277	10.9 Exercises

10.9E xe rc i s e s

1Starting with ds2 = ηαβ dxα dxβ , show that the coordinate transformation r = (x2 + y2 + z2)1/2, θ = arccos(z/r), φ = arctan(y/x) leads to Eq. (10.1), ds2 = −dt2 + dr2 + r2(dθ 2 + sin2 θ dφ2).

2In deriving Eq. (10.5) we argued that if et were not orthogonal to eθ and eφ , the metric would pick out a preferred direction. To see this, show that under rotations that hold t and r fixed, the pair (gθ t, gφt) transforms as a vector field. If these don’t vanish, they thus define a vector field on every sphere. Such a vector field cannot be spherically symmetric unless it vanishes: construct an argument to this effect, perhaps by considering the discussion of parallel-transport on the sphere at the beginning of § 6.4.

3The locally measured energy of a particle, given by Eq. (10.11), is the energy the same particle would have in SR if it passed the observer with the same speed. It therefore

contains no information about gravity, about the curvature of spacetime. By referring to Eq. (7.34) show that the difference between E and E in the weak-field limit is, for

particles with small velocities, just the gravitational potential energy.

4Use the result of Exer. 35, § 6.9 to calculate the components of Gμν in Eqs. (10.14)– (10.17).

5Show that a static star must have Ur = Uθ = Uφ = 0 in our coordinates, by examining the result of the transformation t → −t.

6(a) Derive Eq. (10.19) from Eq. (10.18).

(b)Derive Eqs. (10.20)–(10.23) from Eq. (4.37).

7Describe how to construct a static stellar model in the case that the equation of state has the form p = p(ρ, S). Show that we must give an additional arbitrary function, such

as S(r) or S(m(r)).

8(a) Prove that the expressions Tαβ ;β for α = t, θ , or φ must vanish by virtue of the assumptions of a static geometric and spherical symmetry. (Do not calculate the expressions from Eqs. (10.20)–(10.23). Devise a much shorter argument.)

(b)Derive Eq. (10.27) from Eqs. (10.20)–(10.23).

(c)Derive Eq. (10.30) from Eqs. (10.14), (10.20), (10.29).

(d)Prove Eq. (10.31).

(e)Derive Eq. (10.39).

9(a) Define a new radial coordinate in terms of the Schwarzschild r by

r = r¯(1 + M/2r¯)2.

(10.88)

Notice that as r → ∞, r¯ → r, while at the horizon r = 2M, we have r¯ = 12 M. Show that the metric for spherical symmetry takes the form

= −

$1

− M/2r

!

2 dt2

+ $

+

2r

!

4

¯

+ ¯

ds2

1

M/2r¯

1

M

[dr2

r2 d 2].

(10.89)

+ ¯

¯

(b) Define quasi-Cartesian coordinates by the usual equations x = r¯ cos φ sin θ , y = r¯ sin φ sin θ , and z = r¯ cos θ so that (as in Exer. 1), dr¯2 + r¯2 d 2 = dx2 + dy2 + d z2.

278	Spherical solutions for stars

Thus, the metric has been converted into coordinates (x, y, z), which are called isotropic coordinates. Now take the limit as r¯ → ∞ and show

ds2 = −

$1 − r

+ 0

r2 ! dt2

+ $1 +

r

+ 0

r2 !

(dx2

+ dy2 + dz2).

2M

1

2M

1

¯

¯

¯

¯

This proves Eq. (10.38).

10Complete the calculation for the uniform-density star.

(a)Integrate Eq. (10.48) to get Eq. (10.49) and fill in the steps leading to Eqs. (10.50)– (10.52) and (10.54).

(b)Calculate e and the redshift to infinity from the center of the star if M = 1 M = 1.47 km and R = 1 R = 7 × 105 km (a star like the Sun), and again if M = 1 M and R = 10 km (typical of a neutron star).

(c)Take ρ = 10−11 m−2 and M = 0.5 M , and compute R, e at surface and center, and the redshift from the surface to the center. What is the density 10−11 m−2 in kg m−3?

11Derive the restrictions in Eq. (10.57).

12Prove that Eqs. (10.60)–(10.63) do solve Einstein’s equations, given by Eqs. (10.14)– (10.17) and (10.20)–(10.23) or (10.27), (10.30), and (10.39).

13Derive Eqs. (10.66) and (10.67).

14A Newtonian polytrope of index n satisfies Eqs. (10.30) and (10.44), with the equation of state p = Kρ(1+1/n) for some constant K. Polytropes are discussed in detail by Chandrasekhar (1957). Consider the case n = 1, to which Buchdahl’s equation of state reduces as ρ → 0.

(a)Show that ρ satisfies the equation

r2		dr	r2 dr		+ K		ρ = 0,	(10.90)
1		d		dρ		2π
and show that its solution is
ρ = αu(r),			u(r) =		sin Ar		, A2 =	2π
									,
					Ar			K

where α is an arbitrary constant.

(b)Find the relation of the Newtonian constants α and K to the Buchdahl constants β and p by examining the Newtonian limit (β → 0) of Buchdahl’s solution.

(c)From the Newtonian equations find p(r), the total mass M and the radius R, and show them to be identical to the Newtonian limits of Eqs. (10.62), (10.67), and

(10.65).

15Calculations of stellar structure more realistic than Buchdahl’s solution must be done numerically. But Eq. (10.39) has a zero denominator at r = 0, so the numerical calcu-

lation must avoid this point. One approach is to find a power-series solution to Eqs. (10.30) and (10.39) valid near r = 0, of the form

279		10.9 Exercises
		m(r) = j		mjrj,
		p(r) = j	pjrj,
		ρ(r) = j		ρjrj.	(10.91)
		Assume that the equation of state p = p(ρ) has the expansion near the central density ρc
		p = p(ρc) + (pc c/ρc)(ρ − ρc) + · · · ,			(10.92)

where c is the adiabatic index d(lnp)/d(lnρ) evaluated at ρc. Find the first two nonvanishing terms in each power series in Eq. (10.91), and estimate the largest radius r at which these terms give an error no larger than 0.1% in any power series. Numerical integrations may be started at such a radius using the power series to provide the initial values.

16(a) The two simple equations of state derived in § 10.7, p = kρ4/3 (Eq. (10.81)) and p = ρ/3 (Eq. (10.87)), differ in a fundamental way: the first has an arbitrary dimen-

sional constant k, the second doesn’t. Use this fact to argue that a stellar model constructed using only the second equation of state can only have solutions in

which ρ = μ/r2 and m = νr, for some constants μ and ν. The key to the argument is that ρ(r) may be given any value by a simple change of the unit of length, but there are no other constants in the equations whose values are affected by such a change.

(b) Show from this that the only nontrivial solution of this type is for μ = 3/(56π ), ν = 3/14. This is physically unacceptable, since it is singular at r = 0 and it has no surface.

(c)Do there exist solutions which are nonsingular at r = 0 or which have finite surfaces?

17(This problem requires access to a computer) Numerically construct a sequence of stellar models using the equation of state

p =	kρ4/3,		ρ	(27 k3)−1		,	(10.93)
	1	ρ,	ρ		(27 k3)−1	,
	3

where k is given by Eq. (10.81). This is a crude approximation to a realistic ‘stiff’ neutron-star equation of state. Construct the sequence by using the following values for ρc : ρc/ρ = 0.1, 0.8, 1.2, 2, 5, 10, where ρ = (27 k3)−1. Use the power series developed in Exer. 15 to start the integration. Does the sequence seem to approach a limiting mass, a limiting value of M/R, or a limiting value of the central redshift?

18Show that the remark made before Eq. (10.80), that the nuclei supply little pressure, is true for the regime under consideration, i.e. where me < p2f /3 kT < mp, where k is Boltzmann’s constant (not the same k as in Eq. (10.81)). What temperature range is this for white dwarfs, where n ≈ 1037 m−3?

280	Spherical solutions for stars

19Our Sun has an equatorial rotation velocity of about 2 km s−1.

(a)Estimate its angular momentum, on the assumption that the rotation is rigid (uniform angular velocity) and the Sun is of uniform density. As the true angular velocity is likely to increase inwards, this is a lower limit on the Sun’s angular momentum.

(b)If the Sun were to collapse to neutron-star size (say 10 km radius), conserving both mass and total angular momentum, what would its angular velocity of rigid rotation be? In nonrelativistic language, would the corresponding centrifugal force exceed the Newtonian gravitational force on the equator?

(c)A neutron star of 1 M and radius 10 km rotates 30 times per second (typical of young pulsars). Again in Newtonian language, what is the ratio of centrifugal to gravitational force on the equator? In this sense the star is slowly rotating.

(d)Suppose a main-sequence star of 1 M has a dipole magnetic field with typical strength 1 Gauss in the equatorial plane. Assuming flux conservation in this plane, what field strength should we expect if the star collapses to radius of 10 km? (The Crab pulsar’s field is of the order of 1011 Gauss.)