On the compatibility of nonlinear electrodynamics models with Robinson–Trautmangeometry

T. Tahamtan∗Institute of Theoretical Physics, Faculty of Mathematics and Physics,

Charles University, Prague, V Holešovičkách 2, 180 00 Prague 8, Czech Republic(Dated: March 24, 2021)

Robinson–Trautman solutions with nonlinear electrodynamics are investigated for both L(F )and L(F,G) Lagrangians and presence of electric and magnetic charges as well as electromagneticradiation is assumed. Particular interest is devoted to models representing regular black holes forspherically symmetric situations. The results show clear uniqueness of Maxwell electrodynamics withrespect to compatibility with the Robinson–Trautman class. Additionally, regular black hole modelsare clearly not suited to this class while the famous Born–Infeld model illustrates an importantdistinction between L(F ) and L(F,G) as regards the electric field we obtain when the magnetic fieldis nontrivial.

PACS numbers: 04.20.Jb, 04.70.Bw

I. INTRODUCTION

Nonlinear Electrodynamics (NE) is a generalization ofthe linear Maxwell theory to a nonlinear theory. Suchtheory was initially developed to solve the problem ofthe divergent field of a point charge (see, e.g. [1]) alsoyielding a reasonable self-energy of a charged particle.The best-known and frequently used model of NE wasproposed by Born and Infeld in 1934 [2]. An excellentoverview of nonlinear electrodynamics and its main fea-tures was given in a book by Plebański [3]. Einstein grav-ity coupled with nonlinear electrodynamics has attractedattention of many researchers in the literature and thiscarried over to modified theories as well (for example[4]). Considerable interest is specifically devoted to grav-itating NE models which resolve (remove) the spacetimesingularity in the center of a black hole. Such solutionsare called regular black holes and their association withNE started with the model proposed in [5].

Generally, the Lagrangian L of nonlinear electrody-namics is supposed to be a scalar function of the invari-ants F = FµνFµν and G = Fµν∗Fµν = 12�

µναβFµνFαβ(in fact one should consider only G2 to eliminate thepseudoscalar nature of G). If one applies the NE theo-ries in static spherically symmetric spacetimes then theform of the Lagrangian reduces to L(F ), which is also theone used most frequently in the literature. The L(F,G)form of the Lagrangian is used for studying light prop-agation in the geometric–optics approximation [6], par-ticularly for comparisons with the linear Maxwell theory.For example in [7], by analyzing Fresnel equations of wavepropagation, the authors showed that there is no birefrin-gence in the Born–Infeld model, but the velocity of light(as waves of NE) is different from c and always less thanor equal to c (in Maxwell limit). Paper [8] derived theconditions for causal propagation in L(F,G) theories and

∗Electronic address: tahamtan@utf.mff.cuni.cz

demonstrated that with the L(F ) Lagrangian any theoryother than the standard Maxwell vacuum one necessarilyviolates the causality conditions for some allowed back-ground fields.Apart from studies of the general physical properties

of NE (L(F,G)) there are not many nontrivial (F 6= 0,G 6= 0) exact solutions of Einstein gravity coupled withthese models of nonlinear electrodynamics. One excep-tion is a solution describing the Born–Infeld model inflat spacetime, which was obtained using the Newman–Penrose formalism already in [9].This lack of solutions for the L(F,G)–type model is

the initial motivation for our research. Since this meansstepping out of the spherically symmetric situation wedecided to consider the Robinson–Trautman class, whichessentially contains deformations of spherically symmet-ric situations. Additionally, this class contains generi-cally dynamical solutions, which settle down to a sym-metric situation by radiation of gravitational waves. Thisdynamical nature of the selected geometry has the addi-tional benefit of making it potentially suitable for non-linear stability studies of the NE solutions. Knowingwhether a given static or stationary solution is stableor not yields another perspective of the spacetime. Un-stable solutions usually are of less physical significancethan stable ones since they are likely to decay to a stableconfiguration.However, one of the most frequent methods used to ex-

plore the stability of solutions is linear stability analysis.In [10], Moreno and Sarbach presented a study of dynam-ical stability of black–hole solutions in self-gravitatingnonlinear electrodynamics with respect to arbitrary lin-ear fluctuations of the metric and electromagnetic field.They established stability for several specific models ofNE— particularly those corresponding to regular blackholes such as the Bardeen black hole— based on someconditions on the electromagnetic Lagrangian. Basedon the pulsation master equations obtained in [10], pa-per [11] obtained the fundamental Quasi-Normal (QN)modes associated with the gravitational and electrody-

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namic perturbations of black holes in NE theories. Par-allel to this line of investigation there are several otherstudies of QN modes for NE [12] but their results applyonly to test fields propagating in the fixed geometry ofsuch black holes, unlike in [11].

An alternative study of linear stability [13] based onthe sign of the effective energy shows that, except for theBorn–Infeld model, most other models (and especiallythose for regular black holes) are unstable.

This disagreement concerning the linear stability es-tablished using different methods or imposing variousconditions is exacerbated by an additional issue for reg-ular black holes. Stability analyses are predominantlyinvolving the region above the horizon. This is crucialfor regular black holes since the “removed” singularity isbelow the horizon and the exterior solutions tend to bequite similar to other charged black hole solutions. It isthus important to specifically understand the stability ofregular black holes below their horizon.

These issues might be resolved by finding exact solu-tions of Einstein gravity coupled to NE that would corre-spond to a dynamical spacetime with no symmetries andanalysing their non–linear stability. Base on our past re-search [14–16], the Robinson–Trautman (RT) spacetimesprovide a potentially suitable candidate for such study.

Vacuum Robinson–Trautman spacetimes [17–19] rep-resent deformations of spherically symmetric geometriesand by radiating gravitational waves they asymptoti-cally settle down to the Schwarzschild black hole. Wehope to see a similar behavior when they are coupled toNE, this time settling down to the spherically symmet-ric NE configuration after possibly emitting gravitationaland electromagnetic radiation. Such an approach wasalready successful in showing nonlinear stability of theSchwarzschild thin shell wormhole [16]. In [15], we foundvery restricted Robinson–Trautman solutions (involvingan electric charge only but no radiation) with NE sourcesfor several specific models of the NE Lagrangian (bothwith and without a Maxwell limit). The solutions weregenerated from spherically symmetric ones. In all NEmodels we considered, the singularity of the electromag-netic field is resolved as in the static spherically symmet-ric cases. However, the models resolving the curvaturesingularity in spherically symmetric spacetimes could notbe generalized to the Robinson–Trautman geometry us-ing the approach of our previous paper [15]. Here we wantto consider this problem in full generality and prove theimpossibility of the generalization more rigorously.

The Robinson–Trautman family of solutions has beenused in situations more directly related to astrophysicsas well. In [20], Rezzolla et al. showed an analytic expla-nation of an “antikick” appearing in numerical models ofasymmetric binary black hole mergers by modelling thepost-merger deformed horizon using Robinson–Trautmangeometry and computed associated asymptotic momen-tum caused by directional gravitational radiation emis-sion. Exact formulas for directional pattern of gravi-tational radiation in RT class show its direct relation

to locations of higher horizon deformation. Evidently,RT family can provide useful idealized model for gravita-tional radiation from (slowly rotating) isolated sources,e.g., final stages of compact objects mergers. Subse-quently, solutions coupled to electromagnetic fields canprovide models containing electromagnetic counterpartto gravitational radiation. This is indeed the case forRobinson–Trautman–Maxwell solutions which in generalcontain both gravitational and electromagnetic radiation.Our aim is to analyze the situation when the RT geome-try is coupled to NE. If general solutions do not containelectromagnetic radiation in this case this would providepotential clear observational difference compared to pureMaxwell case and suggest that if NE should play a rolein the strong fields around compact objects there wouldbe no strong electromagnetic counterpart to gravitationalradiation in late merger stage. However, RT class is stillnot completely general to describe these situations in fulland further study would be necessary to make predictionsfor general merger scenarios.Previously, we found a solution containing exact grav-

itational waves for a specific NE Lagrangian (square rootmodel) in Kundt class of geometries which is closely re-lated to the RT class, see [21]. This provided hope thatsuch general solutions can be found in RT class as well.Although vacuum RT spacetimes asymptotically de-

cay to Schwartzschild solution as shown analytically in[22, 23] and also RT–scalar field spacetimes settle downto spherically symmetric solutions [14], the RT–Maxwellsystem was shown to be ill-posed in [24, 25]. This raisesa question whether RT–NE system suffers from the sameproblem and thus might not be physically relevant anduseful for stability analysis and other investigations. Af-ter briefly reviewing the study of ill-posedness of RT–Maxwell system we perform the analysis for the explicitRT–NE model obtained in this paper and show that itis well-posed. We plan to study well-posedness for gen-eral NE model in RT class in the future with the help ofresults established in [26].

II. VACUUM ROBINSON–TRAUTMANMETRIC

The vacuum Robinson–Trautman spacetime can be de-scribed by the line element [17–19]

ds2 = −2H du2 − 2 dudr + r2

P 2(dx2 + dy2), (2.1)

where

2H = K − 2r( lnP ),u − 2m/r (2.2)

with K = ∆( lnP ) and

∆ ≡ P 2(∂xx + ∂yy). (2.3)

The metric generally contains two functions, P (u, x, y)and m(u) . The functionm(u) might be set to a constant

3

by a suitable coordinate transformation [19] and we as-sume this is the case for the coordinates of (2.1). Einsteinequations then reduce to a single nonlinear PDE — theRobinson–Trautman equation

∆∆( lnP ) + 12m( lnP ),u = 0 . (2.4)

These spacetimes are then of algebraic type II.As required by the definition of the Robinson–

Trautman family the spacetime admits a geodesic, shear-free, twistfree and expanding null congruence generatedby l = ∂r with r an affine parameter along this congru-ence, u the retarded time and u = const null hypersur-faces. It shows the propagation of the tetrad componentsof the Weyl tensor along u, from null surface to null sur-face. Spatial coordinates x, y span the transversal 2-spacewhich has the Gaussian curvature (for r = 1)

K(u, x, y) ≡ ∆( lnP ) . (2.5)

For general r = const and u = const, the Gaussian cur-vature is K/r2 so that, as r →∞, these 2-spaces becomelocally flat. As usual, we will assume that the transver-sal 2-spaces are compact and connected which leads toa subclass that contains the Schwarzschild solution (as-suming a vanishing cosmological constant for simplicity)corresponding to K = 1 (consistent with spherical sym-metry). This subclass thus represents its generalizationto a nonsymmetric dynamical situation.

To analyse the Robinson–Trautman equation (2.4), itis useful to introduce the following parametrization

P = f(u, x, y)P0 , (2.6)

where f is a function on a 2-sphere S2, corresponding toP0 = 1 + 14 (x

2 + y2) (such choice gives K = 1). By rig-orously analysing equation (2.4) and substituting (2.6),Chruściel [22, 23] proved that, for an arbitrary, suffi-ciently smooth initial data f(x, y, ui) on an initial hy-persurface u = ui, Robinson–Trautman type II vacuumspacetimes (2.1) exist globally for all u ≥ ui. Moreover,they asymptotically converge to the Schwarzschild–(anti-)de Sitter metric with the corresponding mass m andcosmological constant Λ as u→ +∞. This convergenceis exponentially fast because f behaves asymptotically as

f =∑i,j≥0

fi,juje−2iu/m , (2.7)

where fi,j are smooth functions of the spatial coordinatesx, y. For large retarded times u, the function P given by(2.6) exponentially approaches P0 which describes thecorresponding spherically symmetric solution.

III. ROBINSON–TRAUTMAN SOLUTIONCOUPLED TO ELECTROMAGNETISM

We consider the following action, describing an electro-magnetic field in the form of nonlinear electrodynamics

minimally coupled to gravity,

S = 12

∫d4x√−g [R + L(F,G)] , (3.1)

where R is the Ricci scalar for the metric gµν (we useunits in which c = ~ = 8πG = 1). L(F,G) is the La-grangian of the nonlinear electromagnetic field which weassume to be an arbitrary function of the invariants Fand G constructed from a closed Maxwell 2-form. Byvarying the action (3.1) with respect to the metric, weget Einstein equations

Gµν = Tµν . (3.2)

To keep the original form of the Robinson–Trautmanspacetime, we assume the following metric

ds2 = −(2H+Q) du2− 2 dudr+ R2

P 2(dx2 + dy2), (3.3)

where Q = Q(u, r, x, y) and R = R(u, r) and subse-quently we assume u, r, x, y coordinate ordering. Themetric function H is presented in (2.2) where—to makeour equations simpler— we assume m to be zero. How-ever, one can always recover it as a m/r term in Q.In the next section we will find all the field equations

for a general nonlinear electrodynamic Lagrangian. Todo that, we will first obtain the modified Maxwell equa-tions in the following subsection and then we will sum-marize Einstein equations.

A. Modified Maxwell equations

The Lagrangian of the nonlinear electrodynamics isgenerally supposed to be a scalar function of the invari-ants F = FabF ab and G = FabF ∗ab. The electromagneticfields obey the (generally) modified Maxwell (NE) fieldequations. The source–free nonlinear Maxwell equationsare obtained in the standard way from the variationalprinciple and they read

F[µν,λ] = 0, (3.4)(√−g LF Fµν +

√−g LG ∗Fµν

),µ

= 0, (3.5)

where we use the abbreviations LF = ∂ L∂F , LG =∂ L∂G ,

LFF = ∂2L

∂2F , etc. We consider Fµν to be the only funda-

mental variable as shown by Plebański [3].The nonzero electromagnetic field components in the

Robinson–Trautman class are Fur, Fux, Fuy and Fxy.Since the metric (3.3) can accommodate only the out-going rays aligned with the principle null direction, weassume that Fxr, Fyr or Fux, Fuy are zero since theywould otherwise correspond to rays in the opposite nulldirection (ingoing). This is related to fixing the initialconditions for the evolution of the Robinson–Trautmangeometry. The electromagnetic field invariant is then

F = Fµν Fµν = 2 (fm − fe) ,

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where

fm = gxx gyy (Fxy)2, fe = (Fur)2

and the pseudoscalar invariant would be

G = Fµν ∗Fµν =12�µναβFµνFαβ ,

where �µναβ is the totally antisymmetric Levi-Civita ten-sor with �urxy = −1√−g . Therefore, G can be written inthe following form

G = − 4√−g

FurFxy.

The duality relations between the electromagnetic fieldcomponents take this form

∗F xy = − Fur√−g

, ∗Fur = − Fxy√−g

,

∗F xr = Fuy√−g

, ∗F yr = Fux√−g

. (3.6)

Maxwell equations (3.4) have the following componentsin our case

∂uFxy + ∂yFux + ∂xFyu = 0, (3.7)∂rFxy = 0, (3.8)

∂xFru + ∂rFux = 0, (3.9)∂yFru + ∂rFuy = 0. (3.10)

It is possible to simplify the form of the electromagneticfield using the above equations. From (3.8), we can inte-grate (we selected a convenient form to express the result)

Fxy =B(u, x, y)P (u, x, y)2 (3.11)

and we also introduce a notation Fur = −E(u, r, x, y).Taking derivative of (3.7) with respect to r, we obtainanother useful relation

(Fux),yr = (Fuy),xr . (3.12)

The second modified Maxwell equation (3.5) for themetric (3.3) has the following components (introducingthe notation Ξ =

√−g LF and Ξ̃ =

√−g LG)(

ΞF ru + Ξ̃ ∗F ru),r

= 0 , (3.13)(ΞF rx + Ξ̃ ∗F rx

),r

+(ΞF yx + Ξ̃ ∗F yx

),y

= 0 , (3.14)(ΞF ry + Ξ̃ ∗F ry

),r

+(ΞF xy + Ξ̃ ∗F xy

),x

= 0 (3.15)

and the last equation is(Ξ F ur + Ξ̃ ∗F ur

),u

+(

Ξ F xr + Ξ̃ ∗F xr)

,x+(

Ξ F yr + Ξ̃ ∗F yr)

,y= 0.

(3.16)

The above equations are not yet simplified— to do that,we start with equation (3.13). Using the duality relationsbetween electromagnetic fields in (3.6), equation (3.13)can be written as

ΞFur + LGFxy = C̃(u, x, y). (3.17)

Applying now the above expression, we simplify equation(3.16) further to obtain

(LF Fux + LGFuy),x + (LF Fuy + LGFux),y −(C̃),u

= 0

(3.18)

with C̃ independent of r. Taking derivative of (3.18)with respect to r, we get the following constraint relationwhich will be useful later on to find the expressions forradiative fields,

(LF Fux + LGFuy),xr = − (LF Fuy + LGFux),yr .

The remaining Maxwell equations from the second set,namely (3.14) and (3.15), can be cast in the followingform using our notation for electric and magnetic fieldsand the duality relation (3.6)

(LF Fux + LGFuy),r + Ω,y = 0, (3.19)(LF Fuy + LGFux),r − Ω,x = 0, (3.20)

where Ω =(LGE + LF Br2

). Using (3.19) and (3.20),

we can find the radiative fields Fux(u, r, x, y) andFuy(u, r, x, y) for any nonlinear electrodynamic La-grangian

Fux = −LF (∫

Ω,y dr−�0)+LG(∫

Ω,x dr+�1)L2

F−L2

G

, (3.21)

Fuy =LF (∫

Ω,x dr+�1)+LG(∫

Ω,y dr−�0)L2

F−L2

G

, (3.22)

where �0(u, x, y) and �1(u, x, y) are integration constants.Note that if we assume the electromagnetic fields to beindependent from the transversal directions (E(u, r) andB(u)) then (3.21) and (3.22) imply

Fux =LF �0 − LG �1

L2F − L2G, (3.23)

Fuy =LF �1 − LG �0

L2F − L2G. (3.24)

Applying (3.9) and (3.10) to this special case, we get ra-diative fields that are r-independent. However, analysingthe above expressions, we conclude that this is only pos-sible for a Lagrangian of the Maxwell type.

B. Field equations

After analysing the electromagnetic field equations weturn our attention to Einstein equations, both their geo-metrical and source parts.

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The electromagnetic energy momentum tensor is de-fined as

Tµν =12{δ

µν (L−GLG)− 4 (FνλFµλ)LF }. (3.25)

Before proceeding to evaluate individual components ofthe energy momentum tensor we will use one componentof Einstein equations, Gur = Tur, to simplify the generalmetric (3.3). Since Fxr, Fyr are zero due to our generalsetting this Einstein equation simplifies considerably to

Gur = 2R,rrR

= 0 (3.26)

and gives R that is linear in r

R = U1(u)r + U2(u) .

One can show that the above expression can be simplifiedinto R = r with a coordinate transformation, which wewill assume from now on. It has been shown in [27] thatthe above condition on energy momentum tensor compo-nents leads to this result for any static spherically sym-metric spacetime of four or more dimensions whereas nowwe provided its generalization to the dynamical space-time under consideration.

The nonzero components of the energy momentum ten-sor for electromagnetic fields are

Tuu = T rr =(L−GLG)

2 + 2 feLF , (3.27)

T xx = T yy =(L−GLG)

2 − 2 fmLF (3.28)

and, additionally, because we include radiation there arenonzero off–diagonal terms; i.e.,

T ru = −2 (FuλF rλ)LF (3.29)

= 2LFP 2

r2{

(Fux)2 + (Fuy)2},

T rx = −2 (FxλF rλ)LF (3.30)

= 2LF{FurFux + FuyFxy

P 2

r2

},

T ry = −2 (FyλF rλ)LF (3.31)

= 2LF{FurFuy + FuxFyx

P 2

r2

}and also the following relations between the remainingenergy momentum components,

T xu = −P 2

r2T rx , T

yu = −

P 2

r2T ry . (3.32)

The geometrical part of the remaining Einstein fieldequations is given by the following expressions

Grr = Guu =Q,rr

+ Qr2, (3.33)

Gxx = Gyy =Q,rr

2 +Q,rr

(3.34)

and the off–diagonal terms are

Grx = −12 Q,xr, (3.35)

Gry = −12 Q,yr (3.36)

together with the relations similar to (3.32)

Gxu = −P 2

r2Grx, G

yu = −

P 2

r2Gry. (3.37)

The last off–diagonal term—called the Robinson–Trautman equation in the vacuum case—is

Gru =−12r2 ∆ (K +Q)−

1r

[(lnP ),u (rQ,r − 2Q) +Q,u] .

(3.38)

All these field equations are obtained for the most generalset up which can be used for any particular nonlinearelectrodynamic Lagrangian.

IV. CONSISTENCY OF THE FIELDEQUATIONS

In this section, we check the consistency of Einsteinequations with general NE in the Robinson–Trautmanclass described in section II. Using one of the Einsteinequations (3.2), Guu = Tuu with Guu given by (3.33),we obtain

(rQ),r = r2 Tuu (4.1)

Employing the expression for Tuu from (3.27) and sub-stituting the resulting equation in Gxx = T xx with therespective sides expressed using (3.34) and (3.28), we ar-rive at

G2 LGG(rE,r − 2E

4E

)−GLFG (2rE,rE + F )

= −4E2LFF(rE,rE +

2B2

r4

)+ LF

(rE,rE + 2E2

),

(4.2)

which can be alternatively expressed as

rE,rE[2F LGG + 4E2 (LFF + LGG)− (2GLFG + LF )

]+G

2

2 [LFF − LGG]− F GLFG − 2E2 LF = 0.

(4.3)

The above expression is equivalent to a component of theMaxwell equations (3.13). If the Lagrangian would be inthe form L(F ) this equivalence is still satisfied and theform of the relation (4.3) simplifies considerably

LFF =LF (1 + ζ)

4(B2

r4 + E2 ζ) ,

(4.4)

6

where ζ = r E,r2E . One can arrive at this result by simplyputting all G derivatives of Lagrangian to zero in (4.3).If we apply the same procedure for equationGrx = T rx

and substitute for Q using (4.1), we obtain an expres-sion that, in the case of L(F ) model, corresponds to acomponent of the Maxwell equation (3.9). But for theL(F,G) model these two are not equivalent and equationGrx = T rx has to be satisfied independently. We showthe relevant computations in Appendix A.

A. Example I: Maxwell theory

When studying nonlinear electrodynamics it is alwaysworth it to first review the results for the linear theory,namely the Maxwell Lagrangian, in order to provide acomparison. The Maxwell theory corresponds to the La-grangian L = −F which is also frequently considereda desirable weak-field limit (when F and G are small)of a general nonlinear Lagrangian. Note the overview[19] of a complete solution for the Maxwell theory in theRobinson–Trautman class. The analysis therein was per-formed in the NP formalism. Here, we briefly reviewthe Maxwell solution in the RT class using the tensorialformalism developed in preceding sections.

The independence of the Lagrangian on G significantlysimplifies most of the Maxwell equations. The magneticfield is still the same , B(u, x, y) = P 2 Fxy, see (3.11). Tofind the electric field, we use equation (3.17) and obtainthe following expression for the electric field

E(u, r, x, y) = A(u, x, y)r2

, (4.5)

where A(u, x, y) = P 2 C̃. To find the radiative fields, weuse equations (3.21) and (3.22),

Fux =B,yr− �0, (4.6)

Fuy = −B,xr− �1. (4.7)

From equations (3.18) and (3.12) we obtain the followingrelations

C̃,u − (�0),x − (�1),y = 0, (4.8)B,xx +B,yy = 0. (4.9)

Since we are interested in black-hole solutions we assumethe transversal spaces to be compact and thus ∆B = 0(see (4.9)) means that B should be a constant in x and ywhich further leads to Fux and Fuy being r-independentdue to (4.6) and (4.7). Using this result together with(3.9) and (3.10), we conclude that E,x = 0 and E,y =0. Therefore we have B(u), E(u, r) and the MaxwellLagrangian is independent of x and y.From (4.5) and the electromagnetic energy momentum

tensor components ((3.27), (3.28)), the diagonal energy

momentum components reduce to

Tuu = T rr = −(A2 +B2

r4

), (4.10)

T xx = = T yy(A2 +B2

r4

)(4.11)

and the off–diagonal energy momentum components to

T ru = −2P 2

r2{

(Fux)2 + (Fuy)2}, (4.12)

T rx =2r2{AFux −B Fuy} , (4.13)

T ry =2r2{AFuy +B Fux} . (4.14)

Using Guu = Tuu and Gxx = T xx, the metric functionQ takes the form

Q(u, r, x, y) = A(u)2 +B(u)2

r2− 2m(u, x, y)

r, (4.15)

where we denoted the constant of integration −2m toyield the proper Schwarzschild limit. Substituting themetric solution in Grx − T rx = 0 and using (3.35) and(4.13), we get the following result

m,x = 2 (−AFux +B Fuy) (4.16)

and similarly for Gry − T ry with (3.36) and (4.14), weobtain

m,y = −2 (AFuy +B Fux) . (4.17)

Combining these equations with relation (3.7) for thefields and equation (4.8), we can derive an expressionfor m

∆m = 2P 2 (B (Fxy),u +A C̃,u).

It can be written more explicitly as

∆m = 2(BB,u− 2(lnP ),u (A2 +B2) +AA,u). (4.18)

And finally, the last equation Gru = T ru (see (3.38) and(4.12)) can be split into two equations for terms of dif-ferent orders in r

∆K + 12m (lnP ),u − 4m,u =4P 2{

(�0)2 + (�1)2},

(4.19)∆m+ 4(lnP ),u(A2 +B2) =(A2 +B2),u, (4.20)

where equation (4.20) is identical to (4.18). All equationsfor the Maxwell theory in the RT class are satisfied pro-vided we solve the evolution equation (4.19). It is clearthat if one considers a vanishing electric charge (whichcorresponds to A = 0) the resulting solution can still be anontrivial radiative one and the same holds when B = 0.

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B. Example II: Born–Infeld theory

The Born–Infeld Lagrangian is one of the most at-tractive NE models which in the weak limit goes to theMaxwell case and in the strong regime goes to the squareroot model. This model was used in many different ar-eas of research, from the flat spacetime with the aim toremove a point-charge singularity, to the string theorywhere it appears as a low–energy model. Also, there aremany studies of its properties in GR, investigating, e.g.,its stability, quasi normal modes, etc. and most recentlythe interaction with a scalar field [28]. In [9], the au-thors studied the Born–Infeld model in the flat spacetimeusing the NP formalism and included the other electro-magnetic invariant G. We aim to find an exact solutionfor the Born–Infeld model in a dynamical spacetime (theRobinson–Trautman class). We pay special attention tothe radiative fields for BI model in this geometry. Notethat most of the papers on nonlinear electrodynamicsconcern static or stationary spacetimes and studies ofthe dynamical behavior of this theory in exact form isabsent in the literature. In this subsection, we use thefield equations from the preceding sections to see whethersuch dynamical solutions exist in the RT class or not.

The Born–Infeld Lagrangian has the following form

L(F,G) = 4β2(

1−

√1 + F2β2 −

G2

16β4

), (4.21)

where β is a constant which has the physical interpre-tation of a critical field strength. To solve the modifiedMaxwell equations for this specific Lagrangian, we startwith (3.17) and obtain an expression for the electric fieldE(u, r, x, y) in terms of the magnetic field B(u, x, y)

E(u, r, x, y) = ± C β√β2 r4 +B2 + C2

, (4.22)

where C = P 2 C̃ and B (3.11) are functions of u, x, y.Compared to previous results concerning only the L(F )Lagrangian, the above electric field reproduces themwhen B = 0, for example in paper [15] which investi-gated this model in the RT class. Curiously, when bothE and B are nonzero the L(F ) Lagrangian model pro-duces an electric field that differs from the result obtainedfor L(F,G) and, moreover, the field is no longer regularat the origin. This illustrates the importance of includ-ing the invariant G when considering the magnetic fieldas well. The next step is to find the radiative fields using(3.21) and (3.22) for the BI Lagrangian

Fux = Σ[Γ(∫

Ω,y dr − �0)− C B(∫

Ω,x dr + �1)],

(4.23)Fuy = −Σ

[Γ(∫

Ω,x dr + �1)− CB(∫

Ω,y dr − �0)],

(4.24)

where

Ω = − β B√β2 r4 +B2 + C2

,

Γ = βr2√β2r4 +B2 + C2,

Σ = β2 r4 +B2

B2 (β2 r4 − C2) + β2 r4(β2 r4 + C2) .

After calculating the electromagnetic fields we obtain themetric (3.3), which is primarily determined by the metricfunction Q. Applying (4.1), it takes the form

Q(u, r, x, y) = −mr

+ 2β2

3 r2 − 2β

r

∫ √β2 r4 +B2 + C2,

(4.25)

where B , C and now also m are functions of u, x, y. Notsurprisingly, the above solution is of the same form (apartfrom dependencies on x, y) as that found in [15] with theonly difference being the presence of B. Since the elec-tric fields found here and in [15] are similar, one wouldexpect the similarity in the metric solution as well. How-ever, one should as well note that we have also nontrivialradiative fields present in our current situation which wasnot the case in [15]. With all the electromagnetic fieldsand the metric solution known, one should check whetherthe rest of the field equations are satisfied. Arriving atthe equation Grx = T rx, we realize that it cannot be sat-isfied. Therefore, it is not possible to have a nontrivialelectric and magnetic fields together with radiative fieldsfor the Born–Infeld model in the RT class.Note that with the assumption of vanishing radiative

fields, one finds an exact solution in this theory withboth electric and magnetic point charges, similar to [29].Moreover, it is clear that the functions C and B appear-ing in (4.22) can only depend on u then.

C. Other models

Besides the Born–Infeld model, the above result is validfor several other models of NE — it is not possible to findconsistent solutions for NE in the RT class with nontriv-ial radiative terms. Although some Einstein equationsare equivalent to Maxwell equations (as shown at thebeginning of section IV), the additional field equationscontaining radiative terms cannot be satisfied in the RTclass.

V. ROBINSON–TRAUTMAN SOLUTIONS FORL(F )

Due to the absence of nontrivial exact solutions of Ein-stein gravity coupled with NE Lagrangian when we as-sumed electric and magnetic fields in the RT class, inthis section, we study the case when there exists only a

8

magnetic (or an electric) field — by this we mean non-null fields. Note that to consider “pure magnetic field”(the same would apply to “pure electric field”) meansthe other invariant G ∼ E ·B vanishes identically, sothe form of Lagrangian is effectively L(F ). We alreadyfound solutions for several models of NE for the L(F )Lagrangian in [15] in this spacetime while involving onlyan electric point charge without having electromagneticradiation. In this section we concentrate on solutions fora magnetic charge. First, we find general formula for un-known metric function Q in our metric ansatz (3.3) inthe presence of “magnetic field”. Then we evaluate con-ditions for finding regular black holes in the RT class andextend the solution to contain electromagnetic radiationas well. Furthermore, in the subsequent section, we usethe same method to find an exact solution for “electricfield”.

A. Magnetic field

Recently, studying magnetic fields in NE became sub-ject of substantial interest as a means to find regularblack holes in static spherically symmetric spacetimes.By regular black holes we mean geometries lacking singu-larity at the center of a black hole determined by the pres-ence of a horizon. Thus all scalar invariants are regulareverywhere in the spacetime. The first regular black holewas introduced by Bardeen [30] as a solution generatedby certain stress energy tensor without clear physical in-terpretation. More recently, number of models for regularblack holes have been proposed together with physicallymotivated matter content needed for their explanation[31].

The idea of constructing regular black holes by nonlin-ear electrodynamics as a source was introduced by Ayón-Beato and García [5]. The same authors showed that thecorresponding source of the Bardeen black hole can be as-sociated with a specific model of nonlinear electrodynam-ics Lagrangian coupled to gravity [32]. Soon after thatBronnikov [33] proved a theorem which says that thereis no spherically symmetric solution with a globally reg-ular metric coupled to nonlinear electrodynamics satisfy-ing correct weak–field limit and having nonzero electriccharge. Note that all these solutions and most of the non-linear electrodynamics models representing regular blackhole spacetimes presented in literature are static spher-ically symmetric [34]. Some of these regular black-holesolutions have been extended to stationary spacetimes[35].

Here, we are mainly interested in the possibility of hav-ing regular black holes in the RT class by utilizing suit-able NE model as a source. As we already mentionedearlier based on [33], regular black holes in static spher-ically symmetric (SSS) situations need only magneticfield. Here, we study the same situation although ourspacetime is not static (not even stationary) nor spheri-cally symmetric. But this approach may provide the pos-

sibility of extending those SSS solutions to the dynamicalspacetime.In general, no matter what model of NE is applied,

the equation (3.8) always holds which means that themagnetic field Fxy is independent of r. Since we arestudying the case when there is only a magnetic field weput the electric field E(u, r, x, y) and the radiative terms,Fux, Fuy, to zero. The modified Maxwell equations aregetting significantly shorter and simpler.Using (3.7) we conclude that Fxy is u-independent and

from (3.14) and (3.15) we obtain the following expression

Fxy(x, y) =r2 C0(u, r)P 2LF

. (5.1)

With no radiative terms, Fux = 0 and Fuy = 0, all theoff–diagonal energy momentum tensor components arevanishing. Therefore, Einstein equations Grx = T rx andGry = T ry lead to Q independent of x, y. Turning to theremaining Einstein equations (e.g., Guu = Tuu) we have

(rQ),r = r2L/2, (5.2)

which can be written in the following form

Q(u, r) = 12r

∫r2L(F )dr − m(u)

r,

where ” − m” is an integration constant. It is clearfrom the above equation that the Lagrangian should beL(u, r). Therefore F , the electromagnetic scalar invari-ant, should be F (u, r). Since from the definition we haveF = 2F

2xy P

4

r4 then the function P has necessarily a sepa-rated form, namely

P = qm(u)√Fxy

. (5.3)

Then the electromagnetic scalar invariant is F = 2 q4m

r4

and is obviously singular at r = 0. The above expressionfor P is also consistent with (5.1).Let us recall that the aim of using only the magnetic

field here is to find regular black hole solutions definedby regularity of scalar curvature invariants such as Ricciand Kretschmann scalars. These two scalar quantitiesfor our metric (3.3) can be expressed as

Kretschmann = (Q,rr)2 + (2Q,rr

)2 + (2Qr2

)2, (5.4)

Ricci = −Q,rr −4Q,rr− 2Q

r2. (5.5)

Let us note that the form of these two quantities is notchanging even for more general case when Q is a functionof x, y as well. For having regular solution the mass termm/r must be excluded from Q(u, r). The main equation

9

in the Robinson–Trautman class which determines its dy-namics is the so-called RT equation (3.38) (with T ru = 0now)

Gru = −1

2r2 ∆K −1r

[(lnP ),u (rQ,r − 2Q) +Q,u] ,

(5.6)

where K(u, x, y) is K = ∆( lnP ) as in the vacuum RT.Using (5.3) it can be written as following

q4m κ02r + ( ln qm),u{rQ,r − 2Q}+Q,u = 0, (5.7)

where κ0 = ∆K|qm=1 is necessarily a constant. The so-lution for the above equation is

Q = q2m

2

(2 f(r/qm)−

qm κ0r

∫qmdu

)(5.8)

where f is an arbitrary function. If we look at this so-lution for Q with nonzero κ0 (when κ0 = 0 the solutionreduces from algebraic type II to type D — in effect lead-ing to spherically symmetric case only) and plug it intoscalar invariants (5.5) and (5.4), we see that it is notpossible to have a regular solution. Eventhough therecan be certain solutions for arbitrary nonlinear electro-dynamics Lagrangian none of them can correspond to aregular black hole.

We can impose some assumptions on the magnetic fieldor the corresponding metric solution to see whether insome special cases it is possible to have regular solution.

One such assumption is that the so-called “magneticcharge”, qm, is a constant. Then from F = 2 q

4m

r4 the La-grangian is now only r-dependent. So the metric functionQ (5.2) can be written in the following form

Q(u, r) = R(r)− m(u)r

. (5.9)

For checking the regularity we again compute two scalarinvariant quantities, (5.4) and (5.5). It is clear that pa-rameter m in (5.9) has to vanish as previously. Let usassume that there exists some form of R(r) that makesthe two scalar invariant quantities regular. But now weneed to make sure that with this assumption the space-time is still of type II, namely not spherically symmetric.From the main equation determining the dynamics of RTspacetime (3.38) we have

Gru = −1

2r2 ∆∆( lnP ) = 0, (5.10)

which obviously shows that the spacetime is no longertype II but rather type D only and thus spherically sym-metric.

If we consider an electric charge instead of a magneticone we will arrive at a form of Q corresponding to (5.8)and obtain type of solutions already discussed in [15].These cannot give rise to spherically symmetric regular

black hole solutions due to already discussed results of[33].The above negative results for regular black holes stem

from two crucial facts. First, the necessity of assumingvanishing mass parameter m to preserve regularity andfrom the absence of radiative terms that lead to restrictedform of Q. This means that there are no relevant sourcesfor nontrivial Gaussian curvature (K 6= const.) in RTequation (3.38)

B. Magnetic charge and radiation

In this section, we continue the study with same as-sumptions as in preceding subsection with the differencethat we allow for radiative terms, i.e., E = 0 but Fxy,Fux, Fuy are assumed nonzero.From (3.9) and (3.10), the radiative field components

Fux, Fuy are independent of r like for Fxy. Using thedefinition (3.11) and equations (3.19) and (3.20), we getthe following relations for these two radiative fields

Fux = −(LF B),y(LF B),x

Fuy. (5.11)

Recalling that the electromagnetic scalar invariant is F =2B2r4 , the above equation will simplify to

Fux = −B,yB,x

Fuy (5.12)

and from (3.18) and also (5.12), we get

(Fux),x = −(Fuy),y. (5.13)

The solution (5.2) for Einstein equations from the pre-ceding subsection still holds here and we can write

L = 2r2

(r Q),r. (5.14)

By taking "x" derivative from both sides and consideringF = 2B

2

r4 , we get

2LF B,xB = r2(r Q),rx (5.15)

and similarly by using y derivative. From Grx = T rx((3.35), (3.30)) and Gry = T ry ((3.36), (3.31)), we fur-ther obtain

−4BLF Fuy = r2Q,xr, (5.16)−4BLF Fux = r2Q,yr. (5.17)

With the help of (5.15) and the y-version of it we arriveat

−2 (r Q),rxFuyB,x

= Q,xr, (5.18)

−2 (r Q),ryFuxB,y

= Q,yr. (5.19)

10

Combining these two pairs of equations together and ap-plying (5.12), we get

2 r FuxB,y

(Q,x −Q,y)− (Q,x +Q,y) = C0, (5.20)

2 r FuxB,y

(Q,x +Q,y)− (Q,x −Q,y) = C1, (5.21)

where C0(u, x, y) and C1(u, x, y) are integration con-stants. By solving for Q,x and Q,y from the above equa-tions and assuming Q,xy = Q,yx one can show that Qshould be independent of x, y. By further using Ein-stein field equations, for example Grx − T rx = 0 andGry − T ry = 0, we immediately conclude that the radia-tive fields must vanish. Therefore we end up with thecase where only a magnetic field exists, like in the previ-ous section.

Note that we are generalizing the solutions based onthe vacuum RT solution (containing m(u)). If we con-sider ”− m(u,x,y)r ” term in the vacuum RT solution withthe new assumption that our Lagrangian is of the formL(u, r) then from Eqs (5.16) and (5.17) (instead of Q oneshould put the vacuum metric function H (2.2) now), wewill see that only for the Maxwell case LF = −1 it ispossible to have radiation consistent with m(u, x, y) 6= 0.

C. Electric charge and Radiation

To compare with the preceding sections, we study thecase when there exists only an electric field and electro-magnetic radiation terms. In [15], we found several so-lutions for different models of nonlinear electrodynamicsfor an electric point charge without considering electro-magnetic radiation. This part would be a generalizationof [15]. As usual we start with modified Maxwell equa-tions, from (3.17) we have

LF Fur =A

r2, (5.22)

where A(u, x, y) = P 2 C̃ and C̃(u, x, y) is an integrationconstant. From (3.19) and (3.20) (note that we assumeFxy = 0), we obtain

LF Fux = C̃1, (5.23)LF Fuy = C̃2, (5.24)

where C̃1 and C̃2 are integration constants. From (3.7)we get

(Fux),y = (Fuy),x. (5.25)

By checking the Einstein equations component Guu =Tuu, we arrive at

(r Q),r = r2 Tuuwhich means that

Q(u, r, x, y) = 1r

∫r2 Tuu dr −

2m(u, x, y)r

. (5.26)

Since we are looking for the possibility to have radia-tive terms, we check equations Grx = T rx (see (3.35)and (3.30)) and Gry = T ry (see (3.36) and (3.31)) using(5.22) to arrive at the following expressions

−4AFux = r2Q,xr, (5.27)−4AFuy = r2Q,yr. (5.28)

By taking x, r and y, r derivatives of Q (5.26), namelyQ,xr and Q,yr, and putting them back to the above equa-tions we find

−4AFux = r3 (Tuu),x−∫r2 (Tuu),x dr+2m,x. (5.29)

We can express the relevant energy momentum tensorcomponent from (3.27)

(Tuu),x =LF2 F,x +

2r2

[A,xFur +A(Fur),x] ,(5.30)

F,x = −4 (Fur) (Fur),x, (5.31)

where fe = (Fur)2. Finally, we arrive at

(Tuu),x =2A,xr2

Fur. (5.32)

Now (5.29) would be (using (5.23) and also (5.22))(2 C̃1A,x

r2 + r)Fur =

∫Fur dr −

m,xA,x

(5.33)

and by r derivative of the above equation while remem-bering Fur = −E we obtain

E(u, r, x, y) = − qe(u, x, y)(2 C̃1A,x

r + 1)2 (5.34)

and similarly from the same procedure, starting with y, rderivatives before (5.29), we get alternative form of E.Comparing these two, we get this constraint equation

C̃1A,x

= C̃2A,y

.

From the expression for E and using (3.9), we can expressradiative term in the following way

Fux = −∫E,xdr + �̃0, (5.35)

where �̃0(u, x, y) is an integration constant. On the otherhand by substituting LF from (5.22) into (5.23), one ex-pects that both expressions (5.23) and (5.35) for Fux arethe same leading to equation

− Ar2E

(−∫E,xdr + �̃0) = C̃1. (5.36)

From this equality and the expression for the electricfield (5.34), we get the following relations

qe(u, x, y) = qe(u), �̃0 =qeA

2,x

4 C̃1A, C̃1 =

aA,x√A,

11

where a(u) is an integration constant. Applying this pro-cedure to Fuy we find the relations

�̃1 =qeA

2,y

4 C̃2A, C̃2 =

aA,y√A,

where �̃1(u, x, y) is an integration constant coming from(3.10) in a similar way to �̃0 in (5.35).Using the above results we can re-express the electric

field from (5.34) in a simpler way

E(u, r, x, y) = − qe(2 a√Ar + 1

)2 . (5.37)One expects that this electric field corresponds to somespecific NE Lagrangian. To find the Lagrangian, we firstfind r in terms of F (F = −2E2 ) and substitute it in(5.22) to obtain the following expression

LF =4a2

qe

1(−1 +

(− F2q2e

)1/4)2 . (5.38)Such form clearly means that both qe and a have to beconstants and represent parameters of the Lagrangianmodel rather than, e.g., interpreting qe as a charge whichis anyway not its proper interpretation.

Then we can integrate to obtain corresponding La-grangian

LNew = 32qe a2[u3 + 3u2 − 4u− 2

2(1− u) − 3 ln (1− u)− L0],

(5.39)

where u =√√

−F2q2e

and L0 is an integration constant.To have the exact Maxwell limit one should set L0 = −1but we consider L0 = 0 in the following calculations.Due to the logarithmic term in the Lagrangian we arelimited to field strength corresponding to u < 1 whichcan be fine-tuned using qe to accommodate high valuesof F . The plots of this Lagrangian together with Maxwelland Born–Infeld models are shown in Figure 1. Energyconditions for this new Lagrangian are investigated inAppendix B.

Since the electric field and the Lagrangian are clear wecan find the metric function by using Guu = Tuu andGxx = T xx which leads to

Q(u, r, x, y) = µr

+ K̃(u, x, y)− 8qe a√A(u, x, y) r

+16Λr2 + 48Λr2 ln(

2a r2a r +

√A(u, x, y)

),

(5.40)

where K̃ = 2qeA, Λ = −a2qe3 and the modified mass term

is

µ(u, x, y) = −2m(u, x, y) + qe3aA32

(7− 6 ln(2

√A)).

2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00F

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

L(F

)

LMaxwellLBorn InfeldLNew

Figure 1: Plot of Maxwell, Born–Infeld and the new La-grangian in terms of F . The constant parameters are setto β = 1, a = 1, L0 = −1 and qe = −16.

This solution has the following form asymptotically(for r →∞)

Q→ 16Λ r2 + 3aµ+ 2qeA32

3ar −qeA

2

4a2 r2 +qeA

52

10 a3r3 +O(1r4

)(5.41)

To satisfy further Einstein equations Gry = T ryand Grx = T rx the function µ(u, x, y) should be x, y-independent (µ(u)). The last equation to satisfy isGru = T ru (expressed using (3.38) and (3.29)) whichleads to these dynamical equations

∆(K + K̃

)− 6µ(lnP ),u + 2µ,u = 0, (5.42)

∆A+A2 ∆(lnA) + A32

aqe

(2K̃ (lnP ),u − K̃,u

)= 0,

(5.43)

Provided we solve the above two coupled nonlinearparabolic type PDEs (5.42) and (5.43) for unknown func-tions P and A, we have a complete specific solution forour model. Such a solution is generally radiative (bothin terms of electromagnetic and gravitational waves), hasthe Maxwell-type limit and contains a term mimickingthe cosmological constant (see (5.41)). The metric so-lution is singular at r = 0 while the electric field is not,provided a is positive. Strangely, this NE model seems tobe a unique one compatible with general solutions con-taining electromagnetic radiation in the RT class.If we assume from the beginning that the electric field

has restricted dependence on coordinates (E(u, r)) thenusing (3.9) and (3.10), we derive that Fux (and similarlyFuy) is r-independent and immediately from (5.23) (or(5.24)) one sees that LF has to be a constant leading toMaxwell theory.

12

VI. LINEAR PERTURBATIONS AROUNDSTATIONARY SOLUTION

In the vacuum case the Robinson–Trautman space-times have well defined evolution to the future [22, 23].However, this is not the case for Robinson–Trautman–Maxwell system. In [24], Lun and Chow study linear per-turbations around Reissner–Nordström solution withinthe RT–Maxwell system and show their exponential di-vergence. This result was later confirmed by [25] leadingto limited significance of RT–Maxwell solutions due toits ill-posedness.

First, we review the linear perturbation analysis ofthe Robinson–Trautman–Maxwell system using the equa-tions derived in section IVA. Subsequently, we apply thesame procedure to RT–NE system based on the NE La-grangian derived in section VC.

We will linearize the evolution equations using assump-tions already made in [24]. The function P determiningthe geometry of transversal two-spaces is considered inthe following form, see (2.6),

P = P0(1 + g) , (6.1)

in which P0 corresponds to spherical symmetry and thefunction g(u, x, y) is a perturbation. Therefore it is pos-sible to linearize the field equations by neglecting termswith higher orders of g and its spatial derivatives.In order to investigate the behavior of Robinson–

Trautman–Maxwell system, first we derive two time evo-lution equations for perturbations around stationary so-lution using equations (4.19) and (4.20). If we assumeA = B in (4.20) are constants equal to q0 (following theassumption made in [24, 25]), then the equation wouldsimplifies into

(lnP ),u = −∆m8 q20

. (6.2)

By substituting the above expression in (4.19) we arriveat the second time evolution equation,

m,u =∆K

4 −3

8 q20∆m− P 2

{(�0)2 + (�1)2

}. (6.3)

Based on (6.1), the following linearization holds

K = ∆ lnP ' 1 + 2g + ∆̃g

where ∆̃ ≡ P 20 (∂xx+∂yy). For the subsequent analysis weassume that perturbations can be expanded into spheri-cal harmonics and work with individual modes, therebyobtaining

∆̃ g = −l(l + 1)g .

Similar to 6.1, we assume linear perturbation for m, i.e.,

m = m0 + ψ

and

∆̃ψ = −l(l + 1)ψ .

Neglecting all higher-order terms in g, ψ and their deriva-tives, the equations 6.2 and 6.3 lead to this linearizedsystem

ψ,u =3 l(l + 1)m0

8 q20ψ + l(l + 1) [l(l + 1)− 2]4 g ,

g,u =l(l + 1)

8 q20ψ . (6.4)

Such system provides exponential solution for both g andψ (∼ eγu) determined by eigenvalues

γ = l(l + 1)16 q20

(3m±

√9m2 + 8 q20 [l(l + 1)− 2]

).

(6.5)The solution contains two branches corresponding toeigenvalues of opposite signs for any l ≥ 1, thus the lin-earized equations are unstable. For the negative sign,the limit q0 → 0, goes to the vacuum linearized RT so-lution and this is the stable branch. The stability andwell–posedness for RT class with Maxwell is discussed indetail in [24, 25]. These studies concluded that linearizedRT with Maxwell is unstable and ill-posed. However, [24]observed using Lyapunov functional approach that space-times including past apparent horizon automatically cor-respond to the stable branch.We will perform the same procedure for the only

nonlinear electrodynamics radiative solution, (5.42) and(5.43), we have. First, we substitute (lnP ),u from (5.43)to (5.42) to have two time evolution equations as follow-ing

∆ (K + 2qeA)− 6µ(lnP ),u + 2µ,u = 0 , (6.6)

∆ (K + 2qeA)− 3µ(lnA),u + 2µ,u +3aµ2√A×[

∆AA2

+ ∆(lnA)]

= 0 . (6.7)

For studying linear perturbation , apart from (6.1), weassume function A to be

A = A0(1 + ξ) . (6.8)

A0 is a constant in spherically symmetric limit and

∆̃ ξ = −l(l + 1) ξ .

Applying linear perturbation to equations (6.6) and (6.7)while considering µ = −µ0 as a constant, we obtain thefollowing equations

ξ,u = 2ω {Π ξ − [l(l + 1)− 2] g} , (6.9)

g,u = ω {σ ξ − [l(l + 1)− 2] g} , (6.10)

13

where the constants ω, σ,Π are ω = l(l+1)6µ0 , σ = 2A0 qe

and Π = σ − 3 aµ02

(1+A0A

320

).

In this case the eigenvalues are

γ = l(l + 1)12µ20(2 (1 + Π)− l(l + 1)

±√

(l(l + 1)− 2)2 + 4 (l(l + 1)− 2) (Π− 2σ) + 4 Π2)

(6.11)

and considering that qe is negative while all the otherconstants are positive one can make both eigenvalues neg-ative by adjusting the value of parameter a controllingthe strength of the Lagrangian. Thus the linearized equa-tions are stable for any l ≥ 1. Additionally, for fixed timeand l → ∞ the perturbations vanish as well. This spe-cific RT–NE model is well-posed (since a norm on theinitial data can be constructed that would bound normof the solution at later times from above due to exponen-tial decay ultimately leading to continuous dependenceon initial data).

Surprisingly, there is a big difference in the stabil-ity of solutions between Robinson–Trautman–Maxwelland Robinson–Trautman with specific model of nonlinearelectrodynamics. Although the nonlinear electrodynam-ics model found in the section VC has Maxwell limit butthe corresponding solution in Robinson–Trautman classis well posed while Robinson–Trautman–Maxwell is not.

VII. CONCLUSION AND DISCUSSION

The main purpose of this paper was to derive condi-tions for the existence of NE solutions in the RT class thatwould contain electromagnetic radiation. Since generalRT solutions represent nonlinear deformations of spheri-cally symmetric situations, this would provide a potentialbasis for investigation of nonlinear stability of sphericallysymmetric NE solutions. This would be especially in-teresting for regular black–hole solutions sourced by NE.The presented results show that the Maxwell theory is theunique one providing general solutions including electro-magnetic radiation while NE models suffer from seriousrestrictions— with models featuring regular black holesaffected in particular. The only partial exception to thisis a NE model derived in section VC which provides asolution with an electric charge and radiation which hasa Maxwell limit and involves a singular geometry butregular electromagnetic field.

Our results specifically mean that even the Born–Infeldmodel cannot support magnetic and electric charges ac-companied by radiative terms in the general RT geom-etry. This model also offers an interesting justificationfor considering the L(F,G) Lagrangian instead of justL(F ) when B 6= 0 and E 6= 0 are involved. Namely, theregularity of the electric field is destroyed for L(F ).

Further results for a magnetic charge only (facilitatingan effective transition L(F,G) → L(F )) with or with-out radiative terms explicitly show the impossibility ofgeneralizing regular black–hole solutions to the RT class.This means one cannot hope to perform nonlinear stabil-ity analysis within this class but it also raises the ques-tion whether this means that such solutions are only lim-ited to highly symmetric cases and thus do not representastrophysically relevant situations arising from genericconditions. This represents additional complications forregular black–hole models which already suffer from anon-Maxwellian weak–field limit and, in the strong–fieldlimit, both the value of their Lagrangian and energy mo-mentum tensor attain constant values (this applies to,e.g., Lagrangians corresponding to the Bardeen and Hay-ward models).As noted in Introduction, RT geometry can be used for

analysis of late stage of compact objects merger in cer-tain scenarios. Viewed from this perspective, the clearlack of radiative solutions for RT–NE system in compar-ison with RT–Maxwell one suggests that one should notexpect strong electromagnetic counterparts to gravita-tional radiation in these situations even when the sourcesof electromagnetic field are present if NE plays a role.Since it was previously shown that RT–Maxwell system

is unstable when linearized around stationary solutionwe have applied such analysis to the specific NE modelderived in section VC and concluded that this RT–NEsystem is perturbatively stable and well-posed. Thus itdoes not suffer from limited physical significance as inthe case of RT–Maxwell.Naturally, one should pursue investigation in more gen-

eral geometries to confirm the above results there. Espe-cially spacetimes where radiation is not confined domi-nantly along one direction. This might prove to be essen-tial for NE since general electromagnetic perturbationsdo not propagate along null directions here [8]. Further-more, the issue of well-posedness of general NE withinthe RT geometry can be tackled using results in [26]. Inthe future we plan to clarify how the general RT solu-tion for the specific model of NE settles to the stationaryspacetime (a NE counterpart to Reissner–Nordström so-lution).

Acknowledgments

The author is grateful to Otakar Svítek and DavidHilditch for valuable comments and discussions and alsothanks Martin Žofka for carefully reading the paper. Itis our informal contribution to theoretical investigationsrelated to gravitational waves within the LISA Consor-tium. This work was supported by the research grantGAČR 21-11268S.

14

Appendix A: L(F ) vs. L(F,G) model

Here, we will consider the distinction between the L(F )and the L(F,G) models with respect to Einstein equa-tions component Grx = T rx and its relation to theMaxwell equations. This equation can be cast in thefollowing form

− 12 Q,xr = 2LF{Fur Fux + Fuy

B

r2

}(A1)

with T rx and Grx being given in (3.30) and (3.35). From(4.1), we can find

Q,xr = r (Tuu),x −1r2

∫r2 (Tuu),xdr (A2)

and by substituting the above expression into (A1) weobtain

r (Tuu),x −1r2

∫r2 (Tuu),xdr

= −4LF{Fur Fux + Fuy

B

r2

}. (A3)

To remove the integral we take r-derivative of this equa-tion

r3 (Tuu),xr + 2r2 (Tuu),x = −4 {AFux + Fuy LF B},r .(A4)

Upon further simplification, using (3.27), we have thefollowing

r3 (Tuu),xr + 2r2 (Tuu),x =

LF(r2 F,x −

8r2BB,x

)− 4r2B2 F,x LFF (A5)

and by substituting the above expression into (A4) weget

LF(r2 F,x −

4r2BB,x

)= −4 (AFux),r (A6)

where we used (3.20) and (5.22). With further straight-forward simplification, we obtain a result equivalent toone component of the Maxwell equations (3.9). Thusfor L(F ) it is straightforward to satisfy both the con-sidered Einstein equations component and the Maxwellequations component since one is an integrability condi-tion of the other.

If we repeat the same procedure for the L(F,G) modelwe obtain the following expression

Fux = (A7)(r5

8B

) r2ELF (E,x + (Fux),r) +BLG ((Fux),r − E,x)B r2LGG (2E − rE,r) + LFG (2B2 + r5EE,r)

.

The presence of terms related to G-derivatives of La-grangian makes the above equation completely indepen-dent from the Maxwell equations (unlike in the L(F )case) which results in a serious constraint on finding so-lutions to a coupled system. The final expression clearlysignificantly simplifies for the L(F ) model and only theboxed terms survive leading to previous result.

Appendix B: Energy conditions

In this section, we review the energy conditions for ournew Lagrangian (5.39), the other model of NE consideredin this paper is Born–Infeld which was studied in greatdetail in literature. The essential condition for avoid-ing exotic matter is satisfying the weak energy condition(WEC) for any timelike vector nµ. The WEC is given by

Tµν nµnν ≥ 0. (B1)

For NE energy momentum tensor (3.25) the above rela-tion leads to the following expression

− 12(L−GLG)− 2 (Fνλ nν Fµλ nµ)LF ≥ 0 , (B2)

which in the absence of G (as is the case for solutionconsidered in section VC) leads to the following obser-vation. If both Lagrangian L and its derivative LF areeverywhere negative, WEC holds for arbitrary solution.This is not the case for our situation of Lagrangian (5.39)since by considering constant qe to be negative, then L ispositive (Figure 1) and LF (5.38) is negative. Therefore,one needs to analyze the situation for a given specificsolution. Since the energy momentum tensor is not di-agonal in our coordinate system the investigation is notstraightforward.The time like vector in the most general form should

be (nu, nr, nx, ny) with nµnµ = −1. For simplicity, letus consider the timelike vector with only nu componentbeing nontrivial. For our non-diagonal metric anzats,(3.3), the corresponding covector has form (nu, nr, 0, 0)and clearly nr = −nu. With these assumptions nu canbe expressed as

nu = 1√1− guu

where the denominator is positive and greater than 1 inthe exterior of the horizon. The WEC for such timelikevector, nµ, is

Tµν nµnν = T ru nrnu + Tuu nunu ≥ 0. (B3)

and using the explicit expressions for components of ourtimelike vector (covector) the WEC simplifies into

Tuu − T ru1− guu

− Tuu ≥ 0. (B4)

15

10 8 6 4 2 0F

5

4

3

2

1

0 T

u u

Figure 2: Plot for the energy momentum tensor componentTuu corresponding to the new Lagrangian (5.39) in terms ofF . The constant parameters are set to a = 1, L0 = −1 andqe = −16.

The energy momentum tensor component T ru, (3.29),

for the new Lagrangian (5.39) is

T ru =2 qeP 2

(C̃21 + C̃22

)A(

2 a√Ar + 1

)2and considering negative qe the above expression is al-ways negative. The component Tuu is also negative asconfirmed by its plot for specific values of constants inFigure:2 and this holds whenever qe is negative. There-fore, the WEC energy condition (B4) is satisfied for thisobserver. Moreover, this result holds for arbitrary ob-server in the absence of radiative fields as well.The Null Energy Condition (NEC) is satisfied for any

null vector kµ provided

Tµν kµkν ≥ 0, (B5)

which for our case means

−2 (Fνλ kν Fµλ kµ)LF ≥ 0

and by assuming qe negative we see from (5.38) that LFis always negative. Therefore, NEC is automatically sat-isfied for the new Lagrangian.

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I IntroductionII Vacuum Robinson–Trautman metricIII Robinson–Trautman solution coupled to electromagnetismA Modified Maxwell equationsB Field equations

IV Consistency of the field equationsA Example I: Maxwell theoryB Example II: Born–Infeld theoryC Other models

V Robinson–Trautman solutions for L(F)A Magnetic fieldB Magnetic charge and radiationC Electric charge and Radiation

VI Linear perturbations around stationary solutionVII Conclusion and discussion AcknowledgmentsA L(F) vs. L(F,G) modelB Energy conditions References