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The Hawking Mass in Kerr Spacetime
The Hawking Mass in Kerr Spacetime
Tillämpad matematik, Linköpings tekniska högskola.
Jonas Jonsson Holm
LITH-MAT-EX--04/10--SE
Examensarbete
Nivå:
Examinator:
Linköping
20p
D
Göran Bergqvist
Matematiska institutionen
Tillämpad matematik
Linköpings tekniska högskola
2004-09-16
Datum
Date
Avdelning, Institution
Division, Department
2004-09-16
Matematiska Institutionen
581 83 LINKÖPING
SWEDEN
Språk
Language
Rapporttyp
Report category
Svenska/Swedish
×
Engelska/English
Licentiatavhandling
×
ISBN
ISRN
LITH-MAT-EX--04/10--SE
Examensarbete
C-uppsats
Serietitel och serienummer
D-uppsats
Title of series, numbering
ISSN
0348-2960
Övrig rapport
URL för elektronisk version
http://www.ep.liu.se/exjobb/mai/2004/tm/010/
Titel
Title
Hawkingmassan i Kerr-rumtiden
The Hawking Mass in Kerr Spacetime
Författare
Author
Jonas Jonsson Holm
Sammanfattning
Abstract
In this thesis we calculate the Hawking mass numerically for surfaces in Kerr spacetime. The Hawking mass is a useful tool for proving the Penrose inequality and the
result does not contradict the inequality. It also does not contradict the assumption that the Hawking mass should be monotonic for surfaces in Kerr spacetime.
The Hawking mass is quasi-local and defined by the spin coefficents of Newman and
Penrose, so first we give a discussion about quasi-local quantities and then a short
description of the Newman-Penrose formalism.
Nyckelord
Keyword
Hawking mass, black holes, quasi-local mass, Newman-Penrose formalism
Abstract
In this thesis we calculate the Hawking mass numerically for surfaces
in Kerr spacetime. The Hawking mass is a useful tool for proving the
Penrose inequality and the result does not contradict the inequality. It
also does not contradict the assumption that the Hawking mass should
be monotonic for surfaces in Kerr spacetime. The Hawking mass is quasilocal and defined by the spin coefficents of Newman and Penrose, so first
we give a discussion about quasi-local quantities and then a short description of the Newman-Penrose formalism.
Contents
1 Introduction
5
2 Quasi-local Mass
2.1 Asymptotically Flat Spacetimes . . . . . . . . . . . . . . . . . . .
2.2 Construction of Quasi-local Quantities . . . . . . . . . . . . . . .
2.3 General Expectation on a Quasi-local Quantity . . . . . . . . . .
6
6
7
7
3 The Newman-Penrose Formalism
3.1 The Null Tetrad . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Spin Coefficents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
9
10
4 The Hawking Mass
12
5 Black Holes
14
5.1 The Penrose Inequality . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 The Hawking Mass for Static Black Holes
16
6.1 Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Reissner-Nordström . . . . . . . . . . . . . . . . . . . . . . . . . 17
7 The Hawking Mass in Kerr Spacetime
18
7.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7.2 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . 23
A Maple Worksheet
24
Chapter 1
Introduction
In this thesis we study the Hawking mass, which is a quasi-local mass. This
means that it is defined on an extended region of the spacetime and only depends
on the data on the surface. In General Relativity one can not define energy
density pointwise and then integrate like in classical physics. The Hawking
mass is an important tool for proving the Penrose inequality and agrees with the
expectations in spherical symmetric spacetime. There are some cases however
where it does not give the expected result. The Hawking mass is also often
called the Hawking energy and some people think it should be interpreted as
energy rather than mass. No completly ’correct’ expression for quasi-local mass
is known in General Relativity.
In this thesis we use the sign convention (+ − −−) and the notations from
Chandrasekhar [2]. We also use the geometrized units, which are described in
Wald [10].
Chapter 2
Quasi-local Mass
In classical physics the concept of energy is central, but in General Relativity
this is not well defined. Energy is associated with symmetry, but a spacetime
does not need to have any symmetries at all in general. In classical physics it is
possible to define a energy density locally, then integrate over the volume to get
the total energy. But in General Relativity it is not possible to define energy
density locally. The physical reason is that due to a dimensional argument, an
energy density should be constructed from the first derivatives of the metric.
The equivalance principle, however, allows a change to a frame where these are
zero. The energy-momentum tensor Tab can not be used to define mass/energy
because it just contains the matter contribution, but the gravitational field itself
contains energy, and that field is not included in Tab . We are thus forced to do
a quasi-local definition of energy, due to the fact that the quantities that can be
set to zero by an appropriate transformation, can not be transformed to zero on
a region of the spacetime which is bigger than a point (unless the spacetime is
flat). Quasi-locality means the energy contained within an extended but finite
region of the spacetime. The theory of quasi-locality in General Relativity is far
from complete. The basic problem: ’Find a suitable quasi-local expression for
energy/mass in General Relativity’ is not yet solved despite many suggestions.
Examples of suggestions are the Bartnik mass, the Hawking energy, the Geroch
energy and the Penrose mass. But none of these seem to be the right one.
2.1
Asymptotically Flat Spacetimes
Despite the fact that it is hard to define an energy-momentum locally it is
possible to define energy-momentum for the whole space-time, if it is asymptotically flat [10]. It is often from ’quasi-localization’ of these quantities the
quasi-local expression is achieved. At spatial infinity the mass is represented
by the ADM (Arnowitt-Deser-Misner) energy-momentum. This approach is
based on a Hamiltonian analysis of General Relativity and a 3+1 split of spacetime. At null infinity the mass is represented by the Bondi energy. This energy
decreases with time, meaning that gravitational energy carries away positive
energy. Both of these expressions have been proven positive [9].
2.2 Construction of Quasi-local Quantities
2.2
7
Construction of Quasi-local Quantities
There are two ways to construct a quasi-local quantity [9]. The first is called the
systematic approach and the second is quasi-localization of global quantities.
There are two procedures in the systematic approach. The first is the Lagrangian approach. It has the advantage of being Lorentz-covariant, but its
potential is not unique, meaning that a choice of gauge, translations and boostrotation, should be made. The second systematic procedure is the Hamiltonian
approach. If we are not interested in the structure of the quasi-local spacetime,
we can use the Hamilton-Jacobi method. This results in a 2-surface integral.
But the Hamiltonian approach has the same disadvantages as the Lagrangian
approach.
The most natural way to define a quasi-local quantity is by quasi-localization
of the global expressions that give the results for the energy-momentum in
an asymptotically flat spacetime. Since the global energy-momentum can be
written as 2-surface integrals at infinity, the 2-surface observables are expected
to have importance in defining a quasi-local expression. Summarizing these
procedures these three things must be specified to define a resonable quasi-local
energy-momentum:
• a 2-surface integral,
• a gauge choice,
• a definition for the quasi-symmetries of the 2-surface.
2.3
General Expectation on a Quasi-local Quantity
No completly general definition of mass m(S), for a 2-surface S, is known, although there are many suggestions. A good definition should satisfy these properties [3].
1. A point in a spacetime must have zero mass, meaning that m(S) → 0
when S shrinks towards a point.
2. A metric 2-sphere in Minkowski spacetime should have m(S) = 0
3. In spherical symmetric spacetime there is an invariant mass function,
m(S), that any definition of mass should reduce to in the special case
of spherical symmetry. In particular, in Schwarzschild spacetime of mass
M , we expect the result m(S) = M .
4. In an asymptotically flat spacetime, with any radial coordinate r, we
should have
lim m(S(r)) = mADM ,
r→∞
where mADM is the ADM mass.
5. In an asymptotically flat spacetime, with any radial null coordinate r, we
should have
lim m(S(r)) = mB ,
r→∞
where mB is the Bondi mass.
2.3 General Expectation on a Quasi-local Quantity
8
6. If S 0 is ’bigger’ than S, in the sense that S is completly contained in
the interia of S 0 . Then m(S 0 ) ≥ m(S). This is the requirement of local
positivity.
In particular, the Hawking mass satisfies 1-5 but not 6. However, 6 can be
relaxed to a restricted form of local positivity ρρ0 ≤ 0 (see section 3), which the
Hawking mass satisfies. In addition the Hawking mass satisfies the so called
irreducible mass, that is, if S is an apparent horizon (the outermost surface
where ρρ0 = 0) then
r
Area(S)
.
(2.1)
m(S) =
16π
Chapter 3
The Newman-Penrose
Formalism
The Newman-Penrose formalism was introduced in 1962 by Newman and Penrose. The main idea is to choose a basis of null vectors. The two vectors l and
n’s direction are choosen to be light cone directions and the vectors m and m̄
are complex null vectors choosen so that (l, n, m, m̄) forms a full null basis.
The Hawking mass is defined via the spin coefficents in this formalism and thus
it is very important in this thesis. A short introduction will follow here.
3.1
The Null Tetrad
A null tetrad can be constructed from an ordinary ON-tetrad (t,x,y,z) by
1
la = √ (ta + z a ),
2
1
na = √ (ta − z a ),
2
1 a
a
m = √ (x + iy a ),
2
1
m̄a = √ (xa − iy a ).
2
(3.1)
(3.2)
(3.3)
(3.4)
l and n are real while m and m̄ are complex conjugates of each other. On a
spacelike 2-surface l and n are usally choosen as the null normals. The set of
vectors (l, n, m, m̄) is called a null tetrad and has the following properties [8]
la la = na na = ma ma = m̄a m̄a = 0,
(3.5)
la ma = la m̄a = na ma = na m̄a = 0.
(3.6)
Often a normalization condition is introduced
la na = −ma m̄a = 1,
(3.7)
but this is not always necessary. Directional derivatives are also introduced as
D = l a ∇a ,
∆ = n a ∇a ,
δ = m a ∇a ,
δ̄ = m̄a ∇a .
(3.8)
3.2 Spin Coefficents
10
There are three types of transformations that can be applied to the tetrad which
conserves the properties (3.5)-(3.7) [2]
1. null rotation of class I, leaving the vector l unchanged,
2. null rotation of class II, leaving the vector n unchanged and
3. spin-boost transformation (class III), leaving the directions of l and n
unchanged but rotates m and m̄ by an angel of θ in the (m, m̄)-plane.
The explicit formulas for these rotations are
1. l → l, m → m + Al, m̄ → m̄ + Āl and n → n + Ām + Am̄ + AĀl,
2. n → n, m → m + Bn, m̄ → m̄ + B̄n and l → l + B̄m + B m̄ + B B̄n,
3. l → λ−1 l, n → λn, m → eiθ m and m̄ → e−iθ m̄,
(3.9)
where A and B are complex functions and λ and θ are real functions. The bar
denotes the complex conjugate.
3.2
Spin Coefficents
The spin coefficents are introduced [8] via the directional derivatives (3.8)

κ = ma Dla ² = 21 (na Dla − m̄a Dma ) τ 0 = m̄a Dna








 τ = ma ∆la γ = 12 (na ∆la − m̄a ∆ma ) κ0 = m̄a ∆na
(3.10)

1
a
a
a
0
a

σ
=
m
δl
β
=
(n
δl
−
m̄
δm
)
ρ
=
m̄
δn

a
a
a
a
2






ρ = ma δ̄la α = 12 (na δ̄la − m̄a δ̄ma )
σ 0 = m̄a δ̄na .
Let us look more closely at the spin coefficents ρ and ρ0
ρ = ma δ̄la = ma m̄b ∇b la = ma m̄b la;b
(3.11)
ρ0 = m̄a δna = m̄a mb ∇b na = m̄a mb na;b
As can be seen in the right hand side of equation (3.11) the change of the light
cone vectors are projected on to the (m, m̄)-plane thus measuring the expansion
of the light cone in relation to the surface.
In some cases one can make a geometrical interpretation of the spin coefficents, this is done by analyzing the identity [8]
Dz = −ρz − σz̄.
(3.12)
The interpretation of z is the projection of z a onto the spacelike plane spanned
by (ma ,m̄a ). Now put z = x + iy and look at a few cases.
1. Suppose that σ = 0 and that ρ is real. Then Dz = −ρz or
Dx = −ρx,
Dy = −ρy.
(3.13)
This describes an isotropic magnification of the separation of nearby geodesics
at the rate −ρ.
3.2 Spin Coefficents
11
2. Suppose that σ = 0 and that ρ = −iω (ω real), then
Dx = −ωy,
Dy = ωx.
(3.14)
This means that nearby geodesics rotate with an angular velocity ω.
3. Suppose that ρ = 0 and that σ is real then
Dx = −σx,
Dy = σy.
(3.15)
This means that nearby geodesics undergo a volume-preserving shear.
In general a superposition of these cases occur.
Chapter 4
The Hawking Mass
The Hawking mass with respect to a spacelike 2-surface S of area A is defined
by [5]
s
r
Z
Z
´
´
A ³
A ³
1
0
0
2π
+
ρρ
dS
.
(4.1)
mH (S) =
1+
ρρ dS =
16π
2π S
(4π)3
S
ρ and ρ0 are real if l and n are null normals to S. It was proposed by Stephen
Hawking in 1968 after studying perturbations in k = −1 Friedmann-RobertsonWalker spacetimes. One can motivate the Hawking mass [9] by supposing that
the mass inside a 2-sphere should be a measure of the bending of in- and outgoing
light rays summarized over a surface containing the object. The quantity ρρ0
is invariant under a spin-boost transformation (la → λla , na → λ−1 na ) since
ρ → λρ and ρ0 → λ−1 ρ0 under this transformation. It is necessary to have a
quantity that is invariant under this transformation because a transformation
that leaves the surface unchanged should not change the mass. Thus the mass
must have the form
Z
C + D ρρ0 dS,
(4.2)
S
where the coefficents C and D can be determined under special situations. In
Minkowski spacetime, where ρ = −1/r and ρ0 = 1/2r (in spherical coordinates and with respect to a standard choice of la and na ), the mass should be
zero, that is
D = C/2π.
(4.3)
On the horizon of Schwarzschild black hole, where ρρ0 = 0, the result should
be M. The area of the Schwarzschild horizon is A = 4πr2 and on the horizon
r = 2M which gives A = 16πM 2 , thus
C 2 = A/16π
(4.4)
which gives (4.1). The expression (4.1) tends to the Bondi mass at future null
infinity and it tends to the ADM mass at spatial infinity in asymptotically flat
spacetimes. It also gives the correct expression for metric spheres. However it
can give negative results, e.g., for concave 2-surfaces in Minkowski spacetime [9].
The Hawking mass is neither positive nor monotonic in general [9] (and
references therein), but under special conditions both of these properties can
13
be satisfied. First if the surface S is ’round enough’, then it can be shown that
the Hawking mass is positive definite. It can also be shown to be monotonic in
special cases. For a special family of spacelike surfaces Sr and a special choice
of the coordinate r, it has been shown that mH (Sr ) is non-decreasing with r,
if the dominant energy condition holds. The dominant energy condition can be
interpreted by saying that the speed of energy flow of matter is always less than
the speed of light [10].
Chapter 5
Black Holes
A black hole is a region of spacetime where the gravity is so strong that no
observer or light ray can escape. It consist of a singularity and an event horizon.
A horizon is a surface on which the expansion of the light cone is precisely zero.
Then we know from (3.13) that ρ (or ρ0 ) = 0. It is believed that no ’naked’
singularity can exist, i.e., no singularity can exist without an event horizon.
This is called the cosmic censor conjecture [10]. However there are no evidence
for or against this conjecture. A singularity is a place in spacetime where the
curvature becomes infinite.
A black hole is formed by the gravitational collapse of a star or by an entire
cluster of stars. It is hard to avoid the conclusion that many black holes are
formed by the first process. The number of supernova explosions in our galaxy
that have occured is estimated to 108 , so 108 black holes may have been formed
in our galaxy. However this estimate may be to high since not all supernova
explosion results in a black hole, or it may be to low since a black hole also can
be formed by a violent blowing off of the outer layers of a star.
5.1
The Penrose Inequality
When Penrose studied the cosmic censor conjecture he came to the conclusion
that the conjecture will be false if not the Penrose inequality is true in any
asymptotically flat spacetime. Let m be the ADM mass and let A be the area
of the horizon of the black hole, then the Penrose inequality states that [7]
r
A
m≥
.
(5.1)
16π
If this inequality should be false, then the cosmic censor conjecture would not
hold. But proving it true does not mean that the cosmic censor conjecture must
be true. The inequality has not been proven in the most general case, but there
are some special cases where it can be proven. In particular it has been proven
for spherical symmetric spacetimes using the Hawking mass [9]. There are ideas
to prove the Penrose inequality
p in the general case by using the Hawking mass.
Since we know that mH = A/16π on the horizon (surface where ρρ0 = 0) and
mH → m when r → ∞, one has to prove that there exist 2-surfaces (Sr ) so
that mH (Sr ) increases with r. Finding these kind of surfaces would prove (5.1).
5.2 Metrics
15
This is a very difficult mathematical problem and it has not yet been solved.
But in this thesis we test if mH (Sr ) increases for a natural family of 2-surfaces
(Sr ) in the Kerr-solution.
5.2
Metrics
A stationary black hole is completely described by three parameters: mass, M ,
angular momentum, a, and electric charge, e. The spacetime in the vicinity
of a black hole is described by the metric, which determines length of curves.
There are four different stationary black hole metrics: Schwarzschild (M 6= 0,
a = e = 0), Reissner-Nordström (M 6= 0,e 6= 0, a = 0), Kerr (M 6= 0,a 6= 0,
e = 0) and Kerr-Newman (M 6= 0,e 6= 0,a 6= 0). The most general of these is
the Kerr-Newman metric, the others are only the limit when a or e (or both)
goes to zero. The metrics in these different cases are in coordinates (t, r, θ, ϕ)
[2], [10]
Schwarzschild
ds2 =
³
2M ´ 2 ³
2M ´−1 2
1−
dt − 1 −
dr
r
r
−r2 dθ2 − r2 sin2 θ dϕ2 .
(5.2)
Reissner-Nordström
³
³
e2 ´
2M
e2 ´−1 2
2M
+ 2 dt2 − 1 −
+ 2
dr
ds2 = 1 −
r
r
r
r
−r2 dθ2 − r2 sin2 θ dϕ2 .
(5.3)
Kerr
ds2 =
∆
sin2 θ 2
(dt − a sin2 θ dϕ)2 −
[(r + a2 ) dϕ − a dt]2
Σ
Σ
−
Σ 2
dr − Σ dθ2 ,
∆
(5.4)
where
Σ = r2 + a2 cos2 θ,
2
(5.5)
2
∆ = r − 2M r + a .
(5.6)
Kerr-Newman
ds2 =
−
³ ∆ − a2 sin2 θ ´
Σ
dt2 +
2a sin2 θ(r2 + a2 − ∆)
dt dϕ
Σ
h (r2 + a2 )2 − ∆a2 sin2 θ i
Σ
sin2 θ dϕ2 −
Σ 2
dr − Σ dθ2 , (5.7)
∆
where
Σ = r2 + a2 cos2 θ,
(5.8)
∆ = r2 − 2M r + a2 + e2
(5.9)
Chapter 6
The Hawking Mass for
Static Black Holes
Here we calculate the Hawking mass in Schwarzschild and Reissner-Nordström
spacetimes for surfaces Sr parametrised with θ and ϕ. The spin coefficents ρ
and ρ0 are real, so no rotation is needed in these cases. The calculation of the
Hawking mass then becomes very straightforward. In the standard tetrad [2]

1 (r2 , ∆, 0, 0)
la = ∆







na = 12 (r2 , −∆, 0, 0)
2r






1 (0, 0, 1, i )
 ma = √
sin θ
r 2
(6.1)
the spin coefficents becomes

1

 ρ = −r

 ρ0 = ∆ .
2r3
(6.2)
Note that m1 = m2 = 0 so the plane (ma , m̄a ) is tangent to surfaces where
r = constant and t = constant. In Schwarzschild spacetime the metric is given
by (5.2) and the Reissner-Nordström metric is given by (5.3).
6.1
Schwarzschild
In Schwarzschild spacetime ∆ = r2 − 2M r which gives
ρρ0 = −
r2 − 2M r
2r4
(6.3)
and the area element becomes
√
gθθ gϕϕ = r2 sin θ,
(6.4)
6.2 Reissner-Nordström
17
which gives the area
Z
2π Z π
r2 sin θ dθdϕ = 4πr2
(6.5)
2π(r2 − 2M r)
(r2 − 2M r)r2 sin θ
dθdϕ = −
.
4
2r
r2
(6.6)
A(Sr ) =
0
and
Z
Z
ρρ0 dS = −
S
0
2π Z π
0
0
This gives this result for the Hawking mass
s
Z
´
A ³
r 4πM
0
mH (Sr ) =
= M.
2π
+
ρρ
dS
=
3
4π r
(4π)
S
(6.7)
Thus the Hawking mass in Schwarzschlid spacetime is M, which of course is
very natural since neither charge nor rotation is present here.
6.2
Reissner-Nordström
The calculation of the Hawking mass in Reissner-Nordström spacetime is very
similar. The only difference is that ∆ = r2 − 2M r + e2 which gives
ρρ0 = −
r2 − 2M r + e2
.
2r4
(6.8)
The area element is still (6.4), which of course gives the area (6.5), and the
integral now becomes
Z
Z 2π Z π 2
(r − 2M r + e2 )r2 sin θ
0
ρρ dS = −
dθdϕ
2r4
S
0
0
=
−
2π(r2 − 2M r + e2 )
.
r2
(6.9)
This now gives the Hawking mass
s
Z
´
A ³
e2
r ³ 4πM
2πe2 ´
mH (Sr ) =
2π
+
ρρ0 dS =
− 2
= M − . (6.10)
3
4π
r
2r
(4π)
r
S
This result agrees with most of the other definitions of quasi-local mass and is
thought of as the most reasonable result [1].
Chapter 7
The Hawking Mass in Kerr
Spacetime
The main goal of this thesis is to calculate the Hawking mass for surfaces
parametrised with θ and ϕ and for different values of r and a in Kerr spacetime.
The covariant form of the Kerr metric is with coordinates (t,r,θ,ϕ) [2]


2aM r sin2 θ
r
0
0
1 − 2M
Σ
Σ






Σ


0
0
0
−∆


 , (7.1)
gij = 




0
0
−Σ
0






2 ¤
£
2
2
2aM r
2a
M
r
sin
θ
2
2
0
0
− (r + a ) +
sin θ
Σ∆
Σ
where
∆ = r2 − 2M r + a2 and Σ = r2 + a2 cos2 θ.
(7.2)
The Hawking mass on the (Kerr)-horizon was calculated in [1], but here we
will calculate it outside the horizon. Unfortunately this turns out to be too
complicated to do analytically, thus we are forced to do a numerical evaluation.
The standard tetrad in Kerr spacetime is [2]

1 2
2
i

 l = ∆ (r + a , ∆, 0, a)





1 (r2 + a2 , −∆, 0, a)
ni = 2Σ
(7.3)






 mi = √1 (iasinθ, 0, 1, i ),
sinθ
2ρ̃
where
ρ̃ = r + ia cos θ.
(7.4)
Note that l and n are not orthogonal to Sr , which means that we have to rotate
the coordinate system so that
m1 = m2 = 0.
(7.5)
19
Then ρ and ρ0 become real. To achieve this we have to do two rotations of the
null tetrad. One of type I and one of type II (3.9). First we do a type II rotation
n → n, m → m + Bn,
m̄ → m̄ + B̄n and l → l + B̄m + B m̄ + B B̄n. (7.6)
Then a type I rotation
l → l,
m → m + Al,
m̄ → m̄ + Āl and n → n + Ām + Am̄ + AĀl. (7.7)
Together these two give this total transformation of the tetrad

 n → n + B̄(m + An) + B(m̄ + Ān) + B B̄(l + Ām + Am̄ + AĀn)
l → l + B̄m + B m̄ + B B̄n

m → (1 + ĀB)m + AB m̄ + Bl + A(1 + ĀB)n
(7.8)
The complex functions A and B are here to be determined so that condition
(7.5) is satisfied. This produces these two equations
½
0 = (1 + ĀB)m1 + AB m̄1 + Bl1 + A(1 + ĀB)n1
(7.9)
0 = B + A(1 + ĀB)n2 ,
which can be solved analytically to give
q

2
2


√
r
+
a
±
(r2 + a2 )Σ + 2M ra2 sin2 θ

¯
 A = −i 2ρ̃
a∆sinθ



A∆
 B=
.
2Σ − AĀ∆
(7.10)
Here we choose the − sign in A, since this causes A → 0 as a → 0. This is
necessary because if a = 0, we have the Schwarzschild metric, where no rotation
is needed, i.e., A = 0.
We have now completed the rotation of the tetrad and we turn to the calculations of the spin coefficents. Here we use the fact that the spin coefficents
can be calculated using only partial derivatives (denoted by commas) [2]. Then
the coefficents needed in our calculations (ρ = γ314 and ρ0 = −γ243 ) can be
calculated by
1
γijk = [λijk + λkij − λjki ],
(7.11)
2
where
4 X
4
X
λijk =
[ejµ,ν − ejν,µ ]ei µ ekν ,
(7.12)
µ=1 ν=1
and
e1 = l, e2 = n, e3 = m and e4 = m̄.
(7.13)
The λ’s should of course be calculated with the the rotated tetrad (7.8). We
also have to lover the indices of our new tetrad with the metric (7.1). If we
expand the sum (7.12) we get after noticing that we don’t have any dependence
of t or ϕ
λijk = ej1,2 ei 1 ek2 + ej1,3 ei 1 ek3 − ej1,2 ei 2 ek1 + (ej2,3 − ej3,2 )ei 2 ek3
− ej4,2 ei 2 ek4 − ej1,3 ei 3 ek1 + (ej3,2 − ej2,3 )ei 3 ek2 − ej4,3 ei 3 ek4
+ ej4,2 ei 4 ek2 + ej4,3 ei 4 ek3 .
(7.14)
20
By calculating λ314 , λ431 , λ143 , λ243 , λ432 and λ324 and adding them by (7.11)
we obtain the expressions for the spin coefficents. These expressions are several
pages long and very complicated so there is no point of writing them out. To
this point, however, the calculations have been exact (Maple). From now on
the calculations will be numerical. The two spin coefficents (ρ and ρ0 ) should
be multiplied and integrated. First we calculate the area by this formula
Z
Z 2πZ π
√
A =
dS =
gθθ gϕϕ dθdϕ
S
2πZ π
Z
=
0
Z
= 2π
0
π
0
0
q
sin θ (r2 + a2 )2 − a2 ∆ sin2 θ dθdϕ
q
sin θ (r2 + a2 )2 − a2 ∆ sin2 θ dθ
(7.15)
0
Then we calculate the integral
Z
Z 2πZ π
Z
0
0√
ρρ dS =
ρρ gθθ gϕϕ dθdϕ = 2π
S
0
π
√
ρρ0 gθθ gϕϕ dθ
(7.16)
b−a
[f0 + fn + 2(f2 + f4 + . . . + fn−2 )
3n
+ 4(f1 + f3 + . . . + fn−1 )],
(7.17)
0
0
numerically by using Simpson’s fomula [6]
Z
b
f (x) dx =
a
where fk = f (a + k/n) and with a remaining term
R=−
(b − a)5 (4)
f (ξ)
180 n4
(7.18)
for some ξ ∈ [a, b]. The convergence of (7.17) was checked by increasing the
number of calculation points from 50 to 100. The effect on the result was about
10−6 so the integral has converged at n = 50.
7.1 Results
7.1
21
Results
The results of the calculation is presented in numerical plots. In my calculation
I have choosen M = 2, but one can choose any number. The Hawking mass
(mH ) is plotted versus the angular momentum (a/M ) and the radius (r).
2
1.9
m
H
1.8
1.7
1.6
1.5
1.4
1
0.8
20
0.6
a/M
15
0.4
10
0.2
5
0
r
0
Figure 7.1: Hawking mass plotted versus rotation and radius. Every x marks
a calculation. The curve in the foreground is the horizon and the other curves
have constant a.
As can be seen in this plot, the results do not contradict the assumption that
the Hawking mass should be monotonic and it does not violate the Penrose
inequality. The range of r is
r+ ≤ r ≤ 20,
(7.19)
√
2
2
where r+ = M + M − a (r does not have the same value on the horizon
for different a). When r > 20 nothing more interesting happens to mH , it just
gets closer to M . In the special case where a = 0 the results agrees
p with the
Schwarzschild case and when r = r+ the result agrees with m = A/16π, the
irreducible mass. What happens when r → ∞ is hard to see in this plot, but
by calculating mH for large r one can see that mH → M in this case.
The result can also be plotted in 2-D plots to make the results a bit clearer.
In these two figures the monotonicity becomes a bit more easy to see. Note that
the horizon depends on a, so r = constant does not mean constant distance to
the horizon.
7.1 Results
22
2.1
2
1.9
mH
1.8
1.7
1.6
1.5
1.4
2
4
6
8
10
12
14
16
18
20
r
Figure 7.2: mH plotted with a/M constant. + corresponds to a/M = 0, x to
a/M = 1/2 and o to a/M = 1.
2.05
2
mH
1.95
1.9
1.85
1.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a/M
Figure 7.3: mH plotted with r constant. ◦ corresponds to r = 10, x to r = 7.5,
∗ to r = 5 and + to r = 4.
7.2 Discussion and Summary
7.2
23
Discussion and Summary
In this thesis we have calculated the Hawking mass in Schwarzschild, ReissnerNordström and Kerr spacetimes. It has been confirmed that the Hawking mass
is monotonic with r and a and it does not contradict the Penrose inequality in
these cases. It might be interesting to see the result in Kerr-Newman spacetime
too, but there will be one more parameter, e, which is believed to be small
compared to M and a. It might also be interesting to have an exact result in
Kerr spacetime, but this would probably require very powerful computers.
Appendix A
Maple Worksheet
This is the Maple Worksheet I used in my calculations.
restart:
assume(r,real,a,real,u,real,M,real): additionally(r,positive,a,positive,M,positive):
additionally(M>a,a>0): additionally(0<u,u<2*Pi):
Delta:=rˆ2-2*M*r+aˆ2:
rho1:=r+I*a*cos(u):
rho:=sqrt(rˆ2+aˆ2*cos(u)ˆ2):
N:=array(1..4,[(1)=simplify((r*r+a*a)/(2*rhoˆ2)),(2)=simplify(-Delta/(2*rhoˆ2)),
(3)=0,(4)=simplify(a/(2*rhoˆ2))]):
L:=array(1..4,[(1)=simplify((r*r+a*a)/Delta),(2)=1,
(3)=0,(4)=simplify(a/Delta)]):
M1:=array(1..4,[(1)=I*a*sin(u)/(sqrt(2)*rho1),(2)=0,
(3)=1/(sqrt(2)*rho1),(4)=simplify(I/(sqrt(2)*rho1*sin(u)))]):
M2:=array(1..4,[(1)=-I*a*sin(u)/(sqrt(2)*conjugate(rho1)),(2)=0,
(3)=1/(sqrt(2)*conjugate(rho1)),(4)=simplify(-I/(sqrt(2)*conjugate(rho1)*sin(u)))]):
A1:=conjugate(A):
B1:=conjugate(B):
n1:=N[1]+B1*(M1[1]+A*N[1])+B*(M2[1]+A1*N[1])
+B*B1*(L[1]+A1*M1[1]+A*M2[1]+A*A1*N[1]):
n2:=N[2]+B1*(M1[2]+A*N[2])+B*(M2[2]+A1*N[2])
+B*B1*(L[2]+A1*M1[2]+A*M2[2]+A*A1*N[2]):
n3:=N[3]+B1*(M1[3]+A*N[3])+B*(M2[3]+A1*N[3])
+B*B1*(L[3]+A1*M1[3]+A*M2[3]+A*A1*N[3]):
n4:=N[4]+B1*(M1[4]+A*N[4])+B*(M2[4]+A1*N[4])
+B*B1*(L[4]+A1*M1[4]+A*M2[4]+A*A1*N[4]):
n:=array(1..4,[(1)=n1,(2)=n2,(3)=n3,(4)=n4]):
l1:=L[1]+A1*M1[1]+A*M2[1]+A*A1*N[1]:
l2:=L[2]+A1*M1[2]+A*M2[2]+A*A1*N[2]:
l3:=L[3]+A1*M1[3]+A*M2[3]+A*A1*N[3]:
l4:=L[4]+A1*M1[4]+A*M2[4]+A*A1*N[4]:
24
25
l:=array(1..4,[(1)=l1,(2)=l2,(3)=l3,(4)=l4]):
m1:=M1[1]*(1+B*A1)+M2[1]*A*B+L[1]*B+N[1]*A*(1+B*A1):
m2:=M1[2]*(1+B*A1)+M2[2]*A*B+L[2]*B+N[2]*A*(1+B*A1):
m3:=M1[3]*(1+B*A1)+M2[3]*A*B+L[3]*B+N[3]*A*(1+B*A1):
m4:=M1[4]*(1+B*A1)+M2[4]*A*B+L[4]*B+N[4]*A*(1+B*A1):
m:=array(1..4,[(1)=m1,(2)=m2,(3)=m3,(4)=m4]):
m21:=M2[1]*(1+B1*A)+M1[1]*A1*B1+L[1]*B1+N[1]*A1*(1+B1*A):
m22:=M2[2]*(1+B1*A)+M1[2]*A1*B1+L[2]*B1+N[2]*A1*(1+B1*A):
m23:=M2[3]*(1+B1*A)+M1[3]*A1*B1+L[3]*B1+N[3]*A1*(1+B1*A):
m24:=M2[4]*(1+B1*A)+M1[4]*A1*B1+L[4]*B1+N[4]*A1*(1+B1*A):
mc:=array(1..4,[(1)=m21,(2)=m22,(3)=m23,(4)=m24]):
A:=I*(-sqrt(2)*conjugate(rho1)*(rˆ2+aˆ2-sqrt((rˆ2+aˆ2)*rhoˆ2+2*M*r*aˆ2*sin(u)ˆ2))/
/(Delta*a*sin(u))):
B:=A*Delta/(2*rhoˆ2-A*A1*Delta):
g:=array(1..4,1..4,[(1,1)=1-2*M*r/rhoˆ2,(1,2)=0,(1,3)=0,
(1,4)=2*a*M*r*sin(u)ˆ2/rhoˆ2,(2,1)=0,(2,2)=-rhoˆ2/Delta,
(2,3)=0,(2,4)=0,(3,1)=0,(3,2)=0,(3,3)=-rhoˆ2,(3,4)=0,(4,1)=2*a*M*r*sin(u)ˆ2/rhoˆ2,
(4,2)=0,(4,3)=0,(4,4)=-((rˆ2+aˆ2)+2*aˆ2*M*r*sin(u)ˆ2/rhoˆ2)*sin(u)ˆ2]):
nn1:=n[1]*g[1,1]+n[4]*g[4,1]:
nn2:=n[2]*g[2,2]:
nn3:=n[3]*g[3,3]:
nn4:=n[1]*g[1,4]+n[4]*g[4,4]:
nn:=array(1..4,[(1)=nn1,(2)=nn2,(3)=nn3,(4)=nn4]):
ll1:=l[1]*g[1,1]+l[4]*g[4,1]:
ll2:=l[2]*g[2,2]:
ll3:=l[3]*g[3,3]:
ll4:=l[1]*g[1,4]+l[4]*g[4,4]:
ll:=array(1..4,[(1)=ll1,(2)=ll2,(3)=ll3,(4)=ll4]):
mm1:=m[1]*g[1,1]+m[4]*g[4,1]:
mm2:=m[2]*g[2,2]:
mm3:=m[3]*g[3,3]:
mm4:=m[1]*g[1,4]+m[4]*g[4,4]:
mm:=array(1..4,[(1)=mm1,(2)=mm2,(3)=mm3,(4)=mm4]):
mmc1:=mc[1]*g[1,1]+mc[4]*g[4,1]:
mmc2:=mc[2]*g[2,2]:
mmc3:=mc[3]*g[3,3]:
mmc4:=mc[1]*g[1,4]+mc[4]*g[4,4]:
mmc:=array(1..4,[(1)=mmc1,(2)=mmc2,(3)=mmc3,(4)=mmc4]):
lambda314:=-diff(ll[4],u)*m[3]*mc[4]+diff(ll[4],u)*m[4]*mc[3]:
lambda431:=-diff(mm[1],u)*mc[3]*l[1]+diff(mm[3],r)*mc[3]*l[2]
26
-diff(mm[4],u)*mc[3]*l[4]+diff(mm[4],r)*mc[4]*l[2]:
lambda143:=diff(mmc[1],u)*l[1]*m[3]-diff(mmc[3],r)*l[2]*m[3]
-diff(mmc[4],r)*l[2]*m[4]+diff(mmc[4],u)*l[4]*m[3]:
s1:=1/2*(lambda314+lambda431-lambda143):
lambda324:=-diff(nn[4],u)*m[3]*mc[4]+diff(nn[4],u)*m[4]*mc[3]:
lambda432:=-diff(mm[1],u)*mc[3]*n[1]+diff(mm[3],r)*mc[3]*n[2]
-diff(mm[4],u)*mc[3]*n[4]+diff(mm[4],r)*mc[4]*n[2]:
lambda243:=diff(mmc[1],u)*n[1]*m[3]-diff(mmc[3],r)*n[2]*m[3]
-diff(mmc[4],r)*n[2]*m[4]+diff(mmc[4],u)*n[4]*m[3]:
s2:=-1/2*(lambda243+lambda324-lambda432):
Q:=sqrt(sin(u)ˆ2*((rˆ2+aˆ2)ˆ2-aˆ2*Delta*sin(u)ˆ2)):
Q1:=eval(Q,[r=?,a=?,M=2]):
Ar:=evalf(Int(Q1,u=0..Pi)):
q1:=eval(s1,[r=?,a=?,M=2]):
q2:=eval(s2,[r=?,a=?,M=2]):
q:=q1*q2*Q1:
for n from 1 by 1 to 49 do J[n]:=2*evalf(eval(q,[u=2*n*Pi/100]))*Pi/(3*50) end
do:
for n from 1 by 1 to 50 do K[n]:=4*evalf(eval(q,[u=(2*n-1)*Pi/100]))*Pi/(3*50)
end do:
S1:=evalf(Pi*sum(J[i],i=1..49)):
S2:=evalf(Pi*sum(K[i],i=1..50)):
S:=S1+S2:
mh:=evalf(sqrt(Ar*2*Pi/(4*Pi)ˆ3)*(2*Pi+S)):
Bibliography
[1] G. Bergqvist, Quasilocal Mass for Event Horizons, Class. Quantum Grav.,
9, 1753-1768, 1992
[2] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, 1983
[3] D.M. Eardley, Global Problems in Numerical Relativity, in Smarr, L.L.,
ed., Sources of Gravitational Radiation, Proceedings of the Battele Seattle
Workshop, July 27 - Aug. 4, 1978, 127-138, Cambridge University Press,
Cambridge, 1979
[4] P. Ekström, M. Goliath and U. Nilsson, The Concept of Energy and Mass
in General Relativity, Stockholm University Report, 1996
[5] S. W. Hawking, Gravitational Radiation in an Expanding Universe, J.
Math. Phys., 9, 598-604, 1968
[6] V. I. Krylov, Approximate Calculation of Integrals, The Macmillan Company, New York, 1959
[7] E. Malec, Isoperimetric Inequalities in the Physics of Black Holes, Acta
Phys. Pol. 22, 829, 1991
[8] J. Stewart, Advanced General Relativity, Cambridge University Press, 1990
[9] L. B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in
GR: A Review Article, Living Rev. Relativity, 7, (2004), 4. [Online Article]:
cited [2004-03-30], http://www.livingreviews.org/lrr-2004-4
[10] R. M. Wald, General Relativity, The University of Chicago Press, 1984
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