1. University of Southampton - Mathematics
Math3031 Project
Equations of Stellar Structure
Author: Matthew John Grifferty
A Project Report Submitted for the Award of B.Sc. in
Mathematics
May 13, 2015
Supervisor: Dr. Ian Jones
1
3. Abstract
This report outlines the Newtonian and relativistic equations of stellar
structure. In the Newtonian instance, polytropes are focussed on which leads
to the derivation of the Lane-Emden equation with suitable boundary con-
ditions. A solution for three particular polytropic indices is derived. In the
relativistic approach a suitable metric is first derived which then leads to the
derivation of the Einstein and stress-energy tensor. To proceed further an
argument for a suitable equation of state is proposed which leads to the equa-
tions of motion and Einstein equations. The Tolman-Oppenheimer-Volkov
equation then directly follows as do two particular solutions to the equation.
3
7. 1 Introduction
The following report comprises two main sections: section 2, the Newtonian approach and
section 3, the Relativistic approach to the equations of stellar structure. The Newtonian
approach begins by determining a hydrostatic equilibrium equation for the star. The
definition of a polytrope then follows which leads to the derivation of the dimensionless
Lane-Emden equation. The Lane-Emden equation has explicit solutions for only three
polytropic indices, which are all found, whilst the remaining polytropic indices must be
numerically integrated.
The relativistic approach first needs a suitable metric to work with. This is carefully
derived using arguments of symmetry and also sensibility. With this found it is possible
to calculate the Einstein and stress-energy tensors - though their calculations are omitted
since they are lengthy. An argument is then made for an equation of state which has
the pressure and density related, much like the polytropes from the Newtonian approach.
With these tools in hand, the equations of motion and Einstein equations are derived and
everything required to calculate the external and internal structure of the star is ready.
The Schwarzschild metric describes the external structure whilst the internal structure is
described by the main result of this section, the Tollman-Oppenheimer-Volkov equation.
Two solutions for this equation are explored, credited to Schwarzschild and Buchdahl.
2 Newtonian Equation for the Structure of Stars
In the following model stars are assumed to be spherically symmetric and in a steady
state, that is, the stars mass and volume do not change with time (Shapiro and Teukolsky,
2004).
2.1 Hydrostatic Equilibrium
For such a star the mass inside a radius r is
m(r) =
r
0
ρ4πr2
dr (2.1.1)
dm(r)
dr
= 4πr2
ρ (2.1.2)
as given by Shapiro and Teukolsky (2004). From the steady state assumption the pressure
forces on the star will balance the gravitational forces (Schutz, 2009). Therefore, to derive
the hydrostatic equilibrium equation the pressure forces on the star must be calculated.
Consider an infinitesimally small fluid element between r and r + dr with an area of dA
perpendicular to the radial direction, as shown in Figure 1. Then the mass of this element
is
dm = ρdAdr = ρ4πr2
dr (2.1.3)
and the pressure of the stellar material on the element’s lower face is
Outward force = PdA. (2.1.4)
7
8. On the elements upper face there is both a pressure exerted on it and force from the
gravitational attraction of the stellar material lying within r. The inward force is
Inward force = P(r + dr)dA +
Gm(r)
r2
dm. (2.1.5)
Equating the inward and outward forces, and noting that for an infinitesimally small
element
P(r + dr) − P(r)
dr
=
dP(r)
dr
(2.1.6)
the hydrostatic equilibrium equation (2.1.7) is found to be
dP
dr
= −
Gm(r)ρ
r2
(2.1.7)
as given in Shapiro and Teukolsky (2004).
dr
r
dA
Figure 1: An infinitesimally small cubic fluid element lying between r and r+dr
with its faces of area dA. The balancing of the inward and outward forces leads
to the hydrostatic equilbrium equation (2.1.7).
Using the hydrostatic equilibrium equation, it is possible to calculate the gravitational
potential energy of a star. Let ψ(r) denote the work done, then if W is the gravitational
potential
dW = ψ(r)dm = −
Gm(r)
r
dm (2.1.8)
So, using equation (2.1.3) and substituting in equation (2.1.7)
R
0
dW = −
R
0
Gm(r)
r
ρ4πr2
dr =
R
0
dP
dr
4πr3
dr (2.1.9)
where R is the total radius of the star. Using integration by parts methods this can be
simplified to
W = − P4πr3 R
0
− 3
R
0
P4πr2
dr = −3
R
0
P4πr2
dr (2.1.10)
since the pressure at the centre of the star is 0, as given in Shapiro and Teukolsky (2004).
8
9. 2.2 Energy Density of a Star
Suppose the gas of a star was characterised by the following adiabatic equation of state
P = KρΓ
(2.2.1)
Where K and Γ are constants. Let be the energy density of the gas, including rest mass
energy, per unit volume.; then ρ
is the energy per unit mass. An increase in this energy
is equivalent to the amount of work done per unit mass, that is d ρ
.
P
x
A
Figure 2: A small cubic fluid element with a pressure exerted in the positive x
direction and whose faces have area A.
Work done, denoted here by E, is the product of force and distance, in this case the
force is the pressure P times the area and the distance is a small change in the positive
x direction, as shown in Figure 2. Therefore
dE = PAdx (2.2.2)
Furthermore, the mass is m = ρV and the mass does not change, hence
dm = (dρ)V + ρdV = 0. (2.2.3)
It is also clear that the change in volume is
dV = −Adx. (2.2.4)
Combining equations (2.2.3) and (2.2.4) gives
Adx = −dV =
V
ρ
dρ (2.2.5)
Then, using equation (2.2.2) and m = ρV the work done per unit mass is found to be
dE
m
= P
V
ρ2V
dρ = −Pd
1
ρ
(2.2.6)
and this is equivalent to the increase in the energy per unit mass:
d
ρ
= −Pd
1
ρ
(2.2.7)
Substituting in the equation of state (2.2.1) and integrating gives the total energy density
of the star as
= ρc2
+
P
Γ − 1
, (2.2.8)
9
10. which agrees with Shapiro and Teukolsky (2004). Hence the energy density of the star
excluding rest-mass energy is
0 =
P
Γ − 1
(2.2.9)
which also agrees with Shapiro and Teukolsky (2004). The total internal energy of the
star is then found by substituting equation (2.2.9) into equation (2.1.10) to give:
W = −3(Γ − 1)
R
0
04πr2
dr (2.2.10)
where
R
0
04πr2
dr is the total internal energy of the star (Shapiro and Teukolsky, 2004).
The gravitational potential energy of the star, provided equation (2.2.1) holds everywhere
inside the star, can also be expressed in terms of only the mass and radius of the star
and Γ. This can be found by using equation (2.1.2) to rewrite (2.1.10) as
W = −3
M
0
P
ρ
dm(r) (2.2.11)
and then, since,
P
ρ
dm(r) = d
P
ρ
m − d
P
ρ
m, (2.2.12)
this becomes
W = −3
M
0
d
P
ρ
m
=0
−
M
0
d
P
ρ
m
. (2.2.13)
So, integrating by parts and using
d
P
ρ
=
Γ − 1
Γ
Gm(r)d
1
r
(2.2.14)
and finally integrating by parts once more gives the result:
W = −
3(Γ − 1)
5Γ − 6
GM2
R
(2.2.15)
which is the gravitational energy of the star expressed in terms of the mass, radius and Γ
(Shapiro and Teukolsky, 2004). Equation (2.2.14) can be derived from equations (2.1.7)
and (2.2.1).
2.3 Polytropes and the Lane-Emden Equation
Hydrostatic equilibrium equations with the equation of state given by equation (2.2.1)
are called polytropes (Shapiro and Teukolsky, 2004). Define
Γ =
n + 1
n
, (2.3.1)
10
11. where n is called the polytropic index; a gas with such an equation of state will yield
solutions to the Lane-Emden equation (2.3.10) (Shapiro and Teukolsky, 2004). The
derivation of the Lane-Emden equation begins by combining equations (2.1.7) and (2.1.2)
as follows:
dP
dr
= −
−m(r)
r2
ρG (2.3.2)
r2
ρ
= −m(r)G (2.3.3)
d
dr
r2
ρ
dP
dr
= −
dm(r)
dr
G (2.3.4)
1
r2
d
dr
r2
ρ
dP
dr
= −4πGρ (2.3.5)
where the final step makes use of equation (2.1.2) (Shapiro and Teukolsky, 2004).
This equation can be made dimensionless using the following definitions:
ρ = ρcθn
, (2.3.6)
r = aξ, (2.3.7)
a =
(n + 1)Kρ
1
n −1
c
4πG
(2.3.8)
as given in Shapiro and Teukolsky (2004) and where ρc = ρ(0) is the central density.
Substituting these definitions into equation (2.3.5) and using equation (2.2.1) gives:
1
a3ξ
d
dξ
a2
ξ2
ρcθna
d
dξ
(K(ρcθn
)Γ
) = −4πGρcθn
. (2.3.9)
Then substituting Γ = 1 + 1
n
, cancelling ρc and implicitly differentiating gives the Lane-
Emden equation, which describes the structure of a star with the polytropic equation of
state, and polytrope index n:
1
ξ2
d
dξ
ξ2 dθ
dξ
= −θn
, (2.3.10)
as found in Shapiro and Teukolsky (2004).
The boundary conditions for a star governed by equation (2.3.10) are given at the centre
of the star by Shapiro and Teukolsky (2004) as:
θ(0) = 1 (2.3.11)
θ (0) = 0 (2.3.12)
The first condition (2.3.11) is evident from equation (2.3.6). For the second condition
(2.3.12) note that around r = 0, the centre of the star, the mass is given approximately
by
m(r) ≈
4πρcr3
3
, (2.3.13)
which, using the hydrostatic equilibrium equation (2.1.7), gives
dP
dr
= 0 =
dρ
dr
(2.3.14)
which then leads to condition (2.3.12) (Shapiro and Teukolsky, 2004).
11
12. 2.4 Solutions of the Lane-Emden Equation
2.4.1 Mass-Radius Relationship
Explicit solutions for θ seem to exist only for n = 0, 1, 5; otherwise, the Lane-Emden
equation (2.3.10) can be integrated numerically (Chandrasekhar, 2010). Starting with
ξ = 0, with the polytropic index n < 5, the solutions decrease monotonically with a zero
at a finite value θ(ξ1) = 0. The point ξ = ξ1 is the surface of the star, where P = ρ = 0
and the radius of the star can be determined to be
R = aξ1 =
(n + 1)K
4πG
1
2
ρ
1−n
2n
c ξ1. (2.4.1)
So, using equations (2.3.6), (2.3.7), (2.3.10) and (2.3.8) along with dr = adξ the mass is
be found to be
M =
R
0
4πr2
ρ dr (2.4.2)
= 4πa3
ρc
ξ1
0
ξ2
θn
dξ (2.4.3)
= −4πa3
ρc
ξ1
0
d
dξ
ξ2 dθ
dξ
dξ (2.4.4)
= 4πa3
ρcξ2
1|θ (ξ1)| (2.4.5)
= 4π
(n + 1)K
4πG
3
2
ρ
3−n
2n
c ξ2
1|θ (ξ1)| (2.4.6)
Finally, using equations (2.4.6) and (2.4.1) we get a mass-radius relationship for poly-
tropes:
M = 4πR
3−n
1−n
(n + 1)K
4πG
n
n−1
ξ
3−n
1−n
1 ξ2
1|θ (ξ1)|. (2.4.7)
which is as found in Shapiro and Teukolsky (2004).
2.4.2 n = 0 Polytrope
For an n = 0 polytrope the Lane-Emden equation (2.3.10) becomes:
d
dξ
ξ2 dθ
dξ
= −ξ2
, (2.4.8)
which can be solved using separation of variables
d ξ2 dθ
dξ
= − ξ2
dξ =⇒ ξ2 dθ
dξ
= A −
ξ3
3
(2.4.9)
12
13. where A is some constant. Rearranging and again using separation of variables:
dθ =
A − ξ3
3
ξ2
dξ =⇒ θ = B −
A
ξ−1
−
ξ2
6
, (2.4.10)
where B is also some constant. Then applying the boundary condition θ(0) = 1:
1 = B −
A
0
−
02
6
(2.4.11)
Hence, A = 0 to avoid dividing by 0 and B = 1. Thus for a polytrope with polytropic
index n = 0:
θ(ξ) = 1 −
ξ2
6
; (2.4.12)
which has its first zero at ξ1 =
√
6, agreeing with Figure 3.
2.4.3 n = 1 Polytrope
The solution to a polytrope of index n = 1 can be found using the substitution Ξ(ξ) =
ξθ(ξ). This simplifies the Lane-Emden equation (2.3.10) to
Ξ + Ξ = 0 (2.4.13)
which has harmonic solutions. Applying the boundary condition (2.3.11) the solution is
found to be
θ(ξ) =
sin(ξ)
ξ
(2.4.14)
and this does satisfy the second boundary condition (2.3.12) since, using l’Hˆopital’s rule:
lim
ξ→0
ξ cos(ξ) − sin(ξ)
ξ2
= lim
ξ→0
−ξ sin(ξ) + cos(ξ) − cos(ξ)
2ξ
= −
1
2
lim
ξ→0
sin(ξ) = 0 (2.4.15)
For positive ξ the first zero is found at ξ = π. As shown in Figure 3.
Using the radius equation (2.4.1) and mass equation (2.4.6) the central density of a
polytrope with polytropic index n = 1, a mass of 1.4 solar masses and a radius of 10km
can be shown to be ρc = 2.187 × 1027
kg km−3
.
2.4.4 n = 5 Polytrope
The solution for a polytrope of index n = 5 is explicitly shown in Chandrasekhar (2010,
p. 93-94). Here, it is briefly outlined since the solution is of interest. For the Lane-Emden
equation (2.3.10) make the change of variables:
θ = Axω
z, where ω =
2
n − 1
. (2.4.16)
Then the Lane-Emden equation (2.3.10) becomes
d2
z
dt2
+ (2ω − 1)
dz
dt
+ ω(ω − 1)z + An−1
zn
= 0, (2.4.17)
13
14. which simplifies to
d2
z
dt2
=
1
4
z(1 − z4
) (2.4.18)
Manipulating this expression and considering it in the infinite limits leads, eventually, to
a solution:
θ(ξ) =
1
1 + 1
3
ξ2
1
2
(2.4.19)
However, unlike in the n = 0 and n = 1 cases this has no finite value ξ1 where θ(ξ1) = 0
despite an explicit solution existing. Instead, this case is such that
lim
ξ→∞
θ(ξ) = 0 (2.4.20)
which can be seen in Figure 3 below.
Figure 3: The numerically integrated solutions to the Lane-Emden equation for
n = 0, 1, 2, 3, 4, 5 which are blue, red, green, orange, purple, brown respectively.
As expected, the first zero for the n = 0 case is found at ξ1 =
√
6, the first zero
for the n = 1 case it at ξ1 = π and the n = 5 polytrope case does not have
a zero. The plots for the n = 0, n = 1 and n = 5 cases reflect the solutions
for these cases which were found algebraically - for example the influence of the
sin function is evident in the n = 1 plot. The code to reproduce this graph
in Wolfram Mathematica is found in Appendix A (due to the factor of 1
ξ2 the
model is limited with the boundary condition (2.3.11) defined as θ 1
1010000 = 1
and so, with a suitable range, the n = 5 case in this model will have a zero).
14
15. 3 Relativistic Equation for the Structure of Stars
To begin working toward the Tollman-Oppenheimer-Volkoff equation a metric must be
decided upon.
3.1 Spherically Symmetric Spacetime Metric
3.1.1 Line Element
To begin, consider the line element of Minkowski space
ds2
= −dt2
+ dr2
+ r2
(dθ2
+ sin2
θ dφ2
), (3.1.1)
with the usual (r, θ, φ) coordinates. If both r and t are constant the line element of
Minkowski space (3.1.1) defines a two-dimensional spherical surface (Schutz, 2009). The
line element on such a two-sphere is given by equation (3.1.1) with dt = dr = 0
dl2
= r2
(dθ2
+ sin2
θ dφ2
) := r2
dΩ2
. (3.1.2)
Then any two-surface whose line element is equation (3.1.2) and where r2
is independent
of θ and φ has the geometry of a two-sphere (Schutz, 2009).
To call spacetime spherically symmetric then, is to infer that every point of spacetime is
on a two-surface which is in fact a two-sphere, so it has line element
dl2
= f(r , t)(dθ2
+ sin2
θ dφ2
) (3.1.3)
where the function f(r , t) is an unknown function. The area of each particular sphere
is 4πf(r , t) (Schutz, 2009). For the spherical geometry under consideration define its
radial coordinate r such that f(r , t) = r2
. Hence, as above, any surface with r and t
constant is a two sphere and has area 4πr2
and circumference 2πr. The coordinate r is
not defined in such a way that there is an assumed relationship between itself and the
proper distance from the centre of the two-sphere to its circumference. Indeed r is defined
by the properties of the particular sphere itself and hence the centre of the sphere does
not necessarily have to be a point on the sphere itself - in fact there may not be a point
at the centre of the sphere (Schutz, 2009). Consider Figure 4, for example, which shows
a space consisting of two planes joined by a circular throat (Schutz, 2009). This space
consists of circles but their ‘centres’, which here are axis around which there is symmetry,
are not part of the space. However, if φ is the angle about the axis through the centre
then the line element for each circle is r2
dφ2
and then r is a constant which defines each
circle. This coordinate r has exactly the same definition as will be used in the spherical
spacetime (Schutz, 2009).
3.1.2 Metric
Consider two two-spheres, one at r and one at r + dr, who both use the same coordinate
system (θ, φ). At present there is no relationship between them, however, it would make
15
16. Figure 4: Space made up of two planes connected by a throat from Schutz (2009,
p. 257). There is a circular symmetry about the throat but the centres of the
two-spheres, circles in this case, do not belong to the space.
sense if there were some consistency between how the spheres are oriented. Hence, a
line where θ and φ are constant is orthogonal to the two-spheres. A two dimensional
illustration of this is pictured in Figure 5. By definition this line will have er as a tangent.
Now, eθ and eφ both lie in the two-spheres, therefore it is required that er · eθ = 0 and
er · eφ = 0. It follows that grθ = grφ = 0 (Schutz, 2009). Hence, the metric has been
restricted to the following
ds2
= g00 dt2
+ 2g0r drdt + 2g0θ dθdt + 2g0φ dφdt + grr dr2
+ r2
dΩ2
(3.1.4)
In order to avoid a preferred direction in space it must be the case that, since the whole
spacetime is spherically symmetric, there is a line where r, θ and φ are constant which
is orthogonal to the two-spheres. Similarly to above, this brings the conclusion that
gtθ = gtφ = 0 (Schutz, 2009). So the metric (3.1.4) becomes
ds2
= g00dt2
+ 2g0r drdt + grr dr2
+ r2
dΩ2
(3.1.5)
3.2 Static Spherically Symmetric Spacetime Metric
A static spacetime is defined as one in which the time coordinate t satisfies the following
two properties as given in Schutz (2009, p. 258)
i) all metric components are independent of t
ii) the geometry is unchanged by time reversal, t → −t
Should only the first property be satisfied then the spacetime is called stationary (Schutz,
2009). The coordinate transformation under the second condition is defined by Λ
¯0
0 = −1
and Λi
j = δi
j and therefore
g¯0¯0 = (Λ0
¯0)2
g00 = g00 (3.2.1)
g¯0¯r = Λ0
¯0Λr
¯rg0r = −g0r (3.2.2)
g¯r¯r = (Λr
¯r)2
grr = grr (3.2.3)
16
17. t
y
x
et
eφ
r = constant, φ = constant
rφ
Figure 5: A two sphere with coordinate system (t, r, φ). By ensuring that the
line φ = constant is orthogonal to the two spheres it ensures all the poles that
define the two spheres lie in the same orientation. If this was not the case it
would be hard to define relationships between different two spheres.
as given in Schutz (2009). Then, since the geometry under the second condition must
remain unchanged, such that gαβ = g¯α¯β, g0r ≡ 0, the metric for a static spherically
symmetric spacetime is
ds2
= −e2ϕ
dt2
+ e2Λ
dr2
+ r2
dΩ2
(3.2.4)
Where, as in Schutz (2009), the functions ϕ(r) and Λ(r) have been introduced. These
are in place of the unknowns g00 and grr and this is an acceptable replacement as long
as g00 < 0 and grr > 0.
Far from a star it would be expected that spacetime is flat. Such a requirement gives rise
to the conditions
lim
r→∞
ϕ(r) = lim
r→∞
Λ(r) = 0, (3.2.5)
in order that the metric for Minkowski spacetime, equation (3.1.1), arises. Under these
conditions spacetime is called asymptotically flat (Schutz, 2009).
3.3 Physical Meaning of the Metric
As explained in Schutz (2009) the metric components have physical interpretations since
they are built upon physical symmetries. The first of these is the proper radial distance,
from one radius r1 to another r2:
l12 −
r2
r1
eΛ
dr (3.3.1)
because the distance from one radius to another has constant t, θ and φ. Since the metric
is independent of time Chapter 7 of Schutz (2009) explains that any particle following a
17
18. geodesic has a constant momentum p0 which is defined here as:
p0 := −E (3.3.2)
Suppose however, there was a local inertial observer momentarily at rest, then their four
velocity will have
Ui
=
dxi
dτ
= 0 (3.3.3)
and the normalisation condition U · U = −1 gives U0
= e−ϕ
. Therefore the energy that
the inertial observer will measure is given as:
E∗
= −U · p = e−ϕ
E (3.3.4)
Therefore a particle with a geodesic characterised by a constant E has energy e−ϕ
E,
relative to a local inertial observer who is momentarily at rest. From the conditions (3.2.5)
the e−ϕ
factor becomes 1 far away so that E is the energy of the particle measured by
an observer when the particle moves far away - called the energy at infinity. Everywhere
else e−ϕ
> 1, so the particle will have larger energy to close inertial observers it passes.
This additional energy is the kinetic energy it gains from falling in a gravitational field.
Schutz (2009) explains that this gain in energy is significant when examining photons.
Take a photon emitted at r1 and received far away; suppose its frequency in the local
inertial frame is vem, then its local energy is hvem (where h is Planck’s constant) and
its conserved constant E is hvemeϕ(r1)
. When it reaches the far away observer its energy
is measured as E and hence its frequency is E
h
= vrec = vemeϕ(r1)
. The redshift of the
photon:
z =
λrec − λem
λem
=
vem
vrec
− 1 (3.3.5)
is given by
z = e−ϕ(r1)
− 1 (3.3.6)
which attaches physical meaning to eϕ
(Schutz, 2009).
3.3.1 Einstein Tensor
By calculating the Ricci tensor and Ricci scalar the Einstein tensor for the static spher-
ically symmetric tensor can be found. The Einstein tensor is defined by Schutz (2009)
as:
Gαβ
≡ Rαβ
−
1
2
gαβ
R, (3.3.7)
where R is the Ricci scalar and Rαβ
the Ricci tensor as given in Schutz (2009):
Rαβ = Rµ
αµβ = Γµ
αβ,µ − Γµ
αµ,β + Γµ
γµΓγ
αβ − Γµ
γβΓγ
αµ (3.3.8)
18
19. Thus, the Einstein tensor has only the following four components since all others vanish
(as a result of the metric being diagonal, which means the Ricci tensor is diagonal also):
G00 =
1
r2
e2ϕ d
dr
[r(1 − e−2Λ
)] (3.3.9)
Grr = −
1
r2
e2Λ
(1 − e−2Λ
) +
2
r
ϕ (3.3.10)
Gθθ = r2
e−2Λ
ϕ + (ϕ )2
+
ϕ
r
− ϕ Λ −
Λ
r
(3.3.11)
Gφφ = sin2
θGθθ (3.3.12)
where ϕ := dϕ
dr
etc. (Schutz, 2009).
3.3.2 Stress-Energy Tensor
Since the stars under consideration are static the only non-zero component of the four-
velocity is U0
. The normalisation condition, U · U = −1, for the four velocity then
implies:
U0
= e−ϕ
and U0 = −eϕ
. (3.3.13)
The stress energy tensor, defined in Schutz (2009) as
T = (ρ + p)U ⊗ U + pg−1
(3.3.14)
then has only four components:
T00 = ρe2ϕ
(3.3.15)
Trr = pe2Λ
(3.3.16)
Tθθ = r2
p (3.3.17)
Tφφ = sin2
θTθθ (3.3.18)
3.3.3 Equation of State
The pressure and density, which appear in the stress-energy tensor, can be related by an
equation of state; like the polytrope equation (2.2.1) in the Newtonian model. Given a
simple fluid in local thermodynamic equilibrium the pressure can be expressed in terms
of the energy density and specific entropy:
p = p(ρ, S). (3.3.19)
In most cases the entropy can be assumed constant so that:
p = p(ρ) (3.3.20)
A relation of this form is assumed to exist, though it will differ for different fluids (Schutz,
2009).
19
20. 3.3.4 Equations of Motion
The equation of motion for the fluid comes from the conservation laws as given in Schutz
(2009):
Tαβ
;β = 0 (3.3.21)
Out of the four equations, one for each value of α, only one does not vanish, which is
when the free index α = r. Even when α = r there are many terms to consider:
Trβ
;β = ∂βTrβ
+ Γr
µβTµβ
+ Γβ
µβTrµ
(3.3.22)
= ∂βTrβ
+
1
2
grγ
(gγβ,µ + gγµ,β − gβµ,γ)Tµβ
(3.3.23)
+
1
2
gβη
(gηβ,µ + gηµ,β − gβµ,η)Trµ
(3.3.24)
However, the first term vanishes if β = r, the second term vanishes if µ = β and γ = r
whilst the third term vanishes when µ = r and β = η. Using these to simplify calculations
the following equation is derived:
(ρ + p)
dϕ
dr
= −
dp
dr
(3.3.25)
This equation (3.3.25) describes the pressure gradient required to keep the fluid static in
the surrounding gravitational field dϕ
dr
(Schutz, 2009).
3.3.5 Einstein Equations
The only interesting information from the Einstein equations comes from the (0, 0) and
(r, r) components. The (θ, θ) and (φ, φ) components are almost the same, which upon
consideration of the same components in the Einstein and stress-energy tensors makes
sense (see equations (3.3.11), (3.3.12), (3.3.17) and (3.3.18)). And in fact the Bianchi iden-
tities ensure that these components are a direct result from equations (3.3.21), (3.3.30)
and (3.3.32) (Schutz, 2009).
It is now important to define a new function m(r), which will replace Λ(r) (Schutz, 2009):
m(r) :=
1
2
r(1 − e−2Λ
) (3.3.26)
and this then gives a new definition for the grr term of the metric:
grr = e2Λ
= 1 −
2m(r)
r
−1
. (3.3.27)
Einstein’s equations are given by Schutz (2009, p. 187) as
Gαβ = 8πTαβ. (3.3.28)
Therefore, using equations (3.3.9) and (3.3.15), the (0, 0) component of the Einstein
equations is
1
r2
e2ϕ d
dr
[r(1 − e−2Λ
)] = 8πρe2ϕ
, (3.3.29)
20
21. which gives
dm(r)
dr
= 4πr2
ρ. (3.3.30)
This is very similar to equation (2.1.2) from the Newtonian model. In the Newtonian
model m(r) is called the mass inside a sphere of radius r. Here, m(r) is called the mass
function as well, however, it is not interpreted as the mass of the star since energy is not
localisable in general relativity (Schutz, 2009).
The (r, r) component of the Einstein equations then, using equations (3.3.10) and (3.3.16),
is
−
1
r2
e2Λ
(1 − e−2Λ
) +
2
r
ϕ = 8πpe2Λ
(3.3.31)
which rearranged and using equation (3.3.27), gives
dϕ
dr
=
m(r) + 4πr3
p
r(r − 2m(r))
(3.3.32)
There are now four equations to allow us to find the four unknowns ϕ, m, p, ρ provided
the equation of state is of the form as given in equation (3.3.20). The four equations are
(3.3.20), (3.3.25), (3.3.30) and (3.3.32).
3.4 Schwarzschild Metric
Outside the star the pressure and density are zero and so equations (3.3.30) and (3.3.32)
respectively become:
dm
dr
= 0 (3.4.1)
dϕ
dr
=
m
r(r − 2m)
(3.4.2)
These can be directly solved, applying the requirement that limr→∞ ϕ = 0 (Schutz, 2009).
Their respective solutions are
m(r) = M = constant (3.4.3)
e2ϕ
= 1 −
2M
r
(3.4.4)
Applying equations (3.3.27) and (3.4.4) the exterior metric takes the form:
ds2
= − 1 −
2M
r
dt2
+ 1 −
2M
r
−1
dr2
+ r2
dΩ2
. (3.4.5)
This metric is called the Schwarzschild metric (Schutz, 2009).
3.5 The Tolman-Oppenheimer-Volkov Equation
The main result of this section is the Tolman-Oppenheimer-Volkov (T-O-V) equation.
This equation gives the internal structure of a static spherically symmetric star. Substi-
tuting equation (3.3.25) into equation (3.3.32) gives:
−1
ρ + p
dp
dr
=
m(r) + 4πr3
p
r(r − 2m(r))
(3.5.1)
21
22. which, upon rearrangement, gives the T-O-V equation:
dp
dr
= −
(ρ + p)(m + 4πr3
p
r(r − 2m)
. (3.5.2)
The T-O-V equation (3.5.2) combined with equation (3.3.30) and an equation of state
that does not depend on entropy, i.e. of similar form to equation (3.3.20), provides enough
information to calculate the unknowns m, ρ, p. Then equation (3.3.25) can be used to
calculate ϕ (Schutz, 2009).
3.6 Solutions to the T-O-V Equation
There are two well known solutions to the T-O-V equation.
3.6.1 Schwarzschild Constant-Density Solution
By making the assumption that ρ is constant the solution of equations (3.3.30) and (3.5.2)
becomes simplified. This assumption has no physical justification, though dense neutron
stars have almost uniform density so it does provide some insight (Schutz, 2009).
Equation (3.3.30) can be integrated instantly under this assumption:
m(r) =
4πρr3
3
, r R (3.6.1)
where R is the star’s radius which is yet to be found. Outside of the star, i.e. for r R, the
density vanishes and so m(r) becomes constant (Schutz, 2009). m(r) must be continuous
at R upon demanding that grr be continuous. Therefore
m(r) =
4πρR3
3
:= M, r R (3.6.2)
where M is the Schwarzschild mass. Using equation (3.6.1) the T-O-V equation (3.5.2)
becomes
dp
dr
= −
4
3
πr
(ρ + p)(ρ + 3p)
1 − 8πr2ρ
4
(3.6.3)
Using partial fractions to integrate this from a central pressure pc at r = 0 to some radius
r and pressure p gives
ρ + 3p
ρ + p
=
ρ + 3pc
ρ + pc
1 −
2m
r
1
2
(3.6.4)
which is as found in Schutz (2009). Consider equation (3.6.4) at r = R:
1 =
ρ + 3pc
ρ + pc
1 −
2M
R
1
2
(3.6.5)
Then using equation (3.6.2) an explicit expression for the radius of the star is found:
R2
=
3
8πρ
1 −
(ρ + pc)2
(ρ + 3pc)2
. (3.6.6)
22
23. This can be rearranged to give an explicit expression for pc:
pc = ρ
1 − (1 − 2M
R
)
1
2
3(1 − 2M
R
)
1
2 − 1
. (3.6.7)
From this is can be seen that as M
R
→ 4
9
the central density ρc → ∞ which is a general
limit on M
R
(Schutz, 2009).
Substituting equation (3.6.7) into equation (3.6.4) gives:
pc = ρ
(1 − 2Mr2
R3 )
1
2 − (1 − 2M
R
)
1
2
3(1 − 2M
R
)
1
2 − (1 − 2Mr2
R3 )
1
2
(3.6.8)
ϕ can be found from equation (3.3.25) and noting that the value of ϕ at R is implied by
the continuity of g00 (Schutz, 2009):
g00(R) = 0 1 −
2M
R
(3.6.9)
Using results from this section ϕ is then solved as:
eϕ
=
3
2
1 −
2M
R
1
2 −
1
2
1 −
2Mr2
R3
2
1
, r R (3.6.10)
As r increases ϕ and m monotonically increase, whilst p monotonically decreases (Schutz,
2009).
3.6.2 Buchdahl’s Interior Solution
Another solution, which has no physical consequence, was discovered by Buchdahl (1981)
for the equation of state:
ρ = 12(p∗p)
1
2 − 5p (3.6.11)
where p∗ is some arbitrary constant. As noted in Schutz (2009, p. 267) this solution has
two properties which make it particularly interesting:
i) it can be made causal everywhere in the star by demanding that the local sound speed
dp
dρ
1
2
be less than 1
ii) for small p it reduces to
ρ = 12(p∗p)
1
2 (3.6.12)
The second condition is similar to the Newtonian polytrope for n = 1, see equation
(2.2.1). Hence, Buchdahl’s solution can be seen as a relativistic version of the n = 1
polytrope case, adding to its intrigue. Equation (3.6.11) along with the first condition
above implies
dρ
dp
=
6p
1
2
∗
p
1
2
− 5 =⇒
6p
1
2
∗
p
1
2
− 5
−1
< 1 =⇒ p < p∗ (3.6.13)
23
24. Figure 6: Plot of equation (3.6.11). Its maximum is at p = 6
5
p∗
2
whilst its
zero is at p = 144
25
p∗.
Consider the plot of equation (3.6.11) in Figure 6. The function is monotonic and has a
maximum at p = 6
5
p∗
2
whilst its zero is at p = 144
25
p∗. These facts, combined with the
inequality p < p∗ lead to:
ρ < 12(p∗p∗)
1
2 − 5p∗ = 7p∗ (3.6.14)
Hence, causality leads to the conditions:
p < p∗ and ρ < 7p∗ (3.6.15)
To find a solution a new radial coordinate r is introduced which is defined implicitly in
terms of r. To do so requires another arbitrary constant and a new function:
u(r ) := β
sin Ar
Ar
, A2
:=
288πp∗
1 − 2β
(3.6.16)
Then the new radial coordinate is defined as:
r(r ) = r
1 − β + u(r )
1 − 2β
(3.6.17)
with definitions take from Schutz (2009). Similarly to the solution for an n = 5 polytrope
in section 2.4.4, Buchdahl’s solution is not calculated but written down by Schutz (2009,
p. 268) in terms of the metric (3.2.4) as:
e2ϕ
= (1 − 2β)(1 − β − u)(1 − β + u)−1
(3.6.18)
e2Λ
= (1 − 2β)(1 − β + u)(1 − β − u)−1
(1 − β + β cos Ar )−2
(3.6.19)
p(r) = A2
(1 − 2β)u2
[8π(1 − β + u)2
]−1
(3.6.20)
ρ(r) = 2A2
(1 − 2β)u 1 − β −
3
2
u [8π(1 − β + u)2
]−1
(3.6.21)
24
25. The surface, where the pressure is 0, is at u = 0, which is when r = π
A
≡ R . At this
point it is an immediate consquence that
e2ϕ
= e−2Λ
= 1 − 2β (3.6.22)
from equations (3.6.18) and (3.6.19). Equation (3.6.17) at R then directly gives
R ≡ r(R ) =
π(1 − β)
A(1 − 2β)
(3.6.23)
just as in Schutz (2009). The mass of the star is given by Schutz (2009) as:
M =
πβ(1 − β)
A(1 − 2β)
=
π
288p∗(1 − 2β)
1
2
β(1 − β) (3.6.24)
Hence, β is the ratio of the mass to the radius at the surface. Due to equations (3.6.22)
and (3.3.6), β is related to the redshift of the star through the relationship:
zs =
1
(1 − 2β)
1
2
− 1 (3.6.25)
Clearly as β → 0 the redshift goes to 0 and is the nonrelativistic limit. β then, when
varied, will create changes in the model and it is appropriate to find limits on it. As
already noted its lower limit is above zero. To find an upper limit note that equations
(3.6.20) and (3.6.21) give:
p
ρ
=
1
2
u
1 − β − 3
2
u
, (3.6.26)
which has maximum value when evaluated at the centre of the star r = 0
pc
ρc
=
β
2 − 5β
. (3.6.27)
Directly from the causality requirements (3.6.15) p
ρ
< 1
7
and this is true at the maximum
value pc
ρc
also. So, with some rearranging, β must satisfy:
0 < β <
1
6
(3.6.28)
This range of β spans from Newtonian models, when β ≈ 0, to ultra relativistic models
(Schutz, 2009).
25
27. 4 Conclusion
In conclusion there are two main results of this report. The first is the Newtonian stellar
structure equation for stars that have a polytropic equation of state:
1
ξ2
d
dξ
ξ2 dθ
dξ
= −θn
.
This has explicit solutions when the polytropic index is 0, 1 or 5. In other cases the
equation must be integrated numerically. The solutions for n = 0, 1, 2, 3, 4 and 5 are
plotted in Figure 3. If the polytropic index is known, along with the mass, radius and the
value ξ1 (which is the first zero of the solution, i.e. θ(ξ1) = 0) then the central density
can be calculated using the mass-radius relationship from section 2.4.1.
The second main result is the Tolman-Oppenheimer-Volkov equation:
dp
dr
= −
(ρ + p)(m + 4πr3
p
r(r − 2m)
.
This was arrived at by first defining a metric for a spherically symmetric static spacetime.
Using this metric, the Einstein tensor and the stress-energy tensor were calculated (see
sections 3.3.1 and 3.3.2). This allowed the calculation of the equations of motion to
be performed, which combined with the Einstein equation, led to the T-O-V equation.
Along the way an exterior metric for the star was calculated, called the Schwarzschild
metric. Schwarzschild also derived a constant-density solution to the T-O-V equation.
Whilst this has no particular physical support it is interesting since neutron stars have
near uniform density. The solution to this provided a general limit on the ratio M
R
, which
was 4
9
for a star in this spherically symmetric static spacetime. A second solution was also
explored which was derived by Buchdahl (1981), whilst more complicated, and therefore
only stated, the solution can be seen as a relativistic version of an n = 1 polytrope from
the Newtonian model.
27
29. Reference List
Buchdahl, H.A. (1981). Seventeen Simple Lectures on General Relativity Theory. John
Wiley & Sons Inc. isbn: 978-0471096849.
Chandrasekhar, S. (2010). An Introduction to the Study of Stellar Structure. Dover Pub-
lications. isbn: 978-0486604138.
Schutz, Bernard (2009). A First Course in General Relativity. Second edition. Cambridge
University Press. isbn: 978-0-521-88705-2.
Shapiro, Stuart L. and Saul A. Teukolsky (2004). Black Holes, White Dwarfs, and Neutron
Stars: The Physics Of Compact Objects. First edition. Wiley-VCH Verlag GmbH & Co.
KGaA, Weinheim. isbn: 978-0-47 1-873 16-7.
29
