 1. For convenience,the portion of the horizon between Σ ∩ H [ ξ ] and Σ ∩ H [ ξ ] may be denoted by H . Westart from an initial stationary black hole, then perturb it by throwing a small amount ofcharged matter and wait until it eventually settles to a final stationary state. Formally,the charged matter is described by fields with compact support which intersects Σ and H [ ξ ], but is disjoint from Σ ∩ H [ ξ ] (matter is initially away from the black hole) and Σ (after the process is over and matter has fallen into the black hole, there is no remainingmatter on the final Cauchy hypersurface). In addition we assume that outward pointingvector field n a and corresponding induced orientation ˆ ǫ = i n ǫ has been introduced on eachof these hypersurfaces, as described in the appendix B.Sources are now described by the electromagnetic 4-current j a and the total energy-momentum tensor T (tot) ab , which is a sum of the electromagnetic contribution T ab and the– 20 – Σ H Figure 1 . Spacetime diagram of infalling matter, denoted by gray area. non-electromagnetic contribution e T ab . This generalizes the gNLE field equations to G ab − πT ab = 8 π e T ab , ∇ b Z ab = 4 πj a . (5.53)We assume that ( g ab , A ) is a solution of the source-free gNLE equations (2.3)-(2.5), whilethe perturbations ( δg ab , δ A ) are solutions of the linearized equations with sources δ e T ab and δj a , δ ( G ab − πT ab ) = 8 πδ e T ab , δ ( ∇ b Z ab ) = 4 πδj a . (5.54)Luckily, we do not need to start from scratch, as the expressions for generic variations havebeen prepared within the covariant phase space formalism above. Taking into account that δ ( A a ∇ r Z re ) = ( δA a ) ∇ r Z re + A a δ ∇ r Z re = 0 − πA a δj e (5.55)we see that the variation of the auxiliary 4-form C (5.46) does not vanish on-shell but δC abcd ≈ (cid:16) δ e T ea + A a δj e (cid:17) ǫ ebcd . (5.56)Now, using (5.23) and assumption that field perturbations vanish at Σ ∩ H [ χ ], the equation(5.15) for the Killing vector field ξ a = χ a leads to an on-shell relation δM − Ω H δJ − K iχ δβ i = − Z (Σ , − ˆ ǫ ) ⋆ α χ . (5.57)Here we have introduced an auxiliary 1-form α ξ , defined by α ξ := ⋆ ( i ξ δ C ) = δ e T ( ξ ) + ( i ξ A ) δ j (5.58)for any Killing vector field ξ a . Note that the orientation of the hypersurface Σ , emphasizedin (5.57), is opposite of the induced Stokes’ orientation ˆ ǫ . Symmetry inheritance of all fieldsand perturbations lead to conservation of α ξ in a sense thatd ⋆ α ξ = d i ξ δ C = ( £ ξ − i ξ d) δ C = 0 . (5.59)For simplicity, we may suppress the additional index on α in what follows. Using theStokes’ theorem (B.6) on 4-dimensional submanifold bounded by hypersurfaces Σ and Σ , The 4-form C in [49] is written as C a , but “ a ” is the last index, C bcda . – 21 –orizon portion H and some timelike hypersurface S on which perturbations δ e T ab and δj a vanish (far away from the black hole), we have0 = Z (Σ , ˆ ǫ ) (˜ n a α a )ˆ ǫ + Z ( H , ˆ ǫ ) ( − ℓ a α a )ˆ ǫ . (5.60)As we shall deal with the Raychaudhuri equation, a convenient choice for the null vectorfield ℓ a is ℓ a = ζ a , a vector field tangent to the affinely parametrized null generators of theunperturbed Killing horizon H [ ξ ]. Taking into account all these remarks, we may “shift”the integral in (5.57) from Σ to the black hole horizon, − Z (Σ , − i n ǫ ) ⋆ α = − Z (Σ , − i n ǫ ) ( − n a α a )( i n ǫ ) = Z ( H ,i n ǫ ) ( ζ a α a )( i n ǫ ) . (5.61)In other words, with assumed induced orientation of the horizon, we have δM − Ω H δJ − K iχ δβ i = Z H ζ a α a ˆ ǫ . (5.62)There are two contributions to this integral, electromagnetic and non-electromagnetic. Forthe evaluation of the former one we use the gauge choice in which both Φ and A are zeroat infinity, so that Φ = 0 and − i ξ A = Φ H on the horizon. For the causal, future directed δj a we have ζ a δj a ≤ δQ ≥ δM − Ω H δJ − Φ H δQ − K iχ δβ i = Z H ζ a χ b ( δ e T ab ) ˆ ǫ . (5.63)It remains to be shown that the right hand side is the area term in the first law.The Raychaudhuri equation for the null congruence ζ a = χ a / ( κV ), with the corre-sponding affine parameter V ,d θ d V = − θ − σ ab σ ab − R ab ζ a ζ b , (5.64)in combination with vanishing of the expansion ( θ = 0) and shear ( σ ab = 0) for thestationary background, and Einstein field equation, reduces tod θ d V = − π (cid:16) T ab + e T ab (cid:17) ζ a ζ b . (5.65)In order to extract the change in area of the black hole horizon, we need to look at theperturbed Raychaudhuri equation. Diffeomorphism freedom allows us to make the gaugechoice such that null generators of the unperturbed and perturbed black hole horizonscoincide, which amounts to δζ a ∼ ζ a on the horizon. Thus, using the fact [116] that R ab ζ a ζ b | H = 0, the perturbed Raychaudhuri equation [49] isd δθ d V = − π (cid:16) δT ab + δ e T ab (cid:17) ζ a ζ b (cid:12)(cid:12) H . (5.66) For the extremal Killing horizon H [ χ ], with κ = 0, the Killing vector field χ a is automatically tangentto the affinely parametrized null geodesic generators of the horizon, thus ζ a = χ a . – 22 –he first term on the right hand side consists of δT ab ζ a ζ b = − δ L F ) T (Max) ab ζ a ζ b − L F δT (Max) ab ζ a ζ b + 14 δ ( T g ab ) ζ a ζ b . (5.67)Using the fact that ζ a is null both in the unperturbed and perturbed spacetimes, the lastterm is immediately zero on the horizon, while4 πT (Max) ab ζ a ζ b | H = ( κV ) − E a E a | H = 0 (5.68)4 πδT (Max) ab ζ a ζ b | H = ( κV ) − δ ( E a E a ) | H = 0 (5.69)due to fact, established for the zeroth law, that the electric field E a is null on the horizon.The remaining perturbed Raychaudhuri equation (5.66) may be put in the following form κV d δθ d V = − πζ a χ b δ e T ab (cid:12)(cid:12) H . (5.70)Integral of the left hand side along the horizon portion H leads [117] to the change in area δ A , Z H ζ a χ b ( δ e T ab ) ˆ ǫ = κ π δ A . (5.71)Putting all this together, we have obtained the “physical process” first law if the black holemechanics, δM = κ π δ A + Ω H δJ + Φ H δQ + K iχ δβ i , (5.72)consistent with (5.50). The problem of generalization of the Smarr formula for rotating (stationary axially sym-metric) black holes in the FG -class NLE theories has been recently solved [53], with aninterim result of the form M = κ π A + 2Ω H J + Φ H Q H + Ψ H P H + 12 Z Σ T ⋆ χ . (6.1)This relation follows directly from the Bardeen–Carter–Hawking mass formula and is inprinciple independent of the first law. Additional last term on the right hand side is clearlyabsent in the Maxwell’s electrodynamics, for which T = 0, but does not vanish in generalNLE theory. Furthermore, as was observed in [53], if the NLE Lagrangian is of the form L = σ − f ( σ F , σ G ), with some parameter σ and a real function f , then the trace of theenergy-momentum tensor may be written as T = ( − σ/π ) ∂ σ L , allowing us to write theadditional NLE term, at least formally, as a product of a conjugate pair of thermodynamicvariables.Nevertheless, once the first law is obtained, we may deduce the Smarr formula usinga particular choice of perturbation, that is a path through the phase space of solutionsdefined by the carefully chosen scaling of fields [109]. This approach has been used byZhang and Gao [128] for the F -class NLE theories, along a bit of meandering procedure asthey derive the first law by variation of the mass formula, approach (1a) mentioned in thesection 5. We shall rederive the Smarr formula from the first law (5.50) in order to checkthe consistency of the complete procedure.– 23 – .1 Smarr formula from the first law Let ( g ab , A ) be an initial solution of the gNLE field equations. Our first aim is to finda family of rescaled field configurations ( λ g ab , λ ν A ) with a real parameter λ and a realconstant ν chosen such that rescaled fields are again solutions. Of course, there is no apriori guarantee that such simple construction is possible, but we shall prove that thisis indeed the case. Also, note that Smarr’s formula is sometimes obtained via Euler’stheorem for homogeneous functions [79], under the assumption that the black hole mass M ( A , J, Q, . . . ) is a differentiable homogeneous function of degree 1. Eulerian approachwas, in fact, used in the original Smarr’s derivation [103] for the Kerr–Newman blackhole. However, any generalization in this approach demands a careful justification of thehomogeneity of the black hole mass function for a theory under consideration, as it doesnot hold in general [67].Let us now carefully examine the scaling of all objects appearing in our analysis of thespacetime. Metric rescaling g ab → λ g ab immediately implies corresponding rescaling forthe inverse metric, g ab → λ − g ab , volume form, ǫ → λ ǫ , area of the black hole horizon, A → λ A , as well as the Riemann tensor and its contractions, R abcd → R abcd , R ab → R ab , R → λ − R , G ab → G ab Killing vector k a is normalized at infinity via g ab k a k b = −
1, so that k a → λ − k a and k → λ k . Killing vector m a is normalized along its closed orbits C , I C m a m a m = 2 π (6.2)so that m a → m a and m → λ m . In order to have consistent combination χ a = k a + Ω H m a we need Ω H → λ − Ω H . Also, using the geodesic equation for the Killing vector field ξ a generating the Killing horizon, ξ b ∇ b ξ a = κξ a , we have κ → λ − κ for the surface gravity κ .Consequently, via Komar integrals (5.1), we know that M → λM and J → λ J .Now we turn to the gauge sector. Given that the gauge field scales as A → λ ν A and using the metric scaling described above, we immediately have F → λ ν F , ⋆ F → λ ν ⋆ F , as well as F → λ ν − F and G → λ ν − G . For the electric and magnetic 1-forms defined with respect to the Killing vector field χ a we have E → λ ν − E and B → λ ν − B , so that the associated scalar potentials scale as Φ → λ ν − Φ and Ψ → λ ν − Ψ.Einstein’s field equation G ab = 8 πT ab imply that the energy-momentum tensor should bescale invariant, T ab → T ab , and from (2.9), we see that one consistent choice is L → λ − L . By demanding that this scaling is universal, that is valid for all electromagneticLagrangians, basic Maxwell’s case imply ν = 1. This choice tacitly implies that couplingparameters in a NLE Lagrangian will have some specific scaling, say β i → λ b i β i for somereal exponents b i . For example, we have b → λ − b in the Born–Infeld theory and α → λα in the Euler–Heisenberg theory. Consequently, from (5.2), we have Q → λQ and P → λP ,while (5.51) leads to K i → λ − b i K i . All definite scaling exponents deduced above arecollected in table 1. We stress that these are not some necessary scaling transformations,– 24 –caling exponent − g ab , R , F , G − κ , Ω H R abcd , R ab , G ab , E , B , Φ, Ψ1 M , k , A , F , ⋆ F , Q , P g ab , m , A , J ǫ Table 1 . A summary of scaling exponents for various fields and charges. rather a consistent (and convenient) choice which allows us to apply the first law of blackhole mechanics.All quantities varied in the first law of black hole mechanics are functions of theparameter λ of the form Q ( λ ) = λ q Q (1) , (6.3)with some scaling exponent q . Hence, we have δ Q = d Q ( λ )d λ (cid:12)(cid:12)(cid:12) λ =1 = q Q , (6.4)where we have used abbreviation Q = Q (1) for the initial, unperturbed quantity. Puttingall this together we can recover the generalized Smarr formula M = κ π A + 2Ω H J + Φ H Q + X i b i K iχ β i . (6.5)Again, as was remarked under the equation (5.50), absence of the Ψ H P term in this proce-dure is a consequence of its absence in our form of the first law. On the other hand, directderivation of the generalized Smarr formula [53], being independent of the first law, evadesthese obstacles and contains the magnetic potential-charge term.The authors in [128] argue that the Smarr formula obtained via scaling argumentis of greater generality since it may treat the NLE Lagrangians with multiple couplingparameters. However, the only such example known to us is the ABG Lagrangian (C.4)and even here, as was already remarked in [53], one might write the Lagrangian in a form L = ˜ µα − f ( α F ), with ˜ µ = µ/g and α = g . Then, as the parameters scale as µ → λµ and g → λg , the parameter ˜ µ is scale invariant, implying that ABG case is still covered bythe procedure presented in [53]. Even more generally, one might argue that any physicallysensible NLE theory should in some weak field limit be of the form L = − F + σ (cid:0) c F + 2 c FG + c G (cid:1) + O ( σ ) (6.6)with dominant Maxwell’s term and expansion in some coupling parameter σ (dimensionlessconstants c ij are irrelevant here). Then, a simple algebraic manipulation, L = 1 σ (cid:18) −
14 ( σ F ) + c ( σ F ) + 2 c ( σ F )( σ G ) + c ( σ G ) + O ( σ ) (cid:19) (6.7)brings such Lagrangian in a form which was discussed in [53]. Note that in this case thescaling of the coupling parameter is σ → λ σ .– 25 – .2 Linearity of the Smarr formula Finally, we turn to the question about the (non-)linearity of the Smarr formula: For whichNLE theories the generalized Smarr formula may be brought to the form c M = c κ A + c Ω H J + c Φ H Q + c Ψ H P + c Φ H P + c Ψ H Q (6.8)with some real constants { c , . . . , c } ? A systematic approach to the problem is to look intoterms which would, upon integration of the 3-form T ⋆ χ over Σ, produce such products ofpotentials and charges,d(Φ ⋆ Z ) = − E ∧ ⋆ Z = 12 ⋆ R ( χ ) + (2 πT − L ) ⋆ χ (6.9)d(Ψ F ) = − H ∧ F = 12 ⋆ R ( χ ) + L ⋆ χ (6.10)d(Φ F ) = 12 i χ ( F ∧ F ) = − G ⋆ χ (6.11)d(Ψ ⋆ Z ) = − i χ ( ⋆ Z ∧ ⋆ Z ) = 4 (cid:0) L F L G F + ( L G − L F ) G (cid:1) ⋆ χ (6.12)These equations deserve a brief explanation. The first is obtained from the Einstein fieldequation, energy-momentum tensor in the form (2.9) and identity ⋆i E Z = − E ∧ ⋆ Z . Thesecond is obtained by combining the first one with contraction of (A.9) with the Killingvector field χ a . The remaining two equations are obtained by contractions of (A.8) and(A.11) with χ a .Upon inspection, it is suggestive, although we do not have a watertight argument, thata necessary condition for the linearity of the Smarr formula is L = a ( L F F + L G G ) + b (cid:0) L F L G F + ( L G − L F ) G (cid:1) + c G (6.13)for some real constants a , b and c . Namely, this allows one to turn a linear combination of ⋆ R ( χ ) and T ⋆ χ into a linear combination of terms d(Φ ⋆ Z ), d(Ψ F ), d(Φ F ) and d(Ψ ⋆ Z ),with cancellation of the remaining terms. As the term G is non-dynamical, we can disposeof it and set c = 0. The remaining condition may be interpreted as a nonlinear partialdifferential equation for the Lagrangian (as a function of two variables, F and G ), butunfortunately we do not know its complete, general solution.One possible simplification may be obtained if we restrict the analysis to NLE theorieswhich are invariant with respect to electromagnetic duality rotation, defined by F → F cos α + ⋆ Z sin α and Z → Z cos α + ⋆ F sin α with a real angle α . It known [51] thatnecessary and sufficient condition for such invariance to hold is that difference ⋆Z ab Z ab − G be constant for any field configuration, which translates into constancy of combination2 L F L G F + ( L G − L F ) G + ( G / b = 0 = c case. Characteristics ofthe partial differential equation L = a ( L F F + L G G ) in the F - G plane, defined by the system( ˙ F , ˙ G ) = ( F , G ), are nothing but lines through the origin. The partial differential equationis reduced, along a characteristic, to the ordinary differential equation a ˙ L − L = 0. Hence,on a domain where F = 0 we can write the general solution in a form L = F /a f ( G / F ),– 26 –hile on a domain where G = 0 in a form L = G /a g ( F / G ), with some differentiablereal functions f and g . A prominent class of examples are all NLE theories with tracelessenergy-momentum tensor, solutions of the ( a, b, c ) = (1 , ,
0) case, a member of which isrecently introduced ModMax theory [7, 45]. Also, for constant f and a = 1 /s we have thepower-Maxwell class of NLE theories (linearity of the corresponding Smarr formula hasbeen already confirmed in [53]).Another, pragmatic approach is to insist that the NLE Lagrangian should behave closeto the Maxwell’s in a weak field limit. More precisely, let us assume that Lagrangian L is defined on an open subset O ⊆ R , such that (a) (0 , ∈ O , (b) L : O → R is a C function, and (c) L F (0 ,
0) = − / L G (0 ,
0) = 0. Then partial derivatives of (6.13)with respect to F and G imply −
14 = L F (0 ,
0) = − a and 0 = L G (0 ,
0) = − b , (6.14)so that ( a, b ) = (1 , V ⊆ O be an open set star-shaped with respect to the origin (forall x ∈ V the line segment from the origin to x is contained in V ), in which we analyseproblem along the lines defined by G = p F , with a real parameter p . If the solution iswritten in a form L = F f ( G / F ), then along these lines we have L F = f ( p ) − pf ′ ( p ) and L G = f ′ ( p ), while conditions (b) and (c) above imply that f ( p ) = − / p ∈ R .Have we used the other form of the solution, L = G g ( F / G ), and lines defined by F = p G ,analogous reasoning would lead us to the equivalent conclusion that g ( p ) = − p/ p ∈ R . In other words, given that (6.13) is indeed a necessary condition for the linearityof the Smarr formula (which yet has to be proven rigorously), the only NLE theory withMaxwellian weak field limit and linear Smarr formula is the Maxwell’s electrodynamicsitself, at least on some neighbourhood of the origin of F - G plane. The elaborate web of connections between the gravitational theories and thermodynamicsneeds to be tested against all physically motivated modifications of the classical Einstein–Maxwell theory, hoping that this might lead us to some novel insights about the microscopicpicture of the spacetime. Main goal of this paper was to complete the basic architectureof the black hole thermodynamics in the presence of NLE fields, above all the first law ofblack hole mechanics, along with all auxiliary results that allow us to deduce such relation.Some of the generalizations that wait ahead are pretty much straightforward. Forexample, inclusion of the cosmological constant, with the additional V dΛ term in the firstlaw, can be achieved according to an established procedure [70–72, 78]. Extensions of thefirst law for gravitational theories beyond the canonical general relativity, as long as theelectromagnetic field is minimally coupled to gravitation, are in principle covered by the co-variant phase space formalism procedures [15, 69, 110, 120], although a concrete evaluationof the corrections may be a formidable task. Nontrivial contributions to the gravitational– 27 –instein–Hilbert action can appear, for instance, due to quantization (in a sense of an ef-fective theory) [25, 28, 39] or quantum gravity (via spectral triple) [29, 87]. One line of thefuture developments are generalizations for the lower and higher dimensional spacetimes,with caveat that invariant G has to be excluded or replaced with something else, as F and its Hodge dual ⋆ F have equal orders only in four spacetime dimensions. Considerablybigger challenge is to generalize all these results for NLE Lagrangians which also includeterms with covariant derivatives of the 2-form F , as well as nonminimal coupling to grav-itation and other matter fields. Such corrections to the Maxwell’s Lagrangian could beproduced via generalized uncertainty principle [18] or induced from the noncommutativefield theories [30, 55, 57, 82].It remains to be seen if extension of the phase space and additional variations of theLagrangian parameters in the first law are a mere algebraic, bookkeeping device, or animportant hint for understanding of the thermodynamic features of the spacetime. A Important identities
Here we collect several basic definitions and identities with differential forms, used through-out the paper. Suppose that ω is a p -form on a smooth Lorentzian 4-dimensional manifold.Then the Hodge dual, contraction with vector X a and exterior derivative d are, respectively,defined as ( ⋆ω ) a p +1 ...a = 1 p ! ω a ...a p ǫ a ...a p a p +1 ...a (A.1)( i X ω ) a ...a p − = X b ω ba ...a p − (A.2)(d ω ) a ...a p +1 = ( p + 1) ∇ [ a ω a ...a p +1 ] (A.3)Hodge dual twice applied is identity up to sign, ⋆⋆ ω = ( − p (4 − p )+1 ω (plus for odd p and minus for even p ). We immediately have ⋆ ǫ and ⋆ ǫ = −
1. Particularly usefulloperation is so-called “flipping over the Hodge”, i X ⋆ ω = ⋆ ( ω ∧ X ) (A.4)where X is the associated 1-form, X a = g ab X b . Note that i X ǫ = ⋆ X . For any 1-form α wehave ⋆ d ⋆ α = −∇ a α a and d ⋆ α = ( ∇ a α a ) ǫ .For any 2-form F we have two essential identities F ac F cb − ⋆F ac ⋆F cb = − F g ab , (A.5) F ac ⋆F cb = ⋆F ac F cb = − G g ab . (A.6)– 28 –urthermore, using the identity 2 α ∧ ⋆ β = ( α ab β ab ) ǫ , valid for any 2-forms α and β , it isstraightforward to derive the following identities F ∧ ⋆ F = 12 F ǫ (A.7) F ∧ F = − G ǫ (A.8) F ∧ ⋆ Z = − F L F + G L G ) ǫ (A.9) F ∧ Z = − F L G − G L F ) ǫ (A.10) ⋆ Z ∧ ⋆ Z = 8 (cid:0) ( L F − L G ) G − L F L G F (cid:1) ǫ (A.11)Finally, taking into account ⋆F ab ∇ c F ab = F ab ∇ c ⋆F ab and assuming that d F = 0, we have F ac ∇ a F bc = 14 ∇ b F (A.12)and ⋆F ac ∇ a F bc = F ac ∇ a ⋆F bc = 14 ∇ b G . (A.13) B Stokes’ theorem on Lorentzian manifolds
Suppose that M is an orientable smooth m -manifold with boundary ∂ M and inclusion ı : ∂ M ֒ → M . An orientation on M is fixed by choice of a nowhere vanishing m -form ǫ ,while corresponding induced (Stokes) orientation on the boundary is defined as ˆ ǫ = i N ǫ ,with the outward pointing nonvanshing vector field N a on ∂ M . For any smooth, compactlysupported ( m − α on M the Stokes’ theorem [79] states that Z ( M , ǫ ) d α = Z ( ∂ M , ˆ ǫ ) ı ∗ α , (B.1)where we have, for clarity, emphasized orientations for both the manifold and its boundary.Although the Stokes’ theorem does not rely on any additional structure on the manifold,such as metric or connection, it admits some practical corollaries on (pseudo)-Riemannianmanifolds. Suppose that M is a smooth Lorentzian manifold and N ⊆ M its embeddedcompact m -dimensional submanifold with boundary ∂ N , inclusion : ∂ N ֒ → N andan outward pointing, nonvanishing vector field n a on ∂ N . The corresponding inducedorientation on the boundary ∂ N is ˆ ǫ = i n ǫ . Then for any smooth vector field v a on N the Stokes’ theorem implies Z ( N , ǫ ) ( ∇ a v a ) ǫ = Z ( N , ǫ ) d i v ǫ = Z ( ∂ N , ˆ ǫ ) ∗ ( i v ǫ ) . (B.2)Let us, for concreteness, assume that the boundary of N consists of two spacelike hyper-surfaces Σ and Σ ′ , timelike hypersurface S and a null hypersurface (portion of a black holehorizon) H , ∂ N = Σ ∪ Σ ′ ∪ S ∪ H , as sketched in the figure 2. For each part of the boundary it is convenient to have corre-sponding decomposition of the volume form ǫ :– 29 – ΣΣ ′ H Sℓn n n n
Figure 2 . Submanifold N with four parts of the boundary (spacelike hypersurfaces Σ and Σ ′ ,timelike hypersurface S , null hypersurface H ) and corresponding outward pointing vector field n a . (i) The non-null part of the boundary. We assume that n a is normalized such that n a n a = ±
1. Also, following the convection in [116], we introduce the auxiliary vectorfield ˜ n a := ( n b n b ) n a , so that ˜ n a is outward pointing for spacelike n a and inwardpointing for timelike n a . Then n ∧ ˆ ǫ = f ǫ for some function f and contraction with n a leads to the decomposition ǫ = ( n a n a ) n ∧ ˆ ǫ = ˜ n ∧ ˆ ǫ . (B.3)(ii) The null part of the boundary, generated by the future directed vector field ℓ a . Forthe outward pointing vector field we take a future directed null vector field n a on H ,normalized such that n a ℓ a = −
1. Then ℓ ∧ ˆ ǫ = f ǫ for some function f and contractionwith n a leads to the decomposition ǫ = − ℓ ∧ ˆ ǫ . (B.4)These decompositions imply ∗ ( i v ǫ ) = ( (˜ n a v a )ˆ ǫ on non-null part of ∂ N − ( ℓ a v a )ˆ ǫ on null part of ∂ N (B.5)so that Z N ( ∇ a v a ) ǫ = Z Σ (˜ n a v a )ˆ ǫ + Z Σ ′ (˜ n a v a )ˆ ǫ + Z S (˜ n a v a )ˆ ǫ + Z H ( − ℓ a v a )ˆ ǫ , (B.6)where each component of the boundary ∂ N is oriented with the corresponding inducedStokes’ orientation ˆ ǫ . It is understood that choice of the vector field ℓ a comes with ambi-guity, ℓ a → ℓ ′ a = λℓ a for some positive real function λ , leading to redefinitions n ′ a = λ − n a and ˆ ǫ ′ = i n ′ ǫ , but the integrand above remains unchanged, as ℓ a v a ˆ ǫ = ℓ ′ a v a ˆ ǫ ′ . C A sample of NLE Lagrangians
Comprehensive list of all NLE Lagrangians proposed in the literature would be enormousand not quite enlightening. Therefore we single out just a several most significant ones.– 30 –
Born–Infeld [16, 17], L (BI) = b − r F b − G b ! , (C.1)with real parameter b >
0, corresponding to the strength of the maximal field. Exper-imental constraints [41, 90] for the parameter b give us b & (GeV) . Born–InfeldLagrangian is sometimes truncated, for F ≫ ( G /b ) , to L (tBI) = b − r F b ! . (C.2) • Euler–Heisenberg [62] (see also [40]), in weak field expansion L (EH) = − F + α m e (cid:0) F + 7 G (cid:1) + O ( α ) , (C.3)where α is the fine-structure constant and m e is the mass of the electron. • Ay´on-Beato–Garc´ıa [2, 4], L (ABG) = 3 µg g √ F g √ F ! . (C.4)It is important to stress that coupling parameters µ and g are only a posteriori identified as mass and magnetic charge for some specific solution, such as the Bardeenblack hole. • Power-Maxwell [58, 59], L (pM) = C F s , (C.5)with some real constants C = 0 and s = 0. • ModMax theory [7, 77], L (MM) = 14 (cid:16) − F cosh γ + p F + G sinh γ (cid:17) , (C.6)defined with one real parameter γ , is a unique class of NLE theories which is bothconformally invariant (it has vanishing energy-momentum tensor) and invariant withrespect to electromagnetic duality rotations [51]. Acknowledgments
We would like to thank Prof. Robert M. Wald for his invaluable remarks on paper [49] andProf. Dmitri Sorokin for drawing our attention to the novel ModMax model. The researchhas been supported by Croatian Science Foundation project IP-2020-02-9614.– 31 – eferences [1] M. Aaboud et al. Evidence for light-by-light scattering in heavy-ion collisions with theATLAS detector at the LHC.
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