1
General Relativity and Compact Objects – Neutron Stars and Black Holes
We confine attention to spherically symmetric configurations. The metric for the static case can generally be written
2
λ(r) 2
2
2
2
2
ds = e
dr + r dθ + sin θdφ − eν(r) dt2 .
(1)
Einstein’s equations for this metric are:
1 λ′ (r)
−λ
8πρ (r) = 2 1 − e
+ e−λ
,
r
r
ν ′ (r)
1 −λ
+ e−λ
,
8πp (r) = − 2 1 − e
r
r
p (r) + ρ (r) ′
ν (r) .
p′ (r) = −
2
(2)
Derivatives with respect to the radius are denoted by ′ . We employ units
in which G = c = 1, so that 1 M⊙ is equivalent to 1.475 km. The first of
Eq. (2) can be exactly integrated. Defining the constant of integration so
obtained as m(r), the enclosed gravitational mass, one finds
Z r
−λ
ρr′2 dr′ .
(3)
e = 1 − 2m (r) /r,
m (r) = 4π
0
The second and third of Einstein’s equations form the equation of hydrostatic equilibrium, also known as the Tolman-Oppenheimer-Volkov (TOV)
equation in GR:
−p′ (r)
ν ′ (r)
m (r) + 4πr3 p (r)
=
=
.
ρ (r) + p (r)
2
r (r − 2m (r))
r≤R
(4)
Near the origin, one has ρ′ (r) = p′ (r) = m(r) = 0. Outside the distribution
of mass, which terminates at the radius R, there is vacuum with p(r) =
ρ(r) = 0, and Einstein’s equations give
m (r) = m (R) = M,
eν = e−λ = 1 −
2M
,
r
r≥R
(5)
the Schwarzschild solution. The black hole limit is seen to be R = 2M ,
which is 2.95 km for 1 M⊙.
2
From thermodynamics, if there is uniform entropy per nucleon, the first
law gives
ρ
1
0=d
+ pd
n
n
where n is the number density. If e is the internal energy per nucleon, we
have ρ = n(m + e). From the above, p = n2 de/dn, so that
d (log n) =
1 dρ
dρ
=−
dν,
ρ+p
2 dP
dn =
dρ
,
h
where h = (ρ + p)/n is the enthalpy per nucleon or the chemical potential.
The constant of integration for the number density can be established from
conditions at the surface of the star, where the pressure vanishes (it is not
necessary that the energy density or the number density also vanish there).
If n = no , ρ = ρo and e = eo when P = 0, one finds ρo − mno = no eo and
mn (r) = (ρ (r) + p (r)) e(ν(r)−ν(R))/2 − no eo .
(6)
Another quantity of interest is the total number of nucleons in the star,
N . This is not just M/m (m being the nucleon mass) since in GR the
binding energy represents a decrease of the gravitational mass. The nucleon
number is
−1/2
Z R
Z R
2m
(r)
4πr2 n (r) 1 −
4πr2 eλ/2n (r) dr =
N=
dr,
(7)
r
0
0
and the total binding energy is
BE = N m − M.
(8)
Analytic Solutions to Einstein’s Equations
It turns out there are many analytic solutions to Einstein’s equations.
However, there are only 3 that satisfy the criteria that the pressure and
energy density vanish on the boundary R, and that the pressure and energy
density decrease monotonically with increasing radius. Many others, in fact
an infinite number, are known that have vanishing pressure, but not energy
density, at R.
3
Among the simplest analytic solutions is the so-called Schwarzschild interior solution for an incompressible fluid, ρ(r) = constant. In this case,
4π 3
ρr ,
e−λ = 1 − 2β (r/R)2 ,
3
2
q
p
1
3
1 − 2β −
1 − 2β (r/R)2 ,
eν =
2
2
q
√
1 − 2β − 1 − 2β (r/R)2
3β
q
,
p (r) =
√
4πR2
2
1 − 2β (r/R) − 3 1 − 2β
m (r) =
ρ = n (m + e) = constant,
(9)
n = constant.
Here, β ≡ M/R. Clearly, β < 4/9 or else the central pressure will become
infinite. It can be shown that this limit to β holds for any star. This solution
is technically unphysical for the reasons that the energy
pdensity does not
vanish on the surface, and that the speed of sound, cs = ∂p/∂ρ is infinite.
The binding energy for the incompressible fluid is analytic (taking e = 0):
√
3β 9β 2
3 sin−1 2β p
BE
√
− 1 − 2β − 1 ≃
=
+
+ ···
(10)
M
4β
5
14
2β
In the case that e/m is finite, the expansion becomes
e −1
3β 9β 2
BE e
≃ 1+
+
+ ··· .
− +
M
m
m
5
14
(11)
In 1967, Buchdahl discovered an extension of the Newtonian n = 1
polytrope into GR that has an analytic solution. He assumed an equation
of state
√
(12)
ρ = 12 p∗ p − 5p
and found
eν = (1 − 2β) (1 − β − u) (1 − β + u)−1 ;
eλ = (1 − 2β) (1 − β + u) (1 − β − u)−1 1 − β + β cos Ar′
8πp = A2 u2 (1 − 2β) (1 − β + u)−2 ;
8πρ = 2A2u (1 − 2β) (1 − β − 3u/2) (1 − β + u)−2 ;
r √
1 p 3/2
mn = 12 pp∗ 1 −
;
3 p∗
−2
r
−1
p
∗
c2s = 6
.
−5
p
;
(13)
4
Here, p∗ is a parameter, and r′ is, with u, a radial-like variable
−1
sin Ar′ ;
u = β Ar′
r′ = r (1 − β + u)−1 (1 − 2β) ;
(14)
A2 = 288πp∗ (1 − 2β)−1 .
For this solution, the radius, central pressure, energy and number densities,
and binding energy are
r
π
R = (1 − β)
;
288p∗ (1 − 2β)
pc = 36p∗ β 2 ;
ρc = 72p∗β (1 − 5β/2) ;
nc mn c2 = 72βp∗ (1 − 2β)3/2 ;
β β 2 3β 3
BE
−1/2
−1
= (1 − 1.5β) (1 − 2β)
(1 − β) − 1 ≈ +
+
+ ···.
M
2
2
4
(15)
This solution is limited to values of β < 1/6 for cs,c < 1.
In 1939, Tolman discovered that the simple density function ρ = ρc [1 −
(r/R)2 ] has an analytic solution. It is known as the Tolman 7 solution:
e−λ = 1 − βx (5 − 3x) ,
eν = (1 − 5β/3) cos2 φ,
q
β
1
(ρ + P ) cos φ
−λ tan φ − (5 − 3x) ,
P =
n
=
,
3βe
2
m cos φ1
4πR2
φ = (w1 − w) /2 + φ1 ,
φc = φ (x = 0) ,
p
φ1 = φ (x = 1) = tan−1 β/ [3 (1 − 2β)],
q
w = log x − 5/6 + e−λ / (3β) ,
w1 = w (x = 1) .
In the above, x = (r/R)2 .
sound speed c2s,c are
r
2 tan φc 3
P −
=
ρ c
15
β
(16)
The central values of P/ρ and the square of the
1
,
3
c2s,c = tan φc
r !
1
β
tan φc +
.
5
3
(17)
There is no analytic result for the binding energy, but in expansion
BE
11β 7187β 2 68371β 3
≈
+
+
+ ··· .
(18)
M
21
18018
306306
This solution is limited to φc < π/2, or β < 0.3862, or else Pc becomes
infinite. For causality cs,c < 1 if β < 0.2698.
5
In 1950, Nariai discovered yet another analytic solution. It is known as
the Nariai 4 solution, and is expressed in terms of a parametric variable r′ :
!2
!2
′ 2
′
2
p
cos g r
r
ν = (1 − 2β) e
′
,
e
tan
f
r
,
e−λ = 1 − 3β
R′
c2 cos f (r′ )
r "
r "
′ 2 #
′ 2 #
r
r
3β
3β
f r′ = cos−1 e +
, g r′ = cos−1 c +
,
1−
1−
′
4
R
2
R′
p
e
r′
r=
1 − 2β,
c cos f (r′ )
′ 2 p h√
cos
f
r
c
′ tan g r ′ 3β
2
cos
f
r
p r′ =
′2
2
" 4πR e #
#
"
′ 2
2
i
p
p
r′
r
3
′
′
′
3β
1 − 3β
tan f r
tan f r
,
− sin f r 2 −
R′
2
R′
√
3β
c2
′
√
ρ r =
4πR′2 1 − 2β e2
"
#
r 2 ′
3β r
2 f r′ 3 sin f r′ cos f r′ −
3
−
cos
,
4 R′
"
#
r 2
′
′3
′
p
3β r
r e tan f r
3β
(1
−
2β)
1
−
m r′ = ′2
tan f r′ .
′
′
4 R
R c cos f (r )
(19)
The quantities e and c are
√
1 − 2β
2
+
β
+
2
e2 = cos2 f R′ =
4 + β/3
2e2
c2 = cos2 g R′ =
−1 .
2
2
2
2
7e − 3 5e − 3
2e + 1 − e
The pressure-density ratio and sound speed at the center are
1 √
Pc
=
2 cot f (0) tan g (0) − 2 ,
ρc
3
1
2
2
2
2 tan g (0) − tan f (0) .
cs,c =
3
The central pressure and sound speed become infinite when cos g(0) = 0
or when β = 0.4126, and the causality limit is β = 0.223. This solution is
quite similar to Tolman 7.
6
Neutron Star Maximum Mass
The TOV equation can be scaled by introducing dimensionless variables:
√
√
p = qρo , ρ = dρo , m = z/ ρo , r = x/ ρo ,
(q + d) z + 4πdx3
dq
dz
=−
,
= 4πdx2 dx.
(20)
dx
x (x − 2z)
dx
Rhoades and Ruffini showed that the causally limiting equation of state
p = po + ρ − ρo
ρ > ρo
(21)
results in a neutron star maximum mass that is practically independent of
the equation of state for ρ < ρo , and is
p
(22)
Mmax = 4.2 ρs /ρo M⊙ .
Here ρs = 2.7 · 1014 g cm−3 is the nuclear saturation density. One also finds
for this equation of state that
p
βmax ≃ 0.33 .
(23)
Rmax = 18.5 ρs /ρo km,
Since the most compact configuration is achieved at the maximum mass,
this represents the limiting value of β for causality.
Some analytic motivation for the above results was given by Nauenberg
and Chapline. They assumed that in the interior of a star both n and ρ
were constant, so P is also because of the first law. The TOV equation is
not satisfied for this assumption, however, so the results of this analysis are
very approximate. The baryon number for fixed n and ρ is
Z
3 3/2 χ 2
p
sin θdθ
N = 4π
= 4πn
8πρ
1 − 8πρr2 /3
0
0
(24)
3/2
3
= 2πn
(χ − sin χ cos χ) ,
8πρ
p
√
where sin θ = 2m(r)/r and sin χ = 2β. In terms of χ, we can write the
gravitational mass as
r
3
sin3 χ.
M=
32πρ
Z R
nr2 dr
7
As χ increases, n, ρ and p in the star increase, and the mass M reaches a
maximum for χ < π/2. To guarantee stability, the total nucleon number N
must also be maximized, which is equivalent to the equation
∂M = 0.
∂χ N
This results in a pair of equations:
cos χ
dρ
=6
dχ,
ρ
sin χ
4 sin2 χdχ = (χ − sin χ cos χ)
dρ
dn
3 −2
.
ρ
n
Combining this with the first law dρ/dn = (ρ + p)/n, we obtain
p
6 cos χ (χ − sin χ cos χ)
.
=
ρ 9χ cos χ − 9 sin +7 sin3 χ
The condition that p/ρ < ∞ limits sin χ < 0.985, and p/ρ < 1 limits
sin χ < 0.956. The further condition dp/dρ < 1 limits sin χ < 0.90, which is
equivalent to β < 0.405. Note this value is significantly larger than the limit
obtained above, because of the less restrictive conditions. Nevertheless, we
can now derive a maximum mass by employing the maximal equation of
state Eq. (21). Rewriting this equation as
ρ>
ρo − Po
,
1 − P/ρ
and applying it to the mean density of the star ρ = 3M/(4πR3 ), using
β = M/R, we find
s
s
3
3β
3β 3
M=
<
(1 − p/ρ).
4πρ
4πρo
It is valid to have taken po << ρo . In geometrized units, the nuclear
−1/2
= 70
saturation density ρs = 2.7 · 1014 g cm−3 has the equivalence ρs
km or 45.5 M⊙ . Therefore,
s
3β 3 ρs
M < 45.5
(1 − p/ρ) M⊙.
4π ρo
p
With β = 0.405 and p/ρ = 0.364, the limiting mass is M < 4.57 ρs /ρo
M⊙ .
8
For the Buchdahl solution at the causal limit, β = 1/6 and p/ρ = β/(2 −
5β), which lead to
s
p
πβ 3 (1 − 5β/2)
M = (1 − β)
< 2.14 ρs /ρc M⊙ .
4 (1 − 2β) ρc
√
For the Tolman 7 solution at the causal limit, β ≃ 0.27 and p/ρ = 2/( 75β) ≃
0.44, which lead to
s
p
15β 3
< 4.9 ρs/ρc M⊙.
M=
8πρc
Finally, for the Nariai 4 solution at the causal limit, β ≃ 0.228 and p/ρ ≃
0.246, which lead to
s
p
β
33/2 β 1/2 sin f (0) cos f (0)
M=
<
3.4
ρs /ρc M⊙ .
cos f (R′ )
4πρc
Maximal Rotation Rates for Neutron Stars
The absolute maximum rotation rate is set by the “mass-shedding” limit,
when the rotational velocity
p at the equatorial radius (R) equals the Keplerian orbital velocity Ω = GM/R3, or
10 km 3/2 M 1/2
ms.
Pmin = 0.55
R
M⊙
(25)
However, the actual limit on the period is larger because rotation induces
an increase in the equatorial radius. In the so-called Roche model, one
treats the rotating star as being highly centrally compressed. For an n = 3
polytrope, ρc /ρ̄ ≃ 54, so this would be a good approximation. In more
realistic models, such as ρ = ρc [1 − (r/R)2 ], for which ρc /ρ̄ = 5/2, and an
n = 1 polytrope, for which ρc /ρ̄ = π 2 /3, this approximation is not as good.
Using it anyway, the gravitationl potential near the surface is ΦG = −GM/r
and the centrifugal potential is Φc = −(1/2)Ω2 r2 sin2 θ, and the equation
of hydrostatic equilibrium is
(1/ρ) ∇P = ∇h = −∇ΦG − ∇Φc ,
(26)
9
R
where h = dP/ρ is the enthalpy per unit mass. Integrating this from the
surface to an interior point along the equator, one finds
h (r) − GM/r − (1/2) Ω2 r2 = K = −GM/re − (1/2) Ω2 re2 ,
where re is the equatorial radius and h(re ) = 0. We assume K = −GM/R,
the value obtained for a non-rotating configuration. The potential Φ ≡
ΦG + Φc is maximized at the point where ∂Φ/∂r|rc = 0, or where rc3 =
gM/Ω3 and Φ = −(3/2)GM/rc . Thus, re has the largest possible value
when re = rc = 3R/2, or
3
GM
2
GM
2
Ω = 3 =
.
(27)
3
rc
R3
The revised minimum period then becomes
10 km 3/2 M 1/2
ms.
Pmin = 1.0
R
M⊙
(28)
Calculations including general relativity show that the minimum spin period
for an equation of state can be accurately expressed in terms of its maximum
mass and the radius at that maximum mass as:
10 km 3/2 Mmax 1/2
Pmin ≃ 0.82
ms.
(29)
Rmax
M⊙
It is interesting to compare the rotational kinetic energy T = IΩ2 /2
with the gravitational potential energy W at the mass-shedding limit. I is
the moment of inertia about the rotation axis:
Z
8π R 4
I=
r ρdr
3 0
for Newtonian stars. (In GR, one must take into account frame-dragging as
well as volume and redshift corrections.) Using Ω2 = (2/3)3 GM/R3, we can
write T = α(2/3)3 GM 2 /R and |W | = βGM 2 /R. We have α = 1/5, β = 3/5
for an incompressible fluid; α = 1/3 − 2/π 2, β = 3/4 for an n = 1 polytrope;
α = 0.0377, β = 3/2 for an n = 3 polytrope; α = 1/7, β = 5/7 for ρ =
ρc [(1−(r/R)2 ]. We therefore find that T /|W | is 0.0988, 0.0516, 0.00745 and
0.0593, respectively, for these four cases, at the mass-shedding limit. For
comparison, an incompressible ellipsoid becomes secularly (dynamically)
unstable at T /|W | = 0.1375(0.2738), much larger values.