# On Twisted Spacetimes: a new class of Galilean cosmological models

Daniel de la Fuente
Departamento de Matemáticas
Universidad de Oviedo, 33003 Gijón, Spain
<EMAIL_ADDRESS>Rafael María Rubio
Departamento de Matemáticas
Universidad de Córdoba, 14071 Córdoba, Spain
<EMAIL_ADDRESS>Jose Torrente
Departamento de Matemáticas
Universidad de Córdoba, 14071 Córdoba, Spain
<EMAIL_ADDRESS>
(February 2024)

###### Abstract

Within the generalized Newton-Cartan theory, Galilean Twisted spacetimes are
introduced as dual models of the well-known relativistic twisted spacetimes.
As a natural generalization, torqued vector fields in Galilean spacetimes are
defined, showing that the local structure of a Galilean spacetime admitting a
timelike torqued vector field is given by a Twisted spacetime. In addition,
several results assuring the global splitting as Twisted spacetime are
obtained. On the other hand, completeness of free falling observers is
studied, as well as general geodesic completeness.

##  1 Introduction

The geometric formulation of Newton’s theory of gravity began with E. Cartan
in the early twentieth century [10, 11]. But it was only in the second half of
the century that the Newton-Cartan theory was given significant attention by
various authors. In the 1960s Malament studied the Lorentzian and Galilean
groups, proving that Newtonian gravity constitutes a limit of General
Relativity and how free particles are represented by geodesic curves of
certain connections in both theories. Additionally, Trautman showed the
existence of a symmetric connection in both theories [39], arising due to the
local nature of physical laws and simultaneously encoding the inertia
principle and gravitational forces. Concretely, classical Newtonian
gravitation is formulated as a covariant theory, showing that certain results
previously considered characteristic or singular of the theory of Relativity
are also shared by the (geometric) Newton-Cartan Gravitational Theory. In
fact, Newtonian gravity is explained by the curvature of a connection in the
spacetime, although in this case it no longer arises from any semi-Riemannian
metric. Furthermore, in the geometric formulation of Newtonian Gravity Theory,
the spacetime structure is dynamic in the sense that it participates in the
development of physics rather than being a fixed backdrop (see [24] and
references therein).

Later in that century, several steps were taken towards the generalization of
this theory, such as the introduction of new Newtonian models satisfying the
cosmological principle [30], obtaining the Galilean analogue of relativistic
Robertson-Walker spacetimes. In [18], Newtonian theory achieves the status of
a gauge theory and is shown to be a limit of the Einstein-Klein-Gordon theory.
This view of the theory as a limit of General Relativity continued to develop
during these years [22], concluding also that the results obtained from
Newton-Cartan cosmology were similar on not too large scales to those derived
from Einstein’s theory [34, 35].

In the last few decades, the simplicity of certain Newtonian models in Newton-
Cartan theory, in contrast to the computational complexity of General
Relativity, has led to new applications of this non-relativistic theory in
Cosmology [8], Hydrodynamics [19], quantum collapse [32], AdS/CFT
correspondence [23], condensed matter systems [9], quantum Hall effect [20]
and other related phenomena. The many applications of Newton-Cartan theory
have motivated a review of the geometric structures associated with
generalized Newton-Cartan theory [7] and new extensions such as Newton-Cartan
Supergravity [3] and non-relativistic Strings [2, 5]. Furthermore, new
Newtonian models have been obtained, including Galilean Generalized Robertson-
Walker spacetimes [16], the Newtonian Gödel spacetime [14], standard
stationary Galilean spacetimes [15], as well as the embedding of these
geometries in 5-dimensional Lorentzian manifolds [4, 27, 28, 29].

In this manuscript, a wide family of Galilean spacetimes is presented, the
Galilean Twisted spacetimes (Sect. 3), which constitute a generalization of
the classical Galilean Robertson-Walker spacetimes (RW), and even of the
Galilean Generalized Robertson-Walker spacetimes (GGRW). In fact, Definition
3.1 states

> Given a real interval $I\subseteq\mathbb{R}$, a Riemannian manifold $(F,h)$
> and a positive smooth function $f\in C^{\infty}(I\times F)$, a Galilean
> Twisted (GT) spacetime is given by the tuple $(I\times
> F,\Omega=\mathrm{d}t,g=f^{2}\,\pi_{F}^{*}h,\nabla)$ where $t\coloneq\pi_{I}$
> and $\nabla$ is the only symmetric Galilean connection on $I\times F$ such
> that $\nabla_{\operatorname{\partial_{t}}}\operatorname{\partial_{t}}=0$ and
> $\operatorname{Rot}\operatorname{\partial_{t}}=0$.

According to the cosmological principle, RW spacetimes model a spatially
homogeneous and isotropic universe composed of matter (galaxies) subject only
to the action of its own gravity. Thus, the “absolute space at each moment in
absolute time” must be locally Euclidean with constant sectional curvature.
Furthermore, as large-scale experimental measurements (redshift) indicate, the
expansion is assumed to be homogeneous, i.e., the expansion rate is the same
at all points at a fixed (absolute) time. However, while the hypothesis of
spatial homogeneity and isotropy may be a reasonable first approximation to
the large scale structure of the universe, it could not be appropriate when we
consider a more accurate scale. This led to the introduction of the family of
GGRWs spacetimes, where the absolute space is modelled by an arbitrary
Riemannian manifold, which does not necessarily have to be of constant
curvature. Finally, the Twisted spacetimes presented here assume a non-
homogeneous expansion rate, which means that it depends not only on the moment
in time but also on the point in space. In other words, the “Hubble constant”
is no longer constant.

Twisted products have been studied in the relativistic setting [12, 13, 25,
36]. They have been analyzed in relation to geometric models of fluids,
resulting in a characterization in terms of the stress-energy tensor of an
imperfect fluid [26] and obtaining the properties of static perfect fluids in
Twisted Lorentzian spacetimes [21]. Similar results could be explored in the
non-relativistic setting introduced here.

A Galilean Twisted (GT) spacetime possesses an infinitesimal symmetry given by
the existence of a timelike irrotational and (spatially) conformally
Leibnizian field of observers. Several geometrical properties, as well as
physical interpretations of this family of spacetimes, are given in Sections 4
and 5. We concretely focus on the completeness of its free falling observers.
Our main result (Theorem 4.8) asserts

> _L_ et $(F,h)$ be a complete Riemannian manifold and let $f\in
> C^{\infty}(\mathbb{R}\times F)$ be a positive function. If $\inf f>0$ and
> $\inf(f_{t}/f)>-\infty$ along finite times, then the GT spacetime
> $(M=\mathbb{R}\times
> F,\Omega=\mathrm{d}\pi_{\mathbb{R}},g=f^{2}\,\pi_{F}^{*}h,\nabla)$ is
> geodesically complete.

From a physical perspective, this result is relevant since the completeness of
the timelike trajectories is crucial in order to have a consistent theory
(without observers or with observers that disappear or appear suddenly, it
would not be possible to do Physics).

Section 6 is devoted to the study of Galilean spacetimes admitting a torqued
vector field (see Definition 6.2), which is closely related to the previously
introduced family of irrotational conformally Leibnizian vector fields [16].
The principal result in this section is Theorem 6.5, which gives

> Let $(M,\Omega,g,\nabla)$ be a Galilean spacetime. If it admits a torqued
> vector field $K\in\Gamma(TM)$, then for each $p\in M$, there exists an open
> neighborhood of p, $\mathcal{U}$, and a Galilean diffeomorphism $\Psi\colon
> N\to\mathcal{U}$, where $N$ is a GT spacetime.

Finally, Section 7 is devoted to face some splitting type problems, i.e.,
under which geometrical assumptions a Galilean spacetime decomposes globally
into a Galilean Twisted spacetime. Theorem 7.5 states

> Let $(M,\Omega,g,\nabla)$ be a simply connected Galilean spacetime. It
> admits a global decomposition as a GT spacetime if and only if there exists
> a Galilean torqued vector field $K\in\Gamma(TM)$ such that the associated
> field of observers, $Z\coloneq K/\Omega(K)$, is a uniform global generator.

Moreover, if the leaves of the foliation induced by $\Omega$ are compact, we
can ensure again a global splitting result (Theorem 7.6).

##  2 Premilinaries

Let $M$ be a $(n+1)$-dimensional smooth manifold. As usual, it is assumed
connected, Hausdorff and paracompact. A Leibnizian spacetime [7] is a triad
$(M,\Omega,g)$ where $(\Omega,g)$ is a Leibnizian structure defined by a
global non-vanishing one-form on $M$, $\Omega\in\Lambda^{1}(M)$, and a
positive definite metric $g$ on the vector bundle defined by the kernel of
$\Omega$. Concretely, if $\operatorname{\text{An}\Omega}\coloneq\\{v\in
TM\;:\;\Omega(v)=0\\}$ is the n-distribution induced by $\Omega$, then $g$ is
a smooth, bilinear, symmetric and positive definite 2-covariant tensor

$g\colon\Gamma(\operatorname{\text{An}\Omega})\times\Gamma(\operatorname{\text{An}\Omega})\to
C^{\infty}(M)\,,$

where $\Gamma(\operatorname{\text{An}\Omega})$ denote the subset of sections
$V\in\Gamma(TM)$ such that $V_{p}\in\operatorname{\text{An}\Omega}$, $\forall
p\in M$.

Given a Leibnizian spacetime $(M,\Omega,g)$, the points $p\in M$ are called
events. If $p\in M$, $\Omega_{p}$ is known as the absolute clock at $p$ and
$\big{(}\text{An}(\Omega_{p}),g_{p}\big{)}$ is called the absolute space at
$p$ (see for example [6]). Adopting relativistic terminology, a tangent vector
$v\in T_{p}M$ is said to be spacelike (resp. timelike) if $\Omega_{p}(v)=0$
(resp. $\Omega_{p}(v)\neq 0$). Moreover, a timelike vector $v$ is called
future-pointing (resp. past-pointing) if $\Omega_{p}(v)>0$ (resp.
$\Omega_{p}(v)<0$).

On the other hand, an observer is defined as a smooth curve $\gamma\colon
I\subseteq\mathbb{R}\to M$, being $I$ an interval, such that its velocity is a
normalized timelike future-pointing vector (standard timelike unit [6]), i.e.,
$\Omega_{\gamma(s)}(\gamma^{\prime}(s))=1,\,\forall s\in I$. The parameter
$s\in I$ is the proper time of the observer $\gamma$. A field of observer
$Z\in\Gamma(TM)$ is a smooth vector field whose integral curves are observers,
equivalently, $\Omega(Z)=1$.

In this Leibnizian setting, the synchronizability of observers is intrinsic to
the Leibnizian structure, in contrast to the relativistic notions, which is
relative to each field of observers. In fact, if
$\Omega\wedge\mathrm{d}\Omega=0$ (i.e., the smooth distribution
$\operatorname{\text{An}\Omega}$ is integrable), the spacetime is called
locally synchronizable. In this case, Frobenius Theorem (see [40]) states that
there exists a foliation consisting of spacelike hypersurfaces. Moreover,
locally $\Omega=\lambda\mathrm{d}T$, for certain smooth functions $\lambda>0$
and $T$, and the hypersurfaces of the foliation are exactly
$\\{T=\text{constant}\\}$. Thus, every observer can locally synchronize its
proper time with the so-called "compromise time" $T$. If the absolute clock is
closed, $\mathrm{d}\Omega=0$, the Leibnizian spacetime is said to be proper
time locally synchronizable. Under this assumption, locally we can express
$\Omega=\mathrm{d}T$, for certain smooth function $T$, and the observers can
synchronize their proper times up to a constant. In the case
$\Omega=\mathrm{d}T$, for $T\in C^{\infty}(M)$, we will suppose every observer
is parameterized by the absolute time $T$. Notice that, if $M$ is simply-
connected and $\Omega$ is closed, we can ensure the existence of an absolute
time function making use of the Poincaré Lemma (see, for instance, [37]).

Let $(M,\Omega,g)$ and $(\widetilde{M},\widetilde{\Omega},\widetilde{g})$ be
two Leibnizian spacetimes. A diffeomorphism $\lambda\colon M\to\widetilde{M}$
is called Leibnizian if it preserves the absolute clock and space, that is,
$\lambda^{*}\widetilde{\Omega}=\Omega$ and $\lambda^{*}\widetilde{g}=g$.

To give complete physical meaning to a Leibnizian spacetime, the principle of
inertia has to be included via an affine connection $\nabla$. It is necessary
to consider $\nabla$ compatible with the absolute clock $\Omega$ and the
absolute space $(\operatorname{\text{An}\Omega},g)$, i.e.,

  1. (1)

$\nabla\Omega=0$, equivalently, $X(\Omega(Y))=\Omega(\nabla_{X}Y)$, for all
$X,Y\in\Gamma(TM)$,

  2. (2)

$\nabla g=0$, that is, $Z(g(V,W))=g(\nabla_{Z}V,W)+g(V,\nabla_{Z}W)$, for all
$Z\in\Gamma(TM)$ and $V,W\in\Gamma(\operatorname{\text{An}\Omega})$.

Note that, thanks to condition (1), the right-hand side of item (2) is well
defined. Any such $\nabla$ is called a Galilean connection. Consequently, the
Leibnizian spacetime does not determine a canonical Galilean connection. The
tuple $(M,\Omega,g,\nabla)$ is known as a Galilean spacetime. Recall that if
its torsion tensor vanishes, a connection is said to be symmetric, i.e.,

$\text{Tor}(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y]=0,\quad\forall
X,Y\in\Gamma(TM).$

From a physical viewpoint, it is often convenient to choose a symmetric
connection because it is determined by the associated geodesics [38, Add. 1,
Ch. 6]. Observe that the closeness of the absolute clock and the torsion
tensor are related [7, Lemma 13]

$\Omega\circ\operatorname{Tor}=\mathrm{d}\Omega\,.$ (1)

From now on, we will only consider symmetric Galilean connections, then using
equation (1), $\mathrm{d}\Omega=0$.

Given two Galilean spacetimes $(M,\Omega,g,\nabla)$ and
$(\widetilde{M},\widetilde{\Omega},\widetilde{g},\widetilde{\nabla})$, a
diffeomorphism $\lambda\colon M\to\widetilde{M}$ is called Galilean if it is
Leibnizian and additionally $\lambda^{*}\widetilde{\nabla}=\nabla$, i.e.,
$\widetilde{\nabla}_{\mathrm{d}\lambda(X)}\mathrm{d}\lambda(Y)=\nabla_{X}Y$,
for all $X,Y\in\Gamma(TM)$.

On the other hand, given a Galilean spacetime $(M,\Omega,g,\nabla)$ and a
field of observers $Z\in\Gamma(TM)$, it is well-known that the Galilean
connection $\nabla$ can be characterized using $Z$. The gravitational field
induced by $\nabla$ in $Z$ is the spacelike vector field
$\mathcal{G}=\nabla_{Z}Z$ and similarly, the vorticity or Coriolis field is
the 2-form $\omega=\tfrac{1}{2}\operatorname{Rot}(Z)$. The space of symmetric
Galilean connections is mapped bijectively onto
$\Gamma(\operatorname{\text{An}\Omega})\times\Lambda^{2}(\operatorname{\text{An}\Omega})$,
giving $\nabla\mapsto(\mathcal{G}_{Z},\omega_{Z})$ (see [7, Cor. 28]).
Moreover, the converse is obtained through a formula ’à la Koszul’. If we
consider

$\nabla_{X}Y=P^{Z}\left(\nabla_{X}Y\right)+X(\Omega(Y))Z,\quad\forall
X,Y\in\Gamma(TM)$

where $P^{Z}X=X-\Omega(X)Z$ is the natural spacelike projection for $Z$, then
for each $V\in\Gamma(\operatorname{An}(\Omega))$ [7, eq. (13)],

$\displaystyle 2g\left(P^{Z}\left(\nabla_{X}Y\right),V\right)=$ $\displaystyle
X\left(g\left(P^{Z}Y,V\right)\right)+Y\left(g\left(P^{Z}X,V\right)\right)-V\left(g\left(P^{Z}X,P^{Z}Y\right)\right)$
(2)
$\displaystyle+2\Omega(X)\Omega(Y)g\left(\mathcal{G}^{Z},V\right)+2\Omega(X)\,\omega_{Z}(P^{Z}Y,V)+2\Omega(Y)\,\omega_{Z}(P^{Z}X,V)$
$\displaystyle+\Omega(X)\left(g\left(\big{[}Z,P^{Z}Y\big{]},V\right)-g\left([Z,V],P^{Z}Y\right)\right)$
$\displaystyle-\Omega(Y)\left(g\left(\big{[}Z,P^{Z}X\big{]},V\right)+g\left([Z,V],P^{Z}X\right)\right)$
$\displaystyle+g\left(\big{[}P^{Z}X,P^{Z}Y\big{]},V\right)-g\left(\big{[}P^{Z}Y,V\big{]},P^{Z}X\right)-g\left(\big{[}P^{Z}X,V\big{]},P^{Z}Y\right).$

###  2.1 Spatially conformally Leibnizian spacetimes

Let us recall the notion of spatially conformally Leibnizian vector field,
which appears in several classes of cosmological models in the context of the
generalized Newton-Cartan theory (see [16]) and will be key to obtaining some
of our results.

######  Definition 2.1.

Let $(M,\Omega,g)$ be a Leibnizian spacetime and $K$ a vector field satisfying

$\Omega([K,V])=0\quad\mathrm{for}\;\mathrm{all}\quad
V\in\Gamma(\mathrm{An}(\Omega)).$ (3)

The vector field $K$ is called _spatially conformally Leibnizian_ if the Lie
derivative of the absolute space metric satisfies

${\cal L}_{{}_{K}}g=2\rho\,g,$ (4)

for some smooth function $\rho\in C^{\infty}(M)$. The function $\rho$ is
called conformal factor of $g$ associated to $K$.

Notice that condition (3) ensures that the previous notion is well defined. As
a direct consequence, we have

$K(g(V,W))=2\rho\,g(V,W)+g([K,V],W)+g([K,W],V),\,$ (5)

for all $V,W\in\Gamma(\operatorname{\text{An}\Omega})$.

The next result provides another condition to guarantee that (3) holds for a
vector field.

######  Proposition 2.2.

Let $(M,\Omega,g,\nabla)$ be a Galilean spacetime with symmetric connection
$\nabla$. Then, a vector field $K$ satisfies equation (3) if and only if the
function $\Omega(K)$ is spatially invariant, i.e., $V(\Omega(K))=0$, $\forall
V\in\Gamma\big{(}\mathrm{An}(\Omega)\big{)}$.

Finally, $K\in\Gamma(TM)$ is called a spatially Leibnizian vector field if the
conformal factor $\rho$ vanishes identically, or equivalently, if its local
flows $\Phi_{s}$ preserve the absolute space, i.e., $\Phi_{s}^{*}g=g$. If, in
addition, they also preserve the absolute clock, i.e.,
$\Phi_{s}^{*}\Omega=\Omega$, then $K$ is called Leibnizian.

###  2.2 Spacelike differential map and spacelike gradient

To conclude the preliminaries section, let us introduce some notions that will
be used later. Given a smooth function $\varphi\in C^{\infty}(M)$ on a
Leibnizian spacetime $(M,\Omega,g)$, we define the spacelike gradient of
$\varphi$, as the section
$\operatorname{grad}\varphi\in\Gamma(\operatorname{\text{An}\Omega})$,
determined by

$g(\operatorname{grad}\varphi,V)=\mathrm{d}\varphi(V),\quad\forall
V\in\Gamma(\operatorname{\text{An}\Omega})\,.$

Implicitly $\Omega(\operatorname{grad}\varphi)=0$. This notion is a
generalization of the gradient in semi-Riemannian manifolds.

When a smooth manifold $M$ splits as a product $M=I\times F$, for
$I\subseteq\mathbb{R}$ an open interval and $F$ certain smooth manifold, the
differential of a function $\varphi\in C^{\infty}(I\times F)$ can be
canonically split. We can identify $T_{(t,p)}(I\times F)\equiv T_{t}I\times
T_{p}F\equiv\mathbb{R}\times T_{p}F$. Then, define

$\begin{split}\varphi_{t}\in C^{\infty}(I\times
F),\quad{}&\varphi_{t}(t,p)\coloneq\mathrm{d}\varphi_{(t,p)}(\operatorname{\partial_{t}})\,,\\\
\mathrm{d}^{F}\varphi_{(t,p)}\colon
T_{p}F\to\mathbb{R},\quad{}&\mathrm{d}^{F}\varphi_{(t,p)}(v)\coloneq\mathrm{d}\varphi_{(t,p)}(\overline{v})\,,\end{split}$

where $\operatorname{\partial_{t}}$ is the canonical vector field associated
to the projection onto the first component $t=\pi_{I}$ and
$\overline{v}\in\mathbb{R}\times T_{p}F$ is the horizontal lift of a vector
$v\in T_{p}F$. If $(F,h)$ is a Riemannian manifold, the usual Riemannian
gradient $\operatorname{grad}^{h}$ applies. Thus, we will denote
$\mathrm{d}^{h}=\mathrm{d}^{F}$ to unify the notation although the
differential map only depends on the differential structure. From now on, the
horizontal lift of a vector field $W\in\Gamma(TF)$ to a product $I\times F$
will be denoted by $\overline{W}\in\Gamma(T(I\times
F))\equiv\Gamma(\mathbb{R}\times TF)$. Moreover, the induced differential on
$F$ can be lifted as

$\overline{\mathrm{d}^{h}}\varphi_{(t,p)}\colon\mathbb{R}\times
T_{p}F\to\mathbb{R},\quad\overline{\mathrm{d}^{h}}\varphi_{(t,p)}(a\operatorname{\partial_{t}}+\overline{v})\coloneq\mathrm{d}^{h}\varphi_{(t,p)}(v)\,,$

for all $a\in\mathbb{R}$ and $v\in T_{p}F$.

On the other hand, we say that $\varphi\in C^{\infty}(I\times F)$ is bounded
from below (resp. from above) along finite times if,
$\forall[s_{1},s_{2}]\subset I$, there exists a constant
$C_{s_{1},s_{2}}\in\mathbb{R}$ such that $\varphi>C_{s_{1},s_{2}}$ (resp.
$\varphi<C_{s_{1},s_{2}}$) in $[s_{1},s_{2}]\times F$. A smooth function
$\varphi$ is bounded along finite times if it is bounded from below and from
above along finite times. Obviously, $\inf\varphi>A$, $A\in\mathbb{R}$, along
finite times is equivalent to $\varphi$ being bounded from below along finite
times for constants greater than $A$.

##  3 Galilean Twisted Spacetimes

We introduce a new class of Galilean spacetimes, which extends the family of
cosmological models known as Galilean Generalized Robertson-Walker spacetimes
(see [16, 30]).

######  Definition 3.1.

Let $I\subseteq\mathbb{R}$, $0\in I$, be an open real interval, $(F,h)$ be an
n-dimensional connected Riemannian manifold, and $f\in C^{\infty}(I\times F)$
be a positive smooth function. A Galilean spacetime $(M,\Omega,g,\nabla)$ is
called _Galilean Twisted_ (GT) spacetime if $M=I\times F$,
$\Omega=\mathrm{d}\pi_{I}$, $g$ is the restriction to the vector bundle
$\operatorname{\text{An}\Omega}$ of the following (degenerate) metric on M:

$\overline{g}=f^{2}\pi_{F}^{*}h\,,$ (6)

where $\pi_{I},\pi_{F}$ are the canonical projections onto the open interval I
and the manifold $F$, respectively, and $\nabla$ is the only symmetric
Galilean connection on $M$ such that

$\nabla_{\partial_{t}}\partial_{t}=0\quad\text{ and
}\quad\operatorname{Rot}\partial_{t}=0\,,$ (7)

where $\partial_{t}=\partial/\partial t$ is the global coordinate vector field
associated with $t\coloneq\pi_{I}$.

Abusing the notation, we will denote $g=f^{2}\pi_{F}^{*}h$. Analogously to the
Riemannian twisted product, the Riemannian manifold $(F,h)$ is called fiber.

The vector field $\operatorname{\partial_{t}}$ is a field of observers in $M$,
and the observers in $\partial_{t}$ are called commovil observers in analogy
with both the Galilean and relativistic Generalized Robertson-Walker
spacetimes [1, 16]. Notice that the conditions (7) determine the Galilean
connection $\nabla$ (see equation (2)).

######  Example 3.2.

Consider the GT spacetime
$(\mathbb{R}\times\mathbb{R}^{n},\mathrm{d}\pi_{1},g=f^{2}\pi_{2}^{*}\langle\,,\rangle,\nabla)$,
where $\pi_{1}\colon\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ and
$\pi_{2}\colon\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ are the
canonical projections onto the first and second factors, respectively,
$\langle\,,\rangle$ is the usual Euclidean product on $\mathbb{R}^{n}$ and
$f\in C^{\infty}(\mathbb{R}\times\mathbb{R}^{n})$. If $f$ does not depend on
the fiber $F=\mathbb{R}^{n}$, i.e., $f=f(t)$, $t\in\mathbb{R}$, then we have a
GGRW spacetime [16]. In the simplest case where $f=constant$, the Galilean
connection is just the standard flat connection of $\mathbb{R}^{n+1}$.

##  4 Geodesic Completeness of GT spacetimes

In this section we are going to study the completeness of the geodesics in a
GT spacetime. In particular, this will allow us to know if the free falling
observers live forever.

######  Remark 4.1.

_From the compatibility condition of a Galilean connection $\nabla$ with the
absolute clock $\Omega$ in a Galilean spacetime, it is clear that the
geodesics $\gamma$ maintain their causal character. That is,
$\Omega(\gamma^{\prime})$ is constant along the trajectory of $\gamma$. Then,
the two relevant cases to consider will be spacelike geodesics
($\Omega(\gamma^{\prime})=0$) and free falling observers
($\Omega(\gamma^{\prime})=1$). _

Given a geodesic $\gamma:J\longrightarrow M$ in a GT spacetime $(M=I\times
F,\Omega,g,\nabla)$, where $J\subset\mathbb{R}$ denote an interval in the real
line, we will be able to derive the corresponding equation of its projection
$\pi_{F}(\gamma)$ on the fiber $F$. The following lemma gives us the necessary
computations.

######  Lemma 4.2.

Let $(M=I\times F,\Omega=\mathrm{d}\pi_{I},g=f^{2}\pi_{F}^{*}h,\nabla)$ be a
GT spacetime. If $W_{1},W_{2}\in\Gamma(TF)$, then

$\begin{split}\nabla_{\operatorname{\partial_{t}}}\overline{W}={}&\nabla_{\overline{W}}\operatorname{\partial_{t}}=u_{t}\overline{W}\,,\\\
\nabla_{\overline{W_{1}}}\overline{W_{2}}={}&\overline{\nabla^{h}_{W_{1}}W_{2}}+\mathrm{d}^{h}u(W_{1})\overline{W_{2}}+\mathrm{d}^{h}u(W_{2})\overline{W_{1}}-h(W_{1},W_{2})\overline{\operatorname{grad}^{h}u}\,,\end{split}$
(8)

where $u\coloneq\log f$ and $\operatorname{grad}^{h}u$ denotes the usual
gradient on the Riemannian manifold $(F,h)$.

###### Proof.

It is enough to apply equation (2) for $V=\overline{U}$, $U\in\Gamma(TF)$,
since a local reference frame $\\{E_{i}\\}_{i=1}^{n}$ in $TF$ lift to a local
reference frame
$\\{\operatorname{\partial_{t}}\\}\cup\\{\overline{E_{i}}\\}_{i=1}^{n}$ in
$TM$. ∎

The previous formulae (8) allow us to obtain the equations of the projections
of the geodesics on the fiber of a GT spacetime.

######  Remark 4.3.

Let $\gamma$ be a geodesic $\gamma:J\longrightarrow M$ in a GT spacetime
$(M=I\times F,\Omega,g,\nabla)$, where $J\subset\mathbb{R}$ denote an interval
in the real line. If $\gamma$ is a free falling observer, we have

$\gamma(s)=(s,\sigma(s))\quad\text{and}\quad\gamma^{\prime}(s)=(\operatorname{\partial_{t}}\circ\gamma)(s)+\overline{\sigma^{\prime}}(s)\,,$

where $\sigma$ is a smooth curve in $F$. Consider $X$ a smooth vector field,
which extends $\sigma^{\prime}$ on a neighbourhood of $\sigma(s)\in F$. Then
the lift $\overline{X}$ extends $\overline{\sigma^{\prime}}$ and satisfies
$L_{\operatorname{\partial_{t}}}\overline{X}=[\operatorname{\partial_{t}},\overline{X}]=0$.
Thus, the geodesic equation turns out to be

$\begin{split}0={}&\nabla_{\gamma^{\prime}}\gamma^{\prime}=\nabla_{\gamma^{\prime}}(\operatorname{\partial_{t}}+\overline{X})|_{\gamma}\\\
={}&\big{(}\nabla_{\operatorname{\partial_{t}}}\operatorname{\partial_{t}}+\nabla_{\overline{\sigma^{\prime}}}\operatorname{\partial_{t}}+\nabla_{\gamma^{\prime}}\overline{X}\big{)}|_{\gamma}\\\
={}&\big{(}2\nabla_{\operatorname{\partial_{t}}}\overline{X}+\nabla_{\overline{\sigma^{\prime}}}\overline{X}\big{)}|_{\gamma}\,,\end{split}$

where we have
$\nabla_{\operatorname{\partial_{t}}}\operatorname{\partial_{t}}=0$. Using
(8), the free falling observers $\gamma(s)=(s,\sigma(s))$ are characterized by
the following equation on $(F,h)$,

$0=\mbox{\footnotesize$\displaystyle\frac{D^{h}\sigma^{\prime}}{ds}$}+2[u_{t}+\mathrm{d}^{h}u(\sigma^{\prime})]\sigma^{\prime}-|\sigma^{\prime}|^{2}_{h}\operatorname{grad}^{h}u\,.$
(9)

In the case $\gamma(s)=(c,\sigma(s))$, $c\in I$, is a spacelike geodesic, then
$\gamma^{\prime}=\overline{\sigma^{\prime}}$. The corresponding equation on
$(F,h)$ is given by

$0=\mbox{\footnotesize$\displaystyle\frac{D^{h}\sigma^{\prime}}{ds}$}+2\mathrm{d}^{h}u(\sigma^{\prime})\sigma^{\prime}-|\sigma^{\prime}|^{2}_{h}\operatorname{grad}^{h}u\,.$
(10)

Note that this is now an autonomous equation in the Riemannian manifold
$(F,h)$. $\diamond$

For the rest of the section we suppose $I=\mathbb{R}$ since it is necessary to
ensure the completeness of commovil observers.

Let $(M,\Omega,g,\nabla)$ be a GT spacetime with $M=\mathbb{R}\times F$ and
$g=f^{2}\,\pi_{F}^{*}h$. We can give a unique vector field $G$ on
$T(\mathbb{R}\times F)$ such that, the timelike geodesic in $M$ are
represented by integral curves of $G$ in $T(\mathbb{R}\times F)$. Indeed, let
$\gamma(s)=(s,\sigma(s))$ be a free falling observer, $s_{0}\in J$ and
$(U,x^{1},...,x^{n})$ a coordinate system in $F$, such that $\sigma(s_{0})\in
U$. If we identify $x(\sigma(s))\equiv x(s)=(x^{1}(s),...,x^{n}(s))$, equation
(9) in $\mathbb{R}\times U$ is given by

$\ddot{x}^{k}(s)+\sum_{i,j}\dot{x}^{i}(s)\dot{x}^{j}(s)\Gamma_{ij}^{k}(x(s))+2\big{[}u_{t}(s,x(s))+\sum_{l}\mbox{\footnotesize$\displaystyle\frac{\partial
u}{\partial x^{l}}$}(s,x(s))\dot{x}^{l}(s)\big{]}\dot{x}^{k}(s)\\\
-\Big{(}\sum_{\alpha,\beta}h_{\alpha,\beta}(x(s))\dot{x}^{\alpha}(s)\dot{x}^{\beta}(s)\Big{)}\Big{(}\sum_{\alpha}h^{\alpha
k}(x(s))\mbox{\footnotesize$\displaystyle\frac{\partial u}{\partial
x^{\alpha}}$}(s,x(s))\Big{)}=0,$ (11)

where $k=1,...,n$ and the $\Gamma$’s denote the Christoffel symbols of the
Riemannian metric $h$. Making use of the standard procedure, we can introduce
auxiliary variables $v^{i}=\dot{x}^{i}$ to convert the previous second-order
system, in the following equivalent first-order system in twice the number of
variables,

$\left\\{\begin{aligned} \dot{x}^{k}(s)={}&v^{k}(s)\,,\\\
\dot{v}^{k}(s)={}&-\sum_{i,j}v^{i}(s)v^{j}(s)\Gamma_{ij}^{k}(x(s))-2\big{[}u_{t}(t,x(s))-\sum_{l}\mbox{\footnotesize$\displaystyle\frac{\partial
u}{\partial x^{l}}$}(s,x(s))v^{l}(s)\big{]}v^{k}(s)\\\
{}&+\Big{(}\sum_{\alpha,\beta}h_{\alpha,\beta}(x(s))v^{\alpha}(s)v^{\beta}(s)\Big{)}\Big{(}\sum_{\alpha}h^{\alpha
k}(x(s))\mbox{\footnotesize$\displaystyle\frac{\partial u}{\partial
x^{\alpha}}$}(s,x(s))\Big{)}\,,\end{aligned}\right.$ (12)

for $k=0,...,n$.

Treating $(x^{1},...,x^{n},v^{1},...,v^{n})$ as coordinates on
$T(\mathbb{R}\times U)$, we recognize (12) as the equations for the flow of
the vector field $G\in\mathfrak{X}(\mathbb{R}\times U)$ given by

$G{(s,x,v)}=\sum_{k}\Big{(}v^{k}\mbox{\footnotesize$\displaystyle\frac{\partial}{\partial
x^{k}}$}{\Big{|}_{(s,x,v)}}-\Big{\\{}\sum_{i,j}v^{i}v^{j}\Gamma_{ij}^{k}(x)+2\big{[}u_{t}(s,x(s))+\sum_{l}\mbox{\footnotesize$\displaystyle\frac{\partial
u}{\partial x^{l}}$}(s,x(s))v^{l}\big{]}v^{k}\\\
-\Big{(}\sum_{\alpha,\beta}h_{\alpha,\beta}(x(s))v^{\alpha}v^{\beta}\Big{)}\Big{(}\sum_{\alpha}h^{\alpha
k}(x(s))\mbox{\footnotesize$\displaystyle\frac{\partial u}{\partial
x^{\alpha}}$}(s,x(s))\Big{)}\Big{\\}}\mbox{\footnotesize$\displaystyle\frac{\partial}{\partial
v^{k}}$}{\Big{|}_{(t,x,v)}}\Big{)}\,.$ (13)

To show that $G$ define a global well-defined vector field on
$T(\mathbb{R}\times F)$, it is enough to see that $G$ acts on a smooth
function $\varphi$ by

$G(\varphi)(p,v)=\tfrac{\mathrm{d}}{\mathrm{d}s}{\big{|}_{s=0}}\varphi(\gamma_{v}(s),\dot{\gamma}_{v}(s))$

where $\gamma_{v}(s)$ denote a geodesic satisfying $\gamma(0)=p$ and
$\dot{\gamma}_{v}(0)=v$, $p\in\mathbb{R}\times F$ and $v\in
T_{p}(\mathbb{R}\times F)$. This last statement is clear from the previous
coordinates expressions.

Making use of [31, Lem. 1.56], we can assert that an integral curve $\xi$ of
$G$ on an interval $[0,b[\subset\mathbb{R}$, for $b<+\infty$, can be extended
to $b$ (as an integral curve) if and only if there exists a sequence in
$[0,b[$, $\\{s_{n}\\}\nearrow b$, such that $\\{\xi(s_{n})\\}$ converges.
Therefore, we obtain the following technical lemma.

######  Lemma 4.4.

A free falling observer $\gamma\colon[0,b[\to M$, $0<b<+\infty$, in a GT
spacetime $(M,\Omega,g,\nabla)$ can be extended to $b$ as a geodesic of
$\nabla$ if and only if there exists a sequence in $[0,b[$,
$\\{s_{n}\\}\nearrow b$, such that $\\{\gamma^{\prime}(s_{n})\\}$ converges in
$TM$.

The following result gives natural conditions to ensure completeness of the
free falling observers in a GT spacetime.

######  Theorem 4.5.

Let $(F,h)$ be a complete Riemannian manifold and let $f\in
C^{\infty}(\mathbb{R}\times F)$ be a positive function. If $\inf f>0$ and
$\inf(f_{t}/f)>-\infty$ along finite times, then the trajectories of free
falling observers in the GT spacetime $(M=\mathbb{R}\times
F,\Omega=\mathrm{d}\pi_{\mathbb{R}},g=f^{2}\,\pi_{F}^{*}h,\nabla)$ are
complete.

###### Proof.

Given $0<b<+\infty$, consider a free falling observer

$\gamma\colon[0,b[\to\mathbb{R}\times F,\quad\gamma(s)=(s,\sigma(s))\,,$

where $\sigma\colon[0,b[\to F$ is a smooth curve in $F$. The conditions on the
function $f$ admit to applying the following argument for past extensibility
analogously. Hence, if we show $\gamma$ is extensible towards $b$, the proof
will be complete.

From equation (9) we have

$\tfrac{\mathrm{d}}{\mathrm{d}s}h(\sigma^{\prime},\sigma^{\prime})=-2\big{[}2u_{t}+\mathrm{d}^{h}u(\sigma^{\prime})\big{]}h(\sigma^{\prime},\sigma^{\prime})\,.$

Note that

$\tfrac{\mathrm{d}}{\mathrm{d}s}(u\circ\gamma)=\mathrm{d}u(\gamma^{\prime})=\mathrm{d}u\big{(}\operatorname{\partial_{t}}+\overline{\sigma^{\prime}}\big{)}=u_{t}+\mathrm{d}^{h}u(\sigma^{\prime})\,.$

Therefore, the square $h-$norm of $\sigma^{\prime}$ is given by

$\begin{split}h(\sigma^{\prime},\sigma^{\prime})={}&A\exp\Big{(}-2\int_{0}^{s}u_{t}(\gamma(r))\mathrm{d}r\Big{)}\exp\big{(}-2u(\gamma(s))\big{)}\\\
={}&\tfrac{A}{f^{2}(\gamma(s))}\exp\Big{(}-2\int_{0}^{s}u_{t}(\gamma(r))\mathrm{d}r\Big{)}\,,\end{split}$
(14)

for some $A\in\mathbb{R}$ determined by the initial values $\gamma(0)$ and
$\gamma^{\prime}(0)$. Hence, given $T>0$ such that $[0,b[\subset[0,T]$, the
hypothesis on $f$ imply $h(\sigma^{\prime},\sigma^{\prime})$ is bounded.
Calling Lemma 4.4, we get $\gamma$ is extensible. ∎

######  Remark 4.6.

In our case (finite dimension), it is not necessary to endow the tangent fiber
bundle with a Riemannian metric. Indeed, it is no difficult to see, under the
assumptions of Lemma 4.4, that each integral curve of the vector field $G$ can
be included in a compact subset of $TM$.

In the case of spacelike geodesics, we obtain analogous results.

######  Theorem 4.7.

Let $(F,h)$ be a complete Riemannian manifold and let $f\in
C^{\infty}(\mathbb{R}\times F)$ be a positive function. If $\inf f>0$ along
finite times, then the spacelike geodesics in the GT spacetime
$(M=\mathbb{R}\times
F,\Omega=\mathrm{d}\pi_{\mathbb{R}},g=f^{2}\,\pi_{F}^{*}h,\nabla)$ are
complete.

###### Proof.

Given a spacelike geodesic $\gamma(s)=(c,\sigma(s))$, we can reason
analogously by defining a suitable vector field on $TF$, whose integral curves
are determined by equation (10). Now,

$\tfrac{\mathrm{d}}{\mathrm{d}s}h(\sigma^{\prime},\sigma^{\prime})=-2\mathrm{d}^{h}u(\sigma^{\prime})h(\sigma^{\prime},\sigma^{\prime})\,.$

and
$\frac{\\!\mathrm{d}}{\\!\mathrm{d}s}(u\circ\gamma)=\mathrm{d}^{h}u(\sigma^{\prime})$.
Therefore,

$h(\sigma^{\prime},\sigma^{\prime})=\tfrac{A}{f^{2}(\gamma(s))},\quad
A\in\mathbb{R}\,.$ (15)

Using now $\inf f>0$ on $[0,T]\supset[0,b[$, we have
$h(\sigma^{\prime},\sigma^{\prime})$ is bounded. An easy adaptation of Lemma
4.4 imply $\gamma$ is extensible. ∎

Note that, if the fiber $(F,h)$ is complete, Theorem 4.7 assures the geodesic
completeness of any Riemannian manifold $(F,g_{t})$, for
$g_{t}=f^{2}(t,\cdot)\pi_{F}^{*}h$, under a natural assumption on $f$.

We can ensure the geodesic completeness of a GT spacetime by unifying the
results on the completeness of free falling observers and spacelike geodesics.

######  Theorem 4.8.

Let $(F,h)$ be a complete Riemannian manifold and let $f\in
C^{\infty}(\mathbb{R}\times F)$ be a positive function. If $\inf f>0$ and
$\inf(f_{t}/f)>-\infty$ along finite times, then the GT spacetime
$(M=\mathbb{R}\times
F,\Omega=\mathrm{d}\pi_{\mathbb{R}},g=f^{2}\,\pi_{F}^{*}h,\nabla)$ is
geodesically complete.

If $F$ is compact, the hypothesis for $f\in C^{\infty}(\mathbb{R}\times F)$ in
Theorem 4.8 are fulfilled trivially.

######  Corolary 4.9.

Let $(F,h)$ be a compact Riemannian manifold and $f\in
C^{\infty}(\mathbb{R}\times F)$ a positive function. Then, the GT spacetime
$(M=\mathbb{R}\times
F,\Omega=\mathrm{d}\pi_{\mathbb{R}},g=f^{2}\,\pi_{F}^{*}h,\nabla)$ is
geodesically complete.

The following example prove that the conditions in Theorem 4.8 are indeed
optimal.

######  Example 4.10.

Consider the Riemannian manifold $(F,h)=(\mathbb{R},\cdot)$, to be the one-
dimensional Euclidean space and $f\in C^{\infty}(\mathbb{R}^{2})$ a smooth
positive function. They define the GT spacetime

$(\mathbb{R}\times\mathbb{R},\Omega=\mathrm{d}t,g=f^{2}\pi_{2}^{*}(\cdot)\equiv
f^{2}\mathrm{d}x^{2},\nabla)\,,$

where $t\coloneq\pi_{1}$ and $x\coloneq\pi_{2}$. Then, free falling observers
$\gamma(s)=(s,\sigma(s))$ satisfy the following second-order nonlinear
differential equation (see (9)),

$0=\mathrm{d}\pi_{2}\big{(}\tfrac{D\gamma^{\prime}}{ds}\big{)}=\sigma^{\prime\prime}+2\tfrac{f_{t}}{f}(s,\sigma(s))\,\sigma^{\prime}+\tfrac{f_{x}}{f}(s,\sigma(s))(\sigma^{\prime})^{2}\,.$

From (14) (which can be integrated explicitly in this case), we know

$\sigma^{\prime}(s)=\mbox{\footnotesize$\displaystyle\frac{C}{f(s,\sigma(s))}$}\exp\Big{(}-\int_{0}^{s}u_{t}(r,\sigma(r))\mathrm{d}r\Big{)},\quad
C\in\mathbb{R}\,,$ (16)

for a constant
$C=\big{\|}\overline{\sigma^{\prime}}\big{\|}_{\\{0\\}\times\mathbb{R}}=f(0,\sigma(0))\,\sigma^{\prime}(0)$
(see equation (18)).

If we consider an incomplete geodesic
$\gamma=(\cdot,\sigma(\cdot))\colon[0,b[\to\mathbb{R}\times\mathbb{R}$, for
$b>0$, then $\lim_{s\rightarrow b^{-}}|\sigma^{\prime}(s)|=\infty$. For
example, if $\inf f>0$ along finite times, then it is necessary
$\lim_{s\rightarrow b^{-}}u_{t}(s,\sigma(s))=-\infty$. Thus,

$\lim_{s\rightarrow
b^{-}}\sigma(s)=\pm\infty\quad\text{and}\quad\lim_{x\rightarrow\pm\infty}u_{t}(b,x)=-\infty\,.$

On the other hand, if $\inf u_{t}>-\infty$ along finite times, then
$\lim_{s\rightarrow b^{-}}f(s,\sigma(s))=0$. Hence,

$\lim_{s\rightarrow
b^{-}}\sigma(s)=\pm\infty\quad\text{and}\quad\lim_{x\rightarrow\pm\infty}f(b,x)=0\,.$

Note that the conditions of boundedness from below on $f$ and $f_{t}/f$ arise
naturally when there are incomplete geodesics.

In the case $f\colon F=\mathbb{R}\to\mathbb{R}$ is time-independent, free
falling observers $\gamma(s)=(s,\sigma(s))$ can be integrated as

$\sigma(s)=G^{-1}(Cs+D),\quad C,D\in\mathbb{R}\,,$ (17)

where $G$ is a primitive function of $f$. Note that $G$ is a diffeomorphism on
$\mathbb{R}$ onto its image since $G^{\prime}=f>0$.

For instance, if $f(x)=(1+x^{2})^{-1}$, free falling observers satisfy
$\gamma(s)=(s,\sigma(s))=(s,\tan(Cs+D))$, where $C,D\in\mathbb{R}$ are
constants fixed by $\sigma(0)$ and $\sigma^{\prime}(0)$. These are incomplete
observers.

##  5 Physical interpretation of geodesics

Several interesting physical observations can be obtained from the expressions
in the previous section.

######  Remark 5.1.

From equation (14), in the conditions of previous theorems, the free falling
observers $\gamma$ satisfy

$\|P^{\operatorname{\partial_{t}}}\gamma^{\prime}(s)\|_{\mathcal{F}_{s}}=\|P^{\operatorname{\partial_{t}}}\gamma^{\prime}(0)\|_{\mathcal{F}_{0}}\,\,e^{-\int_{0}^{s}u_{t}(\gamma)}\,,$
(18)

where $\|\cdot\|_{\mathcal{F}_{s}}$ is the norm induced by $g$ on the
spacelike slice $\mathcal{F}_{s}=\\{s\\}\times F$, for all $s\in\mathbb{R}$.
For a spacelike geodesic $\gamma=(c,\sigma(\cdot))$, equation (15) imply

$\|P^{\operatorname{\partial_{t}}}\gamma^{\prime}(s)\|_{\mathcal{F}_{c}}=\|P^{\operatorname{\partial_{t}}}\gamma^{\prime}(0)\|_{\mathcal{F}_{c}}\,.$
(19)

This last property is obvious from the fact that
$\sigma=P^{\operatorname{\partial_{t}}}\gamma^{\prime}$ is a geodesic in the
Riemannian manifold $(\mathcal{F}_{c},g_{c})$.

Now, equation (18) allows us to heuristically deduce the behaviour of the
complete timelike geodesics in a GT spacetime in relation to certain
properties of the twister function. Consider two simultaneous events
$(s_{0},p),(s_{0},q)\in M$. Their spacelike distance (in the absolute space)
is given by

$\text{dist}_{\mathcal{F}_{s_{0}}}(({s_{0}},p),({s_{0}},q))=\inf_{\xi\in
C_{F}([0,1],p,q)}\int_{0}^{1}f\big{(}{s_{0}},\xi(r)\big{)}\,\big{|}\xi^{\prime}(r)\big{|}_{h}\mathrm{d}r\,,$
(20)

where $C_{F}([0,1],p,q)$ is the set of admissible curves in $F$ from $p$ to
$q$ (see, for instance, [17]). Note that the function
$\text{dist}_{\mathcal{F}_{s_{0}}}(({s_{0}},p),({s_{0}},q))$ determine the
distance between the commovil observer $\theta_{p}({s_{0}})=({s_{0}},p)$ and
$\theta_{q}({s_{0}})=({s_{0}},q)$ in the absolute space $\mathcal{F}_{s_{0}}$.
Moreover, if $\beta(r)=(s_{0},\xi(r))$, $r\in[0,1]$, is a minimizing geodesic
in $\mathcal{F}_{s_{0}}$ from $p$ to $q$, then we have

$\text{dist}_{\mathcal{F}_{s_{0}}}(({s_{0}},p),({s_{0}},q))=\int_{0}^{1}\|\beta^{\prime}\|\mathrm{d}r=\|\beta^{\prime}\|\,,$
(21)

where $\|\cdot\|$ is the norm induced by $g$ in $\mathcal{F}_{s_{0}}$.

Obviously, if $f_{t}>0$ (resp. $f_{t}<0$) the absolute space is expanding
(resp. contracting) over time. If for instance $f_{t}>0$, then the spacelike
component of the velocity $\gamma^{\prime}(s)$ of a free falling observer (no
commovil) $\gamma$, relative to the corresponding instant commovil observer,
decreases in norm. Taking this into account, if we assume that the twister
function $f_{t}$ tends to $+\infty$ for $t\nearrow+\infty$, each free falling
observer $\gamma$ evolves asymptotically to commovil observer directions (see
fig. (a)). If otherwise $f_{t}<0$ and we assume $f_{t}$ tends to $-\infty$ for
$t\nearrow+\infty$ and $\inf f>0$, then every free falling observer $\gamma$
evolves asymptotically to some spacelike direction (see fig. (b)).

$\mathcal{F}_{s_{0}}=\\{s_{0}\\}\times
F$$\theta_{q}(s)$$\theta_{p}(s)$$(s_{0},q)$$(s_{0},p)$$\partial_{t}$$\partial_{t}$$\gamma^{\prime}(s_{0})$$\gamma$
(a) $f_{t}(s_{0},p)>0$

$\mathcal{F}_{s_{0}}=\\{s_{0}\\}\times
F$$\theta_{q}(s)$$\theta_{p}(s)$$(s_{0},q)$$(s_{0},p)$$\partial_{t}$$\partial_{t}$$\gamma^{\prime}(s_{0})$$\gamma$
(b) $f_{t}(s_{0},p)<0$

Now, it is natural to ask how one free falling observer measures the speed of
another one. If two points in the fiber $p,q\in F$ are closely enough, free
falling observers passing through one of them can measure the velocity of the
others. If the leaves of the foliation induced by $\Omega$ are affine
Euclidean spaces, the classical notion of relative velocity of an observer
with respect to another is determined by the existence of a position vector
between them. As a generalization, we can give the following,

######  Definition 5.2.

Let $\gamma,\sigma$ be two observers in a (symmetric) Galilean spacetime and
$\mathcal{F}_{s_{0}}$ the spacelike leaf in the absolute time
$t=s_{0}\in\mathbb{R}$. If there exists a spacelike geodesic
$\beta\colon[0,1]\to\mathcal{F}_{s_{0}}$ from $\gamma(s_{0})$ to
$\sigma(s_{0})$ in $\mathcal{F}_{s_{0}}$, the _relative spacelike velocity_ of
$\gamma$ with respect to $\sigma$ at $t=s_{0}$,
$\textbf{V}\big{(}\gamma(s_{0}),\sigma(s_{0})\big{)}$, is defined as the
spacelike projection (with respect to $\sigma^{\prime}$) of the parallel
transport along $\beta$ of $\gamma^{\prime}(s_{0})$, i.e.,

$\textbf{V}\big{(}\gamma(s_{0}),\sigma(s_{0})\big{)}\coloneq
P^{\sigma^{\prime}}\big{(}\mathcal{P}^{\beta}_{\gamma(s_{0}),\sigma(s_{0})}\gamma^{\prime}(s_{0})\big{)}\,,$

where $\mathcal{P}^{\beta}$ denotes the parallel transport along $\beta$.

Recall that in a Riemannian manifold, the existence and uniqueness of
geodesics between two given points is locally ensured. Globally, the existence
result holds in the complete case. Moreover, the no existence of conjugate
points guarantees globally the uniqueness.

Let us deduce some interesting geometric properties of this definition in a GT
spacetime. The fixed reference observer will be taken as a commovil observer.
Indeed, given a free falling observer $\gamma$ in a GT spacetime, denote
$\gamma^{\prime}(s_{0})=\theta_{p}(s_{0})+v$, for
$v\in(\operatorname{\text{An}\Omega})_{(s_{0},p)}$. The parallel transport
$\mathcal{P}^{\beta}_{(s_{0},p),(s_{0},q)}(\gamma^{\prime}(s_{0}))=V(1)$ is
defined by a parallel vector field $V$ along $\beta$ with
$V(0)=\gamma^{\prime}(s_{0})$. This can be split into

$V(r)=a(r)\operatorname{\partial_{t}}|_{(s_{0},\beta(r))}+U(r)\,,$

where $a$ is a smooth function on $[0,1]$ and $U$ is a spacelike vector field
along $\beta$. The initial value condition $V(0)=\gamma^{\prime}(s_{0})$ turns
out

$a(0)=1,\quad U(0)=v\,.$

From the fact that $V$ is parallel and $\beta$ can be lifted from a curve in
$F$, it is easy to obtain that $a^{\prime}=0$ and
$\mbox{\footnotesize$\displaystyle\frac{DU}{\mathrm{d}r}$}=-u_{t}\beta^{\prime}$.
Therefore, $a\equiv 1$ and $U$ satisfy the initial value problem

$\mbox{\footnotesize$\displaystyle\frac{DU}{\mathrm{d}r}$}=-u_{t}\beta^{\prime},\quad
U(0)=v\,.$

The modulus of the relative spacelike velocity of $\gamma^{\prime}(s_{0})$
measured by $\theta_{q}(s_{0})$,
$\big{\|}\textbf{V}\big{(}\gamma^{\prime}(s_{0}),\theta_{q}(s_{0})\big{)}\big{\|}=\|U(1)\|=g(U(1),U(1))$,
can be explicitly obtained in this case. It is enough to compute

$\tfrac{\mathrm{d}}{\mathrm{d}r}\|U\|^{2}=-2u_{t}g(U,\beta^{\prime}),\quad\tfrac{\mathrm{d}}{\mathrm{d}r}g(U,\beta^{\prime})=-u_{t}\|\beta^{\prime}\|^{2}\,,$

using $\beta$ is a geodesic. Obviously, $\|\beta^{\prime}\|$ is a constant.
Taking into account the initial values, we can integrate both two differential
equations, obtaining

$\big{\|}\textbf{V}\big{(}\gamma^{\prime}(s_{0}),\theta_{q}(s_{0})\big{)}\big{\|}=\Big{\|}\Big{(}\int_{0}^{1}u_{t}(s_{0},\beta(r))\mathrm{d}r\Big{)}w-v\Big{\|}_{\mathcal{F}_{s_{0}}}\,,$
(22)

where $v=P^{\operatorname{\partial_{t}}}\gamma^{\prime}(0)$ and the vector
$w\coloneq\beta^{\prime}(0)$. This expression, takes into account the
geometric structure of the slice $\mathcal{F}_{s_{0}}$, as well as, the
distance between the points $(s_{0},p)$ and $(s_{0},q)$ (see equations (20)
and (21)). In particular, if $\gamma$ is a commovil observer, the velocity of
$\theta_{p}(s_{0})$ measured by $\theta_{q}(s_{0})$ is

$\begin{split}\big{\|}\textbf{V}\big{(}\theta_{p}(s_{0}),\theta_{q}(s_{0})\big{)}\big{\|}={}&\Big{|}\int_{0}^{1}u_{t}(s_{0},\beta(r))\mathrm{d}r\Big{|}\big{\|}\beta^{\prime}\big{\|}\\\
={}&\Big{|}\int_{0}^{1}u_{t}(s_{0},\beta(r))\mathrm{d}r\Big{|}\,\text{dist}_{\mathcal{F}_{s_{0}}}\big{(}(s_{0},p),(s_{0},q)\big{)}\,.\end{split}$
(23)

where $|\cdot|$ is the absolute value function. We see that it is proportional
to the spacelike distance between the points $(s_{0},p)$ and $(s_{0},q)$.

Finally, if the GT spacetime can be reduced to a GGRW spacetime, $f=f(t)$,
then expression (23) turns out
$\|\textbf{V}\big{(}{\theta_{p}(s_{0})},\theta_{q}(s_{0})\big{)}\|=u_{t}(s_{0})\,\text{dist}_{\mathcal{F}_{s_{0}}}\big{(}(s_{0},p),(s_{0},q)\big{)}$.

##  6 Galilean Torqued Vector Fields

Along this section, we will see that there exists a natural generalization of
GT spacetimes in a rich geometrical structure characterized by the existence
of a distinguished vector field on the Galilean spacetime. In fact, the main
theorem will state that these Galilean spacetimes are locally GT spacetimes.

######  Proposition 6.1.

Let $(M=I\times F,\Omega=\mathrm{d}t,g=f^{2}\pi_{F}^{*}h,\nabla)$ be a GT
spacetime. Then, the vector field $K=f\operatorname{\partial_{t}}$ satisfies

$\nabla_{X}K=f_{t}X+\alpha(X)K\quad\text{and}\quad\alpha(K)=0,\quad\forall
X\in\Gamma(TM)\,,$ (24)

where $\alpha=\overline{\mathrm{d}^{h}}(\log f)\in\bigwedge^{1}(M)$ (see
subsection 2.2).

###### Proof.

Let $X\in\Gamma(TM)$ be a vector field. Consider $\\{E_{i}\\}_{i=1}^{n}$ a
local reference frame in $TF$ that lifts to a local reference frame
$\\{\operatorname{\partial_{t}}\\}\cup\\{\overline{E_{i}}\\}_{i=1}^{n}$ in
$TM$. Then, there exist local smooth functions $g_{0},g_{1},\ldots,g_{n}$ such
that

$X=g_{0}\operatorname{\partial_{t}}+\sum_{i}g_{i}\overline{E_{i}}\,.$

Then, if $u\coloneq\log f$, we have

$\begin{split}\nabla_{X}K={}&X(f)\operatorname{\partial_{t}}+f\nabla_{X}\operatorname{\partial_{t}}\\\
={}&X(u)K+f\sum_{i}g_{i}u_{t}\overline{E_{i}}\\\
={}&X(u)K+fu_{t}(X-g_{0}\operatorname{\partial_{t}})\\\
={}&f_{t}X+\overline{\mathrm{d}^{h}}u(X)K\,.\end{split}$

∎

As we announced, a generalization of GT spacetimes is defined by means of a
vector field satisfying an analogous expression to equation (24).

######  Definition 6.2.

Let $(M,\Omega,g,\nabla)$ a Galilean spacetime. A future-pointing timelike
vector field $K\in\Gamma(TM)$ satisfying

$\nabla_{X}K=\rho X+\alpha(X)K\quad\text{and}\quad\alpha(K)=0,\quad\forall
X\in\Gamma(TM)\,,$ (25)

for some $\rho\in C^{\infty}(M)$ and $\alpha\in\bigwedge^{1}(M)$, is called a
Galilean torqued vector field. The function $\rho$ is called the torqued
function and $\alpha$ the torqued form.

The torqued function and form are fully determined by the function
$\Omega(K)$, as we prove in the following lemma.

######  Lemma 6.3.

Let $(M,\Omega,g,\nabla)$ be a spacetime and suppose it admits a Galilean
torqued vector field $K\in\Gamma(TM)$ with torqued function $\rho\in
C^{\infty}(M)$ and torqued form $\alpha\in\bigwedge^{1}(M)$. Then,

$\alpha(V)=\tfrac{1}{\Omega(K)}V\big{(}\Omega(K)\big{)},\quad\forall
V\in\Gamma(\operatorname{\text{An}\Omega})\,.$ (26)

and

$\rho=\tfrac{1}{\Omega(K)}K\big{(}\Omega(K)\big{)}\,.$ (27)

###### Proof.

Consider the field of observers $Z=\frac{1}{\Omega(K)}K$. Then, for any
$V\in\Gamma(\operatorname{\text{An}\Omega})$,

$\begin{split}\nabla_{V}Z={}&V\big{(}\tfrac{1}{\Omega(K)}\big{)}K+\tfrac{1}{\Omega(K)}\nabla_{V}K\\\
={}&\tfrac{\rho}{\Omega(K)}V+\left[\tfrac{\alpha(V)}{\Omega(K)}+V\big{(}\tfrac{1}{\Omega(K)}\big{)}\right]K\,.\end{split}$
(28)

Now, making use of the compatibility of the connection with $\Omega$ we have
$\Omega(\nabla_{V}Z)=V(\Omega(Z))=0$. Then, from (28) we get (26).

On the other hand, from equation (25) we have $\nabla_{K}K=\rho K$.
Consequently,

$K(\Omega(K))=\Omega(\nabla_{K}K)=\rho\Omega(K)\,.$

Thus, it follows expression (27). ∎

######  Remark 6.4.

Let $K$ be a Galilean torqued vector field in a Galilean spacetime
$(M,\Omega,g,\nabla)$ with torqued function $\rho\in C^{\infty}(M)$ and
torqued form $\alpha\in\bigwedge^{1}(M)$. Consider the associated field of
observers $Z\coloneq\tfrac{1}{\Omega(K)}K$. Note that we have obtained in the
proof of the lemma the following identity,

$\nabla_{V}Z=\tfrac{\rho}{\Omega(K)}V,\quad\forall
V\in\Gamma(\operatorname{\text{An}\Omega})\,.$ (29)

Since $V(\Omega(Z))=0$ for all $V\in\Gamma(\operatorname{\text{An}\Omega})$,
from equation (29) we can compute

$\begin{split}L_{Z}g(V,W)={}&Zg(V,W)-g([Z,V],W)-g(V,[Z,W])\\\
={}&g(\nabla_{V}Z,W)+g(V,\nabla_{W}Z)\\\
={}&2\tfrac{\rho}{\Omega(K)}g(V,W)\,,\end{split}$

for all $V,W\in\Gamma(\operatorname{\text{An}\Omega})$ (see subsection 2.1).
Therefore, $Z$ is a spatially conformally Leibnizian vector field.

$\diamond$

Finally, we obtain the main theorem of this section, which determines the
local structure of Galilean spacetimes admitting a Galilean torqued vector
field.

######  Theorem 6.5.

Let $(M,\Omega,g,\nabla)$ be a Galilean spacetime. If it admits a torqued
vector field $K\in\Gamma(TM)$, then for each $p\in M$, there exists an open
neighborhood of p, $\mathcal{U}$, and a Galilean diffeomorphism $\Psi\colon
N\to\mathcal{U}$, where $N$ is a GT spacetime.

###### Proof.

This is in fact a generalization of the corresponding proof of the local
structure Theorem for ICL spactimes [16, Th. 12]. In order to make the paper
self-contained we give the details.

Suppose there exists a torqued vector field $K$. Let $\rho\in C^{\infty}(M)$
and $\alpha\in\bigwedge^{1}(M)$ be the associated torqued function and one-
form, and $Z\coloneq\frac{1}{\Omega(K)}K$ be the associated field of
observers. Given an event $p\in M$, take a neighbourhood $U_{p}$ of $p$ in the
leaf $\mathcal{F}_{p}$ of the foliation induced by $\Omega$, and
$I\subset\mathbb{R}$, $0\in I$, a suitable interval such that the local flow
of $Z$,

$\begin{split}\Psi\colon I\times U_{p}\to{}&M\\\
(s,q)\longmapsto{}&\phi_{s}(q)\,,\end{split}$

is well-defined and one-to-one. It satisfies

$\mathrm{d}\Psi_{(s,q)}(1,0)=Z_{\phi_{s}(q)},\quad\mathrm{d}\Psi_{(s,q)}(0,v)=\big{(}\mathrm{d}\phi_{s}\big{)}_{q}(v)\,,$

for all $(s,q)\in I\times U_{p}$ and $v\in T_{p}U_{p}$. Now, if we identify
$t\equiv\pi_{I}$ and $\operatorname{\partial_{t}}\equiv(1,0)$, it follows by
construction

$\Omega(\mathrm{d}\Psi(\operatorname{\partial_{t}}))=\Omega(Z)=1=\mathrm{d}t(\operatorname{\partial_{t}})\,.$

Moreover, the flow $\phi_{s}$ of $Z$ preserves $\Omega$ on spacelike vector
fields, $L_{Z}\Omega(V)=0$, $\forall
V\in\Gamma(\operatorname{\text{An}\Omega})$. Then,

$\Omega\left(\mathrm{d}\Psi_{(s,q)}(0,v)\right)=\Omega\left((\mathrm{d}\phi_{s})_{q}(v)\right)=(\phi_{s}^{*}\Omega)_{q}(v)=\Omega_{q}(v)=0.$

Consequently, $\Psi^{*}\Omega=\mathrm{d}t$, and each level set of $t$
corresponds with certain open set $\mathcal{U}\cap\mathcal{F}_{a}$ of some
leaf of the foliation $\mathcal{F}$ induced by $\Omega$.

From Remark 6.4, $Z$ is spatially conformally Leibnizian:

$L_{Z}g(V,W)=\tfrac{2\rho}{\Omega(K)}g(V,W),\quad\forall
V,W\in\Gamma(\operatorname{An}(\Omega))\,.$ (30)

Consider $v,w\in T_{q}U_{p}$. If we define

$\eta_{q}(s)\coloneq\Psi^{*}g_{(s,q)}((0,v),(0,w))=g_{\phi_{s}(q)}\left(\big{(}\mathrm{d}\phi_{s}\big{)}_{q}(v),\big{(}\mathrm{d}\phi_{s}\big{)}_{q}(w)\right),$

it follows from equation (30),

$\eta_{q}^{\prime}(s)=\left.\tfrac{2\rho}{\Omega(K)}\right|_{\phi_{s}(q)}\eta_{q}(s)\,.$

Then, taking

$f(s,q)\coloneq\exp\left(\int_{0}^{s}\frac{\rho}{\Omega(K)}\left(\phi_{l}(q)\right)\mathrm{d}l\right)\,,$
(31)

we obtain $\eta_{q}(s)=f^{2}(s,q)g_{q}(v,w)$. Therefore,
$\Psi^{*}g(s,q)=f^{2}(s,q)g_{q}$.

Finally, from equation (29), and any arbitrary tensor field
$V\in\Gamma(\operatorname{\text{An}\Omega})$, we have

$\operatorname{Rot}(Z)(V,W)=g\left(\nabla_{V}Z,W\right)-g\left(\nabla_{W}Z,V\right)=0\,,$

and

$\nabla_{Z}Z=\tfrac{\rho}{\Omega(K)}Z+Z\big{(}\tfrac{1}{\Omega(K)}\big{)}K=0\,.$

Thus, $Z$ is an irrotational field of observers. Since
$\mathrm{d}\Psi(\operatorname{\partial_{t}})=Z$, from [7, Cor. 28], the
Galilean connection $\nabla$ is $\Psi-$related to the corresponding GT
connection $\nabla^{GT}$ in $I\times U_{p}$. Therefore, $\Psi$ is a Galilean
diffeomorphism on

$\big{(}I\times
U_{p},\mathrm{d}\pi_{I},f^{2}\,\pi_{U_{p}}^{*}g,\nabla^{GT}\big{)}\,.$

∎

Note that the conclusions are similar when we consider ICL spacetimes [16]. In
this case, the distinction lies in the fact that the torqued factor is not
necessarily spatially uniform.

##  7 Global splitting

This section is devoted to give natural geometric conditions to characterize a
Galilean spacetime as a global GT spacetime. We will use strongly the
geometric properties satisfied by a Galilean torqued vector field.

In the following Lemma, we define an auxiliary semi-Riemannian metric in terms
of the Galilean structure of a Galilean spacetime. The spacelike gradient
defined in subsection 2.2 allows us to write the associated Levi-Civita
connection in terms of the Galilean connection.

######  Lemma 7.1.

Let $(M,\Omega,g,\nabla)$ be a Galilean spacetime and let $Z\in\Gamma(TM)$ be
an irrotational spatially conformally Leibnizian field of observers. Given a
smooth function $\varphi\in C^{\infty}(M)$ and $\epsilon\in\\{-1,+1\\}$,
define on $M$ the semi-Riemannian metric

$\overline{g}_{\epsilon,\varphi}\coloneq\epsilon\Omega\otimes\Omega+e^{2\varphi}g(P^{Z}\cdot,P^{Z}\cdot)\,.$

Then, the associated Levi-Civita connection $\overline{\nabla}$ of
$\overline{g}_{\epsilon,\varphi}$ is given by

$\displaystyle\overline{\nabla}_{X}Y=$
$\displaystyle\nabla_{X}Y+\mathrm{d}\varphi(X)P^{Z}Y+\mathrm{d}\varphi(Y)P^{Z}X-\Omega(X)\Omega(Y)\mathcal{G}^{Z}$
(32)
$\displaystyle-\overline{g}_{\epsilon,\varphi}(P^{Z}X,P^{Z}Y)\operatorname{grad}\varphi-\epsilon\big{(}Z(\varphi)+\tilde{\rho}\big{)}\overline{g}_{\epsilon,\varphi}\left(P^{Z}X,P^{Z}Y\right)Z\,,$

for all $X,Y\in\Gamma(TM)$, where $\mathcal{G}^{Z}$ is the gravitational field
of $Z$, $\tilde{\rho}$ is the conformal factor associated to $Z$ and
$\operatorname{grad}\varphi$ is the spacelike gradient of $\varphi$ in the
Galilean structure.

###### Proof.

Consider $X,Y\in\Gamma(TM)$ and $V\in\Gamma(\operatorname{\text{An}\Omega})$.
Koszul formula [31, Th. 11] gives

$\displaystyle
2\overline{g}_{\epsilon,\varphi}\left(\overline{\nabla}_{X}Y,V\right)=$
$\displaystyle
X\left(\overline{g}_{\epsilon,\varphi}(Y,V)\right)+Y\left(\overline{g}_{\epsilon,\varphi}(X,V)\right)-V\left(\overline{g}_{\epsilon,\varphi}(X,Y)\right)$
(33)
$\displaystyle+\overline{g}_{\epsilon,\varphi}([X,Y],V)-\overline{g}_{\epsilon,\varphi}([X,V],Y)-\overline{g}_{\epsilon,\varphi}([Y,V],X)$
$\displaystyle=$ $\displaystyle
e^{2\varphi}\left\\{2X(\varphi)g\left(P^{Z}Y,V\right)+X\left(g\left(P^{Z}Y,V\right)\right)+2Y(\varphi)g\left(P^{Z}X,V\right)\right.$
$\displaystyle+Y\left(g\left(P^{Z}X,V\right)\right)-2V(\varphi)g\left(P^{Z}X,P^{Z}Y\right)-V\left(g\left(P^{Z}X,P^{Z}Y\right)\right)$
$\displaystyle+g\left(\big{[}P^{Z}X,P^{Z}Y\big{]},V\right)-\Omega(X)g\left(\big{[}P^{Z}Y,Z\big{]},V\right)+\Omega(Y)g\left(\big{[}P^{Z}X,Z\big{]},V\right)$
$\displaystyle-g\left(\big{[}P^{Z}X,V\big{]},P^{Z}Y\right)+\Omega(X)g\left([V,Z],P^{Z}Y\right)$
$\displaystyle\left.-g\left(\big{[}P^{Z}Y,V\big{]},P^{Z}X\right)+\Omega(Y)g\left([V,Z],P^{Z}X\right)\right\\}$

Considering $\omega(Z)=0$, we can derive from equation (2) the following
expression:

$\displaystyle 2\overline{g}_{\epsilon,\varphi}\left(\nabla_{X}Y,V\right)={}$
$\displaystyle 2e^{2\varphi}g\left(P^{Z}\left(\nabla_{X}Y\right),V\right)$
(34) $\displaystyle={}$ $\displaystyle
e^{2\varphi}\left\\{X\left(g\left(P^{Z}(Y),V\right)\right)+Y\left(g\left(P^{Z}(X),V\right)\right)-V\left(g\left(P^{Z}(X),P^{Z}(Y)\right)\right.\right.$
$\displaystyle+2\Omega(X)\Omega(Y)g\left(\mathcal{G}^{Z},V\right)+\Omega(X)\left(g\left(\big{[}Z,P^{Z}(Y)\big{]},V\right)-g\left([Z,V],P^{Z}(Y)\right)\right)$
$\displaystyle-\Omega(Y)\left(g\left(\big{[}Z,P^{Z}(X)\big{]},V\right)+g\left([Z,V],P^{Z}(X)\right)\right)+g\left(\big{[}P^{Z}(X),P^{Z}(Y)\big{]},V\right)$
$\displaystyle-g\left(\big{[}P^{Z}(Y),V\big{]},P^{Z}(X)\right)\left.-g\left(\big{[}P^{Z}(X),V\big{]},P^{Z}(Y)\right)\right\\}\,.$

Substituting (34) in (33), we obtain

$\displaystyle\overline{g}_{\epsilon,\varphi}\left(\overline{\nabla}_{X}Y,V\right)={}$
$\displaystyle\overline{g}_{\epsilon,\varphi}\left(\nabla_{X}Y,V\right)+\mathrm{d}\varphi(X)\overline{g}_{\epsilon,\varphi}(Y,V)+\mathrm{d}\varphi(Y)\overline{g}_{\epsilon,\varphi}(X,V)$
$\displaystyle-e^{2\varphi}V(\varphi)g\left(P^{Z}X,P^{Z}Y\right)-\Omega(X)\Omega(Y)\overline{g}_{\epsilon,\varphi}\left(\mathcal{G}^{Z},V\right)\,.$

On the other hand, applying (2) again and the relation
$L_{Z}g=2\tilde{\rho}g$, we have

$\displaystyle
2\overline{g}_{\epsilon,\varphi}\left(\overline{\nabla}_{X}Y,Z\right)={}$
$\displaystyle
X\left(\overline{g}_{\epsilon,\varphi}(Y,Z)\right)+Y\left(\overline{g}_{\epsilon,\varphi}(X,Z)\right)-Z\left(\overline{g}_{\epsilon,\varphi}(X,Y)\right)$
(35)
$\displaystyle+\overline{g}_{\epsilon,\varphi}([X,Y],Z)-\overline{g}_{\epsilon,\varphi}([X,Z],Y)-\overline{g}_{\epsilon,\varphi}([Y,Z],X)$
$\displaystyle={}$ $\displaystyle\epsilon X(\Omega(Y))+\epsilon
Y(\Omega(X))-\epsilon Z(\Omega(X))\Omega(Y)-\epsilon\Omega(X)Z(\Omega(Y))$
$\displaystyle-2e^{2\varphi}Z(\varphi)g\left(P^{Z}X,P^{Z}Y\right)-e^{2\varphi}Z\left(g\left(P^{Z}X,P^{Z}Y\right)\right)+\epsilon\Omega([X,Y])$
$\displaystyle-\epsilon\Omega([X,Z])\Omega(Y)-e^{2\varphi}g\left(P^{Z}[X,Z],P^{Z}Y\right)$
$\displaystyle-\epsilon\Omega([Y,Z])\Omega(X)-e^{2\varphi}g\left(P^{Z}[Y,Z],P^{Z}X\right)$
$\displaystyle={}$ $\displaystyle 2\epsilon
X(\Omega(Y))-2e^{2\varphi}Z(\varphi)g\left(P^{Z}X,P^{Z}Y\right)-2e^{2\varphi}\tilde{\rho}g\left(P^{Z}X,P^{Z}Y\right).$

Using
$\overline{g}_{\epsilon,\varphi}\left(\nabla_{X}Y,Z\right)=\epsilon\Omega\left(\nabla_{X}Y\right)=\epsilon
X(\Omega(Y))$, equation (35) gives

$\displaystyle\overline{g}_{\epsilon,\varphi}\left(\overline{\nabla}_{X}Y,Z\right)={}$
$\displaystyle\overline{g}_{\epsilon,\varphi}\left(\nabla_{X}Y,Z\right)-e^{2\varphi}Z(\varphi)g\left(P^{Z}X,P^{Z}Y\right)-e^{2\varphi}\tilde{\rho}g\left(P^{Z}X,P^{Z}Y\right)\,.$

∎

We will describe how the semi-Riemannian induced structure on a Galilean
spacetime can be split into a semi-Riemannian twisted product
$I\times_{f}\mathcal{F}$, for an interval $I$, a Riemannian manifold
$\mathcal{F}$ and certain positive smooth function $f\in
C^{\infty}(I\times\mathcal{F})$. This requires a slight adaptation of [33, Th.
1], in which the geodesic completeness of the leaves $L$ is replaced in the
following special case.

Consider a smooth $r$-dimensional manifold $N$ and suppose $N$ admits two
smooth transverse foliations $\mathcal{L}$ and $\mathcal{K}$, $1$-dimensional
the first one and $(r-1)$-dimensional the second one. We can assume that
$\mathcal{L}$ is defined by a smooth vector field $X_{\mathcal{L}}$.

######  Definition 7.2.

The foliation $\mathcal{L}$ admits a _uniform global generator_
$X_{\mathcal{L}}$ if there exists a transverse leaf
$\mathcal{K}_{0}\in\mathcal{K}$ and an open real interval $I$, such that the
maximal flow $\phi$ of $X_{\mathcal{L}}$ is well-defined and onto on
$I\times\mathcal{K}_{0}$.

The proof of [33, Th. 1] remains valid in the case of the following lemma.

######  Lemma 7.3.

Let $(M,g)$ be a simply connected semi-Riemannian manifold endowed with two
orthogonal foliations $\mathcal{L}$ and $\mathcal{K}$. Suppose $\mathcal{L}$
is one-dimensional and admits a uniform global generator $X\in\Gamma(TM)$ with
associated interval $I\subset\mathbb{R}$ and transverse leaf
$\mathcal{K}_{0}\in\mathcal{K}$. If $\mathcal{L}$ is totally geodesic and
$\mathcal{K}$ is totally umbilic, then $(M,g)$ is isometric to a (semi-
Riemannian) twisted product $I\times_{f}\mathcal{K}_{0}$.

The above lemma allows us to prove

######  Theorem 7.4.

Let $(M,\Omega,g,\nabla)$ be a simply connected Galilean spacetime and let
$K\in\Gamma(TM)$ be a Galilean torqued vector field. Given a smooth function
$\varphi\in C^{\infty}(M)$ and $\epsilon\in\\{-1,+1\\}$, if the field of
observers $Z\coloneq K/\Omega(K)$ is a uniform global generator with
associated interval $I\subseteq\mathbb{R}$ and transverse leaf $\mathcal{F}$
in the foliation induced by $\Omega$, then the semi-Riemannian manifold

$(M,\overline{g}_{\epsilon,\varphi}),\quad\text{with}\quad\overline{g}_{\epsilon,\varphi}=\epsilon\Omega\otimes\Omega+e^{2\varphi}g(P^{Z}\cdot,P^{Z}\cdot)\,,$

is isometric to the (semi-Riemannian) twisted product

$(I\times\mathcal{F},\overline{g}),\quad\text{with}\quad\overline{g}=\epsilon\,\pi_{I}^{*}\mathrm{d}t^{2}+f^{2}\pi_{\mathcal{F}}^{*}\big{(}g|_{\mathcal{F}}\big{)}\,,$

where $g|_{\mathcal{F}}$ is the restriction of $g$ to $\mathcal{F}$ and $f$ is
a positive smooth function on $I\times\mathcal{F}$.

Moreover, $Z$ is a spatially conformal vector field of
$\overline{g}_{\epsilon,\varphi}$.

###### Proof.

Note first that we are in the hypothesis of Lemma 7.1 with a conformal factor
$\tilde{\rho}=\rho/\Omega(K)$ (cf. equation (29)).

The induced foliation by $Z$ is parameterized by its integral curves
$\phi_{p}(t)$, for $t\in I$ and $p\in\mathcal{F}$. From equation (32) and
using $\nabla_{Z}Z=0$, we obtain $\overline{\nabla}_{Z}Z=0$. Therefore, the
integral curves $\phi_{p}(t)$ are geodesics for both $\nabla$ and
$\overline{\nabla}$, that is, the associated foliation is totally geodesic.

Moreover, for all vector fields
$V,W\in\Gamma(\operatorname{\text{An}\Omega})$, we have

$\begin{split}2\overline{g}_{\epsilon,\varphi}(\overline{\nabla}_{V}W,Z)={}&-2\overline{g}_{\epsilon,\varphi}(W,\overline{\nabla}_{V}Z)\\\
={}&-Z\overline{g}_{\epsilon,\varphi}(V,W)-\overline{g}_{\epsilon,\varphi}(V,[W,Z])+\overline{g}_{\epsilon,\varphi}(W,[Z,V])\\\
={}&e^{2\varphi}\big{[}-2Z(\varphi)g(V,W)-g(V,\nabla_{W}Z)-g(\nabla_{V}Z,W)\big{]}\\\
={}&-2\big{(}Z(\varphi)+\tfrac{\rho}{\Omega(K)}\big{)}e^{2\varphi}g(V,W)\\\
={}&-2\big{(}Z(\varphi)+\tfrac{\rho}{\Omega(K)}\big{)}\overline{g}_{\epsilon,\varphi}(V,W)\,.\end{split}$

Then,
$\overline{II}(V,W)=-\tfrac{K(\varphi)+\rho}{\Omega(K)}\,\overline{g}_{\epsilon,\varphi}(V,W)Z$;
hence, the leaves of the foliation induced by $\operatorname{\text{An}\Omega}$
are totally umbilic for the semi-Riemannian metric
$\overline{g}_{\epsilon,\varphi}$. From Lemma 7.3,
$(M,\overline{g}_{\epsilon,\varphi})$ is isometric to a (semi-Riemmanian)
twisted product $I\times_{f}\mathcal{F}$.

Furthermore, from equation (32) we have

$\overline{\nabla}_{V}Z=\tfrac{1}{\Omega(K)}(\rho+K(\varphi))V\,,$

for all spacelike vector fields $V$. Then,

$\displaystyle L_{Z}\overline{g}_{\epsilon,\varphi}(V,W)={}$
$\displaystyle\overline{g}_{\epsilon,\varphi}(\overline{\nabla}_{V}Z,W)+\overline{g}_{\epsilon,\varphi}(V,\overline{\nabla}_{W}Z)$
$\displaystyle={}$
$\displaystyle\tfrac{2}{\Omega(K)}(\rho+K(\varphi))\overline{g}_{\epsilon,\varphi}(V,W)\,,$

for all spacelike vector fields $V,W$, that is, $Z$ is spatially conformal for
$\overline{g}_{\epsilon,\varphi}$.

∎

Finally, we are in the position to obtain a converse for Proposition 6.1.

######  Theorem 7.5.

Let $(M,\Omega,g,\nabla)$ be a simply connected Galilean spacetime. It admits
a global decomposition as a GT spacetime if and only if there exists a
Galilean torqued vector field $K\in\Gamma(TM)$ such that the associated field
of observers, $Z\coloneq K/\Omega(K)$, is a uniform global generator relative
to the foliation defined by $\operatorname{\text{An}\Omega}$.

###### Proof.

From Proposition 6.1, the necessary condition is obvious. Now, if the flow of
$Z$ is well-defined and onto on $I\times\mathcal{F}$, for $I\subset\mathbb{R}$
an open interval and $\mathcal{F}$ a leaf of the foliation induced by
$\Omega$, Theorem 7.4 states that there exists a semi-Riemannian isometry map

$\Psi\colon\big{(}M,\overline{g}_{-1,0}\big{)}\longrightarrow\big{(}I\times\mathcal{F},-\mathrm{d}t^{2}+f^{2}\,g|_{\mathcal{F}}\big{)}\,,$

where we take $\epsilon=-1$ and $\varphi=0$. Consider $\Psi$ as a map between
Galilean spacetimes,

$\Psi\colon\big{(}M,\Omega,g,\nabla\big{)}\longrightarrow\big{(}I\times\mathcal{F},\mathrm{d}t,f^{2}\,g|_{\mathcal{F}},\nabla^{0}\big{)}\,,$

where $\nabla^{0}$ is the corresponding GT connection. It suffices to prove
that $\Psi$ is a Galilean diffeomorphism.

First, $\Psi$ maps the foliation induced by $Z$ to the canonical foliation
induced by $\operatorname{\partial_{t}}$ in $I\times\mathcal{F}$, with
$t=\pi_{I}\colon I\times\mathcal{F}\to I$. Consequently, if $\phi_{q}(s)$ is
the flow of $Z$ for $q\in\mathcal{F}$, we have

$\Psi(\phi_{q}(s))=(\beta(s),q^{\prime}),\quad\forall s\in I\,,$

for certain $q^{\prime}\in\mathcal{F}$ and a smooth function $\beta\in
C^{\infty}(I)$. Since $\Psi$ preserves the causal character of vector fields
for the semi-Riemannian structures,
$\mathrm{d}\Psi(Z)=\frac{\mathrm{d}}{\mathrm{d}s}(\Psi\circ\phi_{q})$ must be
timelike unitary. Thus, $(\beta^{\prime})^{2}=1$.

On the one hand, if $\beta^{\prime}=-1$, we may redefine the isometry map as
follows,

$\widetilde{\Psi}\colon M\to I\times\mathcal{F},\quad
p\longmapsto\widetilde{\Psi}(p)=\big{(}-\psi_{1}(p),\psi_{2}(p)\big{)}\,,$

for all $p\in M$, where $\psi_{i}$ are the component functions of $\Psi$,
$\Psi(p)=\big{(}\psi_{1}(p),\psi_{2}(p)\big{)}$ with $\psi_{1}(p)\in I$ and
$\psi_{2}(p)\in\mathcal{F}$. Hence, the orientation of the isometry $\Psi$ is
reversed and the previous argument gives
$\mathrm{d}\widetilde{\Psi}(Z)=\widetilde{\beta^{\prime}}=1$.

Now, if $\beta^{\prime}=1$, we have

$(\Psi\circ\phi_{q})(s)=(s+c,q^{\prime}),\quad\forall s\in I\,,$

with $c\in I$, and $\mathrm{d}\Psi(Z)=\operatorname{\partial_{t}}$. The
manifold $M$ is simply connected and $\mathrm{d}\Omega=0$, then there exists a
function $T\in C^{\infty}(M)$, determined except for a constant, such that
$\Omega=\mathrm{d}T$. It turns out $(t\circ\Psi)(\phi_{q}(s))=s+c$ and
$T(\phi_{q}(s))=s+\tilde{c}$, with $\tilde{c}\in\mathbb{R}$. If necessary,
redefine $T$ via a translation to get $t\circ\Psi=T$. As a consequence,
$\Psi^{*}\mathrm{d}t=\mathrm{d}T$. Using $\Psi$ preserves the semi-Riemannian
structures, from Theorem 7.4 we obtain $\Psi^{*}(f^{2}g|_{\mathcal{F}})=g$.

Note that $\mathrm{d}\Psi(Z)=\operatorname{\partial_{t}}$ and take into
account

$\mathrm{d}\Psi(P^{Z}X)=\mathrm{d}\Psi(X)-\Omega(X)\mathrm{d}\Psi(Z)=P^{\operatorname{\partial_{t}}}\mathrm{d}\Psi(X)\,,$

it is straightforward that formula ’à la Koszul’ (2) gives

$2g|_{\mathcal{F}}(P^{\operatorname{\partial_{t}}}\nabla^{0}_{\mathrm{d}\Psi(X)}\mathrm{d}\Psi(Y),\mathrm{d}\Psi(V))=2g(P^{Z}\nabla_{X}Y,V)\,,$

for all $X,Y\in\Gamma(TM)$ and all
$V\in\Gamma(\operatorname{\text{An}\Omega})$. Therefore $\nabla$ and
$\nabla^{0}$ are $\Psi-$related. ∎

When the spacelike leaves are compact, we can assure the existence of a global
splitting.

######  Theorem 7.6.

Let $(M,\Omega,g,\nabla)$ be a simply connected Galilean spacetime. If there
exists a Galilean torqued vector field $K\in\Gamma(TM)$ and the leaves of the
distribution induced by $\operatorname{\text{An}\Omega}$ are compact, then $M$
is a GT spacetime.

###### Proof.

It is enough to show that the vector field $Z=\frac{K}{\Omega(K)}$ is a
uniform global generator. Indeed, let $\Phi\colon\mathcal{D}\to M$ be the
maximal local flow of $Z$.

For any $p\in\mathcal{F}$, we have $I_{p}\times U_{p}\subset\mathcal{D}$, with
$I_{p}\subset\mathbb{R}$ an open interval and $U_{p}$ an open set in
$\mathcal{F}$. Since $\mathcal{F}$ is compact,
$\mathcal{F}=\bigcup_{i=1}^{k}U_{i}$, $k\in\mathbb{N}$. Then, $\Phi$ is well-
defined on some domain $I\times\mathcal{F}$, for $I\subset\mathbb{R}$ maximal
interval.

In the case $I=\mathbb{R}$, the flow is complete and the proof is ended using
Theorem 7.5. Suppose now $I=]a,b[$, with $b<+\infty$. Given $q\in\mathcal{F}$
and $\epsilon>0$, if $\Phi(\cdot,q)$ is defined on $]a,b+\epsilon[$, then
there exists $\delta>0$ such that
$]-\delta,\delta[\times\mathcal{F}_{\Phi(b,q)}\subset\mathcal{D}$. Therefore,
the flow can be extended,

$\overline{\Phi}(t,q)=\left\\{\begin{array}[]{lcc}\Phi(t,q)&\text{ if }\quad
t\in(a,b-\delta),\\\
\Phi\big{(}t-b+\tfrac{\delta}{2},\Phi(b-\tfrac{\delta}{2},q)\big{)}&\text{ if
}\quad t\in(b-\delta,b+\delta).\end{array}\right.$

This is a contradiction.

Finally, we prove that $\Phi\colon I\times\mathcal{F}\to M$ is onto. Suppose
again by contradiction, that there exists $q\in
M\setminus\Phi(I\times\mathcal{F})$ and consider the maximal interval
$J\subset\mathbb{R}$ where $\Phi\colon J\times\mathcal{F}_{q}\to M$ is
defined. If $\Phi(I\times\mathcal{F})\cap\Phi(J\times\mathcal{F}_{q})$ is not
empty, the flow could be extended towards $I$. Therefore, since $I$ is
maximal, $\Phi(I\times\mathcal{F})$ is open and closed in a connected manifold
$M$. Hence, $\Phi(I\times\mathcal{F})=M$, and the proof is ended using Theorem
7.5. ∎

## Acknowledgements

The three authors are partially supported by Spanish MICINN project
PID2021-126217NB-I00.

## Data availability

Not applicable.

## References

  * [1] L. J. Alías, A. Romero, and M. Sánchez.  Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes.  Gen. Relat. Gravit., 27:71–84, 1995.
  * [2] R. Andringa, E. Bergshoeff, J. Gomis, and M. de Roo.  Stringy’ Newton–Cartan gravity.  Classical Quant. Grav., 23:1–44, 2012.
  * [3] R. Andringa, E. Bergshoeff, J. Rosseel, and E. Sezgin.  3D Newton-Cartan supergravity.  Classical Quant. Grav., 30:1–19, 2013.
  * [4] X. Bekaert and K. Morand.  Embedding nonrelativistic physics inside a gravitational wave.  Phys. Rev. D, 88:1–61, 2013.
  * [5] E. A. Bergshoeff, J. Gomis, J. Rosseel, C. Simsek, and Z. Yan.  String theory and string Newton-Cartan geometry.  J. Phys. A, 53:1–38, 2020.
  * [6] A. N. Bernal, M. López, and M. Sánchez.  Fundamental units of length and time.  Found. Phys., 32:77–108, 2002.
  * [7] A. N. Bernal and M. Sánchez.  Leibnizian, Galilean and Newtonian structures of space-time.  J. Math. Phys., 32:77–108, 2002.
  * [8] U. Brauer, A. Rendall, and O. Reula.  The cosmic no hair theorem and the nonlinear stability of homogeneous Newtonian cosmological models.  Classical Quant. Grav., 11:2283, 1994.
  * [9] T. Brauner, S. Endlich, A. Monin, and R. Penco.  General coordinate invariance in quantum many-body systems.  Phys. Rev. D, 90:1–14, 2014.
  * [10] E. Cartan.  Les variètés a conexion affine.  Ann. Ec. Norm. Sup., 40:1–25, 1923.
  * [11] E. Cartan.  Les variètés a conexion affine (suite).  Ann. Ec. Norm. Sup., 41:325–412, 1924.
  * [12] B-Y Chen.  Totally umbilical submanifolds.  Soochow J. Math., 5:9–37, 1979.
  * [13] B-Y Chen.  Differential Geometry of Warped Product Manifolds and Submanifolds.  World Scientific, 2017.
  * [14] L.F. Costa and J. Natário.  The Coriolis field.  Am. J. Phys., 84:1–9, 2016.
  * [15] D. de la Fuente, J.A.S. Pelegrín, and R. M. Rubio.  On the geometry of stationary Galilean spacetimes.  Gen. Relat. Gravit., 53(8):1–15, 2021.
  * [16] D. de la Fuente and R. M. Rubio.  Galilean Generalized Robertson-Walker spacetimes: a new family of Galilean geometrical models.  J. Math. Phys., 59(2):022903, 2018.
  * [17] M. P. do Carmo.  Riemannian Geometry.  Mathematics: Theory & Applications. Birkhäuser Boston, MA, 1 edition, 1992.
  * [18] C. Duval and H. P. Künzle.  Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation.  Gen. Relat. Gravit., 16:333–347, 1984.
  * [19] M. Geracie, K. Prabhu, and M.M. Roberts.  Fields and fluids on curved non-relativistic spacetimes.  J. High Energy Phys., 8:1–39, 2015.
  * [20] M. Geracie, D.T. Son, C. Wu, and S.F. Wu.  Spacetime symmetries of the quantum Hall effect.  Phys. Rev. D, 91:1–18, 2015.
  * [21] S. Güler, U. C. De, and B. Ünal.  Geometry of Twisted Products and Applications on Static Perfect Fluid Spacetimes.  International Electronic Journal of Geometry, 16(2):598–607, 2023.
  * [22] H. Leihkauf.  On Newton-Cartan-theory.  Ann. Physik, 46:312–314, 1989.
  * [23] F.L. Lin and S.Y. Wu.  Non-relativistic holography and singular black hole.  Phys. Lett. B, 679:65–72, 2009.
  * [24] D. B. Malament.  Topic in the Formulations of General Relativity and Newtonian Gravitation Theory.  Chicago lectures in Physics, University of Chicago Press, University of Chicago Press, 2012.
  * [25] C. A. Mantica and L. G. Molinari.  Generalized Robertson–Walker spacetimes — A survey.  International Journal of Geometric Methods in Modern Physics, 14(03):1730001, 2017.
  * [26] C. A. Mantica and L. G. Molinari.  Twisted Lorentzian manifolds: a characterization with torse-forming time-like unit vectors.  Gen Relativ Gravit, 49(51), 2017.
  * [27] E. Minguzzi.  Classical aspects of lightlike dimensional reduction.  Class. Quant. Grav., 23:7085, 2006.
  * [28] E. Minguzzi.  Eisenhart’s theorem and the causal simplicity of Eisenhart’s spacetime.  Class. Quant. Grav., 24:2781, 2007.
  * [29] E. Minguzzi.  A connection between Lorentzian distance and mechanical least action.  Talk at ’Noncommutative structures and nonrelativistic (super)symmetries’ (Tours), June 2010.  Slides available online at http://www.lmpt.univ-tours.fr/~bekaert/Conference_2010/Minguzzi.pdf.
  * [30] F. Muler-Hoissen.  The Cosmological Principle and a Generalization of Newton’s Theory of Gravitation.  Gen. Relativ. Gravitation, 15:1051–1066, 1983.
  * [31] B. O’Neill.  Semi-Riemannian Geometry with Applications to Relativity.  ISSN. Academic Press, 1983.
  * [32] R. Penrose.  On gravity’s role in quantum state reduction .  Gen. Relat. Gravit., 8:581–600, 1996.
  * [33] R. Ponge and H. Reckziegel.  Twisted products in pseudo-Riemannian geometry.  Geometriae Dedicata, 48:15–25, 1993.
  * [34] W.A. Rodrigues, Q.A. de Souza, and Y. Bozhkov.  The mathematical structure of Newtonian spacetime: classical dynamics and gravitation.  Found. Phys., 25:871–924, 1995.
  * [35] C. Rüede and N. Straumann.  On Newton-Cartan cosmology.  Helv. Phys. Acta, 70:318–335, 1997.
  * [36] A. Soria.  Spacelike Hypersurfaces in Twisted Product Spacetimes with Complete Fiber and Calabi-Bernstein-type Problems.  J. Geom., 114(16), 2023.
  * [37] M. Spivak.  A Comprehensive Introduction to Differential Geometry, Vol. 1.  Publish or Perish, Houston, Texas, 1979.
  * [38] M. Spivak.  A Comprehensive Introduction to Differential Geometry, Vol. 2.  Publish or Perish, Houston, Texas, 1979.
  * [39] A. Trautman.  Comparison of Newtonian and relativistic theories of space-time. .  Perspectives in Geometry, Indiana Univ. Press, Bloomington, Indiana, 1966.
  * [40] F. W. Warner.  Foundations of Differentiable Manifolds and Lie Groups.  Graduate Texts in Mathematics. Springer-Verlag, New York, 1983.