# A Geometrical Setting for Constructing Nonlinear Lorentz Transformations of the Albano-Dresden Type

Authors: Alexandre Lyra; Marcelo Carvalho
Journal: Physics Research International
 (2011)

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## Abstract

We investigate a nonlinear realization of the Lorentz transformation defined in 4-dimensional
spacetime, and the corresponding invariant interval. This new interval is not a quadratic form in
the standard Minkowski spacetime, then we search for a description where it becomes a quadratic
form in an infinite dimensional vector space. In this extended space, we show how to generate two
classes of nonlinear transformations, which form a group and interpret the non-linear transformation
as compatible relations between atlas of a Lyra manifold.


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## Body

## 1. Introduction

Spacetime structure is characterized at the classical level by the four-dimensional intervalI=ημνxμxν (or dI=ημνdxμdxν, when we consider spacetime modelled as a manifold instead of a vector space) and the corresponding Lorentz transformation xμ→x′μ=Λνμxν. Since no experiment can probe spacetime at very small distances, it is possible to consider that at these scales spacetime is described by a modified interval Iq, and the Lorentz invariance is broken in the sense of not leaving Iq invariant. This conjecture was considered by some authors in several different contexts (see [1–3] and references therein).In our work, we investigate a model proposed originally by Albano and Dresden [4–7] where the standard interval I and the Lorentz transformation are replaced by (1)Iq=ημνxμxν+f̂,(2)x′μ=hΛΛνμxν.
Here, f̂ is an arbitrary function depending on the ratio xμ/x0, and hΛ is a function whose form is determined by imposing the nonlinear transformation leaves Iq invariant.The purpose of our work is twofold. First, to provide a general mathematical construction for spacetime models that are based on (1) and (2) and that include, as a particular case, the original Albano-Dresden model. Second, to clarify and revisit some aspects of the Albano-Dresden model, specially those concerning the interpretation of the nonlinear transformation. Like Albano and Dresden, we present no extensive analysis on the possible quantum effects that could justify a change in the spacetime interval, for example, I→Iq, as proposed in (1).Our mathematical construction is based on the attempt to interpretIq as a quadratic form in a certain space. Due to the arbitrariness of f, it is evident that Iq is not a quadratic form in ordinary spacetime. The solution is to consider a family of vector spaces {Ex}x∈M and to introduce sections s∈∏x∈MEx, that is, s:U⊆M→∪x∈MEx, in terms of which we define an extended space M⊕∏x∈MEx where Iq is a quadratic form. Then, assuming the nonlinear transformation x′=hΛΛx and suitable transformations for the sections we deduce, upon imposing invariance of Iq, two forms for hΛ, one determining the original Albano-Dresden transformation and the other a new transformation not contemplated in their model.The interpretation of the nonlinear transformation needs a careful analysis. Albano and Dresden show in [6, 7] that by introducing new variables ξ, ξ′ as ξμ:=xμ1+(1/x2)f(xi/x0), ξ′μ:=x′μ1+(1/x′2)f(x′i/x′0) the nonlinear transformation becomes linear, that is, ξ→ξ′=Λξ. Then, one can associate the violation of Lorentz symmetry to the failure on defining the variables ξ, ξ′. This occurs for points satisfying 1+(1/x2)f(xi/x0)<0 that defines a distinguished region of spacetime. However, theoretical and observational aspects indicate there is no preferred region like this. Therefore, it is necessary to clarify the meaning of the nonlinear transformation in a different way. We achieve this by treating the nonlinear transformation as a passive transformation between charts of two compatible atlas that are employed by two different observers. The possibility to linearize the nonlinear transformation, that is, to reduce it to the standard Lorentz form, is seen as an active transformation between charts of two compatible atlas used by the same observer. With this view, we are able to avoid inferences on the existence of specific spacetime regions where Lorentz violation occurs. Instead of that, we show that the violation can be considered as an effect related to specific choices of atlas and, therefore, it concerns the observer in its relation to the spacetime rather than being an intrinsic property of spacetime.Our work is organized as follows. In Section2 we build an infinite dimensional vector space where we define Iq and derive the nonlinear transformations as transformations that leave Iq invariant. We also show that they obey a group structure. In Section 3 we assume spacetime is endowed with a Lyra structure and interpret the linear and the nonlinear transformations in terms of compatible relations between atlas used by two observers. Finally, in Section 4 we give a heuristic interpretation to the function f̂ showing how it arises as a consequence of the uncertainty principle in the measurement of position. We also indicate how the shift from linear to nonlinear Lorentz transformation may form a theoretical framework for the fluctuation of the spacetime metric.Henceforth, we refer the intervalIq as the Albano-Dresden form. The nonlinear transformation (2) is called Albano-Dresden transformation despite the form of hΛ we consider.


## 2. The Nonlinear Transformation and the Spacetime Structure

### 2.1. Defining the Extended Space and the Albano-Dresden FormIq

Let us assume spacetime is a pair(M,g) with M a 4-dimensional real vector space and g a bilinear form that is symmetric, nondegenerate, and of rank 2. Let {e0,e1,e2,e3} be an orthonormal basis for M in terms of which we write g(eμ,eν)=ημν, with ημν=(-,+,+,+) or (+,-,-,-). The associated quadratic form is written as I(x)=ημνxμxν where x≡x(BA):=xB-xA will always refer to the difference between the coordinates of two events xA=(xAμ) and xB=(xBμ). If we consider an event xA=(xAμ) and write I(x)=ημνxμxν we understand that x≡x(AO):=xA-xO with xO=(0,0,0,0).Now, letΩ≡{Ex}x∈M be a family of real vector spaces indexed by spacetime M. We assume Ex is 4-dimensional with basis {θμx}. The direct product space ∏x∈MEx consists of maps s:M→∪x∈MEx where s(x)∈Ex. Following the notation of [8], we write s=(sx)x∈M, with sx:=s(x)=sxμθμx∈Ex (sxμ∈R), and define the sum of elements s,s′∈∏x∈MEx as (3)s+s′≡(sx)x∈M+(sx′)x∈M:=(sx+sx′)x∈M
while the product of s∈∏x∈MEx by a scalar λ∈R is defined as (4)λs≡λ(sx)x∈M:=(λsx)x∈M.
In this way, ∏x∈MEx becomes a vector space. Let us consider the elements θμ[y]:=(δxyθμx)x∈M which allows us to write s=∑y∈Msyμθμ[y]. In fact, from the sum and product by scalar defined in ∏x∈MEx we have (5)∑y∈Msyμθμ[y]=∑y∈M(syμδxyθμx)x∈M=((∑y∈Msyμδxy)θμx)x∈M=(sxμθμx)x∈M=s.
We notice that s=∑y∈Msyμθμ[y] is understood as a formal expression that attains a rigorous meaning as a finite sum if we consider s having finite support and, in this case, the exterior product becomes the external direct sum ⊕x∈MEx. (In the general case, the meaning given to s=∑y∈Msyμθμ[y] must be understood as similar to the same formal meaning one gives to the expression representing a divergent series of real numbers, or to the path integral measure 𝒟ϕ.)Consider now the exterior direct sum ofM and ∏x∈MEx, that is, M⊕(∏x∈MEx)∋(x,s). Henceforth, we denote (x,s) as x⊕s. Let us introduce elements ẽμ:=(eμ,0), θ̃μy:=(0,θμ[y]), in terms of each we write x⊕s=xμẽμ+∑y∈Msyμθ̃μy. Any finite set of the form {θ̃μ1x1,…,θ̃μrxr}⊂∏x∈MEx is linearly independent (with xi≠xj), and we define a bilinear form in M⊕∏x∈MEx as follows:(6)Q(ẽμ,ẽν)=ημν,Q(ẽμ,θ̃μx)=0,Q(θ̃μx,θ̃νx)=ημνf2(x),Q(θ̃μx,θ̃νy)=0(ifx≠y),
with f an arbitrary nonnull real-valued function of spacetime. (The fact that in (6) we write f2 instead of f will become clear in Section 3.1). By linearity, we extend the bilinear form Q to the whole space M⊕∏x∈MEx as follows: (7)Q(x⊕s,y⊕q)=Q(x,y)+Q(s,q)=ημνxμyν+∑z,wδzwf2(z)ημνszμqwν=ημνxμyν+∑zf2(z)ημνsμ(z)qν(z).
Define now a map (8)Δx:∏x∈MEx⟶∏x∈MEx,s⟶Δxs:=sxμθμ[x]
and the subset (9)M⊕Δ∏x∈MEx:={(x,Δxs)∈M⊕∏x∈MEx:x∈M,s∈∏x∈MEx}.
Restricted to these elements, the bilinear form assumes the value (10)Q(x⊕Δxs,y⊕Δyq)=Q(x,y)+Q(Δxs,Δyq)=ημνxμyν+δxyf2(x)ημνsμ(x)qν(y).
In particular, we obtain Iq as (11)Iq(x,s):=Q(x⊕Δxs,x⊕Δxs)=ημνxμxν+f2(x)ημνsμ(x)sν(x)
which leads us to identify the function f̂ of the original Albano-Dresden transformation [4, 5] as (12)f̂(x)≡f2(x)ημνsμ(x)sν(x).
We identify M with M⊕Δ∏x∈MEx|s=0 by the correspondence M∋x↔(x,0)∈M⊕Δ∏x∈MEx|s=0. Then, I=x2 becomes a particular case of Iq=x2+s2f2 when we take s=0.


### 2.2. Deriving Transformations that LeaveIq Invariant

The analysis of the invariance of the intervalIq=x2+f̂ must take into account transformations of both x and s, that is, (x,s(x))→(x′,s′(x′)). In what follows, we analyze two types of transformations for the sections s that will determine two classes of nonlinear transformations.


#### 2.2.1. Type I Transformations

Let us assume transformations(x,s(x))→(x′,s′(x′)) having the form (13)x′μ:=hΛ(x)Λνμxν,s′μ(x′):=Λνμsν(x).
Imposing invariance of Iq on M⊕Δ∏x∈MEx(14)Q(x′⊕Δx′s′,x′⊕Δx′s′)=Q(x⊕Δxs,x⊕Δxs),∴x2hΛ2(x)+s2(x)f′2(x′)=x2+f2(x)s2(x),hΛ2(x)=1+s2(x)x2(f2(x)-f′2(x′)).
In order to be able to determine hΛ from this last equation, it is sufficient to impose that f(x) depends on the ratio xμ/x0, that is, f≡f(xμ/x0), which gives (15)hΛ(x,s(x))=1+s2(x)x2(f2(x̂)-f′2(Λx̂)),
with type I transformations assuming the form (16)x′μ=Λνμxν1+s2(x)x2(f2(x̂)-f′2(Λx̂)),s′μ(x′)=Λνμsν(x),
where we have used the notation (17)x̂μ:=xμx0,(Λx̂)μ:=ΛνμxνΛα0xα.
In order to exhibit the group structure of type I transformations we consider two consecutive transformations (x,s(x))→L(Λ(1))(x′,s′(x′))→L(Λ(2))(x′,s′′(x′′)): (18)x′=hΛ(1)(x)Λ(1)x=1+s2(x)x2(f2(x̂)-f′2(Λ(1)x̂))Λ(1)x,s′(x′)=Λ(1)s(x),x′′=hΛ(2)(x′)Λ(2)x′=1+s′2(x′)x′2(f′2(x̂′)-f′′2(Λ(2)x̂′))Λ(2)x′,s′′(x′′)=Λ(2)s′(x′).
Using that (19)1+s′2(x′)x′2(f′2(x̂′)-f′′2(Λ(2)x̂′))=1hΛ(1)(x)1+s2(x)x2(f2(x̂)-f′′2(Λ(2)Λ(1)x̂))
we obtain x′′=hΛ(2)Λ(1)(x)Λ(2)Λ(1)x, while for the sections it is immediate that s′′(x′′)=Λ(2)Λ(1)s(x). Thus, we conclude that L(Λ(2))∘L(Λ(1))=L(Λ(2)Λ(1)). In particular, the inverse of transformation (16) is written as (20)x=hΛ-1(x′)Λ-1x′=1+s′2(x′)x′2(f′2(x̂′)-f2(Λ-1x̂′))s(x)=Λ-1s′(x′)
and one verifies that hΛ-1(x′)=1/hΛ(x). We have then showed the group structure of type I transformations.


#### 2.2.2. Type II Transformations

These transformations are defined by(21)x′μ:=hΛ(x)Λνμxν,s′μ(x′):=hΛ(x)Λνμsν(x).
Now, imposing invariance of Iq we obtain hΛ2(x)(x2+s2(x)f′2(x′))=x2+s2(x)f2(x), and in order to determine hΛ we again consider sections whose argument is  x̂μ:=xμ/x0, with (Λx̂)μ:=Λνμxν/Λα0xα. We obtain (22)hΛ(x,s(x))=x2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂)
and type II transformations assume the form (23)x′μ:=Λνμxνx2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂),s′μ(x′):=Λνμsν(x)x2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂).
From these expressions it is straightforward to check the group structure of type II transformations.Remarks 2.2.2.
(1) For both type I and II transformations we notice thatf depends on the ratio xμ/x0, which has no dimension. From the expression of Iq=x2+s2f2(x̂) it is then necessary to introduce a unit of length in s which implies the existence of a fundamental length in the model [4, 5].
(2)s(x)∈Ex, s′(x′)∈Ex′ are vectors belonging to different vector spaces, and each one is characterized numerically by the components sμ(x),s′μ(x′)∈R. Therefore, transformation sμ(x)→s′μ(x′) given in (16) and (23) is seen as a transformation in the space of real maps defined on M.
(3) For a fixedf and its transformed function f′(x′), the nonlinear transformations of type I and II are defined only in a restricted spacetime region [6, 7]. Therefore, the only way to have nonlinear transformations defined through all spacetime is to employ several functions fa:Ua⊂M→R such that in the domains Va⊆Ua where hΛ is defined we have ∪Va=M.
(4) If we considers∈∏x∈MEx satisfying s2(x)=1, type I transformations become
(24)x′μ=Λνμxν1+1x2(f2(x̂)-f′2(Λx̂))
that reduce to the standard form proposed originally by Albano and Dresden [4–7]. This choice, however, is not compatible with type II transformations as it fixes hΛ=1.


## 2.1. Defining the Extended Space and the Albano-Dresden FormIq

Let us assume spacetime is a pair(M,g) with M a 4-dimensional real vector space and g a bilinear form that is symmetric, nondegenerate, and of rank 2. Let {e0,e1,e2,e3} be an orthonormal basis for M in terms of which we write g(eμ,eν)=ημν, with ημν=(-,+,+,+) or (+,-,-,-). The associated quadratic form is written as I(x)=ημνxμxν where x≡x(BA):=xB-xA will always refer to the difference between the coordinates of two events xA=(xAμ) and xB=(xBμ). If we consider an event xA=(xAμ) and write I(x)=ημνxμxν we understand that x≡x(AO):=xA-xO with xO=(0,0,0,0).Now, letΩ≡{Ex}x∈M be a family of real vector spaces indexed by spacetime M. We assume Ex is 4-dimensional with basis {θμx}. The direct product space ∏x∈MEx consists of maps s:M→∪x∈MEx where s(x)∈Ex. Following the notation of [8], we write s=(sx)x∈M, with sx:=s(x)=sxμθμx∈Ex (sxμ∈R), and define the sum of elements s,s′∈∏x∈MEx as (3)s+s′≡(sx)x∈M+(sx′)x∈M:=(sx+sx′)x∈M
while the product of s∈∏x∈MEx by a scalar λ∈R is defined as (4)λs≡λ(sx)x∈M:=(λsx)x∈M.
In this way, ∏x∈MEx becomes a vector space. Let us consider the elements θμ[y]:=(δxyθμx)x∈M which allows us to write s=∑y∈Msyμθμ[y]. In fact, from the sum and product by scalar defined in ∏x∈MEx we have (5)∑y∈Msyμθμ[y]=∑y∈M(syμδxyθμx)x∈M=((∑y∈Msyμδxy)θμx)x∈M=(sxμθμx)x∈M=s.
We notice that s=∑y∈Msyμθμ[y] is understood as a formal expression that attains a rigorous meaning as a finite sum if we consider s having finite support and, in this case, the exterior product becomes the external direct sum ⊕x∈MEx. (In the general case, the meaning given to s=∑y∈Msyμθμ[y] must be understood as similar to the same formal meaning one gives to the expression representing a divergent series of real numbers, or to the path integral measure 𝒟ϕ.)Consider now the exterior direct sum ofM and ∏x∈MEx, that is, M⊕(∏x∈MEx)∋(x,s). Henceforth, we denote (x,s) as x⊕s. Let us introduce elements ẽμ:=(eμ,0), θ̃μy:=(0,θμ[y]), in terms of each we write x⊕s=xμẽμ+∑y∈Msyμθ̃μy. Any finite set of the form {θ̃μ1x1,…,θ̃μrxr}⊂∏x∈MEx is linearly independent (with xi≠xj), and we define a bilinear form in M⊕∏x∈MEx as follows:(6)Q(ẽμ,ẽν)=ημν,Q(ẽμ,θ̃μx)=0,Q(θ̃μx,θ̃νx)=ημνf2(x),Q(θ̃μx,θ̃νy)=0(ifx≠y),
with f an arbitrary nonnull real-valued function of spacetime. (The fact that in (6) we write f2 instead of f will become clear in Section 3.1). By linearity, we extend the bilinear form Q to the whole space M⊕∏x∈MEx as follows: (7)Q(x⊕s,y⊕q)=Q(x,y)+Q(s,q)=ημνxμyν+∑z,wδzwf2(z)ημνszμqwν=ημνxμyν+∑zf2(z)ημνsμ(z)qν(z).
Define now a map (8)Δx:∏x∈MEx⟶∏x∈MEx,s⟶Δxs:=sxμθμ[x]
and the subset (9)M⊕Δ∏x∈MEx:={(x,Δxs)∈M⊕∏x∈MEx:x∈M,s∈∏x∈MEx}.
Restricted to these elements, the bilinear form assumes the value (10)Q(x⊕Δxs,y⊕Δyq)=Q(x,y)+Q(Δxs,Δyq)=ημνxμyν+δxyf2(x)ημνsμ(x)qν(y).
In particular, we obtain Iq as (11)Iq(x,s):=Q(x⊕Δxs,x⊕Δxs)=ημνxμxν+f2(x)ημνsμ(x)sν(x)
which leads us to identify the function f̂ of the original Albano-Dresden transformation [4, 5] as (12)f̂(x)≡f2(x)ημνsμ(x)sν(x).
We identify M with M⊕Δ∏x∈MEx|s=0 by the correspondence M∋x↔(x,0)∈M⊕Δ∏x∈MEx|s=0. Then, I=x2 becomes a particular case of Iq=x2+s2f2 when we take s=0.


## 2.2. Deriving Transformations that LeaveIq Invariant

The analysis of the invariance of the intervalIq=x2+f̂ must take into account transformations of both x and s, that is, (x,s(x))→(x′,s′(x′)). In what follows, we analyze two types of transformations for the sections s that will determine two classes of nonlinear transformations.


### 2.2.1. Type I Transformations

Let us assume transformations(x,s(x))→(x′,s′(x′)) having the form (13)x′μ:=hΛ(x)Λνμxν,s′μ(x′):=Λνμsν(x).
Imposing invariance of Iq on M⊕Δ∏x∈MEx(14)Q(x′⊕Δx′s′,x′⊕Δx′s′)=Q(x⊕Δxs,x⊕Δxs),∴x2hΛ2(x)+s2(x)f′2(x′)=x2+f2(x)s2(x),hΛ2(x)=1+s2(x)x2(f2(x)-f′2(x′)).
In order to be able to determine hΛ from this last equation, it is sufficient to impose that f(x) depends on the ratio xμ/x0, that is, f≡f(xμ/x0), which gives (15)hΛ(x,s(x))=1+s2(x)x2(f2(x̂)-f′2(Λx̂)),
with type I transformations assuming the form (16)x′μ=Λνμxν1+s2(x)x2(f2(x̂)-f′2(Λx̂)),s′μ(x′)=Λνμsν(x),
where we have used the notation (17)x̂μ:=xμx0,(Λx̂)μ:=ΛνμxνΛα0xα.
In order to exhibit the group structure of type I transformations we consider two consecutive transformations (x,s(x))→L(Λ(1))(x′,s′(x′))→L(Λ(2))(x′,s′′(x′′)): (18)x′=hΛ(1)(x)Λ(1)x=1+s2(x)x2(f2(x̂)-f′2(Λ(1)x̂))Λ(1)x,s′(x′)=Λ(1)s(x),x′′=hΛ(2)(x′)Λ(2)x′=1+s′2(x′)x′2(f′2(x̂′)-f′′2(Λ(2)x̂′))Λ(2)x′,s′′(x′′)=Λ(2)s′(x′).
Using that (19)1+s′2(x′)x′2(f′2(x̂′)-f′′2(Λ(2)x̂′))=1hΛ(1)(x)1+s2(x)x2(f2(x̂)-f′′2(Λ(2)Λ(1)x̂))
we obtain x′′=hΛ(2)Λ(1)(x)Λ(2)Λ(1)x, while for the sections it is immediate that s′′(x′′)=Λ(2)Λ(1)s(x). Thus, we conclude that L(Λ(2))∘L(Λ(1))=L(Λ(2)Λ(1)). In particular, the inverse of transformation (16) is written as (20)x=hΛ-1(x′)Λ-1x′=1+s′2(x′)x′2(f′2(x̂′)-f2(Λ-1x̂′))s(x)=Λ-1s′(x′)
and one verifies that hΛ-1(x′)=1/hΛ(x). We have then showed the group structure of type I transformations.


### 2.2.2. Type II Transformations

These transformations are defined by(21)x′μ:=hΛ(x)Λνμxν,s′μ(x′):=hΛ(x)Λνμsν(x).
Now, imposing invariance of Iq we obtain hΛ2(x)(x2+s2(x)f′2(x′))=x2+s2(x)f2(x), and in order to determine hΛ we again consider sections whose argument is  x̂μ:=xμ/x0, with (Λx̂)μ:=Λνμxν/Λα0xα. We obtain (22)hΛ(x,s(x))=x2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂)
and type II transformations assume the form (23)x′μ:=Λνμxνx2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂),s′μ(x′):=Λνμsν(x)x2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂).
From these expressions it is straightforward to check the group structure of type II transformations.Remarks 2.2.2.
(1) For both type I and II transformations we notice thatf depends on the ratio xμ/x0, which has no dimension. From the expression of Iq=x2+s2f2(x̂) it is then necessary to introduce a unit of length in s which implies the existence of a fundamental length in the model [4, 5].
(2)s(x)∈Ex, s′(x′)∈Ex′ are vectors belonging to different vector spaces, and each one is characterized numerically by the components sμ(x),s′μ(x′)∈R. Therefore, transformation sμ(x)→s′μ(x′) given in (16) and (23) is seen as a transformation in the space of real maps defined on M.
(3) For a fixedf and its transformed function f′(x′), the nonlinear transformations of type I and II are defined only in a restricted spacetime region [6, 7]. Therefore, the only way to have nonlinear transformations defined through all spacetime is to employ several functions fa:Ua⊂M→R such that in the domains Va⊆Ua where hΛ is defined we have ∪Va=M.
(4) If we considers∈∏x∈MEx satisfying s2(x)=1, type I transformations become
(24)x′μ=Λνμxν1+1x2(f2(x̂)-f′2(Λx̂))
that reduce to the standard form proposed originally by Albano and Dresden [4–7]. This choice, however, is not compatible with type II transformations as it fixes hΛ=1.


## 2.2.1. Type I Transformations

Let us assume transformations(x,s(x))→(x′,s′(x′)) having the form (13)x′μ:=hΛ(x)Λνμxν,s′μ(x′):=Λνμsν(x).
Imposing invariance of Iq on M⊕Δ∏x∈MEx(14)Q(x′⊕Δx′s′,x′⊕Δx′s′)=Q(x⊕Δxs,x⊕Δxs),∴x2hΛ2(x)+s2(x)f′2(x′)=x2+f2(x)s2(x),hΛ2(x)=1+s2(x)x2(f2(x)-f′2(x′)).
In order to be able to determine hΛ from this last equation, it is sufficient to impose that f(x) depends on the ratio xμ/x0, that is, f≡f(xμ/x0), which gives (15)hΛ(x,s(x))=1+s2(x)x2(f2(x̂)-f′2(Λx̂)),
with type I transformations assuming the form (16)x′μ=Λνμxν1+s2(x)x2(f2(x̂)-f′2(Λx̂)),s′μ(x′)=Λνμsν(x),
where we have used the notation (17)x̂μ:=xμx0,(Λx̂)μ:=ΛνμxνΛα0xα.
In order to exhibit the group structure of type I transformations we consider two consecutive transformations (x,s(x))→L(Λ(1))(x′,s′(x′))→L(Λ(2))(x′,s′′(x′′)): (18)x′=hΛ(1)(x)Λ(1)x=1+s2(x)x2(f2(x̂)-f′2(Λ(1)x̂))Λ(1)x,s′(x′)=Λ(1)s(x),x′′=hΛ(2)(x′)Λ(2)x′=1+s′2(x′)x′2(f′2(x̂′)-f′′2(Λ(2)x̂′))Λ(2)x′,s′′(x′′)=Λ(2)s′(x′).
Using that (19)1+s′2(x′)x′2(f′2(x̂′)-f′′2(Λ(2)x̂′))=1hΛ(1)(x)1+s2(x)x2(f2(x̂)-f′′2(Λ(2)Λ(1)x̂))
we obtain x′′=hΛ(2)Λ(1)(x)Λ(2)Λ(1)x, while for the sections it is immediate that s′′(x′′)=Λ(2)Λ(1)s(x). Thus, we conclude that L(Λ(2))∘L(Λ(1))=L(Λ(2)Λ(1)). In particular, the inverse of transformation (16) is written as (20)x=hΛ-1(x′)Λ-1x′=1+s′2(x′)x′2(f′2(x̂′)-f2(Λ-1x̂′))s(x)=Λ-1s′(x′)
and one verifies that hΛ-1(x′)=1/hΛ(x). We have then showed the group structure of type I transformations.


## 2.2.2. Type II Transformations

These transformations are defined by(21)x′μ:=hΛ(x)Λνμxν,s′μ(x′):=hΛ(x)Λνμsν(x).
Now, imposing invariance of Iq we obtain hΛ2(x)(x2+s2(x)f′2(x′))=x2+s2(x)f2(x), and in order to determine hΛ we again consider sections whose argument is  x̂μ:=xμ/x0, with (Λx̂)μ:=Λνμxν/Λα0xα. We obtain (22)hΛ(x,s(x))=x2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂)
and type II transformations assume the form (23)x′μ:=Λνμxνx2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂),s′μ(x′):=Λνμsν(x)x2+s2(x)f2(x̂)x2+s2(x)f′2(Λx̂).
From these expressions it is straightforward to check the group structure of type II transformations.Remarks 2.2.2.
(1) For both type I and II transformations we notice thatf depends on the ratio xμ/x0, which has no dimension. From the expression of Iq=x2+s2f2(x̂) it is then necessary to introduce a unit of length in s which implies the existence of a fundamental length in the model [4, 5].
(2)s(x)∈Ex, s′(x′)∈Ex′ are vectors belonging to different vector spaces, and each one is characterized numerically by the components sμ(x),s′μ(x′)∈R. Therefore, transformation sμ(x)→s′μ(x′) given in (16) and (23) is seen as a transformation in the space of real maps defined on M.
(3) For a fixedf and its transformed function f′(x′), the nonlinear transformations of type I and II are defined only in a restricted spacetime region [6, 7]. Therefore, the only way to have nonlinear transformations defined through all spacetime is to employ several functions fa:Ua⊂M→R such that in the domains Va⊆Ua where hΛ is defined we have ∪Va=M.
(4) If we considers∈∏x∈MEx satisfying s2(x)=1, type I transformations become
(24)x′μ=Λνμxν1+1x2(f2(x̂)-f′2(Λx̂))
that reduce to the standard form proposed originally by Albano and Dresden [4–7]. This choice, however, is not compatible with type II transformations as it fixes hΛ=1.


## 3. Interpreting the Nonlinear Transformation

Here, we identityEx=TxM, and spacetime becomes a particular case of M⊕ΔΓ(TM) when we take s=0, that is, M~M⊕ΔΓ(TM)|s=0. In order to interpret the nonlinear transformations we assume spacetime is a Lyra manifold, and each observer describes spacetime by choosing one appropriate Lyra structure, perhaps among many possible ones.


### 3.1. Redefining Spacetime in Terms of a Lyra Manifold

Following [9], we define a Lyra manifold as a set M together with 𝒜:={(Ua,φa,fa;xa)}a∈Ω such that (i)
{(Ua,φa;xa)}a∈Ω is a C∞ atlas in M,(ii)
fa:Ua→R-{0} is a C∞ map (fa is called the gauge function on Ua),(iii)
𝒜 is maximal.Since a maximal atlas is an equivalence class of compatible atlas, it is enough to define a Lyra manifold in terms of an atlas satisfying conditions (i) and (ii). In this sense we say the atlas𝒜 defines a Lyra structure for M.Givenp∈M, let (Ua,φa,fa;xa)∈𝒜 with p∈Ua. Locally, we have φa(p)=xa≡(xaμ)∈R4 and we define a basis for TpM as {∂/∂xaμ}. Assume now there is defined a metric on M according to (25)g(∂∂xaμ,∂∂xaν)=agμν(xa).
Introduce a new basis for TpM as {θaμp}, with (26)aθμp:=fa(xa)∂∂xaμ,
where fa(xa) denotes fa∘φa-1. Then (27)g(aθμp,aθνp)=fa2(xa)agμν(xa).
In this expression, gμν are the components of the metric relative to the original basis {∂/∂xaμ}, therefore it transforms as (28)a1gμν(xa1)=∂xa2α∂xa1μ∂xa2β∂xa1νa2gαβ(xa2).
From (26) and (28) we obtain that for all p∈Ua1∩Ua2(29)g(a1θμp,a1θνp)=fa12(xa1)a1gμν(xa1)⟺g(a2θμp,a2θνp)=fa22(xa2)a2gμν(xa2)
which shows that (28) is compatible throughout the charts. One should also observe that relation Q(θ̃μx,θ̃νx)=ημνf2(x) is a consequence of (27) when we take Ep≡TpM and consider gμν=ημν.


### 3.2. The Transformations

Definition 1 (nonlinear Lorentz transformation).
Consider extended spacetimeM⊕ΔΓ(TM), with M endowed with Lyra structures 𝒜𝒪:={(Ua,φa,fa;xa)}, 𝒜𝒪′:={(Ub′,φb′,fb′;xb′)} relative to observers 𝒪 and 𝒪′. Let the spacetime metric (introduced in (27)) be gμν′=gμν=ημν. For every p∈Ua∩Ub′ one defines nonlinear Lorentz transformation as the compatible relations between the charts that have the form
(30)xb′μ=hΛ(xa)Λνμxaν=1+sa2(xa)xa2(fa2(x̂a)-fb′2(Λx̂a))Λνμxaν,(Type I),xb′μ=hΛ(xa)Λνμxaν=xa2+sa2(xa)fa2(x̂a)xa2+sa2(xa)fb′2(Λx̂a)Λνμxaν,(Type II).Definition 2 (Lorentz transformation).
Consider spacetimeM⊕ΔΓ(TM)|s=0, with M endowed with single Lyra structures 𝒜𝒪:={(M,φ,f;x)}, 𝒜𝒪′:={(M,φ′,f′;x′)} relative to 𝒪 and 𝒪′. Let the spacetime metric be gμν′=gμν=ημν. For every p∈M one defines Lorentz transformations as the compatible relations between the charts 𝒜𝒪, 𝒜𝒪′ that have the form x′=Λx.In the case of Lorentz transformation the gauge functionsf and f′ play no distinguished role, and we could have assumed single differentiable atlas 𝒜𝒪:={(M,φ;x)}, 𝒜𝒪′:={(M,φ′;x′)} instead of Lyra atlas.


### 3.3. Reducing the Nonlinear Lorentz Transformation to Its Linear Lorentz Form

At this point, the linear and nonlinear Lorentz transformations were set as passive transformations arising as compatible relations between atlas used by two Lorentz-related observers corresponding to single or multichart atlas. A natural question we must analyze is ifan observer obtains equivalent descriptions for spacetime whether he uses single or multichart atlas. Thus, we search for compatible atlas 𝒜2={(Ua,φa,fa;xa)}≃𝒜1={(M,φ,f;x)} for 𝒪, and 𝒜2′={(Ub′,φb′,fb′;xb′)}≃𝒜1′={(M,φ′,f′;x′)} for 𝒪′ such that we have (31)(31)
Relative to (Ua,φa,fa;xa) the invariant interval (11) has the form Ia=xa2+sa2(xa)fa2(x̂a), while relative to (M,φ;x) its form is I=x2. Let us assume that xaμ=Θ(x)xμ, and that under this transformation Ia reduces to I. Then, we obtain Ia=Θ2x2+sa2(xa)fa2(x̂a)≡x2, which imposes Θ2(x)=1-(sa2(xa)/x2)fa2(x̂a). In order to obtain Θ(x) it is sufficient to assume that the section depends on x̂a, since x̂a=x̂ whenever xa is defined. Then sa(x̂a)=sa(x̂) and we obtain Θ(x)=1-(sa2(x̂)/x2)fa2(x̂), that is, (32)xμ⟶xaμ:=xμ1-sa2(x̂)x2fa2(x̂),x′μ⟶xb′μ:=x′μ1-sb′2(x′̂)x′2fb′2(x′̂)
whose inverse reads as (33)xaμ⟶xμ:=xaμ1+sa2(x̂a)xa2fa2(x̂a),xb′μ⟶x′μ:=xb′μ1+sb′2(x̂b′)xb′2fb′2(x̂b′).
In order to check if these equations reduce transformations (30) to a linear form we write (34)xb′μ=x′μ1+(sb′2(x̂b′)/xb′2)fb′2(x̂b′),xaμ=xμ1+(sa2(x̂a)/xa2)fa2(x̂a).
Then, for both type I and II transformation we have (35)xb′μ=hΛ(xa)Λνμxaν⟹x′μ=ΛνμxνhΛ(xa)xb′2+sb′2(x̂b′)fb′2(x̂b′)xa2+sa2(x̂a)fa2(x̂a)⋅xa2xb′2=Λνμxνxb′2+sb′2(x̂b′)fb′2(x̂b′)xa2+sa2(x̂a)fa2(x̂a),
and since the nonlinear transformation is built in such way that xb′2+sb′2fb′2=xa′2+sa2fa2, we end up with x′μ=Λνμxν.In the context of Lyra manifolds, local coordinates are introduced on everyUa⊂M provided we have a chart (Ua,φa,fa;xa). However, depending on the choice of the sections and on the form of the gauge function, the transformation x=xa1+(sa2(x̂a)/xa2)fa2(x̂a)≡φ∘φa-1(xa) may not be defined in all φa(Ua)⊂R4. Besides that, the transformation x=φ∘φa-1(x) is C∞ only if (1+(sa2(x̂a)/xa2)fa2(x̂a))≠0. Let us assume that this happens in a domain Va⊆Ua. If ∪aVa=M, we restrict the original chart (Ua,φa,fa;xa) to (Va,φa|Va,fa|Va;xa) obtaining another atlas 𝒜3:={(Va,φa|Va,fa|Va;xa)} that is now compatible with 𝒜1={(M,φ,f;x)}. Assuming the other observer also defines an atlas 𝒜3′ compatible with 𝒜1′, we obtain a complete equivalence between linear and nonlinear Lorentz transformations, the latter reducing to the Lorentz transformation by the use of (33). If ∪aVa≠M, some nonlinear transformation may fail to reduce to the Lorentz transformation. This would sign a breaking of Lorentz invariance in the corresponding domain. In our interpretation, this failure is associated with the noncompatibility between differentiable structures, which is quite possible to occur in R4. Its origin is related to how observer describes spacetime, that is, it refers to the choice of the differentiable structure, therefore it is not due to an intrinsic property of spacetime. Albano and Dresden suggest the failure on linearizing the transformation is possibly due to quantum mechanical effects. Here, combining their view with our interpretation, we should analyze the extent to which quantum mechanical effects can originate changes in the differentiable structure of spacetime. (For another perpective on this see [10].)


## 3.1. Redefining Spacetime in Terms of a Lyra Manifold

Following [9], we define a Lyra manifold as a set M together with 𝒜:={(Ua,φa,fa;xa)}a∈Ω such that (i)
{(Ua,φa;xa)}a∈Ω is a C∞ atlas in M,(ii)
fa:Ua→R-{0} is a C∞ map (fa is called the gauge function on Ua),(iii)
𝒜 is maximal.Since a maximal atlas is an equivalence class of compatible atlas, it is enough to define a Lyra manifold in terms of an atlas satisfying conditions (i) and (ii). In this sense we say the atlas𝒜 defines a Lyra structure for M.Givenp∈M, let (Ua,φa,fa;xa)∈𝒜 with p∈Ua. Locally, we have φa(p)=xa≡(xaμ)∈R4 and we define a basis for TpM as {∂/∂xaμ}. Assume now there is defined a metric on M according to (25)g(∂∂xaμ,∂∂xaν)=agμν(xa).
Introduce a new basis for TpM as {θaμp}, with (26)aθμp:=fa(xa)∂∂xaμ,
where fa(xa) denotes fa∘φa-1. Then (27)g(aθμp,aθνp)=fa2(xa)agμν(xa).
In this expression, gμν are the components of the metric relative to the original basis {∂/∂xaμ}, therefore it transforms as (28)a1gμν(xa1)=∂xa2α∂xa1μ∂xa2β∂xa1νa2gαβ(xa2).
From (26) and (28) we obtain that for all p∈Ua1∩Ua2(29)g(a1θμp,a1θνp)=fa12(xa1)a1gμν(xa1)⟺g(a2θμp,a2θνp)=fa22(xa2)a2gμν(xa2)
which shows that (28) is compatible throughout the charts. One should also observe that relation Q(θ̃μx,θ̃νx)=ημνf2(x) is a consequence of (27) when we take Ep≡TpM and consider gμν=ημν.


## 3.2. The Transformations

Definition 1 (nonlinear Lorentz transformation).
Consider extended spacetimeM⊕ΔΓ(TM), with M endowed with Lyra structures 𝒜𝒪:={(Ua,φa,fa;xa)}, 𝒜𝒪′:={(Ub′,φb′,fb′;xb′)} relative to observers 𝒪 and 𝒪′. Let the spacetime metric (introduced in (27)) be gμν′=gμν=ημν. For every p∈Ua∩Ub′ one defines nonlinear Lorentz transformation as the compatible relations between the charts that have the form
(30)xb′μ=hΛ(xa)Λνμxaν=1+sa2(xa)xa2(fa2(x̂a)-fb′2(Λx̂a))Λνμxaν,(Type I),xb′μ=hΛ(xa)Λνμxaν=xa2+sa2(xa)fa2(x̂a)xa2+sa2(xa)fb′2(Λx̂a)Λνμxaν,(Type II).Definition 2 (Lorentz transformation).
Consider spacetimeM⊕ΔΓ(TM)|s=0, with M endowed with single Lyra structures 𝒜𝒪:={(M,φ,f;x)}, 𝒜𝒪′:={(M,φ′,f′;x′)} relative to 𝒪 and 𝒪′. Let the spacetime metric be gμν′=gμν=ημν. For every p∈M one defines Lorentz transformations as the compatible relations between the charts 𝒜𝒪, 𝒜𝒪′ that have the form x′=Λx.In the case of Lorentz transformation the gauge functionsf and f′ play no distinguished role, and we could have assumed single differentiable atlas 𝒜𝒪:={(M,φ;x)}, 𝒜𝒪′:={(M,φ′;x′)} instead of Lyra atlas.


## 3.3. Reducing the Nonlinear Lorentz Transformation to Its Linear Lorentz Form

At this point, the linear and nonlinear Lorentz transformations were set as passive transformations arising as compatible relations between atlas used by two Lorentz-related observers corresponding to single or multichart atlas. A natural question we must analyze is ifan observer obtains equivalent descriptions for spacetime whether he uses single or multichart atlas. Thus, we search for compatible atlas 𝒜2={(Ua,φa,fa;xa)}≃𝒜1={(M,φ,f;x)} for 𝒪, and 𝒜2′={(Ub′,φb′,fb′;xb′)}≃𝒜1′={(M,φ′,f′;x′)} for 𝒪′ such that we have (31)(31)
Relative to (Ua,φa,fa;xa) the invariant interval (11) has the form Ia=xa2+sa2(xa)fa2(x̂a), while relative to (M,φ;x) its form is I=x2. Let us assume that xaμ=Θ(x)xμ, and that under this transformation Ia reduces to I. Then, we obtain Ia=Θ2x2+sa2(xa)fa2(x̂a)≡x2, which imposes Θ2(x)=1-(sa2(xa)/x2)fa2(x̂a). In order to obtain Θ(x) it is sufficient to assume that the section depends on x̂a, since x̂a=x̂ whenever xa is defined. Then sa(x̂a)=sa(x̂) and we obtain Θ(x)=1-(sa2(x̂)/x2)fa2(x̂), that is, (32)xμ⟶xaμ:=xμ1-sa2(x̂)x2fa2(x̂),x′μ⟶xb′μ:=x′μ1-sb′2(x′̂)x′2fb′2(x′̂)
whose inverse reads as (33)xaμ⟶xμ:=xaμ1+sa2(x̂a)xa2fa2(x̂a),xb′μ⟶x′μ:=xb′μ1+sb′2(x̂b′)xb′2fb′2(x̂b′).
In order to check if these equations reduce transformations (30) to a linear form we write (34)xb′μ=x′μ1+(sb′2(x̂b′)/xb′2)fb′2(x̂b′),xaμ=xμ1+(sa2(x̂a)/xa2)fa2(x̂a).
Then, for both type I and II transformation we have (35)xb′μ=hΛ(xa)Λνμxaν⟹x′μ=ΛνμxνhΛ(xa)xb′2+sb′2(x̂b′)fb′2(x̂b′)xa2+sa2(x̂a)fa2(x̂a)⋅xa2xb′2=Λνμxνxb′2+sb′2(x̂b′)fb′2(x̂b′)xa2+sa2(x̂a)fa2(x̂a),
and since the nonlinear transformation is built in such way that xb′2+sb′2fb′2=xa′2+sa2fa2, we end up with x′μ=Λνμxν.In the context of Lyra manifolds, local coordinates are introduced on everyUa⊂M provided we have a chart (Ua,φa,fa;xa). However, depending on the choice of the sections and on the form of the gauge function, the transformation x=xa1+(sa2(x̂a)/xa2)fa2(x̂a)≡φ∘φa-1(xa) may not be defined in all φa(Ua)⊂R4. Besides that, the transformation x=φ∘φa-1(x) is C∞ only if (1+(sa2(x̂a)/xa2)fa2(x̂a))≠0. Let us assume that this happens in a domain Va⊆Ua. If ∪aVa=M, we restrict the original chart (Ua,φa,fa;xa) to (Va,φa|Va,fa|Va;xa) obtaining another atlas 𝒜3:={(Va,φa|Va,fa|Va;xa)} that is now compatible with 𝒜1={(M,φ,f;x)}. Assuming the other observer also defines an atlas 𝒜3′ compatible with 𝒜1′, we obtain a complete equivalence between linear and nonlinear Lorentz transformations, the latter reducing to the Lorentz transformation by the use of (33). If ∪aVa≠M, some nonlinear transformation may fail to reduce to the Lorentz transformation. This would sign a breaking of Lorentz invariance in the corresponding domain. In our interpretation, this failure is associated with the noncompatibility between differentiable structures, which is quite possible to occur in R4. Its origin is related to how observer describes spacetime, that is, it refers to the choice of the differentiable structure, therefore it is not due to an intrinsic property of spacetime. Albano and Dresden suggest the failure on linearizing the transformation is possibly due to quantum mechanical effects. Here, combining their view with our interpretation, we should analyze the extent to which quantum mechanical effects can originate changes in the differentiable structure of spacetime. (For another perpective on this see [10].)


## 4. Some Physical Insights

### 4.1. A Heuristic Interpretation forf̂

The functionf̂ was interpreted by Dresden as a contribution to the classical interval I=c2t2-x⃗2 due to quantum mechanics effects [4, 5]. As an example, they considered a particular situation of two atoms in interaction and showed how the uncertainty in the position of the atoms determines a change in the classical interval. However, it is not clear in their analysis how the function f̂ relates, for example, to the wave function of the two atoms. Here, we will show how this arises.Let us consider two eventsA=(tA,x⃗A), B=(tB,x⃗B) as measured by an observer 𝒪. Event A corresponds to the emission of a quanta by an atom at instant tA and at position x⃗A. Event B corresponds to the absorption of the quanta by another atom at instant tB and at position x⃗B. The 4-interval associated to the events (36)I(tB-tA,x⃗B-x⃗A)=c2(tA-tB)2-|x⃗A-x⃗B|2
is an invariant for any observer that is Lorentz related to 𝒪. However, this expression assumes observer 𝒪 can determine precisely the positions of both atoms. From a quantum mechanical perspective there is an uncertainty on the position of each atom, therefore we must replace the term |x⃗A-x⃗B|2 by an average value. Heuristically, let us conceive each atom as oscillating around fixed positions, for example, x⃗A and x⃗B, with ξ⃗A, ξ⃗B representing small deviations from these values. Then, the effective position of each atom may be written as x⃗A+ξ⃗A and x⃗B+ξ⃗B. Let us assume the wave function of the system at the instant t  (tA≤t≤tB) is described by (37)Ψ(x⃗A,x⃗B;ξ⃗A,ξ⃗B,t)≡ΨA(x⃗B-x⃗A;ξ⃗A,t)×ΨB(x⃗B-x⃗A;ξ⃗B,t)
and normalized such that (38)∫dξ⃗A|ΨA(x⃗B-x⃗A;ξ⃗A,t)|2=1,∫dξ⃗B|ΨB(x⃗B-x⃗A;ξ⃗B,t)|2=1
with all values of ξAj, ξBj being equally probable, that is, (39)∫dξ⃗AξAi|ΨA|2=∫dξ⃗BξBi|ΨB|2=0.
Since x⃗A, x⃗B correspond to fixed positions their role on the wave function is just of being parameters, the functional dependence of Ψ being determined by ξ⃗A, ξ⃗B. This justify the normalization of Ψ as integrals on ξ⃗A, ξ⃗B. Besides that, admitting space is homogeneous and isotropic the wave function should depend on |x⃗B-x⃗A|. Now, in the expression for the interval (36), we replace |x⃗A-x⃗B|2 by the average value of |(x⃗B+ξ⃗B)-(x⃗A+ξ⃗A)|2, that is,(40)∫dξ⃗Adξ⃗Bδij(xBi+ξBi-(xAi+ξAi))×(xBj+ξBj-(xAj+ξAj))|Ψ(x⃗A,x⃗B;ξ⃗A,ξ⃗B,t)|2
which gives (41)Iq(tB-tA,x⃗B-x⃗A)=I(tB-tA,x⃗B-x⃗A)-δij[∫dξ⃗AξAiξAj|ΨA|2+∫dξ⃗BξBiξBj|ΨB|2]
with (42)f̂≡-δij[∫dξ⃗AξAiξAj|ΨA|2+∫dξ⃗BξBiξBj|ΨB|2].
In this expression, the function f̂ is related to the wave function of the system, therefore, it is seen as a quantum effect that is not present in the classical theory. Some criticism could arise if someone inadvertently claims a dependence of f̂ on the instant t used to calculate (40). This would result in an interval Iq(tB-tA,x⃗B-x⃗A;t) whose dependence on t would be meaningless. In fact, this is not the case as there is no distinguished cause affecting the atoms while the radiation propagates, therefore, it is reasonable to assume that the average (40) does not depend on t or, equivalently, that the wave function is stationary.


### 4.2. A Physical Interpretation fors and the Effective Metric for Spacetime

In our model, the sections are the main elements in the transition from ordinary spacetimeM to extended spacetime M⊕ΔΓ(TM), and the most significant aspect of this transition is the change of the interval from I=x2 to Iq=x2+s2f2. In the previous section, we showed in a heuristic way how f̂ could be thought as related to the deviations δx⃗=ξ⃗ in the measurement of the position x⃗ of a physical system. Now, as an intrinsic property of spacetime, we axiomatize that spacetime is described by coordinates and sections s(x), which accounts for the fluctuation of the spacetime coordinates, for example, x→x+s when, by any means, we wish to measure it. With this interpretation, ordinary spacetime M is obtained when s=0 that physically means the interval is I=x2 and there is no fluctuation on the spacetime coordinates, in agreement with the classical regime.We can analyze other aspects brought by the sections, for example, the possibility of interpreting the sections as the source of an effective metric for spacetime. We consider two cases.


#### 4.2.1. Case 1

Let us consider local sectionss:Ua⊆M→TM, ∪aUa=M, that are analytic in their domain. Then, its Taylor series around a certain x0∈Ua is written as (43)sμ(x)=sμ(x0)+s,λμ(x0)(xλ-x0λ)+12s,λτi(x0)(xλ-x0λ)(xτ-x0τ)+⋯
with s,λμ(xa)=(∂sμ(x)/∂xλ)|xa. From (11) we obtain after collecting the terms (44)Iq(x⊕Δxs)=[ηλτ+ημνf2(sμ(x0)s,λτν(x0)+s,λμ(x0)s,τν(x0))]×xλxτ+h(x,x0,s(x0)…),
where h(x,x0,s(x0)…) includes all non-quadratic contributions on x. The quadratic term in x plays the role of an effective metric for spacetime M that we identify as (45)gλτ(x)=ηλτ+ημνf2(sμ(x0)s,λτν(x0)+s,λμ(x0)s,τν(x0)).
Another possibility to arrive at this metric is by considering the embedding problem as described in [11, 12], that is, we search for a smooth manifold (N,ĝ) with dimN≥dimM and such that (M,g) is embedded into N. Lets us consider N as an 8-dimensional manifold with local coordinates denoted by (YA)=(Yα,Yα+4),α=1, 2, 3, 4. Then, the problem is to find YA=YA(xα) such that (46)gμν=ĝAB∂YA∂xμ∂YB∂xν.
Defining (YA(x)):=(xα,sα(x)) with s=(sα):U→R4 an embedding, and (47)(ĝAB)=(ηαβ00ĝαβ)
with (48)ĝλτ(x):=12ημνf2∂xα∂sλ∂xβ∂sτ(∂α∂β(sμsν)|x=x0)
it is immediate to check that (46) determines (45). Then (M,g) is seen as isometrically embedded into (N,ĝ). By an appropriate choice of sections and of f̂, it is possible to obtain an effective metric that has a nonvanishing curvature tensor.


#### 4.2.2. Case 2

Consider now a curve joining two spacetime pointsxA=x(τA), xB=x(τB). Let us identify the classical interval as (49)I=∫τAτBημνdxμdτdxνdτdτ.
The quantum fluctuation Iq represents a deviation of I that we identify in two equivalent ways. First, from the interpretation we gave to the sections we consider that at the quantum level there are fluctuations on the coordinates that change the original path joining the points xA,xB to a different path given now by (50)x̃μ:=xμ+sμ(x)
with the flat metric ημν being kept fixed, and in terms of which we write (51)Iq=∫τAτBημνdx̃μdτdx̃νdτdτ=∫τAτB(ημν+ημλs,νλ+ηνλs,μλ+ηλτs,μλs,ντ)dxμdτdxνdτdτ.
Second, we assume that Iq originates from a change of the flat metric ημν to another metric gμν with the path still given by xμ(τ), that is, (52)Iq=∫τAτBgμν(x)dxμdτdxνdτdτ.
These two forms being equivalent representations for Iq, they must give the same value, therefore we must have (53)gμν(x)=ημν+ημλs,νλ+ηνλs,μλ+ηλτs,μλs,ντ.
Here, when we identify (51) and (52) as two equivalent forms for Iq we are implicitly assuming that the net effect of the fluctuations of the spacetime coordinates x→x+s is to change the flat metric to an effective metric gμν.


## 4.1. A Heuristic Interpretation forf̂

The functionf̂ was interpreted by Dresden as a contribution to the classical interval I=c2t2-x⃗2 due to quantum mechanics effects [4, 5]. As an example, they considered a particular situation of two atoms in interaction and showed how the uncertainty in the position of the atoms determines a change in the classical interval. However, it is not clear in their analysis how the function f̂ relates, for example, to the wave function of the two atoms. Here, we will show how this arises.Let us consider two eventsA=(tA,x⃗A), B=(tB,x⃗B) as measured by an observer 𝒪. Event A corresponds to the emission of a quanta by an atom at instant tA and at position x⃗A. Event B corresponds to the absorption of the quanta by another atom at instant tB and at position x⃗B. The 4-interval associated to the events (36)I(tB-tA,x⃗B-x⃗A)=c2(tA-tB)2-|x⃗A-x⃗B|2
is an invariant for any observer that is Lorentz related to 𝒪. However, this expression assumes observer 𝒪 can determine precisely the positions of both atoms. From a quantum mechanical perspective there is an uncertainty on the position of each atom, therefore we must replace the term |x⃗A-x⃗B|2 by an average value. Heuristically, let us conceive each atom as oscillating around fixed positions, for example, x⃗A and x⃗B, with ξ⃗A, ξ⃗B representing small deviations from these values. Then, the effective position of each atom may be written as x⃗A+ξ⃗A and x⃗B+ξ⃗B. Let us assume the wave function of the system at the instant t  (tA≤t≤tB) is described by (37)Ψ(x⃗A,x⃗B;ξ⃗A,ξ⃗B,t)≡ΨA(x⃗B-x⃗A;ξ⃗A,t)×ΨB(x⃗B-x⃗A;ξ⃗B,t)
and normalized such that (38)∫dξ⃗A|ΨA(x⃗B-x⃗A;ξ⃗A,t)|2=1,∫dξ⃗B|ΨB(x⃗B-x⃗A;ξ⃗B,t)|2=1
with all values of ξAj, ξBj being equally probable, that is, (39)∫dξ⃗AξAi|ΨA|2=∫dξ⃗BξBi|ΨB|2=0.
Since x⃗A, x⃗B correspond to fixed positions their role on the wave function is just of being parameters, the functional dependence of Ψ being determined by ξ⃗A, ξ⃗B. This justify the normalization of Ψ as integrals on ξ⃗A, ξ⃗B. Besides that, admitting space is homogeneous and isotropic the wave function should depend on |x⃗B-x⃗A|. Now, in the expression for the interval (36), we replace |x⃗A-x⃗B|2 by the average value of |(x⃗B+ξ⃗B)-(x⃗A+ξ⃗A)|2, that is,(40)∫dξ⃗Adξ⃗Bδij(xBi+ξBi-(xAi+ξAi))×(xBj+ξBj-(xAj+ξAj))|Ψ(x⃗A,x⃗B;ξ⃗A,ξ⃗B,t)|2
which gives (41)Iq(tB-tA,x⃗B-x⃗A)=I(tB-tA,x⃗B-x⃗A)-δij[∫dξ⃗AξAiξAj|ΨA|2+∫dξ⃗BξBiξBj|ΨB|2]
with (42)f̂≡-δij[∫dξ⃗AξAiξAj|ΨA|2+∫dξ⃗BξBiξBj|ΨB|2].
In this expression, the function f̂ is related to the wave function of the system, therefore, it is seen as a quantum effect that is not present in the classical theory. Some criticism could arise if someone inadvertently claims a dependence of f̂ on the instant t used to calculate (40). This would result in an interval Iq(tB-tA,x⃗B-x⃗A;t) whose dependence on t would be meaningless. In fact, this is not the case as there is no distinguished cause affecting the atoms while the radiation propagates, therefore, it is reasonable to assume that the average (40) does not depend on t or, equivalently, that the wave function is stationary.


## 4.2. A Physical Interpretation fors and the Effective Metric for Spacetime

In our model, the sections are the main elements in the transition from ordinary spacetimeM to extended spacetime M⊕ΔΓ(TM), and the most significant aspect of this transition is the change of the interval from I=x2 to Iq=x2+s2f2. In the previous section, we showed in a heuristic way how f̂ could be thought as related to the deviations δx⃗=ξ⃗ in the measurement of the position x⃗ of a physical system. Now, as an intrinsic property of spacetime, we axiomatize that spacetime is described by coordinates and sections s(x), which accounts for the fluctuation of the spacetime coordinates, for example, x→x+s when, by any means, we wish to measure it. With this interpretation, ordinary spacetime M is obtained when s=0 that physically means the interval is I=x2 and there is no fluctuation on the spacetime coordinates, in agreement with the classical regime.We can analyze other aspects brought by the sections, for example, the possibility of interpreting the sections as the source of an effective metric for spacetime. We consider two cases.


### 4.2.1. Case 1

Let us consider local sectionss:Ua⊆M→TM, ∪aUa=M, that are analytic in their domain. Then, its Taylor series around a certain x0∈Ua is written as (43)sμ(x)=sμ(x0)+s,λμ(x0)(xλ-x0λ)+12s,λτi(x0)(xλ-x0λ)(xτ-x0τ)+⋯
with s,λμ(xa)=(∂sμ(x)/∂xλ)|xa. From (11) we obtain after collecting the terms (44)Iq(x⊕Δxs)=[ηλτ+ημνf2(sμ(x0)s,λτν(x0)+s,λμ(x0)s,τν(x0))]×xλxτ+h(x,x0,s(x0)…),
where h(x,x0,s(x0)…) includes all non-quadratic contributions on x. The quadratic term in x plays the role of an effective metric for spacetime M that we identify as (45)gλτ(x)=ηλτ+ημνf2(sμ(x0)s,λτν(x0)+s,λμ(x0)s,τν(x0)).
Another possibility to arrive at this metric is by considering the embedding problem as described in [11, 12], that is, we search for a smooth manifold (N,ĝ) with dimN≥dimM and such that (M,g) is embedded into N. Lets us consider N as an 8-dimensional manifold with local coordinates denoted by (YA)=(Yα,Yα+4),α=1, 2, 3, 4. Then, the problem is to find YA=YA(xα) such that (46)gμν=ĝAB∂YA∂xμ∂YB∂xν.
Defining (YA(x)):=(xα,sα(x)) with s=(sα):U→R4 an embedding, and (47)(ĝAB)=(ηαβ00ĝαβ)
with (48)ĝλτ(x):=12ημνf2∂xα∂sλ∂xβ∂sτ(∂α∂β(sμsν)|x=x0)
it is immediate to check that (46) determines (45). Then (M,g) is seen as isometrically embedded into (N,ĝ). By an appropriate choice of sections and of f̂, it is possible to obtain an effective metric that has a nonvanishing curvature tensor.


### 4.2.2. Case 2

Consider now a curve joining two spacetime pointsxA=x(τA), xB=x(τB). Let us identify the classical interval as (49)I=∫τAτBημνdxμdτdxνdτdτ.
The quantum fluctuation Iq represents a deviation of I that we identify in two equivalent ways. First, from the interpretation we gave to the sections we consider that at the quantum level there are fluctuations on the coordinates that change the original path joining the points xA,xB to a different path given now by (50)x̃μ:=xμ+sμ(x)
with the flat metric ημν being kept fixed, and in terms of which we write (51)Iq=∫τAτBημνdx̃μdτdx̃νdτdτ=∫τAτB(ημν+ημλs,νλ+ηνλs,μλ+ηλτs,μλs,ντ)dxμdτdxνdτdτ.
Second, we assume that Iq originates from a change of the flat metric ημν to another metric gμν with the path still given by xμ(τ), that is, (52)Iq=∫τAτBgμν(x)dxμdτdxνdτdτ.
These two forms being equivalent representations for Iq, they must give the same value, therefore we must have (53)gμν(x)=ημν+ημλs,νλ+ηνλs,μλ+ηλτs,μλs,ντ.
Here, when we identify (51) and (52) as two equivalent forms for Iq we are implicitly assuming that the net effect of the fluctuations of the spacetime coordinates x→x+s is to change the flat metric to an effective metric gμν.


## 4.2.1. Case 1

Let us consider local sectionss:Ua⊆M→TM, ∪aUa=M, that are analytic in their domain. Then, its Taylor series around a certain x0∈Ua is written as (43)sμ(x)=sμ(x0)+s,λμ(x0)(xλ-x0λ)+12s,λτi(x0)(xλ-x0λ)(xτ-x0τ)+⋯
with s,λμ(xa)=(∂sμ(x)/∂xλ)|xa. From (11) we obtain after collecting the terms (44)Iq(x⊕Δxs)=[ηλτ+ημνf2(sμ(x0)s,λτν(x0)+s,λμ(x0)s,τν(x0))]×xλxτ+h(x,x0,s(x0)…),
where h(x,x0,s(x0)…) includes all non-quadratic contributions on x. The quadratic term in x plays the role of an effective metric for spacetime M that we identify as (45)gλτ(x)=ηλτ+ημνf2(sμ(x0)s,λτν(x0)+s,λμ(x0)s,τν(x0)).
Another possibility to arrive at this metric is by considering the embedding problem as described in [11, 12], that is, we search for a smooth manifold (N,ĝ) with dimN≥dimM and such that (M,g) is embedded into N. Lets us consider N as an 8-dimensional manifold with local coordinates denoted by (YA)=(Yα,Yα+4),α=1, 2, 3, 4. Then, the problem is to find YA=YA(xα) such that (46)gμν=ĝAB∂YA∂xμ∂YB∂xν.
Defining (YA(x)):=(xα,sα(x)) with s=(sα):U→R4 an embedding, and (47)(ĝAB)=(ηαβ00ĝαβ)
with (48)ĝλτ(x):=12ημνf2∂xα∂sλ∂xβ∂sτ(∂α∂β(sμsν)|x=x0)
it is immediate to check that (46) determines (45). Then (M,g) is seen as isometrically embedded into (N,ĝ). By an appropriate choice of sections and of f̂, it is possible to obtain an effective metric that has a nonvanishing curvature tensor.


## 4.2.2. Case 2

Consider now a curve joining two spacetime pointsxA=x(τA), xB=x(τB). Let us identify the classical interval as (49)I=∫τAτBημνdxμdτdxνdτdτ.
The quantum fluctuation Iq represents a deviation of I that we identify in two equivalent ways. First, from the interpretation we gave to the sections we consider that at the quantum level there are fluctuations on the coordinates that change the original path joining the points xA,xB to a different path given now by (50)x̃μ:=xμ+sμ(x)
with the flat metric ημν being kept fixed, and in terms of which we write (51)Iq=∫τAτBημνdx̃μdτdx̃νdτdτ=∫τAτB(ημν+ημλs,νλ+ηνλs,μλ+ηλτs,μλs,ντ)dxμdτdxνdτdτ.
Second, we assume that Iq originates from a change of the flat metric ημν to another metric gμν with the path still given by xμ(τ), that is, (52)Iq=∫τAτBgμν(x)dxμdτdxνdτdτ.
These two forms being equivalent representations for Iq, they must give the same value, therefore we must have (53)gμν(x)=ημν+ημλs,νλ+ηνλs,μλ+ηλτs,μλs,ντ.
Here, when we identify (51) and (52) as two equivalent forms for Iq we are implicitly assuming that the net effect of the fluctuations of the spacetime coordinates x→x+s is to change the flat metric to an effective metric gμν.


## 5. Conclusion

We investigated a particular type of nonlinear transformation and the corresponding invariant intervalIq=x2+f̂ (f̂=s2f2). This interval was conceived as a change of the standard spacetime interval I=x2. One natural question that arises is on the physical principles that determine a change like I→Iq. The original argument proposed by Dresden [4, 5] that we revisited in Section 4.1 gives just a partial answer since it is excessively qualitative and does not contemplate causes other than the uncertainty principle in the position. In our model, we build f̂ from sections s:U⊆M→TM, then to answer that question we must first give a physical interpretation for the sections. Here, the assumption of extra structure in spacetime is not new [13–15], for example, in his model for the ether, Dirac [13] assumed that in spacetime there is defined a velocity distribution vμ that is associated to the gauge field Aμ, for example, k-1Aμ=vμ. In another direction, Caianiello et al. [15] assumed there is defined a field of velocities in spacetime corresponding to the movement of test particles of mass m. This, of course, introduces an unnecessary element in spacetime (the test particles) and the problem is still open if we intend to rely only on elements that are intrinsic to the spacetime.The nonlinear transformation also raises a question about causality. In fact, Zeeman showed that causality implies the usual (linear) Lorentz transformations [16]. Then, we would expect any model that incorporates nonlinear transformations violates causality. However, this is not necessarily true for the nonlinear transformations because Zeeman's result is verified for the spacetime M, not for the extended space M⊕Γ(TM) where the nonlinear transformations were defined, therefore, Zeeman's argument may need some modification or not even apply.Our investigation may be extended to the case spacetime is a manifold. Now, the relevant spacetime structure becomes the fundamental metric formI:=ημνdxμdxν, and we are interested in changes of I=ημνdxμdxν→Iq:=ημνdxμdxν+ημνdsμdsν. Here, the problem is to find the analogue of M⊕Γ(TM) and the corresponding transformations that leave Iq invariant.Nowadays, nonlinear transformations are being studied intensively in the context of the Double Special Relativity (DSR) models, where the nonlinear transformation is defined in the momenta space. A related problem in these models is to find the corresponding coordinate transformation. In [3], it was shown that the Fock transformation [1] was the counterpart in the coordinates of a certain transformation in the momenta. Then, in a similar way, it may be worth to investigate if the Albano-Dresden transformation corresponds to a nonlinear transformation in the momenta space of some DSR model.In our work, the group of nonlinear transformations of the Albano-Dresden type was realized in an infinite dimensional spaceM⊕∏x∈MEx. The infinite dimensionality of the realization stands for the space ∏x∈MEx where the sections were defined. We notice that Albano and Dresden obtained heuristically their transformation in a finite 4-dimensional spacetime M with no recourse to sections. However, as we have shown at the end of Section 2, their transformation is a particular case of type I transformations when we take sections s satisfying s2=1. Due to the heuristic approach followed by Albano and Dresden, it is not clear the details behind the process they employed to construct their realization. Therefore, it is not clear if other nonlinear realization of the Lorentz transformations may be made in a finite dimensional space, or even if other infinite dimensional realizations are related to the same infinite dimensional space we have employed.Finally, it is natural to interpret the functionf present in the Albano-Dresden interval Iq=x2+s2f2 as the gauge function of a Lyra manifold. In fact, for a Lyra manifold the gauge function is intrinsically defined in the manifold as part of the local chart (U,φ) that is then written as (U,φ,f) [9]. Charts like this were also of fundamental importance in the interpretation of the nonlinear transformation and its linearization in terms of compatible atlas as given in Section 3. For a Weyl manifold, the analogue of the gauge function becomes a map λ:M→R that appears related to the gauge transformation of the metric [17, 18]. Then, the role of the gauge function in Weyl and Lyra geometries are quite different. It seems there is no impossibility to define in the Weyl manifold intrinsic functions fi:Ui→R, or even to use λ to introduce a new basis for TpM, repeating the same construction that was made in Section 3 for the Lyra manifold. However, this would extend considerably the definition of the Weyl manifold introducing by hand extra structure that is not typical of this kind of space.



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*Source: 302164-2011-02-10.xml*