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Self-Consistent Solutions for Bulk Gravity-Matter Systems Coupled to Lightlike Branes

We study self-consistent D=4 gravity-matter systems coupled to a new class of Weyl-conformally invariant lightlike branes (WILL}-branes). The latter serve as material and charged source for gravity and electromagnetism. Further, due to the natural coupling to a 3-index antisymmetric tensor gauge field, the WILL-brane dynamically produces a space-varying bulk cosmological constant. We find static spherically-symmetric solutions where the space-time consists of two regions with black-hole-type geometries separated by the WILL-brane which"straddles"their common event horizon and, therefore, provides an explicit dynamical realization of the"membrane paradigm"in black hole physics. Finally, by matching via WILL-brane of internal Schwarzschild-de-Sitter with external Reissner-Nordstrom-de-Sitter (or external Schwarzschild-de-Sitter)geometries we discover the emergence of a potential"well"for infalling test particles in the vicinity of the WILL-brane (the common horizon) with a minimum on the brane itself.


I. INTRODUCTION
Higher-dimensional extended objects gained in recent years a dramatically increasing importance due to various developments in string theory, gravity, astrophysics and cosmology.
In non-perturbative string theory (for a background, see refs. [1]) there arise several types of higher-dimensional membranes (p-branes, Dp-branes) which play a crucial role in the description of string dualities, microscopic physics of black holes, gauge theory/gravity correspondence [2], large-radius compactifications of extra dimensions, cosmological brane-world scenarios [3], model building in high-energy particle phenomenology [4], etc..
Lightlike membranes are of particular interest in general relativity as they describe impulsive lightlike signals arising in various violent astrophysical events, e.g., final explosion in cataclysmic processes such as supernovae and collision of neutron stars [5]. Lightlike membranes are basic ingredients in the so called "membrane paradigm" theory [6] which appears to be a quite effective treatment of the physics of a black hole horizon. Furthermore, the thin-wall description of domain walls coupled to gravity [7,8] is able to provide neat models for many cosmological and astrophysical effects.
In refs. [7,8] lightlike membranes in the context of gravity and cosmology have been extensively studied from a phenomenological point of view,i.e., by introducing them without specifying the Lagrangian dynamics from which they may originate. Recently in a series of papers [10,11] we have developed a new field-theoretic approach for a systematic description of the dynamics of lightlike branes starting from concise Weyl-conformally invariant actions.
The latter are related to, but bear significant qualitative differences from, the standard Nambu-Goto-type p-brane actions [20] (here (p + 1) is the dimension of the brane worldvolume, see Sect.II below). Furthermore, our Weyl-conformally invariant actions are not related to the the p-brane actions previously proposed in the literature [12] where the standard Weyl-conformally noninvariant Nambu-Goto p-brane actions and their supersymmetric counterparts were reformulated in a formally Weyl-invariant form by means of introducing auxiliary non-dynamical fields with a non-trivial transformation properties under Weyl-conformal symmetry appropriately tuned up to compensate for the Weyl non-invariance with respect to the original dynamical degrees of freedom. Namely, one immediately observes that the latter formally Weyl-invariant p-brane actions do not change the dynamical content of the standard Nambu-Goto p-branes (describing inherently massive modes). This is in sharp contrast to the presently discussed Weyl-conformally invariant p-brane models, which describe intrinsically lightlike p-branes for any even p (i.e., for any odd-dimensional world-volume). In what follows we will use for the latter the acronym WILL-branes (Weyl-invariant lightlike branes).
Our approach is based on the general idea of employing alternative non-Riemannian integration measures (volume-forms) in the actions of generally-covariant (reparametrizationinvariant) field theories instead of (or, more generally, on equal footing with) the standard Riemannian volume forms. Namely, instead of (or alongside with) the standard Riemannian integration measure density √ −g with g = det g µν being the determinant of the corresponding Riemannian metric, one can employ the equally well suited non-Riemannian integration measure density: where ϕ i (i = 1, . . . , D) denote auxiliary scalar fields. This idea has been first proposed and applied in the context of four-dimensional theories involving gravity [13] by introducing a new class of "two-measure" gravitational models. It has been demonstrated that the latter models are capable to provide plausible solutions for a broad array of basic problems in cosmology and particle physics, such as: (i) scale invariance and its dynamical breakdown; (ii) spontaneous generation of dimensionful fundamental scales; (iii) the cosmological constant problem; (iv) the problem of fermionic families; (v) applications to dark energy problem and modern cosmological brane-world scenarios. For a detailed discussion we refer to the series of papers [13,14].
Subsequently, the idea of employing an alternative non-Riemannian integration measure was applied systematically to string, p-brane and Dp-brane models [15]. The main feature of these new classes of modified string/brane theories is the appearance of the pertinent string/brane tension as an additional dynamical degree of freedom beyond the usual string/brane physical degrees of freedom, instead of being introduced ad hoc as a dimensionful scale. The dynamical string/brane tension acquires the physical meaning of a world-sheet electric field strength (in the string case) or world-volume (p + 1)-form field strength (in the p-brane case) and obeys Maxwell (Yang-Mills) equations of motion or their higher-rank antisymmetric tensor gauge field analogues, respectively. As a result of the latter property the modified-measure string model with dynamical tension yields a simple classical mechanism of "color" charge confinement [15].
The above mentioned modified-measure p-brane and Dp-brane models [15] share the same drawback as ordinary Nambu-Goto p-branes, namely that Weyl-conformal invariance is lost beyond the simplest string case (p = 1). On the other hand, the form of the action of the modified-measure string model with dynamical tension suggested a natural way to construct explicitly the new class of Weyl-conformally invariant p-brane models for any p [10,11].
The present paper has a two-fold objective. First, in Section II we briefly review (and extend) the construction of WILL-branes, including a Kalb-Ramond-type coupling of the latter to a space-time (p + 1)-rank antisymmetric tensor gauge field. In the second main part we derive systematically spherically-symmetric solutions for the coupled system of bulk Einstein-Maxwell interacting with a WILL-brane (Sections III,IV), the latter serving as a matter and charged source and, in addition, producing a space-varying dynamical cosmological constant. Finally, in Section V we describe a physically interesting effect of creation of a potential "well" around the WILL-brane, which devides space-time as a common horizon into two separate regions with different black hole geometries and whose matching is explicitly given in terms of the free WILL-brane parameters (electric charge and Kalb-Rammond coupling constant).
The above action is invariant under Weyl (conformal) symmetry for any p: with Jacobian det ∂ϕ ′ i ∂ϕ j = ρ.
Rewriting the action (2) in the following equivalent form: we see that the composite field χ plays the role of a dynamical (variable) brane tension. Let us note the following differences of (2) (or (5)) w.r.t. the standard Nambu-Goto p-branes (in the Polyakov-like formulation) : • New non-Riemannian integration measure density Φ(ϕ) instead of the usual √ −γ, and • Auxiliary world-sheet gauge field A a in a "square-root" Maxwell term [21]. As discussed in our previous papers on the subject [10,11], the appearance of this "squareroot" Maxwell term is naturally required for consistency of the WILL-brane dynamics.
• Possibility for natural couplings of auxiliary A a to external world-volume ("color" charge) currents J a .
• Weyl-invariant for any p; describes intrinsically light-like p-branes for any even p (i.e., odd-dimensional world-volume). Let us also note that there are NO quantum conformal anomalies in odd xdimensions!.
Employing the short-hand notations: the equations of motion w.r.t. measure-building auxiliary scalars ϕ i and γ ab read, respectively: Taking the trace in (8) implies M = 0 in Eq. (7).
Next, we get the equations of motion w.r.t. auxiliary gauge field A a : 9) and the equations of motion w.r.t. X µ : is the affine connection corresponding to the external space-time metric G µν .
Consider the γ ab -equations of motion (8); in fact, the latter are constraints analogous to the (classical) Virasoro constraints in ordinary string theory. Since F ab is anti-symmetric (p + 1) × (p + 1) matrix, then F ab is not invertible in any odd (p + 1) -it has at least one zero-eigenvalue vector-field V a (F ab V b = 0). Therefore, for any odd (p + 1) the induced metric (∂ a X∂ b X) on the world-volume of the Weyl-invariant brane is singular (as opposed to the ordinary Nambu-Goto brane (!)) : where ∂ V ≡ V a ∂ a and ∂ ⊥ are derivates along the tangent vectors in the complement of V a .
Thus, we arrive at the following important conclusion: every point on the world-surface of the Weyl-invariant p-brane (2) (for odd (p + 1)) moves with the speed of light in a timeevolution along the zero-eigenvalue vector-field V a of F ab . Therefore, we will name (2) (for odd (p + 1)) by the acronym WILL-brane (Weyl-Invariant Lightlike-brane) model.

Remark.
In what follows we will use a natural ansatz for the world-volume electric field Henceforth we will explicitly consider the special case p = 2 of (2), i.e., the Weyl-invariant lightlike membrane model: Invariance under world-volume reparametrizations allows to introduce the following standard (synchronous) gauge-fixing conditions: The residual τ ≡ σ 0 -independent reparametrization invariance allows for further conformally-flat gauge-fixing of the space-like part of γ ab : with γ ij a standard reference 2D metric on the membrane surface.

C. Coupling to Maxwell and Rank-3 Antisymmetric Tensor Gauge Field
We can extend straightforwardly the WILL-brane model via couplings to external spacetime electromagnetic field A µ and, furthermore, to external space-time rank 3 gauge potential A µνλ (Kalb-Ramond-type coipling) keeping manifest Weyl-invariance: The second Chern-Simmons-like term in (21) is a special case of a class of Chern-Simmonslike couplings of extended objects to external electromagnetic fields proposed in ref. [17].
Let us recall the physical significance of A µνλ [18]. In D = 4 when adding kinetic term for A µνλ coupled to gravity (see Eq.(40) below), its field-strength: with a single independent component F produces dynamical (positive) cosmological constant: The constraints (17)- (18) (gauged-fixed equations of motion w.r.t. γ ab remain unaltered for the action (21). Using the same gauge choice (γ 0i = 0, γ 00 = −1) and ansatz for the world-volume gauge field-strength (F 0i (A) = 0), the equations of motion w.r.t. A a now acquire the form: Eqs.(24) tell us that consistency of charged WILL-brane dynamics implies that the external space-time Maxwell field must have zero magnetic component normal to the brane, as well as that the projection of the external electric field along the brane must be proportional to the gradient of the brane tension. Finally, the X µ equations of motion for (21) read: where F ρκλν is given as in (22) and: where γ (2) = det γ ij (recall Eqs. (14)- (15)).
As a consequence of of the constraints (17)- (18) and the equations of motion for X µ (25) we also obtain:
We will use the following ansatz: Substituting (34)-(35) into the WILL-brane equations of motion one gets: • Equations for r(τ, σ 1 , σ 2 ) from the lightlike (17) and Virasoro-type (18) constraints: • A strong restriction on the gravitational background itself coming from (27): Eq.(37) tells us that the (squared) sphere radius R 2 ≡ C(r, t) must remain constant along the WILL-brane trajectory. For static backgrounds R 2 ≡ C(r) Eqs. (37), (36) imply: Eq.(38) is of primary importance as it shows that the WILL-brane automatically positions itself on the event horizon.
• WILL-brane equations of motion (25) for X 0 ≡ t and X 1 ≡ r turn out to be proportional to each other and reduce to an equation for the conformal factor a(τ ) of the internal membrane metric (35) : where F is the independent component of the rank 4 field-strength (22) (here again one sets at the end t = τ, r = r(τ )).

A. Action and Equations of Motion
Let us now consider the following coupled Einstein-Maxwell-WILL-brane system adding also a coupling to a rank 3 gauge potential: Here (22), and the WILL-brane action is the same as in (21): The equations of motion for the WILL-membrane subsystem are the same as (17)- (18) and (24)-(25), whereas the equations for the space-time fields read: where in the last equation we have used relation (22). The explicit form of the energymomentum tensors read: √ −γ -the brane tension, cf. (5)). The charge current in (43) produced by the WILL-brane is given by:

B. Static Spherically Symmetric Solutions. Matching Accross Horizon
We find the following static spherically symmetric solutions for the bulk gravity-matter system coupled to a charged WILL-brane (40). The bulk space-time consists of two regions separated by the WILL-brane as a common horizon materialized by the WILL-brane: where the subscript (−) refers to the region inside, whereas the subscript (+) refers to the region outside the horizon at r = r 0 ≡ r horizon with A (∓) (r 0 ) = 0. We have Schwarzschildde-Sitter space-time inside horizon: and Reissner-Norström-de-Sitter space-time outside horizon: with Reissner-Norström (squared) charge given by Q 2 = 8πq 2 r 4 0 . The dynamically induced (due to the presence of the rank 3 tensor gauge potential) cosmological constant is different inside and outside the horizon: where F (±) are the corresponding constant values of the rank 4 tensor field-strength (22) according to Eq.(44).
As already discussed in Section III (cf. Eqs.(36)-(38)), the WILL-membrane locates itself automatically on ("straddles") the common event horizon at r = r 0 : Besides inducing the jump (52) in the space-varying cosmological constant, the WILLbrane also causes the following discontinuity in the normal derivative of the metric component A(r) resulting from the WILL-brane contribution (47) Eq.(54) is easily obtained upon using the simple expressions for the components of the Ricci tensor corresponding to the metric (28) R 0 0 = R 1 1 = − 1 2r 2 ∂ ∂r r 2 ∂ ∂r A(r) [19] and upon substituting (53) in (47).
For the Maxwell subsystem (43),(48) we have Coulomb field outside horizon generated by the surface charge of the WILL-brane: and no electric field inside horizon: Apart from the matching conditions for the metric components (54) and the induced cosmological constant (52) when crossing the WILL-brane hypersurface, WILL-brane dynamics imposes an important third matching condition. Namely, the remaining non-trivial WILL-brane Eq.(39) reduces now to the following two equations because of the two different space-time geometries inside and outside the horizon: Eqs.(55),(56) should be yield the same solution for the conformal factor a(τ ) of the internal brane metric, i.e., we have the following additional matching condition: The matching conditions (54) and (57) allow to express all physical parameters of our solution in terms of 3 free parameters (q, β, F ) where: (a) q -WILL-brane surface electric charge density; (b) β -WILL-brane (Kalb-Rammond-type) charge w.r.t. rank 3 space-time gauge potential A λµν ; (c) F (−) -vacuum expectation value of F κλµν (22) in the interior region.
The corresponding explicit expressions read: • Horizon radius: • Schwarzschild mass: • Reissner-Nordström mass: • Reissner-Nordström charge: • Space-varying dynamically induced cosmological constant: • Brane tension: To determine the type of the common horizon materialized by the WILL-brane, we need the expressions for the slopes of the metric coefficients A (±) (r) at r = r 0 . Using expressions (58)-(63) we find: Therefore, in view of Eqs.(64)-(65) (henceforth for definiteness we assume β > 0 [22]) : Remark. In the area of parameter space (i), although the brane tension χ is negative, the brane can be stable due to negative pressure caused by the difference between the "inside" and "outside" dynamically induced cosmological constants F (−) > F (+) (cf. Eq.(62)).
for r < r 0 and r > r 0 , respectively, where: In this special case the brane tension becomes χ = − 1 √ 2πG N 3 4 β. Let us also mention the simple special case β = F (−) = 0, i.e., matching interior purely Schwarzschild black hole region with exterior purely Reissner-Nordström black hole region. This case has already been discussed in [11].
The results in the present Section show that the Einstein-Maxwell-WILL-brane system (40) can be viewed as the first explicit dynamical realization of the "membrane paradigm" in black hole physics [6]. Indeed, the brane dynamics determined by the action (41) dictates that the brane must be inherently lightlike and that it necessarily has to locate itself on a black hole event horizon. Moreover, its properties dynamically determine the nature of the surrounding bulk gravity and matter.

V. TRAPPING POTENTIAL WELL AROUND COMMON HORIZON
Consider planar motion of a (charged) test patricle with mass m and electric charge q 0 in a gravitational background given by the solutions in Section IV, namely, internal Schwarzschild-de-Sitter region matched with external Reissner-Norström-de-Sitter region along a common event horizon materialized by Weyl-conformally invariant lightlike brane, which simultaneously serves as material and charge source for gravity and electromagnetism.
Conservation of energy yields (E, J -energy and orbital momentum of the test particle; prime indicates proper-time derivative): where A (∓) are the same as in (50) and (51).
Taking into account (64)-(65) we see that in the parameter interval: the (squared) effective potential V 2 ef f (r) acquires a potential "well" in the vicinity of the WILL-brane (the common horizon) with a minimum on the brane itself.
Let us illustrate graphically the simplest physically interesting case with q = 0, F (−) = β and β -arbitrary, i.e., matching of Schwarzschild-de-Sitter interior (with dynamically  (67)). The typical form of V 2 ef f (r) is shown in Fig.2. Thus, we conclude that if a test particle moving towards the common event horizon loses energy (e.g., by radiation), it may fall and be trapped by the potential well, so that it neither falls into the black hole nor can escape back to infinity. In this way one could form a "cloud" of trapped particles around the horizon.

VI. CONCLUSIONS AND OUTLOOK
The present paper as well as the previous ones [10,11] show that modifying of worldsheet (world-volume) integration measure (cf. Eq. (3)) significantly affects string and p-brane dynamics.
• Acceptable dynamics in the new class of string/brane models (2) naturally requires the introduction of auxiliary world-sheet/world-volume gauge fields.
• By employing square-root Yang-Mills actions for the auxiliary world-sheet/worldvolume gauge fields one achieves Weyl conformal symmetry in the new class of p-brane theories for any p.
• Here the string/brane tension is not a constant scale given ad hoc, but rather an additional dynamical degree of freedom beyond the ordinary string/brane degrees of freedom.
• When put in a gravitational black hole background, the WILL-brane automatically positions itself on ("materializes") the event horizon.
• Coupled Einstein-Maxwell-WILL-membrane system possesses a self-consistent solution where the WILL-membrane serves as a source for gravity and electromagnetism.
Moreover, it generates dynamical cosmological constant and automatically "straddles" the common event horizon for a Schwarzschild-de-Sitter space-time region (in the interior) and Reissner-Nordström-de-Sitter space-time region (in the exterior). This model is the first explicit dynamical realization of the "membrane paradigm" in black hole physics.
• It is very reasonable that negative surface tension on the horizon, produced by the WILL-brane "sitting" on it, is the cause of a stable configuration where the positive cosmological constant inside the horizon is larger than the one outside the horizon (once again due to the presence of the WILL-brane). The space-varying cosmological constant causes a pressure difference that gives rise to an inward force on the horizon and this is compensated by the negative surface tension due to the WILL-brane, which pushes the horizon surface outwards.
• The non-trivial matching of different black hole regions along the common horizon "straddled" by the WILL-brane produces a trapping potential "well" for infalling test particles in the vicinity of the common horizon "guarding" the inner Schwarzschild-like horizon of the interior Schwarzschild-de-Sitter region.