5-dimensional braneworld with gravitating Nambu-Goto matching conditions

# 5-dimensional braneworld with gravitating Nambu-Goto matching conditions

Georgios Kofinas Research Group of Geometry, Dynamical Systems and Cosmology
Department of Information and Communication Systems Engineering
University of the Aegean, Karlovassi 83200, Samos, Greece
Vasilios Zarikas Department of Electrical Engineering, ATEI Lamias, 35100 Lamia, Greece
July 14, 2019
###### Abstract

We continue the investigation of a recent proposal on alternative matching conditions for self-gravitating defects which generalize the standard matching conditions. The reasoning for this study is the need for consistency of the various codimension defects and the existence of a meaningful equation of motion at the probe limit, things that seem to lack from the standard approach. These matching conditions arise by varying the brane-bulk action with respect to the brane embedding fields (and not with respect to the bulk metric at the brane position) in a way that takes into account the gravitational back-reaction of the brane to the bulk. They always possess a Nambu-Goto probe limit and any codimension defect is seemingly consistent for any second order bulk gravity theory. Here, we consider in detail the case of a codimension-1 brane in five-dimensional Einstein gravity, derive the generic alternative junction conditions and find the symmetric braneworld cosmology, as well as its bulk extension. Compared to the standard braneworld cosmology, the new one has an extra integration constant which accounts for the today matter and dark energy contents, therefore, there is more freedom for accommodating the observed cosmic features. One branch of the solution possesses the asymptotic linearized LFRW regime. We have constrained the parameters so that to have a recent passage from a long deceleration era to a small today acceleration epoch and we have computed the age of the universe, consistent with current data, and the time-varying dark energy equation of state. For a range of the parameters it is possible for the presented cosmology to provide a large acceleration in the high energy regime.

## I Introduction

Distributional (thin) branes in an appropriate dimensional spacetime model the dynamics of various physical systems, as it is the universe itself. A classical infinitely thin test brane with tension (probe) moving in a given background spacetime is governed at lowest order by the Nambu-Goto action CGC (). Variation of this action with respect to the brane embedding fields gives the Nambu-Goto equations of motion which are geometrically described by the vanishing of the trace of the extrinsic curvature, and therefore, the worldsheet swept by the brane is extremal (minimal). When the gravitational field of the defect is taken into account both the bulk metric and the brane position become dynamical. Here, we consider throughout that the bulk metric is regular (finite and continuous) at the brane position. The standard method for obtaining the equations of motion of a back-reacting brane is to consider the bulk field equations with all the localized energy-momentum tensor included and to isolate and integrate out the distributional terms. A discontinuous extrinsic curvature or a conical singularity can source such delta functions of suitable codimension. While for thin shells Israel matching conditions are well-established Israel 1966 (), when the support of a generic distributional stress-energy tensor is higher-codimensional, it does not make sense to consider solutions of Einstein’s equations Israel 1977 (), Geroch (), Garfinkle (), cline () (a pure brane tension is a special situation which is consistent Vilenkin (), Frolov ()).

In Ruth 2004 (), geometric junction conditions for a codimension-2 conical defect in six-dimensional Einstein-Gauss-Bonnet theory were derived with the hope that the above inconsistency is not due to the defect construction, but due to the inability of Einstein gravity to describe complicated distributional solutions. In CKP () the consistency of the whole set of junction plus bulk field equations was explicitly shown for an axially symmetric codimension-2 cosmological brane in six-dimensional EGB gravity, and it is likely that the consistency will remain for non-axial symmetry. Analogously, e.g. a 5-brane in eight dimensions is again of codimension-2 and EGB theory would suffice, but for a 4-brane in eight dimensions (codimension-3) the third Lovelock density Lovelock () would need for consistency. However, e.g. a 2-brane in six dimensions is of codimension-3 and it is probably inconsistent since the spirit of the proposal is to include higher Lovelock densities to accommodate higher codimension defects and there is no higher than the second Lovelock density in six dimensions. In brief, the generalization of the proposal is that in a -dimensional spacetime the maximal Lovelock density should be included (possibly along with lower Lovelock densities) and the branes with codimensions should be consistent according to the standard treatment; for yet higher codimensions the situation is not clear and probably inconsistent. In four dimensions the absence of higher Lovelock densities does not allow the existence of generic codimension-2 or 3 defects, but even if four dimensions are not the actual spacetime dimensionality, at certain length and energy scales it has been tested that four-dimensional Einstein gravity represents effectively the spacetime to high accuracy, so a consistent four-dimensional framework would at least be desirable. Beyond the above (possible) shortcomings, there are extra difficulties with handling distributional sources inside an equation. For example, for a codimension-2 brane there are two kinds of distributions involved, and , where is the radial coordinate from the brane. If both distributions are used to derive two matching conditions Soda (), then an unnatural and undesirable inconsistency for certain boundary conditions arises CKP (). Moreover, there is the problem of the regularization of the distributional equation, since multiplying by only one distributional term remains, while multiplying by all distributions vanish. If however one considers the corresponding variational problem of brane-bulk action (variation with respect to the bulk metric at the brane position), the volume element of integration vanishes the distribution and only one matching condition arises, consistent with bulk dynamics.

Having stated the question of consistency of “high” codimension defects in either or 4-dimensional spacetime, we now pass to the question of the probe limit. We note that the standard equations of motion of a self-gravitating defect do not obey the natural condition of continuous deformation from the probe limit equation of motion (which is the Nambu-Goto equation). Indeed, the Israel matching conditions under vanishing of the brane energy-momentum tensor give vanishing extrinsic curvature (geodesic motion), and similarly the probe limit of a codimension-1 brane in EGB gravity Germani () is another equation of motion, while the codimension-2 matching condition in EGB theory Ruth 2004 (), CKP (), Charmousis () is a third equation of motion. To set an analogy, the linearized equation of motion of a point particle in four dimensions MiSaTa (), QuWa () (which is not involved in our discussion since in this case the bulk metric diverges on the brane) is a correction of the geodesic equation of motion on a given background (of course, for a 0-brane the geodesic equation coincides with the Nambu-Goto) and for a two-body system the probe limit is realized when one mass is much smaller than the other. However, in an analogous case it has been shown GerochJang (), Ehlers () that a probe point mass moves on the geodesic of a background-solution of any gravitational theory, so the above variety of probe equations of motion for different gravitational theories (or also the dependence of the equation of motion on the codimension of the defect) maybe is not acceptable. Additionally, the matching conditions in EGB or Lovelock gravity Germani (), Ruth 2004 (), Charmousis () are cubic or quadratic algebraic equations in the extrinsic curvature with the total brane energy-momentum tensor on their right-hand side. Switching off this brane content the probe limit arises which is a cubic or quadratic equation, therefore, in general, it possesses a multiplicity of probe solutions. This means that these theories do not predict according to the standard approach a unique equation of motion at the probe level 111this argument was mentioned to us by J. Zanelli. In this spirit, a correct probe limit equation of motion should be linear in the extrinsic curvature and such are the geodesic or the Nambu-Goto equations.

We would like to finish the discussion on the standard approach mentioning another possible deficiency. The variation of a bulk action with respect to the bulk metric, beyond the main bulk terms gives as usual additional “garbage” dimensional terms. With the exception of codimension-1 case where the inclusion of the Gibbons-Hawking term on the hypersurface cancels these terms, for all higher codimension defects such terms cannot cancel whatever terms are added on the defect. The only possible thing one could imagine is to consider a “tube” around the defect (two planes for codimension-1, a tube for codimension-2, a sphere for codimension-3, etc.), convert the unpleasant terms into “tube” terms, make the cancelation on the “tube”, and take the shrink limit. The variation of the brane-bulk action in the interior of the “tube”, considering also the relevant distributional terms and integrating out around the defect, will give the brane equation of motion. Therefore, the metric variation outside the “tube” has to be independent in order to get the bulk equations of motion, and also it has to be independent on the defect to get the brane equation of motion. For the “tube” terms to cancel, some condition on the metric variation has to be assumed on the “tube” (either Dirichlet-like if generalized Gibbons-Hawking terms Germani (), robin () are included on the “tube”, or Newmann-like). It seems that in the shrink limit these “tube” conditions will be inconsistent with the independence of the brane metric variation.

A criticism against the standard approach in the lines of the above discussion was performed in kof-ira (), together with a proposal for obtaining alternative matching conditions called “gravitating Nambu-Goto matching conditions”. These arise by varying the brane-bulk action with respect to the brane position variables (embedding fields). Although the brane energy-momentum tensor is still defined by the variation of the brane action with respect to the induced metric, however, this tensor enters the new matching conditions in a different way than before. Here, the distributional terms are still present, not inside a distributional differential equation leading directly to inconsistencies at certain cases, but rather smoothed out inside an integration. In kof-ira (), it was shown in particular the consistency of the codimension-2 defect in EGB gravity according to these alternative junction conditions, while the consistency of the codimension-2 limit of Einstein gravity KofTom () was also discussed. Gravitating matching conditions aim to satisfy all the previous shortcomings of the standard conditions. Four-dimensional Einstein gravity seems to be consistent for any codimension brane and the same seems also true either for Einstein or any Lovelock extension for all higher spacetime dimensions (since the inclusion of the maximal Lovelock density now is not crucial). These alternative matching conditions always have the Nambu-Goto probe limit, independently of the gravitational theory considered, the dimensionality of spacetime or the codimensionality of the defect. Finally, since the proposed equation of motion for the defect is decoupled from the bulk metric variation, the “outside” problem (outside the “tube”) is well-posed with a boundary Dirichlet or Newmann type of variation on the “tube” (the central line is defined by an independent variation with respect to the embedding fields). The proposed matching conditions generalize the standard matching conditions, and so, all the solutions of the bulk equations of motion plus the conventional matching conditions are still solutions of the current system of equations.

In the present work we study the codimension-1 case in Einstein gravity. Our approach is reminiscent of the “Dirac style” variation performed in Davidson (), however our resulting matching conditions in the general case do not coincide with the matching conditions derived in Davidson (). We have applied various methods in order to confirm the derived result. Our main conceptual point, in view of the above arguments of consistency of the various codimension defects and their probe limit, is that the gravitating Nambu-Goto matching conditions may be close to the correct direction for deriving realistic matching conditions. We have applied these matching conditions for a cosmological brane without imposing any restriction about the bulk. The cosmology derived is different than the cosmology derived in Davidson () (where the bulk was assumed to be AdS) and the bulk space found here is not AdS. The set-up of the paper is as follows: In section II the method is introduced as an extension of the Nambu-Goto variation for any codimension, so that the contribution from the gravitational back-reaction is included. In section III the generic alternative junction conditions of a codimension-1 brane in five-dimensional Einstein gravity are derived and manipulated together with the remaining effective equations on the brane. Analogous equations hold for other codimension-1 branes in other spacetime dimensions, but we choose the 3-brane as it can represent our world in the braneworld scenario. In section IV we specialize to the cosmological configuration, integrate the brane system of equations and find the brane cosmology. This cosmology has richer structure compared to the cosmology derived according to the standard conditions. In section V we find the bulk extension of the brane cosmology. In section VI we investigate the cosmological equations which provide interesting and realistic cosmological evolutions and study their phenomenological implications. Finally, in section VII we conclude.

## Ii A brief introduction of the method

In order to get an idea how the proposed variation with respect to the embedding fields of the brane position is performed we give in this section a brief account of the method for any codimension (for more details see kof-ira ()). However, the exact derivation of section III in 5-dimensional spacetime is independent of this section. We start with a general four-dimensional action of the form

 s4=∫Σd4χ√|h|L(hij) (1)

in a -dimensional spacetime, where is any scalar on built up from the induced metric . The brane coordinates are ( are coordinate indices on the brane) and the bulk coordinates are ( are -dimensional indices). In the present paragraph the bulk metric is fixed and non-dynamical, while the treatment of a back-reacted metric will be given in the next paragraph of this section. The embedding fields are the external (bulk) coordinates of the brane, so they are some functions . Let the brane is deformed to another position described by the displacement vector and the corresponding variation of the various quantities is denoted by . The variation of the tangent vectors on the brane is . Since the bulk coordinates do not change, the variation of is

 ¯δx\textslgμν=\textslgμν,λ¯δxλ (2)

and the variation of is

 ¯δxhij=\textslgμν,λxμ,ixν,j¯δxλ+\textslgμνxμ,i¯δxν,j+\textslgμνxν,j¯δxμ,i=xμ,ixν,j(\textslgμν,λ¯δxλ+\textslgμλ¯δxλ,ν+\textslgνλ¯δxλ,μ). (3)

The variation of becomes

 ¯δxs4=∫Σd4χ√|h|τij¯δxhij, (4)

where . Substituting , integrating by parts and imposing , we get

 ¯δxs4 = −2∫Σd4χ√|h|\textslgμσ[(τijxμ,i)|j+τijΓμνλxν,ixλ,j]¯δxσ (5) = −2∫Σd4χ√|h|\textslgμσ(τij|jxμ,i−τijKαijnμα)¯δxσ,

since , , where are the extrinsic curvatures on the brane and () form a basis of normal vectors to the brane. The covariant differentiations and correspond to and respectively, while are the Christoffel symbols of . Due to the arbitrariness of it arises

 τij|jxμ,i−τijKαijnμα=0 (6)

and since the vectors , are independent, two sets of equations arise

 τij|j=0,τijKαij=0⇔τij(xμ;ij+Γμνλxν,ixλ,j)=0. (7)

Note that the previous equivalence of the two expressions, one with free index and the other with free index is due to that the vectors are normal to the brane. The variation described so far is the same with the one leading to the Nambu-Goto equation of motion. Indeed for , it is and the first equation is empty, while the second becomes which is the Nambu-Goto equation of motion. Note again that the previous equivalence of the two expressions for the Nambu-Goto equation, one with free index and the other with free index is due to that the vector is normal to the brane. Similarly, the Regge-Teitelboim equation of motion Teitelboim () is a generalization where collects the four-dimensional terms of (14), i.e. . It is , so the first equation becomes the standard conservation and the second .

In order to express the back-reaction of the brane onto the bulk and vice-versa, we consider a general higher-dimensional action of the form

 sD=∫MdDx√|\textslg|L(\textslgμν), (8)

where is any scalar on built up from the metric , e.g. . Under an arbitrary variation of the bulk metric the variation of is , where , and the stationarity under arbitrary variations gives the bulk field equations . The boundary terms arising from this variation in the presence of a defect disappear by a suitable choice for the boundary condition of , usually by choosing a Dirichlet boundary condition for . However, in the presence of the defect, inside , beyond the regular terms which obey , in general there are also non-vanishing distributional terms making the variation not identically zero. The bulk action knows about the defect through these distributional terms. Since , where is the -dimensional delta function with support on the defect, it is , so only the variation of the bulk metric at the brane position contributes to , as expected. More precisely, these distributional terms always appear in the parallel to the brane components and if , the variation gets the form

 δsD=∫MdDx√|\textslg|kijδ(δ)δhij=∫Σd4χ√|h|kijδhij. (9)

Therefore, there is an extra variation of the bulk metric at the brane position (which in the adapted frame coincides with the variation of the induced metric ) which is independent of the bulk metric variation and this extra variation determines the brane equation of motion. The corresponding variation of the total action at the brane position is

 δ(sD+s4)|brane=∫Σd4χ√|h|(kij+τij)δhij. (10)

In particular, if all the components of the variation are independent from each other, the stationarity of the total action at the brane position gives the standard matching conditions (or standard brane equations of motion) , where , are parallel to the brane tensors. Under a variation of the embedding fields, the variations of , were given in the previous paragraph and the corresponding variation of the total action at the brane position will be

 ¯δx(sD+s4)|brane=∫Σd4χ√|h|(kij+τij)¯δxhij. (11)

Following the same steps as in the previous paragraph with replaced by , the stationarity gives, due to the arbitrariness of , the brane equations of motion

 (kij+τij)|j=0,(kij+τij)Kαij=0⇔(kij+τij)(xμ;ij+Γμνλxν,ixλ,j)=0. (12)

These can be called “gravitating Nambu-Goto matching conditions” since they collect also the contribution from bulk gravity and they form a schematic summary of our proposal. Nothing ab initio assures their consistency with the bulk field equations. However, for the non-trivial case of a codimension-2 defect in six-dimensional EGB gravity the consistency has been shown in kof-ira (). In order to describe the previous variation of the brane position there is also the equivalent passive viewpoint of a bulk coordinate change . Now the brane does not change position but it is described by different coordinates and the change of the embedding fields is . Of course, only the value of the variation on the brane, and not the values away from the brane, is expected to influence the corresponding variation of the brane-bulk action at the brane position. Although a bulk action is invariant under coordinate transformations, the presence of the defect, i.e. of the distributional terms inside , make . Let the tensor fields transform according to their functional variation which is the change in their functional form,

 δxϕμν=ϕ′μν(xρ)−ϕμν(xρ)=ϕλν(xρ)δxμ,λ−ϕμλ(xρ)δxλ,ν−ϕμν,λδxλ=−\poundsδxϕμν, (13)

i.e. transform according to the Lie derivative with generator the infinitesimal coordinate change. Therefore, , while , , thus . The stationarity under arbitrary variations of the total brane-bulk action gives again the same matching conditions as before.

## Iii Five-dimensional setup and alternative matching conditions

Let us consider the general system of five-dimensional Einstein gravity coupled to a localized 3-brane source. The domain wall splits the spacetime into two parts and the two sides of are denoted by . The unit normal vector points inwards . The total brane-bulk action is

 S=12κ25∫Md5x√|\textslg|(R−2Λ5)+∫Σd4χ√|h|(rc2κ25R−λ)−1κ25∫Σ±d4χ√|h|K+∫Md5xLmat+∫Σd4χLmat, (14)

where is the (continuous) bulk metric tensor and is the induced metric on the brane ( are five-dimensional coordinate indices). The bulk coordinates are and the brane coordinates are ( are coordinate indices on the brane). The symbol in an integral means the contribution from both sides of the surface. The calligraphic quantities refer to the bulk metric, while the regular ones to the brane metric. The brane tension is (denoted also by in the next sections concerning cosmology) and the induced-gravity term deffayet (), if present, has a crossover length scale , where . , are the matter Lagrangian densities of the bulk and of the brane respectively. The contribution on each side of the wall of the Gibbons-Hawking term will also be necessary here as in the standard treatment. is the trace of the extrinsic curvature (the covariant differentiation corresponds to ).

Varying (14) with respect to the bulk metric we get the bulk equations of motion

 Gμν=κ25Tμν−Λ5\textslgμν, (15)

where is the bulk Einstein tensor and is a regular bulk energy-momentum tensor. We are mainly interested in a bulk with a pure cosmological constant , but for the present we leave a non-vanishing . More precisely, we define the variation of the bulk metric to vanish on the defect. In this variation, beyond the basic terms proportional to which give (15), there appear, as usually, extra terms proportional to the second covariant derivatives which lead to a surface integral on the brane with terms proportional to . Adding the Gibbons-Hawking term, the normal derivatives of , i.e. terms of the form , are canceled. The remaining boundary terms are either terms proportional to or terms containing . These last terms lead (up to an irrelevant integration on ) again to terms proportional to , and more precisely, all the boundary terms together are basically of the known form . Finally, considering as boundary condition for the variation of the bulk metric its vanishing on the brane (Dirichlet boundary condition for ), there is nothing left beyond the terms in equation (15). The Gibbons-Hawking term will again contribute in a while in another independent variation performed in order to obtain the brane equations of motion.

According to the standard method, the interaction of the brane with the bulk comes from the variation at the brane position of the action (14), which is equivalent to adding on the right-hand side of equation (15) the term , where . is the brane energy-momentum tensor, the brane Einstein tensor and the one-dimensional delta function with support on the defect. This approach leads to the (generalized due to ) Israel matching conditions, it has been analyzed in numerous papers and discussed in the Introduction.

Here, we discuss an alternative approach where the interaction of the brane with bulk gravity is obtained by varying the total action (14) with respect to , the embedding fields of the brane position kof-ira (), Davidson (). The embedding fields are some functions and their variations are . While in the standard method the variation of the bulk metric at the brane position remains arbitrary, here the corresponding variation is induced by as explained in section II, it is given by

 δ\textslgμν=δx\textslgμν=\textslg′μν(xρ)−\textslgμν(xρ)=−(\textslgμν,λδxλ+\textslgμλδxλ,ν+\textslgνλδxλ,μ)=−\poundsδx\textslgμν, (16)

and is obviously independent from the variation leading to (15). The induced metric enters the localized terms of the action (14) and depends explicitly and implicitly (through ) on the embedding fields. Also the bulk terms of (14) contribute implicitly to the brane variation under the variation of the embedding fields. The result of variation gives, as we will see, as coefficient of a combination of vectors parallel and normal to the brane, therefore, two sets of equations will finally arise as matching conditions instead of one. Instead of directly expressing , in terms of , it is convenient to include the constraints in the action and vary independently (however, we will also perform the direct calculation). So, the first constraint implies the independent variation of . The variation affects the variation of the parallel to the brane vectors which in turn influences the variation of the normal vector . So, the additional constraints , have to be added, and is another independent variation. Finally, the third variation depends on by (16). Therefore, are independent from , , but the various components are not all independent from each other, so in the end they have to be expressed in terms of which are independent. If , , are the Lagrange multipliers corresponding to the above constraints, the constraint action added to is

 Sc=∫Σ±d4χ√|h|[λij(hij−\textslgμνxμ,ixν,j)+λinμxμ,i+λ0(\textslgμνnμnν−1)]. (17)

In general, the various Lagrange multipliers are different among the two sides . Moreover, since there are two normals , there are two independent variations with respect to the normals at the two sides. These two variations are independent since one could consider, for example, the case where only the half space exists. Although the bulk metric is continuous across the defect, the variation is different among the two sides . Indeed, the extrinsic curvature is in general discontinuous on the brane and from equation (16), contains derivatives of the metric. Therefore, can be expressed in terms of quantities on either side of the defect, but not simultaneously in terms of quantities on both sides. Finally, there is the extra independent variation with respect to the brane metric .

Variation of with respect to at the brane gives

 δ(S+Sc)∣∣brane = ∫Σ±d4χ√|h|(λixμ,i+2λ0nμ−1κ25Knμ)δnμ (18) + − − 12κ25∫Md5x√|\textslg|{Gμν−κ25Tμν+Λ5\textslgμν}δ\textslgμν∣∣brane−12κ25∫Σ±d4χ√|h|hμνnλ[(δ\textslgμλ);ν−(δ\textslgμν);λ] + 12κ25∫Σ±d4χ√|h|hμνnλ[2(δ\textslgμλ);ν−(δ\textslgμν);λ].

When , one should add in (14) the integral of the extrinsic curvature of (if is not empty) to cancel some terms from the variation ; this, in general, does not affect the dynamics of guven ().

The basic root of difficulty for deriving (18) is the Gibbons-Hawking term . Its treatment, due to the imposition of the constraints (17), is different from the conventional treatment of the variation described after equation (15). In the first line of (18), the terms arise from the appropriate terms of (17) and the term of (14), where the identities , have been used ( are the Christoffel symbols of ). These identities show that is a function of the independent variables , but it does not depend on the independent variable . Next, the terms in the second line of (18) arise again from the appropriate terms of (17) and all the four-dimensional terms of (14). Finally, the variation on the brane is more difficult. The first contribution comes from the appropriate terms of (17) and the derived terms are those of the third line of (18), where special care is needed for the variation of since is kept fixed and not . Second, it arises from the five-dimensional terms of (14) and the derived terms are those of the fourth line of (18). Third, it arises from the term of (14) and the derived terms are those of the fifth line of (18). In order to get these last terms, the identity was used, which arises from and .

The last term of the fourth line of (18) cancels the last term of the fifth line, so the normal derivatives of cancel. Then, use is made of the identity , where denotes covariant differentiation with respect to (or ), to convert the remaining terms of (18) to terms (up to an irrelevant integration on ). Finally, the quantity in curly brackets appearing in the fourth line of (18) vanishes since it coincides with equation (15) which is also valid on the brane. The variation (18) takes the form

 δ(S+Sc)∣∣brane = ∫Σ±d4χ√|h|(λixμ,i+2λ0nμ−1κ25Knμ)δnμ (19) + −

As explained above, are independent variations, but depends on which are also independent. So, gives

 λixμ,i+(2λ0−1κ25K)nμ=0 (20) λij++λij−+1κ25(Kij−K2hij)∣∣++1κ25(Kij−K2hij)∣∣−+12(Tij−λhij)−rc2κ25Gij=0 (21) ∫Σ±d4χ√|h|[λijxμ,ixν,j+λ0nμnν+12κ25(Kμν−Knμnν)]δ\textslgμν=0, (22)

where obeys (16). Equation (20) holds separately for each side . Since the vectors  ,  are independent, equation (20) implies for any side separately , . Equation (21) contains the combination and it will be seen that the matching conditions contain the same combination, so the matching conditions will be unambiguously determined. Then, equation (22), with satisfying (21), takes the form

 ∫Σ±d4χ√|h|(12κ25Kμν+λijxμ,ixν,j)δ\textslgμν=0. (23)

Since , equation (23) is written as

 ∫Σ±d4χ√|h|μijxμ,ixν,jδ\textslgμν=0,μij=12κ25Kij+λij. (24)

Contrary to the present situation, had we considered all independent, equations (20), (21) would still arise. Equation (24) would be written as providing . Then, using equation (21), the Israel matching condition would arise.
In our approach has to be expressed via (16) in terms of quantities on either side and equation (24) becomes

 ∫Σ+orΣ−d4χ√|h|Mijxμ,ixν,jδ\textslgμν=0,Mij=μij++μij−, (25)

or equivalently

 ∫Σ+orΣ−d4χ√|h|Mij(\textslgμν,λxμ,ixν,jδxλ+2\textslgμνxμ,ixλ,jδxν,λ)=0. (26)

After an integration of (26) by parts and imposing , we get

 ∫Σ+orΣ−d4χ√|h|\textslgμσ[Mij|jxμ,i+Mij(xμ;ij+Γμνλxν,ixλ,j)]δxσ=0 (27)

and since the extrinsic curvature satisfies  , equation (27) becomes

 ∫Σ+orΣ−d4χ√|h|\textslgμσ(Mij|jxμ,i−MijKijnμ)δxσ=0. (28)

Due to the arbitrariness of it holds

 Mij|jxμ,i−MijK+ijnμ+=0,Mij|jxμ,i−MijK−ijnμ−=0, (29)

therefore, two sorts of matching conditions arise

 MijK+ij=0 , MijK−ij=0 (30) Mij|j = 0. (31)

Substituting of from (21), we get the matching conditions of codimension-1 Einstein gravity

 [(Kij+−K+hij)+(Kij−−K−hij)+