Appendix: Reduction of the Conformal Field Equations

6 Appendix: Reduction of the Conformal Field Equations

In this appendix we show how to perform the reduction process for the conformal field equations to obtain the symmetric hyperbolic system of evolution equations and the constraints. We assume that we are given a time-like unit vector $ta$ with respect to which the reduction is done. Any vector $va$ may be decomposed into parts perpendicular and parallel to $ta$ ,

va = ˆva + vta, with ˆvat = 0 and v = vat, a a

and similarly for one-forms $wa$ . We call a vector spatial with respect to $a t$ if it is orthogonal to $ta$ . In particular, the metric $gab$ gives rise to a spatial metric $hab$ by the decomposition

The volume four-form $𝜀abcd$ that is defined by the metric also gives rise to a decomposition as follows:

The covariant derivative operator $∇a$ is written analogously,

∇a = Da + taD,

thus defining two derivative operators: $Da$ with $taDa = 0$ and $D = ta∇a$ . The covariant derivative of $ta$ itself is an important field. It gives rise to two component fields defined by

Note that $b χa$ is spatial in both its indices and that there is no symmetry implied between the two indices. Similarly, $χb$ is automatically spatial.

It is useful to define two new derivative operators $∂$ and $∂a$ by the following relations:

These operators have the property that they are compatible with the spatial metric $hab$ and that they annihilate $ta$ and $ta$ . If $ta$ is the unit normal of a hypersurface, i.e. if $ta$ is hypersurface orthogonal, then $χ ab$ is symmetric, $h ab$ is the induced (negative definite) metric on the hypersurface, and $∂ a$ is its Levi–Civita connection. In general this is not the case and so the operator $∂a$ possesses torsion. In particular, we obtain the following commutators (acting on scalars and spatial vectors):

2∂[a∂b]f + 2χ [ab]∂f = 0, (43 ) 2∂[a∂b]vc + 2χ[ab]∂vc + harhbshctRrstdvd − 2χ [a|c|χb]eve = 0, (44 ) c [∂,∂a ]f + χa∂f + χa ∂cf = 0, (45 ) [∂,∂a]vb + χa∂vb + χac∂cvb − χbχaeve + χabχeve + tchaehbfRcefdvd = 0. (46 )

These commutators are obtained from the commutators between the derivative operators $Da$ and $D$ by expressing them in terms of $∂a$ and $∂$ on the one hand, and by the four-dimensional connection $∇a$ on the other hand. This procedure yields two equations for the derivatives of $ta$ ,

The information contained in the commutator relations and in the Equations (47

) and (48

) is completely equivalent to the Cartan equations for $∇a$ that define the curvature and torsion tensors.

This completes the preliminaries and we can now go on to perform the splitting of the equations. Our intention is to end up with a system of equations for all the spatial parts of the fields. In order not to introduce too many different kinds of indices, all indices refer to the four-dimensional space-time, but they are all spatial, i.e. any transvection with $t a$ and $ta$ vanishes. If we introduce hypersurfaces with normal vector $a t$ , then there exists an isomorphism between the tensor algebra on the hypersurfaces and the subalgebra of spatial four-dimensional tensors.

We start with the tensorial part of the equations. To this end we decompose the fields into various spatial parts and insert these decompositions into the conformal field equations defined by Equations (17 , 18 , 19 , 20 , 21 ). The fields are decomposed as follows:

e e ef Kabcd = 4t[aEb][ctd] + 2t[aBb] 𝜀ecd − 2𝜀ab Be[ctd] + 𝜀abeE 𝜀fcd, (49 ) Φ = ϕ + 2t ϕ + t tϕ, (50 ) ab ab (a b) a b Σa = σa + taσ. (51 )

The function $ϕ$ is fixed in terms of $ϕ ab$ because $Φ ab$ is trace-free.

Inserting the decomposition of $Kabcd$ into Equation (18 ), decomposing the equations into various spatial parts, and expressing derivatives in terms of the operators $∂a$ and $∂$ yields four equations:

∂aBac + 𝜀eadEce χad + 𝜀ecdEaeχad = 0, (52 ) a e ad e ad ∂ Eac − 𝜀eadBc χ − 𝜀ecdBa χ = 0, (53 ) ∂Ebc + 𝜀ae(b∂aBec ) + 2χEbc − 2χa(bEc)a + hbcEadχad − Ea (bχc )a + 2𝜀ea(bBec )χa = 0, (54 ) a e a ad a e a ∂Bbc − 𝜀ae(b∂ E c) + 2χBbc − 2χ (bBc )a + hbcBadχ − Ba (bχc ) − 2𝜀ea(bE c)χ = 0. (55 )

Here we have defined $χ = χaa$ . Treating the other fields and equations in a similar way, we obtain Equation (17

) in the form of four equations:

∂ϕa − ∂aϕ − χeϕae + 2χaeϕe + χa ϕ + ∂aΛ + Ead σd = 0, (56 ) 1 ∂[cϕa] − χ[ceϕa]e + χ [ca]ϕ − -𝜀caeBed σd = 0, (57 ) 2 ∂ϕab − ∂(aϕb) + 2 χ(aϕb) + χ(aeϕb)e − χ (ab)ϕ − hab∂Λ − Eabσ − B(ae𝜀b)deσd = 0, (58 ) 1 1 ∂ [cϕa]b − χ [ac]ϕb + ϕ[aχc]b − hb[a∂c]Λ +--𝜀caeBebσ − -𝜀caeEef𝜀fbdσd = 0. (59 ) 2 2

The equation (19

) for the conformal factor is rather straightforward. We obtain

while Equation (20

) yields four equations:

∂ σ + χ σ − h S + Ω ϕ = 0, (62 ) a b ab ab ab ∂bσ − χbeσe + Ω ϕb = 0, (63 ) ∂σ + χ σ + Ω ϕ = 0, (64 ) be b b ∂ σ − χ σe − S + Ω ϕ = 0. (65 )

Finally, the equation (21

) for $S$ gives two equations

This completes the gauge-independent part of the equations. In order to deal with the gauges we now have to introduce an arbitrary tetrad and arbitrary coordinates. We extend the time-like unit vector to a complete tetrad $(ta,eai)$ with $taeai = 0$ for $i = 1,2,3$ . Let $xμ$ with $μ = 0,1,2,3$ be four arbitrary functions, which we use as coordinates. Application of $∂$ and $∂a$ to the coordinates yields

The four functions $cμ$ and the four one-forms $cμa$ may be regarded as the 16 expansion coefficients of the tetrad vectors in terms of the coordinate basis $∂μ = ∂∕∂x μ$ because of the identity

In a similar spirit, we apply the derivative operators to the tetrad and obtain

Again, transvection with $ta$ on any index of $Λa b$ and $Γ a c b$ vanishes. Furthermore, both $Λ ab$ and $Γ cab$ are antisymmetric in their (last two) indices. Together with the 12 components of $χa$ and $χab$ these fields provide an additional 12 components, which account for the 24 connection coefficients of the four-dimensional connection $∇a$ with respect to the chosen tetrad.

Note that these fields are not tensor fields. They do not transform as tensors under the change of tetrad. Since we will keep the tetrad fixed here, we may, however, regard them as defining tensor fields whose components happen to coincide with them in the specified tetrad.

In order to extract the contents of the first of Cartan’s structure equations, one needs to apply the commutators (43 ) and (45 ) to the coordinates to obtain

Similarly, the second of Cartan’s structure equations is exploited by applying the commutators to the tetrad vectors. Equation (16

) is then used to substitute for the Riemann tensor in terms of the gravitational field, the trace-free part of the Ricci tensor, and the scalar curvature. Apart from the Equations (47

) and (48

), which come from acting on $ta$ , this procedure yields

∂Γ − ∂ Λ − 2 Γ eΛ + χ Λ + χ eΓ + 2χ χ + ΩB e𝜀 + 2h ϕ = 0, (75 ) abc a bc a[b c]e a bc a ebc a[b c] a ebc a[b c] ∂aΓ bcd − ∂bΓ acd + 2χ [ab]Λcd + 2Γ [ae|c|Γ b]ed + 2χ[a|dχb]c + Ω 𝜀abeEef 𝜀fcd − 2h ϕ + 2h ϕ + 4h h Λ = 0. (76 ) c[a b]d d[a b]c c[a b]d

Now we have collected all the equations that can be extracted from the conformal field equations and Cartan’s structure equations. What remains to be done is to separate them into constraints and evolution equations. Before doing so, we notice that we do not have enough evolution equations for the tetrad components and the connection coefficients. The remedy to this situation is explained in Section 3.2. It amounts to adding appropriate “divergence equations”. We obtain these by computing the “gauge source functions”. The missing equations for the coordinates are obtained by applying the d’Alembert operator to the coordinates. Expressing the wave operator in terms of $∂$ and $∂a$ yields the additional equations

In order to find the missing equations for the tetrad we need to compute the gauge source functions

In a similar way as explained above, we may regard these functions as components of tensor fields $Fab$ and $Fa$ whose components happen to agree with them in the specified basis. Thus,

Fik = Fabeaiebk, F0i = Faeai.

Computing these tensor fields from (78 , 79 ) gives

Now we are ready to collect the constraints:

∂acμ − ∂bcμ + 2χ [ab]cμ = 0, (82 ) e be af ∂aΓ bcd − ∂bΓ acd + 2χ [ab]Λcd − 2Γ [a|c| Γ b]ed + 2χ[a|dχb]c + Ω 𝜀ab Eef 𝜀 cd − 2hc [aϕb]d + 2hd[aϕb]c + 4hc[ahb]dΛ = 0, (83 ) e ∂aχbc − ∂bχac + 2 χ[ab]χc − Ω 𝜀ab Bec − 2hc[aϕb] = 0, (84 ) ∂cϕa − ∂aϕc − 2χ [ceϕa]e + 2χ[ca]ϕ − 𝜀caeBedσd = 0, (85 ) e ef d ∂cϕab − ∂aϕcb − 2χ [ac]ϕb + 2ϕ[aχc]b − 2hb[a∂c]Λ + 𝜀caeB bσ − 𝜀caeE 𝜀fbdσ = 0, (86 ) ∂aBac + 𝜀eadEceχad + 𝜀ecdEaeχad = 0, (87 ) a e ad e ad ∂ Eac − 𝜀eadBc χ − 𝜀ecdBa χ = 0, (88 ) ∂a Ω − σa = 0, (89 ) ∂ σ + χ σ − h S + Ωϕ = 0, (90 ) a b ab abe ab ∂bσ − χb σe + Ω ϕb = 0, (91 ) ∂ S + ϕ σb + ϕ σ − Ω ∂ Λ − 2Λ σ = 0. (92 ) a ab a a a

Finally, we collect the evolution equations:

∂cμ − ∂ cμ = − χ cμ − χ bcμ, (93 ) a a a a b ∂cμ + ∂acμa = χacμa − χcμ + F μ. (94 ) ∂ Γ − ∂ Λ = 2Γ eΛ − χ Λ − χ eΓ − 2χ χ − ΩB e𝜀 − 2h ϕ , (95 ) abc ea bc ea[b c]e a bc a ebc a[b c] a ebc a[b c] ∂ Λab + ∂ Γ eab = χ Γ eab − χ Λab + Fab, (96 ) ∂χa + ∂eχea = χeχea − χaχee − χecΓ eca − χcΛca − Fa, (97 ) c ∂χab − ∂aχb = − χa χcb − χaχb − ΩEab − ϕab − hab(ϕ − 2Λ ), (98 ) ∂ϕa + ∂b ϕab = χeϕae − χcaϕc − ϕaχ + χaϕcc − 3∂aΛ, (99 ) e e d ∂ϕab − ∂(aϕb) = − 2χ (aϕb) − χ(a ϕb)e + χ(ab)ϕ + hab∂ Λ + Eabσ + B (a𝜀b)deσ , (100 ) ∂Ebc + 𝜀ae(b∂aBec ) = − 2χEbc + 2χa(bEc)a − hbcEad χad + Ea(bχc )a − 2𝜀ea(bBec )χa, (101 ) a e a ad a e a ∂Bbc − 𝜀ae(b∂ E c) = − 2χBbc + 2χ (bBc)a − hbcBad χ + Ba(bχc) + 2𝜀ea(bE c)χ , (102 ) ∂Ω = σ, (103 ) ∂σb = − χbσ − Ωϕb, (104 ) ∂σ = χeσe + S − Ω ϕ, (105 ) ∂S = − ϕ σb − ϕ σ + Ω ∂Λ + 2σ Λ. (106 ) b

This is the complete system of evolution equations that can be extracted from the conformal field equations. As it is written, this system is symmetric hyperbolic. This is not entirely obvious but rather straightforward to verify. It is important to keep in mind that with our conventions the spatial metric $h ab$ is negative definite. Altogether these are 65 equations.