The Hawking Mass in Kerr Spacetime
Till¨ampad matematik, Link¨opings tekniska h¨ogskola.Jonas Jonsson Holm LITH-MAT-EX--04/10--SE
Examensarbete 20p
Niv˚a: D
Examinator: G¨oran Bergqvist
Matematiska institutionen Till¨ampad matematik Link¨opings tekniska h¨ogskola Link¨oping 2004-09-16
Matematiska Institutionen 581 83 LINK ¨OPING SWEDEN 2004-09-16 × × http://www.ep.liu.se/exjobb/mai/2004/tm/010/ LITH-MAT-EX--04/10--SE Hawkingmassan i Kerr-rumtiden The Hawking Mass in Kerr Spacetime
Jonas Jonsson Holm
In this thesis we calculate the Hawking mass numerically for surfaces in Kerr space-time. 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 assump-tion 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.
Hawking mass, black holes, quasi-local mass, Newman-Penrose formalism
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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 descrip-tion of the Newman-Penrose formalism.
Contents
1 Introduction 5
2 Quasi-local Mass 6
2.1 Asymptotically Flat Spacetimes . . . 6 2.2 Construction of Quasi-local Quantities . . . 7 2.3 General Expectation on a Quasi-local Quantity . . . 7
3 The Newman-Penrose Formalism 9
3.1 The Null Tetrad . . . 9 3.2 Spin Coefficents . . . 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-Nordstr¨om . . . 17
7 The Hawking Mass in Kerr Spacetime 18
7.1 Results . . . 21 7.2 Discussion and Summary . . . 23
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 Tabcan 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 asymp-totically 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 space-time. 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 7
2.2
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 La-grangian approach. It has the advantage of being Lorentz-covariant, but its potential is not unique, meaning that a choice of gauge, translations and boost-rotation, 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
Quan-tity
No completly general definition of mass m(S), for a 2-surface S, is known, al-though there are many suggestions. A good definition should satisfy these prop-erties [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
r→∞m(S(r)) = mADM,
where mADM is the ADM mass.
5. In an asymptotically flat spacetime, with any radial null coordinate r, we should have
lim
r→∞m(S(r)) = mB,
2.3 General Expectation on a Quasi-local Quantity 8
6. If S0 is ’bigger’ than S, in the sense that S is completly contained in
the interia of S0. Then m(S0) ≥ 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
m(S) = r
Area(S)
Chapter 3
The Newman-Penrose
Formalism
The Newman-Penrose formalism was introduced in 1962 by Newman and Pen-rose. 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 ¯m are complex null vectors choosen so that (l, n, m, ¯m) 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 la = √1 2(t a+ za), (3.1) na = √1 2(t a− za), (3.2) ma = √1 2(x a+ iya), (3.3) ¯ ma = √1 2(x a− iya). (3.4)
l and n are real while m and ¯m 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, ¯m) is called a null tetrad and has the following properties [8]
lala = nana= mama = ¯mam¯a= 0, (3.5)
lama= lam¯a= nama = nam¯a= 0. (3.6)
Often a normalization condition is introduced
lana = −mam¯a = 1, (3.7)
but this is not always necessary. Directional derivatives are also introduced as D = la∇
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 ¯m by an angel of θ in the (m, ¯m)-plane. The explicit formulas for these rotations are
1. l → l, m → m + Al, m → ¯¯ m + ¯Al and n → n + ¯Am + A ¯m + A ¯Al, 2. n → n, m → m + Bn, m → ¯¯ m + ¯Bn and l → l + ¯Bm + B ¯m + B ¯Bn, 3. l → λ−1l, n → λn, m → eiθm and ¯m → e−iθm,¯
(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 = maDl a ² = 12(naDla− ¯maDma) τ0= ¯maDna τ = ma∆l a γ = 12(na∆la− ¯ma∆ma) κ0= ¯ma∆na σ = maδl a β = 12(naδla− ¯maδma) ρ0 = ¯maδna ρ = maδl¯ a α = 12(naδl¯a− ¯ma¯δma) σ0= ¯maδn¯ a. (3.10)
Let us look more closely at the spin coefficents ρ and ρ0
ρ = maδl¯
a= mam¯b∇bla= mam¯bla;b
ρ0 = ¯maδn
a= ¯mamb∇bna= ¯mambna;b
(3.11)
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, ¯m)-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 coeffi-cents, this is done by analyzing the identity [8]
Dz = −ρz − σ¯z. (3.12)
The interpretation of z is the projection of za onto the spacelike plane spanned
by (ma, ¯ma). 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] mH(S) = r A 16π ³ 1 + 1 2π Z S ρρ0dS ´ = s A (4π)3 ³ 2π + Z S ρρ0dS ´ . (4.1)
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-Robertson-Walker 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−1na) since
ρ → λρ and ρ0 → λ−1ρ0 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
C + D Z
S
ρρ0dS, (4.2)
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
coordi-nates 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 = 16πM2, thus
C2= 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 108black 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]
m ≥ r
A
16π. (5.1)
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 in the general case by using the Hawking mass. Since we know that mH =
p
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
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-Nordstr¨om (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 = ³1 − 2M r ´ dt2−³1 − 2M r ´−1 dr2 −r2dθ2− r2sin2θ dϕ2. (5.2) Reissner-Nordstr¨om ds2 = ³ 1 − 2M r + e2 r2 ´ dt2− ³ 1 − 2M r + e2 r2 ´−1 dr2 −r2dθ2− r2sin2θ dϕ2. (5.3) Kerr ds2 = ∆ Σ(dt − a sin 2θ dϕ)2−sin2θ Σ [(r 2+ a2) dϕ − a dt]2 −Σ ∆dr 2− Σ dθ2, (5.4) where Σ = r2+ a2cos2θ, (5.5) ∆ = r2− 2M r + a2. (5.6) Kerr-Newman ds2 = ³ ∆ − a2sin2θ Σ ´ dt2+2a sin2θ(r2+ a2− ∆) Σ dt dϕ − h (r 2+ a2)2− ∆a2sin2θ Σ i sin2θ dϕ2−Σ ∆dr 2− Σ dθ2, (5.7) where Σ = r2+ a2cos2θ, (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-Nordstr¨om 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] la = 1 ∆ (r2, ∆, 0, 0) na= 1 2r2(r2, −∆, 0, 0) ma= 1 r√2(0, 0, 1, i sin θ ) (6.1)
the spin coefficents becomes ρ = − 1r ρ0 = ∆ 2r3. (6.2)
Note that m1 = m2 = 0 so the plane (ma, ¯ma) is tangent to surfaces where
r = constant and t = constant. In Schwarzschild spacetime the metric is given by (5.2) and the Reissner-Nordstr¨om 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 √
6.2 Reissner-Nordstr¨om 17
which gives the area
A(Sr) = Z 2π 0 Z π 0 r2sin θ dθdϕ = 4πr2 (6.5) and Z S ρρ0dS = − Z 2π 0 Z π 0 (r2− 2M r)r2sin θ 2r4 dθdϕ = − 2π(r2− 2M r) r2 . (6.6)
This gives this result for the Hawking mass
mH(Sr) = s A (4π)3 ³ 2π + Z S ρρ0dS´= r 4π 4πM r = M. (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-Nordstr¨
om
The calculation of the Hawking mass in Reissner-Nordstr¨om spacetime is very similar. The only difference is that ∆ = r2− 2M r + e2which gives
ρρ0= −r
2− 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 S ρρ0dS = − Z 2π 0 Z π 0 (r2− 2M r + e2)r2sin θ 2r4 dθdϕ = −2π(r 2− 2M r + e2) r2 . (6.9)
This now gives the Hawking mass
mH(Sr) = s A (4π)3 ³ 2π + Z S ρρ0dS´= r 4π ³ 4πM r − 2πe2 r2 ´ = M −e 2 2r. (6.10) 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]
gij= 1 − 2M rΣ 0 0 2aM r sinΣ 2θ 0 − Σ∆ 0 0 0 0 −Σ 0 2aM r Σ∆ 0 0 − £ (r2+ a2) + 2a2M r sin2θ Σ ¤ sin2θ , (7.1) where ∆ = r2− 2M r + a2 and Σ = r2+ a2cos2θ. (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]
li= 1 ∆ (r2+ a2, ∆, 0, a) ni= 1 2Σ (r2+ a2, −∆, 0, a) mi= √1 2˜ρ(iasinθ, 0, 1, isinθ ), (7.3) where ˜ ρ = 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
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, m → ¯¯ m + ¯Bn and l → l + ¯Bm + B ¯m + B ¯Bn. (7.6) Then a type I rotation
l → l, m → m + Al, m → ¯¯ m + ¯Al and n → n + ¯Am + A ¯m + A ¯Al. (7.7) Together these two give this total transformation of the tetrad
n → n + ¯B(m + An) + B( ¯m + ¯An) + B ¯B(l + ¯Am + A ¯m + A ¯An) l → l + ¯Bm + B ¯m + B ¯Bn
m → (1 + ¯AB)m + AB ¯m + Bl + A(1 + ¯AB)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 + ¯AB)m1+ AB ¯m1+ Bl1+ A(1 + ¯AB)n1
0 = B + A(1 + ¯AB)n2, (7.9)
which can be solved analytically to give A = −i√2¯˜ρ r 2+ a2± q (r2+ a2)Σ + 2M ra2sin2θ a∆sinθ B = A∆ 2Σ − A ¯A∆. (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 cal-culations 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
γijk= 1
2[λijk+ λkij− λjki], (7.11) where λijk= 4 X µ=1 4 X ν=1 [ejµ,ν− ejν,µ]eiµekν, (7.12) and e1= l, e2= n, e3= m and e4= ¯m. (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 = ej1,2ei1ek2+ ej1,3ei1ek3− ej1,2ei2ek1+ (ej2,3− ej3,2)ei2ek3
− ej4,2ei2ek4− ej1,3ei3ek1+ (ej3,2− ej2,3)ei3ek2− ej4,3ei3ek4
20
By calculating λ314, λ431, λ143, λ243, λ432 and λ324and 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
A = Z S dS = Z 2π 0 Z π 0 √ gθθgϕϕdθdϕ = Z 2π 0 Z π 0 sin θ q (r2+ a2)2− a2∆ sin2θ dθdϕ = 2π Z π 0 sin θ q (r2+ a2)2− a2∆ sin2θ dθ (7.15)
Then we calculate the integral Z S ρρ0dS = Z 2π 0 Z π 0 ρρ0√g θθgϕϕdθdϕ = 2π Z π 0 ρρ0√g θθgϕϕdθ (7.16)
numerically by using Simpson’s fomula [6] Z b
a
f (x) dx = b − a
3n [f0+ fn+ 2(f2+ f4+ . . . + fn−2)
+ 4(f1+ f3+ . . . + fn−1)], (7.17)
where fk = f (a + k/n) and with a remaining term
R = −(b − a)
5
180 n4 f
(4)(ξ) (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 21
7.1
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).
0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 1.4 1.5 1.6 1.7 1.8 1.9 2 r a/M m H
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)
where r+ = M +
√
M2− a2 (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 with the Schwarzschild case and when r = r+ the result agrees with m =
p
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 4 6 8 10 12 14 16 18 20 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 r mH
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.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.8 1.85 1.9 1.95 2 2.05 a/M mH
Figure 7.3: mH plotted with r constant. ◦ corresponds to r = 10, x to r = 7.5,
7.2 Discussion and Summary 23
7.2
Discussion and Summary
In this thesis we have calculated the Hawking mass in Schwarzschild, Reissner-Nordstr¨om 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:
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