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From p-branes to Cosmology

We study the relationship between static p-brane solitons and cosmological solutions of string theory or M-theory. We discuss two different ways in which extremal p-branes can be generalised to non-extremal ones, and show how wide classes of recently discussed cosmological models can be mapped into non-extremal p-brane solutions of one of these two kinds. We also extend previous discussions of cosmological solutions to include some that make use of cosmological-type terms in the effective action that can arise from the generalised dimensional reduction of string theory or M-theory.


Introduction
A considerable literature exists on the subject of p-brane solutions of the low-energy limits of string theories and M-theory. These include the BPS-saturated classes of extremal p-branes, which preserve some fraction of the spacetime supersymmetry, and also their non-extremal generalisations, where supersymmetry is completely broken. A characteristic feature of all these solitonic solutions is that they are static (or at least stationary, if one allows rotating solutions as well). Other studies have focussed on a different class of solutions to the lowenergy effective equations of motion, namely cosmological solutions in which the metric evolves in time. Typically, these solutions describe the evolution of a universe from a small initial radius to a large radius at late times.
Although they are ostensibly very different in their interpretation, it has been observed that there are actually many similarities between some of the p-brane solutions and the cosmological solutions [1]. Indeed in some examples, an exact correspondence has been established [2]. The most obvious difference between the two classes of solution is that the fields of the cosmological solutions depend upon time, while in the p-brane solitons they depend instead on the coordinates of the transverse space (in fact, in the cases of relevance here, they depend just on the radial distance from the origin in the transverse space).
However, this distinction is not such a profound one; it is, for example, well known that in the interior region of the Schwarzschild black hole the usual time coordinate becomes spacelike, while the usual radial coordinate acquires a new interpretation as the time coordinate [3].
Thus one way, at least, in which an equivalence can arise is that the interior solution of a black p-brane may be re-interpreted as a cosmological model. As we shall see later, not all of the mappings between p-brane and cosmological solutions are of this kind, and sometimes it is necessary to make Wick rotations in order to interchange the timelike and spacelike character of coordinates in the two cases.
The purpose of this paper is to investigate the mapping between p-brane solutions and cosmological solutions in a somewhat general framework. In order to do so, we shall begin by first obtaining a large class of cosmological solutions. In general, our starting point can be taken to be a theory describing the coupling of gravity to a dilatonic scalar field and an antisymmetric tensor field strength of degree n. A special case of this, which has not been extensively discussed in the literature, is when the degree n is actually zero. In this case the "field strength" is really just a constant; if there were no coupling to a dilaton, it would in fact correspond precisely to a pure cosmological constant term. In section 2, we begin by setting up appropriate ansätze for the metric and field strength that enable us to study rather broad classes of cosmological solutions, including in particular the ones utilising a 0-form field strength, which we examine in section 3.
In section 4, we examine the relationship between the cosmological solutions obtained in the previous sections, and non-extremal p-brane solitons. In order to do this, it is necessary to distinguish between two different kinds of generalisation of the standard BPSsaturated extremal p-branes. In the first of these, which we shall refer to as type 1 nonextremal p-branes, the form of the D-dimensional metric remains the same as in the extremal case, namely ds 2 = e 2A dx µ dx µ + e 2B (dr 2 + r 2 dΩ 2 ), where A and B are functions of r. However, whereas in the extremal case the function X ≡ dA +dB is equal to zero, here in the non-extremal generalisation it becomes non-vanishing [4]. (We have defined The other kind of non-extremal generalisation, which we shall refer to as type 2, begins from a modified form for the metric, namely ds 2 = e 2A (−e 2f dt 2 +dx i dx i )+e 2B (e −2f dr 2 +r 2 dΩ 2 ) [5,6]. In this case, the relation dA+dB = 0 is still maintained. Although both the type 1 and type 2 non-extremal generalisations introduce an additional function, namely X or f , the way in which they enter the metric ansatz is quite different, although the two become equivalent when p = 0. As we shall see, both types of non-extremal p-brane generalisation can be mapped over into cosmological solutions. The more standard cosmological solutions actually correspond to the type 1 black p-branes; for this reason, we shall refer to such cosmological solutions as type 1. These are the kind that are constructed in sections 2 and 3. We also discuss some of the cosmological solutions associated with the type 2 black p-branes; some examples of this type have been considered in [2]. Finally in section 4, we also show how certain special cases of the type 1 solutions can in fact be obtained by a dimensional reduction of type 2 solutions, where the original time coordinate is used for the compactification process.

General cosmological solutions in string or M theories
The study of the cosmological consequences of string theory has been an area of much active research in the past [7][8][9][10][11]. Recently, considerable attention has been directed to string-inspired cosmology in various dimensions [12][13][14][15][16][17][18][19][20][21][22]. Cosmological models are described by solutions of the low-energy effective theory in which the metric tensor, and the other fields, are time dependent. We shall consider cosmological metrics of the form where U , A and B are functions only of t, and ds 2 q and ds 2 q denote metrics on maximallysymmetric spaces of positive, negative or zero curvature, with dimensions q andq respectively, i.e.
where dΩ 2 and d Ω 2 are the metrics on unit (q − 1)-and (q − 1)-spheres respectively. For later convenience, we define d = q − 1 andd =q and hence d +d = D − 2. The constant k andk can be taken to be equal to 0, 1, or −1 independently, corresponding to flat, spherical or hyperboloidal spatial sections respectively. A large class of cosmological solutions with k = 0 were obtained in [19]. The solutions include some realistic models, where the (q + 1)dimensional spacetime expands at large time, while theq-dimensional space contracts and becomes unobservable at large time. Note that whenk = 0, the solution can be compactified on anyq-dimensional Ricci-flat space with metric ds 2 q . In this section, we generalise the previously-known solutions to include the cases where k = 0, and briefly discuss their cosmological characteristics.
In the vielbein basis e 0 = e U dt, e a = e Aēa , e α = e Bēα , we find that the curvature 2-forms are given by where a dot denotes a derivative with respect to the time coordinate t, Θ a b is the curvature 2-form for the metric ds 2 q =ē aēa in the vielbein basisē a , and Θ α β is the curvature 2-form for the metric ds 2 q =ē αēα in the vielbein basisē α . It follows that the tangent-space components of the Ricci tensor for the metric (2.1) are given by whereR ab andR αβ denote the tangent-space components of the Ricci tensor for the qdimensional andq-dimensional spatial metrics. In all the cases we shall consider, these metrics will be Einstein, and we may writeR ab = k(q − 1)δ ab andR αβ =k(q − 1)δ αβ .
We begin by solving for the simplest cosmological solutions in D dimensions, which involve only the metric, a dilaton and an n-rank antisymmetric field strength F n . The Lagrangian is given by where the constant a can be conveniently parameterised as  [25]. The equations of motion from the Lagrangian where S M N is a symmetric tensor given by There are two types of ansätze for the field strength F n that are compatible with the symmetries of the metric (2.1), giving rise to "electric" and "magnetic" cosmological solutions. In the electric solutions, the ansatz for the antisymmetric tensor is given in terms of its potential, and in a coordinate frame takes the form and hence where f is a function of t only. Here and throughout this paper ǫ M ···N and ǫ M ···N are taken to be tensor densities of weights −1 and 1 respectively, with purely numerical components ±1 or 0. Note in particular that they are not related just by raising and lowering indices using the metric tensor. For electric solutions, we have q = D − n andq = n − 1.
For the magnetic cosmological solutions, the ansatz for the tangent-space components for the antisymmetric tensor is F a 1 a 2 ···aq = λe −qA ǫ a 1 a 2 ···aq , (2.11) where λ is a constant. Thus we have q = n andq = D − n − 1. The form of the exponential prefactor is determined by the requirement that F n satisfy the Bianchi identity dF n = 0.
Substituting the ansätze for the metric and the field strength into the equations of motion (2.7), we findφ where ǫ = 1 for the electric case and ǫ = −1 for the magnetic case. In the electric case, the constant λ arises as the integration constant for the function f in (2.9).
Let us first consider the equations of motion (2.12) for single-scalar cosmological solutions in D dimensions. It is convenient to make the gauge choice U = qA +qB, and to define The equations of motion for Z 1 , Z 2 and φ becomë 14) φ = 1 2 ǫaλ 2 e 2qZ 2 −ǫ∆φ/a , together with the first integral: For generic values of k,k and the charge parameter λ, the equations of motion (2.14) cannot be solved in a closed form. The following are special cases where solutions can be obtained:

Case 1:
A special class of solutions were obtained in [19], namely those withk = 0, but with k arbitrary. In this case, a further redefinition of fields was performed, namely These fields satisfy the following equations: It follows from (2.15) that the first integral corresponds to a conserHamiltonian for the above equations, Thus X and Φ both satisfy Liouville equations. The manifest positivity of the left-hand side of (2.18) (assuming ∆ > 0) shows that the HamiltonianẊ 2 + k(q − 1) 2 e 2X for X must be positive, and hence the appropriate form of the solution is where c and δ are constants. Note that in taking the square root of e 2X , the positive root should be chosen in the expression for e −X . The HamiltonianΦ 2 + ∆λ 2 e Φ for Φ is also manifestly positive, and so the solution can be written as where β and γ are constants. The solution for Y may be taken to be simply The constraint (2.18) therefore implies that In terms of the original functions A, B and U appearing in the metric (2.1), and the dilaton φ, the solution takes the form together with U = qA +qB. In a case where there is no dilaton, the solutions for A and B are again given by (2.23), with µ = 0. If insteadq = 0, we have from (2.6) that a 2 = ∆; the solution for φ follows from (2.23) by settingq = 0, and A is given by A = X/(q − 1), with X given by (2.19).
The solutions presented above are single-scalar solutions, involving a single field strength carrying an electric or a magnetic charge. In fact, whenk = 0 one can also obtain general solutions for multi-scalar, or dyonic cosmological models [19]. We shall see in section 4 that these solutions are closely related to static p-brane solutions.

Case 2:
In this case we havek = 0, but with k = 0. The solution for the function Z 1 is easily found to be Z 1 = µt. To simplify the equations of motion for Z 2 and φ, we define The first order equation (2.15) becomeṡ The Liouville equation (2.26) with the first-order constraint (2.27) can be easily solved, giving finally where c is given above equation (2.26). It follows from (2.27) that the constant β is given by In terms of the original functions A, B and U appearing in the metric (2.1), the solution is given by It is of interest to examine the cosmological features of these solutions. Note that unlike thek = 0 case, there exists a limit where β = 0 = µ. In this limit, all the functions A, B and φ depend linearly on log t. When β and µ are non-vanishing, the time dependence of these functions is more complicated. Let us consider β and µ to be both positive. At the beginning, when sinh(βt + α) vanishes, both the scale factors e A and e B diverge. As the coordinate t approaches infinity, e A and e B behave quite differently. For ∆ > 0, the scale factor e B shrinks to zero, whilst the behaviour of the scale factor e A depends on the sign factor e A thus starts from infinity at the beginning and shrinks to a minimum and then grows indefinitely at late times. The behaviour of the dilaton depends on the sign of ǫ.

Case 3:
In this case, we take both k andk to be non-vanishing. As we mentioned earlier, the equations (2.14) cannot be solved in general. However there exists a special solution, where the first derivatives of the functions Z 1 , Z 2 and φ are proportional to one another. The equations (2.14) can be reduced to a single Liouville equation if we have where the constants x 1 , x 2 and x 3 are given by (2.32) The Liouville equation is given byü − e 2u = 0, and the first-order equation becomesu 2 − e 2u = 0, with the solution u = −log t + c. Here c is a constant of integration. Thus using the freedom to choose c, we can relax one of the relations in (2.32). In this case, the behaviour of the metric components is 33) and so e A and e B decrease as t increases. For electric solutions (ǫ = 1) the dilaton φ tends to infinity for large t, while for magnetic solutions (ǫ = −1) the dilaton tends −∞ as t increases. It is also not hard to express the scale factors in terms of the comoving time τ : (2.34)

Cosmological solutions with cosmological terms
In the previous section, we obtained cosmological solutions using n-form field strengths.
Such solutions arise in massless supergravities in D-dimensions. Supergravities can also admit one or more cosmological terms, which can be viewed as the special case where n = 0. Such theories are in general massive, but they can nevertheless be obtained by the Kaluza-Klein reduction of massless supergravities in higher dimensions. For example, gauged massive supergravity in D = 4 [26] or D = 7 [27] can be obtained by the dimensional reduction of 11-dimensional supergravity [30] on a 7-sphere [28] or a 4-sphere [29].
The highest dimensional massive supergravity is the massive type IIA theory in D = 10 [31], which seems to indicate the possible existence of a 13-dimensional theory [32]. In fact, the number of massive supergravities in lower dimensions is far greater than that of massless supergravities, at least in the maximally supersymmetric case. It was shown recently that certain generalised Kaluza-Klein compactifications of the low-energy limits of string theory or M-theory can give rise to massive supergravities with cosmological terms in D-dimensions [33,32,34,35]. The generalised reduction involves making an ansatz for a rank (n − 1) potential such that its field strength contains a term that is a constant multiple of an harmonic n-form on the compactifying manifold. Such a phenomenon was also discussed earlier from a group theoretic point of view in [36]. In general, such massive supergravities contain cosmological terms. We are interested in cases where the bosonic Lagrangian can be consistently truncated to one containing just the metric, dilatonic scalar fields and cosmological terms, of the form: where φ = (φ 1 , φ 2 , . . .) denotes a set of dilatonic scalar fields in the theory and c α are constant vectors. In the toroidal compactification, consistency of the truncation requires that the dilatonic vectors c α satisfy the following dot product relations [24] In this case, the Lagrangian (3.1) can be further truncated to a single-scalar Lagrangian, of the form [24] where The where the constants β and µ are related by In thisk = 0 case, the generalisation of the solution (3.7) to a multi-scalar solution with the dilaton vectors c α satisfying the dot products (3.2) is straightforward, and the solution is given by  In this case, the coupling constant g e runs from infinity to zero as the coordinate t goes from −∞ to ∞, and the coupling constant g m behaves in exactly the opposite way (here without loss of generality we assume that µ is positive). Similarly, we can discuss the behaviour at large |t| for the scale factor R = e B . In this case, we need to compare the value of β with a 2 (D − 2)/2µ: Again the behaviour of the scale factor R depends only on the sign of ∆. If ∆ is negative, the scale factor diverges at large |t|. On the other hand, if ∆ is positive, the scale factor runs from zero to infinity as the t coordinate runs from ∞ to −∞.
In order to discuss the evolution of the solutions, it is useful to introduce the comoving time coordinate τ , defined by ds 2 = −dτ 2 + e 2B ds 2 q , which implies that τ = t e (D−1)B dt. Thus at large t, we have τ ∼ exp((D − 1)(β|t| − 1 2 (D − 2)a 2 t)/(2(D − 2))). If ∆ is positive, the universe starts at finite τ 0 when the scale factor is zero, and then expands to infinity as τ goes to infinity. In this case, since we have R = τ 1/(D−1) , the expansion of the universe slows down at large τ , but there is no inflation in the early stages. If ∆ is negative, the situation is different. In this case, the comoving time τ runs from infinity to a finite value and then to infinity again as t goes from −∞ to ∞. Thus we can also consider that the universe starts at a finite τ 0 when its size is minimal, and then it expands to infinity as τ goes to ∞.
The physical properties of the cosmological solutions with cosmological terms were discussed above in the context in the D-dimensional theories themselves. Since these theories are dimensional reductions of more fundamental theories such as string theory or M-theory, their cosmological features can be more appropriately analysed in ten or eleven dimen- In this 11-dimensional metric, we see that the scale factor for the 3-dimensional space R = e B+φ √ 7/6 ∼ (cosh(βt + α)) −1/2 now shrinks to zero at large |t|. On the other hand, the scale factor R ′ = e −φ/(3 √ 7) for the 7-dimensional internal space runs from infinity to a minium size and then to infinity again as the coordinate t runs from −∞ to ∞. Note that the comoving time coordinate τ runs from 0 to τ 0 , where R ′ reaches its minimum, and then goes to infinity. Thus the 11-dimensional metric effectively describes an 8-dimensional expanding universe which starts at time τ 0 , where it has a Plankian size. Note that the time interval (0, τ 0 ) is of the Plankian scale too. Since the scale factor R ′ has only one minimum, at τ = τ 0 , it implies thatṘ ′ ≡ dR ′ /dτ > 0 for all times τ 0 < τ < ∞. It also implies that close to the beginning of the universe, τ ∼ τ 0 , we haveR ′ > 0 and hence an inflationary expansion. On the other hand as τ → ∞ we have R ′ ∼ τ (β+3µ)/(β+21µ) , and henceR ′ < 0, implying a slowing down of the expansion rate of the universe. Now let us consider a 4-dimensional magnetic cosmological solution in D = 4 using a cosmological term. The 4-dimensional theory in question is obtained [35] from a generalised compactification of M-theory on a 7-torus or on a 7-dimensional Joyce manifold [41]. The solution involves 7 cosmological terms, and hence has a 2 = 25/7 and ∆ = 4/7 (see [35] for the configuration of the cosmological terms). We consider only the case where all the cosmological constants are equal, for which the solution is given by 14) where β = 5 3/7µ. Again this solution can be oxidised to D = 11, in fact giving the same result (3.13) as in the electric solution described above, except that now B and φ are given by (3.14) rather than (3.12). In this case, one can easily verify that R shrinks and R ′ expands, effectively describing an 8-dimensional expanding universe. As in the example above, at the beginning of the universe τ = τ 0 there is an inflationary period, whilst at large times the expansion rate slows done. infinity as the 4-dimensional space expands while the 6-dimensional space contracts. Since at the starting point of the universe τ = τ 0 the scale factor R ′ is a minimum, we have thaẗ R ′ > 0 in the early universe. At large τ , we have R ′ ∼ τ (β+6µ)/(β+24µ) , implying thatṘ ′ > 0 butR ′ < 0.
We can alternatively oxidise the solution to D = 10 instead. In this case, it will effectively describe a 4-dimensional expanding universe. To see this, we note that the metric in D = 10 is given by Again the scale factor for the six-dimensional space with metric ds 2 6 shrinks whilst the 3dimensional space with metric ds 2 3 expands, effectively describing a 4-dimensional expanding universe, with an inflationary period at early times, and a slowing down of the expansion rate these days. In fact this is a solution in D = 10 which is supported by a 3-form field strength. General cosmological solutions involving 3-form field strengths in D = 10 were also discussed in [1,19]. It is interesting to study the behaviour of the string coupling in D = 10. There are two maximally supersymmetric string theories in D = 10, namely the type IIA and type IIB strings. If we oxidise the D = 7 solution to D = 10, where the D = 7 theory is taken to be embedded in the type IIA theory, we find that the corresponding string coupling constant is given by g = e − √ 5/2φ/4 , which diverges at large |t|. If on the other hand we view the D = 7 theory as being embedded in the type IIB string, and oxidise it to D = 10, there are two possible outcomes. If the charge is carried by the NS-NS 3-form, then the coupling constant is the same of the type IIA case, and diverges at large |t|; on the other hand, if the charge is carried by the R-R 3-form, then the coupling constant is the inverse of that in the type IIA case, and hence goes to zero at large |t|.

Relation between cosmological and p-brane solutions
It has been observed recently that there is a similarity in certain cases between the cosmological solutions of the kind we are discussing in this paper, and type 1 black p-brane solutions in string theory [1]. In this section, we shall explore this in a more general setting, and show that there is an exact one-to-one map between the k = 1,k = 0 type 1 cosmological solutions and the type 1 black p-brane solutions that were discussed in [4]. In the special case when p = 0, these latter solutions are the same as standard non-extremal black holes. Thus a subset of the type 1 cosmological solutions can be mapped into standard black-hole solutions. However, when p is greater than zero, the type 1 black p-brane solutions obtained in [4] are not ordinary type 2 [6] black p-branes. Thus in cases where the cosmological solutions map into p-brane solutions with p > 0, it is to these non-standard type 1 black p-branes, rather than the standard type 2 ones, that the mapping occurs.
To see this in detail, we begin by recalling the structure of the type 1 black p-branes constructed in [4]. They have metrics which take the form except that there the further restriction d A +d B = 0 is imposed. In [4], more general solutions, which are non-extremal, were obtained by relaxing this assumption, so that there is an additional variable X ≡ d A +d B appearing in the equations of motion. This has the solution e X = 1 −κ 2 ρ 2 , whereκ is a constant and ρ ≡r −d . It is natural to introduce a further redefined radial coordinate ξ, such that e X ∂/∂ρ = ∂/∂ξ, and thusκρ = tanhκξ.
The general type 1 black p-brane solutions take the form [4] e − (D−2)∆ 2d The various constants of integration are subject to the constraintdβ 2 = 2(d + 1) The metrics described by these solutions have an outer event horizon where the function e X = 1 −κ 2 ρ 2 vanishes. In the case when p = 0, they describe precisely the standard nonextremal black-hole solutions of string theory, withκ being a non-extremality parameter that vanishes in the extremal limit. However, when p is greater than zero, (4.1) is quite different from the metric of a standard type 2 black p-brane, for which one would have [6] ds 2 = e 2Â (−e 2f dt 2 + dx i dx i ) + e 2B (e −2f dr 2 + r 2 dΩ 2 ) , To see this, it is instructive first to write the metric (2.1) for k = 1 andk = 0 in the following form: where ν is a constant. Now, using these relations in the explicit forms of A, B, A and B, it is easy to relate the other parameters that appear in the p-brane and cosmological solutions: (4.8) The above demonstration illustrates that for the case k = 1 andk = 0, the type 1 cosmological solutions are identical to those of the type 1 non-extremal p-brane solitons obtained in [4], up to general coordinate transformations that may include Wick rotations.
In fact once the equivalence of the type 1 p-brane ansatz (4.1) and the cosmological ansatz So far we have seen that the single-scalar purely electric or purely magnetic type 1 cosmological solutions can be one-to-one mapped to the associated type 1 non-extremal p-branes. The generalisation of this mapping to multi-scalar or dyonic solutions is straightforward. The corresponding non-extremal p-brane and cosmological solutions can be found in [4] and [19] respectively.
As we discussed in section 2, cosmological solutions also exist for other values of k andk. As shown above whenk = 0, the corresponding flat space with metric ds 2 q can be reinterpreted as the world volume of the p-brane. It seems that such an interpretation breaks down whenk = 0, and such cosmological solutions would map into solutions that do not have an interpretation as p-branes. On the other hand, there is no such restriction on k, which defines the structure of the transverse space of the corresponding p-brane. When k = 1, we have seen that the cosmological solutions can be mapped to isotropic p-branes.
By the same token, we can obtain new non-extremal p-brane solutions by mapping the cosmological solutions with other values of k. In the case of k = 0, the corresponding p-brane solutions allow extremal limits, and were obtained in [42]. These supersymmetric solutions describe configurations of p-branes uniformly distributed over a q-dimensional hyperplane. They become domain walls after compactifying these q coordinates [34].
So far, we related type 1 cosmological solutions and the type 1 non-extremal p-branes. As was shown in [5,6], there is another universal way to blacken extremal p-branes, namely the type 2 p-branes given by (4.3) and (4.4). It was observed in [2] that in the region where e 2f is negative, the signature of the t and r coordinates reverses, leading to a natural interpretation of the interior p-brane solution, inside the horizon at e 2f = 0, as a cosmological solution.
In this case there is no need to perform any Wick rotation, since the sign reversal of the function e 2f automatically gives the original r coordinate a timelike interpretation, while the original time coordinate becomes spatial. For the case p = 0, as we have already remarked, the standard and the non-standard black p-branes coincide, and the description of the mapping to the cosmological solutions is identical. Let us therefore consider the remaining cases, when p > 0. It seems to be natural to consider the universe as beginning at r = 0, since this is a point where the curvature is singular. The universe will then evolve to the horizon at r = κ 1/d . At r = 0 the scale factors for dx i dx i and for dΩ 2 are zero, expanding to a finite size at the horizon. In this interval, the comoving time τ = i r e B−f dr covers a finite range. The scale factor for dt 2 at the horizon is zero, whilst it is proportional to rd (2a 2 /d−1) at small r. Thus to give a satisfactory cosmological model we must have a 2 ≥ d/2, since there should not be spatial directions that grow to infinite size at a finite comoving time.
To summarise, we have seen that there are two types of non-extremal p-branes solutions: the non-standard (type 1) with the metric (4.1), and the standard (type 2) with the metric (4.3). Correspondingly there are type 1 and type 2 cosmological solutions. We observed that the type 1 black p-branes are identical to the type 1 cosmological solutions that were obtained in section 2, with k = 1 andk = 0. In general, the mapping between these p-brane metrics and cosmological metrics requires the Wick rotation of one or more coordinates.
It is also straightforward to observe that the interior of the type 2 black p-branes can be interpreted as type 2 cosmological models. In this case, no Wick rotation of coordinates is necessary. It is of interest to study the inter-relationship of these solutions Since the time coordinate t of a type 2 black p-brane becomes spacelike inside the horizon, while the original r coordinate becomes timelike, one can choose to perform a Kaluza-Klein reduction of the interior metric in which t is compactified. This gives rise to a lower-dimensional cosmological solution with the general form of the type 1 models. There is however a slight complication in general, since the lower-dimensional solution will acquire an additional scalar degree of freedom. However, if we start with a higher-dimensional type 2 solution with no dilaton excitation, its compactification gives precisely a type 1 cosmological solution. Since this solution can be mapped to a type 1 black p-brane by Wick rotation, it implies that under appropriate circumstances type 1 black p-branes can also obtained from type 2 black p-branes by compactifying the time coordinate, made spacelike by virtue of the Wick rotation. We shall now illustrate this, taking the type 2 black membrane in D = 11 as an example. Its metric is given by where e 2f = 1 − κr −6 and H = 1 + κr −6 sinh 2 µ. If we simply compactify the solution on one of the world-volume spatial coordinates x i , it reduces to a type 2 black string in D = 10. However we can instead perform Wick rotations to make the coordinate t spacelike, and the metric dx i dx i to be Minkowskian, and then compactify the Wick rotated t coordinate to get a different dimensionally-reduced 10-dimensional metric ds 2 10 , given by ds 2 11 = e ϕ/6 ds 2 10 + e −4ϕ/3 d(it) 2 . Thus we find Thus we see that the dimensional reduction of the Wick-rotated time coordinate of a dilatonfree type 2 black p-brane gives a special case of a type 1 black (p − 1)-brane in one lower dimension.
A special case of the type 2 −→ type 1 reductions discussed above is when one begins with a black hole in the higher dimension. In order to make an exact correspondence with the type 1 solutions, we should again consider examples where there is no dilaton involved in the higher-dimensional solution. A simple example is provided by considering black-hole solutions in the N = 1 string in six dimensions, where the charge is carried by a Yang-Mills field. Including the effects of loop corrections, it was shown in [43] that the relevant part of the effective Lagrangian is given by where v andṽ are constants. This admits a black-hole solution [44] where φ is a constant, given by e φ √ 2 = v/ṽ, and the metric is given by In the exterior region e 2f > 0 this describes a non-extremal instanton with a Euclidean signature; in the interior region it can be viewed as a cosmological solution, since the coordinate r becomes timelike. Note that the metric is independent of the electric charge, which appears only in the harmonic function H. In fact this is a general phenomenon that will occur whenever a black-hole solution is compactified in the time direction. Thus starting from the black-hole metric