# Spacetime symmetries and and geometric diffusion

Marc Basquens  Marc Basquens, Department of Energy Technology, Royal Institute
of Technology (KTH), 10044 Stockholm, Sweden<EMAIL_ADDRESS>,  Antonio Lasanta
Antonio Lasanta, Departamento de Álgebra, Facultad de Educación, Economía y
Tecnología de Ceuta, Universidad de Granada, Cortadura del Valle, s/n, 51001
Ceuta, Spain. Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, E-18071 Granada, Spain. Nanoparticles Trapping
Laboratory, Universidad de Granada, Granada, Spain<EMAIL_ADDRESS>,
Emanuel Mompó  Emanuel Mompó, Departamento de Matemática Aplicada, Grupo de
Dinámica No Lineal, Universidad Pontificia Comillas, C. de Alberto Aguilera
25, 28015 Madrid, Spain. Instituto de Investigación Tecnológica (IIT),
Universidad Pontificia Comillas, 28015 Madrid, Spain<EMAIL_ADDRESS>,
Valle Varo  Valle Varo, Institute for Research in Technology, Universidad
Pontificia Comillas, C. de Sta. Cruz de Marcenado, 26, 28015 Madrid, Spain
<EMAIL_ADDRESS>and  Eduardo J S Villaseñor  Eduardo J S Villaseñor,
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la
Universidad 30, 28911 Leganés, Spain. Grupo de Teorías de Campos y Física
Estadística, Unidad Asociada al IEM-CSIC, Serrano 123, 28006 Madrid, Spain
<EMAIL_ADDRESS>
# Spacetime symmetries and geometric diffusion

Marc Basquens  Marc Basquens, Department of Energy Technology, Royal Institute
of Technology (KTH), 10044 Stockholm, Sweden<EMAIL_ADDRESS>,  Antonio Lasanta
Antonio Lasanta, Departamento de Álgebra, Facultad de Educación, Economía y
Tecnología de Ceuta, Universidad de Granada, Cortadura del Valle, s/n, 51001
Ceuta, Spain. Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, E-18071 Granada, Spain. Nanoparticles Trapping
Laboratory, Universidad de Granada, Granada, Spain<EMAIL_ADDRESS>,
Emanuel Mompó  Emanuel Mompó, Departamento de Matemática Aplicada, Grupo de
Dinámica No Lineal, Universidad Pontificia Comillas, C. de Alberto Aguilera
25, 28015 Madrid, Spain. Instituto de Investigación Tecnológica (IIT),
Universidad Pontificia Comillas, 28015 Madrid, Spain<EMAIL_ADDRESS>,
Valle Varo  Valle Varo, Institute for Research in Technology, Universidad
Pontificia Comillas, C. de Sta. Cruz de Marcenado, 26, 28015 Madrid, Spain
<EMAIL_ADDRESS>and  Eduardo J S Villaseñor  Eduardo J S Villaseñor,
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la
Universidad 30, 28911 Leganés, Spain. Grupo de Teorías de Campos y Física
Estadística, Unidad Asociada al IEM-CSIC, Serrano 123, 28006 Madrid, Spain
<EMAIL_ADDRESS>
###### Abstract.

We examine relativistic diffusion through the frame and observer bundles
associated with a Lorentzian manifold $(M,g)$. Our focus is on spacetimes with
a non-trivial isometry group, and we detail the conditions required to find
symmetric solutions of the relativistic diffusion equation. Additionally, we
analyze the conservation laws associated with the presence of Killing vector
fields on $(M,g)$ and their implications for the expressions of the geodesic
spray and the vertical Laplacian on both the frame and the observer bundles.
Finally, we present several relevant examples of symmetric spacetimes.

###### 2020 Mathematics Subject Classification:

53B30, 53Z05

This work has been supported by the Spanish Ministerio de Ciencia Innovación y
Universidades-Agencia Estatal de Investigación Grant No.
AEI/PID2020–116567GB-C22 and PID2021-128970OA-I00. E. J. S. V. is supported by
the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual
Agreement with UC3M in the line of Excellence of University Professors
(EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research
and Technological Innovation). A.L. was also partly supported by FEDER/Junta
de Andalucía Consejería de Universidad, Investigación e Innovación, and by the
European Union, through Grants No. A-FQM-644-UGR20, and No. C-EXP-251-UGR23.
The authors want to thank F. Barbero for many interesting discussions and
comments.

##  1\. Introduction

Relativistic kinetic theory has been studied almost since the inception of
general relativity, and its formulation has been developed by many scientists
over the years (see [sarbach2013relativistic, sarbach2014geometry,
acuna2022introduction] for a historical overview). The kinetic theory of
relativistic gases, initially proposed by Synge in 1934 [synge1934energy],
gained significance after the 1960s due to technological advancements and
discoveries such as quasars and the cosmic microwave background radiation.
Although most of the underlying geometric ideas had been developed in the
1960s by Berger [berger1965lectures, besse2012manifolds] in the Riemannian
context, the work of J. Ehlers in the 1970s [ehlers, ehlers1973survey] was
particularly important, providing a coherent and robust mathematical framework
where relativistic kinetic theory takes place on appropriate submanifolds of
the tangent bundle of a spacetime (see [sarbach2014geometry] for a recent
review).

The Fokker–Planck (FP) equation describes diffusion processes, with
applications in various domains of physics and engineering. In astrophysics,
for instance, they are employed to model the theory of cosmic rays
[lasuik2019]. Similarly, Fokker–Planck equations are utilized in plasma
physics to analyze the effects of near-miss encounters between ions (heavy
particles) and electrons [peeters2008].

However, unlike relativistic kinetic theory, the field of relativistic
diffusion is still in its early stages. The covariant description poses
significant challenges. In particular, there are numerous approaches to
deriving a Fokker–Planck equation from a system of stochastic differential
equations. Due to these ambiguities, multiple models exist in the literature
under the label ‘relativistic Fokker–Planck equation’; for further details, we
refer the readers to [debbasch2007relativistic, chevalier2008relativistic,
dunkel2009]. Besides the theoretical problems, experiments to test the
proposed theories are difficult to perform.

In the context of differential stochastic equations on a Riemannian manifold,
the orthonormal frame bundle proves to be particularly useful. This framework,
developed by [Elworthy1988] and [Hsu2002], is central to the construction of
Brownian motion in Riemannian manifolds. In the context of Minkowski
spacetime, the relativistic description of a Brownian process was pioneered by
Dudley in 1966 [dudley1966lorentz]. Nevertheless, the generalization of
Dudley’s work to the framework of general relativity (i.e., to a generic
Lorentzian manifold $(M,g)$) had to await the research of J. Franchi and Y. Le
Jan in 2005 [franchi_jan]. Their diffusion process, initially defined at the
level of pseudo-orthonormal frames $\mathcal{SO}^{+}(M)$, incorporates
Brownian noise only in the vertical directions and projects onto a diffusion
process on the pseudo-unit tangent bundle (unit observer bundle $UM$). The
infinitesimal generator of their $\mathcal{SO}^{+}(M)$-valued Stratonovitch
stochastic differential equation decomposes into the sum of the vertical
Laplacian and the horizontal vector field generating the geodesic flow. This
infinitesimal generator allows us to write a relativistic Fokker–Planck
equation. This Fokker–Planck equation was used by Calogero et al. to describe
diffusion in different cosmological settings [felix2010relativistic,
calogero2011kinetic, calogero2012cosmological, felix2013newtonian,
calogero2013cosmology, felix2014spatially, alho2015dynamics].

Building upon these ideas, A. Franchi and Y. Le Jan [franchi_jan] utilize the
bundle $\mathcal{SO}^{+}(M)$ of direct pseudo-orthonormal frames, with fibers
modelled on the special Lorentz group (and having their first element in the
positive half of the unit pseudo-sphere in the tangent space) to extend the
concept of relativistic diffusion to general Lorentzian manifolds. This
approach defines a Stratonovich stochastic differential equation that takes
values in the $\mathcal{SO}^{+}(M)$ group, similar to a Langevin equation.
When projected, this equation naturally generates a diffusion process on the
mass shell. Moreover, by following what is done in the Riemannian case, we can
induce a pure diffusion on $M$ through a pullback operation, which results in
a Fokker–Planck-type equation.

In this paper, we show that the perspective provided by the frame bundle
$\mathcal{SO}^{+}(M)$ helps to incorporate spacetime symmetries and define
symmetric solutions to the Fokker–Planck equations both in the frame and
observer bundle description. The careful way of incorporating symmetries into
the equations has been exemplified in the case of the Vlasov equation—as
demonstrated in works such as [sarbach2013relativistic] by Sarbach and
Zannias—particularly on the mass shell. Here, we present a generalization that
encompasses the previous result as a special case while also enabling the
incorporation of diffusion.

The structure of the paper is as follows: After this introduction, in Section
2, we present the necessary geometric structures necessary to describe
diffusion processes in the frame bundle of a spacetime. We discuss the
infinitesimal generator $\mathcal{L}$ of the Franchi–Le Jan-process, and we
prove that, when the isometry group of the spacetime is non-trivial, its
symmetry-reduced version is consistent. We also elucidate the connection of
$\mathcal{L}$ with the corresponding FP equation (symmetric or not) on the
$UM$ bundle. In Section 3, following a reasoning similar to that of Elhers in
[ehlers, ehlers1973survey], we present a justification of the FP equations
based on the conservation of the average number of particle world lines
crossing any Cauchy hypersurface. We also define the particle current density,
entropy current, and the energy-momentum tensor, along with some of their most
important properties. In Section 4, we provide the explicit equations in
several relevant spacetimes: flat Friedmann–Lemaître–Robertson–Walker
cosmologies, the exterior Schwarzschild spacetime, and the Nariai spacetime.
Finally, in Section 5, we present our conclusions and outline some future
lines of research.

##  2\. Diffusion in the orthonormal frame bundle

###  2.1. Geometric framework

The orthonormal frame bundle provides the natural arena for understanding
stochastic phenomena like Brownian motion on a Riemannian manifold [Hsu2002].
As discussed by [franchi_jan, franchi2012hyperbolic], this also holds for
Lorentzian manifolds of dimension $1+n$.

Let $(M,g)$ be an $(1+n)$-dimensional oriented, and time-oriented Lorentzian
manifold of signature $(-,+,\cdots,+)$. A frame $u\in\mathcal{F}_{x}(M)$ at a
point $x\in M$ is a linear isomorphism $u:\mathbb{R}^{1+n}\rightarrow T_{x}M$.
The set

$\mathcal{F}(M)=\bigcup_{x\in M}\mathcal{F}_{x}(M)\ ,$

of all frames on $M$ constitutes a principal $GL(1+n,\mathbb{R})$-bundle over
$M$. This principal bundle is referred to as the frame bundle of $M$. In terms
of local coordinates $x^{\mu}$ on $M$, a frame $u$ can be written in the form

$u(e_{I})=e^{\mu}_{I}\frac{\partial}{\partial
x^{\mu}}\,,\quad\det{e^{\mu}_{I}}\neq 0\,,$

where $e_{I}$, $0\leq I\leq n$, denotes the canonical basis of
$\mathbb{R}^{1+n}$. Hence $(x^{\mu},e^{\mu}_{I})$ are local bundle coordinates
for $\mathcal{F}(M)$. If we focus on the set of $g$-orthonormal frames instead
of all linear frames we can construct the $O(1+n)$-principal bundle of
orthonormal frames denoted by $\mathcal{O}(M)$. Furthermore, an additional
reduction of $\mathcal{F}(M)$ can be considered by changing the gauge group
from $GL(1+n,\mathbb{R})$ to the restricted (proper, orthochronous) Lorentz
group $SO^{+}(1,n)$:

$\mathcal{SO}^{+}(M)=\bigcup_{x\in M}\mathcal{SO}^{+}_{x}(M),$

where

$\displaystyle\mathcal{SO}^{+}_{x}(M)$
$\displaystyle=\\{u\in\mathrm{Hom}^{+}(\mathbb{R}^{1,n},T_{x}M)\,:\,g_{x}(u(e_{I}),u(e_{J}))=\langle
e_{I},e_{J}\rangle_{1,n}\,,0\leq I,J\leq n\,,$ $\displaystyle\hskip
150.79968ptu(e_{0})\textrm{ future pointing }\\}\ .$

Here and in the following we will assume that
$e_{I}\in\mathbb{R}^{1+n}=\mathbb{R}^{1,n}$ satisfy

$\langle
e_{I},e_{J}\rangle_{1,n}=\eta_{{}_{IJ}}=\left\\{\begin{array}[]{rcl}-1&\mathrm{if}&\,I=J=0\\\
1&\mathrm{if}&\,I=J=i\neq 0\\\ 0&&\,\mathrm{otherwise}\end{array}\right.,$

where $\eta_{IJ}$ represents the components of the Minkowski metric on
$\mathbb{R}^{1,n}$.

It is straightforward to show that $\mathcal{SO}^{+}(M)$ is a manifold of
dimension $1+n+\binom{1+n}{2}$. A frame $u\in\mathcal{SO}_{x}^{+}(M)$ is an
isometry from the $(1+n)$-dimensional Minkowski space
$(\mathbb{R}^{1,n},\langle\cdot,\cdot\rangle_{1,n})$ to $(T_{x}M,g_{x})$ that
preserves orientation and time-orientation. We will denote by $\pi$ be
canonical projection $\pi:\mathcal{SO}^{+}(M)\rightarrow M$ such that, given
$u\in\mathcal{SO}_{x}^{+}(M)$, $\pi(u)=x$. Notice also that, by construction,
$u(e_{I})$ may be interpreted as the $I$-th vector of a basis for the tangent
space $T_{x}M$, effectively serving as the $I$-th vector of the frame.

We will denote by $\Phi$ the canonical right action of $SO^{+}(1,n)$ on
$\mathcal{SO}^{+}(M)$,

$\displaystyle\Phi:\mathcal{SO}^{+}(M)\times SO^{+}(1,n)$
$\displaystyle\rightarrow\mathcal{SO}^{+}(M),$ $\displaystyle(u,A)$
$\displaystyle\mapsto\Phi(u,A)=\Phi_{u}(A)=\Phi^{A}(u)=uA,$

where $uA$ denotes the map
$(uA)(e_{I}):=u(Ae_{I})=u(e_{J}A^{J}_{\,I})=u(e_{J})A^{J}_{\,I}$. This action
enables the introduction of the so called fundamental vector fields on
$\mathcal{SO}^{+}(M)$. These vector fields are given by the image of the
mapping
$\lambda:\mathfrak{so}(1,n)\rightarrow\mathfrak{X}(\mathcal{SO}^{+}(M))$,
defined by

$(\lambda X)(u):=T_{e}\Phi_{u}(X),\quad X\in\mathfrak{so}(1,n),$

highlighting that each fundamental vector field is inherently a vertical
field, satisfying

$T\pi(\lambda X)=0,$

and obeys the relation $T\Phi^{A}(\lambda X)=\lambda(\mathrm{Ad}_{A^{-1}}X)$.

In aligning our methodology with the results of [franchi_jan], we endow the
Lie algebra $\mathfrak{so}(1,n)$ with the trace form $B$ defined by

$B(X,Y)=-\frac{1}{2}\mathrm{tr}(XY)\ .$

The $\mathfrak{so}(1,n)$-basis $X_{IJ}$, for $0\leq I<J\leq n$, defined by

$(X_{IJ})^{K}\,_{L}=\delta^{K}\,_{I}\eta_{JL}-\delta^{K}\,_{J}\eta_{IL},$

is $B$-orthonormal

$B(X_{I_{1}J_{1}},X_{I_{2}J_{2}})=\eta_{I_{1}I_{2}}\eta_{J_{1}J_{2}}.$

Notice that $X_{0i}$ and $X_{ij}$ correspond to generators of boosts and
rotations, respectively. The $\mathrm{ad}$-invariant inner product $B$ allows
us to equip $SO^{+}(1,n)$ with a bi-invariant semi-Riemannian metric, denoted
as $(\\!\\!(\cdot,\cdot)\\!\\!)$, which is defined via the pullback of $B$ by
left-translations, $(\\!\\!(\cdot,\cdot)\\!\\!)_{A}=L^{*}_{A}B$. The Levi-
Civita connection associated with this metric, satisfies the relation,

$D_{X}Y=\frac{1}{2}[X,Y]\,,\quad\forall X,Y\in\mathfrak{so}(1,n)\,.$

Hence, in terms of the map $\lambda$, the quadratic Casimir element
$\mathcal{C}:=c(B)$ associated with $B$ acts as a second-order differential
operator on $C^{\infty}(\mathcal{SO}^{+}(M))$ through

$\mathcal{C}f:=c(B)f=\mathrm{div}(Df)=\sum_{0\leq I<J\leq
n}\eta_{II}\eta_{JJ}(\\!\\!(X_{IJ},D_{X_{IJ}}Df)\\!\\!)=\sum_{0\leq I<J\leq
n}\eta_{II}\eta_{JJ}X_{IJ}(X_{IJ}f)\,.$

Following again the notation of reference [franchi_jan], we set

$V_{i}:=\lambda(X_{0i})\,,\quad 1<i<n\,,\quad
V_{ij}:=\lambda(X_{ij})\quad\textrm{ for }1\leq i<j\leq n,$

and we will write

$\displaystyle\mathcal{C}=\sum_{0\leq I<J\leq
n}\eta_{II}\eta_{JJ}\lambda(X_{IJ})^{2}=\sum_{1\leq i<j\leq
d}V_{ij}^{2}-\sum_{i=1}^{d}V_{i}^{2}\,.$

Notice that $SO^{+}(1,n)$ is not compact. The non-compactness is responsible
for the fact that the bi-invariant metric $(\\!\\!(\cdot,\cdot)\\!\\!)$ is not
Riemannian, and the operator $\mathcal{C}$ is neither positive nor negative
definite. We will see that this fact does not prevent us from defining a
diffusion operator through $\mathcal{C}$.

Finally, using the Levi-Civita connection $\nabla$ on $(M,g)$, the tangent
space $T\mathcal{SO}^{+}(M)$ can be decomposed in the form

$T\mathcal{SO}^{+}(M)=\mathcal{V}\oplus\mathcal{H}\ ,$

where the fibers $\mathcal{V}_{u}$ of the vertical bundle $\mathcal{V}$
satisfy $\mathcal{V}_{u}=\ker T_{u}\pi$ and, on the other hand, the horizontal
bundle

$\mathcal{H}_{u}=\\{H_{\mathrm{v}}(u)\,:\,\mathrm{v}\in\mathbb{R}^{1+n}\\},$

is defined in terms of the horizontal vector fields $H_{\mathrm{v}}$ that are
characterized by the following construction: given
$\mathrm{v}\in\mathbb{R}^{1+n}$ and $u\in\mathcal{SO}^{+}(M)$, there is a
unique horizontal vector field $H_{\mathrm{v}}$ satisfying

$\theta(H_{\mathrm{v}}(u))=\mathrm{v}\,,$

where $\theta=e_{I}\theta^{I}$ is the $\mathbb{R}^{n+1}$-valued soldering one-
form (canonically) defined by

$T_{u}\pi(v)=u(\theta(v))=\theta^{I}(v)u(e_{I})\,,\quad v\in
T_{u}\mathcal{SO}^{+}(M)\,.$

It is also easy to show that
$H_{\alpha_{1}\mathrm{v}_{1}+\alpha_{2}\mathrm{v}_{2}}=\alpha_{1}H_{\mathrm{v}_{1}}+\alpha_{2}H_{\mathrm{v}_{2}}$
and $T\Phi^{A}(H_{\mathrm{v}})=H_{A^{-1}\mathrm{v}}$. This implies that, not
being right invariant, the vector fields $H_{\mathrm{v}}$ are not horizontal
lifts of vector fields on $M$. In the case $\mathrm{v}=e_{I}$, we will use the
notation $H_{I}:=H_{e_{I}}$ to make contact with [franchi_jan].

The fields $H_{\mathrm{v}}$ are often called basic vector fields and satisfy
the following [Bishop]:

######  Theorem 1 (Geodesics).

Let $\tilde{\gamma}$ be the horizontal lift through $u\in\pi^{-1}(\gamma(0))$
of a smooth curve $\gamma$ in $M$ to $\mathcal{SO}^{+}(M)$. Then,
$\gamma=\pi\circ\tilde{\gamma}$ is a geodesic if and only if there exists a
vector $\mathrm{v}=\mathrm{v}^{I}e_{I}\in\mathbb{R}^{1+n}$ such that
$\tilde{\gamma}$ is an integral curve of the horizontal vector field
$H_{\mathrm{v}}$. This condition is satisfied if and only if

(1) $\tilde{\gamma}^{\prime}(s)=\mathrm{v}^{I}H_{I}(\tilde{\gamma}(s)),$

where $\tilde{\gamma}^{\prime}(s)$ denotes the derivative of $\tilde{\gamma}$
with respect to the parameter $s$.

######  Theorem 2 (Parallelization of $\mathcal{SO}^{+}(M)$).

The vector fields $H_{I}$, $V_{i}$, and $V_{ij}$ provide a parallelization of
$\mathcal{SO}^{+}(M)$; that is, for every $u\in\mathcal{SO}^{+}(M)$, the
tangent vectors $H_{I}(u)$, $V_{i}(u)$, and $V_{ij}(u)$ form a basis of
$T_{u}\mathcal{SO}^{+}(M)$.

In terms of adapted coordinates $(x^{\mu},e^{\mu}_{I})$ of the frame bundle
$\mathcal{F}(M)$, the vector fields $H_{I}$ and $\lambda(X_{IJ})$ can be
written as

$\displaystyle H_{I}$ $\displaystyle=e^{\alpha}_{I}\frac{\partial}{\partial
x^{\alpha}}-e^{\mu}_{I}e_{J}^{\nu}\Gamma^{\alpha}_{\mu\nu}\frac{\partial}{\partial
e^{\alpha}_{J}}\,,\quad\lambda(X_{IJ})=(\eta_{JK}e^{\mu}_{I}-\eta_{IK}e^{\mu}_{J})\frac{\partial}{\partial
e^{\mu}_{K}}\ ,$

where $\Gamma^{\alpha}_{\mu\nu}$ denotes the Christoffel symbols (i.e. the
connection coefficients of the Levi-Civita connection associated with $g$,
expressed in the coordinate basis $x^{\mu}$:
$\nabla_{\partial_{x^{\mu}}}\partial_{x^{\nu}}=\Gamma^{\alpha}_{\mu\nu}\partial_{x^{\alpha}}$).
The coordinate expression for $\lambda(X_{IJ})$ can be equivalently written as

$\displaystyle V_{i}=e^{\mu}_{0}\frac{\partial}{\partial
e^{\mu}_{i}}+e^{\mu}_{i}\frac{\partial}{\partial e^{\mu}_{0}}\ ,\quad
V_{ij}=e^{\mu}_{i}\frac{\partial}{\partial
e^{\mu}_{j}}-e^{\mu}_{j}\frac{\partial}{\partial e^{\mu}_{i}}\ .$

###  2.2. Fokker–Planck equation

The geometric objects introduced in the previous section allowed J. Franchi
and Y. Le Jan in [franchi_jan] to define an $\mathcal{SO}^{+}(M)$-valued
Stratonovitch stochastic differential equation,

$\mathrm{d}\Psi_{s}=\left(H_{0}(\Psi_{s})\,\mathrm{d}s+\sigma\sum_{i=1}^{n}V_{i}(\Psi_{s})\circ\mathrm{d}W^{i}_{s}\right)\
,$

for $\Psi=(\Psi_{s})\in\mathcal{SO}^{+}(M)$. Here $W=(W^{i}_{s})$ is a
$\mathbb{R}^{n}$-valued Brownian motion, and $\sigma>0$ is a constant that
measures the strength of the diffusion process. This equation describes a
natural geometric diffusion process in $\mathcal{SO}^{+}(M)$. The generator of
this process is

$\mathcal{L}:=H_{0}+\frac{\sigma^{2}}{2}\sum_{i=1}^{n}V^{2}_{i}\ ,$

whose adjoint $\mathcal{L}^{*}$ allows us to write the Fokker–Planck equation

(2)
$\displaystyle\left(H_{0}-\frac{\sigma^{2}}{2}\sum_{i=1}^{n}V^{2}_{i}\right)F=0\
.$

In section 3, we will justify equation (2) generalizing the line of reasoning
followed by J. Elhers in reference [ehlers] to derive the Vlasov (Liouville)
equation.

###  2.3. Symmetries

In numerous physically relevant contexts, the spacetime isometry group
$\mathrm{Iso}(M,g)$ is not trivial and its Lie algebra is characterized
through the (complete) Killing vector fields. In other words, the flow of a
Killing vector field $\xi\in\mathfrak{X}(M)$ is an isometry of $(M,g)$ and,
hence

(3) $\pounds_{\xi}g=0.$

It is well known that the presence of Killing fields leads to conservation
laws (see, for example, [o1983semi]): if $\xi$ is a Killing vector field, the
smooth functions

$C^{\xi}_{I}:\mathcal{F}(M)\rightarrow\mathbb{R}\,,\quad u\mapsto
C^{\xi}_{I}(u)=g(u(e_{I}),\xi),\quad 0\leq I\leq n\,,$

satisfy

(4) $\pounds_{H_{I}}C^{\xi}_{I}=0\,.$

In particular $\pounds_{H_{0}}C^{\xi}_{0}=0$.

In addition to these conservation laws, as discussed in [KN, mok1979complete],
given any vector field $\xi\in\mathfrak{X}(M)$, there exists a complete lift
(natural lift) of $\xi$ to $\mathcal{F}(M)$ which we will denote as
$\xi^{\mathrm{c}}\in\mathfrak{X}(\mathcal{F}(M))$. The complete lift is
characterised by the following three properties:

  1. (1)

$\xi^{\mathrm{c}}$ remains invariant under right translations, i.e.,
$\Phi^{A*}\xi^{\mathrm{c}}=\xi^{\mathrm{c}}$;

  2. (2)

The Lie derivative of the soldering form with respect to $\xi^{\mathrm{c}}$
vanishes, $\pounds_{\xi^{\mathrm{c}}}\theta=0$;

  3. (3)

$\xi^{\mathrm{c}}$ is $\pi$-related to $\xi$, such that
$T_{u}\pi(\xi^{\mathrm{c}})=\xi(\pi(u))$ for all $u\in\mathcal{F}(M)$.

In bundle coordinates, the complete lift is represented as

(5) $\xi=\xi^{\mu}\frac{\partial}{\partial
x^{\mu}}\mapsto\xi^{\mathrm{c}}=\xi^{\mu}\frac{\partial}{\partial
x^{\mu}}+e^{\nu}_{I}\frac{\partial\xi^{\mu}}{\partial
x^{\nu}}\frac{\partial}{\partial e^{\mu}_{I}},$

Moreover, if $\xi$ is a Killing vector field, then $\xi^{\mathrm{c}}$
tangentially aligns with the bundle $\mathcal{SO}^{+}(M)$. This follows from
the fact that

$\displaystyle\pounds_{\xi^{c}}F_{IJ}=0\ ,$

where $F_{IJ}:\mathcal{F}(M)\rightarrow\mathbb{R}^{(2+n)(1+n)/2},\,u\mapsto
F_{IJ}(u)=g(u(e_{I}),u(e_{J}))$.

The previous results allow us to prove the following proposition that relates
the complete lift of Killing fields with the fundamental and basic vector
fields:

######  Proposition 3.

The complete lift $\xi^{\mathrm{c}}$ of a Killing vector field $\xi$ to
$\mathcal{SO}^{+}(M)$ commutes with both fundamental and basic vector fields,
specifically with $V_{i}$, $V_{ij}$, and $H_{I}$.

###### Proof.

Given $\xi\in\mathfrak{X}(\mathcal{M})$ with $\pounds_{\xi}g=0$, it suffices
to verify that the commutator of $\xi^{\mathrm{c}}$ with each basis vector
field $(V_{i},V_{ij},H_{I})$ vanishes. Employing local coordinates,

$\displaystyle[V_{i},\xi^{\mathrm{c}}]=$
$\displaystyle\left[e^{\mu}_{0}\frac{\partial}{\partial
e^{\mu}_{i}}+e^{\mu}_{i}\frac{\partial}{\partial
e^{\mu}_{0}},\xi^{\alpha}\frac{\partial}{\partial
x^{\alpha}}+e^{\alpha}_{K}\frac{\partial\xi^{\beta}}{\partial
x^{\alpha}}\frac{\partial}{\partial e^{\beta}_{K}}\right]$ $\displaystyle=$
$\displaystyle\frac{\partial\xi^{\beta}}{\partial
x^{\alpha}}\left(\left[e^{\mu}_{0}\frac{\partial}{\partial
e^{\mu}_{i}},e^{\alpha}_{K}\frac{\partial}{\partial
e^{\beta}_{K}}\right]+\left[e^{\mu}_{i}\frac{\partial}{\partial
e^{\mu}_{0}},e^{\alpha}_{K}\frac{\partial}{\partial
e^{\beta}_{K}}\right]\right)=0$ $\displaystyle[V_{ij},\xi^{\mathrm{c}}]=$
$\displaystyle\left[e^{\mu}_{j}\frac{\partial}{\partial
e^{\mu}_{i}}-e^{\mu}_{i}\frac{\partial}{\partial
e^{\mu}_{j}},\xi^{\alpha}\frac{\partial}{\partial
x^{\alpha}}+e^{\alpha}_{K}\frac{\partial\xi^{\beta}}{\partial
x^{\alpha}}\frac{\partial}{\partial e^{\beta}_{K}}\right]$ $\displaystyle=$
$\displaystyle\frac{\partial\xi^{\beta}}{\partial
x^{\alpha}}\left(\left[e^{\mu}_{j}\frac{\partial}{\partial
e^{\mu}_{i}},e^{\alpha}_{K}\frac{\partial}{\partial
e^{\beta}_{K}}\right]-\left[e^{\mu}_{i}\frac{\partial}{\partial
e^{\mu}_{j}},e^{\alpha}_{K}\frac{\partial}{\partial
e^{\beta}_{K}}\right]\right)=0$ $\displaystyle[H_{I},\xi^{\mathrm{c}}]=$
$\displaystyle\left[e^{\mu}_{I}\frac{\partial}{\partial
x^{\mu}}-e^{\gamma}_{I}e_{J}^{\nu}\Gamma^{\mu}_{\gamma\nu}\frac{\partial}{\partial
e^{\mu}_{J}},\xi^{\alpha}\frac{\partial}{\partial
x^{\alpha}}+e^{\alpha}_{K}\frac{\partial\xi^{\beta}}{\partial
x^{\alpha}}\frac{\partial}{\partial e^{\beta}_{K}}\right]$ $\displaystyle=$
$\displaystyle e^{\mu}_{I}e^{\nu}_{J}\left(\frac{\partial^{2}\xi}{\partial
x^{\mu}\partial
x^{\nu}}+\xi^{\beta}\frac{\partial\Gamma^{\alpha}_{\mu\nu}}{\partial
x^{\beta}}+\frac{\partial\xi^{\beta}}{\partial
x^{\mu}}\Gamma^{\alpha}_{\beta\nu}+\frac{\partial\xi^{\beta}}{\partial
x^{\nu}}\Gamma^{\alpha}_{\mu\beta}-\frac{\partial\xi^{\alpha}}{\partial
x^{\beta}}\Gamma^{\beta}_{\mu\nu}\right)\frac{\partial}{\partial
e^{\alpha}_{J}}$ $\displaystyle=$ $\displaystyle
e^{\mu}_{I}e^{\nu}_{J}\left(\pounds_{\xi}\Gamma\right)^{\alpha}_{\mu\nu}\frac{\partial}{\partial
e^{\alpha}_{J}}=0\,,$

In the last line we have used that $\pounds_{\xi}g=0$ and, thus,
$\pounds_{\xi}\Gamma=0$. ∎

Note that in the case of the vertical vectors $V_{i},V_{ij}$, we have not used
the fact that the vector field $\xi^{\mathrm{c}}$ is the lift of a Killing
vector field. Hence, for vertical linear operators, the vanishing of the
commutator holds for lifts of arbitrary vector fields on $M$ but in this case
$\xi^{\mathrm{c}}$ is not tangent to $\mathcal{SO}^{+}(M)$.

######  Corollary 4.

The complete lift $\xi^{\mathrm{c}}$ of $\xi$ commutes with the second-order
differential operators $\lambda(X_{IJ})^{2}$; in particular, it commutes with
$V^{2}_{i}$, $V^{2}_{ij}$, and with the Casimir $\mathcal{C}$.

###### Proof.

Using the fact that

$\pounds_{\xi^{\mathrm{c}}}\pounds_{\lambda(X_{IJ})}-\pounds_{\lambda(X_{IJ})}\pounds_{\xi^{\mathrm{c}}}=\pounds_{[\xi^{\mathrm{c}},\lambda(X_{IJ})]}=0,$

and taking into account that the second-order operator
$\lambda(X_{{}_{IJ}})^{2}$ can be written in the form
$\lambda(X_{IJ})^{2}f=\pounds_{X_{IJ}}(\pounds_{X_{IJ}}f)$, we have

$\pounds_{\xi^{\mathrm{c}}}\pounds_{X_{IJ}}(\pounds_{X_{IJ}}f)=\pounds_{X_{IJ}}(\pounds_{X_{IJ}}(\pounds_{\xi^{\mathrm{c}}}f))\,.$

∎

With this in mind, we can finally delineate the conditions that symmetric
solutions must fulfill to satisfy a Fokker–Planck type equation.

######  Proposition 5.

The symmetric solutions for the FP diffusion equation in $\mathcal{SO}^{+}(M)$
are those functions simultaneously satisfying

(6)
$\displaystyle\left(H_{0}-\frac{\sigma^{2}}{2}\sum_{i=1}^{n}V^{2}_{i}\right)F=0\,,\quad\pounds_{\xi^{\mathrm{c}}}F=0\,,$

for every Killing field $\xi$ of $(M,g)$.

######  Remark 6.

The vanishing of the commutator of $\xi^{\mathrm{c}}$ with the basic and
fundamental vector fields is crucial for the consistency of the symmetric
equations because

$\left(H_{0}-\frac{\sigma^{2}}{2}\sum_{i=1}^{n}V^{2}_{i}\right)F=0\Rightarrow\pounds_{\xi^{\mathrm{c}}}\left(H_{0}-\frac{\sigma^{2}}{2}\sum_{i=1}^{n}V^{2}_{i}\right)F=0,$

but this last equation is trivially satisfied when
$\pounds_{\xi^{\mathrm{c}}}F=0$ due to commutativity mentioned above.

######  Remark 7.

The conclusions of the previous corollary are also satisfied for the equation

$\left(H_{0}+\frac{\sigma^{2}}{2}\mathcal{C}\right)F=0\,.$

As we will see in Lemma 8 of Section 2.4, this equation (which is not a
diffusion equation) induces the same FP equation in the unit-tangent bundle.

###  2.4. Double and timelike unit tangent bundles in Lorentzian manifolds

The bundle $\mathcal{SO}^{+}(M)$ is very useful from the point of view of both
writing stochastic differential equations and working with symmetries as we
have seen so far. However, from a physical point of view, the frames have too
many degrees of freedom, and one would like to work in the configuration
space. This is the approach directly taken by [ehlers], developing the
equations with the structures of the physically relevant configurations,
corresponding to the (unit mass) observer subbundle $UM$ of the tangent bundle
$TM$. The introduction of $UM$ by J.A. Thorpe, as referenced in [Th1] and
further discussed in [Beem], underpins the study of space-time singularities
by providing a mathematically rigorous platform. The purpose of this
subsection is to present these natural structures and to establish a
connection with the structures introduced previously.

In the following we will need to consider the double tangent bundle, denoted
as $TTM$, of a semi-Riemannian manifold $(M,g)$. This bundle can be decomposed
into vertical and horizontal sub-bundles, represented as:

(7) $TTM=\mathcal{V}TM\oplus\mathcal{H}TM.$

For any tangent vector $v\in TM$, the vertical subspace $\mathcal{V}_{v}TM$
and the horizontal subspace $\mathcal{H}_{v}TM$ at $v$ are naturally
isomorphic to the tangent space at the base point $\mathrm{p}(v)$ of $v$, with
$\mathrm{p}:TM\to M$ being the canonical projection.

Given a vector $X\in T_{\mathrm{p}(v)}M$, one can select a curve $\gamma(t)$
in $M$ where $\gamma^{\prime}(0)=X$. Utilizing the Levi-Civita connection, we
construct a curve $c$ in $TM$ satisfying $\gamma=\mathrm{p}\circ c$ and
$c(0)=v$, with the condition $\nabla_{\gamma^{\prime}}c=0$. Through this
construction, $v$ is mapped to $c^{\prime}(0)$, a process referred to as the
horizontal lift of $X$. In local coordinates, this horizontal lift is
expressed as:

$T_{\mathrm{p}(v)}M\ni X=X^{\mu}\frac{\partial}{\partial x^{\mu}}\mapsto
X^{\mathcal{H}}=X^{\mu}\frac{\partial}{\partial
x^{\mu}}-\Gamma^{\mu}_{\alpha\beta}v^{\alpha}X^{\beta}\frac{\partial}{\partial
v^{\mu}}\in\mathcal{H}_{v}M\,.$

where $\Gamma^{\mu}_{\alpha\beta}$ are the Christoffel symbols of the Levi-
Civita connection. Therefore, the identification of vectors in the horizontal
subspace $\mathcal{H}_{v}TM$ with vectors in $T_{\mathrm{p}(v)}M$ is
facilitated by the tangent map of the projection $\mathrm{p}:TM\to M$,
restricted to $\mathcal{H}TM$.

On the other hand, concerning the vertical component of a vector in the double
tangent bundle $TTM$, for any vector $v$ in the tangent bundle $TM$, the space
$T_{\mathrm{p}(v)}M$ is a vector space. This fact establishes a natural
isomorphism $\imath_{v}:T_{v}T_{\mathrm{p}(v)}M=\mathcal{V}_{v}TM\rightarrow
T_{\mathrm{p}(v)}M$. In local coordinates, this isomorphism takes the form

$\imath_{v}(V^{\mu}\frac{\partial}{\partial
v^{\mu}})=V^{\mu}\frac{\partial}{\partial x^{\mu}}\,.$

The reverse process is facilitated by the so-called vertical lift, which for
any $X\in T_{\mathrm{p}(v)}M$, is given by

$T_{\mathrm{p}(v)}M\ni X=X^{\mu}\frac{\partial}{\partial x^{\mu}}\mapsto
X^{\mathcal{V}}=X^{\mu}\frac{\partial}{\partial v^{\mu}}\in
T_{v}T_{\mathrm{p}(v)}M\,.$

To decompose any vector $W\in TTM$ into its vertical and horizontal
components,

$W=W^{\mathrm{ver}}+W^{\mathrm{hor}},$

an additional construct, the connection map $K$ associated with the Levi-
Civita connection, is introduced, where $K:TTM\rightarrow\mathcal{V}TM$. The
decomposition in coordinates is described as

$W^{\mathrm{ver}}=K_{v}(W)\,,\quad W^{\mathrm{hor}}=W-K_{v}(W)\,,$

and specifically,

$K_{v}\left(X^{\mu}\frac{\partial}{\partial
x^{\mu}}+V^{\mu}\frac{\partial}{\partial
v^{\mu}}\right)=\left(V^{\mu}+\Gamma^{\mu}_{\alpha\beta}v^{\alpha}X^{\beta}\right)\frac{\partial}{\partial
v^{\mu}}\,,\quad W\in T_{v}TM\,.$

These structures enable the definition of a natural horizontal vector field
$L\in\mathfrak{X}(TM)$, mapping $v\mapsto L(v)\in T_{v}TM$ and determined by
the requirements

$T\mathrm{p}(L(v))=v\,,\quad K_{v}(L(v))=0\,,\quad\forall v\in TM\,,$

or, equivalently, $L(v)^{\mathrm{hor}}=v^{\mathcal{H}}$,
$L(v)^{\mathrm{ver}}=0$. This vector field, $L$, known as the geodesic spray,
is characterized in local coordinates by

$L(v)=v^{\mu}\frac{\partial}{\partial
x^{\mu}}(v)-\Gamma^{\mu}_{\alpha\beta}v^{\alpha}v^{\beta}\frac{\partial}{\partial
v^{\mu}}(v)\,.$

In the following, we will also use the canonically vertical vector field $A$
in $TM$ (the Liouville vector field) that is, the generator of the dilations

$a^{t}:TM\rightarrow TM\,,\quad v\mapsto a^{t}(v)=tv\,,\quad
t\in\mathbb{R}\setminus\\{0\\}.$

In bundle coordinates

$A(v)=v^{\mu}\frac{\partial}{\partial v^{\mu}}(v)\,.$

These canonical isomorphisms can be also used to define the Sasaki metric g on
$TM$ associated with the metric $g$ on $M$. The Sasaki metric is formulated by
designating the vertical bundle $\mathcal{V}_{v}TM$ and the horizontal bundle
$\mathcal{H}_{v}TM$ as orthogonal components. Mathematically, this is
expressed as

(8)
$\textsf{g}(V,W)=g(T_{v}\mathrm{p}(V),T_{v}\mathrm{p}(W))+g(\imath_{v}K_{v}(V),\imath_{v}K_{v}(W))\,,\quad
V,W\in T_{v}TM\,.$

In local coordinates, the Sasaki metric can be written in detail as

(9)
$\textsf{g}=g_{\mu\nu}\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}+g_{\mu\nu}(\mathrm{d}v^{\mu}+\Gamma^{\mu}_{\alpha\beta}v^{\alpha}\mathrm{d}x^{\beta})\otimes(\mathrm{d}v^{\mu}+\Gamma^{\mu}_{\gamma\delta}v^{\gamma}\mathrm{d}x^{\delta})\,.$

It is important to note that if the signature of $g$ is $(1,n)$, then,
correspondingly, the signature of g is $(2,2n)$, reflecting the doubled
dimensionality and the preservation of the manifold’s metric properties within
its tangent bundle.

As we will see, the pullback of g to the unit future observer bundle,

$\displaystyle UM=U^{1}M=\\{v\in TM\mid g(v,v)=-1,v\text{ is future
directed}\\}\subset TM\,,$

is important for the analysis of the FP equation. $UM$ is a codimension one
submanifold of $TM$ and the canonical projection $\tau:UM\rightarrow M$ equips
it as a subbundle of $TM$.

One notable property of $UM$ is that any tangent vector within $\mathcal{H}TM$
also belongs to the tangent space of $UM$ ($\mathcal{H}_{v}TM\subset
T_{v}UM$). This inclusion is substantiated by the behavior of the function
$E:TM\rightarrow\mathbb{R}$, defined by $v\mapsto E(v)=g(v,v)/2$, which
maintains a constant value for all vectors in $\mathcal{H}TM$, as indicated by
$\pounds_{W}E=\mathrm{d}E(W)=0$. This feature ensures that the geodesic flow,
generated by the vector field $L$, preserves the structure of $UM$, hence
maintaining its integrity under dynamical evolution.

The Sasaki metric g on $TM$ induces a Lorentzian metric $\mathrm{g}$ of
signature $(1,2n)$ on $UM$. This metric equips the $n$-dimensional
submanifolds $\mathbb{H}_{v}=\tau^{-1}(\tau(v))\subset UM$ with a Riemannian
(hyperbolic) metric, where each $\mathbb{H}_{v}$ represents a fiber of the
projection $\tau:UM\rightarrow M$ through $v$. The Riemannian metric,
characterized by its positive signature and hyperbolic nature, lays the
groundwork for defining the vertical Laplacian $\Delta^{\\!\mathrm{ver}}$, a
second-order differential operator acting on $C^{\infty}(UM)$. The vertical
Laplacian is expressed as

$(\Delta^{\\!\mathrm{ver}}f)(v)=(\Delta^{\mathbb{H}_{v}}f\upharpoonright\mathbb{H}_{v})(v)\,,$

where $f\upharpoonright\mathbb{H}_{v}$ denotes the restriction of $f$ to
$\mathbb{H}_{v}$, and $\Delta^{\mathbb{H}_{v}}$ is the Laplace-Beltrami
operator associated with the metric induced on $\mathbb{H}_{v}$ by
$(UM,\mathrm{g})$. The local coordinate representation of
$\Delta^{\\!\mathrm{ver}}$ highlights its dependency on the geometric
structure of $UM$ and the dynamics of space-time, as given by

$\Delta^{\\!\mathrm{ver}}f=\left(n\,v^{i}\frac{\partial}{\partial
v^{i}}+\left(v^{i}v^{j}+g^{ij}\right)\frac{\partial^{2}}{\partial
v^{i}\partial v^{j}}\right)f\,.$

We already have all the ingredients to establish the form of the Fokker–Planck
equation on $UM$ and its relationship with the one introduced in
$\mathcal{SO}^{+}(M)$. This is achieved through the following:

######  Lemma 8.

Let $\pi_{0}:\mathcal{SO}^{+}(M)\rightarrow UM\,,u\mapsto\ u(e_{0})$ and
$f\in\mathcal{C}^{\infty}\left(UM\right)$, then

  * (1)

$T\pi_{0}(V_{ij})=0$ and $V_{ij}\left(\pi_{0}^{*}f\right)=0$,

  * (2)

$T\pi_{0}(H_{0})=L$ and
$H_{0}\left(\pi_{0}^{*}f\right)=\pi_{0}^{*}\left(Lf\right)$,

  * (3)

$\displaystyle\mathcal{C}\left(\pi_{0}^{*}f\right)=-\sum_{i=1}^{n}V_{i}^{2}\left(\pi_{0}^{*}f\right)=-\pi_{0}^{*}\left(\Delta^{\\!\mathrm{ver}}f\right)$

where $\Delta^{\\!\mathrm{ver}}$ is the vertical Laplacian and $L$ is the
geodesic spray of $UM$.

###### Proof.

Properties (1) and (2) follow almost directly:

$\displaystyle V_{ij}(\pi_{0}^{*}f)$
$\displaystyle=\mathrm{d}(\pi_{0}^{*}f)(V_{ij})=\pi_{0}^{*}(\mathrm{d}f)(V_{ij})=(\mathrm{d}f)(T\pi_{0}V_{ij})\circ\pi_{0}=0\,,$
$\displaystyle H_{0}(\pi_{0}^{*}f)$
$\displaystyle=\mathrm{d}(\pi_{0}^{*}f)(H_{0})=\pi_{0}^{*}(\mathrm{d}f)(H_{0})=(\mathrm{d}f)(T\pi_{0}H_{0})\circ\pi_{0}=(\mathrm{d}F)(L)\circ\pi_{0}=\pi_{0}^{*}(Lf)\,,$

where we have used that $T\pi_{0}V_{ij}=0$ and $T\pi_{0}H_{0}=L$.

Property (3) can be derived in local coordinates for any function $F\in
C^{\infty}(\mathcal{SO}^{+}(M))$,

$\displaystyle\sum_{i=1}^{n}V_{i}^{2}F=\sum_{i=1}^{n}\left(e^{j}_{i}e^{k}_{i}\frac{\partial}{\partial
e_{0}^{j}}\frac{\partial}{\partial
e_{0}^{k}}+e^{j}_{i}\frac{\partial}{\partial
e_{i}^{j}}+ne_{0}^{j}\frac{\partial}{\partial
e_{0}^{j}}+2e_{0}^{j}e_{i}^{k}\frac{\partial}{\partial
e_{0}^{k}}\frac{\partial}{\partial
e_{i}^{j}}+e_{0}^{j}e_{0}^{k}\frac{\partial}{\partial
e_{i}^{j}}\frac{\partial}{\partial e_{i}^{k}}\right)F\ .$

Hence, when $F=\pi_{0}^{*}f$, the derivatives with respect to $e_{i}^{j}$
vanish, and, identifying $e_{0}^{i}=v^{i}$,

$\displaystyle\mathcal{C}\left(\pi_{0}^{*}f\right)=-\sum_{i=1}^{n}V^{2}_{i}\left(\pi_{0}^{*}f\right)=-\left(n\
v^{i}\frac{\partial}{\partial
v^{i}}+\left(v^{i}v^{j}+g^{ij}\right)\frac{\partial}{\partial
v^{i}}\frac{\partial}{\partial v^{j}}\right)f.$

Therefore

$\displaystyle\mathcal{C}\left(\pi_{0}^{*}f\right)=-\pi_{0}^{*}\left(\Delta^{\textrm{ver}}f\right)\
.$

∎

Similar to the framework in $\mathcal{SO}^{+}(M)$, $TM$ supports the complete
lift of vector fields from $M$. A differential 1-form
$\omega\in\Omega^{1}(M)$, may be viewed as a scalar function
$F^{\omega}:TM\rightarrow\mathbb{R}$ with $v\mapsto F^{\omega}(v)=\omega(v)$.
The complete lift $\xi^{\mathrm{ct}}\in\mathfrak{X}(TM)$ of a vector field
$\xi\in\mathfrak{X}(M)$ is uniquely defined by the property

$\pounds_{\xi^{\mathrm{ct}}}F^{\omega}=F^{\pounds_{\xi}\omega},\quad\forall\omega\in\Omega^{1}(M)\,.$

In local bundle coordinates $(x^{\mu},v^{\mu})$ of $TM$, this lift is
expressed as

$\xi=\xi^{\mu}\frac{\partial}{\partial
x^{\mu}}\mapsto\xi^{\mathrm{ct}}=\xi^{\mu}\frac{\partial}{\partial
x^{\mu}}+v^{\nu}\frac{\partial\xi^{\mu}}{\partial
x^{\nu}}\frac{\partial}{\partial v^{\mu}}\,.$

When $\xi$ is a Killing field, $\xi^{\mathrm{ct}}$ is tangent to $UM$ and
corresponds to the complete lift $\xi^{\mathrm{c}}$ to $\mathcal{SO}^{+}(M)$
via

$T\pi_{0}(\xi^{\mathrm{c}})=\xi^{\mathrm{ct}}\,.$

This relationship establishes an equivalence with the symmetric equations in
$\mathcal{SO}^{+}(M)$, articulated through the following corollary:

######  Corollary 9.

The diffusion equation in $UM$, represented as the Fokker–Planck equation

$\left(L-\frac{\sigma^{2}}{2}\Delta^{\\!\mathrm{ver}}\right)f=0\,,$

is the equation obtained from the diffusion equation (2) on
$\mathcal{SO}^{+}(M)$ just by imposing that $F=\pi_{0}^{*}f$. Similarly, in
the presence of spacetime symmetries, the equations

$\left(L-\frac{\sigma^{2}}{2}\Delta^{\\!\mathrm{ver}}\right)f=0\
,\quad\pounds_{\xi^{\mathrm{ct}}}f=0\ ,$

are derived from equation (6) on Proposition 5.

Finally, notice that if $\xi$ is a Killing vector field, the function

$C^{\xi}:TM\rightarrow\mathbb{R}\,,\quad v\mapsto C^{\xi}(v)=g(v,\xi)\ ,$

satisfies

(10) $\pounds_{L}C^{\xi}=0\,.$

These conserved quantities are analogous to (4) for $I=0$.

##  3\. Relativistic diffusion on the tangent bundle

The development of relativistic kinetic theory, pivotal for understanding
relativistic thermodynamics, is grounded in the foundational work by J. Ehlers
[ehlers] in the early 1970s and further explored by others such as Sarbach and
Franchi [sarbach2013relativistic, franchi2012hyperbolic, sarbach2014geometry].
This theory extends the classical kinetic theory of gases to a relativistic
framework, thereby providing a more comprehensive model of matter that
incorporates its particle nature. Central to this theory is the concept of the
one-particle distribution function, which, analogous to its non-relativistic
counterpart, quantifies the expected particle density within a defined volume
in the phase space of a single particle.

The objective of this section is to generalize Ehlers’ results to allow for
the presence of diffusion as well as to try to justify, in an approach closer
to physical applications, the FP equation presented in Corollary 9. In the
previous sections we focused on the $m=1$ case but in this section we will
allow for any value $m>0$ for the mass of the particles.

###  3.1. The observer bundle

To define the one-particle phase space for massive particles with arbitrary
masses in a $(1+n)$-dimensional, oriented and time-oriented space-time
$(M,g)$, we define it as the smooth $2(n+1)$-dimensional submanifold (open
subset) of the tangent bundle $TM$:

$\mathcal{P}:=\left\\{v\in TM:g(v,v)<0,\,v\text{ is future
directed}\right\\},$

where $\mathcal{P}$ admits a foliation into mass-shell bundles,

$\mathcal{P}=\bigcup_{m>0}U^{m}M,$

with

$U^{m}M:=\left\\{v\in TM:g(v,v)=-m^{2},\,v\text{ is future
directed}\right\\},$

being a $(2n+1)$-dimensional smooth fiber bundle over $M$ for any $m>0$. In
particular, the case $m=1$ is just the unit observer bundle presented in the
previous section. Each fiber,

$U^{m}_{x}M=\left\\{v\in T_{x}M:g_{x}(v,v)=-m^{2},\,v\text{ is future
directed}\right\\},$

is isometric to the $n$-dimensional hyperbolic Riemannian space
$\mathbb{H}^{n}(m)$ of constant negative curvature $-1/m^{2}$, corresponding
to the $n$-dimensional (future-pointing) $m$-mass hyperboloid in Minkowski
space-time. When convenient, we will write $\mathbb{H}_{x}=U^{m}_{x}M$.

It is straightforward to prove that the Liouville vector field $A$ and the
geodesic spray $L$ are orthogonal and tangent to $U^{m}M$, respectively.
Moreover, the geodesic spray $L$ of $TM$ induces fields on both the open
submanifold $\mathcal{P}\subset TM$ and in $U^{m}M$. The notation $L_{m}$ is
used when referring to the geodesic spray as a vector field on $U^{m}M$.

The one-particle phase space, along with the mass-shell bundles and their
fibers, inherit significant geometric properties from the tangent bundle
$(TM,\textsf{g})$, where g denotes the Sasaki metric defined in equation (8).
As regular, orientable submanifolds of $TM$, the pull-back of the Sasaki
metric g via the canonical inclusion furnishes $\mathcal{P}$, $U^{m}M$, and
$U^{m}_{x}M$ with semi-Riemannian metrics of signatures $(2,2n)$, $(1,2n)$,
and $(0,n)$ respectively. The volume forms corresponding to these metrics are
denoted as

$\mathrm{vol}_{\textsf{g}},\quad\mathrm{vol}_{m},\quad\text{and}\quad\sigma_{x}^{m}\,.$

The volume $\sigma_{x}^{m}$ appears in the definition of many physical
quantities, so it is convenient to keep the following result in mind:

######  Lemma 10.

The volume form $\sigma_{x}^{m}$, evaluated at $v\in U^{m}_{x}M$, satisfies

$\sigma_{x}^{m}(v)=-m\frac{w\lrcorner\mathrm{vol}_{g_{x}}(v)}{g_{x}(w,v)}\,,$

where $\mathrm{vol}_{g_{x}}$ is the volume form induced by $g_{x}$ in $T_{x}M$
(and $\mathrm{vol}_{g_{x}}(v)$ the volume of $T_{v}T_{x}M$), $w\in
T_{v}T_{x}M$ is any vector such that $g_{x}(w,v)\neq 0$. In particular, using
local coordinates and choosing $w=\partial/\partial v^{0}$, we have

$\sigma_{x}^{m}(v)=-m\frac{\sqrt{-\det
g_{x}}}{g_{0\mu}(x)v^{\mu}}\,\mathrm{d}v^{1}\wedge\dots\wedge\mathrm{d}v^{n}\,,$

where $\det g_{x}=\det(g_{\mu\nu}(x))$ and $v^{0}$ is given in terms of
$(v^{1},\dots,v^{n})$ by $g_{\mu\nu}(x)v^{\mu}v^{\nu}=-m^{2}$.

###### Proof.

Notice that $T_{x}M$ is a vector space so $T_{v}T_{x}M$ can be identified with
$T_{x}M$. Given $v\in U^{m}_{x}M$, any basis $(b_{1},\dots,b_{n})$ of
$T_{v}U^{m}_{x}M=v^{\perp_{g_{x}}}\subset T_{v}T_{x}M$,

$\mathrm{vol}_{g_{x}}(w,b_{1},\dots,b_{n})\ ,$

vanishes when $w\in\mathrm{span}(b_{1},\dots,b_{n})$. Hence, there exists a
volume-form $\sigma_{x}^{m}$ on $U^{m}_{x}M$ such that

$\mathrm{vol}_{g_{x}}(w,b_{1},\dots,b_{n})=-m^{-1}g_{x}(w,v)\sigma_{x}^{m}(b_{1},\dots,b_{n})\,,\quad
v\in T_{v}U^{m}_{x}M\,.$

The form

$\sigma_{x}^{m}(b_{1},\dots,b_{n})=-m\frac{\mathrm{vol}_{g_{x}}(w,b_{1},\dots,b_{n})}{g_{x}(w,v)}\
,$

is independent of the choice of $w$ (as long as $g_{x}(w,v)\neq 0$). By
setting $w=m^{-1}v$ it is clear that $\sigma_{x}^{m}=\mathrm{vol}_{x}^{m}$
(recall that the normalized Liouville vector $m^{-1}A$ is the unitary normal
to $U^{m}_{x}M$). On the other hand, by choosing $w=\partial/\partial v^{0}$

$\mathrm{vol}_{x}^{m}=\sigma_{x}^{m}=-m\frac{\partial_{v^{0}}\lrcorner\mathrm{vol}_{g_{x}}}{g_{0\mu}(x)v^{\mu}}=-m\frac{\sqrt{-\det
g_{x}}}{g_{0\mu}(x)v^{\mu}}\mathrm{d}v^{1}\wedge\dots\wedge\mathrm{d}v^{n}=m\frac{\sqrt{-\det
g_{x}}}{|v_{0}|}\mathrm{d}v^{1}\wedge\dots\wedge\mathrm{d}v^{n}\,,$

where $v_{0}:=g_{0\mu}(x)v^{\mu}$. ∎

Although it is not strictly necessary, in many cases it is convenient to use
the spacetime metric to bring the natural structures of the contangent bundle
$T^{*}M$ to the tangent bundle $TM$. This is so since the geodesic spray and
the volume structures derived from the Sasaki metric can be described in terms
of the energy, tautological form and the symplectic structure of $T^{*}M$ (and
also because the charged particle treatment is more natural in $T^{*}M$). We
will follow the approach developed by M. Berger in [berger1965lectures].
Taking advantage of the natural isomorphism between $TM$ and $T^{*}M$ induced
by the metric $g$, analogous structures are derived from the cotangent bundle
$T^{*}M$. The tautological one-form of $T^{*}M$ corresponds to a one-form
$\alpha\in\Omega^{1}(TM)$, with

$\alpha(V)=g(v,Tp(V)),\quad V\in T_{v}TM,\quad p:TM\rightarrow M\,.$

Hence, $\mathrm{d}\alpha$ introduces a symplectic form on $TM$ (non-degenerate
and closed), essential for the formulation of Hamiltonian dynamics. The
Liouville vector field $L$ and the tautological one-form $\alpha$ satisfy

$L\lrcorner\mathrm{d}\alpha=-\mathrm{d}E,$

where the energy function $E:TM\rightarrow\mathbb{R}$ is defined by:

$E(v)=\frac{1}{2}g(v,v)\quad\text{or equivalently}\quad\alpha(L)=2E.$

######  Proposition 11.

Let $\mathrm{vol}_{\textsf{g}}$ denote the volume form on the tangent bundle
$TM$ induced by the Sasaki metric g, and let $\alpha$ represent the
tautological one-form on $TM$ associated with the metric $g$ on the base
manifold $M$. Then, the volume form $\mathrm{vol}_{\textsf{g}}$ and the volume
form $\mathrm{vol}_{m}$ on the mass-shell submanifold $U^{m}M$ are given by:

$\mathrm{vol}_{\textsf{g}}=c(\mathrm{d}\alpha)^{1+n},\quad\mathrm{vol}_{m}=c_{m}\alpha_{m}\wedge(\mathrm{d}\alpha_{m})^{n}\,,$

where $\alpha_{m}$ is the pullback of $\alpha$ to $U^{m}M$ and the constants
$c$ and $c_{m}$ are given by

$c=\frac{(-1)^{\binom{1+n}{2}}}{(1+n)!}\,,\quad
c_{m}=\frac{(-1)^{\binom{1+n}{2}}}{n!m}\,.$

Furthermore, the volume form $\mathrm{vol}_{m}$ can be disintegrated along the
fibers $U^{m}_{x}M$ for each $x\in M$. Denoting by $\sigma_{x}^{m}$ the
canonical volume measure on the hyperbolic space $U_{x}^{m}M$ within the
Minkowski space $(T_{x}M,g_{x})$, we obtain the integral formula:

$\int_{U^{m}M}f\,\mathrm{vol}_{m}=\int_{M}\left(\int_{U^{m}_{x}M}(f\upharpoonright
U^{m}_{x}M)\,\sigma_{x}^{m}\right)\mathrm{vol}_{g},$

where $\mathrm{vol}_{g}$ is the Riemannian volume form on $M$ induced by $g$.

###### Proof.

The same steps followed in the Riemannian case discussed in
[besse2012manifolds] allow us to show that
$\mathrm{vol}_{\textsf{g}}=c(\mathrm{d}\alpha)^{1+n}$. To compute
$\mathrm{vol}_{m}$ notice that $m^{-1}A$ is the unit normal to $U^{m}M$ and

(11)
$m^{-1}A\lrcorner\mathrm{vol}_{\textsf{g}}=m^{-1}cA\lrcorner(\mathrm{d}\alpha)^{1+n}=\frac{c(1+n)}{m}\alpha\wedge(\mathrm{d}\alpha)^{n}\,,$

where we have used that

$A\lrcorner\mathrm{d}\alpha=\alpha\,.$

Therefore, the result follows by pulling-back (11) to $U^{m}M$

$\mathrm{vol}_{m}=\frac{c(1+n)}{m}\alpha_{m}\wedge(\mathrm{d}\alpha_{m})^{n}\,.$

∎

Notice that, in addition to $\mathrm{vol}_{m}$, the manifold $U^{m}M$ is also
equipped with a canonically defined $2n$-form

$\displaystyle\omega_{m}:=L_{m}\lrcorner\mathrm{vol}_{m}\ .$

The previous discussion leads to the following result.

######  Corollary 12.

For a $(1+n)$-dimensional space-time $(M,g)$, the $(2n+1)$-dimensional future
observer bundle $U^{m}M$ is equipped with a volume form and a $2n$-form,

$\mathrm{vol}_{m}=c_{m}\alpha_{m}\wedge(\mathrm{d}\alpha_{m})^{n}\quad\textrm{
and
}\quad\omega_{m}=L_{m}\lrcorner\mathrm{vol}_{m}=-m^{2}c_{m}(\mathrm{d}\alpha_{m})^{n},$

that satisfy

$\pounds_{L_{m}}\mathrm{vol}_{m}=0,\quad\pounds_{L_{m}}\omega_{m}=0,\quad
L_{m}\lrcorner\omega_{m}=0,\quad\mathrm{d}\omega_{m}=0.$

###### Proof.

The assertion
$L_{m}\lrcorner\omega_{m}=L_{m}\lrcorner(L_{m}\lrcorner\mathrm{vol}_{m})=0$
directly follows from the definition. The remaining properties can be derived
from:

$\mathrm{d}\omega_{m}=\mathrm{d}(L_{m}\lrcorner\mathrm{vol}_{m})=\pounds_{L_{m}}\mathrm{vol}_{m}=c_{m}\pounds_{L_{m}}(\alpha_{m}\wedge(\mathrm{d}\alpha_{m})^{1+n})=0,$

where the last equality is obtained by making use of
$\pounds_{L_{m}}\alpha_{m}=0$, justified by:

$L\lrcorner\alpha=2E\,\textrm{ and
}\,L\lrcorner\mathrm{d}\alpha=-\mathrm{d}E\Rightarrow\pounds_{L}\alpha=\mathrm{d}E.$

Given that $L$ is tangent to $U^{m}M$, allowing us to pull back
$\pounds_{L}\alpha=\mathrm{d}E$ to $U^{m}M$, the equation
$\pounds_{L_{m}}\alpha_{m}=0$ holds because $E$ is constant on $U^{m}M$. ∎

###  3.2. Diffusion Equation on the Observer Bundle

In this section, we examine the diffusion equation within the observer bundle
$U^{m}M$, focusing on the integration of the $(2n)$-form $\omega_{m}$ over
hypersurfaces and its implications for volume conservation under the dynamics
defined by $L_{m}$, the Liouville vector field . When $L_{m}$ is tangent to
$\mathcal{S}$, the condition $L_{m}\lrcorner\omega_{m}=0$ ensures

$\int_{\mathcal{S}}\omega_{m}=0\ .$

Conversely, if $L_{m}$ intersects $\mathcal{S}$ transversely, $\omega_{m}$
acts as a volume form on $\mathcal{S}$. The closure of $\omega_{m}$, denoted
by $\mathrm{d}\omega_{m}=0$, guarantees a consistent volume assignment across
any two diffeomorphically related hypersurfaces, $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$, via the flow induced by $L_{m}$. To establish this,
consider a tube $\mathcal{T}$ formed by Lie dragging a compact hypersurface
$\mathcal{S}_{1}$ along $L_{m}$ to a second boundary $\mathcal{S}_{2}$. The
invariance of volume under such transformations follows from:

(12) $\displaystyle
0=\int_{\mathcal{T}}\mathrm{d}\omega_{m}=\int_{\mathcal{S}_{2}}\omega_{m}-\int_{\mathcal{S}_{1}}\omega_{m},$

indicating that the volume enclosed by any hypersurface $\mathcal{S}$ shaped
through this process remains invariant.

The hypersurfaces $\mathcal{S}$ (transverses to $L_{m}$) may be endowed with a
volume form

$m^{-1}f\omega_{m}\ ,$

for any smooth function $f>0$: the so called one-particle distribution
function on $U^{m}M$. The normalization factor $m^{-1}$ comes from the fact
that

$\textsf{g}(L,L)\upharpoonright U^{m}M=-m^{2}\ .$

Physically, the quantity

$N(\mathcal{S}):=m^{-1}\int_{\mathcal{S}}f\omega_{m}\ ,$

provides the average of particle trajectories that pass through $\mathcal{S}$
(that is, trajectories with tangent vectors belonging to $\mathcal{S}$).

We observe that

(13)
$\displaystyle\mathrm{d}(f\omega_{m})=\pounds_{L_{m}}(f\,\text{vol}_{m})=(\pounds_{L_{m}}f)\text{vol}_{m},$

indicating that, irrespective of $f$, the following integral relation holds:

(14)
$\displaystyle\int_{\mathcal{T}}\pounds_{L_{m}}f\,\text{vol}_{m}=\int_{\mathcal{S}_{2}}f\omega_{m}-\int_{\mathcal{S}_{1}}f\omega_{m}=m\Big{(}N(\mathcal{S}_{2})-N(\mathcal{S}_{1})\Big{)}.$

Enforcing $\pounds_{L_{m}}f=0$ guaranteed that the volume form $f\omega_{m}$
consistently assigns identical volumes to all hypersurfaces $\mathcal{S}$
within the tube $\mathcal{T}$:

(15) $\int_{\mathcal{S}_{2}}f\omega_{m}=\int_{\mathcal{S}_{1}}f\omega_{m}.$

Therefore, there is no net change in the average number of particle
trajectories passing through the surfaces $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$. This condition, encapsulated by the equation

$\pounds_{L_{m}}f=0,$

is recognized as the Vlasov (or Liouville) equation. However, adherence to the
conservation law (15) does not strictly necessitate the Vlasov equation; it
suffices for $f$ to satisfy:

(16) $\int_{\mathcal{T}}\pounds_{L_{m}}f\,\text{vol}_{m}=0,$

leveraging theorem 11 to express this as:

$\int_{\mathcal{T}}(\pounds_{L_{m}}f)\,\text{vol}_{m}=\int_{x\in
M}\left(\int_{U^{m}_{x}M}(\pounds_{L_{m}}f\upharpoonright{U^{m}_{x}M})\,\sigma^{m}_{x}\right)\text{vol}_{g}=0,$

where requiring that the integral over $U^{m}_{x}M$ vanishes for every $x\in
M$ guarantees the conservation law (15), i.e.

$\int_{U^{m}_{x}M}(\pounds_{L_{m}}f\upharpoonright{U^{m}_{x}M})\,\sigma^{m}_{x}=0\Rightarrow\int_{\mathcal{S}_{2}}f\omega_{m}=\int_{\mathcal{S}_{1}}f\omega_{m}\,.$

This might be obtained for instance with,

(17) $\pounds_{L_{m}}f=\frac{\sigma^{2}}{2}\Delta^{\mathrm{ver}}_{m}f,$

where the operator $\Delta^{\mathrm{ver}}_{m}$ acts on $f\upharpoonright
U^{m}_{x}M$ similarly to the Laplace-Beltrami operator on $U^{m}_{x}M$,
ensuring the integral’s nullity via integration by parts. This formulation
aligns with the Fokker–Planck equation, underscoring geometric diffusion
processes as developed by Franchi and Jan [franchi_jan], and introduced by
Calogero [calogero2011kinetic].

###  3.3. Current densities and conservation laws

In practice, the hypersurfaces $\mathcal{S}$ appearing in equation (15) that
we will use in the following have the form

(18) $\displaystyle\mathcal{S}_{\Sigma}=\pi^{-1}_{m}(\Sigma)=\\{v\in U^{m}M\
:\ \pi_{m}(v)\in\Sigma\\}\ ,$

and are constructed from spacelike hypersurfaces $\Sigma\subset M$. Notice
that, if $f$ decays rapidly enough, equation (15) holds even when $\Sigma$ is
a (non-compact) Cauchy surface. We can split the integrals in (13) over
$\mathcal{S}_{\Sigma}$ in a similar way as the one in theorem 11 to get

(19)
$\displaystyle\int_{\mathcal{S}_{\Sigma}}f\omega_{m}=\int_{x\in\Sigma}\left(\int_{v\in
U^{m}_{x}M}g(\mathsf{n}_{x},v)f(v)\
\sigma^{m}_{x}(v)\right)\text{vol}_{\Sigma}(x)\ ,$

where $\mathsf{n}$ is the future-directed normal to $\Sigma$. Motivated by the
previous expression, associated with $f$, we define the particle current
density $\mathsf{J}$, i.e. the vector field $\mathsf{J}\in\mathfrak{X}(M)$
[$M\ni x\mapsto\mathsf{J}_{x}\in T_{x}M$]

(20)
$\displaystyle\mathsf{J}(\alpha)=m^{-1}\int_{U^{m}_{x}}F^{\alpha}f\sigma^{m}_{x}=m^{-1}\int_{v\in
U^{m}_{x}}\alpha(v)\ f(v)\ \sigma^{m}_{x}(v)\ ,\quad\forall\alpha\in
T_{x}^{*}M\,,$

where $v\mapsto F^{\alpha}(v)=\alpha(v)$ is a smooth function on $U^{m}_{x}M$
for any $\alpha\in T_{x}^{*}M$. In this way, we can consider the map

$\Sigma\ni x\mapsto
g_{x}(\mathsf{J},\mathsf{n})=\mathsf{J}(\mathsf{n}^{\flat})(x)=\int_{\in
U^{m}_{x}M}F^{\mathsf{n}^{\flat}_{x}}f\sigma^{m}_{x}=\int_{v\in
U^{m}_{x}}g(\mathsf{n}_{x},v)\ f(v)\ \sigma^{m}_{x}(v)\ ,$

where $\mathsf{n}^{\flat}$ is the one form
$\mathsf{n}^{\flat}(\cdot)=g(\mathsf{n},\cdot)$. Therefore, if $\Sigma_{1}$,
$\Sigma_{2}$ are Cauchy hypersurfaces, then denote
$(\mathcal{S}_{\Sigma_{1}},\mathcal{S}_{\Sigma_{2}})_{m}$ the tube
$\mathcal{T}$ generated by $\mathcal{S}_{\Sigma_{1}}$ to
$\mathcal{S}_{\Sigma_{2}}$ by the flow of $L_{m}$. Then the following holds

(21) $\displaystyle
0=\int_{(\mathcal{S}_{\Sigma_{1}},\mathcal{S}_{\Sigma_{2}})_{m}}\pounds_{L_{m}}f\
\text{vol}_{m}=\int_{\Sigma_{2}}g(\mathsf{J},\mathsf{n}_{2})\
\text{vol}_{\Sigma_{2}}-\int_{\Sigma_{1}}g(\mathsf{J},\mathsf{n}_{1})\
\text{vol}_{\Sigma_{1}}\ ,$

where, $\mathsf{n}_{1}$, $\mathsf{n}_{2}$ are the future-directed normals of
$\Sigma_{1}$, $\Sigma_{2}$. In the case $\Sigma_{1}$, $\Sigma_{2}$ are non-
compact, $f$ should decay rapidly enough.

In terms of local coordinates

$\mathsf{J}(x)=m^{-1}\left(\int_{v\in U^{m}_{x}}v^{\mu}\ f(v)\
\sigma^{m}_{x}(v)\right)\frac{\partial}{\partial x^{\mu}}(x)\,.$

######  Theorem 13.

If $f$ satisfies the FP equation
$\left(L_{m}-\frac{\sigma^{2}}{2}\Delta^{\mathrm{ver}}_{m}\right)f=0$ then
$\mathrm{div}\,\mathsf{J}=0$.

###### Proof.

A direct argument (see, for example [ehlers]), in which it suffices to express
Stokes’ theorem in terms of the semi-Riemannian metrics involved, informs us
that if

$\mathsf{J}(x)=\left(\int_{v\in U^{m}_{x}}v^{\mu}\ f(v)\
\sigma^{m}_{x}(v)\right)\frac{\partial}{\partial x^{\mu}}(x)\ ,$

the following identity holds

$\mathrm{div}\mathsf{J}=m^{-1}\int_{U^{m}_{x}M}(\pounds_{L_{m}}f)\,\sigma^{m}_{x}\,.$

Hence, if $f$ satisfies the FP equation

$\displaystyle(\text{div}\,\mathsf{J})(x)=$ $\displaystyle
m^{-1}\int_{U^{m}_{x}M}(\pounds_{L_{m}}f)\,\sigma^{m}_{x}=\frac{\sigma^{2}}{2m}\int_{\mathbb{H}_{x}}(\Delta^{\mathbb{H}_{x}}f)\,\sigma^{m}_{x}=0\,,$

where in the last equality we have used integration by parts. ∎

The property $\mathrm{div}\,\mathsf{J}=0$ is an expression of the conservation
of the average number of particle world lines crossing a Cauchy hypersurface
$\Sigma$:

$\mathsf{N}(\Sigma):=N(\mathcal{S}_{\Sigma})=m^{-1}\int_{\mathcal{S}_{\Sigma}}f\omega_{m}=m^{-1}\int_{\Sigma}g(\mathsf{J},\mathsf{n}_{\Sigma})\
\text{vol}_{\Sigma}\,,$

where $\mathsf{n}_{\Sigma}$ is the (future pointing) unit normal to $\Sigma$.

We can also define an entropy current associated with a solution $f$ of the FP
equation as the vector field [$M\ni x\mapsto\mathsf{S}_{x}\in T_{x}M$]

$\displaystyle\mathsf{S}(\alpha)=-\frac{k_{B}}{m}\int_{U^{m}_{x}M}F^{\alpha}f\log{f}\
\sigma^{m}_{x}=-\frac{k_{B}}{m}\int_{v\in U^{m}_{x}M}\alpha(v)f(v)\log{f(v)}\
\sigma^{m}_{x}(v)\ ,\quad\alpha\in T^{*}_{x}M\,.$

where $k_{B}$ is the Boltzmann constant. In local coordinates

$\displaystyle\mathsf{S}=S^{\mu}\frac{\partial}{\partial x^{\mu}}\,,\quad
S^{\mu}(x)=-\frac{k_{B}}{m}\int_{v\in U^{m}_{x}M}v^{\mu}f(v)\log{f(v)}\
\sigma^{m}_{x}(v)\,.$

######  Theorem 14.

If $f>0$ satisfies the FP equation
$\left(L_{m}-\frac{\sigma^{2}}{2}\Delta^{\mathrm{ver}}_{m}\right)f=0$ then
$\mathrm{div}\,\mathsf{S}\geq 0$.

###### Proof.

$\displaystyle(\text{div}\,\mathsf{S})(x)=$
$\displaystyle-\frac{k_{B}}{m}\int_{U^{m}_{x}M}\pounds_{L_{m}}(f\log{f})\,\sigma^{m}_{x}=-\frac{k_{B}}{m}\int_{U^{m}_{x}M}(\pounds_{L_{m}}f)\left(\log{f}+1\right)\,\sigma^{m}_{x}$
$\displaystyle=$
$\displaystyle-\frac{k_{B}\sigma^{2}}{2m}\int_{\mathbb{H}_{x}}(\Delta^{\mathbb{H}_{x}}f)\left(\log{f}+1\right)\,\sigma^{m}_{x}=-\frac{k_{B}\sigma^{2}}{2m}\int_{\mathbb{H}_{x}}(\Delta^{\mathbb{H}_{x}}f)\log{f}\,\sigma^{m}_{x}$
$\displaystyle=$
$\displaystyle\frac{k_{B}\sigma^{2}}{2m}\int_{\mathbb{H}_{x}}\frac{||\mathrm{grad}\,f||^{2}}{f}\sigma^{m}_{x}\geq
0\ ,$

since the metric on $\mathbb{H}_{x}$ is Riemannian and $f>0$. ∎

The entropy current associates a total entropy $S_{\Sigma}$ to any (oriented)
hypersuface $\Sigma\subset M$ through the expression

$\mathsf{s}(\Sigma):=\int_{\Sigma}g(\mathsf{S},\mathsf{n})\,\mathrm{vol}_{\Sigma}\,,$

where, as before, $\mathsf{n}_{\Sigma}$ denotes the normal to $\Sigma$.

Similarly, one can also introduce the energy-momentum tensor

$\displaystyle\mathsf{T}(\alpha,\beta)=T(\beta,\alpha)=m^{-1}\int_{U^{m}_{x}M}F^{\alpha}F^{\beta}f\sigma^{m}_{x}=m^{-1}\int_{v\in
U^{m}_{x}M}\alpha(v)\ \beta(v)\ f(v)\ \sigma^{m}_{x}(v)\ ,$

for all $\alpha$, $\beta\in T_{x}^{*}M\,.$ In local coordinates,

$\displaystyle\mathsf{T}=T^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\otimes\frac{\partial}{\partial x^{\nu}}\,,\quad
T^{\mu\nu}(x)=m^{-1}\int_{v\in U^{m}_{x}M}v^{\mu}v^{\nu}f(v)\
\sigma^{m}_{x}(v)\,.$

######  Theorem 15.

If $f$ satisfies the FP equation
$\left(L_{m}-\frac{\sigma^{2}}{2}\Delta^{\mathrm{ver}}_{m}\right)f=0$ then
$\mathrm{div}\,\mathsf{T}=\frac{\sigma^{2}}{2}n\mathsf{J}$

###### Proof.

Given any $\beta\in T^{*}_{x}M$, we have that

$\displaystyle\left(\mathrm{div}\,\mathsf{T}\right)(\beta)$
$\displaystyle=m^{-1}\int_{U^{m}_{x}M}(\pounds_{L_{m}}f)F^{\beta}\,\sigma^{m}_{x}=\frac{\sigma^{2}}{2m}\int_{\mathbb{H}_{x}}(\Delta^{\mathbb{H}_{x}}f)F^{\beta}\,\sigma^{m}_{x}$
$\displaystyle=\frac{\sigma^{2}}{2m}\int_{\mathbb{H}_{x}}f\,(\Delta^{\mathbb{H}_{x}}F^{\beta})\,\sigma^{m}_{x}=\frac{n\sigma^{2}}{2m}\int_{\mathbb{H}_{x}}f\,F^{\beta}\,\sigma^{m}_{x}$
$\displaystyle=\frac{n\sigma^{2}}{2}\mathsf{J}(\beta)$

where we have used that $F^{\beta}:U^{m}_{x}M\rightarrow\mathbb{R}$, $v\mapsto
F(v)=\beta(v)$, satisfies $\Delta^{\mathbb{H}_{x}}F^{\beta}=nF^{\beta}$. ∎

######  Corollary 16.

If $f$ satisfies the FP equation
$\left(L_{m}-\frac{\sigma^{2}}{2}\Delta^{\mathrm{ver}}_{m}\right)f=0$ then
$\mathrm{div}(\mathrm{div}\,\mathsf{T})=0$.

##  4\. Some relevant spacetimes

In this section, we use the results obtained in the previous sections to
obtain the Fokker–Planck equations for one-particle distribution functions
preserving the underlying symmetries in two physically relevant spacetimes.

###  4.1. Flat Friedmann–Lemaître–Robertson–Walker cosmologies

The line-element of the flat Friedmann–Lemaître–Robertson–Walker (FLRW) space
time $(\mathbb{R}^{4},g)$ is

$\displaystyle
ds^{2}=-(\mathrm{d}x^{0})^{2}+a^{2}\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}\
,$

where $x^{\mu}$ are Cartesian coordinates of $\mathbb{R}^{4}$ and
$a=a(x^{0})>0$ is the scale factor. Using global bundle coordinates, the
geodesic spray $L_{m}$ and the vertical Laplacian $\Delta^{\mathrm{ver}}_{m}$
on $U^{m}M=\mathbb{R}^{4}\times\mathbb{R}^{3}$ are given by

$\displaystyle L_{m}$
$\displaystyle=\mathrm{v}^{0}\left(\partial_{x^{0}}-2a^{\prime}a^{-1}v^{i}\partial_{v^{i}}\right)+v^{i}\partial_{x^{i}}\
,$ $\displaystyle\Delta^{\mathrm{ver}}_{m}$
$\displaystyle=\left(\frac{\delta^{ij}}{a^{2}}+\frac{v^{i}v^{j}}{m^{2}}\right)\partial_{v^{i}}\partial_{v^{j}}+\frac{3v^{i}}{m^{2}}\partial_{v^{i}}\
,$

where we have defined

$\mathrm{v}^{0}=\mathrm{v}^{0}(x^{0},v^{i}):=\sqrt{m^{2}+a^{2}\mathrm{v}^{2}}\,\quad\mathrm{v}:=\mathrm{v}(v^{i})=\sqrt{\delta_{ij}v^{i}v^{j}}\,.$

The FP equation (17) can be written in the form

(22) $\displaystyle\frac{\partial f}{\partial{x^{0}}}$
$\displaystyle+\frac{v^{i}}{\mathrm{v}^{0}}\frac{\partial
f}{\partial{x^{i}}}=\frac{\sigma^{2}}{2\mathrm{v}^{0}}\left(\left(\frac{\delta^{ij}}{a^{2}}+\frac{v^{i}v^{j}}{m^{2}}\right)\frac{\partial^{2}f}{\partial{v^{i}}\partial{v^{j}}}+\frac{3v^{i}}{m^{2}}\frac{\partial
f}{\partial{v^{i}}}\right)+2\frac{a^{\prime}}{a}v^{i}\frac{\partial
f}{\partial{v^{i}}}\ .$

The isometry group of $(M,g)$ is just the Euclidean group $E(3)$ ($\dim
E(3)=6$). The Killing fields of the metric $g$ are the ones associated with
homogeneity and isotropy (translations and rotations):

$\displaystyle\xi_{i}=\frac{\partial}{\partial x^{i}}\
,\quad\xi_{ij}=x^{i}\frac{\partial}{\partial
x^{j}}-x^{j}\frac{\partial}{\partial x^{i}}\ ,$

whose lifts to $U^{m}M$ are

$\displaystyle\xi^{\mathrm{ct}}_{i}=\frac{\partial}{\partial x^{i}}\
,\quad\xi^{\mathrm{ct}}_{ij}=x^{i}\frac{\partial}{\partial
x^{j}}-x^{j}\frac{\partial}{\partial x^{i}}+v^{i}\frac{\partial}{\partial
v^{j}}-v^{j}\frac{\partial}{\partial v^{i}}\ .$

Forcing $f$ to simultaneously satisfy $\pounds_{\xi^{\mathrm{ct}}_{i}}f=0$ and
$\pounds_{\xi^{\mathrm{ct}}_{ij}}f=0$ we have

$\displaystyle\frac{\partial f}{\partial x^{i}}=0\,,\quad v^{i}\frac{\partial
f}{\partial v^{j}}-v^{j}\frac{\partial f}{\partial v^{i}}=0\,.$

Hence $f=f(x^{0},x^{i},v^{i})$ must be independent of $x^{i}$ and the
dependence on $v^{i}$ is only through $\mathrm{v}$ because
$\pounds_{\xi^{\mathrm{ct}}_{ij}}\mathrm{v}=0$. Therefore, if we are
interested in symmetric solutions to the FP equation of we can consider
$f(x^{0},v^{i})=F(x^{0},\mathrm{v})$ and rewrite the equation (22) in the
simpler form

$\displaystyle\frac{\partial F}{\partial
x^{0}}-2a^{\prime}a^{-1}\mathrm{v}\frac{\partial
F}{\partial\mathrm{v}}=\frac{\sigma^{2}}{2\mathrm{v}^{0}}\left(\left(\frac{1}{a^{2}}+\frac{\mathrm{v}^{2}}{m^{2}}\right)\frac{\partial^{2}F}{\partial\mathrm{v}^{2}}+\left(\frac{2}{a^{2}\mathrm{v}}+\frac{3\mathrm{v}}{m^{2}}\right)\frac{\partial
F}{\partial\mathrm{v}}\right)\ .$

By using the notation introduced in equation (10), the previous equation can
be further simplified if the norm $p$ of the linear momentum,

$p=\delta_{ij}C^{\xi_{i}}C^{\xi_{j}}=a^{2}\mathrm{v}\ ,$

is used instead of $\mathrm{v}$. This is so because

$\pounds_{\xi^{\mathrm{ct}}_{ij}}p=0\quad\textrm{ and
}\quad\pounds_{L_{m}}p=0\,.$

Using $(x^{0},p)$ as reduced coordinates, the FP equation becomes

$\displaystyle\mathrm{v}^{0}\frac{\partial F}{\partial
x^{0}}=\frac{\sigma^{2}}{2}\left(\left(a^{2}+\frac{p^{2}}{m^{2}}\right)\frac{\partial^{2}F}{\partial
p^{2}}+\left(\frac{2a^{2}}{p}+\frac{3p}{m^{2}}\right)\frac{\partial\,\mathrm{F}}{\partial
p}\right)\ ,$

where $\mathrm{v}_{0}=a^{-1}\sqrt{m^{2}a^{2}+p^{2}}$.

Taking into account that

$\sigma^{m}_{x}=m\frac{a^{3}(x^{0})}{\mathrm{v}^{0}(x^{0},\mathrm{v})}\,\mathrm{d}v^{1}\wedge\mathrm{d}v^{2}\wedge\mathrm{d}v^{3}\,,$

the currents associated with the solutions of the FP equation are

$\displaystyle\mathsf{J}^{0}(x^{0},\bm{x})$
$\displaystyle=a^{3}(x^{0})\int_{\mathbb{R}^{3}}\ f(x^{0},\bm{x},\bm{v})\
\mathrm{d}^{3}\bm{v},$ $\displaystyle\mathsf{J}^{i}(x^{0},\bm{x})$
$\displaystyle=a^{3}(x^{0})\int_{\mathbb{R}^{3}}\ \ \frac{v^{i}\
f(x^{0},\bm{x},\bm{v})}{\mathrm{v}^{0}(x^{0},\bm{v})}\,\mathrm{d}^{3}\bm{v},$
$\displaystyle\mathsf{S}^{0}(x^{0},\mathbf{x})$
$\displaystyle=-\frac{k_{B}a^{3}(x^{0})}{m}\int_{\mathbb{R}^{3}}\
f(x^{0},\bm{x},\bm{v})\log{f(x^{0},\bm{x},\bm{v})}\ \mathrm{d}^{3}\bm{v},$
$\displaystyle\mathsf{S}^{i}(x^{0},\mathbf{x})$
$\displaystyle=-\frac{k_{B}a^{3}(x^{0})}{m}\int_{\mathbb{R}^{3}}\ \frac{v^{i}\
f(x^{0},\bm{x},\bm{v})\log{f(x^{0},\bm{x},\bm{v})}}{\mathrm{v}^{0}(x^{0},\bm{v})}\
\mathrm{d}^{3}\bm{v},$

and the conserved average number of occupied trajectories
$\mathsf{N}(\Sigma_{t})$ of the Cauchy hypersurfaces
$\Sigma_{t}=\\{x^{0}=t\\}$ is given by

$\mathsf{N}(\Sigma_{t})=a^{3}(t)\int_{\mathbb{R}^{3}}\mathsf{J}^{0}(t,\bm{x})\,\mathrm{d}^{3}\bm{x}=a^{6}(t)\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}f(t,\bm{x},\bm{v})\
\mathrm{d}^{3}\bm{x}\,\mathrm{d}^{3}\bm{v}\ .$

For symmetric solutions $f(t,\bm{x},\bm{v})=F(t,\mathrm{v})=F(t,p)$ (with the
usual abuse of notation)

$\mathsf{J}^{0}(x^{0})=\frac{4\pi}{a^{3}(x^{0})}\int_{0}^{\infty}F(x^{0},p)\,p^{2}\mathrm{d}p\quad\textrm{
and }\quad\mathsf{J}^{i}(x^{0})=0\,.$

In this case

$0=\mathrm{div}\,\mathsf{J}=\frac{1}{a^{3}(x^{0})}\frac{\partial}{\partial
x^{0}}\big{(}a^{3}(x^{0})\mathsf{J}^{0}(x^{0})\big{)}\ .$

Hence, as pointed out in [calogero2011kinetic],

$a^{3}(t_{1})\mathsf{J}^{0}(t_{1})=a^{3}(t_{2})\mathsf{J}^{0}(t_{2})\,,\quad\forall
t_{1},t_{2}\,.$

Notice that, for symmetric solutions, the integral in the definition of

$\mathsf{N}(\Sigma_{t})=\int_{\Sigma_{t}}g(\mathsf{J},\mathsf{n}_{\Sigma_{t}})\
\text{vol}_{\Sigma_{t}}=\int_{\mathbb{R}^{3}}a^{3}(t)\,\mathsf{J}^{0}(t)\,\mathrm{d}^{3}\bm{x}\
,$

diverges (because $\Sigma_{t}$ is not compact) but one can renormalize the
relevant objects. For example, as we have pointed out,

$\displaystyle\mathsf{n}(\Sigma_{t})=a^{3}(t)\mathsf{J}^{0}(t)=4\pi
a^{6}(t)\int_{0}^{\infty}\mathrm{v}^{2}F(t,\mathrm{v})\,\mathrm{d}\mathrm{v}=4\pi\int_{0}^{\infty}p^{2}F(t,p)\,\mathrm{d}p\
,$

does not depend on $t$.

###  4.2. Exterior Schwarzschild spacetime

The exterior Schwarzschild spacetime $(M,g)$ describes the gravitational field
outside a spherical body of mass $r_{s}/2$ where $r_{s}$ is the Schwarzschild
radius. Topologically

$M=\mathbb{R}\times(r_{s},\infty)\times\mathbb{S}^{2}$

, and, in spherical coordinates, the line-element is given by

$\displaystyle
ds^{2}=-\left(1-\frac{r_{s}}{r}\right)dt^{2}+\left(1-\frac{r_{s}}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\varphi^{2})\
.$

The geodesic flow vector field on $U^{m}M$ is given by

$\displaystyle L_{m}$
$\displaystyle=\sqrt{\frac{r}{r-r_{s}}\left(m^{2}+\mathrm{v}^{2}\right)}\
\partial_{t}+v^{r}\partial_{r}+v^{\theta}\partial_{\theta}+v^{\varphi}\partial_{\varphi}$
$\displaystyle-\frac{1}{2r^{2}}\Big{(}r_{s}m^{2}+(3r_{s}-2r)r^{2}\left((v^{\theta})^{2}+\sin^{2}{\theta}(v^{\varphi})^{2}\right)\Big{)}\partial_{v^{r}}$
$\displaystyle-\frac{1}{r}\left(2v^{r}v^{\theta}-r\sin{\theta}\cos{\theta}(v^{\varphi})^{2}\right)\partial_{v^{\theta}}-\frac{1}{r}\left(2v^{r}v^{\varphi}+2rv^{\theta}v^{\varphi}\cot{\theta}\right)\partial_{v^{\varphi}}\
,$

where we have defined

$\mathrm{v}^{2}:=\frac{r}{r-r_{s}}(v^{r})^{2}+r^{2}\Big{(}(v^{\theta})^{2}+\sin^{2}{\theta}(v^{\varphi})^{2}\Big{)}\,.$

The isometry group of $(M,g)$ is $\mathbb{R}\times O(3)\times\mathbb{Z}_{2}$
(time translations, the orthogonal group in three dimensions, and time
reversal). The (globally defined, smooth) Killing vector fields of this metric
is given, in local coordinates, by

$\displaystyle\xi_{0}$ $\displaystyle=\frac{\partial}{\partial t}\ ,$
$\displaystyle\xi_{1}$
$\displaystyle=\sin{\varphi}\frac{\partial}{\partial\theta}+\cot{\theta}\cos{\varphi}\frac{\partial}{\partial\varphi}\
,$ $\displaystyle\xi_{2}$
$\displaystyle=\cos{\varphi}\frac{\partial}{\partial\theta}-\cot{\theta}\sin{\varphi}\frac{\partial}{\partial\varphi}\
,$ $\displaystyle\xi_{3}$ $\displaystyle=\frac{\partial}{\partial\varphi}\ .$

The vector field $\xi_{0}$ is timelike and the fields $\xi_{i}$, $i=1,2,3$,
are tangent to the spheres $\\{t=t_{0},r=r_{0}\\}$ and generate a
$\mathfrak{so}(3)$ algebra. Notice that

$g(\xi_{1},\xi_{1})=r^{2}(\sin^{2}\varphi+\cos^{2}\theta\cos^{2}\varphi)\,,\,g(\xi_{2},\xi_{2})=r^{2}(\cos^{2}\varphi+\cos^{2}\theta\sin^{2}\varphi)\,,\,g(\xi_{3},\xi_{3})=\sin^{2}\theta.$

Then $\xi_{1}$, $\xi_{2}$, and $\xi_{3}$ vanishes at the intersection of the
sphere with the axis of the rotation that each of them generate. Their lifts
to $U^{m}M$ are given by

$\displaystyle\xi^{\mathrm{ct}}_{0}$ $\displaystyle=\frac{\partial}{\partial
t}\ ,$ $\displaystyle\xi^{\mathrm{ct}}_{1}$
$\displaystyle=\sin{\varphi}\frac{\partial}{\partial\theta}+\cot{\theta}\cos{\varphi}\frac{\partial}{\partial\varphi}+v^{\varphi}\cos\varphi\frac{\partial}{\partial
v^{\theta}}-\left(v^{\theta}(1+\cot^{2}\theta)\cos\varphi+v^{\varphi}\cot\theta\sin\varphi\right)\frac{\partial}{\partial
v^{\varphi}}\ ,$ $\displaystyle\xi^{\mathrm{ct}}_{2}$
$\displaystyle=\cos{\varphi}\frac{\partial}{\partial\theta}-\cot{\theta}\sin{\varphi}\frac{\partial}{\partial\varphi}-v^{\varphi}\sin\varphi\frac{\partial}{\partial
v^{\theta}}+\left(v^{\theta}(1+\cot^{2}\theta)\sin\varphi-v^{\varphi}\cot\theta\cos\varphi\right)\frac{\partial}{\partial
v^{\varphi}}\ ,$ $\displaystyle\xi^{\mathrm{ct}}_{3}$
$\displaystyle=\frac{\partial}{\partial\varphi}\ .$

Hence, imposing

$\pounds_{\xi^{\mathrm{ct}}_{k}}f=0,\quad k=0,1,2,3,$

a straightforward computation implies that the function
$f=f(t,r,\theta,\varphi,v^{r},v^{\theta},v^{\varphi})$ must have the form

$\displaystyle f=F(r,v^{r},\ell)\ ,$

where $\ell$ denotes the angular momentum that appears in the change of chart
$(r,v^{r},\theta,v^{\theta},v^{\varphi})\leftrightarrow(r,v^{r},\theta,\ell,\psi)$
defined though

$\displaystyle\ell^{2}$
$\displaystyle=r^{4}\big{(}(v^{\theta})^{2}+\sin^{2}{\theta}(v^{\varphi})^{2}\big{)}\,,\quad\psi=\arctan\left(v^{\varphi}\sin\theta/v^{\theta}\right).$

Using the notation introduced in (10):

$\ell^{2}=\delta_{ij}C^{\xi_{i}}C^{\xi_{j}}\ ,$

and hence $\ell$ satisfies

$\pounds_{\xi^{\mathrm{ct}}_{k}}\ell=0\,,\quad\pounds_{L_{m}}\ell=0\,.$

Notice that, contrary to what happens when imposing $\pounds_{\xi_{i}}h=0$ on
a function $h:\mathbb{R}^{3}\rightarrow\mathbb{R}$, the conditions
$\pounds_{\xi^{\mathrm{ct}}_{i}}f=0$ eliminate the dependence of $f$ on three
of its arguments.

Using these coordinates, the FP equation reduces to

(23)
$\displaystyle\left(v^{r}\partial_{r}+\frac{(2r-3r_{s})\ell^{2}-m^{2}r_{s}r^{2}}{2r^{4}}\partial_{v^{r}}\right)F=$
$\displaystyle=\frac{\sigma^{2}}{2}\Bigg{(}\left(r^{2}+\frac{\ell^{2}}{m^{2}}\right)\partial^{2}_{\ell}+\left(\frac{r^{2}}{\ell}+\frac{3\ell}{m^{2}}\right)\partial_{\ell}+\left(1-\frac{r_{s}}{r}+\frac{(v^{r})^{2}}{m^{2}}\right)\partial^{2}_{v^{r}}+\frac{2\ell
v^{r}}{m^{2}}\partial_{\ell}\partial_{v^{r}}+\frac{3v^{r}}{m^{2}}\partial_{v^{r}}\Bigg{)}F\
,$

in full agreement (when $m=1$) with corollary 4.2 of reference [franchi_jan].

###  4.3. Nariai spacetime

As discussed in reference [schleich2010simple], if $\dim M=4$, the Birkhoff’s
theorem states that the only locally spherically symmetric solutions to
$\mathrm{Ric}=\Lambda g$ are locally isometric either to one of the
Schwarzschild-de Sitter (anti-de Sitter) family of solutions

(24) $\displaystyle
ds^{2}_{\mathrm{SdS}}=-\left(1-\frac{r_{s}}{r}-\frac{\Lambda
r^{2}}{3}\right)dt^{2}+\left(1-\frac{r_{s}}{r}-\frac{\Lambda
r^{2}}{3}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\varphi^{2})\
,$

or to the Nariai spacetime

$\displaystyle M=\mathbb{R}\times\mathbb{S}^{1}\times\mathbb{S}^{2}\,,\quad
ds^{2}=-dt^{2}+\frac{1}{\Lambda}\cosh^{2}(\sqrt{\Lambda}t)dx^{2}+\frac{1}{\Lambda}(d\theta^{2}+\sin^{2}{\theta}d\varphi^{2})\
,$

for which $\Lambda$ is stricly positive. The Nariai spacetime is the semi-
Riemannian product of the 1+1 de Sitter space $\mathrm{dS}_{2}(\Lambda)$ and a
round sphere of curvature $\Lambda$. Hence, its isometry group is
$O(2,1)\times O(3)$. The (globally defined, smooth) Killing vector fields of
the Nariai metric are given, in local coordinates, by:

$\displaystyle\xi_{1}$
$\displaystyle=\sin{\varphi}\frac{\partial}{\partial\theta}+\cot{\theta}\cos{\varphi}\frac{\partial}{\partial\varphi}\
,$ $\displaystyle\xi_{2}$
$\displaystyle=\cos{\varphi}\frac{\partial}{\partial\theta}-\cot{\theta}\sin{\varphi}\frac{\partial}{\partial\varphi}\
,$ $\displaystyle\xi_{3}$ $\displaystyle=\frac{\partial}{\partial\varphi}\ ,$
$\displaystyle\kappa_{1}$ $\displaystyle=\sin(x)\frac{\partial}{\partial
t}+\sqrt{\Lambda}\cos(x)\tanh(\sqrt{\Lambda}t)\frac{\partial}{\partial x}\ ,$
$\displaystyle\kappa_{2}$ $\displaystyle=-\cos(x)\frac{\partial}{\partial
t}+\sqrt{\Lambda}\sin(x)\tanh(\sqrt{\Lambda}t)\frac{\partial}{\partial x}\ ,$
$\displaystyle\kappa_{3}$ $\displaystyle=\frac{\partial}{\partial x}\ .$

The Lie algebra of the isometry group of the Nariai spacetime is isomorphic to
$\mathfrak{sl}(2,\mathbb{R})\times\mathfrak{so}(3)$. The vector fields
$\xi_{1}$, $\xi_{2}$, and $\xi_{3}$ are responsible for the $\mathfrak{so}(3)$
sector and $\kappa_{1}$, $\kappa_{2}$, and $\kappa_{3}$ correspond to
$\mathfrak{sl}(2,\mathbb{R})$:

$[\kappa_{1},\kappa_{2}]=\Lambda\kappa_{3}\,,\quad[\kappa_{2},\kappa_{3}]=-\kappa_{1}\,,\quad[\kappa_{3},\kappa_{1}]=-\kappa_{2}\,,$

as can be easily seen, for example, using the basis

$b_{1}=\Lambda^{-1/2}\kappa_{2}-\kappa_{3}\,,b_{2}=\Lambda^{-1/2}\kappa_{1}\,,b_{3}=\Lambda^{-1/2}\kappa_{2}+\kappa_{3}\,.$

In addition to the angular momentum of $\mathfrak{so}(3)$, we have the
following conserved quantities derived from $\mathfrak{sl}(2,\mathbb{R})$:

$\displaystyle C^{\kappa_{1}}$
$\displaystyle=-\sin(x)v^{t}+\frac{\cos(x)\sinh(\sqrt{\Lambda}t)\cosh(\sqrt{\Lambda}t)}{\sqrt{\Lambda}}v^{x}\
,$ $\displaystyle C^{\kappa_{2}}$
$\displaystyle=\cos(x)v^{t}+\frac{\sin(x)\sinh(\sqrt{\Lambda}t)\cosh(\sqrt{\Lambda}t)}{\sqrt{\Lambda}}v^{x}\
,$ $\displaystyle C^{\kappa_{3}}$
$\displaystyle=\frac{\cosh^{2}(\sqrt{\Lambda}t)}{\Lambda}v^{x}\ .$

Notice that

$\big{(}C^{\kappa_{1}}-\sqrt{\Lambda}\cos(x)\tanh(\sqrt{\Lambda}t)C^{\kappa_{3}}\big{)}^{2}+\big{(}C^{\kappa_{2}}-\sqrt{\Lambda}\sin(x)\tanh(\sqrt{\Lambda}t)C^{\kappa_{3}}\big{)}^{2}=(v^{t})^{2}\,.$

The lifts to $U^{m}M$ of the Killing fields are given by

$\displaystyle\xi^{\mathrm{ct}}_{1}$
$\displaystyle=\sin{\varphi}\frac{\partial}{\partial\theta}+\cot{\theta}\cos{\varphi}\frac{\partial}{\partial\varphi}+v^{\varphi}\cos\varphi\frac{\partial}{\partial
v^{\theta}}-\left(v^{\theta}(1+\cot^{2}\theta)\cos\varphi+v^{\varphi}\cot\theta\sin\varphi\right)\frac{\partial}{\partial
v^{\varphi}}\ ,$ $\displaystyle\xi^{\mathrm{ct}}_{2}$
$\displaystyle=\cos{\varphi}\frac{\partial}{\partial\theta}-\cot{\theta}\sin{\varphi}\frac{\partial}{\partial\varphi}-v^{\varphi}\sin\varphi\frac{\partial}{\partial
v^{\theta}}+\left(v^{\theta}(1+\cot^{2}\theta)\sin\varphi-v^{\varphi}\cot\theta\cos\varphi\right)\frac{\partial}{\partial
v^{\varphi}}\ ,$ $\displaystyle\xi^{\mathrm{ct}}_{3}$
$\displaystyle=\frac{\partial}{\partial\varphi}\ ,$
$\displaystyle\kappa^{\mathrm{ct}}_{1}$
$\displaystyle=\sin(x)\frac{\partial}{\partial
t}+\sqrt{\Lambda}\cos(x)\tanh(\sqrt{\Lambda}t)\frac{\partial}{\partial x}$
$\displaystyle+v^{x}\cos(x)\frac{\partial}{\partial
v^{t}}+\sqrt{\Lambda}\left(-v^{x}\sin(x)\tanh(\sqrt{\Lambda}t)+\frac{\sqrt{\Lambda}\cos(x)v^{t}}{\cosh^{2}(\sqrt{\Lambda}t)}\right)\frac{\partial}{\partial
v^{x}}\ ,$ $\displaystyle\kappa^{\mathrm{ct}}_{2}$
$\displaystyle=-\cos(x)\frac{\partial}{\partial
t}+\sqrt{\Lambda}\sin(x)\tanh(\sqrt{\Lambda}t)\frac{\partial}{\partial x}$
$\displaystyle+v^{x}\sin(x)\frac{\partial}{\partial
v^{t}}+\sqrt{\Lambda}\left(v^{x}\cos(x)\tanh(\sqrt{\Lambda}t)+\frac{\sqrt{\Lambda}\sin(x)v^{t}}{\cosh^{2}(\sqrt{\Lambda}t)}\right)\frac{\partial}{\partial
v^{x}}\ ,$ $\displaystyle\kappa^{\mathrm{ct}}_{3}$
$\displaystyle=\frac{\partial}{\partial x}\ ,$

Using the previous results, it is easy to show that symmetric FP solutions
have the form $f=F(t,p,\ell)$, where

$p=C^{\kappa_{3}}=\frac{\cosh^{2}(\sqrt{\Lambda}t)}{\Lambda}\,v^{x}\,,\quad\ell^{2}=\delta_{ij}C^{\xi_{i}}C^{\xi_{j}}=\frac{(v^{\theta})^{2}+\sin^{2}\theta(v^{\varphi})^{2}}{\Lambda^{2}}\,.$

The reduced FP equation for $F$ is

$\displaystyle\sqrt{m^{2}+\frac{\Lambda
p^{2}}{\cosh^{2}(\sqrt{\Lambda}t)}+\Lambda\ell^{2}}\,\frac{\partial
F}{\partial t}$
$\displaystyle=\frac{\sigma^{2}}{2}\Bigg{(}\left(\frac{1}{\Lambda}+\frac{\ell^{2}}{m^{2}}\right)\partial^{2}_{\ell}+\left(\frac{1}{\Lambda\ell}+\frac{3\ell}{m^{2}}\right)\partial_{\ell}+\left(\frac{\cosh^{2}(\sqrt{\Lambda}t)}{\Lambda}+\frac{p^{2}}{m^{2}}\right)\partial^{2}_{p}+\frac{2\ell
p}{m^{2}}\partial_{\ell}\partial_{p}+\frac{3p}{m^{2}}\partial_{p}\Bigg{)}F\,.$

######  Remark 17.

It is interesting to notice that in local coordinates, when the spacetime
metric has the form

$ds^{2}=g_{tt}(t,r)dt^{2}+g_{rr}(t,r)dr^{2}+R^{2}(t,r)\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\ ,$

the vertical Laplacian acting on rotational-invariant functions is just

$\Delta^{\mathrm{ver}}_{m}=\left(R^{2}+\frac{\ell^{2}}{m^{2}}\right)\partial^{2}_{\ell}+\left(\frac{R^{2}}{\ell}+\frac{3\ell}{m^{2}}\right)\partial_{\ell}+\left(\frac{1}{g_{rr}}+\frac{(v^{r})^{2}}{m^{2}}\right)\partial^{2}_{v^{r}}+\frac{2\ell
v^{r}}{m^{2}}\partial_{\ell}\partial_{v^{r}}+\frac{3v^{r}}{m^{2}}\partial_{v^{r}}\,.$

For example, for the Schwarzschild spacetime ($\Lambda=0$) we recover (23) and
it is also straightforward to write the corresponding operator in the case of
Schwarzschild-de Sitter (or anti-de Sitter) spacetimes described by (24).

Furthermore, an interesting open submanifold of Nariai spacetime is the
Bertotti-Kasner spacetime $(M_{\mathrm{BK}},g_{\mathrm{BK}})$, that is a
particular example of the Kantowski-Sachs metrics. This spacetime was used by
W. Rindler in [rindler1998birkhoff] to discuss Birkhoff’s theorem (see also
[kasner1925algebraic, bertotti1959uniform]):

$\displaystyle M_{\mathrm{BK}}=\mathbb{R}^{2}\times\mathbb{S}^{2}\,,\quad
ds^{2}_{\mathrm{BK}}=-dt^{2}+e^{2\sqrt{\Lambda}t}dr^{2}+\frac{1}{\Lambda}(d\theta^{2}+\sin^{2}{\theta}d\varphi^{2})\
.$

The $\mathfrak{sl}(2)$ Killing fields are

$X_{1}=\frac{\partial}{\partial r}\ ,\quad
X_{2}=-\frac{1}{\sqrt{\Lambda}}\frac{\partial}{\partial
t}+r\frac{\partial}{\partial r}\ ,\quad
X_{3}=-2\sqrt{\Lambda}r\frac{\partial}{\partial
t}+\left(e^{-2\sqrt{\Lambda}t}+\Lambda r^{2}\right)\frac{\partial}{\partial
r}\ ,$

and the reduced FP equation is

(25)
$\displaystyle\sqrt{m^{2}+e^{-2\sqrt{\Lambda}t}p^{2}+\Lambda\ell^{2}}\,\frac{\partial
F}{\partial t}$
$\displaystyle=\frac{\sigma^{2}}{2}\Bigg{(}\left(\frac{1}{\Lambda}+\frac{\ell^{2}}{m^{2}}\right)\partial^{2}_{\ell}+\left(\frac{1}{\Lambda\ell}+\frac{3\ell}{m^{2}}\right)\partial_{\ell}+\left(e^{2\sqrt{\Lambda}t}+\frac{p^{2}}{m^{2}}\right)\partial^{2}_{p}+\frac{2\ell
p}{m^{2}}\partial_{\ell}\partial_{p}+\frac{3p}{m^{2}}\partial_{p}\Bigg{)}F\ ,$

where, in this case, $p=e^{2\sqrt{\Lambda}t}v^{r}$.

##  5\. Conclusions and comments

In this paper, we have extensively discussed the relationship between the
symmetry properties of a spacetime $(M,g)$ and relativistic diffusion
processes on two relevant bundles over $M$: the (restricted) orthonormal frame
bundle $\mathcal{SO}^{+}(M)$ and the observer bundle $U^{m}M$ of arbitrary
positive mass $m$. We have demonstrated that the complete lift of a Killing
vector field of $(M,g)$ commutes with both fundamental and basic vector fields
within $\mathcal{SO}^{+}(M)$. This observation indicates that the symmetries
inherent to the spacetime $(M,g)$ also preserve the generator of the diffusion
processes on both $\mathcal{SO}^{+}(M)$ and $U^{m}M$. Furthermore, we have
identified the conditions that satisfy symmetric solutions to the
Fokker–Planck diffusion equation on $\mathcal{SO}^{+}(M)$ and $U^{m}M$, and we
have detailed how to establish the connection between them, thereby extending
previous findings in the literature.

To establish connections with numerous physical applications, we have
demonstrated that within the framework provided by $U^{m}M$, the dynamics of
the particle system can be characterized by the one-particle distribution
function $f$, which obeys the Fokker–Planck equation ensuring the conservation
of the average of particle trajectories passing through a given Cauchy
surface. This conservation principle can also be derived through the vanishing
divergence of the current density vector field $\mathsf{J}$ associated with
$f$. Additionally, we have proven that the divergence of the entropy current
$\mathsf{S}$ is non-negative, and that the divergence of the energy-momentum
tensor field $\mathsf{T}$ is proportional to the particle current density,
consistent with the expected behavior of a diffusion process.

While all the results presented in this work can be established without
resorting to symplectic techniques, we have chosen to follow this approach
anticipating future uses. For instance, it streamlines the incorporation of
additional physical information, as is the case of electromagnetic fields: by
simply modifying the symplectic form $\mathrm{d}\alpha$ to
$\mathrm{d}\alpha+qF$, where $q$ represents the particle charge and $F$
denotes the Faraday tensor describing the electromagnetic field in spacetime,
we can seamlessly integrate electromagnetic effects into our analysis.

Finally, we have illustrated the results using several physically relevant
spacetimes, spanning both cosmological and astrophysical applications.

## References