Traversable wormholes and the Brouwer fixed-point theorem

The Brouwer fixed-point theorem in topology states that for any continuous mapping $f$ on a compact convex set into itself admits a fixed point, i.e., a point $x_0$ such that $f(x_0)=x_0$. Under certain conditions, this fixed point corresponds to the throat of a traversable wormhole, i.e., $b(r_0)=r_0$ for the shape function $b=b(r)$. The possible existence of wormholes can therefore be deduced from purely mathematical considerations without going beyond the existing physical requirements.


Introduction
Wormholes are handles or tunnels in spacetime connecting widely separated regions of our Universe or entirely different universes. Morris and Thorne [1] proposed the following line element for the wormhole spacetime: using units in which c = G = 1. Here Φ = Φ(r) is called the redshift function, which must be finite everywhere to prevent the appearance of an event horizon. The function b = b(r) is called the shape function since it determines the spatial shape of the wormhole when viewed, for example, in an embedding diagram [1]. The spherical surface r = r 0 is the throat of the wormhole and is characterized by the following condition: b(r 0 ) = r 0 . Mathematically speaking, r = r 0 is called a fixed point of the function b = b(r) and will play a key role in our discussion. For a traversable wormhole, an important requirement is the flare-out condition b ′ (r 0 ) < 1; also, b(r) < r for r > r 0 . The flare-out condition can only be met by violating the null energy condition (NEC), which states that for all null vectors k α , where T αβ is the stress-energy tensor. Matter that violates the NEC is called "exotic" in Ref. [1]. In particular, for the outgoing null vector (1, 1, 0, 0), the violation has the form T αβ k α k β = ρ + p r < 0.
Here T t t = −ρ is the energy density, T r r = p r is the radial pressure, and T θ θ = T φ φ = p t is the lateral pressure. For completeness, let us also list the Einstein field equations: and .
The purpose of this paper is to make use of fixed-point theory to show that certain physical conditions imply the possible existence of traversable wormholes. To that end, we need the following special case of the Brouwer fixed-point theorem: Theorem [2]. Let f be a continuous function from a closed interval [a, b] on the real line into itself. Then f has a fixed point, i.e., there is a point x 0 such that f (x 0 ) = x 0 .
A function that maps a set into itself is called a self-mapping.

Some consequences of the Brouwer fixed-point theorem
Accrding to Ref. [3], the total mass-energy M of an isolated star is well defined as long as one retains spherical symmetry. In Schwarzschild coordinates, where ρ(r) is the energy density. Moreover, everywhere outside the star, m(r) = M ≡ total mass-energy in the Newtonian limit.
We also have from Eq. (4) that The line element (page 608 in Ref. [3]) is given by where R is the radius of the spherical star. According to Eqs. (7) and (9), b(r) = 2m(r). With our wormhole spacetime in mind, a more convenient form of the line element is and where R is the radius of the star. Now m(r) corresponds to b(r) in line element (1).
To apply the Brouwer fixed-point theorem, suppose we let r = r 1 be a spherical core, i.e., 0 < r 1 < R. If ρ(r) is the mass density of the star for r > r 1 , then the mass becomes Since m * (r 1 ) = 0, the mapping is not a self-mapping. So consider a new mapping where K is a constant slightly bigger than r 1 . So m(r 1 ) = K. Since r = R be the radius of the star, we now draw the important conclusion that m(r) in Eq. (13) maps the closed interval [r 1 , R] into itself, i.e., where provided that R is large enough so that R * ≤ R. This conclusion can be illustrated graphically, as shown in Fig. 1.
We also have m ′ (r) = 4πr 2 ρ(r) ≪ 1, By Eq. (16), b ′ (r 0 ) = m ′ (r 0 ) < 1, so that the flare-out condition is satisfied. In the resulting wormhole spacetime, the region inside the throat r = r 0 is not part of the wormhole, but this region still contributes to the gravitational field. This can be compared to a thin-shell wormhole resulting from a Schwarzschild black hole [4]: while not part of the manifold, the black hole generates the underlying gravitational field.
As actual (quantified) example of the type of wormhole discussed is given in Ref. [5]. It is shown that for a typical neutron star, the possible formation of a wormhole requires a core of quark matter that is approximately 1 m in radius. (Quark matter is believed to exist at the center of neutron stars [6].)

A dark-matter background
In the discussion of dark matter, several models have been proposed for the energy density. The best-known of these is the Navarro-Frenk-White model [7] ρ(r) = ρ s r rs 1 + r rs 2 , where r s is the characteristic scale radius and ρ s is the corresponding density. The Universal Rotation Curve [8] is given by where r c is the core radius of the galaxy and ρ c is the central halo density. Another example is the King model whose energy density is given by [9] ρ(r) = κ where η, κ, r 0 , and λ are constants. All of these models have a low energy density. So if r 1 is the radius of a star, then the star itself becomes the core. Furthermore, since there is no outer boundary, we can choose R large enough so that m(r) in Eq. (13) is a self-mapping. The existence of a fixed point now implies the possible existence of a wormhole in the dark-matter region. This conclusion is independent of the dark-matter model chosen.
Remark: The form of b(r) can be obtained from Eq. (9) in conjunction with one of the above choices of ρ(r). The determination of the integration constant requites an extra condition, however: a physically viable assumption is the usual b(r 0 ) = r 0 . (See, for example, Ref. [10].)

Conclusion
If r = r 1 denotes the radius of the core of the star, then m(r), the effective mass of the star for r > r 1 , is given by Eq. (13). The function m(r) satisfies the hypothesis of the Brouwer fixed-point theorem. The fixed point can be viewed as the radius of the throat of a traversable wormhole since b(r 0 ) = m(r 0 ) = r 0 and b ′ (r 0 ) = m ′ (r 0 ) < 1, thereby satisfying the flare-out condition. This result can also be applied to a dark-matter setting by treating a star of radius r = r 1 as the core. So the possible existence of traversable wormholes follows directly from purely mathematical considerations without going beyond the physical requirements already in place.