PDF Exercises on Gravitational Collapse and Black Holes Conventions

Exercises on Gravitational Collapse and Black Holes

(Ph.D course by A. del Rio at the University of Valencia, 29.09.2020 - 29.10.2020)

Conventions:

We follow Wald’s book conventions. Namely, the metric signature is (−,+,+,+), the Riemann tensor is defined by [∇a,∇b]v_c=:R_abc^dv_d, the Ricci tensor isR_ab:=R^c_acb, and the scalar curvature isR:=g^abR_ab.

Problems:

1. (Compactness parameter)

Given a distribution of massM and characteristic scaleR, define an adimensional quantity that allows you to decide whether general relativistic effects must be taken into account to properly describe the dynamics of the system. Then, apply this criterium to explain why GR effects are important for Neutron stars (M ∼ 1,5M, R ∼10 km), while they can be safely neglected for both the Sun (M = 1M, R∼7×10⁵ km) and White Dwarfs (M ∼0,6M,R∼7000 km).

2. (Chandrasekhar mass limit, rigurous derivation)

(a) To obtain the equation of state of a degenerate, relativistic electron gas, recall from kinetic theory that the distribution functionf(~x, ~p) is defined so that _(2π^g

~)³f(~x, ~p)d³xd³pis the number of particles in a phase-space cell of volume d³xd³p. In the ideal gas approximationf is given by the Fermi–Dirac distribution, that forT = 0 (dead stars) isf = Θ(p_F− |~p|), withp_F the Fermi momentum. Show that

n_e:= 2 (2π~)³

Z

f(~x, ~p)d³p= x³

3π²λ³_e, (1)

P :=1 3

2 (2π~)³

Z

|~p|vf(~x, ~p)d³p= m_ec²

3π²λ³_eφ(x), (2)

where x:=pF/(mec),λe=~/(mec), and φ(x) :=

Z x 0

y⁴dy p1 +y² =3

8

x 2

3x²−1 p

1 +x²+ ln x+p

1 +x²

. (3)

The energy density is given byρ=µ_em_Hn_e, wherem_His the atomic mass unit, and the mean molecular weight µ_e is defined byµ⁻¹_e :=P

iZ_iX_i/Ai in which Z_i is the atomic number of an ion of typei, X_i is the mass fraction of this ion, and A_i is its atomic mass number. The equation of stateP =P(ρ) is determined in this parametric form.

(b) Introducing the parameters:

p0:= m_ec²

3π²λ³_e, ρ0:= µ_em_H

3π²λ³_e, r₀²:= 1 f_c²

p₀

4πGρ²₀, m0:= 4πf_c³ρ0r³₀, (4) and using the above results, show that the Newtonian equations of hydrostatic equilibrium, ^dp_dr =

−ρ^Gm_r2 , ^dm_dr = 4πr²ρ , are transformed to ^dx_dξ = −fc

√1+x² x

µ

ξ², ^dµ_dξ = _f¹3

cξ²x³, where ξ :=r/r0 and µ:=m/m₀. Finally, define a new variableθ by 1 +f_cθ:=√

1 +x²and get dθ

dξ =−µ

ξ², dµ

dξ =ξ²θ^3/2(θ+ 2/fc)^3/2 . (5)

Argue that these ODEs must be solved with boundary conditionsθ(ξ= 0) = 1,µ(ξ= 0) = 0 and are integrated untilξ₁, whereθ(ξ=ξ₁) = 0. This value determines the radius of the star byR=ξ₁r₀, and the mass by M =µ₁m₀ withµ₁:=µ(ξ=ξ₁).

(c) Simplify the above equations in the extreme relativistic regime, f_c>>1. Then, solve them nume- rically to get ξ1 = 6,897,µ1 = 2,018. Together with the approximation, fc ≈xc= (ρc/ρ0)^1/3, deduce the Chandrasekhar mass:M = 1,46_µ⁴2

2

M, which is independent of the central energy density.

3. (General form of the metric in any spherically symmetric spacetime)

(i) Consider the usual metric on the round 2-sphere, h =dθ²+ sin²θdφ², and let ξ =F(θ, φ)_∂θ^∂ + G(θ, φ)_∂φ^∂ a vector field on it. Show that the Killing equation (Lξh)ab= 0 gives

∂θF = 0, F+ tanθ∂φG= 0, ∂φF+ sin²θ∂θG= 0, (6) and that by integrating these equations the general solution is

ξ=asin(φ+b) ∂

∂θ+ [c+acos(φ+b) cotθ] ∂

∂φ (7)

where a, b, c∈R. Thus, a basis for the Lie algebra of KVF on the 2-sphere is ξ₁= ∂

∂φ, ξ₂= cosφ∂

∂θ −cotθsinφ ∂

∂φ, ξ₃= sinφ∂

∂θ+ cotθcosφ ∂

∂φ (8)

(ii) Consider now a spacetime metric gab(x), and a set of coordinates{t, r, θ, φ}, such that the three vector fields above are Killing vector fields. Show that the Killing equation for ξ1 implies that all metric componentsgab(x) are independent ofφ, and then show that the Killing equation forξ2implies:

gtθ =gtφ=grθ =grφ =gθφ = 0, gφφ =gθθsin²θ, and gtt,gtr,grr, gθθ independent of θ. Thus, (the Killing equation forξ3does not give extra information)

ds²=gtt(t, r)dt²+ 2gtr(t, r)dtdr+grr(t, r)dr²+gθθ(t, r)(dθ²+ sin²θdφ²) (9) Now, find suitable coordinate transformations such that the metric above takes the final form

ds²=−e^−2Φ(t,r)dt²+e^2Λ(t,r)dr²+r²(dθ²+ sin²θdφ²) (10) This is the general form of the metric in any spherically symmetric spacetime.

4. (TOV equations).

The metric of any spherically symmetric spacetime can always be written in the form (10), with Φ = Φ(t, r), Λ = Λ(t, r) arbitrary functions. Introduce the so-called relativistic mass-energy function m(t, r) defined by

e^−2Λ=: 1−2Gm(t, r)

r ≡f(t, r) (11)

Assume a body in hydrostatic equilibrium (i.e.m=m(r), Φ = Φ(r)), and that the body is described by a perfect fluid, with energy-momentum tensor

Tab= (ρ+p)uaub+pgab (12)

where ρ,p,u^a are the proper energy density, pressure and 4-velocity field of the fluid, respectively. In the coordinates above the normalized vector isu=√¹

−g⁰⁰

∂

∂t.

(i) Show that the ttcomponent of Einstein’s equationG_ab= 8πGT_ab leads to dm

dr = 4πr²ρ (13)

(ii) Show that the rrcomponent of Einstein’s equationGab= 8πGTab leads to dΦ

dr =− G

r²f(r)(m(r) + 4πr³p(r)) (14)

(iii) Show that ther component of the stress-energy conservation equation∇aT^a_b= 0, together with the above results, leads to

dp

dr =−G(ρ(r) +p(r))

r²f(r) (m(r) + 4πr³p(r)) (15)

These are the Tolman-Oppenheimer-Volkoff equations. They describe the hydrostatic equilibrium of static, spherically symmetric bodies.

5. (Maximum mass for neutron star in the incompressible fluid approximation)

(a) The Tolman-Oppenheimer-Volkoff (TOV) equations describe the hydrostatic equilibrium of static, spherically symmetric relativistic bodies. Given an equation of stateP=P(ρ), they must typically be integrated numerically. The case of an incompressible fluid (ρ =ρ0 constant) is illustrative and can be solved analytically. Solve TOV equations with boundary conditions p(r= 0) =pc,p(r=R0) = 0, m(r= 0) = 0,m(r=R0) =M, and show that

p(r) =ρ0c²

p1−Rr²/R³₀−p

1−R/R₀ 3p

1−R/R₀−p

1−Rr²/R³₀, (16)

where R = 2GM/c². The central pressure is then p_c = ¹⁻

√1−R/R0

3√

1−R/R₀−1. Using this formula show that there is a critical massMcthat cannot be exceeded if the body is to remain in hydrostatic equilibrium, and is given by

Mc = r 3

4πρ0

4c² 9G

^3/2

= 5,7M

4×10¹⁷kg/m³ ρ0

^1/2

. (17)

What is the value of this bound (in solar mass units) for typical neutron star densities (∼10¹⁷kg/m³∼ 10⁻¹³M/m³)? You can use the useful relation ^G_c2M= 1500 m.

(b) Solve TOV equations in the Newtonian limit, still in the case of an incompressible fluid, and find p(r) =−2πG

3 ρ²₀(r²−R²₀). (18)

Does the Newtonian result predict the existence of an upper mass limit? Justify your answer.

6. (Buchdahl’s theorem)

The existence of upper mass limits in GR is not an artifact of having restricted consideration to stars of uniform density. Buchdahl’s theorem states that any reasonable (ρ(r)≥0,dρ/dr≤0) static, spherically symmetric interior solution must haveM ≤^4R_9G in order to remain in equilibrium.

Using the spherically symmetric metric (10) and (11), it can be shown that the equation Grr = 8πG p e^2Λ= ^G_r^θθ₂ e^2Λ leads to

d dr



 q

1−^2mG_r r

d dre^−Φ



 = e^−Φ q

1−^2mG_r d dr

Gm(r)

r³ (19)

(i) Imposing ^dρ_dr ≤0, and using the fact that the interior solution of the star must join smoothly to the Schwarzschild solution, show that this formula implies

q

1−^2mG_r r

d

dre^−Φ≥ M G

R³ (20)

where M andRare the mass and the radius of the star.

(ii) Integrate from 0 toR to find e^−Φ(0)≤

r

1−2M G R −M G

R³ Z R

0

rdr q

1−^2mG_r

(21)

(iii) Argue that ^dρ_dr ≤0 impliesm(r)≥ ^{M Gr}_R3³.

(iv) Show then that e^−Φ(0)≤

r

1−2M G R − M

R³ Z R

0

rdr q

1−^{2M Gr}_R3 ²

=3 2

r

1−2M G R −1

2 (22)

and conclude that necessarily M ≤ ^4R_9G. 7. (Well definite causal character of geodesics)

Show that if a geodesic is timelike, null, spacelike, at a given point of spacetime, then it is timelike, null, spacelike at any other point in the curve.

8. (Maximally symmetric spacetimes)

(a) Let ζ be a Killing vector field. Prove that the set of all Killing vector fields on a spacetime form a vector space on the real numbers. Prove then that ∇_a∇_bζ_c=R_cba^dζ_d, whereR_abcd is the Riemann tensor. A corollary of the latter result is that a Killing vector fieldζ^a is completely determined by the values of ζ^a and L_ab := ∇_aζ_b at any particular point. Argue then that on a spacetime of dimension n there can be at most ⁿ⁽ⁿ⁺¹⁾₂ linearly independent Killing vector fields (and thereby at most an

n(n+1)

2 parameter group of isometries). An-dimensional spacetime that has ⁿ⁽ⁿ⁺¹⁾₂ isometries is called maximally symmetric.

(b) Prove that the Riemann tensor of a maximally-symmetric,n-dimensional spacetime satisfiesRabcd=

R

n(n−1)[gacgbd−gbcgad], where R is the scalar curvature. First, use the previous result and the Ricci identity to write∇_[a∇_b]∇_cξ_d in terms of the Riemann tensor and to find

(R_abc^eδ_d^g+R_abd^gδ_c^e−R_dcb^gδ_a^e+R_dca^gδ_b^e)∇eξ_g= (∇aR_dcb^e− ∇bR_dca^e)ξ_e . (23) Now, if the space is maximally symmetric one can always find, at any point, a basis whose Killing vectors vanish but their derivative does not. This impliesR_abc^[eδ^g]_d+R_abd^[gδ^e]c−R_dcb^[gδ^e]a+R_dca^[gδ_b^e] = 0.

Contracting indices and using properties of the Riemann tensor one gets the desired result.

9. (Symmetries and conserved quantities)

SupposeV^a is a Killing vector field andT_abis any stress-energy tensor.

(i) Show thatJ^a:=T^abVb is a conserved current. InterpretJ^a whenV is timelike.

(ii) Given a spacelike hypersurface Σ, ifT_ab has compact spatial support (i.e. that vanishes outside a closed and bounded set in any spacelike hypersurface), show that the integral

QV :=

Z

Σ

T^a_bV^bnadΣ (24)

where n^a is a unit timelike vector normal to Σ, is independent of the choice of spacelike hypersurface Σ. Thus,Q_V is “conserved in time”.

10. (Birkhoff’s theorem)

The non-vanishing components of the Einstein tensorG_abassociated to the metric (10) of any spheri- cally symmetric spacetime are given by

G_tt = 2e^−2(Λ+Φ)Λ_,r

r +e^−2Φ1−e^−2Λ(t,r)

r² (25)

Gtr = 2Λ,t

r (26)

G_rr = −2Φ_,r

r +1−e^2Λ(t,r)

r² (27)

Gθθ = Gφφ

sin²θ =−r²e^−2Λ(Φ,rr−Φ²_,r−Φ,rΛ,r+Φ,r+ Λ,r

r )−r²e^2Φ(Λ,tt+ Λ²_,t+ Φ,tΛ,t) (28) Show that any vacuum spherically symmetric spacetime is locally isometric to Schwarzschild solution.

11. (Geodesics of Schwarzschild solution)

Consider a test particle that orbits a Schwarzschild black hole of mass M with non-zero angular mo- mentum per unit massL. Without loss of generality, (due to spherical symmetry and parity invariance) we can restrict attention to study equatorial geodesics. Consider k with values k = 0 and k = 1 for massless (null) and massive (timelike) test particles, respectively.

(i) Show that the orbits satisfies:

1

2r˙²+V(r) = 1

2E² (29)

where E is a constant, V(r) = ¹₂(1−^2M_r )(^L_r2² +k). This equation shows that the radial motion of a geodesic is the same as that of a unit mass particle of energy E²/2 in an ordinary one-dimensional effective potential V(r). [Hint: no need to use the geodesic equation, make use of the constants of motion]

(ii) Show that the orbits further satisfies d²u

dφ² +u= M k

L² + 3M u² (30)

where u= 1/randφis the azimutal angle.

(iii) Show that unstable circular geodesics of photons exist at radius R= 3M (this is called the ”light ring”)

(iv) Show that no stable timelike circular geodesics exists forR <6M (R= 6M is called the innermost stable circular orbit (ISCO)).

12. (Light escaping to infinity)

A photon is emitted outward from a pointP outside a Schwarzschild black hole with radial coordinate r in the range 2M < r < 3M. Show that if the photon is to reach infinity the angle ψ its initial direction makes with the radial direction (as determined by a stationary observer atP) cannot exceed

sinψ= s

27M² r²

1−2M

r

(31) [Hint: find a local orthonormal basis at each point, {e^a_i(x)}_i=t,r,θ,φ, that satisfies g_ab = η_ijeⁱ_ae^j_b with η_ij= diag(−1,1,1,1) such thate^a_t is the observer’s 4-velocity (this is called a “4-bein”); ifu^a denotes the tangent vector of the light geodesic, calculate first tanψ= ^u_u^a_a^e_e^φa_r

a

and then use it to obtain sinψ]

13. (Geodesic incompleteness of Schwarzschild black hole)

Show that in region II of the Kruskal manifold one may regard ras a time coordinate and introduce a new spatial coordinatexsuch that

ds²=− 1

2m

r −1dr²+ 2m

r −1

dx²+r²(dθ²+ sin²θdφ²) (32) Hence show that every timelike curve in region II intersects the singularity atr= 0 within a proper time no greater than πM. For what curves is this bound attained?

14. (Vaidya spacetime)

Take Schwarzschild metric in outgoing Eddington-Finkelstein coordinates, and letM be now a function of retarded time, M =M(u). Show that this metric is a solution of non-vacuum Einstein’s equations, and find the correspondingTab. Give a physical interpretation.

15. (Gravitational Redshift formula)

Consider two static observers in Schwarzschild geometry (i.e. observers whose 4-velocity is tangent to the static Killing fieldk= _∂t^∂)O1, O2 whose 4-velocities areu1,u2, respectively. SupposeO1 emits a light signal at eventP1which is received byO2 at eventP2. The light signal travels on a null geodesic, with tangent vector ξ^a. The frequency of emission is ω1 =−ξau^a₁|P₁ while the frequency of reception isω2=−ξau^a₂|P₂. Show that

ω₁ ω2

= s

1−2M/r₂ 1−2M/r1

(33) Given the maximum value Rmax=^9M₄ found in a previous exercise for a static spherically symmetric body in equilibrium, show that the maximum redshift of light emitted from the surface of a star is

z= ω1

ω2

_max

−1 = 2. (34)

This rules out the possibility that observed redshifts greater than 2 (as commonly found for quasars) could arise solely from the gravitational redshift, indicating that the corresponding sources are located at cosmological distances.

16. (Conformal Killing field)

A vector field ξis said to be a conformal Killing vector if

(L_ξg)_ab= Λ²g_ab (35)

for some non vanishing function Λ. It is said that ξ is the generator of conformal transformations of the metric.

(a) If ξ is a Killing vector of g_ab, (L_ξg)_ab = 0, show that ξ is then a conformal Killing vector of the conformally transformed metric Λ²gab for any non-vanishing conformal factor Λ.

(b) Show that the action for amasslesstest particle S[x, e] = 1

2 Z

dλe⁻¹(λ)gab(x(λ)) ˙x^a(λ) ˙x^b(λ) (36) is invariant, to first order in the constantα, under the transformation

x^a→x^a+αξ^a, e→e+α

4eg^ab(Lξg)ab (37)

if and only if ξis a conformal Killing vector. Would the action of a massive test particle be invariant under the same transformation?

(c) In view of the above result, explain why Carter-Penrose diagrams provide accurate descriptions of the causal structure of the spacetime.

17. An observer (not necessarily freely-falling) orbits a Kerr black hole in the equatorial plane (θ=^π₂).

(a) Let his orbit be at constant r. Define Ω = ^dφ_dt to be his angular velocity relative to a distant stationary observer. In terms of Ω,r,M, a, find the (normalized) observer’s 4-velocityu^a andua. (b) Suppose that the circular orbit lies in the ergosphere. Show that the observer cannot remain at rest with respect to the distant observer. That is, show that Ω must be nonzero.

(c) If the observer is in the regionr₋< r < r₊, show that he cannot remain at constant radius.

18. (Kepler law for Kerr spacetime)

Show that the generalized Kepler law for circular, equatorial orbits around a Kerr black hole of mass M and angular momentum per unit massais

Ω²_± :=

dφ dt

²

= M

(±r^3/2+aM^1/2)² (38)

[Hint: focus on thercomponent of the geodesic equation] What are the two signs physically indicating?

19. (Geodesic equation in Kerr)

The equations governing geodesic motion in the Kerr spacetime were given without justification. Here we provide a derivation, which is valid both for timelike and null geodesics.

(a) By definition, a Killing tensor fieldξabis one which satisfies the equationξ_(ab;c)= 0. Show that if ξ_ab is a Killing tensor and u^a satisfies the geodesic equationu^au^b_;a = 0, thenξ_abu^au^b is a constant of the motion.

(b) Verify that ξαβ = ∆k(αlβ)+r²gαβ is a Killing tensor field of the Kerr spacetime. Here l =

r²+a²

∆ ∂t−∂r+_∆^a∂φ andk= ^r2+a_∆ ²∂t+∂r+_∆^a∂φ are the null vectors that were introduced to derive the Kerr metric in Kerr coordinates.

(c) Write the equations E = −t^aua, L = −φ^aua explicitly in terms of u^a = ( ˙t,r,˙ θ,˙ φ). Then invert˙ these relations to obtain the equations for ˙t, ˙φ. [Hint: Make sure to involve the inverse metric.]

(d) The Carter constantQis defined byξabu^au^b=Q+ (L−aE)². By working out the left-hand side, derive the equation for ˙r. [Hint: Expresska andla in terms of the Killing vectors, and thenξabu^au^b in terms ofE ,L, and ˙r². ]

(e) Finally, use the normalization condition gabξ^aξ^b = −a, (where a = 1 for timelike geodesics and a= 0 for null geodesics) to obtain the equation for ˙θ.

20. (Killing horizons and surface gravity of a Kerr BH).

The Kerr metric can be expressed as ds²=−

1−2M r ρ²

dv²+ 2dvdr−2asin²θdrdψ−4M arsin²θ

ρ² dvdψ+ Σ

ρ²sin²θdψ²+ρ²dθ², (39) whereρ²=r²+a²cos²θ, and Σ = r²+a²2

−a²∆ sin²θwith ∆ =r²+a²−2M r. In these coordinates {v, r, θ, ψ}, called “ingoing”, the metric is well behaved on the Future Event Horizon r =r₊ [where r₊=M+√

M²−a²is one of the two roots of ∆(r) = 0].

(a) Show that ξ = ξ^{a ∂}_∂xa = _∂v^∂ + ΩH ∂

∂ψ, where ΩH = _a2+r^a ₊², is a Killing Vector Field of the Kerr metric [no need of any explicit calculation!]. Then show that, on r = r+, we have ξa = gabξ^b = (1−aΩHsin²θ)∇arandξaξ^a = 0. Justify, in light of these results, that the Event Horizon is a Killing Horizon ofξ.

(b) To every Killing Horizon we can associate a quantity called surface gravityκ, defined byξ^b∇bξ^a= κξ^a. Show that this equation is equivalent to ∇a(−ξ^bξb) = 2κξa.

(c) Use this alternative expression to derive the surface gravity of the Kerr BH:

κ=r+−M r²₊+a² =

√M²−a²

r²₊+a² . (40)

You may take the following steps. First, find the expression:

gabξ^aξ^b=gψψ

ΩH+ gψv

gψψ

²

+gvvgψψ−g²_vψ gψψ

=Σ sin²θ ρ²

ΩH−2M ar Σ

²

−ρ²∆

Σ . (41)

Second, prove the identities ∆(r+) = 0 and ΩH= ^{2M ar}_Σ(r ⁺

+), and use them to deduce:

−ξ^bξb

;a|r=r+= ρ²

Σ∆,a. (42)

Finally, use the result of section (b). A few lines of algebra will carry you to the desired result forκ.

21. (Smarr’s Formula)

Check the algebraic relation

M = 2ΩHJ+κA

2π (43)

where Ω_H,M,J,A,κare the angular velocity of horizon, mass, angular momentum, area, and surface gravity of the Kerr black hole. This is known as the Smarr’s formula, and it can be shown to generalize to any Black Hole spacetime.

22. (1st Law of BH mechanics)

Suppose that a Kerr black hole of massM and angular momentumJ is perturbed momentarily so that it ends in another Kerr black hole of parameters becomeM +δM J+δJ. Prove that

κ

8πδA=δM−ΩHδJ . (44)

This is known as the 1st Law of Black hole mechanics, and it can be shown to generalize to any Black Hole spacetime.