# Probing time orientability of spacetime

N.A. Lemos<EMAIL_ADDRESS>Instituto de Física, Universidade Federal
Fluminense, Av. Litorânea, S/N
24210-340 Niterói – RJ, Brazil  D. Müller<EMAIL_ADDRESS>Instituto de Física,
Universidade de Brasília 70919-970 Brasília - DF, Brazil  M.J. Rebouças
<EMAIL_ADDRESS>Centro Brasileiro de Pesquisas Físicas, Rua Dr.
Xavier Sigaud 150
22290-180 Rio de Janeiro – RJ, Brazil

###### Abstract

In general relativity, cosmology and quantum field theory, spacetime is
assumed to be an orientable manifold endowed with a Lorentz metric that makes
it spatially and temporally orientable. The question as to whether the laws of
physics require these orientability assumptions is ultimately of observational
or experimental nature, or the answer might come from a fundamental theory of
physics. The possibility that spacetime is time non-orientable lacks
investigation, and so should not be dismissed straightaway. In this paper, we
argue that it is possible to locally access a putative time non-orientability
of Minkowski empty spacetime by physical effects involving quantum vacuum
electromagnetic fluctuations. We set ourselves to study the influence of time
non-orientability on the stochastic motions of a charged particle subject to
these electromagnetic fluctuations in Minkowski spacetime equipped with a time
non-orientable topology and with its time orientable counterpart. To this end,
we introduce and derive analytic expressions for a statistical time
orientability indicator. Then we show that it is possible to pinpoint the time
non-orientable topology through an inversion pattern displayed by the
corresponding orientability indicator, which is absent when the underlying
manifold is time orientable.

###### pacs:

03.70.+k, 05.40.Jc, 42.50.Lc, 04.20.Gz, 98.80.Jk, 98.80.Cq

##  I Introduction

In relativistic cosmology and quantum field theory spacetime is described as a
four-dimensional differentiable manifold, a topological space with an
additional differentiable structure that permits to locally define
connections, metric and curvature with which the gravitation theories and the
dynamics of other fields are locally formulated. So, two important elements in
the mathematical description of the Universe and in the local dynamics of the
(micro) physical laws are the geometry and the topology of the underlying
spacetime manifold.

Topologically inequivalent manifolds can have the same geometry; conversely, a
fixed topological space can have different geometries. So, for example,
Kasner, Gödel, plane-wave and Minkowski geometries all can have
$\mathbb{R}^{4}$ as underlying manifold, whereas the manifold
$\mathbb{S}^{2}\times\mathbb{R}^{2}$, admits, for example, Reissner-Nordström
and Schwarzschild solutions of Einstein’s field equations, which represent
physically different gravitational fields [2, 3].

Geometry is a local attribute that brings about curvature. Topology is a
global property of manifolds that requires consideration of the entire
manifold. So, for example, compactness is a global property: it cannot be
mathematically formulated in local terms. Simple-connectedness and
orientability are examples of other topological properties of spacetime
manifolds. They do not arise from local mathematical knowledge alone. This is
a mathematical issue. However, if local physics is brought into the scene, it
can play a key role in the local access to the topological properties of
spacetime. This is the main concern of this work, which focuses on the time
orientability of spacetime.

The most primary definition of orientability arises in the context of a
topological space. However, when extra structure is added to the topological
manifold, different formulations of orientability can be given. Thus, for
example, the Möbius band $\mathbb{M}^{2}$ and the Klein bottle
$\mathbb{K}^{2}$ are non-orientable as topological spaces. When endowed with
Lorentz metrics, two possible types of orientability (non-orientability) come
about: spatial or temporal orientability (non-orientability), depending on the
way the manifolds are equipped with a Lorentz metric. These examples make it
apparent that whether a spacetime is time or space orientable may be looked
upon as a joint topological-geometrical property, in the sense that it depends
on the topology of the underlying manifold but also on the specific Lorentz
metric we equip the manifold with [3, 4].

Most illustrations of non-orientability we encounter depict spatial non-
orientability by making use of spatial paths which bring a traveler back to
her/his starting point with left- and right-hand sides exchanged (mirror-
reversed) [5]. So, for example, in the Wikipedia entry “Orientability” [6] the
crab’s left and right claws are reversed when it makes a round (spatial) trip
along the Möbius band. However, the illustration of time non-orientability
would be trickier, as it deals with an inversion (flip) of the time $t$ after
a round-trip over the whole manifold. We are familiar with left and right
exchanges, but not with clocks first running forward then backward in time
after a circumnavigation. Yet, the Möbius band $\mathbb{M}^{2}$ can also be
used as an example of time non-orientability as long as one suitably endows it
with time and space coordinates $t,x$ such that time is inverted (flips) after
a circumnavigation (translation along the $x$-direction). This is a periodic
time inversion of topological origin.

It is generally assumed that the spacetime manifolds one deals with in physics
are orientable as topological manifolds and are additionally endowed with an
appropriate Lorentz metric, making them separately time and space
orientable.111Theoretical arguments in support of these orientability
assumptions combine space and time universality of local physical laws with
the thermodynamically defined local arrow of time, charge conjugation and
parity (CP) violation and CPT invariance [7, 2, 3]. The impossibility of
having globally defined spinor fields on non-orientable spacetime manifolds is
also often used in favor of these orientability assumptions [2, 3, 8, 9, 10,
11]. It should be noted, however, that time universality can be looked upon as
a topological assumption of global time homogeneity. This topological
assumption rules out time non-orientability from the outset. Non-orientability
raises intriguing questions, and it is generally seen as an undesirable
feature in physics. Yet, space and time non-orientability of Lorentzian
spacetime manifolds are concrete mathematical possibilities for the physical
spacetime. On the other hand, severe orientability assumptions could risk
ruling out something that the spacetime topology might be trying to “tell” us,
and it might also discourage further investigations on the interplay of
physics and topology. Thus, it seems reasonable to expect that the ultimate
answer to questions regarding the orientability of spacetime should rely on
local experiments or cosmological observations, or even might come from a
fundamental theory of physics.

In this article we address the question as to whether one can empirically and
locally access the putative topological property of time non-orientability
(orientability) of spacetime manifolds $\mathcal{M}_{4}$.222The nature of time
and the existence of various arrows of time are contested issues, which we
evade in this paper, in which we assume that time is real and passes in all
scales. The literature on this debate is fascinating and vast, but for the
sake of brevity we refer the readers to Refs. [12, 13] and references therein,
including the books [14, 15, 16, 17].

Since the net role played by time orientability is more clearly ascertained in
static flat spacetime, whose dynamical degrees of freedom are frozen, in this
work we focus on Minkowski spacetime. The physical system that turns out to be
suitable to play the revealing role of global time non-orientability is a non-
relativistic charged particle locally subjected to quantum vacuum fluctuations
of the electromagnetic field.333 In an influential work, this system was used
by Yu and Ford [18] in Minkowski spacetime with a conducting plane (nontrivial
spatial topology). They endeavored to shed light on the question as to whether
a test charged particle would perform stochastic motions induced by quantum
fluctuations of the electromagnetic field [19, 20]. Related investigations
have since been made in a number of papers, including Refs. [21, 22, 23, 24,
25, 26, 27, 28, 29].

To show that one can access time orientability, we investigate signatures of
time non-orientability through the stochastic motions of a charged test
particle under electromagnetic quantum fluctuations in Minkowski spacetime
$\mathcal{M}_{4}$ with time non-orientable topology and in its time orientable
counterpart. The time orientability statistical indicator that we introduce
combines geometrical-topological properties with the dynamics of the above
specific physical system engineered to test orientability.

The structure of the paper is as follows. In Section II we set up the notation
and present some key concepts and results regarding topology of manifolds,
which will be needed in the remainder of the paper. In Section III we present
the physical system along with the background geometry and topology. In
Section IV we introduce the time orientability statistical indicator and
derive its expressions for a charged particle under quantum vacuum
electromagnetic fluctuations in Minkowski spacetime equipped with temporally
non-orientable and orientable topologies. We show that a comparison of the
time evolution of our statistical indicators for these cases allows one to
discriminate the orientable from the non-orientable topology. Time non-
orientability can be locally unveiled by the inversion pattern of the curves
of the time orientability statistical indicator for a point charge under
quantum vacuum electromagnetic fluctuations. In Section V we present our main
conclusions and final remarks.

##  II Context and mathematical preliminaries

To make this work to a certain extent self-contained, in this section we
define the notation, give some basic definitions and present a few results
concerning the topology of the manifolds which are used throughout this paper.

We first briefly underscore the context of this article vis-à-vis some recent
works involving spacetimes with a nontrivial topology. For both the spatially
flat Friedmann-Robertson-Walker (FRW) and the Minkowski spacetimes, the
underlying manifold $\mathcal{M}_{4}$ is globally decomposable as
$\mathcal{M}_{4}=\mathbb{R}\times M^{3}$, where $M^{3}$ is the simply-
connected $3$-manifold $\mathbb{R}^{3}$. However, since topologically
inequivalent manifolds can have a given spacetime geometry, the underlying
topology of FRW or Minkowski manifolds can belong to a set of diverse
possibilities. For example, the spatial section $M^{3}$ of both spacetimes can
be any quotient manifold of the form $M^{3}=\mathbb{E}^{3}/\Gamma$, where
$\mathbb{E}^{3}$ is the covering space,444$\mathbb{E}^{n}$ is $\mathbb{R}^{n}$
endowed with the Euclidean metric. and $\Gamma$ is a discrete group of freely
acting isometries of $\mathbb{E}^{3}$, also referred to as the holonomy [30,
31]. 555Refs. [32, 33, 34, 35] give a detailed account of the classification
of flat $3$-dimensional Euclidean topologies. The familiar form
$\mathcal{M}_{4}=\mathbb{R}\times M^{3}$ is very often assumed in cosmology
and quantum field theory, and in particular it is adopted in cosmic topology,
which investigates the spatial topology of the Universe.666For review papers
on cosmic topology and recent observational constraints, see Refs. [36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50].

In the present work, rather than addressing a possible nontrivial spatial
topology of the FRW spacetime, which is the subject matter of cosmic topology,
we focus on a single topological property of the Minkowski spacetime manifold
$\mathcal{M}_{4}$, namely time orientability. Let us first point out that any
simply connected spacetime manifold is both time and space orientable. Thus,
perhaps the simplest way to equip Minkowski spacetime with time non-
orientability is by taking

$\mathcal{M}_{4}=\mathcal{M}^{2}\times\mathbb{E}^{2}\,,$ (1)

where $\mathcal{M}^{2}$ is a two-dimensional time non-orientable quotient
manifold, $\mathcal{M}^{2}=\mathbb{E}^{2,1}/\Gamma$, where $\mathbb{E}^{2,1}$
denotes the simply-connected plane equipped with a Lorentz metric, that is,
Minkowski two-dimensional spacetime. It follows that the orientability of
$\mathcal{M}_{4}$ reduces to the orientability of $\mathcal{M}^{2}\,$.777A
mathematically much more involved way would be by the decomposition
$\mathcal{M}_{4}=\mathbb{K}^{3}\times\mathbb{R}$ with the time $t$ chosen as
the flipped direction of the $3$-dimensional non-orientable Klein space [5,
30, 31, 32, 33, 34, 35].

To make a suitable choice of $\mathcal{M}^{2}$, we start by reviewing some
basic results concerning Euclidean two-dimensional manifolds and then proceed
to choose a non-orientable Lorentzian two-dimensional manifold. A first
question we address is the identification of all topologically inequivalent
Euclidean two-dimensional manifolds.

The simplest example of a two-dimensional Euclidean manifold with nontrivial
topology is the cylinder $\mathbb{C}^{2}$, whose construction as a quotient
manifold, $\mathbb{C}^{2}=\mathbb{E}^{2}/\Gamma$, is such that a point
$P=(x,y)$ of the cylinder is obtained from the covering manifold
$\mathbb{E}^{2}$ by identifying the points that are equivalent under the
action of the elements $\gamma_{i}$ of the covering isometry group $\Gamma$.
Specifically, a point $P=(x,y)$ of the cylinder is the equivalence class of
all points $P^{\prime}$ in the covering space $\mathbb{E}^{2}$ such that
$P^{\prime}=\\{(x+n_{x}a,y)\mid n_{x}\in\mathbb{Z},a={\rm const}\\}$, or
$P\equiv P^{\prime}=\gamma_{i}P$ with $\gamma_{i}\in\Gamma$ being a
translation by $a$ in the direction of the $x$-axis. A convenient way to
visualize a quotient manifold is through the so-called fundamental domain
(cell) or fundamental polygon. For the cylinder $\mathbb{C}^{2}$ the
fundamental cell is a strip of $\mathbb{E}^{2}$ bounded by parallel lines, say
$x=0$ and $x=a$, that are identified through translations to form the cylinder
$\mathbb{C}^{2}$. The cylinder $\mathbb{C}^{2}$ is a surface (two-manifold)
lying in three-dimensional space. This embedded view, however, does not
necessarily work for other Euclidean two-manifolds.

A very important advantage of the quotient construction,
$\mathbb{E}^{2}/\Gamma$, comes from examples of $2$-manifolds that cannot be
properly displayed in the regular three-dimensional space. The twisted
cylinder $\mathbb{C^{*}}^{2}$ is one example of this kind of manifold. It
cannot lie in ordinary three-dimensional space without intersecting itself.
Yet it has a rather simple representation as a quotient manifold. Indeed, it
is obtained from a strip of $\mathbb{E}^{2}$ bounded by parallel lines which
are identified through a glide reflection: translation followed by a flip
(inversion).888 A popular, but not entirely accurate, embedded representation
of the twisted cylinder is by means of the well known Möbius band
$\mathbb{M}^{2}$, which is obtained through the identification of two opposite
sides of a rectangle after a flip (half twist). The Möbius band is visually
useful as it can lie in three-dimensional space, but it is not a regular
manifold since it has a boundary. The twisted cylinder $\mathbb{C^{*}}^{2}$,
on the other hand, is a quotient manifold
$\mathbb{C^{*}}^{2}=\mathbb{E}^{2}/\Gamma$, but cannot lie in
$\mathbb{R}^{3}$. A point $P=(x,y)$ of the twisted cylinder
$\mathbb{C^{*}}^{2}$ represents a set of points in the covering space
$\mathbb{E}^{2}$ of the form $P^{\prime}=\\{(x+n_{x}a,(-1)^{n_{x}}y)\mid
n_{x}\in\mathbb{Z},\,a={\rm const}\\}$ or $P\equiv P^{\prime}=\gamma_{i}P$
with $\gamma_{i}\in\Gamma$ being translation by $a$ in the $x$-direction
followed by an inversion (flip) in the $y$-direction, a single glide
reflection.

Finally the last two-dimensional quotient Euclidean manifolds are the torus
$\mathbb{T}^{2}$ and the Klein bottle $\mathbb{K}^{2}$. Both can be obtained
through identification of opposite sides of a rectangle. The torus and the
Klein bottle are related quite in the same way as the cylinder and the twisted
cylinder. For this reason, the Klein bottle is also known as the twisted torus
[51]. A point $P=(x,y)$ of the torus $\mathbb{T}^{2}$ represents the
equivalence class of all points in the covering manifold $\mathbb{E}^{2}$ such
that $P^{\prime}=\\{(x+n_{x}a\,,\,y+n_{y}b)\mid
n_{x},n_{y}\in\mathbb{Z},\,a,b={\rm consts}\\}$, or
$P\equiv\,P^{\prime}=\gamma_{i}P$ with $\gamma_{i}\in\Gamma$ being a
translation by $a$ in the $x$-direction and by $b$ in the $y$-direction. As
for the Klein bottle, one has $P\equiv
P^{\prime}=\\{(x+n_{x}a,(-1)^{n_{x}}y+n_{y}b)\mid
n_{x},n_{y}\in\mathbb{Z},a,b={\rm consts}\\}$.

It can be shown that, under the assumptions of completeness and connectedness,
there are no other Euclidean two-dimensional manifolds of the form
$\mathbb{E}^{2}/\Gamma$ (quotient manifolds) than the above-mentioned four
[52]. Actually, any other group $\Gamma$ yields a quotient space that is not a
manifold [52]. Moreover, $\mathbb{E}^{2}$ and these four Euclidean quotient
manifolds are connected and complete (Euclidean metric). It turns out that
connectedness and completeness exclude all Euclidean two-dimensional manifolds
except for the previously mentioned quotients $\mathbb{E}^{2}/\Gamma$ and
$\mathbb{E}^{2}$ itself. This is an instance of the Killing-Hopf theorem,
which ensures that, apart from $\mathbb{E}^{2}$, each connected, complete
Euclidean two-dimensional manifold is either a cylinder, twisted cylinder,
torus or Klein bottle [52].

Let us move on to another important topological feature of Euclidean manifolds
we use in this work. In dealing with topological manifolds we have the concept
of global homogeneity of topological origin. A way to characterize global
homogeneity of the quotient manifolds $\mathcal{M}^{2}=\mathbb{E}^{2}/\Gamma$
is through distance functions. Indeed, for any point
$\mathbf{x}\in\mathcal{M}^{2}$ the distance function
$\ell_{\gamma}(\mathbf{x})$ for a given $\gamma\in\Gamma$ is defined by

$\ell_{\gamma}(\mathbf{x})=d(\mathbf{x},\gamma\mathbf{x})\;,$ (2)

where $d$ is the two-dimensional Euclidean metric. The distance function
provides the length of the closed geodesic that passes through $\mathbf{x}$
and is associated with a discrete isometry $\gamma\in\Gamma$. In globally
homogeneous manifolds the distance function for any covering isometry $\gamma$
is constant. In globally inhomogeneous manifolds, in contrast, the length of
the closed geodesic associated with at least one $\gamma$ is non-translational
and depends on the point $\mathbf{x}\in\mathcal{M}^{2}$, and then is not
constant. When the distance between a point $\mathbf{x}$ and its image
$\gamma\mathbf{x}$ is a constant for all points $\mathbf{x}$ and elements
$\gamma\in\Gamma$ then the isometries are translations, that is, all elements
of the covering group $\Gamma$ in globally homogeneous manifolds are
translations. This means that in these quotient manifolds the sides of the
fundamental domain are identified through independent translations. Thus, in
the above set of two-dimensional quotient manifolds, besides $\mathbb{E}^{2}$,
the cylinder and the torus are topologically homogeneous, whereas the twisted
cylinder and the Klein bottle are globally inhomogeneous.

Orientability is another very important global (topological) property of a
manifold that measures whether one can consistently choose a clockwise
orientation for loops in the manifold. A path in a $\mathcal{M}^{2}$ that
brings a traveler back to the starting point mirror-reversed is called an
orientation-reversing path. Manifolds that do not have an orientation-
reversing path are called orientable, whereas manifolds that contain an
orientation-reversing path are non-orientable [5]. The Euclidean plane,
cylinder and torus are orientable, whereas the twisted cylinder and the Klein
bottle are non-orientable two-dimensional manifolds. For two-dimensional
quotient manifolds $\mathbb{E}^{2}/\Gamma$, when the covering group $\Gamma$
contains at least one isometry $\gamma$ that is a reflection (flip) the
corresponding quotient manifold is non-orientable. Therefore, the torus is
orientable but the Klein bottle is non-orientable. Clearly non-orientable
manifolds are necessarily globally inhomogeneous as the covering group
$\Gamma$ contains a reflection, which obviously is a non-translational
covering isometry.

In Table 1 we collect the names, symbols, generators $\gamma\in\Gamma$
together with the topological properties of orientability and global
homogeneity of the two-dimensional manifolds $\mathcal{M}^{2}$, which are
possible factors of the product manifold
$\mathcal{M}_{4}=\mathcal{M}^{2}\times\mathbb{E}^{2}$, which in turn we shall
take as the underlying manifold to be endowed with the Minkowski spacetime
metric $\eta_{\mu\nu}=\mbox{diag}(+1,-1,-1,-1)$.

Manifold | Symbol | Generators $\gamma_{i}\in\Gamma$ | Orientable |  Homogeneous
---|---|---|---|---
Cylinder | $\mathbb{C}^{2}$ | 1 Translation | yes |  yes
Twisted cylinder | $\;\mathbb{C^{*}}^{2}$ | 1 Glide reflection | no |  no
Torus | $\mathbb{T}^{2}$ | 2 Translations | yes |  yes
Klein bottle | $\mathbb{K}^{2}$ |  1 Transl. 1 Glide reflect. | no |  no
Table 1: Names and symbols of two-dimensional Euclidean quotient manifolds
$\mathbb{E}^{2}/\Gamma$ together with the generators $\gamma_{i}$ of the group
$\Gamma$, and two global (topological) properties: orientability and global
homogeneity.

For the product manifold given by equation (1) to be time non-orientable the
factor $\mathcal{M}^{2}$ has to be a Lorentzian non-orientable manifold. The
Lorentz metric, on the other hand, decisively constrains the topology of the
possible underlying manifolds. Thus, not every smooth manifold can be made a
Lorentzian manifold. For example, the two-sphere $\mathbb{S}^{2}$, for which
the Euler characteristic $\chi=2$, cannot have a Lorentz metric (a consequence
of the “hairy ball” theorem), whereas the torus $\mathbb{T}^{2}$, for which
$\chi=0$, can (see, for example, Ref. [59]).

It can be shown that every compact connected smooth manifold with Euler
characteristic $\chi=0$ admits a Lorentz metric [60]. For the case of two-
dimensional manifolds, this ensures that both the torus $\mathbb{T}^{2}$ and
the Klein Bottle $\mathbb{K}^{2}$, for both of which $\chi=0$, can be equipped
with a Lorentz metric. It can also be proved that every $n$-dimensional
noncompact smooth manifold admits a Lorentz metric [60]. This guarantees, in
particular, that both the cylinder $\mathbb{C}^{2}$ and the twisted cylinder
$\mathbb{C^{*}}^{2}$ can be made Lorentzian manifolds.999A smooth
$n$-dimensional manifold $M$ with $n\geq 2$ can be endowed with a Lorentz
metric if and only if either $M$ is noncompact or it is compact and has Euler
characteristic $\chi(M)=0$ [60, 61].

##  III Charged particle under electromagnetic fluctuations

From now on, we consider a point charge under quantum vacuum electromagnetic
fluctuations as the physical system used to locally probe a potential time
non-orientablity of spacetime.

###  III.1 The physical system

Let a nonrelativistic test particle with charge $q$ and mass $m$ be locally
subject to vacuum fluctuations of the electric field ${\bf E}({\bf x},t)$ in a
topologically nontrivial spacetime manifold equipped with the Minkowski metric
$\eta_{\mu\nu}=\mbox{diag}(+1,-1,-1,-1)$.

Locally, the motion of the charged test particle is determined by the Lorentz
force. In the nonrelativistic limit the equation of motion for the point
charge is

$\frac{d{\bf v}}{dt}=\frac{q}{m}\,{\bf E}({\bf x},t)\,,$ (3)

where $\mathbf{v}$ is the particle’s velocity and $\mathbf{x}$ its position at
time $t$. We assume that on the time scales of interest the particle
practically does not move, i.e. it has a negligible displacement, so we can
ignore the time dependence of $\mathbf{x}$. Thus, the particle’s position
$\mathbf{x}$ is taken as constant in what follows [18].101010The corrections
arising from the inexactness of this assumption are negligible in the low
velocity regime. Assuming that the particle is initially at rest, integration
of Eq. (3) gives

${\bf v}({\bf x},t)=\frac{q}{m}\int_{t_{0}}^{t}{\bf E}({\bf
x},t^{\prime})\,dt^{\prime}\,,$ (4)

and the mean squared velocity, velocity dispersion or simply dispersion in
each of the three independent directions $i=x,y,z$ is given by111111By
definition, $\,\bigl{\langle}\,\Delta
v^{2}_{i}(\mathbf{x},t)\,\bigr{\rangle}=\bigl{\langle}\,v^{2}_{i}(\mathbf{x},t)\,\bigr{\rangle}-\bigl{\langle}\,v_{i}(\mathbf{x},t)\bigr{\rangle}^{2}$.

$\Bigl{\langle}\Delta
v^{2}_{i}\Bigr{\rangle}=\frac{q^{2}}{m^{2}}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\Bigl{\langle}E_{i}({\bf
x},t^{\prime})E_{i}({\bf
x},t^{\prime\prime})\Bigr{\rangle}\,dt^{\prime}dt^{\prime\prime}\,.$ (5)

Following Yu and Ford [18], we assume that the electric field is a sum of
classical $\mathbf{E}_{c}$ and quantum $\mathbf{E}_{q}$ parts. Because ${\bf
E}_{c}$ is not subject to quantum fluctuations and $\langle{\bf
E}_{q}\rangle=0$, the two-point function $\langle E_{i}({\bf x},t)E_{i}({\bf
x}^{\prime},t^{\prime})\rangle$ in equation (5) involves only the quantum part
of the electric field [18].

It can be shown [62] that locally

$\Bigl{\langle}E_{i}({\bf x},t)E_{i}({\bf
x}^{\prime},t^{\prime})\Bigr{\rangle}=\frac{\partial}{\partial
x_{i}}\frac{\partial}{\partial{x^{\prime}}_{i}}D({\bf x},t;{\bf
x}^{\prime},t^{\prime})-\,\,\frac{\partial}{\partial
t}\frac{\partial}{\partial t^{\prime}}D({\bf x},t;{\bf
x^{\prime}},t^{\prime})$ (6)

where, in Minkowski spacetime, the Hadamard function $D({\bf x},t;{\bf
x}^{\prime},t^{\prime})$ is given by

$D_{0}({\bf x},t;{\bf x}^{\prime},t^{\prime})=\frac{1}{4\pi^{2}(\Delta
t^{2}-|\Delta\mathbf{x}|^{2})}\,.$ (7)

The subscript $0$ indicates standard Minkowski spacetime
$\mathbb{R}\times\mathbb{E}^{3}$, $\Delta t=t-t^{\prime}$ and
$|\Delta\mathbf{x}|\equiv r$ is the spatial separation for topologically
trivial Minkowski spacetime:

$r^{2}=(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}\,.$ (8)

###  III.2 The spacetime manifold

In this work, the time non-orientable spacetime manifold that we shall
consider is of the form $\mathcal{M}_{4}=\mathcal{M}^{2}\times\mathbb{E}^{2}$
in which the first factor $\mathcal{M}^{2}$ is the non-orientable twisted
cylinder $\mathbb{C}^{*2}$ equipped with a Lorentz metric with time as the
flipped coordinate. This makes $\mathcal{M}^{2}$ time non-orientable and, as a
consequence, the spacetime manifold $\mathcal{M}_{4}$ is also time non-
orientable. In Table 2 we show the time separation $\Delta t$ and the spatial
separation $r$ that enter the spacetime interval $\Delta s^{2}=\Delta
t^{2}-r^{2}$ in $\mathcal{M}_{4}$ for $\mathcal{M}^{2}=\mathbb{C}^{*2}$ and
$\mathcal{M}^{2}=\mathbb{C}^{2}$, the cases treated in this paper. For
$\mathcal{M}^{2}=\mathbb{C}^{*2}$ the non-orientability is associated with the
time coordinate.

Manifold | Time separation $\Delta t$ |  Spatial separation $r^{2}$
---|---|---
$\mathbb{C}^{*2}\times\mathbb{E}^{2}$ | $t-(-1)^{n_{x}}t^{\prime}$ | $(x-x^{\prime}-n_{x}a)^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}$
$\mathbb{C}^{2}\times\mathbb{E}^{2}$ | $t-t^{\prime}$ | $(x-x^{\prime}-n_{x}a)^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}$
Table 2: Time and space separation for the spacetime interval $\Delta
s^{2}=\Delta t^{2}-r^{2}$ in the Hadamard function for the spacetime manifolds
obtained by taking the Cartesian product with $\mathbb{E}^{2}$ of the two-
dimensional manifolds $\mathbb{C}^{*2}$ or $\mathbb{C}^{2}$ with coordinates
$t,x$; the coordinates associated with $\mathbb{E}^{2}$ are $y,z$. With time
always in the first factor, each flat four-manifold thus obtained is endowed
with a Lorentz metric with signature $-2$. In the case of $\mathbb{C}^{*2}$
the identification with inversion is made on the time direction, so that the
corresponding spacetime is not time orientable. The topological compact
spatial scale is denoted by $a$. The number $n_{x}$ is an integer that runs
from $-\infty$ to $\infty$.

Let us try to make our terminology and notation as clear as possible. Being
aware of the abuse of language but striving for simplicity, we henceforward
give the name twisted cylinder, denoted by
$\mathbb{C}^{*2}\times\mathbb{E}^{2}$, to the time non-orientable spacetime
with coordinates $(t,x,y,z)$, whose points are identified as
$(t,x,y,z)\equiv((-1)^{n_{x}}t,x+n_{x}a,y,z)$, and equipped with the Lorentz
metric $\Delta s^{2}=\Delta t^{2}-r^{2}$, where $\Delta t$ and $r$ are given
in the first line of Table 2. Similarly, we give the name cylinder, denoted by
$\mathbb{C}^{2}\times\mathbb{E}^{2}$, to the time orientable spacetime with
coordinates $(t,x,y,z)$, whose points are identified as
$(t,x,y,z)\equiv(t,x+n_{x}a,y,z)$, and equipped with the Lorentz metric
$\Delta s^{2}=\Delta t^{2}-r^{2}$, where $\Delta t$ and $r$ are given in the
second line of Table 2.

##  IV TIME-ORIENTABILITY INDICATOR FOR THE TWISTED CYLINDER

We take up now the study of the stochastic motions of a charged particle under
quantum vacuum electromagnetic fluctuations in the time non-orientable twisted
cylinder $\mathbb{C}^{*2}\times\mathbb{E}^{2}$.

In a topologically nontrivial spacetime, the time interval $\Delta t$ and the
the spatial separation $r$ take new forms that capture the periodic boundary
conditions imposed on the covering space. To obtain the correlation function
for the electric field that is required to compute the velocity dispersion (5)
for the twisted cylinder $\mathbb{C}^{*2}\times\mathbb{E}^{2}$, we replace in
Eq. (6) the Hadamard function $D({\bf x},t;{\bf x}^{\prime},t^{\prime})$ by
its renormalized version given by [21]

$D_{ren}({\bf x},t;{\bf x}^{\prime},t^{\prime})=D({\bf x},t;{\bf
x}^{\prime},t^{\prime})-D_{0}({\bf x},t;{\bf
x}^{\prime},t^{\prime})=\sum\limits_{{n_{x}=-\infty}}^{{\infty\;\;\prime}}\frac{1}{4\pi^{2}(\Delta
t^{2}-r^{2})}\,,$ (9)

where here and in what follows $\sum^{\;{}^{\prime}}$ indicates that the
Minkowski contribution term $n_{x}=0$ is excluded from the summation, and,
according to Table 2, the time and space separations for the time non-
orientable spacetime $\mathbb{C}^{*2}\times\mathbb{E}^{2}$ are

$\Delta t=t-(-1)^{n_{x}}t^{\prime},\qquad
r^{2}=(x-x^{\prime}-n_{x}a)^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}.$ (10)

The term with $n_{x}=0$ in the sum (9) is the Hadamard function $D_{0}({\bf
x},t;{\bf x}^{\prime},t^{\prime})$ for standard Minkowski spacetime. This term
has been subtracted out from the sum because it gives rise to an infinite
contribution to the velocity dispersion.

Thus, from equation (6) the renormalized correlation functions

$\bigl{\langle}E_{i}({\bf x},t)E_{i}({\bf
x}^{\prime},t^{\prime})\bigr{\rangle}_{ren}=\frac{\partial}{\partial
x_{i}}\frac{\partial}{\partial{x^{\prime}}_{i}}D_{ren}({\bf x},t;{\bf
x}^{\prime},t^{\prime})-\frac{\partial}{\partial t}\frac{\partial}{\partial
t^{\prime}}D_{ren}({\bf x},t;{\bf x^{\prime}},t^{\prime})$ (11)

are then given by

$\bigl{\langle}E^{i}({\bf x},t)E^{i}({\bf
x}^{\prime},t^{\prime})\bigr{\rangle}_{ren}^{\mathbb{C}^{*2}}=\sum\limits_{{n_{x}=-\infty}}^{{\infty\;\;\prime}}\frac{[3(-1)^{n_{x}}-1]\Delta
t^{2}+[1+(-1)^{n_{x}}]r^{2}-4r_{i}^{2}}{2\pi^{2}[\Delta t^{2}-r^{2}]^{3}},$
(12)

where $\Delta t$ and $r^{2}$ are given by Eq. (10), while

$r_{1}\equiv r_{x}=x-x^{\prime}-n_{x}a,\quad r_{2}\equiv
r_{y}=y-y^{\prime},\quad r_{3}\equiv r_{z}=z-z^{\prime}.$ (13)

The orientability indicator $\mbox{\large$I$}_{v^{2}_{i}}^{\mathbb{C}^{*2}}$
that we will consider is defined by replacing the electric field correlation
functions in Eq. (5) by their renormalized counterparts:121212To avoid a
cluttered notation, for the superscript on the statistical indicator and on
the electric field correlation functions we write just $\mathbb{C}^{*2}$
instead of $\mathbb{C}^{*2}\times\mathbb{E}^{2}$.

$\mbox{\large$I$}_{v^{2}_{i}}^{\mathbb{C}^{*2}}({\bf
x},t,t_{0})=\frac{q^{2}}{m^{2}}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\Bigl{\langle}E^{i}({\bf
x},t^{\prime})E^{i}({\bf
x},t^{\prime\prime})\Bigr{\rangle}_{ren}^{\mathbb{C}^{*2}}\,dt^{\prime}dt^{\prime\prime}\,.$
(14)

From (9) it is clear that the orientability indicator
$\mbox{\large$I$}_{v^{2}_{i}}^{\mathbb{C}^{*2}}$ is the difference between the
velocity dispersion in $\mathbb{C}^{*2}\times\mathbb{E}^{2}$ and the one in
Minkowski spacetime with trivial topology.

Before moving on to explicit computations, a few words of clarification on the
meaning of the indicator (14) are fitting. From equations (5) and (9) a
general definition of the orientability indicator can be written in the form

$\mbox{\large$I$}_{v^{2}_{i}}^{MC}=\Bigl{\langle}\Delta
v_{i}^{2}\Bigr{\rangle}^{MC}-\;\,\Bigl{\langle}\Delta
v_{i}^{2}\Bigr{\rangle}^{SC},$ (15)

where $\bigl{\langle}\Delta v_{i}^{2}\bigr{\rangle}$ is the mean square
velocity dispersion, and the superscripts $MC$ and $SC$ stand for multiply-
and simply-connected manifolds, respectively. The right-hand side of (15) is
defined by first taking the difference of the two terms with ${\bf
x}^{\prime}\neq{\bf x}$ and then setting ${\bf x}^{\prime}={\bf x}$.

On the face of it, the indicator (15) does not appear to be measurable because
it involves the difference of quantities associated with two different
spacetimes, but the spacetime we live in is unique. However,
$\mbox{\large$I$}_{v^{2}_{i}}^{MC}$ is accessible by measurements performed in
our spacetime, which is to be tested for time non-orientablity, by the
following procedure. First one would measure the velocity correlation function
$\bigl{\langle}\Delta v_{i}({\bf x},t)\Delta v_{i}({\bf
x}^{\prime},t)\bigr{\rangle}^{MC}$ for ${\bf x}\neq{\bf x}^{\prime}$, then one
would subtract out the correlation function $\bigl{\langle}\Delta v_{i}({\bf
x},t)\Delta v_{i}({\bf x}^{\prime},t)\bigr{\rangle}^{SC}$ that has been
theoretically computed for ${\bf x}\neq{\bf x}^{\prime}$ for the corresponding
topologically trivial Minkowski spacetime in the Appendix of Ref. [28].
Finally, the corresponding curve for the difference (15) as a function of time
would be plotted in the coincidence limit ${\bf x}={\bf
x}^{\prime}$.131313This approach is analogous to one used in the search for
spatial topology of the universe from discrete cosmic sources, called cosmic
crystallography [63], in which a topological signature of $3-$space is given
by a constant times the difference
$\Phi_{exp}^{MC}(s_{i})-\Phi^{SC}_{exp}(s_{i})$ of the expected pair
separation histogram (EPSH), and the EPSH for the underlying simply connected
covering manifold [64, 65], which can be theoretically computed in analytical
form [64, 66].

###  IV.1 Indicators for the twisted cylinder

The orientability indicator $\mbox{\large$I$}_{v^{2}_{i}}^{\mathbb{C}^{*2}}$
can be computed with the help of the integrals [21]

$\displaystyle I_{-}$ $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}\int_{t_{0}}^{t}dt^{\prime}dt^{\prime\prime}\frac{1}{[(t^{\prime}-t^{\prime\prime})^{2}-r^{2}]^{3}}$
(16) $\displaystyle=$
$\displaystyle\frac{3(t-t_{0})}{16r^{5}}\ln\left[\left(\frac{t_{0}-t+r}{t_{0}-t-r}\right)^{2}\right]+\frac{1}{4r^{2}\left[(t-t_{0})^{2}-r^{2}\right]}+\frac{1}{4r^{4}}$

and

$\displaystyle J_{-}$ $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}\int_{t_{0}}^{t}dt^{\prime}dt^{\prime\prime}\frac{(t^{\prime}-t^{\prime\prime})^{2}}{[(t^{\prime}-t^{\prime\prime})^{2}-r^{2}]^{3}}$
(17) $\displaystyle=$
$\displaystyle\frac{(t-t_{0})}{16\,r^{3}}\ln\left[\left(\frac{t_{0}-t-r}{t_{0}-t+r}\right)^{2}\right]+\frac{1}{4\left[(t-t_{0})^{2}-r^{2}\right]}+\frac{1}{4r^{2}}$

as well as

$\displaystyle I_{+}$ $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}\int_{t_{0}}^{t}dt^{\prime}dt^{\prime\prime}\frac{1}{[(t^{\prime}+t^{\prime\prime})^{2}-r^{2}]^{3}}=\frac{3t_{0}}{16r^{5}}\ln\left[\left(\frac{2t_{0}-r}{2t_{0}+r}\right)^{2}\left(\frac{t_{0}+t+r}{t_{0}+t-r}\right)^{2}\right]+\frac{1}{8r^{2}\left[4t_{0}^{2}-r^{2}\right]}$
(18)
$\displaystyle+\frac{3t}{16r^{5}}\ln\left[\left(\frac{t_{0}+t+r}{t_{0}+t-r}\right)^{2}\left(\frac{2t-r}{2t+r}\right)^{2}\right]+\frac{1}{8r^{2}\left[4t^{2}-r^{2}\right]}-\frac{1}{4r^{2}\left[(t+t_{0})^{2}-r^{2}\right]}$

and

$\displaystyle J_{+}$ $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t}\int_{t_{0}}^{t}dt^{\prime}dt^{\prime\prime}\frac{(t^{\prime}+t^{\prime\prime})^{2}}{[(t^{\prime}+t^{\prime\prime})^{2}-r^{2}]^{3}}=\frac{t_{0}}{16r^{3}}\ln\left[\left(\frac{2t_{0}+r}{2t_{0}-r}\right)^{2}\left(\frac{t_{0}+t-r}{t_{0}+t+r}\right)^{2}\right]+\frac{1}{8\left[4t_{0}^{2}-r^{2}\right]}$
(19)
$\displaystyle+\frac{t}{16r^{3}}\ln\left[\left(\frac{t_{0}+t-r}{t_{0}+t+r}\right)^{2}\left(\frac{2t+r}{2t-r}\right)^{2}\right]+\frac{1}{8\left[4t^{2}-r^{2}\right]}-\frac{1}{4\left[(t+t_{0})^{2}-r^{2}\right]}$

By using these integrals and equations (12) and (13) in Eq. (14) we find

$\mbox{\large$I$}_{v^{2}_{i}}^{\mathbb{C}^{*2}}({\bf
x},t,t_{0})=\frac{q^{2}}{\pi^{2}m^{2}}\bigg{\\{}\sum_{\mbox{\scriptsize
even}\,\,n_{x}}^{\hskip
11.38092pt\prime}\bigl{[}J_{-}+(r^{2}-2r_{i}^{2})I_{-}\bigr{]}-2\sum_{\mbox{\scriptsize
odd}\,\,n_{x}}\bigl{[}J_{+}+r_{i}^{2}I_{+}\bigr{]}\bigg{\\}}.$ (20)

where, since the coincidence limit ${\bf x}={\bf x}^{\prime}$ has been taken,

$r_{1}=-n_{x}a,\quad r_{2}=r_{3}=0,\quad
r^{2}=r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=n_{x}^{2}a^{2},$ (21)

as follows from Eq. (13) in the coincidence limit.

###  IV.2 Indicators for the cylinder

We are looking for a local way to probe a putative time non-orientability of
spacetime. To this end, let us compare the above results for the twisted
cylinder $\mathbb{C}^{*2}\times\mathbb{E}^{2}$ manifold with those for its
time-orientable counterpart, the cylinder
$\mathbb{C}^{2}\times\mathbb{E}^{2}=\mathbb{R}\times\mathbb{S}^{1}\times\mathbb{E}^{2}$.
The indicators for the cylinder are given in Ref. [28] as141414In equations
(26) and (27) of Ref. [28] the choice $t_{0}=0$ was made. Also, what we call
here the cylinder is denoted there by $E_{16}$.

$\displaystyle I^{\mathbb{C}^{2}}_{v_{x}^{2}}({\bf
x},t,t_{0})=-\frac{q^{2}(t-t_{0})}{4\pi
m^{2}}\sum_{n_{x}}^{\prime}\frac{1}{r^{3}}\ln\frac{(r-t+t_{0})^{2}}{(r+t-t_{0})^{2}},$
(22) $\displaystyle I^{\mathbb{C}^{2}}_{v_{y}^{2}}({\bf
x},t,t_{0})=I^{\mathbb{C}^{2}}_{v_{z}^{2}}({\bf
x},t)=\frac{q^{2}(t-t_{0})}{8\pi
m^{2}}\sum_{n_{x}}^{\prime}\left[\frac{4(t-t_{0})}{r^{2}[(t-t_{0})^{2}-r^{2}]}+\frac{1}{r^{3}}\ln\frac{(r-t+t_{0})^{2}}{(r+t-t_{0})^{2}}\right],$
(23)

where $r=n_{x}a$.

###  IV.3 Locally probing time non-orientability

A significant difference between the case of the time non-orientable twisted
cylinder and that of time-orientable spacetimes is that the indicator (20)
depends both on $t$ and $t_{0}$, and not only on the difference $t-t_{0}$.
This is because the twisted cylinder $\mathbb{C}^{*2}\times\mathbb{E}^{2}$ is
not globally temporally homogeneous.

By means of a picture, we highlight the main consequences of the twisted
cylinder’s lack of global time homogeneity on the statistical indicator (20).
In Fig. 1 we show two components of indicator (20) as functions of $t,t_{0}$,
with the indicator values represented by colors (the $z$-component of the
indicator is not shown because it coincides with its $y$-component). Contour
lines of the indicator are also displayed. Figs. 1 (a) and (b) illustrate the
global temporal inhomogeneity of the twisted cylinder
$\mathbb{C}^{*2}\times\mathbb{E}^{2}$: the patterns encountered as one moves
horizontally along the $t$-axis are different for each fixed $t_{0}$. The
nontrivial contour curves are also quite different from those for
$\mathbb{C}^{2}\times\mathbb{E}^{2}$, which are the straight lines
$t-t_{0}=\mbox{const}$. There are periodic repetition patterns but with
changing absolute value of the indicators. The topological singularity
structure, indicated by the white straight lines, is also much richer than the
one for the time orientable case. For example, the vertical lines
$t=1/2,3/2,5/2,\ldots$ are singularities of topological nature that are not
present in the indicators for the time orientable spacetime
$\mathbb{C}^{2}\times\mathbb{E}^{2}$.

|
---|---
(a) | (b)
Figure 1: Two components of indicator (20) as functions of $t,t_{0}$. The
values of the indicator are represented by colors. Panel (a) shows the
$x$-component of the indicator (20), while panel (b) shows its $y$-component
(the $z$-component is not displayed because it equals the $y$-component).
Contour lines of the indicators are also exhibited. In these plots, $q,\,m$
and $a$ are all set to unity for qualitative reasons only. The white thin
lines are topological singularities that arise for special values of
$t,t_{0}$, such as $t-t_{0}=r$, $t+t_{0}=r$, $t=r/2$, $t_{0}=r/2$, where
$r=|n_{x}|a$. The structure of singularities of topological origin is much
richer than the one in the case of time-orientable spacetimes [21, 28, 29].

Figure 1 illustrates quite vividly how the global inhomogeneity of the twisted
cylider manifests itself through the statistical indicator (20). Next we study
how the indicator (20) for the twisted cylinder bahaves as a function of $t$
for fixed $t_{0}$. Figures 2 and 3 compare components of the indicator for
$\mathbb{C}^{*2}\times\mathbb{E}^{2}$ with those for its time orientable
counterpart $\mathbb{C}^{2}\times\mathbb{E}^{2}$ for two values of $t_{0}$.
Both for $t_{0}=0$ (Fig. 2) and $t_{0}=1.3$ (Fig. 3) there appears an
inversion pattern in the case of $\mathbb{C}^{*2}\times\mathbb{E}^{2}$,
roughly of the form $\cup$ followed by $\cap$. The absence of any inversion
pattern in the case of the time-orientable
$\mathbb{C}^{2}\times\mathbb{E}^{2}$ allows one to pinpoint the temporally
non-orientable case.

|
---|---
(a) | (b)
Figure 2: Comparison of two components of the indicator for the twisted cylinder $\mathbb{C}^{*2}\times\mathbb{E}^{2}$ with those for its time orientable counterpart $\mathbb{C}^{2}\times\mathbb{E}^{2}$. In both plots $t_{0}=0$. Panel (a) shows the $x$-component of the indicator for $\mathbb{C}^{*2}\times\mathbb{E}^{2}$ given by (20) plotted together with the $x$-component of the indicator for $\mathbb{C}^{2}\times\mathbb{E}^{2}$ given by (23). Panel (b) shows the same comparison for the $y$-component of the indicator. In these plots, $q,\,m$ and $a$ are all set to unity for qualitative reasons only. The solid curves for $\mathbb{C}^{*2}\times\mathbb{E}^{2}$ present an inversion pattern, roughly of the form $\cup$ followed by $\cap$, which is absent from the dashed curves for the time-orientable $\mathbb{C}^{2}\times\mathbb{E}^{2}$.  |
---|---
(a) | (b)
Figure 3: The same as in Fig. 2 for $t_{0}=1.3$. Again, the solid curves for
$\mathbb{C}^{*2}\times\mathbb{E}^{2}$ exhibit an inversion pattern, roughly of
the form $\cup$ followed by $\cap$, which is absent from dashed the curves for
its time-orientable counterpart $\mathbb{C}^{2}\times\mathbb{E}^{2}$.

##  V Summary of results and final comments

In relativistic cosmology and quantum field theory, spacetime is assumed to be
orientable as a topological manifold, and is additionally endowed with a
Lorentz metric, making it separately time and space orientable. The question
as to whether the laws of physics require that spacetime manifolds adhere to
these orientability assumptions are among the unsettled issues in this
framework. Although non-orientability is generally seen as an undesirable
feature, non-orientable Lorentzian spacetime manifolds are concrete
mathematical possibilities in physics at different scales. Therefore, the
possibility that spacetime is time non-orientable ought not to be jettisoned
forthwith. Strict orientability assumptions could potentially interfere with
the understanding of fundamental aspects of the interplay of physics and
topology. In fact, we do not know to what extent the topology of the
underlying spacetime manifold might encode, or whether it is necessary to
capture or express, basic features of the physical world in some scale. It is
conceivable that topological properties such as global homogeneity and
orientability of spacetime manifolds might be testable in both micro- and
macro-physical scales. Previous speculations on time orientability violation
have revolved about Gedanken experiments that would allegedly provide
signatures a putative time non-orientability of spacetime [67, 68]. These
ideas and other approaches to testing time orientability have been recently
subjected to a searching critique [69]. Here, however, we have settled on
probing the orientability of spacetime by means of measurable local physical
effects.

Inasmuch as the role played by time orientability is more clearly accessed in
static flat spacetime, whose dynamical degrees of freedom are frozen, in the
present paper, instead of the expanding Friedmann–Lemaître–Robertson–Walker
spacetime (FLRW), we have focused on the time orientability of Minkowski
spacetime, leaving for a forthcoming article [70] some important related
questions regarding the so-called arrow of time — observed time asymmetry in
macrophysics and in the evolution of the universe, despite the time reversal
invariance of the fundamental laws of physics. This includes, for example,
whether temporal orientability of the FLRW universe can be probed and whether
one can understand in a far-reaching context the several existing arrows of
time [12, 16, 17].

In this paper we have argued that a presumed time non-orientability of
Minkowski empty spacetime can be locally probed through physical effects
associated with quantum vacuum electromagnetic fluctuations. To this end, we
have studied the stochastic motions of a charged particle under these
fluctuations in Minkowski spacetime manifold, $\mathcal{M}_{4}$, endowed with
a time non-orientable topology (the twisted cylinder151515We recall that a
visualization of the twisted cylinder is provided by the Möbius band, a
representation that is somewhat inaccurate but can be realized inthree-
dimensional space. $\mathcal{M}_{4}=\mathbb{C}^{*2}\times\mathbb{E}^{2}$) and
its time orientable counterpart (the cylinder
$\mathcal{M}_{4}=\mathbb{C}^{2}\times\mathbb{E}^{2}$) — see Tables 1 and 2. We
have found that the statistical topological indicator given by Eq. (14) is
suitable to bring out the time non-orientability in the twisted cylinder case.
Accordingly, we have derived analytical expressions for the statistical
orientability indicator corresponding to Minkowski spacetime manifold
$\mathcal{M}_{4}$ equipped with time non-orienbtable and its time orientable
counterpart.

The chief conclusion of this work is reached through comparisons between the
stochastic motions of a charged test particle in Minkowski spacetime endowed
with each of the two spacetime topologies. Since the expressions for the
orientability indicators — Eqs. (20), (22) and (23) — are too involved for a
direct comparison, to demonstrate our main result, which is ultimately stated
in terms of patterns of curves for the orientability indicators, we have
performed numerical computations and plotted figures for the components of our
statistical orientability indicator.

Figure 1 aims to illustrate the topological temporal inhomogeneity for
$\mathcal{M}_{4}$ corresponding to the twisted cylinder
$\mathbb{C}^{*2}\times\mathbb{E}^{2}$, a feature not shared by the cylinder
$\mathbb{C}^{2}\times\mathbb{E}^{2}$. The patterns encountered are different
for each fixed $t_{0}$, and the closed contour curves are also quite different
from those for $\mathbb{C}^{2}\times\mathbb{E}^{2}$, which are straight
$45$-degree lines. The topological singularity structure is also markedly
richer than the one for the time orientable case.

However, the answer to the central question of the paper, namely how to
locally probe the time orientability of Minkowski intrinsically, is
accomplished by comparing the time evolution of the orientability indicators
$\mbox{\large$I$}_{v^{2}_{i}}^{\mathbb{C}^{*2}}$ and
$\mbox{\large$I$}_{v^{2}_{i}}^{\mathbb{C}^{2}}$. Indeed, Figs. 2 and 3 show
that it is possible to locally unveil time non-orientability through the
inversion pattern of curves of the time non-orientability indicator for a
charged particle under quantum vacuum electromagnetic fluctuations. This
inversion pattern is a signature of time non-orientability and is absent in
the time orientable case.

It should be noticed that the time non-orientability considered in this paper
is of topological origin. It is different from the existence of closed
timelike curves in Gödel and other spacetime solutions of Einstein’s equations
[71]. In the Gödel model, for example, gravity tilts the local light cones
over extended spacetime regions so as to allow continuous closed timelike
curves — there is no periodic time inversion of topological origin.

To summarize, the main result of our analysis is that it may be feasible to
look into a conceivable topological time non-orientability of Minkowski empty
spacetime by measurable local physical effects associated with quantum vacuum
electromagnetic fluctuations.

###### Acknowledgements.

M.J. Rebouças acknowledges the support of FAPERJ under a CNE E-26/202.864/2017
grant, and thanks CNPq for the grant under which this work was carried out. We
are grateful to A.F.F. Teixeira for a careful reading of the manuscript and
discussions.

## References

  * [1]
  * [2] S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973).
  * [3] R. Geroch and G.T. Horowitz, Global structure of spacetimes in General relativity: An Einstein centenary survey, pp. 212-293, Eds. S. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979).
  * [4] E. Minguzzi, Lorentzian causality theory, Living Rev Relativ 22, 3 (2019).
  * [5] J.R. Weeks, The Shape of Space, 3rd ed. (CRC Press, Boca Raton, FL, 2020).
  * [6] Wikipedia: Orientability, available at: https://en.wikipedia.org/wiki/Orientability (Accessed: 21 August 2022).
  * [7] Ya. B. Zeldovich and I.D. Novikov, JETP Letters 6, 236 (1967).
  * [8] R. Penrose, The Structure of Space-Time, in BattelIe Rencontres in Mathematics and Physics: Seattle, 1967, C. M. DeWitt and J. A. Wheeler, Eds. (Benjamin, New York, 1968).
  * [9] R. Penrose and W. Rindler, Spinors and Space-time, Volume 1, Two-spinor calculus and relativistic fields (Cambridge University Press, Cambridge, 1986).
  * [10] R. Geroch, J. Math. Phys. 9, 1739 (1968).
  * [11] R. Geroch, J. Math. Phys. 9, 343 (1970).
  * [12] G.F.R. Ellis and B. Drossel, Found. Phys. 50, 161 (2020).
  * [13] M. Castagnino, L. Lara and O. Lombardi, Class. Quantum Grav. 20, 369 (2003).
  * [14] H. Reichenbach The Direction of Time, University of California Press, Berkeley 1971.
  * [15] P.C.W. Davies, The Physics of Time Asymmetry, University of California Press, Berkeley- Los Angeles, 1974.
  * [16] J. Barbour, The End of Time: The Next Revolution in Physics, Oxford University Press, Oxford (2001).
  * [17] C. Rovelli, The Order of Time, Riverhead Books, USA 2018.
  * [18] H. Yu and L.H. Ford, Phys. Rev. D 70, 065009 (2004).  DOI 10.1103/PhysRevD.70.065009.  URL http://dx.doi.org/10.1103/PhysRevD.70.065009
  * [19] G. Gour and L. Sriramkumar, Found. Phys. 29, 1917 (1999).
  * [20] M.T. Jaekel and S. Reynaud, Quant. Opt. 4, 39 (1992).
  * [21] C.H.G. Bessa and M.J. Rebouças, Class. Quant. Grav. 37(12), 125006 (2020).  DOI 10.1088/1361-6382/ab848a.  URL http://dx.doi.org/10.1088/1361-6382/ab848a
  * [22] H.w. Yu and J. Chen, Phys. Rev. D 70, 125006 (2004).  DOI 10.1103/PhysRevD.70.125006.  URL https://journals.aps.org/prd/abstract/10.1103/PhysRevD.70.125006
  * [23] L.H. Ford, Int. J. Theor. Phys. 44, 1753 (2005).  DOI 10.1007/s10773-005-8893-z.  URL http://dx.doi.org/10.1007/s10773-005-8893-z
  * [24] H.w. Yu and J. Chen, P.x. Wu, JHEP 02, 058 (2006).  DOI 10.1088/1126-6708/2006/02/058.  URL http://dx.doi.org/10.1088/1126-6708/2006/02/058
  * [25] M. Seriu and C.H. Wu, Physical Review A 77(2) (2008).  DOI 10.1103/physreva.77.022107.  URL http://dx.doi.org/10.1103/PhysRevA.77.022107
  * [26] V. Parkinson and L.H. Ford, Phys. Rev. A 84, 062102 (2011).  DOI 10.1103/PhysRevA.84.062102.  URL http://dx.doi.org/10.1103/PhysRevA.84.062102
  * [27] V.A. De Lorenci, C.C.H. Ribeiro and M.M. Silva, Phys. Rev. D 94(10), 105017 (2016).  DOI 10.1103/PhysRevD.94.105017.  URL http://dx.doi.org/10.1103/PhysRevD.94.105017
  * [28] N.A. Lemos, M.J. Rebouças, Eur. Phys. J. C 81(7), 618 (2021).  DOI 10.1140/epjc/s10052-021-09426-9.  URL http://dx.doi.org/10.1140/epjc/s10052-021-09426-9
  * [29] N.A. Lemos, D. Müller and M.J. Rebouças, Phys. Rev. D 106, 023528 (2022)
  * [30] J.A. Wolf, Spaces of Constant Curvature (McGraw-Hill, New York, 1967).
  * [31] W.P. Thurston, Three-Dimensional Geometry and Topology. Vol.1, Edited by Silvio Levy (Princeton University Press, Princeton, 1997).
  * [32] C. Adams and J. Shapiro, American Scientist 89, 443 (2001).
  * [33] B. Cipra, What’s Happening in the Mathematical Sciences (American Mathematical Society, Providence, RI, 2002).
  * [34] A. Riazuelo, J. Weeks, J.P. Uzan, R. Lehoucq and J.P. Luminet, Phys. Rev. D 69, 103518 (2004).
  * [35] H. Fujii and Y. Yoshii, Astron. Astrophys. 529, A121 (2011).
  * [36] G.F.R. Ellis, Gen. Rel. Grav. 2, 7 (1971).  DOI 10.1007/BF02450512.  URL http://dx.doi.org/10.1007/BF02450512
  * [37] M. Lachièze-Rey and J.P. Luminet, Phys. Rep. 254(3), 135 (1995).  DOI https://doi.org/10.1016/0370-1573(94)00085-H.  URL https://www.sciencedirect.com/science/article/pii/037015739400085H
  * [38] G.D. Starkman, Class. Quant. Grav. 15, 2529 (1998).  DOI 10.1088/0264-9381/15/9/002.  URL http://dx.doi.org/10.1088/0264-9381/15/9/002
  * [39] J. Levin, Phys. Rep. 365(4), 251 (2002).  DOI 10.1016/S0370-1573(02)00018-2.  URL https://www.sciencedirect.com/science/article/pii/S0370157302000182
  * [40] M.J. Rebouças and G.I. Gomero, Braz. J. Phys. 34, 1358 (2004).  DOI 10.1590/S0103-97332004000700012.
URL https://www.scielo.br/j/bjp/a/DPsmZZVfnnqxqb4q8NTPf8q/?lang=en&format=html

  * [41] M.J. Rebouças, A Brief Introduction to Cosmic Topology, in Proc. XIth Brazilian School of Cosmology and Gravitation, eds. M. Novello and S.E. Perez Bergliaffa (Americal Institute of Physics, Melville, NY, 2005).
  * [42] J.P. Luminet, Universe 2(1), 1 (2016).  DOI 10.3390/universe2010001.  URL http://dx.doi.org/10.3390/universe2010001
  * [43] P.A.R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A.J. Banday, et al., Astronom. Astrophys. 571, A16 (2014).  DOI 10.1051/0004-6361/201321591.  URL http://dx.doi.org/10.1051/0004-6361/201321591
  * [44] P.A.R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A.J. Banday, R.B. Barreiro, J.G. Bartlett, et al., Astronom. Astrophys. 594, A13 (2016).  DOI 10.1051/0004-6361/201525830.  URL http://dx.doi.org/10.1051/0004-6361/201525830
  * [45] N.J. Cornish, D.N. Spergel, G.D. Starkman and E. Komatsu, Phys. Rev. Lett. 92, 201302 (2004).  DOI 10.1103/PhysRevLett.92.201302.  URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.201302
  * [46] J. Shapiro Key, N.J. Cornish, D.N. Spergel and G.D. Starkman, Phys. Rev. D 75, 084034 (2007).  DOI 10.1103/PhysRevD.75.084034.  URL https://journals.aps.org/prd/abstract/10.1103/PhysRevD.75.084034
  * [47] P. Bielewicz and A.J. Banday, Mon. Not. Roy. Astron. Soc. 412, 2104 (2011).  DOI 10.1111/j.1365-2966.2010.18057.x.  URL https://academic.oup.com/mnras/article/412/3/2104/1058423
  * [48] P.M. Vaudrevange, G.D. Starkman, N.J. Cornish and D.N. Spergel, Phys. Rev. D 86, 083526 (2012).  DOI 10.1103/PhysRevD.86.083526.  URL http://dx.doi.org/10.1103/PhysRevD.86.083526
  * [49] R. Aurich and S. Lustig, Mon. Not. Roy. Astron. Soc. 433, 2517 (2013).  DOI 10.1093/mnras/stt924.  URL https://academic.oup.com/mnras/article/433/3/2517/1235728
  * [50] G.I. Gomero, B. Mota and M.J. Rebouças, Phys. Rev. D 94(4), 043501 (2016).  DOI 10.1103/PhysRevD.94.043501.  URL http://dx.doi.org/10.1103/PhysRevD.94.043501 .
  * [51] M.A. Henle, Combinatorial Introduction to Topology, pp. 100$-$111\. Dover,New York (1994).
  * [52] J. Stillwell, Geometry of Surfaces, Springer-Verlag New York, Inc. (1992).
  * [53] B.S. DeWitt, C.F. Hart and C.J. Isham, Physica 96A, 197 (1979).
  * [54] J.S. Dowker and R. Critchley, J. Phys. A 9, 535 (1976).
  * [55] P.M. Sutter and T. Tanaka, Phys. Rev. D 74, 024023 (2006).
  * [56] M.P. Lima and D. Müller, Class. Quant. Grav. 24, 897 (2007).
  * [57] D. Müller, H.V. Fagundes and R. Opher, Phys. Rev. D 66, 083507 (2002).
  * [58] D. Müller, H.V. Fagundes and R. Opher, Phys. Rev. D 63, 123508 (2001).
  * [59] D. Martin, Manifold Theory: Introduction for Mathematical Physicists (Horwood Publishing, West Sussex, 2002).
  * [60] B. O’Neill, Semi-Riemannian Geometry with application to Relativity, Academic Press (1983).
  * [61] F.J. Palomo and A. Romero, Certain actual topics on modern Lorentzian geometry in Handbook of differential geometry, vol II, p.pp 516-546 (2006). Eds. F.J.E. Dillen and L.C. A Verstraelen.
  * [62] N.D. Birrel and P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982).
  * [63] R. Lehoucq, M. Lachièze-Rey and J.-P. Luminet, Astron. Astrophys. 313, 339 (1996).
  * [64] G.I. Gomero, M.J. Rebouças and A.F.F. Teixeira, Class. Quantum Grav. 18, 1885 (2001).
  * [65] G.I. Gomero, A.F.F. Teixeira, M.J. Rebouças and A. Bernui, Int. J. Mod. Phys. D 11, 869 (2002).
  * [66] M.J. Rebouças Int. J. Mod. Phys. D 9 561 (2000).
  * [67] M. Hadley, Class. Quantum Grav. 19, 4565 (2002).
  * [68] M. Hadley, Testing the Orientability of Time. Preprints 2018040240 (2018).
  * [69] M. Bielińska and J. Read, Testing Spacetime Orientability, Foundations of Physics (2022) (forthcoming).
  * [70] N.A. Lemos, D. Müller and M.J. Rebouças, Probing time orientability of Friedmann-Robertson-Walker spacetime (in preparation).
  * [71] K. Gödel, Rev. Mod. Phys. 21, 447 (1949).