# The PI Launchpad:
Expanding the base of potential Principal Investigators across space sciences

Erika Hamden 1,∗, Michael H. New 2, D.E. (Betsy) Pugel 3, Michael Liemohn 4,
Randii Wessen 5, Richard Quinn 6, Paul Propster 5, Kirsten Petree 2, Ellen M.
Gertsen2, Paula Evans7, and Nicole Cabrera Salazar 8

###### Abstract

The PI Launchpad attempts to provide an entry level explanation of the process
of space mission development for new Principal Investigators (PIs). In
particular, PI launchpad has a focus on building teams, making partnerships,
and science concept maturity for a space mission concept, not necessarily
technical or engineering practices. Here we briefly summarize the goals of the
PI Launchpad workshops and present some results from the workshops held in
2019 and 2021. The workshop attempts to describe the current process of space
mission development (i.e. space-based telescopes and instrument platforms,
planetary missions of all types, etc.), covering a wide range of topics that a
new PI may need to successfully develop a team and write a proposal. It is not
designed to replace real experience but to provide an easily accessible
resource for potential PIs who seek to learn more about what it takes to
submit a space mission proposal, and what the first steps to take can be. The
PI Launchpad was created in response to the high barrier to entry for early
career or any scientist who is unfamiliar with mission design. These barriers
have been outlined in several recent papers and reports, and are called out in
recent space science Decadal reports.

##  1 Keywords:

Space Missions, Space Sciences, PI training, Inclusion, Workshops

##  2 Introduction

The process of successful space mission development is long, iterative, and
challenging. It can also be extremely rewarding, inspiring, and even fun! Due
in large part to the competitive nature of the proposal process, the behind
the scenes work of developing a new mission and writing the resulting proposal
can be relatively opaque. It is often a challenge for new PIs to break into
this space, finding themselves behind the ball from the start, uncertain of
next steps, and without adequate support and resources to move forward. These
obstacles are borne out by the demographics both of PIs and Science team
members for selected and proposed space missions, which tend to be both very
male and very white (Centrella et al., 2019). A recent report by the National
Academies of Science, Engineering, and Medicine has detailed both the problem
in PI demographics and made recommendations which cover, among other things,
de-mystifying and simplifying the proposal process, supporting potential PIs
with training, building PI training into existing missions, and supporting
underrepresented groups (National Academies of Sciences and Medicine, 2022a).

Here we present the PI Launchpad, a workshop which seeks to address some of
the challenges a new PI will inevitably run into when developing a mission
concept for the first time and give them tools and contacts to address these
challenges with an eye towards mission success. The workshop is jointly funded
by the Heising-Simons Foundation and NASA. The first workshop was held in
November, 2019 in Tucson, Arizona, over three days. A second workshop was held
virtually in June 2021 and took place over two weeks. A third workshop is in
development for July 2023 in Ann Arbor, Michigan. Additional workshops will be
held every two years.

For more detail on the proposal process itself, including NASA’s review and
evaluation process, please see NASA’s webpage for new PIs (New PI Resources)
and a colloquium by Dr. Thomas Zurbuchen which describes the evaluation
process (Link to Youtube). In addition, a presentation from the 2019 PI
Launchpad provides an outline for the NASA evaluation process (Proposal
Process), with a graphic from this presentation shown in Figure 1. For more
detail on best practices for proposal development, with a focus on how to
create compelling science-driven mission concepts, see Wessen et al. (2022).

Figure 1:  A flowchart of the NASA evaluation process for PI lead missions.
Slide taken from 2019 PI Launchpad.

##  3 The 2019 and 2021 Launchpad Workshops

The PI Launchpad workshop addresses the challenges a new PI might face by
providing information at a high level about the typical mission development
process and conveying to interested scientists the recommended timelines and
steps for proposing a mission.

The inaugural PI Launchpad was held in November of 2019, at the University of
Arizona. It was a 2.5 day workshop and was jointly funded by NASA and through
a grant from the Heising-Simons Foundation, which paid for 40 in-person
attendees, supporting travel, housing, food, and transportation. The workshop
was widely regarded as a success, and the participants expressed their overall
enjoyment of the program captured in a report prepared by a STEM-equity
consultant, Movement Consulting. Of particular note was the importance of the
networking opportunities (both informal and formal) that the workshop
provided. To increase outreach and accessibility, we recorded all talks and
panels during the workshop, and posted closed captioned videos and materials
online on a NASA-hosted website (PI Launchpad). The experience attending this
workshop was described by one participant as ”transformational” for them.

A second workshop was held in July 2021, in an all-virtual format of two 90
minute sessions per day, spread out over two weeks, with a mix of panels,
small group activities, lectures, and discussions. The switch to a virtual
format was necessitated by the COVID-19 pandemic. There were again 40
participants. Two highlights were a panel which included all NASA Science
Division Directors moderated by Prof. Erika Hamden (University of Arizona), to
discuss what they were looking for in PI-led missions. This demonstrated both
the buy-in from NASA decision makers for improving PI demographics, and their
commitment to transparency by answering questions frankly and clearly. A
second highlight was a “fireside chat” with NASA Science Associate
Administrator Dr. Thomas Zurbuchen, who spoke at length with Ellen Gertsen
about NASA’s overall objectives with PI-led missions. For both of these
events, participants could ask questions freely of the NASA administration. In
the 2021 workshop, small groups of participants were paired with a mentor
virtually and there were virtual networking sessions. The virtual nature of
the workshop was a hindrance in creating organic networking opportunities. As
with the 2019 workshop, all content was posted online after the workshop for
anyone to freely access. In addition, a report was generated by our STEM-
equity consultant with suggestions for improvements and analysis of the impact
of the workshop on participants.

For both workshops, the number of applications we received far exceeded the
number of participants we could support. This indicates that there is still a
large population of potential NASA mission PIs who want to learn the basics of
building a successful proposal and team. For both workshops, we worked with a
STEM Equity Consultant, Dra. Nicole Cabrera Salazar of Movement Consulting,
Inc., who conducted pre- and post-workshop surveys and assessments, interviews
with participants, and compiled reports which analyzed strengths and areas for
improvement. Based on these reports, we know that most participants found the
workshop to be incredibly valuable to them. Prior to the 2019 workshop, only
13% of participants reported knowing what the next steps were for their
mission, after the workshop, that number jumped to 90%. For the 2021 workshop,
those numbers went from 13% who knew next steps prior to the workshop to 82%.
Anecdotally, the team has heard from numerous people, participants and non-
participants, who felt that the PI Launchpad and the online content was
instrumental in them developing their own mission concept. One participant
e-mailed the following:

> Being a part of the PI Launchpad gave me the confidence to take the reins of
> a project that was wildly beyond my skill set, while also giving me the
> tools to figure out the best way forward for my team and for my science. I’m
> not sure I would have agreed to be PI…without the PI Launchpad.

##  4 Brief overview of the PI Launchpad content

The PI Launchpad works to cover a wide range of topics that are relevant to a
new PI or mission team member. There, of course, are a nearly infinite number
of topics which could be included and thus, the challenge for organizing it is
to ensure that the most important topics are highlighted and given time to be
explored, while also providing resources for a potential PI to continue to
learn and explore the process of mission development on their own. Briefly,
these topics fall into a few categories: Timelines; Mission and Science Team
Roles; Developing a Science Case; Networking and Building Partnerships;
Accessing Resources and Support. Both previously held workshops provided an
overview of these topics, to varying levels of detail.

###  4.1 Timelines

The time required to develop a mission concept to a level of maturity for a
successful proposal varies depending roughly on the cost of the mission. An
Explorer class mission (150-300 M$) will typically take two years of
development prior to submission. A concept that has already been proposed may
not need as much time, since it is relying on work done previously by the
mission team. A smaller mission, such as a Mission of Opportunity (MoO) or
Pioneers-class mission (20-70 M$) may only take a year of development, while
larger missions such as Probes or Discovery Class missions ($>$500 M$ to 1 B$)
may take even longer. Flagships ($>$ 1 B$, which don’t have PIs) are in
process for over a decade. These are rules of thumb, and each particular
experience will be slightly different. But the primary takeaway is that the
earlier a PI starts their mission concept, the better positioned they will be
when the Announcement of Opportunity (AO) is released and the proposal
deadline is set. NASA SMD provides a projection for when they anticipate
various calls for proposals coming out via the Science Office for Mission
Assessments (SOMA) website (SOMA Planning Website). At least one session per
workshop is devoted to discussing timelines and how early to start.

Figure 2:  A mock timeline for a Discovery-class mission development from a
NASA center, with an expected Final AO release in Q3 of 2025. Proposal
development at the center can start as early as 3 years before the expected
AO. Successful missions will be in development for two years prior to the
submission. This graphic was adapted from a graphic presented by JPL at the
2019 PI Launchpad (Timeline presentation).

###  4.2 Mission and Science Team Roles

The role of the PI is just one of many critical roles in mission development.
Other roles include a Deputy PI, Project Scientist, Instrument Scientist,
Project Manager, Leads for various science objectives, and others. In
addition, a PI will need to build a science team made up of many scientists
with complimentary roles and specialities in order to ensure that the science
can be achieved. Building these teams can take some time and should be
approached with care. A PI needs to build a supportive team, identify key team
members early, and provide team members with an understanding of the
expectations for each role and the timelines involved. Team dynamics,
leadership skills, and excellent communication are necessary skills for
building a successful science team. Several sessions for each workshop cover
topics related to this, including diversity in science teams, how to build a
science team, and non-PI roles in the mission.

###  4.3 Developing a Science Case

Developing a science case is the most important aspect of building a
successful mission and proposal, but it can also be one of the most
challenging. A new PI may be uncertain of the maturity of their concept,
uncertain if it is actually a good or a bad idea. It is a long process to turn
an an initial idea into mature “Science Objectives” that can motivate a
mission. The only way to address these concerns, mature the mission, and
determine if the concept will work is to start engaging with other scientists
and get additional input on possible instrument implementations. This process
of development is iterative and will take time. Early on, it may feel like the
science concept is too amorphous to list into objectives, or that it is
difficult to achieve the level of specificity that a science objective
requires. By discussing the concept with more people, and asking them to join
a science team with regular meetings, a new PI can begin to hammer out what
will work and what doesn’t. In particular, focusing on developing a story
about the science concept and building a Science Traceability Matrix (STM) can
help to refine the science case. A large fraction of the PI Launchpad focuses
on this, with sessions focusing on science storytelling, developing a pitch,
refining science objectives, and how to use graphics to tell a story.

###  4.4 Networking and Building Partnerships

Similar to the process highlighted in Section 4.2, a PI will also need to
build partnerships with managing centers, industrial partners, and science
team members. These partnerships take time to solidify and many potential
partners will begin their process of mission development as early as 2 years
before an AO will be released. This means a future PI will need to start
developing a science case and then approaching possible partners between 2 and
1.5 yrs before the AO is released. The science case does not need to be
finalized. In fact, it must be an iterative process that the PI conducts along
with their partners. But a new PI should have an idea of what they want to
explore as they begin to approach potential partners. This step can be
challenging if, with new PIs frequently unsure of who to contact at a possible
industrial partner. Many aerospace companies and NASA Centers have “New
Business” leads who are a good first point of contact. If they aren’t the
right person, they can direct a new PI to the right person. The most important
step is to make an initial contact. The PI Launchpad typically has two “Speed
Networking” sessions so participants can make contacts at a range of
industrial partners and NASA Centers.

###  4.5 Accessing Resources and Support

Finally, developing missions and writing proposals costs money. Many
universities and institutions have funding available for a new PI, if one
knows who to ask. This funding can provide partial salary support for the PI
or team members, pay for engineers to create optical or mechanical designs,
pay for graphics support, and additional support. In addition, institutions
that have proposed missions in the past may have example proposals that can be
shared with a new PI. Each institution is different, and determining what
support is available is critical to secure the seed money needed for a
proposal to be successful. Each workshop has at least one panel focused on
what types of institutional support are available and how to access them.

##  5 Will your mission be selected?

It is important to provide realistic expectations early on the chances of
selection. Most submitted proposals are not selected. Most selected missions
have been proposed multiple times. NASA’s Transiting Exoplanet Survey
Satellite (TESS) provides a good illustration of how long the process can be,
even after a proposal is first submitted (Ricker, 2021). The idea for TESS
came out of the team that worked on the High Energy Transient Explorer-2
(HETE-2). A group out of MIT realized that HETE-2, a UV/high-energy transient
mission, could be repurposed to look for exo-planet transits, and suggested
this to NASA in 2005. NASA declined, citing the upcoming launch of the Kepler
mission, which was better suited to this type of work. The team then
formulated the TESS concept and proposed it as a Small Explorer in 2008. It
was selected for a Phase A study, but ultimately not selected for flight. The
team re-proposed in 2011 as a Medium Explorer, and again was selected for a
Phase A study. In this instance, it was selected for development and was
launched in 2018. Thus, from first conception in 2005, TESS went through
several iterations, two explorer proposal rounds, two Step 2 rounds, and 13
years of development before launch. Kepler itself was proposed 5 times before
being selected in 2000 for launch in 2009. A new PI should anticipate that
their experience may be similar and be prepared for multiple rounds of
proposing to see an idea through.

##  6 Where else to look for information

NASA maintains a website for the PI Launchpad workshop (PI Launchpad Website),
which has pdfs of presentations and recordings of many of the workshop
sessions from 2019 and 2021. In addition, the PI Launchpad Workbook and
additional resources are also available at the same website. The National
Academies report on diversity in PI-led missions (National Academies of
Sciences and Medicine, 2022a) provides a comprehensive overview of the NASA
side of the proposal evaluation process.

##  7 Discussion

The PI Launchpad, after only two workshops, has already had a national impact.
It has been directly called out in the Astro 2020 Decadal Survey (National
Academies of Sciences and Medicine, 2021), and the Planetary Science and
Astrobiology Decadal Survey (National Academies of Sciences and Medicine,
2022b), as the type of program that NASA should support and expand. The
immediate impact on the 80 participants has been captured in their survey
responses, but it remains to be seen what the long term impact will be.
Anecdotally, many PI Launchpad participants have joined or lead science teams
for proposals at all scales of mission sizes. In the long run, the impact will
depend on NASA’s willingness to fund and support new PIs and to require
diversity in science and mission team membership. In the current status quo,
compelling and groundbreaking science ideas may never see the light of day
because of the challenges of being a first-time PI. We are all the poorer for
it.

## Conflict of Interest Statement

Author NCS is employed by Movement Consulting, LLC and author PE was employed
by NASA, but is now employed by LMI Consulting, LLC. The remaining authors
declare that the research was conducted in the absence of any commercial or
financial relationships that could be construed as a potential conflict of
interest.

## Author Contributions

All authors provided substantial contributions to the design and execution of
the PI Launchpad Workshops, either in 2019, 2021, or both. EH wrote the draft
of the manuscript. All authors provided edits and comments on the draft. EH,
MHN, EMG, NCS, BP, RW, PP organized the 2019 workshop. EH, MHN, BP, ML, RW,
RQ, PP, KP, EMG, PE, NCS organized the 2021 workshop.

## Funding

The PI Launchpad is generously supported by a grant from the Heising-Simons
Foundation. Additional support has been provided by NASA SMD.

## Acknowledgments

Some of the research was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National
Aeronautics and Space Administration. The authors acknowledge the contribution
of all of the speakers and participants of the two PI launchpad workshops. In
addition, we acknowledge the support and encouragement of Cyndi Atherton, of
the Heising-Simons Foundation, and the support of Dr. Thomas Zurbuchen, NASA
Associate Administrator for SMD.

## References

  * Centrella et al. (2019) Centrella, J., New, M., and Thompson, M. (2019).  Leadership and Participation in NASA’s Astrophysics Explorer-Class Missions.  In _Bulletin of the American Astronomical Society_. vol. 51, 290
  * National Academies of Sciences and Medicine (2021) National Academies of Sciences, E. and Medicine (2021).  _Pathways to Discovery in Astronomy and Astrophysics for the 2020s_.  10.17226/26141
  * National Academies of Sciences and Medicine (2022a) National Academies of Sciences, E. and Medicine (2022a).  _Advancing Diversity Equity Inclusion, and Accessibility in the Leadership of Competed Space Missions_ (Washington, DC: The National Academies Press).  10.17226/26385
  * National Academies of Sciences and Medicine (2022b) National Academies of Sciences, E. and Medicine (2022b).  _Origins, Worlds, and Life: A Decadal Strategy for Planetary Science and Astrobiology 2023-2032_ (Washington, DC: The National Academies Press).  10.17226/26522
  * Ricker (2021) Ricker, G. (2021).  Tess: A behind-the-scenes look at nasa’s latest planet hunter [Online; posted 25-August-2021]
  * Wessen et al. (2022) Wessen, R. R., Propster, P., Cable, M., Case, K., Guethe, C., Matousek, S., et al. (2022).  Developing compelling and science-focused mission concepts for NASA competed mission proposals.  _Acta Astronautica_ 191, 502–509.  10.1016/j.actaastro.2021.12.002


#  Lagrangian Formulation of the Raychaudhuri Equation in Non-Riemannian
Geometry

Anish Agashe<EMAIL_ADDRESS>
###### Abstract

The Raychaudhuri equation for a congruence of curves in a general non-
Riemannian geometry is derived. A formal connection is established between the
expansion scalar and the cross-sectional volume of the congruence. It is found
that the expansion scalar is equal to the fractional rate of change of volume,
weighted by a scalar factor that depends on the non-Riemannian features of the
geometry. Treating the congruence of curves as a dynamical system, an
appropriate Lagrangian is derived such that the corresponding Euler-Lagrange
equation is the Raychaudhuri equation. A Hamiltonian formulation and Poisson
brackets are also presented.

Keywords: Raychaudhuri equation, Lagrangian, non-Riemannian geometry.

##  1 Introduction

Given a geometry, two independent features can be endowed to it, namely,
metric and connection. A metric formalises the notion of distances, while a
connection formalises the notion of parallelism [1]. General relativity (GR)
employs a geometrical framework in which the connection can be expressed
solely in terms of the metric and its (partial) derivatives. This connection
is both symmetric and metric compatible and is commonly known as Levi-Civita
connection and such a geometry is called Riemannian. On the other hand,
geometries in which any one of the aforementioned conditions (symmetry or
metric-compatibility) is violated are called non-Riemannian. The focus of this
paper is a general non-Riemannian geometry, where both these conditions are
violated.

Using a non-Riemannian geometrical framework, several extensions of GR have
been proposed [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. With
an increasing interest in these theories, the mean kinematics of congruence of
curves in their underlying geometries have also been analysed [18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Given a congruence of curves
(timelike or null), its kinematics can be described in terms of the
irreducible components of the (transverse part of the) gradient of the tangent
to the congruence. These components are known as expansion (the trace part),
rotation (the antisymmetric part), and shear (the symmetric traceless part).
Then, one can derive equations governing the evolution of these variables
purely in terms of geometrical quantities. The equation for the evolution of
expansion of a congruence of timelike curves has been historically known as
the Raychaudhuri equation111Strictly speaking, this is a geometrical identity
rather than an equation. However, it can be converted into an equation by
using it in conjunction with the field equations of a given gravity theory.
[33].

In this paper, we present a Lagrangian (and a Hamiltonian) formulation of the
Raychaudhuri equation (RE). This has been previously done in Riemannian
geometry [34, 35, 36]. Here, we extend this to a general non-Riemannian
geometry. To be able to do this, the RE needs to be written as a second-order
differential equation. In Riemannian geometry, this is done by the virtue of
the fact that the expansion scalar is equal to the fractional rate of change
of the cross-sectional volume of the congruence [36, 34, 35]. However, we find
that not to be the case in a non-Riemannian geometry. Instead, the expansion
scalar is related to the fractional rate of change of a scalar variable that
is a combination of the cross-sectional volume and the non-Riemannian features
of the geometry. Using this variable as the dynamical degree of freedom (dof),
we construct a Lagrangian, the corresponding Euler-Lagrange equation of which,
is the RE. We also provide a Hamiltonian formulation and Poisson brackets for
the RE.

The paper is arranged in the following manner: in section 2, we introduce
basics of non-Riemannian geometry. Then, in section 3, we follow steps
analogous to the Riemannian geometry to derive the generalised RE. We present
the Lagrangian and Hamiltonian formulation of the Raychaudhuri equation in
section 4. Finally, in section 5, we discuss the results and provide
concluding remarks.

The notation and convention used is as follows: Greek indices run from $0$ to
$n-1$ and Latin indices from $1$ to $n-1$. Indices with round brackets
$(~{})$/square brackets $[~{}]$ are symmetrised/anti-symmetrised; and
underlined indices are not included in (anti-)symmetrisation. The partial
derivative is denoted by $\boldsymbol{\partial}$; covariant derivative is
denoted by $\boldsymbol{\nabla}$; the directional derivative along a vector,
$\boldsymbol{X}$, is denoted by
$\boldsymbol{\textrm{D}_{X}}\equiv\boldsymbol{X\cdot\nabla}$; and the Lie
derivative of a vector, $\boldsymbol{Y}$, along another vector,
$\boldsymbol{X}$, denoted by
$\boldsymbol{\mathfrak{L}_{X}Y}\equiv\boldsymbol{X\cdot\partial
Y-Y\cdot\partial X}$. The sign convention followed is the ‘Landau-Lifshitz
Space-like Convention (LLSC)’ [37].

##  2 Basics of non-Riemannian Geometry

Consider an arbitrary $n$-dimensional non-Riemannian manifold with a metric,
$\boldsymbol{g}$, and a connection, $\boldsymbol{\Gamma}$. The connection is
neither metric compatible nor symmetric. These features of the connection are
used to define two quantities – torsion and non-metricity – that take into
account the non-Riemannian nature of the geometry. The torsion,
$\boldsymbol{T}$, is defined as the antisymmetric part of the connection,

${T^{\rho}}_{\alpha\beta}={\Gamma^{\rho}}_{[\alpha\beta]}={\frac{1}{2}}\left({\Gamma^{\rho}}_{\alpha\beta}-{\Gamma^{\rho}}_{\beta\alpha}\right)$
(2.1)

and non-metricity, $\boldsymbol{Q}$, is defined as222The covariant derivative
is defined as,
${\nabla}_{\rho}{V^{\alpha_{1}\alpha_{2}\dots}}_{\beta_{1}\beta_{2}\dots}=\partial_{\rho}{V^{\alpha_{1}\alpha_{2}\dots}}_{\beta_{1}\beta_{2}\dots}+{{\Gamma}^{\alpha_{1}}}_{\sigma\rho}{V^{\sigma\alpha_{2}\dots}}_{\beta_{1}\beta_{2}\dots}+{{\Gamma}^{\alpha_{2}}}_{\sigma\rho}{V^{\alpha_{1}\sigma\dots}}_{\beta_{1}\beta_{2}\dots}+\dots{\\\
}-{{\Gamma}^{\sigma}}_{\beta_{1}\rho}{V^{\alpha_{1}\alpha_{2}\dots}}_{\sigma\beta_{2}\dots}-{{\Gamma}^{\sigma}}_{\beta_{2}\rho}{V^{\alpha_{1}\alpha_{2}\dots}}_{\beta_{1}\sigma\dots}-\dots{}$
(2.2) ,

$Q_{\alpha\beta\sigma}={\frac{1}{2}}\nabla_{\alpha}\textnormal{{g}}_{\beta\sigma}={\frac{1}{2}}\partial_{\alpha}\textnormal{{g}}_{\beta\sigma}-{\frac{1}{2}}{\Gamma^{\rho}}_{\beta\alpha}\textnormal{{g}}_{\rho\sigma}-{\frac{1}{2}}{\Gamma^{\rho}}_{\sigma\alpha}\textnormal{{g}}_{\beta\rho}$
(2.3)

The connection can be decomposed in the following manner,

${\Gamma^{\rho}}_{\alpha\beta}={\overline{\Gamma}^{\rho}}_{\alpha\beta}+{N^{\rho}}_{\alpha\beta}$
(2.4)

where, $\boldsymbol{\overline{\Gamma}}$ is the Riemannian part (Levi-Civita
connection), and $\boldsymbol{N}$ is the so-called distortion tensor that
encapsulates the deviation from the Riemannian nature of geometry. The
distortion is given in terms of the torsion and non-metricity as,

${N^{\rho}}_{\alpha\beta}:=\left({T^{\rho}}_{\alpha\beta}-{T_{\alpha}}^{\rho}\\!_{\beta}-{T_{\beta}}^{\rho}\\!_{\alpha}\right)+\left({Q^{\rho}}_{\alpha\beta}-{Q_{\alpha}}^{\rho}\\!_{\beta}-{Q_{\beta}}^{\rho}\\!_{\alpha}\right)$
(2.5)

The combination of the torsion terms above is called the contorsion tensor,
$\boldsymbol{K}$, given by,

${K^{\rho}}_{\alpha\beta}={T^{\rho}}_{\alpha\beta}-{T_{\alpha}}^{\rho}\\!_{\beta}-{T_{\beta}}^{\rho}\\!_{\alpha}$
(2.6)

and the combination of the non-metricity terms is called deformation (or
disformation) tensor, $\boldsymbol{P}$, given by,

${P^{\rho}}_{\alpha\beta}={Q^{\rho}}_{\alpha\beta}-{Q_{\beta}}^{\rho}\\!_{\alpha}-{Q_{\alpha\beta}}^{\rho}$
(2.7)

The Ricci identity is given by,

$\left({\nabla}_{\alpha}{\nabla}_{\beta}-{\nabla}_{\beta}{\nabla}_{\alpha}\right)V^{\rho}={{R}^{\rho}}_{\sigma\alpha\beta}V^{\sigma}+2{T^{\sigma}}_{\alpha\beta}\nabla_{\sigma}V^{\rho}$
(2.8)

where, ${{R}^{\rho}}_{\sigma\alpha\beta}$ is the Riemann curvature tensor
defined as,

${{R}^{\rho}}_{\sigma\alpha\beta}=\partial_{\alpha}{{\Gamma}^{\rho}}_{\sigma\beta}-\partial_{\beta}{{\Gamma}^{\rho}}_{\sigma\alpha}+{{\Gamma}^{\rho}}_{\mu\alpha}{{\Gamma}^{\mu}}_{\sigma\beta}-{{\Gamma}^{\rho}}_{\mu\beta}{{\Gamma}^{\mu}}_{\sigma\alpha}$
(2.9)

The Ricci tensor and Ricci scalar are defined in the usual manner as the
traces of the Riemann curvature tensor333Besides the usual Ricci tensor above,
there exist two more independent traces of the Riemann tensor,
$\displaystyle{R}_{\alpha\beta}$
$\displaystyle=\textnormal{{g}}^{\rho\sigma}{R}_{\rho\sigma\alpha\beta}={{R}^{\sigma}}_{\sigma\alpha\beta}$
(2.10) $\displaystyle{R}_{\rho\beta}$
$\displaystyle=\textnormal{{g}}^{\sigma\alpha}{R}_{\rho\sigma\alpha\beta}={{R_{\rho}}^{\alpha}}\
\\!_{\alpha\beta}$ (2.11) These are called the homothetic tensor and co-Ricci
tensor, respectively. Although there are three independent traces of the
Riemann tensor, the Ricci scalar is uniquely defined since the trace of the
homothetic tensor vanishes and that of the co-Ricci tensor is simply $-R$.,

$\displaystyle{R}_{\sigma\beta}$
$\displaystyle=\textnormal{{g}}^{\rho\alpha}{R}_{\rho\sigma\alpha\beta}={{R}^{\alpha}}_{\sigma\alpha\beta}$
(2.12) $\displaystyle{R}$
$\displaystyle=\textnormal{{g}}^{\sigma\beta}{R}_{\sigma\beta}={{R}^{\alpha\beta}}_{\alpha\beta}$
(2.13)

##  3 Raychaudhuri Equation

To derive the Raychaudhuri equation, we begin with considering a congruence of
timelike curves. Let these curves be parametrised by a parameter, $t$. Then,
the vector field, $u^{\alpha}=\frac{\partial x^{\alpha}}{\partial t}$, is
tangent to the curves. One of the effects of the non-metricity of the geometry
is that the length of vectors is not preserved under parallel transportation
[30]. Therefore, the norm of $u^{\alpha}$ cannot be normalised like it is done
in a Riemannian geometry. Instead, now we have,

$u^{\alpha}u_{\alpha}=-\ell^{2}$ (3.1)

where, $\ell\equiv\ell(x^{\alpha})$.

The Raychaudhuri equation is simply the evolution equation for the trace of
the transverse component of the $n$-velocity gradient
($\nabla_{\alpha}u_{\beta}$) of the curves. To find the transverse component,
we define a projection operator, given by,

$h_{\alpha\beta}=\textnormal{{g}}_{\alpha\beta}+\frac{1}{\ell^{2}}u_{\alpha}u_{\beta}\;\;\Rightarrow\
{h^{\alpha}}_{\beta}=\delta^{\alpha}_{\beta}+\frac{1}{\ell^{2}}u^{\alpha}u_{\beta}$
(3.2)

It is easy to check that,
$u^{\alpha}h_{\alpha\beta}=0=h_{\alpha\beta}u^{\beta}$;
${h^{\alpha}}_{\sigma}{h^{\sigma}}_{\beta}={h^{\alpha}}_{\beta}$; and
$h^{\alpha\beta}h_{\alpha\beta}={h^{\alpha}}_{\alpha}=n-1$. The transverse
component of the velocity gradient is then given by,

${{}^{{\rm(T)}}}\nabla_{\alpha}u_{\beta}={h^{\rho}}_{\alpha}{h^{\epsilon}}_{\beta}\nabla_{\rho}u_{\epsilon}=\nabla_{\alpha}u_{\beta}+X_{\alpha\beta}$
(3.3)

where, we have defined,

$X_{\alpha\beta}:=\frac{1}{\ell^{2}}u^{\epsilon}\nabla_{\alpha}u_{\epsilon}u_{\beta}+\frac{1}{\ell^{2}}u_{\alpha}a_{\beta}+\frac{1}{\ell^{4}}u^{\epsilon}a_{\epsilon}u_{\alpha}u_{\beta}$
(3.4)

with, $a_{\beta}:=\textrm{D}_{u}u_{\beta}=u^{\alpha}\nabla_{\alpha}u_{\beta}$.

The kinematic variables are defined as the trace (expansion), antisymmetric
(rotation), and symmetric traceless (shear) parts of the velocity gradient,

$\displaystyle\theta$
$\displaystyle:=\textnormal{{g}}^{\alpha\beta}{h^{\rho}}_{\alpha}{h^{\epsilon}}_{\beta}\nabla_{\rho}u_{\epsilon}=h^{\alpha\beta}\nabla_{\alpha}u_{\beta}$
(3.5) $\displaystyle\omega_{\alpha\beta}$
$\displaystyle:={h^{\rho}}_{[\alpha}{h^{\epsilon}}_{\beta]}\nabla_{\rho}u_{\epsilon}$
(3.6) $\displaystyle\sigma_{\alpha\beta}$
$\displaystyle:={h^{\rho}}_{(\alpha}{h^{\epsilon}}_{\beta)}\nabla_{\rho}u_{\epsilon}-\frac{1}{n-1}\theta
h_{\alpha\beta}$ (3.7)

Using equations (3.3) and (3.5), we can write,

$\nabla_{\alpha}u_{\beta}=\frac{1}{n-1}\theta
h_{\alpha\beta}+\omega_{\alpha\beta}+\sigma_{\alpha\beta}-X_{\alpha\beta}$
(3.8)

The first step in deriving the Raychaudhuri equation is to write (using the
Ricci identity(Eq. (2.8)) the following equation,

$\left({\nabla}_{\alpha}{\nabla}_{\beta}-{\nabla}_{\beta}{\nabla}_{\alpha}\right)u_{\rho}h^{\beta\rho}u^{\alpha}=-R_{\alpha\beta}u^{\alpha}u^{\beta}+2T^{\rho\alpha\beta}u_{\alpha}\nabla_{\rho}u_{\beta}+4T^{\rho\alpha\beta}Q_{\rho\alpha\epsilon}u_{\beta}u^{\epsilon}$
(3.9)

Further, a straightforward calculation leads to,

$\nabla_{\alpha}\nabla_{\beta}u_{\rho}h^{\beta\rho}u^{\alpha}=\textrm{D}_{u}\theta+\left(2{Q_{\alpha}}^{\beta\rho}u^{\alpha}-\frac{1}{\ell^{2}}A^{\beta}u^{\rho}\right)\nabla_{\beta}u_{\rho}-\frac{1}{\ell^{2}}A^{\rho}a_{\rho}$
(3.10)
$\nabla_{\beta}\nabla_{\alpha}u_{\rho}h^{\beta\rho}u^{\alpha}=\nabla_{\rho}a^{\rho}+2{Q^{\rho\alpha}}_{\rho}a_{\alpha}+\frac{1}{\ell^{2}}u^{\rho}\textrm{D}_{u}a_{\rho}-\textnormal{{g}}^{\rho\alpha}\textnormal{{g}}^{\epsilon\beta}\left(\nabla_{\alpha}u_{\beta}\right)\left(\nabla_{\epsilon}u_{\rho}\right)\\\
+\left(2Q^{\rho\alpha\beta}u_{\alpha}+\frac{2}{\ell^{2}}{Q_{\epsilon}}^{\alpha\beta}u_{\alpha}u^{\epsilon}u^{\rho}-\frac{1}{\ell^{2}}a^{\beta}u^{\rho}\right)\nabla_{\beta}u_{\rho}$
(3.11)

where, $A^{\beta}:=\textrm{D}_{u}u^{\beta}$ and
$a^{\beta}=\textnormal{{g}}^{\alpha\beta}a_{\alpha}\neq A^{\beta}$. Further,
one can also show that,

$\textnormal{{g}}^{\rho\alpha}\textnormal{{g}}^{\epsilon\beta}\left(\nabla_{\alpha}u_{\beta}\right)\left(\nabla_{\epsilon}u_{\rho}\right)=\frac{1}{n-1}\theta^{2}-\omega^{2}+\sigma^{2}+X^{2}$
(3.12)

where,
$\sigma^{2}=\textnormal{{g}}^{\rho\alpha}\textnormal{{g}}^{\epsilon\beta}\sigma_{\alpha\beta}\sigma_{\rho\epsilon}$;
$\omega^{2}=\textnormal{{g}}^{\rho\alpha}\textnormal{{g}}^{\epsilon\beta}\omega_{\alpha\beta}\omega_{\rho\epsilon}$;
and
$X^{2}=\textnormal{{g}}^{\rho\alpha}\textnormal{{g}}^{\epsilon\beta}X_{\alpha\beta}X_{\epsilon\rho}$.

Using equations (3.10) and (3.11) in Eq. (3.9), we get,

$\textrm{D}_{u}\theta+\left(2{Q_{\alpha}}^{\beta\rho}u^{\alpha}-2Q^{\rho\alpha\beta}u_{\alpha}\right)\nabla_{\beta}u_{\rho}-\frac{1}{\ell^{2}}A^{\rho}a_{\rho}-\nabla_{\rho}a^{\rho}-2{Q^{\rho\alpha}}_{\rho}a_{\alpha}-\frac{1}{\ell^{2}}u^{\rho}\textrm{D}_{u}a_{\rho}\\\
+\frac{1}{n-1}\theta^{2}-\omega^{2}+\sigma^{2}+X^{2}=-R_{\alpha\beta}u^{\alpha}u^{\beta}+2T^{\rho\alpha\beta}u_{\alpha}\nabla_{\rho}u_{\beta}+4T^{\rho\alpha\beta}Q_{\rho\alpha\epsilon}u_{\beta}u^{\epsilon}$
(3.13)

Rearranging the above equation, we finally get the Raychaudhuri equation in a
familiar form,

$\textrm{D}_{u}\theta+\frac{1}{n-1}\theta^{2}=-R_{\alpha\beta}u^{\alpha}u^{\beta}+\omega^{2}-\sigma^{2}+\nabla_{\rho}a^{\rho}+\mathcal{N}$
(3.14)

where, we have combined the extra terms coming from the non-Riemannian nature
of the geometry as a single scalar, $\mathcal{N}$, given by,

$\mathcal{N}:=\left(2T^{\beta\alpha\rho}+4Q^{[\rho\alpha]\beta}\right)u_{\alpha}\nabla_{\beta}u_{\rho}+2{Q^{\rho\alpha}}_{\rho}a_{\alpha}+4T^{\rho\alpha\beta}Q_{\rho\alpha\epsilon}u_{\beta}u^{\epsilon}+\frac{1}{\ell^{2}}\textrm{D}_{u}\left(u^{\rho}a_{\rho}\right)-X^{2}$
(3.15)

Equation (3.14) is the Raychaudhuri equation in a general non-Riemannian
geometry. In the Riemannian limit, we have, $\boldsymbol{Q}=0=\boldsymbol{T}$,
and $u^{\rho}a_{\rho}=0\Rightarrow X^{2}=0$, which makes $\mathcal{N}=0$.

##  4 Lagrangian Formulation

To construct a Lagrangian for the RE, it is useful to first convert it in into
a second-order differential equation. In Riemannian geometry, this is done by
recognising that the expansion scalar, $\theta$, is equal to the fractional
rate of change of the cross-sectional volume of the congruence. A formal proof
of this can be found in [38]. Then, recasting the RE in terms of the volume
element of the congruence’s cross-section, one can construct a Lagrangian
using this volume element as the dynamical degree of freedom. Therefore, in
order to construct a Lagrangian for the non-Riemannian RE, we will first
follow a similar procedure as in [38] to establish a relationship between the
expansion scalar and the cross-sectional volume.

###  4.1 Expansion Scalar and Cross-Sectional Volume

To introduce the notion of a cross-sectional volume, we select a particular
point, $P$, corresponding to a parameter value, $t_{P}$, on any given curve in
the congruence. Then, the cross-section around this curve at point, $P$, can
be constructed by taking a set of points, $P^{\prime}$, on the neighbouring
curves which have the same parameter value, $t_{P}$. Let us call this set of
points $\Sigma_{P}$. To construct a metric on $\Sigma_{P}$, we introduce a
coordinate system, $y^{a}\ (a=1,2,\dots,n-1)$. The vectors,
$n^{\alpha}_{a}=\frac{\partial x^{\alpha}}{\partial y^{a}}$, are tangent to
the cross-section. In other words, $n^{\alpha}_{a}$ is orthogonal to the
congruence, i.e., $n^{\alpha}_{a}u_{\alpha}=0$. Now, for a distance between
two points on the cross-section, we have $\textrm{d}t=0$, and hence, we can
write,

$\displaystyle\textrm{d}s^{2}$
$\displaystyle=\textnormal{{g}}_{\alpha\beta}\textrm{d}x^{\alpha}\textrm{d}x^{\beta}=\textnormal{{g}}_{\alpha\beta}\frac{\partial
x^{\alpha}}{\partial y^{a}}\textrm{d}y^{a}\frac{\partial x^{\beta}}{\partial
y^{b}}\textrm{d}y^{b}$ (4.1) $\displaystyle\Rightarrow\textrm{d}s^{2}$
$\displaystyle=\textnormal{{g}}_{\alpha\beta}n^{\alpha}_{a}n^{\beta}_{b}\textrm{d}y^{a}\textrm{d}y^{b}=h_{ab}\textrm{d}y^{a}\textrm{d}y^{b}$
(4.2)

Therefore, the $(n-1)$-tensor,
$h_{ab}=\textnormal{{g}}_{\alpha\beta}n^{\alpha}_{a}n^{\beta}_{b}$, acts like
the $(n-1)$-metric of the cross-section. The volume element on the cross-
section would then be given by, $\delta V=\sqrt{h}\ \textrm{d}^{n-1}y$, where,
$h=\det h_{ab}$. Since $n^{\alpha}_{a}u_{\alpha}=0$, we can also write,
$h_{ab}~{}=~{}h_{\alpha\beta}n^{\alpha}_{a}n^{\beta}_{b}$.

To calculate a rate of change of volume, we want to compare $\Sigma_{P}$ to
another such set of points, say, $\Sigma_{Q}$, at some other point, $Q$, on
the curve where the parameter value is $t_{Q}$. Since each point,
$P^{\prime}$, on the cross section is also a point on a neighbouring curve,
the coordinate system, $y^{a}$, can also be used as a label for these
neighbouring curves. Then, identifying all the points on $\Sigma_{Q}$ by the
curves passing through them, we automatically get a coordinate system on
$\Sigma_{Q}$ as well. Since we set up the coordinates, $y^{a}$, such that they
remain the same along the curves (that is a curve retains its label), the
change in the volume element as one goes from $P$ to $Q$ is only due to the
change in $\sqrt{h}$. Therefore, one can write,

$\frac{1}{\delta V}\textrm{D}_{u}\delta
V=\frac{1}{\sqrt{h}}\textrm{D}_{u}\sqrt{h}=\frac{1}{2h_{ab}}\textrm{D}_{u}h_{ab}$

To calculate the fractional rate of change of the volume, we start with,

$\displaystyle\textrm{D}_{u}h_{ab}$
$\displaystyle=\textrm{D}_{u}(\textnormal{{g}}_{\alpha\beta}n^{\alpha}_{a}n^{\beta}_{b})=\left(2P_{\rho\alpha\beta}u^{\rho}+2\nabla_{(\alpha}u_{\beta)}\right)n^{\alpha}_{a}n^{\beta}_{b}$
$\displaystyle\Rightarrow\frac{1}{2h_{ab}}\textrm{D}_{u}h_{ab}$
$\displaystyle=\left(P_{\rho\alpha\beta}u^{\rho}+\nabla_{(\alpha}u_{\beta)}\right)h^{\alpha\beta}$
$\displaystyle\Rightarrow\frac{1}{\delta V}\textrm{D}_{u}\delta V$
$\displaystyle=\theta+{P_{\rho\alpha}}^{\alpha}u^{\rho}+\frac{1}{\ell^{2}}P_{\rho\alpha\beta}u^{\rho}u^{\alpha}u^{\beta}=\theta+\phi$
(4.3)

where,
$\phi={P_{\rho\alpha}}^{\alpha}u^{\rho}+\frac{1}{\ell^{2}}P_{\rho\alpha\beta}u^{\rho}u^{\alpha}u^{\beta}$.

Therefore, instead of being exactly equal to it (as in the Riemannian case),
the expansion scalar here is related to the fractional rate of change of the
cross sectional $(n-1)$-volume of the congruence through a scalar, $\phi$,
that depends on the non-metricity. There are three points to be noted here:
(i) due to the scalar, $\phi$, one cannot use the volume element directly as
the dynamical dof in formulating a Lagrangian for RE; (ii) since $\phi$
depends only on the projection of non-metricity along the congruence, there
exists a subclass of non-Riemannian geometries (with
$P_{\rho\alpha\beta}u^{\rho}=0$) for which the volume element does act as a
dynamical dof; and (iii) for non-Riemannian geometries with only torsion
(i.e., $\boldsymbol{Q}=0$), the volume element again acts as the dynamical
variable.

####  4.1.1 A Case when $\phi=0$

The directional derivative of the $n$-velocity is given by,

$A^{\alpha}:=u^{\beta}\nabla_{\beta}u^{\alpha}\;\;\Rightarrow\
2u_{\alpha}A^{\alpha}=u^{\beta}\nabla_{\beta}(-\ell^{2})+P_{\rho\alpha\beta}u^{\rho}u^{\alpha}u^{\beta}$
(4.4)

Therefore, if one considers fixed length curves that are autoparallel with
respect to the non-Riemannian connection $(\boldsymbol{A}=0)$ (or curves whose
path-acceleration ($\boldsymbol{A}$) is orthogonal to the velocity), we get,
$P_{\rho\alpha\beta}u^{\rho}u^{\alpha}u^{\beta}=0$. Further, for a Weyl type
non-metricity [2, 17]
($Q_{\rho\alpha\beta}={Q_{\rho\epsilon}}^{\epsilon}\textnormal{{g}}_{\alpha\beta}$),
this would mean that ${P_{\rho\alpha}}^{\alpha}u^{\rho}=0$. Combined together,
this results in $\phi=0$. We present this case only as a quick example. There
might exist other forms of the non-metricity tensor or particular types of
congruences as well, for which $\phi$ vanishes.

###  4.2 Constructing a Lagrangian

We started with establishing a relation between the cross-sectional volume
element and the expansion scalar with the hope that we would be able to
convert the RE into a second-order differential equation. However, that does
not seem to be as straightforward as in Riemannian geometry. To reconcile this
problem, we introduce another scalar, say, $\psi$, such that,
$\textrm{D}_{u}(\ln\psi)=\phi$. Using this, the equation for the fractional
rate of change of cross-sectional volume becomes,

$\textrm{D}_{u}(\ln\delta
V)=\theta+\textrm{D}_{u}(\ln\psi)\Rightarrow\textrm{D}_{u}\left[\ln\left(\frac{\delta
V}{\psi}\right)\right]=\theta$ (4.5)

Defining a new variable, $q=\frac{\delta V}{\psi}$, the above equation
becomes,

$\frac{1}{q}\textrm{D}_{u}q=\theta$ (4.6)

Therefore, the expansion scalar in a non-Riemannian geometry can be thought of
as being equal to the fractional rate of change of the volume element with a
‘weight’ given by a scalar factor that depends on the non-metricity. It is
this quantity, $q=\psi^{-1}\delta V$, that we will consider as our dynamical
degree of freedom. Using Eq. (4.6) in Eq. (3.14), we get,

$\frac{1}{q}\textrm{D}^{2}_{u}q-m\frac{1}{q^{2}}\left(\textrm{D}_{u}q\right)^{2}=-R_{\alpha\beta}u^{\alpha}u^{\beta}+\omega^{2}-\sigma^{2}+\nabla_{\rho}a^{\rho}+\mathcal{N}$
(4.7)

where, $m=\frac{n-2}{n-1}$. The above equation is a second-order differential
equation which can be derived as the dynamical equation of a Lagrangian. Based
on the analysis in [34], one can construct the following Lagrangian,

$\mathcal{L}={\frac{1}{2}}q^{2m}\left(\textrm{D}_{u}q\right)^{2}-U(q)$ (4.8)

In analogy with a mechanical system, the first term can be considered as a
‘kinetic’ term (square of first differential) and second term as a
‘potential’. The corresponding Euler-Lagrange equations are given by,

$\displaystyle\frac{\partial\mathcal{L}}{\partial
q}-\textrm{D}_{u}\left[\frac{\partial\mathcal{L}}{\partial\left(\textrm{D}_{u}q\right)}\right]$
$\displaystyle=0$ (4.9)
$\displaystyle\Rightarrow\frac{1}{q}\textrm{D}^{2}_{u}q-m\frac{1}{q^{2}}\left(\textrm{D}_{u}q\right)^{2}$
$\displaystyle=\frac{\partial U}{\partial q}q^{-(2m+1)}$ (4.10)

The above equation matches the RE (Eq. (4.7)) with the potential defined in
the following manner,

$\frac{\partial U}{\partial
q}=\left(-R_{\alpha\beta}u^{\alpha}u^{\beta}+\omega^{2}-\sigma^{2}+\nabla_{\rho}a^{\rho}+\mathcal{N}\right)q^{2m+1}$
(4.11)

###  4.3 Hamiltonian Formulation and Poisson Brackets

Once we have the Lagrangian for the RE, constructing the Hamiltonian becomes
straightforward. Using the canonical definition, the Hamiltonian can be
defined as,

$\mathcal{H}=\frac{\partial\mathcal{L}}{\partial\left(\textrm{D}_{u}q\right)}\textrm{D}_{u}q-\mathcal{L}={\frac{1}{2}}q^{2m}\left(\textrm{D}_{u}q\right)^{2}+U(q)$
(4.12)

Then, the first Hamilton’s equation gives the conjugate momentum,
$p=q^{2m}\textrm{D}_{u}q$. The second one is given by,

$\displaystyle\textrm{D}_{u}p$
$\displaystyle=-\frac{\partial\mathcal{H}}{\partial q}$ (4.13)
$\displaystyle\Rightarrow\frac{1}{q}\textrm{D}^{2}_{u}q-m\frac{1}{q^{2}}\left(\textrm{D}_{u}q\right)^{2}$
$\displaystyle=\frac{\partial U}{\partial q}q^{-(2m+1)}$ (4.14)

which is the same as the Raychaudhuri Eq. (4.7) using the definition of the
potential (4.11).

Using the above definition of the Hamiltonian, it is easy to see that the RE
can also be written in terms of Poisson brackets as,

$\\{p,\mathcal{H}\\}=0$ (4.15)

##  5 Discussion

In this paper, we generalised the Raychaudhuri equation to non-Riemannian
geometry and then presented a Lagrangian formulation for it. Treating the
Raychaudhuri equation as the dynamical equation for the expansion of curves,
we first recognised an appropriate dynamical degree of freedom that can be
used to construct a Lagrangian. In Riemannian geometry, this is simply the
volume element of the congruence’s cross-section. However, in non-Riemannian
geometry, we found that this dof is given by a ‘weighted’ volume element with
the weight given by,
$\psi^{-1}=\textrm{e}^{-\int\left({P_{\rho\alpha}}^{\alpha}u^{\rho}+\frac{1}{\ell^{2}}P_{\rho\alpha\beta}u^{\rho}u^{\alpha}u^{\beta}\right)\textrm{d}t}$.
Treating the congruence of timelike curves as a dynamical system with this
weighted volume element as the dynamical degree of freedom, we constructed a
Lagrangian such that the corresponding Euler-Lagrange equation gave us the
Raychaudhuri equation. From this Lagrangian, we constructed a Hamiltonian and
showed that the one of the Hamilton’s equations corresponds to the
Raychaudhuri equation. Further, Poisson brackets for the RE were also
presented.

The Raychaudhuri equation is a geometric evolution equation for the trace part
of the velocity gradient (also called expansion scalar) of a congruence of
timelike curves. In GR, it plays an instrumental role in cosmology, studies of
gravitational collapse, singularity theorems, black hole mechanics, etc. [39].
The Lagrangian formulation of it has been used in studies of geometric flows
[36, 35]. The results in this paper will be useful in studying these issues in
the context of non-Riemannian theories of gravity. They may also prove useful
in studies of quantum gravity (e.g., for the analysis of (quantum) singularity
theorems) within these theories (see [40, 34, 36] and also [41, 42]). Further,
in [35], a connection was established between the expansion scalar and the
geometric entropy. It would be a worthwhile to check if this result can be
generalised to non-Riemannian geometries.

It would also be of interest to extend this analysis to the case of null
curves. However, the non-constancy of the vector norms would mean that a null
curve may not remain null under parallel transportation. However, particular
forms of non-metricity that allow for fixed length vectors can be used after
which, one would essentially follow the same steps as presented in this paper.
Finally, it is important to note that the Raychaudhuri equation only describes
the evolution of the expansion of the congruence. Therefore, to characterise
the congruence (flow) completely and one would also need a Lagrangian
formulation for the other kinematic quantities (rotation and shear).

## Acknowledgements

We thank Sai Madhav Modumudi for useful discussions and comments.

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