Quantum Stabilization of General-Relativistic Variable-Density Degenerate Stars

Trinity College Trinity College Digital Repository

Faculty Scholarship

2012

David Cox Trinity College

Ronald Mallett

Mark P. Silverman Trinity College, [email protected]

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Part of the Physics Commons Journal of Modern Physics, 2012, 3, 561-569 doi:10.4236/jmp.2012.37077 Published Online July 2012 (http://www.SciRP.org/journal/jmp)

Quantum Stabilization of General-Relativistic Variable-Density Degenerate Stars

David Eric Cox1, Ronald L. Mallett1, M. P. Silverman2 1Department of Physics, University of Connecticut, Storrs, USA 2Department of Physics, Trinity College, Hartford, USA Email: [email protected]

Received April 23, 2012; revised May 18, 2012; accepted June 3, 2012

ABSTRACT Research by one of the authors suggested that the critical mass of constant-density neutron stars will be greater than eight solar masses when the majority of their neutrons group into bosons that form a Bose-Einstein condensate, provided the bosons interact with each other and have scattering lengths on the order of a picometer. That analysis was able to use Newtonian theory for the condensate with scattering lengths on this order, but general relativity provides a more fundamental analysis. In this paper, we determine the equilibrium states of a static, spherically-symmetric variable-density mixture of a degenerate gas of noninteracting neutrons and a Bose-Einstein condensate using general relativity. We use a Klein-Gordan Lagrangian density with a Gross-Pitaevskii term for the condensate and an effective field for the neutrons. We show that a new class of compact stars can exist with masses above the Oppenheimer-Volkoff limit, provided the scattering length of the bosons is large enough. These stars have no internal singularities, obey causality, and demonstrate a quantum mechanism consistent with general relativity that could prevent collapsed stars from becoming black holes.

Keywords: Gravity; General Relativity; Neutron Stars; Black Holes; Bose-Einstein Condensates; Numerical Physics

1. Introduction ation and neutron interaction approximately quadrupled the critical mass of an OV star. Assuming it would have The general theory of relativity (GR) predicts that a suf- the same effect on the stars we are about to discuss (and ficiently-massive non-rotating electrically-neutral neu- confirming this is the subject of future inquiries), then if tron star modeled as a noninteracting degenerate gas will the modified critical mass is above 12 M⊙, we would not collapse to a geometric point within an event horizon: a expect the formation of stellar black holes. While this black hole [1]. The critical mass for these Oppen- would not prevent the formation of intermediate and heimer-Volkoff (OV) stars is 0.71 M⊙. Taking rotation super-massive black holes, it would remove one category and particle interaction into account raises the critical of unsettling astronomical objects. mass to at best 3 M⊙ [2]. Surveys of stellar black hole One of the authors proposed a mechanism that lowers candidates with both low- and high-mass companions stellar pressure [5]. If even-numbered groups of neutrons suggest maximum masses of 45 M⊙ [3,4]. The prediction condense into spin-0 bosons in a manner analogous to that massive stars must collapse to nonphysical end states Cooper pairing, and if these bosons formed a Bose- is a consequence of GR’s classical limits. Presumably, Einstein condensate instead of a superfluid, then the once the star has collapsed to the order of the Planck condensate would exert less pressure than the remaining −35 length (1.62·10 m), quantum effects will dominate, neutrons, reducing the star’s gravitational field enough to preventing further collapse. Although this assumption ensure stability. Note that this is a different process than avoids a singularity, the resulting prediction of objects previous models which have shown that pion or kaon with 1057 particles compressed into a 10−105 m3 volume is condensates increase pressure. The presence of these unpalatable to many. exotic Bose-Einstein condensates, which reduce the However, if there is a natural process that consistently critical mass, have been ruled out by recent measure- prevents massive stars from collapsing, then GR is not ments of neutron star masses on the order of two solar inconsistent with physically-sensible expectations. Pres- masses [6]. These measurements do not rule out the sure is a gravitational source in GR, so lowering the possibility of Bose-Einstein condensates consisting of star’s pressure would raise its critical mass. Adding rot- groups of neutrons, provided such condensates increase

Copyright © 2012 SciRes. JMP 562 D. E. COX ET AL. neutron-star critical mass. we remove the restrictions of the previous investigation The proposed model had s neutrons condense into and study variable-density stars using GR. bosons with masses, The fermicon model should not be confused with analyses that have examined effects of an external scalar mQsmb =, (1) field on the critical mass of a neutron star. Such research where m is the neutron mass and Q reflects the is motivated, for example, by the possibility that scalar strength of the neutron-neutron interaction that produces fields can represent dark matter or other exotic quantities the bosons. The remaining neutrons sink to the star’s and explore the consequences of their presence on stellar center while the condensate forms a halo. The two stability. Although scalar fields are present in our models, materials are in hydrostatic and chemical equilibrium. they represent quantities intrinsic to the star, which is This model used two simplifying assumptions. One is assumed to occupy empty space. Furthermore, these that each material has a constant density and the other is fields are the result of neutron grouping to a Bose- that the gravitational field is weak enough to use the Einstein condensate. The grouping of neutrons into Newtonian equation for hydrostatic equilibrium instead superfluid states is believed to occur within neutron stars of the TOV equation. The results for bosons that interact [6], and the possibility that superfluid states can change with each other with scattering lengths on the order of a to Bose-Einstein condensates has been established in picometer justified these assumptions. terrestrial laboratories [11]. As a result, the experimental For chemical equilibrium, basis of our models is more solidly established.

sn= b (2) Our general-relativistic model qualitatively differs must hold true at the boundary between the neutrons and from the original one in two aspects. First, the original the condensate. The chemical potential of a Bose-Eins- model, which gave solutions for what we term type I tein condensate, deduced from low-temperature experi- fermicon stars, assumed that the condensate and re- ments on Bose gases, takes the form, maining neutrons physically separate with the neutrons occupying the core (an “oil-and-water” model), while our 22 bb=14mc π bb an , (3) general-relativistic model, which gives solutions for what we call type II fermicon stars, assumes that the con- with reduced Compton length,    mc, scattering bb densate and remaining neutrons are mixed together (a length, a , and number density, nb [7-10]. Figure 1 shows some of the end-states, called fermicon “salt-water” model). Solutions existed for type I fer- (fermions and a boson condensate) stars. In the Qs =2.2, micon stars because they are two-body problems in Newtonian theory. In GR, the gravitational field counts a =100 am case, there is a critical mass of 3.19 M⊙ as a “body” since it can feel its own effects and the type I compared to the OV critical mass of 0.71 M⊙. The critical masses for fermicon stars with scattering lengths of ±1 fermicon model becomes a three-body problem. The “salt-water” model of type II fermicon stars simplifies pm is greater than 8 M⊙, although it is difficult to say how much greater from the obtainable solutions. the problem back to a two-body (star plus field) one. Unfortunately, the a =100 am case is suspect Second, type I fermicon stars included those whose because the TOV equation would be more appropriate bosons attracted each other. Our initial investigations of since the radii of the stars are comparably close to the self-interacting boson stars using GR suggest that event horizons they would have had if they collapsed to mutually-attractive bosons reduce the critical mass, so black holes (stars with the critical mass have radii only type II fermicon stars only have bosons with repulsive 6% greater than their Schwarzschild radii). In this article self-interaction ( a >0) formed from exothermic grouping ( Q <1). Neutrons with the same momentum eigenstates can pair. These pairs do not need to be adjacent to each other in physical space. The Pauli exclusion principle does not allow more than two neutrons (one spin up and one spin down) to have the same momentum eigenstates, so the majority of neutrons in a system have high-momentum eigenvalues and correspondingly high energies. If the pairs behave like spin-0 bosons, then these bosons can all occupy a ground-state energy, E , with a corresponding ground-state momentum eigenvalue,

pEcm=.2  c2 (4) Figure 1. Mass versus radius in type I fermicon stars. ground b

All of the neutrons which have momentum eigen- is = nbG  . values higher than pground can exist with lower energy Application of the action principle produces the as bosons and find it energetically favorable to associate necessary quantities and relationships. Given the action into bosons. The neutrons with momentum eigenvalues operators, less than p would have to expend energy to ground I=cgx14  d and (15) associate, so they do not become bosons. The details  must wait for a better understanding of the physics of 14 I=cgx  d , (16) high density nuclear material, so that Q , s , and a PP are presently adjustable parameters in our model. the relationship,

μν 4 2. Field Equations and Equations of State 2IcggxP =T μν  d, (17) Consider a neutron star that condenses into a type II defines the energy-momentum tensor operator [14]. The fermicon star. Groups of s neutrons associate into statement, bosons. A spinor operator,  , represents the remaining neutrons. Curved-space Dirac matrices that obey [12], 2I=cgx  ††4 d=0, (18) μν,=2g μν1, (5)  defines the field equation of motion for the condensate. define spin connection matrices and the differential The Einstein field equation follows from operators 2I=cgggx    μν μν d=04 . (19) 1 ν and D . (6) μν4 ;μμμμ The current density operator of the bosons is [13] A scalar field operator,  , represents the Bose-Eins-   tein condensate. Let 0 be the boson vacuum state and μ i bb† † j=   . (20) a and a be the ground-state creation and annihilation b c † μ μ  operators, respectively, for the bosons such that [13]   † From Equation (18), the condensate field equation is a=NNNbbb 11, (7) 2;μ 2 2 † ;μ Qs4π a 0 (21) a=NNNbbb 1, (8)

† Nb The line element within the star is NNbb!=a0 , (9) 222 22222 2 d=sgrctgrrr00   d11   d d rsin  d, N=aa,and† (10) b (22) † a,a =1. (11) Substituting Equation (22) and the separation of vari- ables, The gravitational field depends on the Ricci tensor, μ RreiEt/ aa†  8π , (23) Rμν , and the curvature scalar, Rμ .    Given these relationships, the Lagrangian density into (21), multiplying the result on the right by aa†  operators for the neutrons, condensate, and gravitational and then taking the mean gives field are 1 11 μν 2 RggrR=ln ln2  n =DDicg μν μ ν mc , (12) 11 00 2  22 2 2 3 22 2 2 2 μν ††1 2  gQsNaRE11  2 b  mcg00 (24) b =2 gQμνsmQ smc   2 (13) 2 πmc22 a††  Qs ,and  R  ,

 =2Rμ 1  2 , (14) where a prime indicates a derivative with respect to r . G μ   Following Colpi, Shapiro, and Wasserman [15], the respectively, where  is the neutron reduced Compton identities, length, and  2 is the relativistic gravitational constant, 24 ˆ ˆ 32   8πGc , and 1 is the unit operator in spinor space. rrmm PP a and RmERamc , (25) The Lagrangian density operator for the particles is make Equation (24) dimensionless and show that a Pn=  b and the total Lagrangian density operator

Copyright © 2012 SciRes. JMP 564 D. E. COX ET AL. scattering length at least as large as the neutron reduced 123 PT===12 T T3. Ruffini and Bonazzola Compton length produces a last term at least 2 30 25 3 mmP  10 times larger than the first term and the showed that, as long as   6.56 10 kg/m , the terms left side of the equation. Neglecting these gives with  behave like a perfect fluid and reduce to the

2 form [16], RE=2 12mcgQsNa 2  2 3. (26) 00 b   =sinhan0  ff  d (34) Evaluating Equation (19) and applying the semi- 1 2 1 classical approximation, we obtain Pc=sinh8sinh30  f ff, (35) 3 2 1  2 RRgT = . (27) where  =5.7210 17 kg/m3. The maximum density of μν2  μν μν 0 a stable OV star easily meets Ruffini and Bonazzola’s Substituting Equation (22) into (27) gives conditions (it is 410 18 kg/m3), so these reductions are 21 g00 =1 rg00 g 11 T 1 g 00 g 11 r . (28) valid. We consider f , the exact form of which we obtain g =1ggrrgT 22 0. (29) 11 11 11 11 0 shortly, to be the effective field that describes a large Evaluating Equation (17) and taking the mean gives collection of neutrons. Substituting Equations (23), (34)

000 and (35) into (30) through (33) gives Ticg00=DD 0 0 0 T02=32 N mcπ Qs 22† 0b  2Qsmc g00 2  2 22 2 (30)  4 Qs E mc g00 R 22†† 1    2Qsmg11  Qsmc   2 43g RNaRc 2 24 2 sinh ff. πmc22 a††  Qs . 11 b 0 (36) 111 Ticg=DD  12 11111  TNmcQs1b=32 π  2 †  2  2 22 2   2Qsmg11   4 Qs E mc g00 R (31)   1 (37) 222†† 2 24  2Qsmc g00  Qsmc  43gR NaR 2 11 b  22 †† πmc a  Qs . 112  0cfsinh 8sinh ff 3 . 32 22 Ticr22=DD 222 2311 2 TT23== 0 c sinh8sinh3 f ff 2 †  32   2Qsmg11 (32) 2  Nmcb 32π Qs 222††1 (38)  2Qsmc g00  Qsmc  2   2 22 2 2 4 Qs E mc g R   00 ††  πmc22 a  Qs .  43.gR 2 NaR24 322 11 b  Ticr32222=DD sin   Substituting Equations (25) gives dimensionless forms 2 †     2Qsmg11 showing that the term with R is negligible because it is 15 (33) mmP  10 times the other terms. Neglecting this term 222†1 †and substituting Equation (26) gives  2Qsmc g00  Qsmc  2 02 TcQsa=4π3 †† 00 πmc22 a Qs . 2 Qs2 3 E22 mc g Dots indicate derivatives with respect to time. We will 00 now show that this energy-momentum density behaves 222 2 2 like a perfect fluid. The density of a perfect fluid is Emcg00 Qsc 0 sinh ff .  cT20= and its pressure is 0 (39)

123 cally defines the effective neutron field; Equation (2) TTT123== (40) must hold for every point within the star. 2 The chemical potential of a degenerate ideal gas in flat =4 π0cQsa3 2 spacetime is equal to its Fermi energy [17] and the 222 2 presence of a gravitational field does not change this Emcg00  Qs (41) relationship [18]. The Fermi momentum and f are

112 related by 0cfsinh 8sinh ff 3 . 32 pmcfF =sinh. (47) The only two diagonal energy-momentum tensor com- Combining this with the relativistic relationship, 2 ponents we need are T 0 and T 1 since the star's 22 2 0 1 mc= EFF  p c , gives pressure is now isotropic. 2 Substituting Equations (39) and (41) into (28) and (29) n =coshmc f . (48) gives The chemical potential of the condensate is 1 222  22 g00=8 rg 00 g 11 mcπ Qs a =14mcπ a n . 3  bb  bb (49) 2 222 2 Substituting Equations (20) with the index  =0, Emcg00  Qs  (42) (23), (26), (48), and (49) into (2) gives 1   cf2 sinh 8sinh ff 3 1 2 0  fQEsmcg=23cosh  00  2   (50) ggr1. 222 2  00 11 Emcg00  Qs.  grgmcQsa =222 24π 2 11 11   The inverse hyperbolic cosine is not defined when its 2 parameter is less than one. As r increases, so will g 2 22 00 Qs3 E mc g 2 00  22 (43) until g00 = EQsmc at the star’s radius. This would 2 Emcg22 Qs2 give 00  1 2 fa star  =ifcosh Q Q1. (51) 011cffggsinh  111r . Since Q <1, there must be a radial distance, a< a, such that fa =0. As a result, type II 3. Boundary Conditions and Condition of core star  core  Chemical Equilibrium fermicon stars share a feature with type I fermicon stars: each has a Bose-Einstein condensate halo. The difference 2 22 Beyond the radial distance where g00 = EQsmc, is that type I fermicon stars have a purely neutron core the condensate density becomes negligible. We will show while type II fermicon stars have a core that is a mixture that this is also the star's radius, astar . of neutrons and a Bose-Einstein condensate. Since our At the star’s radius, the interior solutions for g00 and model requires a mixture of the neutrons and the g11 must match the vacuum Schwarzschild solution. In condensate, this is an indication of its limits. Neverthe- particular, in the limit as r approaches astar from the less, we expect results which have negligible halos or left, halos that dominate the star to be valid.

22 lim gMcaga00=1  4π star=1 11 star . (44)  4. Macroscopic Quantities ra star Oppenheimer and Volkoff showed that the enclosed Solving Equation (45) for the enclosed mass gives mass is related to the metric by [1] Mr=4π  22 c r11 g . (52)    11  22 g11 =4πrMrcr    4π . (45) The density and pressure are Substituting M  =4πr 2  and using l’Hôpital’s rule 2 gives =4π 33Qsa Qs2  E22 mc g 00 0 lim g11 =1. (46) (53) r0 2 Emcg22 Qs2  sinh ff . The condition for chemical equilibrium mathemati- 00 0

2 2222 2 dP P=4π00 c3 Qsa E mc g0  Qs v =. (55)  d (54) 112  We can find this speed, since Equations (53) and (54) 0cfsinh 8sinh ff 3 . 32 give expressions for the density and pressure. The maximum speed of sound for a type II fermicon star with 5. Solutions a scattering length of a femtometer is 0.55c . The solutions obey causality. We found numerical solutions for the differential equa- Figure 3 shows another way to compare the two sets tions of the metric components, Equations (42) and (43), of solutions in Figure 2 by plotting mass-vs-core density. by providing a value for g at the center of the star and 00 OV and fermicon stars with densities greater than 4.01 searching for a boson ground-state energy, E , that kg/μm3 and 18.3 kg/μm3, respectively, are unstable. To satisfied the limiting condition, Equation (44), for g 00 see why the fermicon stars have higher densities than OV at the outer border. We abandoned solutions with masses stars, consider the asymptotic behavior of fermicon less than 0.1 M⊙ since neutrons decay into protons, density and pressure as the scattering length gets very electrons, and antineutrinos in these cases. Stellar radii small or very large: are measured using the Schwarzschild radial coordinate   for comparison with the event horizon the star would a  0 have had if it collapsed. 2 4π 33Qsa Qs2  E22 mc g  (56) In both cases, there is a maximum mass and a range of 000 masses where two or more radii exist. Oppenheimer and 2 Emcg22 Qsa2 1. Volkoff showed that the equilibrium solution with a 00 lower density is the stable one and the one with the higher density is unstable to perturbations [1]. sinhf f  . (57) a  0 For example, Figure 2 compares solutions for OV stars and for type II fermicon stars whose bosons have a 2 PcQ 4π0 3 sa repulsive scattering length of one femtometer and that are a  0 2 (58) 2 formed from “strongly-bound” ( Q =0.9) pairs ( s =2) of Emcg22 Qsa2 1. 00  neutrons. The critical masses are 0.710 M⊙ for OV and  0.428 M⊙ for fermicon stars. OV and fermicon stars with 112  radii less than 9.11 km and 4.10 km, respectively, are Pcff0 sinh 8sinh 3f . (59) a  32 unstable. Note the loop at the far left of the fermicon star  plot, which indicates that three equilibrium states exist We found that the boson ground-state energy stays between 0.241 M⊙ and 0.283 M⊙. For each mass in this close to the rest energy of a boson and that g00 does not range, only the state with the largest radius is stable. vary much as initial parameters are changed. All else The speed of sound in a medium is being equal, the density and pressure of fermicon stars with small scattering lengths get larger as the scattering length gets smaller. Recall that the motivation for exploring fermicon stars was that we sought equations of

Figure 2. Q = 0.9, s = 2, a = 1 fm solutions compared with OV solutions. Figure 3. Q = 0.9, s = 2, a = 1 fm mass-vs.-core density.

Copyright © 2012 SciRes. JMP D. E. COX ET AL. 567 state with lower pressure to reduce the gravitational field pm. It only includes the stable solutions for the a =10 and increase stability. We therefore would expect low- pm and a =20 pm stars because the time to find the scattering length fermicon stars to have lower critical unstable solutions was prohibitive. masses than OV stars. Although the a =1 pm type II fermicon stars have In contrast, fermicon stars with large scattering lengths masses higher than OV stars, they are less massive than have expressions for density and pressure identical to type I stars. Type I fermicon stars have higher critical those of OV stars. However, f , which represents the masses because they have constant density. The critical presence of neutrons, is smaller in a fermicon star than in mass for a star with a constant density of  is [19] an OV star because some of the neutrons have condensed 33 3 into bosons. We therefore would expect critical mass to Mc=4 243π G=3.6 M⊙  kgμm. (60) increase as scattering length increases and sufficiently- 3 For  =4 kg/μm , this is a maximum mass of 1.8 M , large scattering lengths to produce more massive stars ⊙ which is 2.6 times larger than an OV star’s critical mass than OV stars. of 0.71 M⊙ at the same core density. These expectations are confirmed in Figures 4 and 5, A type II fermicon star with Q =0.9, s =2, and which show mass-vs-radius graphs for fermicon stars a =20 pm has a critical mass of 13.6 M⊙. Adding with Q =0.9, s =2, and scattering lengths that vary neutron self-interaction and rotation to the OV star model from a femtometer to 20 picometers. Figure 4 includes quadruples its critical mass. If the same increase applies the plot for OV stars. Figure 5 includes the solutions for to type II fermicon stars, then pairing of neutrons into a type I fermicon stars with Q =0.9, s =2, and a =1 Bose-Einstein condensate within neutron stars could

raise their critical masses to about 50 M⊙, preventing collapse to stellar black holes. The maximum speed of sound for the stable solutions is 0.15c ; as the solutions become less dense, the speed of sound decreases. Returning to type II fermicon stars with one-femtometer scattering lengths, Figure 6 shows the relationship between the star’s radius and its core density. Type II fermicon stars have cores consisting of a mixture of neutrons and condensate surrounded by a halo of condensate. The thickness of this halo diminishes as the core density rises. The fermicon star with a critical mass has a 4.10 km radius, 334 meters of which is the condensate halo. The Schwarzschild radius of a black hole with this star’s mass is 1.26 km. Figure 7 shows the relationship between the core den- Figure 4. Type II fermicon stars with femptometer-scale sity and radius for type II fermicon stars with a scattering scattering. All of the fermicon solutions have Q = 0.9 and s = length of a picometer. These stars have larger radii 2. consistent with their lower densities. Their radii are about 200 times larger than their Schwarzschild radii. One feature that is difficult to see in Figure 7 is that the low-density stars are purely condensate and that the

Figure 5. Type II fermicon stars with picometer-scale scattering. All solutions have Q = 0.9 and s = 2. Figure 6. Q = 0.9, s = 2, a = 1 fm radius-vs.-core density.

Copyright © 2012 SciRes. JMP 568 D. E. COX ET AL. stable stars that have neutrons are still dominated in volume by the condensate. Figure 8 shows radii for densities up to 0.00018 kg/μm3. Below 0.000572 kg/μm3, essentially all of the neutrons group into the condensate (more accurately, from the precision of our solutions, these stars have a core with a radius less than a milli- meter). A critical-mass picometer-scaled fermicon star has a core with a radius of nine kilometers and a halo 194 kilometers thick. Type II fermicon stars with scattering lengths com- parable to their reduced Compton lengths do not have the opportunity to condense into esentially-pure condensate stars since such stars have much lower masses (the Figure 9. Q = 0.8, s = 2, a = 1 fm mass-vs.-core density. neutrons in a star below 0.1 M⊙ will decay). If the neutrons pair into bosons with a stronger binding (e.g., Q =0.8), the boson rest energy will be smaller. As Figures 9 and 10 show, these stars are slightly more massive at lower densities than stars with weaker binding, condense completely into condensate stars at low densities, and are slightly larger than stars with weaker binding. Neutrons may also find it energetically favorable to associate in groups of four (or higher even numbers) into bosons. Figures 11 through 12 show features of type II fermicon stars formed when four neutrons strongly group into a single boson ( Q =0.9 and s =4) and these bosons have a repulsive interaction with a scattering length Figure 10. Q = 0.8, s = 2, a = 1 fm radius-vs.-core density. of one femtometer. These stars have lower critical masses 3 at higher densities (0.153 M⊙ at 151 kg/μm ) than their

Figure 11. Q = 0.9, s = 4, a = 1 fm mass-vs.-core density. Figure 7. Q = 0.9, s = 2, a = 1 pm radius-vs.-core density.

Figure 8. Q = 0.9, s = 2, a = 1 pm radii at low densities. Figure 12. Q = 0.9, s = 4, a = 1 fm radius-vs.-core density.

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