Weylian reduction theory for self-similar models

A geometrical method of formulating self-similar models in general relativity or in other gravitational theories is presented. The method consists of two techniques: (1) a Kaluza–Klein-like dimensional reduction technique for self-similar spacetimes, and (2) a systematic method of describing tensor ﬁelds on a self-similar spacetime in terms of ﬁelds on the reduced space. It is shown that the reduced space is a Weyl–Dirac conformal manifold and a self-similar model is formulated as a conformally covariant differential equation system. .


Introduction
Self-similar models in general relativity have been widely studied in recent decades. They are attractive objects of study, mainly because they often play important roles in developing understanding of the dynamical features of general relativity [1][2][3][4], but a more practical reason is that they are relatively easy to study. Specifically, since self-similarity is a continuous symmetry, it (together with other continuous symmetries imposed on the system, such as spherical symmetry) reduces the number of coordinates on which unknown variables depend and, thus, reduces the basic equations to differential equations on a lower dimensional space. We term this process the dimensional reduction of the equation system. For example, the basic equations for a spherically symmetric self-similar model can be dimensionally reduced to an ordinary differential equation system, which is much easier to analyze than spherically symmetric models without self-similarity. The dimensional reduction process described above is usually performed by selecting a coordinate system that is suitably adapted to self-similarity. However, there are a number of criteria for choosing a "preferable" coordinate system adapted to self-similarity, which are dependent on the purpose of the particular work, the physical and mathematical properties of the individual model, and personal preferences. Consequently, diverse formulations have been developed for each major self-similar model. This is not an ideal situation for researchers studying self-similar models, because formulations based on different coordinate systems are often related in a non-trivial way, and this causes considerable difficulty when results obtained from different formulations must be compared. See, e.g., Refs. [5,6].
This situation motivates us to take another approach, i.e., to develop a geometrical dimensional reduction technique that enables us to formulate self-similar models in a completely coordinatefree way. In this paper, we achieve this by introducing ideas from the Kaluza-Klein (KK) theory. In the simplest KK theory on n + 1 dimensions, one assumes homogeneity in the direction of the PTEP 2015, 013E01 M. Yoshikawa

Weyl geometry
Weyl geometry was introduced in 1918 by H. Weyl in his attempt to unify gravitation and electromagnetism; it is now considered to be the prototype of subsequent gauge theories [11,12,14,15]. In Weyl geometry, we study conformal manifolds with an additional structure; an affine connection subject to a compatibility condition, which will be given below. Recall that a conformal manifold is a pair (M c , C) of a differentiable manifold M c and a conformal class C on it, and a conformal class on M c is a collection of pseudo-Riemannian metrics on the manifold of the form C = [γ ] = {e 2ρ γ | ρ ∈ C ∞ (M c )}. We say that C = [γ ] is of Euclidean (resp. Lorentzian) signature if the metric signature of γ is (+, +, . . . , +) (resp. (−, +, . . . , +)). Although one does not focus on a specific metric in C in principle, it is often useful in actual calculations to select a "working metric" γ ∼ γ pq from C, called a gauge. Changing from one working metric to another is called a gauge transformation or a Weyl transformation, and can be written in the form for some ρ ∈ C ∞ (M c ).
A Weyl connection D p on (M c , C) is defined as a torsion-free affine connection on M c compatible with C in the following sense: for any γ pq ∼ γ ∈ C, there is W p ∈ T p (M c ) such that One can safely replace "any γ ∈ C" in the statement with "some γ ∈ C", because, if γ pq satisfies (2), then e 2ρ γ pq satisfies (2) with W p replaced by W p − (dρ) p . A Weyl connection on a conformal manifold is not unique due to the freedom of choosing W p . A Weyl structure on (M c , C) is defined as a map A p : C → T p (M c ) such that the image A p (γ ) transforms as an Abelian gauge field under the Weyl transformation (1), i.e., A p (e 2ρ γ ) = A p (γ ) − (dρ) p , (γ ∈ C, ρ ∈ C ∞ (M c )). ( A Weyl manifold is defined as a conformal manifold equipped with a Weyl structure, and will be denoted by (M c , C, A p ). Note that defining a Weyl structure on a conformal manifold is equivalent to giving a Weyl connection on the same space [16]. Indeed, given any Weyl connection D p on (M c , C), we have a unique Weyl structure, A p : γ → W p , specified by condition (2). Conversely, any Weyl structure A p on (M c , C) uniquely specifies a Weyl connection such that (2) holds for W p = A p (γ ) for any γ ∈ C, which is given by and where ∇ (γ ) p represents the Levi-Civita connection with respect to γ . Note that γ pq is the inverse of γ pq ; γ pr γ rq = γ qr γ r p = δ p q . There are several intrinsic curvature tensors on (M c , C, A p ). First, we have the distance curvature F pq , where We also have the curvature tensor R p qrs with respect to the Weyl connection, i.e., This satisfies the usual identities where the brackets on the indices indicate skew-symmetrization. In contrast to the Riemann curvature tensor, the skew-symmetricity with respect to the first two indices does not always hold. Indeed, it is not difficult to verify that the skew-symmetric part W p qrs ≡ γ pt γ u[t R u q]rs , called the direction curvature, can be written as [12,16] We contract W p qrs to obtain the Weylian counterpart of the Ricci tensor, so that Unlike the Ricci tensor, W pq is not symmetric, and while the contracted Bianchi identity is analogous to the Riemannian counterpart, with Let us introduce some additional concepts that are helpful in working with conformal or Weyl manifolds. A Weyl covariant tensor field on (M c , C) is any map T p 1 ··· p i q 1 ···q j : C → T where ω is a real number called the Weyl weight (or weight for short) of T p 1 ··· p i q 1 ···q j . If ω = 0, it is simply called a Weyl invariant tensor field. We denote by ω W short) the collection of Weyl covariant tensor fields on (M c , C) of Weyl weight ω taking values in T p 1 ··· p i q 1 ···q j (M c ). We can naturally identify T forms an R-vector space with respect to addition and scalar multiplication, defined by Other tensor algebra operations for Weyl covariant tensor fields, i.e., tensor product, contraction, and index permutation, are defined similarly. For example, the tensor product is defined by and this gives an R-bilinear map: The tensor product for the Weyl covariant scalar fields (i = j = k = l = 0) is simply called the multiplication. The contraction and index permutations provide the following R-linear maps: Index Permutations: ω W All these operations coincide with the usual forms when they act on Weyl invariant tensor fields. The Weyl covariant metric q pq : is defined by q pq (γ ) = γ pq , and is an element of 2 W pq (M c ). The contravariant counterpart q pq of q pq is given by q pq (γ ) = γ pq , and belongs to −2 W pq (M c ). The tensor product and contraction operations allow us to define the index raising and lowering operations using q pq and q pq , which change the Weyl weight by −2 and +2, respectively.
It is sometimes useful to consider the R-graded vector space for each valence type p 1 ··· p i q 1 ···q j . An element of these spaces will also be referred to as a Weyl covariant tensor field; it is homogeneous if it has a definite Weyl weight, and inhomogeneous otherwise. We note that not only homogeneous elements, but also inhomogeneous elements of W p 1 ··· p i q 1 ···q j (M c ), can be naturally regarded as maps from C to T p 1 ··· p i q 1 ···q j (M c ). All algebraic operations considered above can be extended straightforwardly to act on these spaces. Evidently, the space of Weyl covariant scalar fields forms a commutative R-graded ring with respect to the addition and multiplication, and every other space W Here, ω is the weight operator, i.e., the linear operator on W p 1 ··· p i q 1 ···q j (M c ) that multiplies each homogeneous summand by its Weyl weight. Clearly, this coincides with the Weyl connection operator D r when it acts on Weyl invariant tensor fields. It is also easy to see that D r preserves the Weyl covariance and the R-grading (Weyl weight), so it gives a graded R-linear map from each Weyl covariant class W . Moreover, it satisfies other standard properties of covariant derivatives: the Leibniz rule and commutativity with contractions and index permutations. Thus, D r is a natural extension of D r to act on Weyl covariant classes. Another important property is that it annihilates the Weyl covariant metric, such that This implies that the Weyl covariant derivative also commutes with the operations of index raising and lowering. Finally, we define WD manifolds. 2 A WD manifold is a quad (M c , C, A p , φ) where the former three elements constitute a Weyl manifold (M c , C, A p ), and the latter φ is a Weyl covariant scalar field on (M c , C, A p ) of Weyl weight +1. We assume that φ is strictly positive-valued, i.e., φ(γ ) is PTEP 2015, 013E01 M. Yoshikawa a strictly positive-valued function on M c for any (or, equivalently, some) γ ∈ C. With the additional scalar field, we have a preferred gauge γ E ∈ C, called the Einstein gauge, specified by the condition φ(γ E ) = 1. This implies that (M c , C, A p , φ) also has the aspect of a pseudo-Riemannian manifold (M c , γ E pq ) equipped with a one-form A E p ≡ A p (γ E ). Due to the redundant structure, (M c , C, A p , φ) has another "natural" affine connection other than the Weyl connection, namely, the Levi-Civita connection ∇ E p in the Einstein gauge. More generally, when other "natural" gauge conditions can be placed using φ or other available fields, we have more "natural" affine connections.

Dimensional reduction of simple homothetic spacetimes
Self-similarity of a spacetime (M , g ab ) is characterized by the existence of a homothety (or a proper homothety, to be precise) acting upon it [18], i.e., a smooth one-parameter transformation group h : where h * τ denotes the pullback by h τ . The generating vector field of a homothety is called a homothetic vector field. Note that ξ ∼ ξ a ∈ T a (M ) is a homothetic vector field on (M , g ab ) iff it is complete and satisfies Throughout this paper, the homothetic vector field generating h will be denoted by ξ h (or ξ a h , ξ b h , · · · ). The main object of study in this paper is a triple (M , g ab , h) of a differentiable manifold M , a smooth metric g ab on M of Lorentzian signature, and a proper homothety h acting on (M , g ab ). We say that the triple (M , g ab , h) is a simple homothetic spacetime if g ab ξ a h ξ b h has constant sign on M with no zero points. Hereafter, (M , g ab , h) is always a simple homothetic spacetime of dimension n + 1 (n ≥ 1), M h = M /h is the orbit space of h, and π is the natural projection from M onto M h . We employ lower-case Latin letters a, b, c, · · · as abstract indices associated with the (co)tangent spaces of M , which are raised and lowered with g ab and g ab as usual.

Basic structure of a simple homothetic spacetime
For a simple homothetic spacetime (M , g ab , h), let ψ be the function on M given by Then, ψ is a smooth strictly positive-valued function on M satisfying or, equivalently, This implies that, along each orbit of h, ψ is a monotonically increasing function onto (0, ∞), so each orbit intersects with the level set N E ≡ ψ −1 ({1}) exactly once. Thus, the projection π restricted to N E , is a bijection. Note that dψ is non-zero at any point of M because of (28) and the strict positivity of ψ, so the level set N E is an n-dimensional closed submanifold of M . Consequently, one can 6 introduce the structure of an n-dimensional differentiable manifold on M h by requiring the map (29) to be a diffeomorphism. 3 We define a reference surface of (M , g ab , h) as a smooth map S : M h → M such that Since the inverse of the diffeomorphism (29) gives a smooth embedding S E : M h → M satisfying (30), (M , g ab , h) has at least one reference surface. Actually, it has as many reference surfaces as elements of C ∞ (M h ), as will be shown later. Any reference surface S is an embedding, because condition (30) implies that S is a homeomorphism into M and that the pushforward Whenever there is no danger of confusion, we will also refer to the submanifold S (M h ) as a reference surface and denote it by S . Given a reference surface S of (M , g ab , h), we have a unique function η S ∈ C ∞ (M ) characterized by the conditions Actually, η S is explicitly given by We call η S the homothetic scale function with respect to S . Let Then, F S is a diffeomorphism, because it has the smooth inverse map . This shows that a simple homothetic spacetime (M , g ab , h) always has the direct product structure It is easy to verify that h ρ S is in R. Thus, each S ∈ R defines a map This map is bijective. Indeed, it is clearly injective, and any S 1 ∈ R can be written as It is also easy to see that the homothetic scale function with respect to h ρ S is related to that with respect to S by PTEP 2015, 013E01 M. Yoshikawa

WD manifolds as dimensionally reduced simple homothetic spacetimes
For any reference surface S of (M , g ab , h), the conformally rescaled metric e −2η S g ab is invariant under the actions of h * τ because, by (24) and (32), any change of e −2η S cancels any change of g ab . Thus, h serves as a 1D isometry group for the conformally rescaled spacetime (M , e −2η S g ab ) whose orbit space is an n-dimensional manifold M h , so the usual KK argument can be applied. Specifically, for any point x of the orbit manifold M h and for a local coordinate system (X 1 , . . . , X n ) on M h around x, the pulled-back coordinate functions X μ o ≡ π * X μ (μ = 1, . . . , n) together with η S form a local coordinate system on M around the orbit π −1 (x). (We call a local coordinate system (X 1 o , . . . , X n o , η S ) obtained in this way a homothetic local coordinate system, in accordance with Ref. [1].) In terms of the local coordinate system, ξ h can be written as and the conformally rescaled metric e −2η S g can be written in the KK form where X o abbreviates the collection (X 1 o , . . . , X n o ), and s represents the sign of g ab ξ a h ξ b h . As φ S , A S μ , and q S μν are n-variable functions independent of η S , they can be considered as local coordinate representations of a scalar field, a one-form, and a symmetric tensor field, respectively, on M h . Let us adopt upper-case Latin letters A, B, C, . . . as abstract indices associated with the (co)tangent spaces of M h , and denote these fields on M h by φ(S ), A A (S ), and q AB (S ), respectively, where "(S )" is attached in order to indicate that they are dependent on the choice of S ∈ R. Specifically, they are locally given by where X abbreviates (X 1 , . . . , X n ). Then, the metric can be written in a coordinate-free form as To elucidate the geometrical aspects of the fields (39)-(41), it is useful to introduce the following fields on M : is the orthogonal projection tensor to the orthogonal complement 8 where the sign of φ is chosen so that φ(S ) > 0. By applying S * to these and using (30) and (31), we arrive at the following geometrical expressions for φ(S ), A A (S ), and q AB (S ): To see the dependence of these fields on the choice of S ∈ R, let us apply (45)-(47) to two arbitrary reference surfaces, S and h ρ S . Using (36) and the injectivity of π * , we obtain These should be contrasted with the corresponding result in KK theory, in which A A follows the same transformation law as above, while φ and q AB are invariant [7]. Needless to say, this difference comes from the existence of the factor e 2η S in (42). One can easily see that the symmetric tensor field q(S ) ∼ q AB (S ) gives a smooth metric on M h of Lorentzian signature (when s = +) or Euclidean signature (when s = −). Thus we deduce from (53) and the bijectivity of (35) that is a conformal class on M h of Lorentzian or Euclidean signature, and can be identified with R by the map R S → q(S ) ∈ C h . With this identification, A A and φ are maps from C h to T A (M h ) and C ∞ (M h ), respectively. Furthermore, by (51) and (52), A A is a Weyl structure on the conformal manifold (M h , C h ), and φ is a Weyl covariant scalar field on (M h , C h ) of Weyl weight 1, which is obviously strictly positive-valued. Consequently, the quad as the reduced WD manifold. Choosing a gauge γ ∈ C h in the reduced WD manifold corresponds to choosing a reference surface S ∈ R, and a Weyl transformation γ → e 2ρ γ in the reduced WD manifold corresponds to the change of reference surface from S to h ρ S . In particular, the Einstein gauge γ E corresponds to the reference surface S E represented by the level set N E .
, then, as seen in the previous subsection, we know that the original M has the direct product structure M ≈ M h × R on which h acts as h τ ( p, η) = ( p, η + τ ). Hence, we can reconstruct g ab from the Weyl-Dirac structure by applying (42) in a suitable gauge γ = q(S ) ∈ C h , where s = + (resp. s = −) when the conformal structure C h is of Lorentzian (resp. Euclidean) signature. Actually, this correspondence between simple homothetic manifolds and WD manifolds is one-to-one in the following sense. Theorem 3.1. The dimensional reduction procedure provides one-to-one correspondence, up to equivalence, between (a) the class of (n + 1)-dimensional simple homothetic spacetimes, and (b) the class of n-dimensional WD manifolds with a conformal structure of Lorentzian or Euclidean signature. Here, two simple homothetic spacetimes (M , g ab , h) and (M , g ab , h ) are defined to be equivalent if and two WD manifolds The proof is straightforward and is therefore omitted.

Correspondence of connections
In KK theory, it is well known that a link exists between the Levi-Civita connections on the full spacetime and on the reduced spacetime, via the concept of "horizontal lift" [7]. In this section, we will follow the same lines to see the correspondence given by Theorem 3.1 from the perspective of connections.
At each point p of (M , g ab , h), the horizontal tangent space H p is defined as the orthogonal complement to ξ h in the tangent space T p M . As was previously mentioned, the orthogonal projection from . . , is defined as the vector field on M given by If we take a local coordinate system X = (X 1 , . . . , X n ) on M h and write V = μ V μ (X ) ∂ ∂ X μ , the horizontal lift ↑ V can be written in terms of the homothetic local coordinate system Thus, the horizontal lift of a smooth vector field is also smooth. The following properties can be readily verified: where f is any smooth function on M h . Our claim is stated in the following theorem. 10 Here, ∇ a is the Levi-Civita connection on (M , g ab ).
To prove this theorem, we first show the following lemma.
Proof It is evident thatD S U V given by (62) is smooth and the map ( (58) and (59) that If D a is torsion-free, the r.h.s. vanishes for any U, V , which means that¯ S = 0.
Proof of Theorem 3.2 By Lemma 3.3, for an arbitrarily chosen S , we have a unique torsion-free for any smooth vector fields U ∼ U A and V ∼ V B on M h . By the dimensional reduction procedure described in Sect. 3.2 and the definition of the horizontal lift, we deduce that where γ ∼ γ AB is the gauge corresponding to S . Using these together with (57), we compute Recall that a WD manifold has other "natural" affine connections than the Weyl connection. The above theorem asserts, nevertheless, that the Levi-Civita connection on the (n + 1)-dimensions singles out the Weyl connection D a . Hence, the Weyl connection is preferable among others when one wants to investigate Riemannian geometric structure on the (n + 1)-dimensions via the reduced geometry.

Self-similar tensor field calculus 4.1. Modules of self-similar tensor fields
In a self-similar model, a suitable self-similarity condition adapted to the homothety, h, of the background spacetime (M , g ab ) is imposed on every fundamental field of the theory. For a tensor field t a 1 ···a r b 1 ···b s , the self-similarity condition usually takes the form or, equivalently, Here, ω is a real number called the self-similarity weight (or weight for short) of t a 1 ··· b 1 ··· . The condition (66) implies that the tensor field t a 1 ··· b 1 ··· should transform as t a 1 ··· b 1 ··· → r ω t a 1 ··· b 1 ··· under the scale transformation g ab → r 2 g ab , so ω is defined such that the physical quantity represented by t a 1 ··· b 1 ··· has the dimension of (length) ω in suitable units. We call a tensor field t a 1 ··· b 1 ··· subject to (66) a self-similar tensor field on (M , g ab , h) of (self-similarity) weight ω. For example, the metric tensor field g ab is a self-similar tensor field of weight 2, and its contravariant counterpart g ab is a selfsimilar tensor field of weight −2. A self-similar tensor field of weight 0 is also said to be h-invariant.
In other words, S a 1 ···a r b 1 ···b s (M ) is R-graded by the self-similarity weight ω. An element of S a 1 ···a r b 1 ···b s (M ) is also referred to as a self-similar tensor field on (M , g ab , h); it is said to be homogeneous if it has a definite weight and inhomogeneous otherwise.
The tensor product, contraction, and index permutation operations provide the maps: Moreover, the following facts can be easily verified: We define a tensor field t ∼ t a 1 ···a r b 1 ···b s on M to be horizontal if any possible contraction of t with ξ h or ξ h vanishes, i.e., Clearly, this is a generalization of the horizontality for vector fields described in Sect. 3 where The tensor product, contraction, and index permutation operations preserve horizontality, so they also provide maps (69)

Dimensional reduction of self-similar tensor fields
We now go on to discuss dimensional reduction of self-similar tensor fields on (M , g ab , h) using a step-by-step process from the simplest case of self-similar scalar fields to more general cases. In what follows in this section, we represent by S γ the reference surface of (M , g ab , h) corresponding to γ ∈ C h in the sense described in Sect. 3.2, and by η γ the homothetic scale function with respect to S γ . Then, (36) yields the Weyl transformation law for η γ , where (a) Self-similar scalar fields.
We claim that f is an element of R W (M h ), the R-graded ring of Weyl covariant scalar fields on the reduced WD manifold. Indeed, if f is a homogeneous element of weight ω, then, for any γ ∈ C h , e −ωη γ f is constant along each orbit of h and, hence, is equal to π * S * γ (e −ωη γ f ). So we have This, together with (80), gives the Weyl transformation law f (e 2ρ γ ) = e ωρ f (γ ). Thus, f is a Weyl covariant scalar field on M h of weight ω, and the claim is proved for homogeneous f . The claim for inhomogeneous f follows from this and the linearity of S * γ . 13 It is clear from the above argument that the map preserves the R-grading. Moreover, this also preserves ring operations, since the pullback map S * γ commutes with addition and multiplication. Therefore, (83) is a graded ring homomorphism. Let us show that (83) where F j represents the homogeneous summand of F of weight ω j . The Weyl transformation law (80) guarantees that the r.h.s. is independent of the choice of γ ∈ C h . Obviously, ↑ F belongs to R S (M ), so we have a well defined map It is clear by construction that ↑ F = F holds for any F ∈ R W (M h ). From (82) and the linearity of (83), ↑ ( f ) = f also holds for any f ∈ R S (M ). Thus, (85) is the inverse of (83), and the claim is proved.
To summarize, we have proved:

Theorem 4.1. The lifting map (85) is a graded ring isomorphism.
Consequently, it is reasonable to define the reduced field of a self-similar scalar field f as the unique Weyl covariant scalar field F ∈ R W (M h ) such that f = ↑ F, which is equal to the f given by (81).
In addition to the ring operations, more scalar field operations can be incorporated in our framework. The following two operations are elementary and would be of importance in actual applications.

i) Substitution into a function ϕ → h(ϕ)
Here Similarly, we can also consider the substitution operation into a multi-variable function This might seem to be a special case of the substitution operation given above, but the difference is that it can act on any homogeneous scalar field. Indeed, this gives a map ω S(M ) → κω S(M ) on the (n + 1)-dimensional side, and ω W (M h ) → κω W (M h ) on the n-dimensional side for each ω ∈ R. When κ < 0, scalar fields having zero points should be avoided. The power operation also commutes with the lifting map  H p labeled by a 1 , . . . , a r and s copies of H  *   p labeled by b 1 , . . . , b s . Then, a self-similar tensor field t a 1 ···a r b 1 ···b s belongs to the horizontal self-similar class H a 1 ···a r b 1 ···b s (M ) iff its value at each p ∈ M is in [H p ] a 1 ···a r b 1 ···b s . Let p : H p → T π( p) M h be the linear isomorphism given in Sect. 3.3. The tensor product of r copies of the inverse map −1 p and s copies of the transpose map t p gives a linear isomorphism where [T π( p) M h ] A 1 ···A r B 1 ···B s represents the tensor product of r copies of T π( p) M h labeled by A 1 , . . . , A r and s copies of T * π( p) M h labeled by B 1 , . . . , B s . When r = s = 0, this is the identity map on R.
which is independent of the choice of γ ∈ C h due to (80); if T A 1 ···A r B 1 ···B s is inhomogeneous, then ↑ T a 1 ···a r b 1 ···b s is the sum of the horizontal lift of each homogeneous summand. This generalizes the definition of horizontal lift for vector fields given in Sect. 3.3, since (89) reduces to (55) when r = 1, s = 0, and ω = 0. Since the transpose of p coincides with the transpose of (π * ) p , for More generally, for where T j B 1 ···B s is the homogeneous summand of T B 1 ···B s of weight ω j . This reduces to (84) when s = 0, so the above definition of horizontal lift is also consistent with that for scalar fields. We proceed to establish basic properties of horizontal lifting for Weyl covariant tensor fields.

Theorem 4.2. The operation of horizontal lifting gives a graded R-linear isomorphism
and commutes with the tensor algebra operations as follows: where where Proof To prove that (92) is well defined, it suffices to verify that ↑ T a 1 ···a r b 1 ···b s belongs to . Using a partition of unity, T A 1 ···A r B 1 ···B s can be written as a locally finite sum where It follows from the definition of horizontal lift and (84), (55), and (90) that For each λ, we have ↑ V a λ,1 , . . . , ↑ V a λ,r ∈ 0 H a (M ); (π * W λ,1 ) a , . . . , (π * W λ,s ) a ∈ 0 H a (M ); and ↑ F λ ∈ R S (M ) from (78), (79), and Theorem 4.1, respectively. Thus, ↑ T a 1 ···a r b 1 ···b s is in H a 1 ···a r b 1 ···b s (M ) and the map (92) is well defined. It is clear from the above argument that (92) is R-linear and preserves R-grading. The properties (93)-(95) follow readily from the definition of horizontal lift and the exponential law.
To complete the proof, we show that (92) As the r.h.s. is smooth, t A 1 ···A r B 1 ···B s is actually a map into T A 1 ···A r B 1 ···B s (M h ). It also follows from (99) and the argument in (a) that the map gives an element of R W (M h ), which implies that t It is now straightforward to confirm that this is the inverse map of (92).
This theorem reduces to Theorem 4.1 when r = s = 0. If we identify the coefficient rings R S (M ) and R W (M h ) by Theorem 4.1, (92) is an isomorphism in the category of graded R W (M h )-modules. As in the scalar field case, we define the reduced field of a horizontal self-similar tensor field t a 1 ···a r b 1 ···b s as the unique Since p a b and p ab are horizontal self-similar tensor fields that can be completely characterized by the properties p a b v b = v a (∀v a ∈ 0 H a (M )) and p ac p cb = p a b , respectively, we deduce from 16 For a horizontal tensor field, one can use p ab and p ab , instead of g ab and g ab , to raise and lower the indices. So (100) and (103) imply the following corollary. (c) General self-similar tensor fields.
From (102) we have For t a 1 ···a r b 1 ···b s ∈ S a 1 ···a r b 1 ···b s (M ), we write t a 1 ···a r b 1 ···b s = δ a 1 c 1 · · · δ a r c r δ d 1 b 1 · · · δ d s b s t c 1 ···c r d 1 ···d s , substitute (104) into each δ · · , and then expand. This yields t a 1 ···a r b 1 ···b s expressed as a sum of tensor products of horizontal self-similar tensor fields and copies of ξ h and ξ h . By applying Theorem 4.2 to each of the horizontal components, we conclude that t a 1 ···a r b 1 ···b s can be uniquely written in the form We call this the horizontal decomposition of t a 1 ···a r b 1 ···b s . In this way, any t a 1 ···a r b 1 ···b s ∈ S a 1 ···a r b 1 ···b s (M ) can be reduced to a series of the Weyl covariant tensor fields (T (0) on the reduced WD manifold. We term this process the dimensional reduction of t a 1 ···a r b 1 ···b s , and the resultant Weyl covariant tensor fields T (0) . . are referred to as the reduced horizontal components. Note that, for a homogeneous self-similar tensor field of self-similarity weight ω, all the reduced horizontal components are also homogeneous and of Weyl weight ω, since ξ a h and ξ h a are h-invariant. Along with the reduced horizontal components, the algebraic expression (105) is also important for the following reasons: (1) it gives us information on how to reconstruct the original self-similar tensor field from the reduced horizontal components, and (2) it can be used, with the aid of Theorem 4.1 and Theorem 4.2, to determine how operations on self-similar tensor fields can be translated into operations on the reduced horizontal components. Thus, when working with reduced horizontal components, it is often convenient to have them embedded in the horizontal decomposition (105), rather than to deal with each of them separately.
Let us see some examples. The horizontal decompositions of the metric tensors g ab and g ab are PTEP 2015, 013E01 M. Yoshikawa while the volume form (dV ) a 1 a 2 ···a n+1 on (M , g ab ) is a self-similar tensor field of weight n + 1, which can be horizontally decomposed as Here, (dv) A 1 ···A n is the Weyl covariant tensor field of weight n such that, for each γ ∈ C h , (dv) A 1 ···A n (γ ) is the volume form on (M h , γ AB ). The sign on the r.h.s. of (108) is determined from the orientations of M and M h . We have another example of horizontal decomposition (104), and also other trivial examples (100)-(103). The dimensional reduction theory for general self-similar tensor fields given above is sufficient for application to self-similar models. However, as in the preceding cases, it is also possible to describe it as an isomorphism between a self-similar class S a 1  a , so that one can consider the the modules generated by tensor products of Weyl covariant tensor fields and copies of these variables. This approach would give us a formally elegant description of the principle of dimensional reduction, but we will not elaborate on it here, since such an abstraction does not seem to provide any advantages in actual use.

Calculating covariant derivatives
The horizontal decomposition (105) implies that, to calculate the Levi-Civita covariant derivative of a self-similar tensor field, we need the horizontal decompositions of the following two types of objects: (1) ∇ a ξ b h and ∇ a ξ h b ; (2) Covariant derivatives of horizontally lifted Weyl covariant tensor fields.
It is useful to introduce the following quantities: which are Weyl covariant tensor fields of weight 0 and 2, respectively. Note that (109) is the Weyl structure A A in the Einstein gauge, presented here in manifestly Weyl invariant form. It is not difficult to verify the following: where F ∈ R W (M h ) and U A , V A ∈ T A (M h ) are arbitrary. The last two identities immediately yield Hence, we obtain the following theorem. 18 For V A ∈ T A (M h ) and W A ∈ T A (M h ), we use Theorem 3.2 and the above formulae to obtain the following derivative formulae for ↑ V a and ↑ W a : where An arbitrary Weyl covariant tensor field T A 1 ·A r B 1 ·B s can be written in the form (96), so we apply (111), (119), and (120) to establish the following.

Theorem 4.5. For any T
Here, By applying Theorem 4.4 and Theorem 4.5 to the horizontal decomposition (105), we can obtain the horizontal decomposition formula for the covariant derivative of any self-similar tensor field. A direct consequence of the result is that the Levi-Civita covariant derivative operator gives a graded R-linear map ∇ c : S a 1 ···a r b 1 ···b s (M ) → S a 1 ···a r cb 1 ···b s (M ) for each valence type a 1 ···a r b 1 ···b s .

Curvature tensors
The derivative formulae obtained above can be used to show that the curvature tensors on (M , g ab , h) are self-similar tensor fields, and to obtain their horizontal decompositions, which are given below.
where the reduced horizontal components are given by Here, the parentheses on the indices indicate symmetrization.
Ricci tensor: The Ricci tensor Ric ab ≡ R c acb is also a self-similar tensor field of weight 0. The horizontal decomposition is given by where the reduced horizontal components are Here, Ricci scalar: The Ricci scalar R ≡ g ab Ric ab is a self-similar scalar field of weight −2, and can be written as

Dimensional reduction of non-self-similar tensor fields
To end this section, we discuss an approach to tensor fields that are not assumed to belong to selfsimilar classes. Briefly, the idea used here is a generalization of the KK expansion theory. We will show below that it can be incorporated into the framework constructed in this section, but with a slight extension. We state that a tensor field t a 1 ···a r b 1 ···b s on (M , g ab , h) has weak (self-similarity) weight ω if, for any h-invariant 1-forms w ( is Fourier transformable in a suitable sense (ordinary, L 2 , distributional, or so on, depending on the given circumstances) with respect to the parameter τ along each orbit of h. For such t a 1 It is easy to verify that each of the complex tensor fieldst (ω+ik) a 1 ···a r b 1 ···b s (k ∈ R) satisfies the selfsimilarity condition (66), but with the complex self-similarity weight ω + ik. Using this and the Fourier inversion formula, we obtain It is straightforward to generalize the entire argument in Sect. 4.2 to complex self-similar tensor fields of complex self-similarity weights so that it can be applied tot (λ) a 1 ···a r b 1 ···b s . Then, we obtain the "generalized" horizontal decomposition Here, the reduced horizontal componentsT (λ,0) . are now complex Weyl covariant tensor fields of complex Weyl weights. The algebraic and differential operations on t a 1 ···a r b 1 ···b s can be translated to those on the reduced horizontal components using the same technique described in Sects. 4.2 and 4.3 for self-similar tensor fields. We can also consider tensor fields that are "inhomogeneous" with respect to the weak weight, similar to the inhomogeneous self-similar tensor fields discussed in the preceding subsections.
To obtain a self-similar model from the theory, we assume a simple homothety h on the spacetime (M , g ab ), with t (1) , . . . , t (m) subject to the self-similarity condition (66) with suitable self-similarity weights. Then E (0) , E (1) , . . . , E (m) are also self-similar tensor fields on (M , g ab ), so (137) hold iff every homogeneous summand of every horizontal component vanishes for each E (0) , E (1) , . . . , E (m) . In this way, the equation system on (n + 1)-dimensions for the tensor fields g, t (1) , . . . , t (m) is reduced to equations on the n-dimensions for the reduced horizontal components of g, t (1) , . . . , t (m) , which are written in a Weyl covariant way.
If equation system (137) is scale covariant, it is reasonable to choose the self-similarity weights of t (1) , . . . , t (m) to be equal to ω 1 , . . . , ω m in condition (138). Then, the self-similar tensor fields E 0 , E (1) , . . . , E (m) are homogeneous of weight 0 , 1 , . . . , m , respectively. This implies that the reduced equation system consists of as many equations as the original equation system (137) and, hence, can be expected to have exactly the required number of equations to determine the unknown functions. In this sense, the scale covariance (138) is a sufficient condition for the theory to give a valid self-similar model. If, on the contrary, there is an inhomogeneous E ( j) , the reduced equation system is overdetermined unless there is degeneracy among the homogeneous summands.
Models with fields that are not subject to self-similarity can be formulated in the same way using the complexified framework introduced in Sect. 4.5. In principle, it is applicable to any model provided one can set a suitable homothetic background, but, practically, it would be most suitable for linear perturbative models.

Variational formulation of self-similar models
It is natural to wonder whether a self-similar model can be formulated within a variational principle, as in KK theory, by imposing self-similarity directly on the action integral. Unfortunately, the answer is negative, as will be explained below.
Then, the equations of motion for the action are also scale covariant, so it gives a valid self-similar model. Under the self-similarity conditions, L dV is a self-similar (n + 1)-form on (M , g ab , h) of weight , so we can use (108) and (46) to horizontally decompose it as where L ∈ −n−1 W (M h ), with ↑ L = L and η the homothetic scale function. Hence, we can write the action as a multiple integral over M ≈ M h × R. However, the self-similarity of the integrand