3+1 formalism and bases of numerical relativity - LUTh ...
3+1 formalism and bases of numerical relativity - LUTh ... 3+1 formalism and bases of numerical relativity - LUTh ...
118 Asymptotic flatness and global quantities Using this property, as well as expression (7.75) of dVµ with the components nµ = (−N,0,0,0) given by Eq. (4.38), we get Vt ∇νA µν dVµ = − Vt ∂ ∂xν √ µν −gA nµ √ γ √ d −g 3 x = Vt ∂ ∂xν √ 0ν γNA d 3 x, (7.78) where we have also invoked the relation (4.55) between the determinants of g and γ: √ −g = N √ γ. Now, since A αβ is antisymmetric, A 00 = 0 and we can write ∂/∂x ν √ γNA 0ν = ∂/∂x i √ γ V i where V i = NA 0i are the components of the vector V ∈ T (Σt) defined by V := −γ(n · A). The above integral then becomes Vt ∇νA µν dVµ = Vt 1 ∂ √ γ ∂xi √ i γV √ 3 γ d x = DiV Vt i√ γ d 3 x. (7.79) We can now use the Gauss-Ostrogradsky theorem to get ∇νA µν dVµ = V i √ 2 si q d y. (7.80) Vt Noticing that ∂Vt = Ht ∪ St (cf. Fig. 7.2) and (from the antisymmetry of A µν ) we get the identity (7.74). ∂Vt V i si = V ν sν = −nµA µν sν = 1 2 Aµν (sµnν − nµsν), (7.81) Remark : Equation (7.74) can also be derived by applying Stokes’ theorem to the 2-form 4 ǫαβµνA µν , where 4 ǫαβµν is the Levi-Civita alternating tensor (volume element) associated with the spacetime metric g (see e.g. derivation of Eq. (11.2.10) in Wald’s book [265]). where Applying formula (7.74) to A µν = ∇ µ k ν we get, in view of the definition (7.71), MK = − 1 ∇ν∇ 4π Vt µ k ν dVµ + M H K , (7.82) M H K := 1 ∇ 8π Ht µ k ν dS H µν will be called the Komar mass of the hole. Now, from the Ricci identity (7.83) ∇ν∇ µ k ν − ∇ µ ∇ ν k ν = =0 4 R µ νk ν , (7.84) where the “= 0” is a consequence of Killing’s equation (7.73). Equation (7.82) becomes then MK = − 1 4 µ R 4π νk Vt ν dVµ + M H K = 1 4 Rµνk 4π Vt ν n µ √ γ d 3 x + M H K . (7.85)
7.6 Komar mass and angular momentum 119 At this point, we can use Einstein equation in the form (4.2) to express the Ricci tensor 4R in terms of the matter stress-energy tensor T. We obtain MK = 2 Tµν − 1 2 Tgµν n µ k ν√ γ d 3 x + M H K . (7.86) Vt The support of the integral over Vt is reduced to the location of matter, i.e. the domain where T = 0. It is then clear on formula (7.86) that MK is independent of the choice of the 2-surface St, provided all the matter is contained in St. In particular, we may extend the integration to all Σt and write formula (7.86) as MK = 2 Σt T(n,k) − 1 √γ 3 H T n · k d x + MK . (7.87) 2 The Komar mass then appears as a global quantity defined for stationary spacetimes. Remark : One may have M H K < 0, with MK > 0, provided that the matter integral in Eq. (7.87) compensates for the negative value of M H K . Such spacetimes exist, as recently demonstrated by Ansorg and Petroff [21]: these authors have numerically constructed spacetimes containing a black hole with M H K < 0 surrounded by a ring of matter (incompressible perfect fluid) such that the total Komar mass is positive. 7.6.2 3+1 expression of the Komar mass and link with the ADM mass In stationary spacetimes, it is natural to use coordinates adapted to the symmetry, i.e. coordinates (t,x i ) such that ∂t = k . (7.88) Then we have the following 3+1 decomposition of the Killing vector in terms of the lapse and shift [cf. Eq. (4.31)]: k = Nn + β. (7.89) Let us inject this relation in the integrand of the definition (7.71) of the Komar mass : ∇ µ k ν dSµν = ∇µkν(s µ n ν − n µ s ν ) √ q d 2 y = 2∇µkν s µ n ν√ q d 2 y = 2(∇µN nν + N∇µnν + ∇µβν)s µ n ν√ q d 2 y = 2(−s µ ∇µN + 0 − s µ βν∇µn ν ) √ q d 2 y = −2 s i DiN − Kijs i β j √ q d 2 y, (7.90) where we have used Killing’s equation (7.73) to get the second line, the orthogonality of n and β to get the fourth one and expression (3.22) for ∇µn ν to get the last line. Inserting Eq. (7.90) into Eq. (7.71) yields the 3+1 expression of the Komar mass: MK = 1 i s DiN − Kijs 4π St i β j √ 2 q d y . (7.91)
- Page 68 and 69: 68 3+1 decomposition of Einstein eq
- Page 70 and 71: 70 3+1 decomposition of Einstein eq
- Page 72 and 73: 72 3+1 equations for matter and ele
- Page 74 and 75: 74 3+1 equations for matter and ele
- Page 76 and 77: 76 3+1 equations for matter and ele
- Page 78 and 79: 78 3+1 equations for matter and ele
- Page 80 and 81: 80 3+1 equations for matter and ele
- Page 82 and 83: 82 3+1 equations for matter and ele
- Page 84 and 85: 84 Conformal decomposition equivale
- Page 86 and 87: 86 Conformal decomposition As an ex
- Page 88 and 89: 88 Conformal decomposition 6.2.4 Co
- Page 90 and 91: 90 Conformal decomposition 6.3.1 Ge
- Page 92 and 93: 92 Conformal decomposition where K
- Page 94 and 95: 94 Conformal decomposition to write
- Page 96 and 97: 96 Conformal decomposition hence Lm
- Page 98 and 99: 98 Conformal decomposition 6.5.2 Ha
- Page 100 and 101: 100 Conformal decomposition discuss
- Page 102 and 103: 102 Conformal decomposition Remark
- Page 104 and 105: 104 Asymptotic flatness and global
- Page 106 and 107: 106 Asymptotic flatness and global
- Page 108 and 109: 108 Asymptotic flatness and global
- Page 110 and 111: 110 Asymptotic flatness and global
- Page 112 and 113: 112 Asymptotic flatness and global
- Page 114 and 115: 114 Asymptotic flatness and global
- Page 116 and 117: 116 Asymptotic flatness and global
- Page 120 and 121: 120 Asymptotic flatness and global
- Page 122 and 123: 122 Asymptotic flatness and global
- Page 124 and 125: 124 Asymptotic flatness and global
- Page 126 and 127: 126 The initial data problem Notice
- Page 128 and 129: 128 The initial data problem where
- Page 130 and 131: 130 The initial data problem 8.2.3
- Page 132 and 133: 132 The initial data problem where
- Page 134 and 135: 134 The initial data problem Figure
- Page 136 and 137: 136 The initial data problem Figure
- Page 138 and 139: 138 The initial data problem In par
- Page 140 and 141: 140 The initial data problem Accord
- Page 142 and 143: 142 The initial data problem Remark
- Page 144 and 145: 144 The initial data problem Remark
- Page 146 and 147: 146 The initial data problem 8.4.1
- Page 148 and 149: 148 The initial data problem Since
- Page 150 and 151: 150 The initial data problem • fo
- Page 152 and 153: 152 Choice of foliation and spatial
- Page 154 and 155: 154 Choice of foliation and spatial
- Page 156 and 157: 156 Choice of foliation and spatial
- Page 158 and 159: 158 Choice of foliation and spatial
- Page 160 and 161: 160 Choice of foliation and spatial
- Page 162 and 163: 162 Choice of foliation and spatial
- Page 164 and 165: 164 Choice of foliation and spatial
- Page 166 and 167: 166 Choice of foliation and spatial
7.6 Komar mass <strong>and</strong> angular momentum 119<br />
At this point, we can use Einstein equation in the form (4.2) to express the Ricci tensor 4R in<br />
terms <strong>of</strong> the matter stress-energy tensor T. We obtain<br />
<br />
MK = 2 Tµν − 1<br />
2 Tgµν<br />
<br />
n µ k ν√ γ d 3 x + M H K . (7.86)<br />
Vt<br />
The support <strong>of</strong> the integral over Vt is reduced to the location <strong>of</strong> matter, i.e. the domain where<br />
T = 0. It is then clear on formula (7.86) that MK is independent <strong>of</strong> the choice <strong>of</strong> the 2-surface<br />
St, provided all the matter is contained in St. In particular, we may extend the integration to<br />
all Σt <strong>and</strong> write formula (7.86) as<br />
<br />
MK = 2<br />
Σt<br />
<br />
T(n,k) − 1<br />
<br />
√γ 3 H<br />
T n · k d x + MK . (7.87)<br />
2<br />
The Komar mass then appears as a global quantity defined for stationary spacetimes.<br />
Remark : One may have M H K < 0, with MK > 0, provided that the matter integral in Eq. (7.87)<br />
compensates for the negative value <strong>of</strong> M H K . Such spacetimes exist, as recently demonstrated<br />
by Ansorg <strong>and</strong> Petr<strong>of</strong>f [21]: these authors have <strong>numerical</strong>ly constructed spacetimes containing<br />
a black hole with M H K < 0 surrounded by a ring <strong>of</strong> matter (incompressible perfect<br />
fluid) such that the total Komar mass is positive.<br />
7.6.2 <strong>3+1</strong> expression <strong>of</strong> the Komar mass <strong>and</strong> link with the ADM mass<br />
In stationary spacetimes, it is natural to use coordinates adapted to the symmetry, i.e. coordinates<br />
(t,x i ) such that<br />
∂t = k . (7.88)<br />
Then we have the following <strong>3+1</strong> decomposition <strong>of</strong> the Killing vector in terms <strong>of</strong> the lapse <strong>and</strong><br />
shift [cf. Eq. (4.31)]:<br />
k = Nn + β. (7.89)<br />
Let us inject this relation in the integr<strong>and</strong> <strong>of</strong> the definition (7.71) <strong>of</strong> the Komar mass :<br />
∇ µ k ν dSµν = ∇µkν(s µ n ν − n µ s ν ) √ q d 2 y<br />
= 2∇µkν s µ n ν√ q d 2 y<br />
= 2(∇µN nν + N∇µnν + ∇µβν)s µ n ν√ q d 2 y<br />
= 2(−s µ ∇µN + 0 − s µ βν∇µn ν ) √ q d 2 y<br />
= −2 s i DiN − Kijs i β j √ q d 2 y, (7.90)<br />
where we have used Killing’s equation (7.73) to get the second line, the orthogonality <strong>of</strong> n <strong>and</strong><br />
β to get the fourth one <strong>and</strong> expression (3.22) for ∇µn ν to get the last line. Inserting Eq. (7.90)<br />
into Eq. (7.71) yields the <strong>3+1</strong> expression <strong>of</strong> the Komar mass:<br />
MK = 1<br />
<br />
i<br />
s DiN − Kijs<br />
4π St<br />
i β j √ 2<br />
q d y . (7.91)
Change language