Jump to ContentJump to Main Navigation
Introduction to 3+1 Numerical Relativity$

Miguel Alcubierre

Print publication date: 2008

Print ISBN-13: 9780199205677

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199205677.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 26 February 2017

(p.410) APPENDIX C BSSNOK WITH NATURAL CONFORMAL RESCALING

(p.410) APPENDIX C BSSNOK WITH NATURAL CONFORMAL RESCALING

Source:
Introduction to 3+1 Numerical Relativity
Publisher:
Oxford University Press

The BSSNOK formulation of the 3+1 evolution equations described in Section 2.8 is based on a conformal rescaling of the metric and extrinsic curvature. However, in its standard version this formulation uses a rescaling of the extrinsic curvature that is in fact not the most natural rescaling for this quantity. Here I will present a version of the BSSNOK equations that uses the natural rescaling for a tracefree symmetric tensor.

We start by summarizing the standard BSSNOK equations. We perfom a conformal transformation of the spatial metric and extrinsic curvature of the form

(C.1)
γ ˜ i j : = e 4 ϕ γ i j ,
(C.2)
A ˜ i j : = e 4 ϕ ( K i j 1 3 γ i j K ) ,

where the conformal factor is taken to be ϕ = 1 12 ln γ , so that the conformal metric γ ˜ i j has unit determinant. In the previous expression K is the trace of K ij, which implies that the tensor à ij is traceless. We also introduces the auxiliary variables

(C.3)
Γ ˜ i = γ ˜ j k Γ ˜ j k i = j γ ˜ i j ,

where the second equality follows from the fact that γ ˜ i j has unit determinant.

The full system of evolution equations then becomes

d d t γ ˜ i j = 2 α A ˜ i j ,
(C.5)
d d t ϕ   = 1 6 α K ,
(C.6)
d d t K = D i D i α + α (   A ˜ i j A ˜ i j + 1 3 K 2 ) + 4 π α ( ρ + S ) ,
(C.7)
d d t A ˜ i j = e 4 ϕ { D i D j α + α R i j + 4 π α [ γ i j ( S ρ ) 2 S i j ] } TF         + α ( K A ˜ i j 2 A ˜ i k A ˜ k j ) ,
(C.8)
d d t Γ ˜ i = γ ˜ j k j k β i + 1 3 γ ˜ i j j k β k 2 A ˜ i j j α        + 2 α ( Γ ˜ j k i A ˜ j k + 6 A ˜ i j j ϕ 2 3 γ ˜ i j j K 8 π e 4 ϕ j i ) ,
(p.411) where TF denotes the tracefree part of the expression inside the brackets, and where we have used the Hamiltonian constraint to eliminate the Ricci scalar from the evolution equation for K, and the momentum constraints to eliminate the divergence of à ij from the evolution equation for Γ ˜ i . In the previous expressions we have d / d t : = t Δ £ , with Δ £ the Lie derivative with respect to the shift that must be calculated for tensor densities: ψ, a scalar density of weight 1/6, and γ ˜ i j and à ij, tensor densities with weight −2/3. Also, even though the Γ ˜ i is strictly speaking not a vector, its Lie derivative is understood as that corresponding to a vector density of weight 2/3. Finally, the Ricci tensor is separated into two contributions in the following way:
(C.9)
R i j = R ˜ i j + R i j ϕ ,

where R ˜ i j is the Ricci tensor associated with the conformal metric γ ˜ i j :

(C.10)
R ˜ i j = 1 2 γ ˜ l m l m γ ˜ i j + γ ˜ k ( i j ) Γ ˜ k + Γ ˜ k Γ ˜ ( i j ) k      + γ ˜ l m ( 2 Γ ˜ l ( i k Γ ˜ j ) k m + Γ ˜ i m k Γ ˜ k l j ) ,

and where R i j ϕ is given by φ:

(C.11)
R i j ϕ = 2 D ˜ i D ˜ j ϕ 2 γ ˜ i j D ˜ k D ˜ k ϕ + 4 D ˜ i ϕ D ˜ j ϕ 4 γ ˜ i j D ˜ k ϕ D ˜ k ϕ ,

with D ˜ i the covariant derivative associated with the conformal metric.

The Hamiltonian and momentum constraints also take the form

(C.12)
R = A ˜ i j A ˜ i j 2 3 K 2 + 16 π ρ ,
(C.13)
j A ˜ i j = Γ ˜ j k i A ˜ j k 6 A ˜ i j j ϕ + 2 3 γ ˜ i j j K + 8 π e 4 ϕ j i .

Now, in Chapter 3 it was shown that the natural conformal transformation for the tracefree extrinsic curvature is in fact Āij10 A ij (Ā ij2 A ij), with φ=ln ψ. We will then consider the following conformal transformation

(C.14)
γ ˜ i j : = e 4 ϕ γ i j ,
(C.15)
A ˜ i j : = e + 2 ϕ ( K i j 1 3 γ i j K ) .

We will furthermore introduce the densitized lapse

(C.16)
α ¯ = α γ 1 / 2 = e 6 ϕ α .

With this new conformal scaling, the evolution equations become instead (do notice that in some places there is α ¯ and in others just α) (p.412)

(C.17)
d d t γ ˜ i j = 2 α ¯ A ˜ i j ,
(C.18)
d d t ϕ = e 6 ϕ 6 α ¯ K ,
(C.19)
d d t K = D i D i α + α ¯ [ e 6 ϕ A ¯ i j A ¯ i j + 1 3 e 6 ϕ K 2 + 4 π e 6 ϕ ( ρ + S ) ] ,
(C.20)
d d t A ¯ i j = e 2 ϕ { D i D j α + α R i j + 4 π α [ γ i j ( S ρ ) 2 S i j ] } TF         2 α ¯ A ¯ i k A ¯ k j ,
(C.21)
d d t Γ ¯ i = γ ˜ j k j k β i + 1 3 γ ˜ i j j k β k 2 A ¯ i j j α ¯        + 2 α ¯ ( Γ ˜ j k i A ˜ j k 2 3 e 6 ϕ γ ¯ i j j K 8 π e 10 ϕ j i ) ,

and the constraints become

(C.22)
R = e 12 ϕ A ¯ i j A ¯ i j 2 3 K 2 + 16 π ρ ,
(C.23)
j A ¯ i j = Γ ¯ j k i A ¯ j k + 2 3 e 6 ϕ γ ¯ i j j K + 8 π e 10 ϕ j i .

Notice that with this new rescaling, a term involving derivatives of the conformal factor φ has disappeared from both the evolution equation for Γ ¯ i and the momentum constraints.