Scalar modes and the linearized Schwarzschild solution [1ex] on a quantized FLRW space-time in Yang-Mills matrix models

UWThPh-2019-34

Harold C. Steinacker

Faculty of Physics, University of Vienna

Boltzmanngasse 5, A-1090 Vienna, Austria

Email: harold.steinacker@univie.ac.at

Abstract

We study scalar perturbations of a recently found 3+1-dimensional FLRW quantum space-time solution in Yang-Mills matrix models. In particular, the linearized Schwarzschild metric is obtained as a solution. It arises from a quasi-static would-be massive graviton mode, and slowly decreases during the cosmic expansion. Along with the propagating graviton modes, this strongly suggests that 3+1 dimensional (quantum) gravity emerges from the IKKT matrix model on this background. For the dynamical scalar modes, non-linear effects must be taken into account. We argue that they lead to non-Ricci-flat metric perturbations with very long wavelengths, which would be perceived as dark matter from the GR point of view.

## 1 Introduction

The starting point of this paper is a recent solution of the IKKT-type matrix models with mass term [1], which is naturally interpreted as 3+1-dimensional cosmological FLRW quantum space-time. It was shown that the fluctuation modes around this background include spin-2 metric fluctuations, as well as a truncated tower of higher-spin modes which are organized in a higher-spin gauge theory. The 2 standard Ricci-flat massless graviton modes were found, as well as some additional vector-like and scalar modes whose significance was not fully clarified.

The aim of the present paper is to study in more detail the metric perturbations, and in particular to see if and how the (linearized) Schwarzschild solution can be obtained. We will indeed find such a solution, which is realized in the scalar sector of the linearized perturbation modes exhibited in [1]. This means that the model has a good chance to satisfy the precision solar system tests of gravity. We will also elaborate and discuss in some detail the extra scalar mode, which is not present in GR. This seems to provide a natural candidate for apparent dark matter.

Since the notorious problems in attempts to quantize gravity arise primarily from the Einstein-Hilbert action, is is very desirable to find another framework for gravity, which is more suitable for quantization. String theory provides such a framework, but the traditional approach using compactifications leads to a host of issues, notably lack of predictivity. This suggests to use matrix models as a starting point, and in particular the IKKT or IIB model [2], which was originally proposed as a constructive definition of string theory. Remarkably, numerical studies in this non-perturbative formulation provide evidence [3, 4, 5] that 3+1-dimensional configurations arise at the non-perturbative level, tentatively interpreted as expanding universe. However, this requires a new mechanism for gravity on 3+1-dimensional non-commutative backgrounds as in [1], which does not rely on compactification. The present paper provides further evidence and insights for this mechanism.

The (linearized) Schwarzschild metric is clearly the benchmark for any viable theory of gravity. There has been considerable effort to find noncommutative analogs of the Schwarzschild metric from various approaches, leading to a number of proposals [6, 7, 8, 9] and references therein, cf. also [10]; however, none is truly satisfactory. The proposals are typically obtained by some ad-hoc modification of the classical solution, without any intrinsic role of noncommutativity, which is put in by hand. In contrast, the quantum structure (or its semi-classical limit) plays a central role in the present framework. Our solution is a deformation of the noncommutative background which respects an exact rotation symmetry, even though there are only finitely many d.o.f. per unit volume. The solution has a good asymptotics at large distances, allowing superpositions corresponding to arbitrary mass distributions. In fact we obtain generic quasi-static Ricci-flat linearized perturbations, which complement the Ricci-flat propagating gravitons found in [1].

This realization of the (linearized) Schwarzschild solution is remarkable and may seem surprising, because the action is of Yang-Mills type, and no Einstein-Hilbert-like action is required111It is well-known that gravity can be obtained from a Yang-Mills-type action by imposing constraints, cf. [11, 12, 13]. However this essentially amounts to a reformulation of classical GR, and the usual problems are expected to arise upon quantization. In contrast, we do not impose any constraints on the Yang-Mills action. Nevertheless, quantum effects are expected to induce Einstein-Hilbert-like terms in the quantum effective action, as discussed in [1]. This may play an important role here as well, but we focus on the classical mechanism. Another interesting possibility was proposed in [14], which has some similarities to the present mechanism but appears to have some issues.. This means that the theory has a good chance to survive upon quantization, which is naturally defined via integration over the space of matrices. The IKKT model is indeed well suited for quantization, and quite clearly free of ghosts and other obvious pathologies. It is background-independent in the sense that it has a large class of solutions with different geometries, and defines a gauge theory for fluctuations on any background.

The price to pay is a considerable complexity of the resulting theory. As explained in [15, 1] the background leads to a higher-spin gauge theory, with a truncated tower of higher spin modes, and many similarities with (but also distinctions from) Vasiliev theory [16]. Since space-time itself is part of the background solution, it is not unreasonable to expect Ricci-flat deformations, cf. [17, 18, 19]. However, Lorentz invariance is very tricky on noncommutative backgrounds. In the present case the space-like isometries of the FLRW space-time are manifest, but invariance under (local) boosts is not. Nevertheless, the propagation of all physical modes is governed by the same effective metric. In particular, the concept of spin has to be used with caution, and would-be spin modes decompose further into sectors governed by the space-like isometry. The tensor fields are accordingly characterized by the transformation under the local stabilizer group, and the term “scalar modes” is understood in this sense throughout the paper. However this complication is in fact helpful to identify physical degrees of freedom in the physical sector, and to understand the absence of ghosts.

Let us describe the new results in some details. We focus on the scalar fluctuation mode which was found in [1], and elaborate the associated metric fluctuations. The main result is that there is a preferred “quasi-static” vacuum solution which leads to the linearized Schwarzschild metric on the FRW background. This strongly suggests that a near-realistic gravity emerges on the background, however only the vacuum solution is considered here. Quasi-static means that the solution is static on local scales at late times, but slowly decays on cosmic scales, in a specific way. This is a somewhat unexpected result, whose significance is not entirely clear. The quasi-static solution is singled out because all other solutions lead to a large diffeo term, which makes the linearized treatment problematic. Hence the Schwarzschild solution is the “cleanest” case, while the generic dynamical scalar modes require non-linear considerations somewhat reminiscent of the Vainshtein mechanism [20]. We offer a heuristic way to understand them, which points to the intriguing - albeit quite speculative - possibility that these non-Ricci-flat scalar modes might provide a geometrical explanation for dark matter at galactic scales.

Along the way, we also find the missing off-shell scalar fluctuation mode, which was missing in [1]. Thus all 10 off-shell metric fluctuation modes for the most general metric fluctuations are realized, and the model is certainly rich enough for a realistic theory of gravity. That theory would clearly deviate from GR at cosmic scale, since the FLRW background solution is not Ricci flat, but requires no stabilization by matter (or energy) and no fine-tuning.

Finally, it should be stressed that even though the model is intrinsically noncommutative, it should be viewed in the spirit of almost-local and almost-classical field theory. Space-time arises as a condensation of matrices rather than some non-local holographic image, with dynamical local fluctuations described by an effective field theory.

The paper is meant to be as self-contained and compact as possible. We start with a lightning introduction to the space-time under consideration, and elaborate only the specific modes and aspects needed to obtain the Schwarzschild solution. For some results we have to refer to [1], but the essential new computations are mostly spelled out. For the skeptical reader, some of the missing steps may be uncovered from the file in the arXive.

## 2 Quantum FLRW space-time M3,1n

The quantum space-time under consideration is based on a particular representation of , which is a lowest weight unitary irreps in the short discrete series known as minireps or doubletons [21, 22]. Those are the unique irreps which remain irreducible under the restriction to . We denote the generators in this representation by , which are Hermitian operators satisfying

where be the invariant metric of . We then define

 Xμ \coloneqqrMμ5,X4:=rM45 Tμ \coloneqqR−1Mμ4μ,ν=0,…,3 . (2.2)

Then the transform as vector operators under , while the are vector operators under . The -invariant fuzzy or quantum space-time is then defined through the algebra of functions generated by the for . The commutation relations (2.1) imply

 [Xμ,Xν] =−ir2Mμν\eqqcoloniΘμν, (2.3a) =i1RημνX4, (2.3b) =−ir2R2Θμν, (2.3c) =−i1RXμ, (2.3d) =−ir2RTμ, (2.3e)

and the irreducibility of under implies the relations [1]

 XμXμ =−R2−X4X4, R2=r24(n2−4) (2.4a) TμTμ =1r2+1r2R2X4X4, (2.4b) XμTμ+TμXμ =0 . (2.4c)

There are some extra constraints involving , which will only be given in the semi-classical version below. Unless otherwise stated, indices will be raised and lowered with or . Apart from the extra constraints, the construction is quite close to that of Snyder [23] and Yang [24].

The proper interpretation of this structure is not obvious a priori, due to the extra generators and . These cannot be dropped, because the full algebra is generated by the alone. A proper geometrical understanding is obtained by considering all the generators of . As explained in [25, 26, 27], these are naturally viewed as quantized embedding functions of a coadjoint orbit . Here is a 6-dimensional noncompact analog of , which is singled out by the constraints satisfied by . Hence the full algebra can be interpreted as a quantized algebra of functions on , dubbed fuzzy . Furthermore, is naturally a bundle over , which is defined by the satisfying (2.4a). Hence the space generated by the , can be viewed as projection of to along , as sketched in figure 1. This is the space-time of interest here, which is covariant under . For similar covariant quantum spaces see e.g. [28, 26, 29, 30, 31, 32].

### 2.1 Semi-classical structure of M3,1

We will mostly restrict ourselves to the semi-classical limit of the above space, working with commutative functions of and , but keeping the Poisson or symplectic structure encoded in . The constraints (2.4) etc. imply the following relations

 xμxμ =−R2−x24=−R2cosh2(η),R∼r2n (2.5a) tμtμ =r−2cosh2(η), (2.5b) tμxμ =0, (2.5c) tμθμα =−sinh(η)xα, (2.5d) xμθμα =−r2R2sinh(η)tα, (2.5e) ημνθμαθνβ =R2r2ηαβ−R2r4tαtβ−r2xαxβ (2.5f)

where . Here is a global time coordinate defined by

 x4=Rsinh(η) , (2.6)

which will be related to the scale parameter of the universe (2.23). Clearly the can be viewed as Cartesian coordinate functions. Similarly, the describe the fiber over as discussed above. On the other hand, the relation (2.3b) implies that the derivations

 −i[Tμ,.]∼{tμ,.} =sinh(η)∂μ (2.7)

act as momentum generators on , leading to the useful relation

 ∂μϕ=β{tμ,ϕ},β=1sinh(η) (2.8)

for . In particular, a -invariant matrix d’Alembertian can be defined as

 □:=[Tμ,[Tμ,.]] ∼ −{tμ,{tμ,.}} . (2.9)

It acts on any , and will play a central role throughout this paper. We also define a globally defined time-like vector field

 τ:=xμ∂μ. (2.10)

To get some insight into the , fix some reference point on , which using invariance can be chosen as

 ξ=(x0,0,0,0),x0=Rcosh(η) . (2.11)

Then (2.5e) provides a relation between the and the generators,

 tμ=−1Rr2x4xνθνμ \lx@stackrelξ= −1Rr21tanh(η)θ0μ , t0\lx@stackrelξ= 0 . (2.12)

Conversely, the self-duality relation on [33]

 ϵabcdeθabxc =nrθde (2.13)

relates the space-like and the time-like components of on , and an explicit expression of in terms of can be derived [1]

 θμν = c(xμtν−xνtμ)+bϵμναβxαtβ (2.14) withc =r2sinh(η)cosh2(η)andb=nr32Rcosh2(η). (2.15)

#### Hyperbolic coordinates.

Now consider the adapted hyperbolic coordinates

 ⎛⎜ ⎜ ⎜ ⎜⎝x0x1x2x3⎞⎟ ⎟ ⎟ ⎟⎠=Rcosh(η)⎛⎜ ⎜ ⎜ ⎜⎝cosh(χ)sinh(χ)sin(θ)cos(φ)sinh(χ)sin(θ)sin(φ)sinh(χ)cos(θ)⎞⎟ ⎟ ⎟ ⎟⎠ . (2.16)

We will see that measures the cosmic time, cf. (2.6), while the space-like distance from the origin on each time slice is measured by . Noting that

 xμxνR2cosh2(η)dxμdxν =R2sinh2(η)dη2 (2.17)

we obtain the induced (flat) metric of in these coordinates

 ds2g =ημνdxμdxν=R2(−sinh2(η)dη2+cosh2(η)dΣ2) (2.18)

where is the metric on the unit hyperboloid ,

 dΣ2 =dχ2+sinh2(χ)dΩ2,dΩ2=dθ2+sin2(θ)dφ2 . (2.19)

However, the effective metric in the present framework is a different one, which is also invariant but not flat.

### 2.2 Effective metric and d’Alembertian

In the matrix model framework considered below, the effective metric on the background under consideration is given by [1]

 Gμν =αγμν = sinh−1(η)ημνα=√1~ρ2|γμν|=sinh−3(η) γαβ =ημνθμαθνβ=sinh2(η)ηαβ . (2.20)

This is an -invariant FLRW metric with signature . Here is an irrelevant constant which adjusts the dimensions. There are several ways to obtain this metric. One is by rewriting the kinetic term in covariant form [1, 19]

 S[ϕ]=Tr[Tμ,ϕ][Tμ,ϕ]∼∫d⁴x√|G|Gμν∂μϕ∂νϕ , (2.21)

and another way is given below by showing (2.26). Using (2.18), this metric can be written as

 ds2G=Gμνdxμdxν =−R2sinh3(η)dη2+R2sinh(η)cosh2(η)dΣ2 =−dt2+a2(t)dΣ2 (2.22)

and we can read off the cosmic scale parameter

 a(t)2 =R2sinh(η)cosh2(η) \lx@stackrelt→∞∼ R2sinh3(η), (2.23) dt =Rsinh(η)32dη . (2.24)

Hence for late times, and the Hubble rate is decreasing as . This is related to the time-like vector field (2.10) via

 ∂∂η =tanh(η)τ,∂∂t=1R1√sinh(η)cosh(η)τ \lx@stackrelt→∞∼ 1Rβτ . (2.25)

As a consistency check, it is shown in appendix 7.6 that the covariant d’Alembertian of a scalar field is indeed given by up to a factor [19],

 −□ϕ =ηαβ{tα,{tβ,ϕ}}=ηαββ−1(∂αβ−1∂βϕ) =β−2(ηαβ∂α∂β−1x24xβ∂β)ϕ =β−3∇α∂αϕ=β−3□G (2.26)

where is the covariant derivative w.r.t. . In particular, we note the useful formula

 ∂α∂αϕ=β2(−□+1R2τ)ϕ . (2.27)

We would like to decompose into time derivatives and the space-like Laplacian on

 −Δ(3)ϕ =∇(3)μ∇(3)μϕ=∂μ(Pμν⊥∂νϕ) (2.28)

using the time-like and space-like projectors

 Pμντ :=1xαxαxμxν,Pμν⊥:=ημν−Pμντ . (2.29)

After some calculations using (2.8) and the formulas in section 7.2, one obtains

 □ϕ=(β−2Δ(3)+1R2τ+sinh2(η)R2cosh2(η)(2+τ)τ)ϕ (2.30)

for scalar fields . This can be checked e.g. for . On the other hand we can use the above hyperbolic coordinates (2.16), where

 Gμν =R2sinh(η)diag(−sinh2(η),cosh2(η),cosh2(η)sinh2(χ),cosh2(η)sinh2(χ)sin2(θ)) (2.31)

so that

 □G =−1√|Gμν|∂μ(√|Gμν|Gμν∂ν) (2.32)

This reduces indeed to (2.30) using and (2.25). The Laplacian (2.28) on the space-like reduces for rotationally invariant functions to

 Δ(3)ϕ(χ) =−1R2cosh2(η)1sinh2(χ)∂χ(sinh2(χ)∂χϕ) . (2.33)

### 2.3 Higher spin sectors and filtration

Due to the extra generators , the full algebra of functions decomposes into sectors which correspond to spin harmonics on the fiber:

 End(Hn)=C=C0⊕C1⊕…⊕CnwithS2|Cs=2s(s+1) (2.34)

Here can be viewed as a spin operator222Since local Lorentz invariance is not manifest, the usual notion of spin cannot be used, and is a substitute. on [33], which commutes with . In the semi-classical limit, the are modules over , and can be realized explicitly in terms of totally symmetric traceless space-like rank tensor fields on

 ϕ(s)=ϕμ1...μs(x)tμ1...tμs,ϕμ1...μsxμi=0 (2.35)

due to (2.5). The underlying structure provides an -invariant derivation

 Dϕ :={x4,ϕ} =r2R21x4tμ{tμ,ϕ}=−1x4xμ{xμ,ϕ} =r2Rtμ1…tμstμ∇(3)μϕμ1…μs(x) (2.36)

where is the covariant derivative along the space-like . Hence relates the different spin sectors in (2.34):

 D=D−+D+: Cs →Cs−1⊕Cs+1,D±ϕ(s)=[Dϕ(s)]s±1 (2.37)

where denotes the projection to defined through (2.34). For example, and . This allows to define a further refinement [1]

 C(s,k)\coloneqqK(s,k)/K(s,k−1),K(s,k)=ker(D−)k+1⊂Cs . (2.38)

Then

 D±:C(s,k) →C(s−1,k−1) . (2.39)

In particular, is the space of divergence-free traceless space-like rank tensor fields on , while encodes the traceless second derivatives of the scalar field . These will play an important role below. Finally, is extended to via [1]

 sinh(η)(τ+s)ϕ(s)=xμ{tμ,ϕ(s)} , (2.40)

which gives (7.13).

#### Averaging.

We will need some explicit formulas for the projection to :

 [tμtν]0 \eqqcoloncosh2(η)3r2Pμν⊥, (2.41)

in terms or the projector (2.29) on the time-slices . This can be viewed as an averaging over . Furthermore, we have [1]

 [tαθμν]0 =13(sinh(η)(ηανxμ−ηαμxν)+xβεβ4αμν), (2.42a) 0 =35([tμ1tμ2][tμ3tμ4]0+[tμ1tμ3][tμ2tμ4]0+[tμ1tμ4][tμ2tμ3]0). 1 =35([tαtβ]0tγ+tα[tβtγ]0+tβ[tαtγ]0) . (2.42b)

As an application, one can derive the following formula

 {xμ,{xμ,ϕ}}0 =r2R23(3−cosh2(η))β2(−□+1R2τ)ϕ+r23(2τ+7)τϕ (2.43)

for . This could be another natural d’Alembertian on which exhibits a transition from a Euclidean to a Minkowski era, as discussed in [25]. However here the effective d’Alembertian will be , which respects the spin sectors (2.34).

## 3 Matrix model and higher-spin gauge theory

Now we return to the noncommutative setting, and define a dynamical model for the fuzzy space-time under consideration. We consider a Yang-Mills matrix model with mass term,

 S[Y] =1g2Tr([Yμ,Yν][Yμ′,Yν′]ημμ′ηνν′+6R2YμYνημν) . (3.1)

This includes in particular the IKKT or IIB matrix model [2] with mass term, which is best suited for quantization because maximal supersymmetry protects from UV/IR mixing [34]. As observed in [1], is indeed a solution of this model333any other positive mass parameter in (3.1) would of course just result in a trivial rescaling. For negative mass parameter, would be a solution [25], but the fluctuations are more difficult to analyze., through

 Yμ=Tμ . (3.2)

Now consider tangential deformations of the above background solution, i.e.

 Yμ=Tμ+Aμ , (3.3)

where is an arbitrary (Hermitian) fluctuation. The Yang-Mills action (3.1) can be expanded as

 S[Y]=S[T]+S2[A]+O(A3) , (3.4)

and the quadratic fluctuations are governed by

 S2[A]=−2g2Tr(Aμ(D2−3R2)Aμ+G(A)2). (3.5)

Here

 D2A=(□−2I)A (3.6)

is the vector d’Alembertian, which involves the scalar matrix d’Alembertian on the background (2.9), (2.26) as discussed before, and the intertwiner

 I(A)μ=−i[[Yμ,Yν],Aν]=ir2R2[Θμν,Aν]\eqqcolon−1r2R2~I(A)μ (3.7)

using (2.3c). As usual in Yang-Mills theories, transforms under gauge transformations as

 δΛA=−i[Tμ+Aμ,Λ]∼{tμ,Λ}+{Aμ,Λ} (3.8)

for any , and the scalar ghost mode

 G(A)=−i[Tμ,Aμ]∼{tμ,Aμ}, (3.9)

should be removed to get a meaningful theory. This can be achieved by adding a gauge-fixing term to the action as well as the corresponding Faddeev-Popov (or BRST) ghost. Then the quadratic action becomes

 S2[A]+Sg.f+Sghost =−2g2Tr(Aμ(D2−3R2)Aμ+2¯¯c□c) (3.10)

where denotes the fermionic BRST ghost; see e.g. [35] for more details.

## 4 Fluctuation modes

All indices will be raised and lowered with in this section. We should expand the vector modes into higher spin modes according to (2.34), (2.35)

 Aμ =Aμ(x)+Aμα(x)tα+Aμαβ(x)tαtβ+… ∈ C0⊕C1⊕C2⊕ … (4.1)

However these are neither irreducible nor eigenmodes of . In [1], three series of spin eigenmodes were found of the form

 A(g)μ[ϕ(s)]={tμ,ϕ(s)}∈Cs,A(+)μ[ϕ(s)]={xμ,ϕ(s)}|Cs+1 ≡{xμ,ϕ(s)}+∈Cs+1,A(−)μ[ϕ(s)]={xμ,ϕ(s)}|Cs−1 ≡{xμ,ϕ(s)}−∈Cs−1 (4.2)

for any , which satisfy

 D2A(g)μ[ϕ] =A(g)μ[(□+3R2)ϕ], (4.3) D2A(+)μ[ϕ(s)] =A(+)μ[(□+2s+5R2)ϕ(s)], (4.4) D2A(−)μ[ϕ(s)] =A(−)μ[(□+−2s+3R2)ϕ(s)]. (4.5)

We provide in appendix 7.1 a simple new derivation for the last two relations. Hence diagonalizing is reduced to diagonalizing on , and we have the on-shell modes for

 A(+)[ϕ(s)] for   (□+2s+2R2)ϕ(s)=0, (4.6) A(−)[ϕ(s)] for  (□+−2sR2)ϕ(s)=0, (4.7) A(g)[ϕ(s)] for  □ϕ(s)=0 . (4.8)

Of course is a pure gauge mode and hence unphysical. Furthermore, the following gauge fixing identities444As a check, consider e.g. . It satisfies due to (2.36), and (7.45) gives and , consistent with (A.33) in [1]. were shown in [1]

 {tμ,A(+)μ[ϕ(s)]} =s+3RD+ϕ(s), (4.9) {tμ,A(−)μ[ϕ(s)]} =−s+2RD−ϕ(s). (4.10)

In particular for , is already gauge fixed555For some linear combinations of and must be taken to obtain a gauge-fixed physical solution. However, this is not our concern here.. This will lead to the physical spin 2 metric fluctuations. According to the discussion in section 2.3, they decompose into the modes and , which we will denote – in slight abuse of language – as helicity 2, 1 and 0 sectors of the would-be massive spin 2 modes, respectively. We will focus on the physical helicity 0 or scalar mode, with on-shell condition

 A(−)μ[D+Dϕ],(□+2R2)ϕ=0,ϕ∈C0 (4.11)

due to (7.3). However, one series of spin (off-shell) eigenmodes of is still missing, and was not known up to now. We will find the missing scalar mode in section 4.2, in terms of

 A(τ)μ[ϕ(s)]=xμϕ(s) . (4.12)

That ansatz was also considered in [1], where it was shown to satisfy

 D2A(τ)μ[ϕ(s)] =A(τ)μ[(□+7R2)ϕ(s)]+2ðμϕ(s) (4.13) {tμ,A(τ)μ[ϕ(s)]} =sinh(η)(4+s+τ)ϕ(s) . (4.14)

Here will be defined in (4.23). We will show in the following that provides the on-shell mode leading to the linearized Schwarzschild metric. Moreover, an ansatz based on will give solutions which are equivalent on-shell, but not off-shell.

### 4.1 Scalar A(−)[D+Dϕ] mode

We need the explicit form of . This is quite tedious to work out and delegated to the appendix 7.5, where we provide an exact expression in (7.32). This simplifies considerably using the on-shell condition (4.11), leading to

 A(−)μ[D+D+ϕ]=2r45(β(tμ+xμtα∂α)−13r2θμγ∂γ(τ+4+β2))(τ+2)ϕ+{tμ,Λ} (4.15)

with given in (7.34). This is a reasonable perturbation of the background , as long as remains bounded. Remarkably, (4.15) can be rewritten via as

 A(−)μ[D+D+ϕ] =25r2R(D(xμϕ′)−R3A(+)μ[(τ+4+β2)(τ+2)ϕ])+{tμ,Λ} =25r2RA(S)μ[ϕ′]+{tμ,Λ′} (4.16)

where is the new mode defined in (4.38), with

 ϕ′=β(τ+2)ϕ,Λ′=Λ+215r2RD(τ+4+β2)(τ+2)ϕ . (4.17)

To see this, the identities

 β(tμ+xμtα∂α)(τ