Embed Size (px)
Citation preview
1
General Relativistic Non-Neutral White Dwarf Stars
Erik Amorim1 and Parker Hund2
1Universitat zu Koln, Institut fur Mathematik, Weyertal 86-90, D - 50931, Cologne, Germany2Department of Mathematics, Rutgers University,
110 Frelinghuysen Rd., Piscataway, NJ 08854, USA∗
We generalize the recent Newtonian two-component charged fluid models for white dwarf stars ofKrivoruchenko, Nadyozhin and Yudin and of Hund and Kiessling to the context of general relativity.We compare the equations and numerical solutions of these models. We extend to the generalrelativistic setting the non-neutrality results and bounds on the stellar charge obtained by Hundand Kiessling.
I. INTRODUCTION
In [1], a two-component system of cold fluids in hydro-static equilibrium with polytropic pressure density rela-tion under Newtonian gravity was introduced and stud-ied. This system is formed of electrons and ions and wasfirst written
µi +miφG + ZeφE = const , (1)
µe +meφG − eφE = const , (2)
where µi is the chemical potential of the ions, µe is thechemical potential of the electrons, φG is the gravita-tional potential, φE is the electric potential, mi and me
are the masses of the ion particles and electrons, and Zeand −e are their charges. To close the system, a poly-tropic equation of state was taken. The usual assumption[6] of local neutrality was not made, and this led to solu-tions which are not globally neutral.
[2] was interested in applying this model to whitedwarfs, in particular using the nonrelativistic ‘5/3’ poly-tropic pressure law derived from Fermi-Dirac statistics.However, they noted that, as the heavier ions in a whitedwarf are bosons, such a pressure law does not applyto them. [2] subsequently studied in greater detail thetwo-fluid model consisting of only electrons and protons,noting that such a model would be an approximation to asmall range of ‘brown dwarf’ stars. In that paper, it wasnoted that, for a wide range of pressure laws including the‘5/3’ and special relativistic laws discussed in [6], thereis a simple set of bounds on the total amount of chargesuch a star can hold in its ground state configuration. InCGS units, this is
1− Gm2p
e2
1 +Gmemp
e2
≤ NeNp≤
1 +Gmemp
e2
1− Gm2e
e2
, (3)
where Ne and Np are the total number of electrons andprotons, and G is Newton’s gravitational constant. Seealso [21] for more details on these bounds. It was spec-ulated that such bounds would also hold in the generalrelativistic case.
In the present paper, we determine the general rela-tivistic version of these bounds, as well as present the
∗ ©(2021) The authors.
general relativistic analogue of system (1) and (2). Todo this, we apply the framework developed by Olson andBailyn in a series of papers [3], [4], and [5]. However, inthe course of our study, we also noticed that an algebraerror had been made in [5], the work most directly re-lated to ours. Subsequently, the system presented thereis incorrect, although it appears the correct system wasused for the numerical computations. In section II wereview this framework and present the corrected systemof equations, in section III compare this system to thespecial relativistic version of (1) and (2), in section IVwe extend the charge bounds results of [2], in section Vwe present the result of numerical calculations, in sectionVI we discuss how these results apply to neutron stars,and in section VII we conclude.
Aside from being explicitly in the vein of [1], [2], [3],[4], and [5], this paper can also be related to other lines ofresearch. There is a large literature on interior solutionsto the exterior Reissner-Nordstrom metric. Much of thiswork focuses on exact solutions, for example, [8], [9], [10],[11], [12], [13], [14], [15], [16], [17], these solutions oftenfound by “electrifying” a noncharged solution. [7] studiesthe effects of charge on aspects like the mass-radius rela-tion of stellar configurations, and even has a version of (3)in their equation (24). Their model is however a single-fluid model where the charge density is proportional tothe energy density, and this allows them to construct con-figurations with very large amounts of charge. [18] un-dertakes a similar study for strange stars, although theyuse a Gaussian for their charge distribution. [19] and[20] investigate a single fluid model with relativistic andnonrelativistic polytropic equations of state and chargedensity proportional to energy density, focusing on howthe charge interacts with properties such as the Buch-dahl, quasiblack hole, and Oppenheimer-Volkov limits.
[22] confirms the impossibility of local charge neutral-ity found in [3], as well as producing more numerical re-sults describing the solution of the equilibrium equations.This was part of a series of papers including [23], [24], and[25] which built models of white dwarfs or neutron starsusing what they termed “generalized Fermi energies”, or“Klein potentials”. The models were taken to be globallyneutral, but locally nonneutral.
arX
iv:2
110.
0145
7v1
[gr
-qc]
4 O
ct 2
021
2
II. THE OLSON-BAILYN APPROACH
In [3], it was noted that, in a multicomponent Einstein-Maxwell system of spherically symmetric fluids, there isone more unknown than equation. It is further noted thatthere are a variety of approaches to closing the system,and one of the more common is to assume local chargeneutrality. Instead of making such an assumption, theauthors of [3] minimize the energy of the system to de-termine the last equation. In doing so, they derive anequation for every component in the fluid, which theycall the “species TOV” equations (their sum is equiva-lent to the Bianchi identity for the stress-energy tensor):
µ′i =qir2Eeλ/2 + µi
λ′2− 4πG
c4reλ
∑j
njµj
, (4)
where the spacetime metric is taken as
ds2 = eνdt2 − eλdr2 − r2(dθ2 + sin2 θdφ2) , (5)
primes mean derivative with respect to the radial coor-dinate, the constant qi is the charge of each particle ofthe species labelled by i (note that i does not mean “ion”here, but is instead a label), ni(r) is the correspondingnumber density at radius r, E(r) is the total charge below
radius r, defined via
E(r) =
∫ r
0
(∑i
qini(s)
)eλ(s)/2s2ds , (6)
and µi(r) is the chemical potential of the fluid of particlespecies i, defined via
µi(r) = c2∂ρm∂ni
(r) , (7)
where ρm(r) is the total mass-energy density. It is alsonoted that, except under very specific circumstances, (4)precludes local charge neutrality.
In [4], a white dwarf star was modelled by applyingthese results to a two-component system formed of a per-fect Fermi gas of electrons and a perfect Bose gas of nu-clei. Since the Bose gas contributes no pressure and hasmass-energy density of
ρmN= mNnN (8)
(N for “nucleon”), they are able to reduce the systemdown to a single second-order ODE, somewhat similar tothe one found by Chandrasekhar in the locally neutralcase.
In [5], the results of [3] were applied to a model of aneutron star consisting of three perfect Fermi gases ofelectrons, protons, and neutrons. The system of equa-tions used is
µ′i =qiE
r2(1− 2m/r)1/2+
µi1− 2m/r
−mr2
+1
2
GE2
c4r3+
4πGr
c4(ρmc
2 −∑
j=e,p,n
njµj)
, (9)
where the label i can be e, p, n for “electron”, “proton”and “neutron” and where m(r) is defined via
e−λ = 1− 2m(r)
r, (10)
together with the equations
m′ =4πGr2
c2
[ρm +
E2
8πc2r4
], (11)
E ′ = 4πr2 qene + qpnp(1− 2m
r
)1/2 . (12)
Note that the usual mass function is related to m by
M(r) = Gm(r)c2 . The mass-energy density function ρm
that appears in equation (11) is assumed to be a knownexpression of the ni, and thus also each µi is determinedby ni. In that paper and also here for us, it is assumed
that ρm = ρme+ ρmp
+ ρmn, with
ρmi=πm4
i c3
h3(2ni + n
1/3i )(n
2/3i + 1)1/2
− ln(n1/3i + (n
2/3i + 1)1/2) , (13)
where
ni =3h3ni
8πm3i c
3. (14)
This is the same energy density term derived in [6] us-ing the special relativistic kinetic energy instead of theNewtonian kinetic energy. It yields the following relationbetween µi and ni:
µi = mic2
√1 + ni
2/3 . (15)
The strategy in [5] to obtain a workable system fornumeric computation is to first eliminate (11) and (12)
3
by solving the electron and proton equations (9) for Eand m. But they have made an error here. With thedefinitions
K = 1− 2rqpµ′e − qeµ′p
qpµe − qeµp, (16)
a =G1/2r
c2µeµ
′p − µ′eµp
qpµe − qeµp, (17)
b =8πGr2
c4(ρmc
2 −∑i
niµi) (18)
(equations (3.7)–(3.9) in [5]), they write that
m =r
2
b+ a2
K + a2, (19)
E =c2ra
G1/2
(K − bK + a2
)1/2
(20)
(equations (3.5) and (3.6) in [5]), while in fact the correctequations are
m =r
2
b+ a2 +K − 1
K + a2, (21)
E =c2ra
G1/2
(1− bK + a2
)1/2
. (22)
Consequently, from (11) and (12) they derive
d
dr
(rb+ a2
K + a2
)=
8πGr2
c2ρm + a2 K − b
K + a2, (23)
d
dr
(r2a2 K − b
K + a2
)=
8πG1/2ar3
c2(qene + qpnp) (24)
(equations (3.10) and (3.11) in [5]), while the correctequations should be
d
dr
(rb+ a2 +K − 1
K + a2
)=
8πGr2
c2ρm + a2 1− b
K + a2,
(25)
d
dr
(r2a2 1− b
K + a2
)=
8πG1/2ar3
c2(qene + qpnp) .
(26)
Note that the corrected equations would not differ fromthe ones they reported if K = 1, while in fact one cansee from (16) that, for small r, the value of K is close tobut not equal to 1.
Next, it is explained in [5] how to produce numericalsolutions to this model by first noting that np − ne willbe very small, about 10−33, so that it can be assumedthat np = ne and (23) can be solved, taking for initialconditions at r = 0 an arbitrary value of the central den-sity np(0) = ne(0) as well as n′p(0) = n′e(0) = 0 (which isdictated by spherical symmetry). The neutron variablesnn and µn, which appear in (23), are determined by theβ-equilibrium condition µn = µp + µe (equation (2.12)in [5]). The result is then plugged into (24) to get anestimate of the total charge.
There are two remarks that we would like to makeabout this procedure. The first is that it appears thatthe numerical calculations in [5] were done using the cor-rect equations. Indeed, the apparent singularity at r = 0is removable for the corrected system (25) and (26), butnot for the originally reported equation (23) (see the ap-pendix for this calculation), so it would not be possibleto choose the central density as initial condition for theequations as they appear in that paper. But we haveredone the computations using the corrected equationsand found identical plots to theirs.
The second remark is that the system comprised ofequations (25) and (26) is not useful when one wants toconsider np 6= ne, as we want to do. This is because thissystem was derived using (9) for both values i = e, p,which is only valid when ni 6= 0. Indeed, it is implicit inthe model that, for each label i, if ni(r0) = 0, then we setni(r) = 0 for all r ≥ r0. In particular, in the case of onlytwo particle species, if r0 is the radius at which one ofthe two particle densities (say, the electrons) first reacheszero, we solve for the protons by numerically solving thesystem given by the protons’ (9), (11), and (12), choosingthe boundary conditions so as to have continuity, but(9) for the electrons is not valid for r ≥ r0. Therefore,outside of the support of one of the particle species, (25)and (26) are not valid.
III. COMPARISONS WITH THENONRELATIVISTIC MODEL
In this section we only consider models with electronsand protons, with the goal of drawing a comparison be-tween the general relativistic model described in sectionII (“the GR model”) and the special relativistic one de-scribed by (1) and (2) using the special relativistic kineticenergy to derive relation (15) between the chemical po-tential and the particle density (“the SR model”). Thelatter model was given in [2] in the form
4
kpy′′p = −kp
2
ry′p +
1
l3p
(1−
Gm2p
q2p
)(y2p − 1)3/2 − 1
l3e
(1 +
Gmpme
q2p
)(y2e − 1)3/2 , (27)
key′′e = −ke
2
ry′e +
1
l3e
(1− Gm2
e
q2p
)(y2e − 1)3/2 − 1
l3p
(1 +
Gmpme
q2p
)(y2p − 1)3/2 . (28)
where li = (3π2)1/3~mic
, ki = mic2
4πq2p, and yi =
√1 + l2i n
2/3i .
So, in fact, the unknowns ye and yp are related to thechemical potentials by µi = mic
2yi. Rewriting this sys-tem in terms of ni and µi, we obtain
(r2µ′p)′ = +4πq2
pr2(np − ne)− 4πGmpr
2(mpnp +mene) ,
(29)
(r2µ′e)′ = −4πq2
pr2(np − ne)− 4πGmer
2(mpnp +mene) .
(30)
Meanwhile, the main equations of the GR model arethe two equations (9) for µp and µe (taking µn = nn =0 on the right hand side), equation (11) for the massfunction m, and equation (12) for the electric charge E ,together with the definitions (13) and (7) of the energydensity ρm = ρme
+ ρmpand the chemical potentials
µi. Instead of considering m, we will work here with thetrue mass function M(r) = c2m(r)/G. We also remarkthat (11) can be used to rewrite (9) in the following form:
r2µ′i =qiE
(1− 2GMc2r )1/2
+µi
1− 2GMc2r
[r2
(GM
c2r
)′− 4πGr3
c4(neµe + npµp)
]. (31)
A. The post-Minkowski approximation
A natural question to ask is whether the SR and GRmodels coincide in low orders of a power series expansionin G. We shall see here that they match only in zerothorder, which is to be expected: on the one hand, whenG = 0, the flat space of special relativity also solves theEinstein equations of general relativity, thus explainingthe match at zeroth order; on the other hand, the SRmodel contains terms of degree 1 in G that are nonrela-tivistic in nature, derived from the Newtonian potential,and there’s no reason to hope that they will result fromany approximation scheme performed from a fully rela-tivistic model.
For G = 0, equations (29) and (30) of the SR modelreduce to
(r2µ′p)′ = +4πq2
pr2(np − ne) , (32)
(r2µ′e)′ = −4πq2
pr2(np − ne) , (33)
while equation (31) of the GR model becomes the follow-ing two when written for i = p and i = e:
r2µ′p = qpE , (34)
r2µ′e = qeE . (35)
Considering also equation (12) for E in the GR model,which when G = 0 reduces to E ′ = 4πr2(qene+qpnp), wesee that equations (32) and (33) yield the same solutionsas (34) and (35) provided the chosen initial conditionsni(0) and n′i(0) = 0 are the same between the models.
To study what happens for G 6= 0, we let n0i and µ0
i
denote the solutions to the G = 0 equations of eithermodel, and we also use a superscript zero on any expres-sion that depends on these solutions, like ρ0
m, E0 etc. Wealso write the solutions of either the SR or the GR modelfor G 6= 0 in the form
µi = µ0i +Gµ1
i +O(G2) , ni = n0i +Gn1
i +O(G2) , (36)
and apply the same notation to any expression that de-pends on ni and µi. Working only to first order in G, itis immediate that the SR model equations imply
(r2(µ1p)′)′ = +4πq2
pr2(n1
p − n1e)− 4πmpr
2(mpn0p +men
0e),
(37)
(r2(µ1e)′)′ = −4πq2
pr2(n1
p − n1e)− 4πmer
2(mpn0p +men
0e).
(38)
Meanwhile, keeping only first order terms in equa-tion (31) of the GR model gives
r2(µ1i )′ = qiE1 +
qiE0M0
c2r
+ µ0i
[r2
(M0
c2r
)′− 4πr3
c4(n0eµ
0e + n0
pµ0p)
], (39)
where E1 andM0 are obtained from the first order version
5
of equations (12) and (11):
(E1)′ = 4πr2
(M0
c2r(qen
0e + qpn
0p) + qen
1e + qpn
1p
), (40)
(M0)′ = 4πr2ρ0m +
E0E1
c2r2. (41)
After differentiating (39), we don’t find agreement withthe SR equations (37) and (38), nor should we expectto, considering that the mass density ρm, which entersthe equations of the GR model but not those of the SRmodel, can at this point still be any arbitrary function ofthe ni.
B. The weak field limit of the GR model
A different type of approximation scheme often per-formed in general relativity is the weak field limit, inwhich the metric is assumed to be close to the flat met-ric of special relativity. One might ask whether the GRmodel reduces to the SR model under this approxima-tion, but we shall see here that the answer is also no,although the discrepancy is small.
Taking into account the definition (10) of the massfunction, we see that, in order to find the weak field limitof the GR model, one must assume that the expressionGM/c2r and its derivatives are small. So let us studywhat the equations of the model become when neglect-ing all occurrences of GM/c2r and its derivative in (12)and (31). This amounts to studying the zeroth orderterms in our equations with respect to an expansion inpowers of GM/c2r. The relevant equations become:
E ′ = 4πr2(qene + qpnp) , (42)
r2µ′i = qiE −4πGµir
3
c4(neµe + npµp) . (43)
Therefore, taking the derivative of (43) and using (42)on the right hand side,
(r2µ′i)′ = 4πqir
2(qene + qpnp)
− 4πG
c4
(µir
3(neµe + npµp)
)′. (44)
This only matches the SR equations (29) and (30) in thecontribution from the electric charge, that is, the firstterm on the right hand side.
C. The weak field nonrelativistic limit
Let us now check that, assuming a weak field and un-der the Newtonian limit c→∞, the GR and SR modelsreduce to the Newtonian model described in [1]. Thismodel is comprised of (1) and (2), but using the nonrela-tivistic form of the kinetic energy to derive the following
relation between the chemical potential and the particledensity:
µNRi =h2
2mi
(3ni8π
)2/3
. (45)
This corresponds to the c-independent term in the ex-pansion with respect to powers of c2 of the relativistic µigiven in (15):
µi = mic2 +
h2
2mi
(3ni8π
)2/3
+O
(1
c2
). (46)
If we disregard the terms containing powers of c−2 fromthis and plug it into the µ′i terms of the SR model equa-tions (29) and (30), the constants mic
2 will drop out andwe easily find the Newtonian model.
As for the GR model, we begin with the equation
eν/2µi + qiφE = const , (47)
where
φE(r) =
∫ r
0
E(s)
s2e(λ(s)+ν(s))/2ds (48)
is the electric potential. Equation (47) was given in sec-tion II of [5] and is equivalent to the STOV equations (9).Now define as in [24] the Newtonian gravitational poten-tial
φG(r) =c2ν(r)
2. (49)
Note that
eν ≈ 1 +2φGc2
. (50)
Hence, in the weak field limit, we may disregard highorder terms in φG. We can find a differential equation forφG by considering the radial component of the Einsteinequations, which reads
φ′G = GM − 4πr3P
c2 − E22rc2
r2(1− 2GM
c2r
) , (51)
where P is the pressure. If we keep in all equations ofthe model only terms of up to first order in φG as wellas GM/c2r, as well as let 1/c2 = 0 in order to obtain theNewtonian limit, then the equation above turns into theusual Newtonian equation
φ′G =GM
r2, (52)
where M now satisfies
M ′(r) = 4πr2(mene +mpnp) . (53)
We also find that the electric potential reduces to
φ′E =Er2
, (54)
6
which is the same as its usual classical equation, consid-ering the weak field form (42) of the E equation. Nowreplace eν/2 by 1 +φG/c
2 and µi by mic2 +µNRi in (47),
multiply out, ignore the ensuing term µNRi /c2 and differ-entiate; we find the Newtonian model equations
(µNRi )′ +miφ′G + qiφ
′E = 0 . (55)
IV. BOUNDS ON THE CHARGE
In this section we find charge bounds analogous to (3)for the GR model of a system composed of electrons and
protons. Assume that we have a solution of the systemand suppose that the support of np is larger than that ofne and is bounded (if it has unbounded support, much ofthe next section can be skipped, although we do need toassume sufficient decay of µ′p). Then multiply the proton
version of (9) by r2 and evaluate at R, the radius ofthe star, which by definition is the smallest r at whichne(r) = np(r) = 0. We get
qpE(R)(1− 2m(R)
R
)1/2− mpc
2m(R)
1− 2m(R)R
+mpc
2
1− 2m(R)R
GE2(R)
2c4R= R2µ′p(R) . (56)
Let Q = E(R) be the total charge and M = c2
Gm(R) thetotal mass. Then (56) can be written as
qpQ(1− 2GM
c2R
)1/2 − mpGM
1− 2GMc2R
+mpc
2
1− 2GMc2R
GQ2
2c4R= R2µ′p(R) .
(57)If it was the electron support which was larger, the sameargument would give us
qeQ(1− 2GM
c2R
)1/2 − meGM
1− 2GMc2R
+mec
2
1− 2GMc2R
GQ2
2c4R= R2µ′e(R)
(58)(notice that now the first term is negative). These equa-tions are the analogues of equations (67) and (70) from[2].
Since (3) gives bounds on the most extreme charges aconfiguration can hold, to make a direct comparison wedo the same. However, since there are two degrees offreedom in the GR model (in both cases the central den-sities), what we call “the most extreme charge” is onlymost extreme under the assumption of some other quan-tity being fixed. In the nonrelativistic case, this quantitywas Np = Np(R), which was an easy quantity to workwith since it was directly related to both M and Q viaM = meNe +mpNp and Q = qeNe + qpNp. It was notedin [2] that the density np or ne must be decreasing at theboundary of the star so that the right hand sides of (67)and (70) in that paper are both nonpositive, which gives
qeQ−meGM ≤ 0 , (59)
qpQ−mpGM ≤ 0 , (60)
and plugging the definitions of M and Q in terms of Neand Np into these equations yields the charge bounds(3). In the GR model, however, M is not defined as asimple linear combination of Ne and Np, so Np is not easyto work with. We therefore seek to make a comparison
to (59) and (60), which we can see already somewhatresemble (57) and (58).
To this end, we first also make the observation as in [2]that µ′p(R) ≤ 0, since µp decreases to mpc
2 close to thesurface of the star. Now assume that we have a positivelycharged configuration, that is, Q > 0. We are interestedin the most charged configurations, so we consider whathappens as we increase Np and hold Ne fixed. If weknew that the radius R diverges faster than Q or M asNp increases, we could apply this to (57) to give us (60).We therefore seek to prove this.
As explained in section III of [3], the special TOV equa-tions are obtained from the condition ∂Λ
∂Ni= 0 for
Λ = M +GQ2
Rc4. (61)
Differentiating this with respect to Ni and taking intoaccount Q = qeNe + qpNp, we can compute that
∂R
∂Ni=
2RqiQ
+c4R2
GQ2
∂M
∂Ni. (62)
Since we are assuming Q > 0 and are increasing Np, thefirst term on the right is positive. So, if we can showthat ∂M
∂Np> 0, we show that the radius is increasing with
increasing Np. We show in the appendix that
∂M
∂Np= mpe
−λ(R)/2 . (63)
Therefore we can conclude that, given Q > 0, as Npincreases, so too does R.
Now we would like to conclude that R increases fasterthan Q and M . First note that, by (62), R grows faster
than M since c4R2
GQ2 > 1. By (10) we have
e−λ(R) = 1− 2M
R, (64)
7
so as R increases, e−λ(R)/2 increases, this allows us tosay that as we increase R, we have a lower bound on
e−λ(R)/2, let us call it e−λ/2. So we can say that in thissituation,
∂M
∂Np≥ mpe
−λ/2 > 0 . (65)
Since Q grows linearly in Np, we can write that
Q(Np) = Q0 + qpNp for an appropriate Q0. Define Rthrough
∂R
∂Np=
c4R2mp
G(Q0 + qpNp)2e−λ/2 . (66)
Comparing (62) and (66), we can see that R > R, as long
as we choose the correct initial condition for R. (66) canbe solved by a separation of variables to give
1
R=
c4mpe−λ
Gqp(Q0 +Npqp)+K . (67)
Now K is negative if
R(0) >GqpQ0
c4mpe−λ/2. (68)
Since we can “start” the differential equation when Q0 =qp > 0, this inequality will be true as the right hand sideis very small, but the left is larger than the radius of aneutral configuration, which is still thousands of kilome-ters. Then
R =Gqp(Q0 +Npqp)
c4mpe−λ/2 +KGqp(Q0 +Npqp). (69)
Since K is negative, R blows up at a finite Np, whichimplies that so does R. We may then conclude that, forfixed Ne, Q is bounded above and R grows faster than Qfor large enough Np.
We can get another lower bound on R by following thesame procedure with the first term on the right in (62).Define R by
∂R
∂Np=
2RqpQ0 +Npqp
. (70)
Solving this we get
R = (Q0 + qpNp)2 +K (71)
for K the integration constant. We can therefore con-clude that R > Q.
Thus we can finally conclude that, as Np grows, Rgrows faster than M and Q, and further that there issome finite N∞p such that R→∞ as Np → N∞p . There-fore, by taking this limit, we can determine that, in themost positively charged cases, (57) reduces to (60). Theexact same argument works to show that, in the mostnegatively charged cases, (58) reduces to (59). So we con-clude that, in terms of Q and M , the relativistic boundson charge are the same as (59) and (60).
V. NUMERICAL RESULTS
We can directly compare the solutions of our systemto the numerical results found in [2], reproduced herefor ease. For reasons which are explained there, it isvery difficult to produce meaningful graphs using the truephysical values of the constants. We therefore adopt thesame approach of using ‘science fiction’ values me/mp =1/10 and Gm2
p/e2 = 1/2 (where e = qp = −qe).
Figures 2 and 3 are sample plots using the Newtonianset of equations (1) and (2) and the special relativisticset of equations (27) and (28), respectively. These, andthe ones like them found in [2], can be compared to theplots in figures 5 and 7. In all cases, the plots were madeby holding the central density of the protons constantwhile slightly altering the central density of the electrons.One can see that the relativistic solutions have the samegeneral structure as the nonrelativistic solutions: as theelectron central density is increased, the electron radiusapproaches infinity, while, as the electron central densityis decreased, the proton radius approaches infinity.
The radius is of course difficult to see since the densitiesthemselves quickly decay to be very close to zero. Figure1 is a plot of the radii themselves as the central protondensity is changed.
Figure 4 represents the same type of plot, but with themass plotted instead of the radius. We say the star hasa “positively charged atmosphere” when the support ofthe protons contains that of the electrons; otherwise, wesay it has a “negatively charged atmosphere”.
Finally, figure 6 plots the ratio of the total charge tothe total mass. In section IV, it was derived that thebounds of this ratio are given by
−meG
e≤ Q
M≤ mpG
e, (72)
which in the science fiction values we are using is givenapproximately by
−0.0707 ≤ Q
M≤ 0.707 , (73)
and we can see from the plot that we can get arbitrarilyclose to these values with numerical solutions.
8
0
5
10
15
20
25
30
55.05 55.1 55.15 55.2 55.25 55.3 55.35 55.4
Len
gth
Electron Central Density
Radius Plot
FIG. 1. Shown are the radii of the configurations as the cen-tral electron density is varied while the central proton densityis held fixed at 100. The red colored part of the curve rep-resents configurations with a positively charged atmospherewhile the green part of the curve represents configurationswith a negatively charged atmosphere. The science fictionvalues of Gm2
p/e2 = 1/2 and me/mp = 1/10 are used, so the
values of length are not meaningful.
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8 10 12 14 16
Den
sity
Radial Distance
Particle Densities for no atmosphere star (5/3 case)
protonelectron
FIG. 2. An example when the kinetic energy law is 5/3.
0
20
40
60
80
100
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Den
sity
Radial Distance
Particle Densities for no atmosphere star (Special Relativistic Case)
protonelectron
FIG. 3. An example when the kinetic energy law is specialrelativistic.
210
220
230
240
250
260
270
55.05 55.1 55.15 55.2 55.25 55.3 55.35 55.4
Mas
s
Electron Central Density
Mass Plot
FIG. 4. Shown are the masses of the configurations as the cen-tral electron density is varied while the central proton densityis held fixed at 100. The red colored part of the curve rep-resents configurations with a positively charged atmospherewhile the green part of the curve represents configurationswith a negatively charged atmosphere. The science fictionvalues of Gm2
p/e2 = 1/2 and me/mp = 1/10 are used, so the
values of mass are not meaningful.
0
10
20
30
40
50
60
70
80
90
100
0 0.5 1 1.5 2 2.5 3
Den
sity
Radial Distance
Particle Densities for a negative atmosphere star (GR case)
ElectronsProtons
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.5 2 2.5 3 3.5 4 4.5
Den
sity
Radial Distance
Particle Densities for a negative atmosphere star (GR case)
ElectronsProtons
FIG. 5. Shown are the density functions for the protons andelectrons of a configuration which is close to the lower boundon the charge and with science fiction values Gm2
p/e2 = 1/2
and me/mp = 1/10. The bottom graph is zoomed in on apart of the top graph.
9
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
55.05 55.1 55.15 55.2 55.25 55.3 55.35 55.4
Electron Central Density
Ratio Plot
Charge to Mass
FIG. 6. Shown are the charge to mass ratios of the configura-tions as the central electron density is varied while the centralproton density is held fixed at 100. The red colored part ofthe curve represents configurations with a positively chargedatmosphere while the green part of the curve represents con-figurations with a negatively charged atmosphere. The sci-ence fiction values of Gm2
p/e2 = 1/2 and me/mp = 1/10 are
used. It should be noted that, in these units, the minimumratio is ≈ −0.0707 while the maximum is ≈ 0.707.
0
10
20
30
40
50
60
70
80
90
100
0 0.5 1 1.5 2 2.5 3
Den
sity
Radial Distance
Particle Densities for a positive atmosphere star (GR case)
ElectronsProtons
0
0.5
1
1.5
2
2.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
Den
sity
Radial Distance
Particle Densities for a positive atmosphere star (GR case)
ElectronsProtons
FIG. 7. Shown are the density functions for the protons andelectrons of a configuration which is close to the upper boundon the charge and with science fiction values Gm2
p/e2 = 1/2
and me/mp = 1/10. The bottom graph is zoomed in on apart of the top graph.
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Den
sity
Radial Distance
No Atmosphere Neutron Star
electronsprotons
neutrons
0
0.05
0.1
0.15
0.2
0.25
0.3
0.74 0.745 0.75 0.755 0.76 0.765 0.77 0.775 0.78 0.785 0.79
Den
sity
Radial Distance
No Atmosphere Neutron Star
electronsprotons
neutrons
FIG. 8. Shown are the density functions for the protons,electrons, and neutrons of a neutron star configuration whichhas no atmosphere of protons or electrons and with sciencefiction valuesGm2
p/e2 = 1/2 andme/mp = 1/10. The bottom
graph is zoomed in on a part of the top graph.
VI. NEUTRON STARS
Everything we have done above applies as well whenwe consider neutron stars. The only difference in theneutron star model is that a third equation of the form(9) is added for the Fermi gas of neutrons. This thirdequation is implied by the equation for β-equilibrium
µn = µp + µe . (74)
Figures 8, 9, and 10 illustrate, using the science-fictionvalues, some possible configurations given by this model.Immediately one notices that these science fiction valuesproduce some rather extreme looking configurations: fora central proton density of 100, the largest the centralelectron density can be for a stable configuration is about6.5. We have kept these same values for consistency,and also because we are interested in how the solutionschange as we change the central densities, and, as in thewhite dwarf case, these science fiction values dramaticallyexaggerate the changes. Compare, for example, theseplots to figure 2 in [22].
Another thing one notices is that, as one approachesthe extremely charged cases, the neutron density has a
10
0
50
100
150
200
250
300
350
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Den
sity
Radial Distance
Negative Atmosphere Neutron Star
electronsprotons
neutrons
0
5
10
15
20
25
0.5 0.55 0.6 0.65 0.7 0.75 0.8
Den
sity
Radial Distance
Negative Atmosphere Neutron Star
electronsprotons
neutrons
FIG. 9. Shown are the density functions for the protons,electrons, and neutron of a neutron star configuration which isclose to the lower bound on the charge and with science fictionvalues Gm2
p/e2 = 1/2 and me/mp = 1/10. The bottom graph
is zoomed in on a part of the top graph.
larger support than one of the electron or proton density.This would seem to indicate a limitation of the model.
VII. CONCLUSION
We have applied the model developed by Olson andBailyn in [3], [4] and [5] to write the equations of a gen-eral relativistic model of a star composed of fluids of elec-trons, protons and (optionally) neutrons. It generalizesthe Newtonian model from [1] and the special relativis-tic model from [2] in the sense that all three make thesame predictions in the Newtonian limit (large c), but wehave also mentioned some differences between them inthe post-Minkowski (small G) and weak field (flat met-ric) approximations. After pointing out an arithmeticerror in the Olson-Bailyn papers which was inconsequen-tial in their numeric calculations but relevant in ours, weshowed how the equations of the model can be used torigorously prove the bounds (72) on the total charge that
the star can hold. The bounds proved were the same asin [2] for the special relativistic model. We then plot-ted numerical solutions that do not assume local chargeneutrality or approximate neutrality, as was the case in
0
50
100
150
200
250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Den
sity
Radial Distance
Positive Atmosphere Neutron Star
electronsprotons
neutrons
0
5
10
15
20
25
0.5 0.55 0.6 0.65 0.7 0.75 0.8
Den
sity
Radial Distance
Positive Atmosphere Neutron Star
electronsprotons
neutrons
FIG. 10. Shown are the density functions for the protons,electrons, and neutrons of a neutron star configuration whichis close to the upper bound on the charge and with sciencefiction valuesGm2
p/e2 = 1/2 andme/mp = 1/10. The bottom
graph is zoomed in on a part of the top graph.
[5]. The graphs obtained behave similarly to the specialrelativistic ones appearing in [2], and they also provideevidence that our charge bounds are close to begin sharp.
APPENDIX
A. Singularity of (23)
One could rewrite equation (23) in terms of the un-known
s =h2
m2ec
2
(3ne8π
)2/3
(75)
in the following form:
11
b+ a2
K + a2+
r
K + a2
(∂b
∂r+∂b
∂ss′ + 2a
(∂a
∂r+∂a
∂ss′ +
∂a
∂s′s′′))
(76)
− r b+ a2
(K + a2)2
(∂K
∂r+∂K
∂ss′ +
∂K
∂s′s′′ + 2a
(∂a
∂r+∂a
∂ss′ +
∂a
∂s′s′′))
=8πGr2
c2ρm + a2 K − b
K + a2
or, solving for s′′,(2ar
K + a2− r b+ a2
(K + a2)2
(∂K
∂s′+ 2a
∂a
∂s′
))s′′ = − b+ a2
K + a2− r
K + a2
(∂b
∂r+∂b
∂ss′ + 2a
(∂a
∂r+∂a
∂ss′))
(77)
+ rb+ a2
(K + a2)2
(∂K
∂r+∂K
∂ss′ + 2a
(∂a
∂r+∂a
∂ss′))
+8πGr2
c2ρm + a2 K − b
K + a2.
A simple heuristic way to see the singularity at r = 0is to consider the asymptotic behavior of these terms asr → 0. We assume s(0) 6= 0, but that s′(0) = 0, althoughwe do not know the rate at which s′ goes to zero: byL’Hopital’s rule we would need to know s′′. Thus wehave the following behaviors near zero: a ∼ rs′, b ∼ r2,K ∼ 1, ∂a
∂s ∼ rs′, ∂a∂r ∼ s′, ∂a
∂s′ ∼ r, ∂b∂s ∼ r2, ∂b
∂r ∼ r,∂K∂r ∼ s
′, ∂K∂s ∼ rs
′, and ∂K∂s′ ∼ r.
One can check that the coefficient of s′′ has leadingorder term behaving like r3s′ or r4: it is possible theseterms could cancel, but then the coefficient would go tozero even faster. On the right side of (77) however, theleading order terms are − b
K+a2 and − rK+a2
∂b∂r which do
not cancel and behave like r2 near zero. So there is noway to remove the singularity: to leading order we haveeither
s′′ =c
r2(78)
or
s′′ =c
rs′(79)
In the former case, s has a singularity at r = 0 from alogarithmic term. In the latter case, one can consider
T (r) = (s′(r))2
2 so that T ′(r) = s′s′′ = cr , and T (0) = 0.
Fixing r0 > 0, and 0 < r < r0, we compute
T (r0)− T (r) =
∫ r0
r
T ′(x)dx = c ln(r0/r) , (80)
which implies that
T (r) = T (r0)− c ln(r0/r) . (81)
So we must have s′ become unbounded around r = 0.
B. Computation of∂M
∂Np
Here we check inequality (63). This calculation is quitesimilar to the computation of δΛ
δNpfound in the appendix
of [3]. We remind the reader that the symbol Ni standsfor Ni(R), the total number of particles of species i. Welet δ denote variation with respect to the functions Ni(r)defined by
Ni(r) = 4π
∫ r
0
nieλ/2s2ds , (82)
assuming that Ni(0) is held fixed. The label i can meane or p. Start with
δm(r) =4πG
c2
∫ r
0
(δρm +
2EδE8πc2s4
)s2ds , (83)
which comes from equation (3.4) in [3]. We have
δρm =∂ρm∂np
δnp +∂ρm∂ne
δne . (84)
Differentiating (82) and taking a variation, we find
δni =1
4πr2(e−λ/2δN ′i +N ′iδe
−λ/2) , (85)
while equation (10) gives
δe−λ/2 = −eλ/2 δm(r)
r. (86)
Plugging this into (84), we obtain
δρm =∂ρm∂np
1
4πr2
(e−λ/2δN ′p −N ′peλ/2
δm(r)
r
)+∂ρm∂ne
1
4πr2
(e−λ/2δN ′e −N ′eeλ/2
δm(r)
r
). (87)
12
Then (83) becomes
δm(r) =4πG
c2
∫ r
0
[∂ρm∂np
1
4π
(e−λ/2δN ′p −N ′peλ/2
δm(s)
s
)+∂ρm∂ne
1
4π
(e−λ/2δN ′e −N ′eeλ/2
δm(s)
s
)]ds (88)
+4πG
c2
∫ r
0
[2E
8πc2s2(qpδNp + qeδNe)
]ds ,
where we have used (6) to write δE in terms of δNe and δNp. Taking a derivative of both sides,
(δm)′ +Geλ/2
c2r
(∂ρm∂np
N ′p +∂ρm∂ne
N ′e
)δm =
G
c2
[e−λ/2
(δN ′p
∂ρm∂np
+ δN ′e∂ρm∂ne
)+Ec2r2
(qpδNp + qeδNe)
]. (89)
As a solution we get
δm(r) = e−D(r)
∫ r
0
eD(s)G
c2
[e−λ/2
(δN ′p
∂ρm∂np
+ δN ′e∂ρm∂ne
)+Ec2s2
(qpδNp + qeδNe)
]ds , (90)
where we defined the integrating factor
D(r) =
∫ r
0
Geλ/2
c2s
(∂ρm∂np
N ′p +∂ρm∂ne
N ′e
)ds . (91)
Since our goal is to find ∂m(R)/∂Np, assume only changes in the proton density from now on. We evaluate the aboveat r = R and use integration by parts:
δm(R) = e−D(R)G
c2
∫ R
0
[eD(s) Eqp
c2s2δNp −
(eD(s)e−λ/2
∂ρm∂np
)′δNp
]ds+
G
c2e−λ(R)/2 ∂ρm
∂np(R)δNp(R) . (92)
The integrand of the first term on the right hand side evaluates to zero with the help of the definition of D in (91)
and of the species TOV equation (9) for (∂ρm∂np)′ =
µ′p
c2 . Therefore
δm(R) =G
c2e−λ(R)/2 ∂ρm
∂np(R)δNp , (93)
which implies that
∂m(R)
∂Np=G
c2e−λ(R)/2 ∂ρm
∂np(R) . (94)
Taking into account M = c2
Gm and mpc2 = µp(R) = c2 ∂ρm∂np
(R), we reach the conclusion that
∂M
∂Np= mpe
−λ(R)/2 > 0 . (95)
ACKNOWLEDGMENT
We thank Michael Kiessling, Shadi Tahvildar-Zadeh,and Eric Ling for helpful discussions.
[1] M. I. Krivoruchenko, D. K. Nadyozhin, and A. V. Yudin,“Hydrostatic equilibrium of stars without electroneutral-ity constraint,” Phys. Rev. D. 97:083016 (2018).
[2] P. Hund and M. K.-H. Kiessling, “On electrically non-neutral ground states of stars,” Phys. Rev. D. 103:043004(2021).
13
[3] E. Olson and M. Bailyn, “Internal structure of multicom-ponent static spherical gravitating fluids,” Phys. Rev. D.12:3030–3036 (1975).
[4] E. Olson and M. Bailyn, “Charge effects in a static,spherically symmetric, gravitating fluid,” Phys. Rev. D.13:2204–2211 (1975).
[5] E. Olson and M. Bailyn, “Internal Structure of multi-component fluids,” Phys. Rev. D. 18:2175–2179 (1978).
[6] S. Chandrasekhar, An Introduction to the Study of StellarStructure, 1st ed. (Dover, Mineola, NY, 1958).
[7] Ray, S., Espindola, A. L., Malheiro, M., Lemos, J. P.,and Zanchin, V. T., “Electrically charged compact starsand formation of charged black holes,” Phys. Rev. D.68(8):084004 (2003).
[8] Nduka, A., “Charged fluid sphere in general relativ-ity,” General Relativity and Gravitation 7(6), 493–499,(1976).
[9] Nduka, A., “Charged static fluid spheres in Einstein-Cartan theory,” General Relativity and Gravitation 8(6),371–377, (1977).
[10] Pant, D. N., and A. Sah, “Charged fluid sphere in gen-eral relativity,” Journal of Mathematical Physics 20(12),2537–2539, (1979).
[11] Patel, L. K., and Koppar, S. S., “A charged analogueof the Vaidya-Tikekar solution,” Australian Journal ofPhysics 40(3), 441–448, (1987).
[12] Murad, M. H., and Fatema, S., “Some exact relativisticmodels of electrically charged self-bound stars,” Interna-tional Journal of Theoretical Physics 52(12), 4342–4359,(2013).
[13] Fatema, S., and Murad, M. H., “ An exact family of Ein-stein–Maxwell Wyman–Adler solution in general relativ-ity,” International Journal of Theoretical Physics 52(7),2508–2529, (2013).
[14] Kiess, T. E., “Exact physical Maxwell-Einstein Tolman-VII solution and its use in stellar models,” Astrophysicsand Space Science 339(2), 329–338, (2012).
[15] Ivanov, B. V., “Static charged perfect fluid spheres ingeneral relativity,” Physical Review D 65(10), 104001,(2002).
[16] Mehra, A. L., “Static charged gas spheres in general rel-ativity,” General Relativity and Gravitation 12(3), 187–193, (1980).
[17] Hajj-Boutros, J., and Sfeila, J., “New exact solutions fora charged fluid sphere in general relativity,” General Rel-ativity and Gravitation 18(4), 395–410, (1986).
[18] Negreiros, R. P., Weber, F., Malheiro, M., and Usov,V., “Electrically charged strange quark stars,” PhysicalReview D 80(8), 083006, (2009).
[19] Arbanil, J. D., Lemos, J. P., and Zanchin, V. T., “Poly-tropic spheres with electric charge: compact stars, theOppenheimer-Volkoff and Buchdahl limits, and quasi-black holes,” Physical Review D 88(8), 084023, (2013).
[20] Arbanil, J. D., and Zanchin, V. T., “Relativistic poly-tropic spheres with electric charge: Compact stars, com-pactness and mass bounds, and quasiblack hole configu-rations,” Physical Review D 97(10), 104045, (2018).
[21] P. Hund and M. K.-H. Kiessling “How much electric sur-charge fits on... a ‘white dwarf’ star?”, American Journalof Physics 89: 291–299 (2021).
[22] Rotondo, M., Rueda, J.A, Ruffini, R., and Xue, S.s.,“The self-consistent general relativistic solution for a sys-tem of degenerate neutrons, protons and electrons in β-equilibrium,” Phys. Lett. B. 701 667–671 (2011).
[23] Rotondo, M., Rueda, J. A., Ruffini, R., and Xue, S.S., “Relativistic Thomas-Fermi treatment of compressedatoms and compressed nuclear matter cores of stellar di-mensions,” Physical Review C 83(4), 045805 (2011).
[24] Rotondo, M., Rueda, J.A, Ruffini, R., and Xue, S.s.,“Relativistic Feynman-Metropolis-Teller theory for whitedwarfs in general relativity,” Physical Review D 84(8)084007 (2011).
[25] Belvedere, R., Pugliese, D., Rueda, J. A., Ruffini, R.,and Xue, S. S., “Neutron star equilibrium configura-tions within a fully relativistic theory with strong, weak,electromagnetic, and gravitational interactions,” NuclearPhysics A 883 1–24 (2012).
