Neutron Star Interiors and Topology Change

Abstract

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.

1. Introduction

Wormholes are handles or tunnels in the spacetime topology connecting different parts of our universe or of different universes. The meticulous analysis in [1] has shown that such wormholes may be actual physical objects permitting two-way passage.

Also introduced in [1] is the metric for a static spherically symmetric wormhole; namely, 𝑑𝑠²=−𝑒^𝜙(𝑟)𝑑𝑡²+𝑒^𝜆(𝑟)𝑑𝑟²+𝑟²𝑑𝜃²+sin²𝜃𝑑𝜑², (1) 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(𝑟₀)=𝑟₀, where 𝑟=𝑟₀ is the radius of the throat of the wormhole and also 𝑏^′(𝑟₀)<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 [4–7], 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 noninteracting 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 class of exact solutions of the Einstein-Maxwell field equations describing a wormhole with an anisotropic matter distribution is discussed in [11].

A combined model consisting of neutron-star matter and of a phantom/ghost scalar field yielding a nontrivial topology is discussed in [12]. The model is referred to as a neutron-star-plus-wormhole system.

2. Basic Equations

For our basic equations we follow [13]. The energy momentum tensor of the two-fluid model is given by 𝑇⁰₀≡𝜌_effective=𝜌+𝜌_𝑞,𝑇¹₁=𝑇²₂≡−𝑝_effective=−𝑝+𝑝_𝑞. (2)(3) In (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 composition.

The Einstein field equations are listed next [13]: 𝑒^−𝜆𝜆^′𝑟−1𝑟²+1𝑟²=8𝜋𝜌+𝜌_𝑞,𝑒^−𝜆𝜙^′𝑟+1𝑟²−1𝑟²=8𝜋𝑝+𝑝_𝑞,𝑒^−𝜆2𝜙^′2−𝜆^′𝜙^′2+𝜙^′−𝜆^′𝑟+𝜙^′′=8𝜋𝑝+𝑝_𝑞, (4)(5)(6) recalling that 𝑇¹₁=𝑇²₂.

In the MIT bag model, the matter equation of state is given by 𝑝_𝑞=13𝜌_𝑞−4𝐵, (7) where 𝐵 is the bag constant, which we take to be 145 MeV/(fm)³ [9, 14]. 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 [11] 𝑝=𝑚𝜌,0<𝑚<1. (8)

Since we are assuming the pressure to be isotropic, the conservation equation is [13] 𝑑𝑝_effective+1𝑑𝑟2𝜙^′𝜌_effective+𝑝_effective=0. (9)

3. Solutions of Basic Equations

As noted in the Introduction, we will concentrate mainly on the noninteracting fluid model. This means that the two fluids, normal matter and quark matter, do not interact. The resulting conservation equations are therefore independent of each other. Using (7) and (8), we have 𝑑𝜌𝑑𝑟+𝜙^′1+𝑚2𝑚𝜌=0,𝑑𝜌_𝑞𝑑𝑟+2𝜙^′𝜌_𝑞−𝐵=0. (10) The solutions are 𝜌=𝜌₀𝑒^{−𝜙(1+𝑚)/2𝑚},𝜌_𝑞=𝐵+𝜌_(𝑞,0)𝑒^−2𝜙, (11)(12) where 𝜌₀ and 𝜌_(𝑞,0) are integration constants. Equations (4), (5), and (6) now give 𝑒^−𝜆−𝜆^′𝜙^′2+𝜙^′22+𝜙^′′+2𝜙^′𝑟=8𝜋𝜌(1+3𝑚)+2𝜌_𝑞−4𝐵. (13) It becomes apparent that this equation is linear in 𝑒^−𝜆 once it is rewritten in the following form: 𝑒^−𝜆′+𝑒^−𝜆𝜙^′+2𝜙^′′𝜙^′+4𝑟=2𝜙^′(8𝜋)𝜌(1+3𝑚)+2𝜌_𝑞−4𝐵. (14) The integrating factor is I.F.=𝑒^{𝜙+2ln𝜙′+4ln𝑟}=𝑒^𝜙𝜙^′2𝑟⁴, (15) leading at once to 𝑑𝑒𝑑𝑟^−𝜆𝑒^𝜙𝜙^′2𝑟⁴=𝑒^𝜙𝜙^′2𝑟⁴2𝜙^′(8𝜋)𝜌(1+3𝑚)+2𝜌_𝑞−4𝐵. (16) 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, 𝐹(𝑟)=𝜌₀𝑒^{−𝜙(1+𝑚)/2𝑚}(1+3𝑚)+2𝜌_(𝑞,0)𝑒^−2𝜙−2𝐵. (17) We now have 𝑒^−𝜆=16𝜋𝑒^𝜙𝜙^′2𝑟^4𝑟𝑟0𝑒^𝜙𝜙^′𝑟⁴₁𝐹𝑟₁𝑑𝑟₁. (18) The lower limit 𝑟=𝑟₀ is the radius of the throat of the wormhole, as we will see.

As noted in Section 1, the shape function 𝑏=𝑏(𝑟) is obtained from e^{−𝜆(𝑟)}, so that 𝑏(𝑟)=𝑟1−𝑒^{−𝜆(𝑟)}. (19) We obtain 𝑏(𝑟)=𝑟1−16𝜋𝑒^𝜙𝜙^′2𝑟^4𝑟𝑟0𝑒^𝜙𝜙^′𝑟⁴₁𝐹𝑟₁𝑑𝑟₁. (20) Note especially that 𝑏(𝑟₀)=𝑟₀, which characterizes the throat of the wormhole.

To study the other requirement, 0<𝑏^′(𝑟₀)<1, we start with 𝑏^′(𝑟)=1−𝑒^{−𝜆(𝑟)}−𝑑+𝑟𝑒𝑑𝑟^{−𝜆(𝑟)}, (21) which leads to 𝑏^′𝑟₀=1+𝑟₀16𝜋𝜙^′𝑟₀−𝜌₀𝑒^{−𝜙(𝑟0)(1+𝑚)/2𝑚}(1+3𝑚)−2𝜌_(𝑞,0)𝑒^{−2𝜙(𝑟0)}+2𝐵. (22) 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^′(𝑟₀) will indeed be less than unity if the expression inside the brackets in (22) is negative and sufficiently small in absolute value. If this is the case, then the flare-out condition is met; that is, 𝑟₀𝑏^′𝑟₀−𝑏𝑟₀2𝑏𝑟₀²<0. (23) Since it indicates a violation of the weak energy condition, the flare-out condition is the primary prerequisite for the existence of wormholes [1]. (The condition 𝑏(𝑟)/𝑟<1 will be checked later.)

Unfortunately, the constants 𝐵 and 𝜌_(𝑞,0) are also large, so that (22) must be examined more closely.

4. The Gradient of the Redshift Function

To analyze (22), we start with the gradient of 𝜙 [15]: 𝑑𝜙=(𝑟)𝑑𝑟2𝐺𝑚𝑐²𝑟²𝐺+8𝜋𝑟𝑐⁴𝑝11−2𝐺𝑚(𝑟)/𝑐²𝑟, (24) where 𝑚(𝑟) is the total mass-energy inside the radial distance 𝑟. (This formula is also used in the derivation of the Tolman-Oppenheimer-Volkoff equation.) In geometrized units we get, for 𝑟=𝑟₀, 𝜙^′𝑟₀=2𝑚𝑟₀/𝑟²₀+8𝜋𝑟₀𝑝𝑟₀1−2𝑚𝑟₀/𝑟₀. (25)

Since we are now concerned with what is going to become the throat at 𝑟=𝑟₀, we need to evaluate 𝜌₀ and 𝜌_(𝑞,0) in (11) and (12) to obtain 𝜌₀=𝜌𝑟₀𝑒^{𝜙(𝑟0)(1+𝑚)/2𝑚},𝜌_(𝑞,0)=𝑒^2𝜙(𝑟0)𝜌_𝑞𝑟₀−𝐵. (26) So (22) becomes 𝑏^′𝑟₀=1+16𝜋𝑟₀−𝜌𝑟₀(1+3𝑚)−2𝜌_𝑞𝑟₀+4𝐵2𝑚𝑟₀/𝑟²₀+8𝜋𝑟₀𝑝𝑟₀/1−2𝑚𝑟₀/𝑟₀. (27) 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(𝑟₀)=4𝐵. Then 𝑏^′𝑟₀𝑟=1+16𝜋−𝜌₀(1+3𝑚)−4𝐵2𝑚𝑟₀/𝑟³₀+8𝜋𝑝𝑟₀/1−2𝑚𝑟₀/𝑟₀. (28) Disregarding 𝑟₀ for the time being, suppose that we determine 𝑚(𝑟₀) so that 2𝑚(𝑟₀) exceeds 16𝜋(4𝐵). We obtain 16𝜋(4𝐵)=16𝜋(4)(145)MeV(fm)³𝐺𝑐⁴m⁻², (29) which leads to 𝑚(𝑟₀)≈10⁻⁸ m. The other term in the denominator, 8𝜋𝑝(𝑟₀), could be close to zero (since 𝑚>0). However, by simply doubling the value of 𝑚(𝑟₀), we can compensate for the normal-matter term −𝜌(𝑟₀)(1+3𝑚) in the numerator, since this term is smaller than the other term. Given that the density of nuclear matter ranges from 6×10¹⁷ kg/m³ to 8×10¹⁷ kg/m³, we may take 𝜌_𝑞 to be about 2×10¹⁸ kg/m³. So in geometrized units, 𝜌_𝑞(𝑟₀) is in fact larger than and reasonably close to 4𝐵, thereby justifying the assumption 𝜌_𝑞(𝑟₀)≈4𝐵.

Remark 1. The assumption 𝜌_𝑞(𝑟₀)≈4𝐵 yields −2𝜌_𝑞(𝑟₀)+4𝐵≈−4𝐵, leading to a negative numerator in (28). It is conceivable that the bag pressure 𝐵 increases with increasing density, but to reverse the sign, the value would have to be more than doubled for a fixed 𝜌_𝑞. More importantly, however, (7) suggests that this increase is likely to be compensated for by a concomitant increase in 𝜌_𝑞.

Next, given the enormous density near the center of a neutron star, 𝑚(𝑟₀)≈10⁻⁸ m would correspond to a fairly small value for 𝑟₀: using 𝜌_𝑞(𝑟₀)≈4𝐵 again, 43𝜋𝑟³₀𝜌_𝑞𝑟₀=10⁻⁸ (30) yields 𝑟₀≈1 m, although larger values would be allowed. So the above estimate holds.

Finally, observe that the other factor in the denominator, 1−2𝑚(𝑟₀)/𝑟₀, is now seen to be very close to unity.

We conclude that 𝑏^′(𝑟₀)<1. So in the neighborhood of the throat, 𝑏(𝑟)/𝑟<1. To show that this inequality holds away from the throat, recall that (1/2)𝑏(𝑟) is the total mass inside a sphere of radius 𝑟. Hence 12𝑏(𝑟)=^𝑟_𝑟04𝜋𝑟²𝜌(𝑟)1𝑑𝑟,2𝑏^′(𝑟)=4𝜋𝑟²𝜌(𝑟). (31) Since 𝜌(𝑟) decreases as 𝑟 increases, 𝑏^′(𝑟) is positive but decreasing, so that 𝑏(𝑟)/𝑟 remains less than unity for 𝑟>𝑟₀.

5. Completing the Wormhole Structure

As we have seen, 𝑚(𝑟₀) is the mass inside the radial distance 𝑟=𝑟₀. Since 𝑟=𝑟₀ is now the throat of the wormhole, it follows from the definition of throat that the interior 𝑟<𝑟₀ 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. This comparison explains why our wormhole requires a minimum throat radius to help produce a sufficiently strong gravitational field.

Since the throat of the wormhole is deep inside the neutron star, it cannot be directly observed. According to [12], 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.

5.1. The Need for Quark Matter

If quark matter is eliminated from the model, then the right-hand side of (13) becomes 8𝜋𝜌(1+3𝑚). As a consequence, the 4𝐵 term in (28) is also eliminated. Since 𝑚>0 and 16>8/3, 16𝜋−𝜌𝑟₀(41+3𝑚)>2×3𝜋𝜌𝑟₀=2𝑚𝑟₀𝑟³₀. (32) So the fraction in (28) exceeds in absolute value the following: −2𝑚𝑟₀/𝑟³₀(1+3𝑚)2𝑚𝑟₀/𝑟³₀+8𝜋𝑚𝜌𝑟₀/1−2𝑚𝑟₀/𝑟₀. (33) Although we are now dealing with nuclear matter (𝜌≈10⁻¹⁰), we still have 1−2𝑚(𝑟₀)/𝑟₀≈1. It is quite evident that for 𝑚 close to zero, the fraction in statement (33) is slightly larger than unity in absolute value; hence 𝑏^′(𝑟₀) is negative, which is absurd. As 𝑚 increases, 𝑏^′(𝑟₀) only decreases further, so that the flare-out condition cannot be met.

It now becomes apparent that our conclusion depends critically on the fact that 𝐵 is a constant: increase the throat size until 2𝑚(𝑟₀) surpasses 16𝜋(4𝐵). So based on our model, a topology change is possible only if there is a sufficient amount of quark matter in the center. More precisely, the assumption 𝜌(𝑟₀)≈4𝐵 implies that the quark matter has to extend at least to 𝑟=𝑟₀.

5.2. The Interacting Case

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 [13, 17]: 𝑑𝜌𝑑𝑟+𝜙^′1+𝑚2𝑚𝜌=𝑄,𝑑𝜌_𝑞𝑑𝑟+2𝜙^′𝜌_𝑞−𝐵=−3𝑄. (34) The quantity 𝑄 expresses the interaction between the two types of matter and falls off rapidly as 𝑟→∞. Furthermore, since the interaction is assumed to be relatively weak, 𝑄 is a very small quantity compared to 𝐵. So in comparing 𝑚(𝑟₀) and 𝜌_𝑞(𝑟₀) in the expression for 𝑏^′(𝑟₀), the presence of the quantity 𝑄 is going to have little effect.

6. 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 may result in a topology change; that is, for certain choices of the matter content in the Einstein field equations, a neutron star could give rise to a wormhole.