1. Introduction

Traditionally in relativity, in order to describe different phenomena in our universe, such as a trajectory of an object, one time and three space are the chosen number of real dimensions ((1 + 3)-dimensions). Yet the physical reasons why our world requires (1 + 3)-dimensions, remains as an open problem. It is evident that from the point of view of number theory the (1 + 3)-world the space and time are not symmetric. The natural question is: why three space and only one time dimension? Looking for the answer of this question one finds that the (5 + 5)-dimensional space-time (five time and five space dimensions) is a common signature to both type IIA strings and type IIB strings [¹]. In fact, versions of M-theory [²-³] lead to type IIA and to type IIB string in space-time of signatures (5 + 5). It turns out that by duality transformations string theories of signatures (5 + 5) are related to other string signatures such as (1 + 9) [³].

Of course, the (5 + 5)-dimensional world is more symmetrical in the number of space and time dimensions than the (1 + 3)-world. Thus, considering seriously the (5 + 5)-world, just as the (1 + 3)-dimensional signature can be considered as a reduced world of the de Sitter (1 + 4)-dimensional or anti-de Sitter (2 + 3)-dimensional signatures via the cosmological constants Λ > 0 and Λ > 0, respectively, here, one may assume that up to two cosmological constants, the (4 + 4)-world emerges from (5 + 5)-dimensional world. In fact, the (4 + 4)-dimensions can be considered as the transverse coordinates of the (5 + 5)-dimensions [⁴-⁵].

Fortunately, there are already a number of works with interesting results in the (4 + 4)-world that can be considered as additional motivation for increasing interest in such a scenario. First, the Dirac equation in (4 + 4)-dimensions is consistent with Majorana-Weyl spinors which give exactly the same number of components as the complex spinor of 1/2-spin particles such as the electron or quarks [⁶,⁷]. Second, the most general Kruskal-Szekeres transformation of a black-hole coordinates in (1 + 3)-dimensions leads to 8-regions (instead of the usual 4-regions), which can be better described in (4 + 4)-dimensions [⁸]. Third, loop quantum gravity in (4 + 4)-dimensions [⁹-¹⁰] admits a self-duality curvature structure analogue to the traditional (1 + 3)-dimensions. It also has been shown [¹¹] that duality

of a Gaussian distribution in terms of the standard deviation σ of 4-space coordinates associated with the de Sitter space (anti-de Sitter) and the vacuum zero-point energy yields to a Gaussian of 4-time coordinates of the same vacuum scenario. Finally, it has been suggested that the mathematical structures of matroid theory [¹²] (see also Refs. [¹³-²⁰] and references therein) and surreal number theory [²¹-²³] (see also Refs [²⁴-²⁵] and references therein) may provide interesting routes for a connection with the (4 + 4)-world.

2. The (4+4)-world black hole

Let us start considering the ansatz

Here, the indices μ^,ν^,... run from 1 to 8 and the matrices g~ij(θ(+)) and g~ab(θ(-)) are defined as

and

respectively. Here, the notation θ(+) and θ(-) means that the angle θ(+) refers to the (1 + 3)-world, while the angle θ(-) corresponds to the (3 + 1)-world.

From the chosen form of gμν it is evident that one is dealing with a spheric symmetric static system in (4 + 4)-dimensions. The only unknown variables will be fr,ρ,hr,ρ, p(r,ρ) and q(r,ρ) which must be determined with the relativistic gravitational field equations in (4 + 4)-dimensions.

The non-vanishing Christoffel symbols associated with (1), involving the indices values of μ, v = 1 to 4, are

while for the values μ, v = 5 to 8 one gets

where A'=∂A/∂r and B˙=∂B/∂ρ for any arbitrary functions A(r,ρ) and B(r,ρ). One still must include the non-vanishing mixture Christoffel symbols

and

In vacuum the gravitational field equations simply establish that the Ricci tensor Rμ^ν^=R μ^α^ν^α^ must vanish, that is one has

Using the Christoffel symbols (4)-(7) one learns that (8) leads to

(where due to (1) and (2) the indices i,j,... run from 3 to 4), and also to

Here, according to the ansatz choice (1) and (3), the indices a,b,... take the values 7 and 8.

3. Black-hole duality solution

Our next step is to look for a black-hole solution of (9) and (10). For this purpose, focusing in the last formula in (9) one observes that assuming the two equations

such a formula can be simplified in the form

A general solution of this equation can be written as

with A(ρ) an arbitrary function of ρ. Following similar steps and assuming

the last equation in (10) leads to

whose solution is

with B(r) an arbitrary function of r.

Our next step is to determine the functions A(ρ) and B(r). For this purpose, one may first focus in the first equation of (9). Considering (11) and (14) one see that such equation reduces to

Since e-h=ef, with (13), one verifies that

Thus, (17) is further reduced to

which can also be written as

with f˙≠0. This expression can be integrated yielding

where ln a is a constant independent of ρ. This means that

From (16) one learns that (22) becomes

Thus, if one sets a = B one sees that

in agreement with (14).

Hence, (19) can be written as

Substituting (13) into this equation one obtains

with r0 also a constant. Therefore,

and consequently one gets

Substituting this result into (13) one discovers the surprising result

Following similar steps one shall obtain that the first equation in (10) leads to the solution

Summarizing, we have derived the black-hole solution in (4 + 4)-dimensions;

In principle r0 and ρ0 are different constants but if one sets

the metric (30) becomes a totally dual black-hole solution. Let us explain in some detail this comment. First from (30) and (31) one sees that the line element in (4 + 4)-dimensions can be written as

where

and

In order for ds(+)2 to describe the usual black-hole element in (1 + 3)-world one must set

with G the Newton gravitational constant and M(+) the mass source in the (1 + 3)-world. Similarly in order for ds(-)2 to describe the usual black-hole element in (3 + 1)-world one must set

where M(-) is the source mass in the (3 + 1)-world. Thus, from the perspective of a (1 + 3)-world observer the combination ξ2/ρ is just a related to the source mass M(+). This means that even if the parameter ρ associated with (3 + 1)-world appears in the line element ds(+)2 is in fact related to source mass M(+) according to (35). In dual form, the parameter r of the (1 + 3)-world is interpreted by a (3 + 1)-world observer as the mass source M(-) according to (36).

Even clearer dual properties of the line element (32)-(34) emerge when one considers the event horizon of (33) and (34). Suppose there are parameters r and ρ such that

Of course, in this case both the terms with dr2 and dρ2 present an apparent singularity. From (37) one sees that this means that

which is clearly a dual relation: A large radius rs of the event horizon of the (1 + 3) black-hole corresponds to a small radius ρs of the (3 + 1) black-hole and vice versa.

At least at the level of black-holes the above result establishes a dual link between the (1 + 3)-world and the (3 + 1)-world that needs to be consider when one looks for a solution of quantum gravity. In fact, thinking about the magnetic monopole g and the electric charge e duality, namely

with ge=nℏ and n=1,2,..., one is tempted to assume that (38) implies a quantum duality of the form

Of course in order to fully understand the consequences of (38) or (40) one needs to clarify the meaning of the constant parameter ξ. At first sight one may propose that ξ=lP, with lP the Plank length. However, in this case (38) implies a smaller black-hole radius than the Planck-length. So, assuming that the Planck length lP is the smallest possible length then one must expect that ξ ∼ 1.

4. Cosmological constant duality

Let us now introduce two cosmological constants; Λ⁺ for the (1 + 3)-world and Λ^- for the (3 + 1)-world. Thus, one assumes that the gravitation field Equation (8) can be splitted as

and

with the indice μ,ν,... runing from 1 to 4 and the indices A, B runing from 5 to 8. Assuming again (11) and (14) we find that the relevant equation in the (1 + 3)-world will be

with a general solution of the form

with A(ρ) and F(ρ) arbitrary functions of ρ. For the corresponding equation for the (3 + 1)-world one shall have

whose a general solution becomes

with B(r) and G(r) arbitrary functions of r. Again, one may choose A(ρ) and B(r) as A=r02/ρ and B=r02/r, respectively. While a dual solution for F(ρ) and G(r) is obtained by setting F=l-2ρ2 and G=l-2r2, with l a dimensional fundamental constant. An intesting aspect of this construction emerges if one chooses a black-hole horizon such that

Of course this formula leads to a cosmological constant duality of the form

This means that a small cosmological constant Λ⁺ in the (1 + 3)-world must lead to a large cosmological constant Λ^- in the (3 + 1)-world and vice versa, as predicted in Ref. [²⁶].

5. Final remarks

The present work opens many possible physical routes for further work. First, it may be interesting to consider a generalized Kruskal-Szekeres transform of the line element (32). This must lead to a connection with the observation [⁸] that in the (4 + 4)-world such a transform implies 8-regions instead of the usual 4-regions. Second, since it has been shown that in (4 + 4)-dimensions there exist a kind of duality of the cosmological constant one wonders what is the relation of such a duality with dual black-hole solution developed in this work (see Ref. [¹¹]). Finally the quantum relation (40) may motive to see the consequences of our dual black-hole solution with quantum gravity theory. At this respect, it is worth mentioning that oriented matroid theory [¹²] (see also Refs. [¹³-¹⁹] and references therein) and surreal number theory (see Ref. [²⁰] and also Refs [²¹-²⁴] and references therein) are two promising underlaying mathematical structures for dealing with the key dual concept in (4 + 4)-dimensions [¹⁸].