Neutron star interiors and topology change

Quark matter is believed to exist in the center of neutron stars. A combined model consisting of quark matter and ordinary matter is used to show that the extreme conditions existing in the center could result in a topology change, that is, in the formation of wormholes.


Introduction
Wormholes are handles or tunnels in the spacetime topology connecting different parts of our Universe or different universes. The meticulous analysis in Ref. [1] has shown that such wormholes may be actual physical objects permitting two-way passage.
Also introduced in Ref. [1] is the metric for a static spherically symmetric wormhole, namely ds 2 = −e φ(r) dt 2 + e λ(r) dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ), where e λ(r) = 1/(1 − b(r)/r). Here b = b(r) is the shape function and φ = φ(r) is the redshift function, which must be everywhere finite to prevent an event horizon. For the shape function we must have b(r 0 ) = r 0 , where r = r 0 is the radius of the throat of the wormhole; also, b ′ (r 0 ) < 1 and b(r) < r to satisfy the flare-out condition [1]. We are using geometrized units where G = c = 1. While Morris and Thorne concentrated on traversable wormholes suitable for humanoid travelers (and possible construction by an advanced civilization), a more general question is the possible existence of naturally occurring wormholes [2]. It has also been suggested that the extreme conditions inside a massive compact star may result in a topology change [3]. It is proposed in this paper that a neutron star is a candidate for such a topology change as a result of quark matter [4,5,6,7], which is believed to exist at the center of neutron stars [8]. To that end, we will use a two-fluid model consisting of ordinary matter and quark matter based on the MIT bag model [9]. Two cases will be considered: the non-interacting and interacting two-fluid models. (We will concentrate mainly on the former case.) The reason for having two cases is that quark matter is a viable candidate for dark matter [10], albeit under different circumstances, and so is likely to be weakly interacting.
A combined model consisting of neutron-star matter and of a phantom/ghost scalar field yielding a nontrivial topology is discussed in Ref. [11]. The model is referred to as a neutron-star-plus-wormhole system.

Basic equations
For our basic equations we follow Ref. [12]. The energy momentum tensor of the two-fluid model is given by In Eqs. (2) and (3), ρ and p correspond to the respective energy density and pressure of the baryonic matter, while ρ q and p q correspond to the energy density and pressure of the quark matter. The left-hand sides are the effective energy density and pressure, respectively, of the combination. The Einstein field equations are listed next [12]: and recalling that T 1 1 = T 2 2 . In the MIT bag model, the matter equation of state is given by where B is the bag constant, which we take to be 145 MeV/(fm) 3 [9,13]. Since quarks are part of the standard model, we assume that p q > 0; hence ρ q > 4B. For normal matter we use the somewhat idealized equation of state [14] p = mρ, 0 < m < 1.
Since we are assuming the pressure to be isotropic, the conservation equation is [12] d(p effective ) dr

Solutions of basic equations
As noted in the Introduction, we will concentrate mainly on the non-interacting fluid model. This means that the two fluids, consisting of normal matter and quark matter, do not interact. The resulting conservation equations are therefore independent of each other. Using Eqs. (7) and (8), we have and The solutions are ρ = ρ 0 e −φ(1+m)/2m (12) and where ρ 0 and ρ (q,0) are integration constants. Equations (4), (5) anf (6) now give It now becomes apparent that this equation is linear in e −λ once it is written in the following form: The integrating factor is leading at once to To write the solution for e −λ , we first define F (r) = ρ(1 + 3m) + (2ρ q − 4B), which becomes, after substituting the solutions ρ and ρ q , The lower limit r = r 0 is the radius of the throat of the wormhole, as we will see.
We obtain Note especially that b(r 0 ) = r 0 , which characterizes the throat of the wormhole.
To study the other requirement, 0 < b ′ (r 0 ) < 1, we start with Given that φ ′ , the gradient of the redshift function, is related to the tidal force, one would expect its magnitude to be extremely large. So b ′ (r 0 ) will indeed be less than unity if the expression inside the brackets in Eq. (18) is negative and sufficiently small in absolute value. If this is the case, then the flare-out condition is met, that is, Since this indicates a violation of the null energy condition, the flare-out condition is the primary prerequisite for the existence of wormholes [1]. (The condition b(r) < r will be checked later.) Unfortunately, the constants B and ρ (q,0) are also large, so that Eq. (18) must be examined more closely.

The gradient of the redshift function
To analyze Eq. (18) we start with the gradient of φ [15]: where m(r) is the total mass energy inside the radial distance. (This formula is also used in the derivation of the Tolman-Oppenheimer-Volkoff equation.) In geometrized units we get for r = r 0 Since we are now concerned with what is going to become the throat r = r 0 , we need to evaluate ρ 0 and ρ (q,0) in Eqs. (12) and (13) to obtain and We already know that p = mρ and that ρ q > 4B, so that the numerator is indeed negative. For the purpose of comparison, we will simply assume for now that ρ q (r 0 ) = 4B. (See Remark 1 below.) Then Given that 1 2 b(r) is the effective mass inside a sphere of radius r, 1 − 2m(r 0 )/r 0 = 0. So near r = r 0 , the fraction in Eq. (24) is small and negative. Observe also that according to Eq. (20), near r = r 0 , φ ′ is indeed large, as expected.
We conclude that b ′ (r 0 ) < 1. So in the neighborhood of the throat, b(r)/r < 1. To show that this inequality holds away from the throat, observe that Since ρ(r) decreases as r increases, b ′ (r) is positive, but decreasing, so that b(r)/r remains less than unity for r > r 0 . Remark 1: Given that the density of nuclear matter ranges from 6 × 10 17 kg/m 3 to 8 × 10 17 kg/m 3 , we may take ρ q to be about 10 18 kg/m 3 . So ρ q (r 0 ) is quite close to 4B, thereby justifying the assumption ρ q (r 0 ) ≈ 4B.

Completing the wormhole structure
Since r = r 0 is the throat of the wormhole, it follows from the definition of throat that the interior r < r 0 is outside the manifold. Although not part of the wormhole spacetime, it still contributes to the gravitational field. This can be compared to a thin-shell wormhole from a Schwarzschild black hole [16]: while not part of the manifold, the black hole generates the gravitational field.
Since the throat of the wormhole is deep inside the neutron star, it cannot be directly observed. According to Ref. [11], however, there may still be observable tell-tale signs: if two neutron stars are connected by a wormhole, they would have identical, or nearly identical, characteristics. An even stronger indication would be any observed variation in the mass of a neutron star.
Remark 2: The quark-matter core, being surrounded by nuclear matter, raises a question regarding the interface region. As noted in the Introduction, quark matter is believed to be weakly interacting. So the most likely result is a drop in the energy density in the neighborhood of the core's surface.
We conclude with some brief comments on the interacting case. If the two fluids are assumed to interact, then the conservation equations take on the following forms [12,17]: and The quantity Q expresses the interaction between the two types of matter and falls off rapidly as r increases. Furthermore, since the interaction is assumed to be relatively weak, Q is a very small quantity compared to B. So the presence of the quantity Q is going to have little effect.

Conclusion
Quark matter is believed to exist in the center of neutron stars. The analysis in this paper is therefore based on a two-fluid model comprising ordinary and quark matter with an isotropic matter distribution. It is shown that the extreme conditions existing in the center could result in a topology change, that is, a neutron star with a quark-matter core could give rise to a wormhole.