# Signal Synchronization Strategies and Time Domain SETI with Gaia DR3

Andy Nilipour Department of Astronomy, Yale University, Steinbach Hall, 52
Hillhouse Ave, New Haven, CT 06511  Radio Astronomy Lab, University of
California, Berkeley, 501 Campbell Hall 3411, Berkeley, CA, 94720, USA  James
R. A. Davenport Department of Astronomy, University of Washington, Box 351580,
Seattle, WA 98195, USA  Steve Croft Radio Astronomy Lab, University of
California, Berkeley, 501 Campbell Hall 3411, Berkeley, CA, 94720, USA  SETI
Institute, Mountain View, California  Andrew P. V. Siemion Radio Astronomy
Lab, University of California, Berkeley, 501 Campbell Hall 3411, Berkeley, CA,
94720, USA  SETI Institute, Mountain View, California  Jodrell Bank Centre for
Astrophysics (JBCA), Department of Physics & Astronomy, Alan Turing Building,
The University of Manchester, M13 9PL, UK  University of Malta, Institute of
Space Sciences and Astronomy

###### Abstract

Spatiotemporal techniques for signal coordination with actively transmitting
extraterrestrial civilizations, without the need for prior communication, can
constrain technosignature searches to a significantly smaller coordinate
space. With the variable star catalog from Gaia Data Release 3, we explore two
related signaling strategies: the SETI Ellipsoid, and that proposed by Seto,
which are both based on the synchronization of transmissions with a
conspicuous astrophysical event. This dataset contains more than 10 million
variable star candidates with light curves from the first three years of
Gaia’s operational phase, between 2014 and 2017. Using four different
historical supernovae as source events, we find that less than 0.01% of stars
in the sample have crossing times, the times at which we would expect to
receive synchronized signals on Earth, within the date range of available Gaia
observations. For these stars, we present a framework for technosignature
analysis that searches for modulations in the variability parameters by
splitting the stellar light curve at the crossing time.

Astrobiology, Astrometry, Search for extraterrestrial intelligence,
Technosignatures

††software:  Python, IPython (Pérez & Granger, 2007), NumPy (Oliphant, 2007),
Matplotlib (Hunter, 2007), SciPy (Virtanen et al., 2020), Pandas (Wes, 2010),
Astropy (Astropy Collaboration et al., 2013)

##  1 Introduction

Searches for technosignatures, the detection of which would be indicative of
extraterrestrial intelligence, must grapple with a vast, multidimensional
parameter space that leaves unknown the nature of the signal, and the spatial
and temporal locations of the transmission (Wright et al., 2018). Because we
are unable to constantly monitor across all possible remote sensing
modalities, we must select when, where, and how to conduct our observations.
An advanced extraterrestrial civilization could infer this searching
difficulty, and may synchronize its signal with some conspicuous astrophysical
event. These source events, which should be easily visible and uncommon, act
as “Schelling” or focal points (Schelling, 1960), allowing for coordination
despite a lack of communication between the two parties.

The distribution of stars from which a transmission synchronized with an event
that would be observable on Earth forms a so-called search for
extraterrestrial intelligence (SETI) Ellipsoid, and it can be used to greatly
constrain the number of technosignature observation candidates. Though this
framework has been described in the past (e.g., Makovetskii, 1977; Lemarchand,
1994), a precise application of the SETI Ellipsoid requires precise stellar
positions and distances, which has only recently been made possible by Gaia.

Gaia Early Data Release 3 (Gaia EDR3; Gaia Collaboration et al., 2021)
contains the full astrometric solution of nearly 1.5 billion stars, which has
allowed for the calculation of precise photogeometric distances (Bailer-Jones
et al., 2021). Decreased uncertainties in stellar distances reduce the timing
uncertainty of observations using the SETI Ellipsoid scheme. However, the Gaia
catalog contains stars up to tens of kpc away, and at these large distances,
the errors in distance translate to impractically high timing uncertainties.

To mitigate the effect of stellar distance uncertainties, Seto (2021)
considers a framework in which an extraterrestrial civilization follows a
certain geometric signaling scheme with a time-dependent directionality,
rather than sending out an isotropic transmission at one point in time, and
observers on Earth follow a closely linked receiving scheme. In this approach,
shown in Figure 1, rather than observing stars lying on an ellipsoid in
Galactic Cartesian coordinates at a given point in time, we would instead
observe along two concentric rings on the celestial sphere, up to a certain
depth. Using this, we do not need to precisely know the distance to the
candidate star, only that it is less than some upper bound, giving an
advantage to the Seto scheme.

Figure 1: The angles of observation for the Seto scheme. Synchronized signals
from stars along the four blue lines, which correspond to two conical shells
when rotated about the horizontal axis, will be observable on Earth now; these
systems are the current technosignature candidates for this signaling
framework. The path from the source event through the potentially transmitting
civilization to Earth, through a pair of red and blue lines, forms a right
angle at a point on the SETI Ellipsoid, where the technosignature candidate is
located.

With the SETI Ellipsoid and the Seto scheme, we can assign each star an
‘Ellipsoid’ and a ‘Seto’ crossing time that indicates when that star should be
observed according to the respective coordination framework. Scheduling future
observations is possible with this methodology, as is looking through archival
and publicly available data. Gaia Data Release 3 (Gaia DR3; Gaia Collaboration
et al., 2023) contains variability analysis and type classification, along
with epoch photometry, for approximately 10.5 million stars. Though these
photometry data are limited, one way we can search for extraterrestrial
intelligence signals is to perform independent variability analysis on the
stellar light curves before and after the stars’ crossing time. Such an
analysis would be primarily sensitive to modulations in a periodic star’s
frequency, phase, and amplitude, as well as any other statistically
significant differences in light-curve properties before and after a star’s
crossing time, which may be a form of information transmitted by a
sufficiently advanced civilization.

In this paper, we use Gaia to explore the SETI Ellipsoid and the Seto scheme
as techniques for analyzing past observations in addition to prioritizing
future technosignature searches.111Source code available at Nilipour et al.
(2023) In Section 2 we present the sample of variable stars identified in Gaia
DR3. In Section 3 we discuss the geometry of, and our methods of target
selection relative to, the SETI Ellipsoid, and in Section 4 we do the same for
the Seto scheme. In Section 5 we explore our candidate sample derived from
both methodologies. In Section 6 we present a novel time domain SETI approach
that we apply to a subset of the periodic variables in the sample. Finally, in
Section 7 we conclude with a summary of our work, its limitations, and
possibilities for future work.

##  2 Gaia Data Release 3

Figure 2:  Two-dimensional histogram of the distance uncertainties for the
variable stars in Gaia DR3, with distances less than 5000 lyr. The median and
mode distance uncertainties are shown in the dashed and solid lines,
respectively. Stars with a distance uncertainty less than 1 lyr are the most
ideal; there are 43,237 such stars total, mostly at low distances.

Gaia DR3 (Gaia Collaboration et al., 2023), the most recent data release of
the Gaia mission, complements the astrometric and photometric data of the
previous EDR3 (Gaia Collaboration et al., 2021) with spectra, radial
velocities, astrophysical parameters, and rotational velocities for subsets of
the full catalog, as well as epoch photometry and variability analysis for
10.5 million stars, which is of particular interest for this work. Each of
these stars is placed into one of 24 variable classifications. Although stars
that have not been classified as a known variable type by Gaia could still
harbor a signal in their light curve, such as a single flux measurement with a
significant deviation observed at the crossing time, the epoch photometry for
these stars is not yet available, and so not considered for this analysis.
However, our approach described in Section 6 can be easily applied to such
stars when their light-curve data are released.

Figure 3: Diagram of the SETI Ellipsoid, which has its foci at the source
event (purple dot) and Earth (green dot). The blue dashed line represents the
information front of the source event; stars outside this sphere (blue dot 1)
have not yet observed the event. Stars outside the ellipsoid (orange dot 2)
have observed the event, and if they transmitted a signal upon observation,
such a signal would arrive at Earth in the future. For stars on the ellipsoid
(pink dot 3), their synchronized transmission would be observable on Earth
now; at any time, these are the current technosignature candidates. Signals
from stars inside the ellipsoid (red dot 4) would have been received on Earth
in the past.

With a prior that factors in Gaia’s magnitude limit and interstellar
extinction, Bailer-Jones et al. (2021) produced photogeometric distance
estimates (using the Gaia parallax, color, and apparent magnitude) for 1.35
billion stars. Bailer-Jones et al. give the median, $r_{\text{med}}$, and the
16th and 84th percentiles, $r_{\text{lo}}$ and $r_{\text{hi}}$, respectively,
of the photogeometric posterior. In this work, the distances used are
$r_{\text{med}}$ and the distance uncertainties used are the symmetrized
uncertainty $\sigma_{r}=(r_{\text{hi}}-r_{\text{lo}})/2$.

Of the 10.5 million total variable stars, 9.3 million have Bailer-Jones
distances; henceforth, we refer to this set as the Gaia catalog of variable
stars. As seen in Figure 2, the distance uncertainties in these measurements
are generally large, which makes timing uncertainty for signal synchronization
via the SETI Ellipsoid, which is directly related to the stellar distance
uncertainty, unfeasible. Around 0.5% of Gaia variable stars have a distance
uncertainty less than 1 ly, with the distances to all of these stars less than
2000 ly.

Gaia epoch photometry has excellent long-term stability, which is useful when
considering the potentially large timing uncertainties on the stellar crossing
times. However, the light curves for individual stars are sparse, which makes
searching for transient signals from extraterrestrial civilizations
challenging. Instead, our search for signals, described in Section 6, relies
on longer-term signals involving the variability of the host star.
Additionally, the epoch photometry in Gaia DR3 only includes data from a
three-year period between 2014 and 2017, and thus contains less than half of
the data collected to date, which limits our target selection accordingly.

##  3 Target Selection via the SETI Ellipsoid

Figure 4: Current SETI Ellipsoid with SN 1987A as the source event, in
galactocentric $Y$ and $Z$ coordinates. The red dots are stars that have not
yet seen SN 1987A; the blue dots have seen it, but are outside the SETI
Ellipsoid. The pink dots are stars inside the SETI Ellipsoid, and the black
dots are those within 0.1 ly of the SETI Ellipsoid.

The SETI Ellipsoid framework is based on the simplest synchronized
communication system, in which an extraterrestrial civilization transmits an
isotropic signal when it observes the source event, and has been previously
proposed as a method to constrain the technosignature search space
(Lemarchand, 1994). The geometry of the SETI Ellipsoid is shown in Figure 3
and described here.

All stars that have observed a source event are contained within the sphere
centered at the event with radius

$\mathcal{R}=\mathcal{D}+\mathscr{T}$

where $\mathcal{D}$ is the distance from the event to Earth and
$\mathscr{T}=cT$ is the current time elapsed since Earth observation of the
source event ($T$) times the speed of light (c). Suppose each of these stars
hosts an extraterrestrial civilization that has transmitted a signal upon its
observation of the source event. The stars from which these emitted signals
reach Earth at any given time lie on the SETI Ellipsoid. Formally, the
ellipsoid, which has foci at Earth and the source event, is defined by

$d_{1}+d_{2}=\mathcal{D}+\mathscr{T}$

where $d_{1}$ is the distance from the event to the star and $d_{2}$ is the
distance from the star to Earth. As can be seen in Figure 3, the linear
eccentricity of the ellipse is

$\mathscr{C}=\frac{\mathcal{D}}{2}$

and the semi-major axis is

$\mathscr{A}=\frac{\mathcal{D}+\mathscr{T}}{2}$

So, the semi-major axis grows at half the speed of light.

Using this schematic, we can categorize all stars into four interest levels,
which correspond to the labels 1–4 in Figure 3:

  1. 1.

Stars outside the sphere of radius $\mathcal{R}$, which have not yet seen the
source event. Synchronized transmissions from these stars will not be
observable until a minimum of $T$ years from now, which, for most events, is
unreasonably far in the future.

  2. 2.

Stars within the radius $\mathcal{R}$ sphere but outside the SETI Ellipsoid,
from which a synchronized signal would have already been sent but not received
on Earth. Signals from these stars will not be observable for a maximum of $T$
years; the stars in this category that are closer to Earth may be potential
candidates for scheduling future technosignature searches.

  3. 3.

Stars on the SETI Ellipsoid. Synchronized signals from these stars would be
arriving at Earth now, making these ideal candidates for immediate
observations.

  4. 4.

Stars inside the SETI Ellipsoid, from which synchronized signals would have
already been received in the past. Archival data or previous observations can
be used to explore these stars.

It may be unreasonable to assume that an extraterrestrial civilization will
send a signal immediately upon observing a conspicuous event. More
realistically, there will be some delay between reception and transmission.
Another possible reason for delay is that the civilization may wait until a
time-resolved light curve of the event is complete before broadcasting a
mimicked copy of the signal. For example, if the source event is a supernova
(SN), they may send a transmission that will produce a light curve of the same
shape and width of the supernova light curve, once the supernova luminosity
has been reduced to some fraction of its maximum value. In this case, stars
inside, but close to, the SETI Ellipsoid are also strong candidates for
technosignature searches.

Figure 5: Crossing time diagram for the SETI Ellipsoid with SN 1987A as the
source event, in galactocentric $Y$ and $Z$ coordinates, with $\lvert
X\rvert<150$ pc. The vast majority of stars will not be on the SETI Ellipsoid
for thousands of years. The contour lines show the growth of the Ellipsoid.
Figure 6: SETI Ellipsoid target selection for SNe 1987A, 1604, 1572, and 1054,
in galactocentric $X$ and $Z$ coordinates. The red dots are the 465 stars with
SETI Ellipsoid crossing times, including error bounds, within the dates of
available Gaia epoch photometry. The light blue dots are background stars.
Table 1:

Distances and Uncertainties to the Four SNe Used for Our Signaling Methods.
SN Distance (kpc) Distance Uncertainty (kpc) Source SN 1987A 51.5 1.2 Panagia
(1999) SN 1604 5.1 +0.8/-0.7 Sankrit et al. (2016) SN 1572 2.75 0.25 Tian &
Leahy (2011) SN 1054 1.93 0.11 Trimble (1973)

Davenport et al. (2022) focus on the SETI Ellipsoid with SN 1987A as the
source event utilizing the Gaia Catalog of Nearby Stars (GCNS), which consists
of 331k stars within the 100 pc neighborhood. SN 1987A was chosen because of
its recency and proximity, but for a Galactic signaling scheme, it may be
preferable to use source events within the Milky Way. We therefore add SETI
Ellipsoids with respect to SNe 1604, 1572, and 1054, which have well-known
dates of observation.

Figure 3 shows that the closest point on the SETI Ellipsoid to Earth is at
distance $\frac{\mathscr{T}}{2}=\frac{cT}{2}$, so for source events with an
age greater than about $650$ yr, the SETI Ellipsoid will be entirely outside
the 100 pc neighborhood, and even for those with age of a few hundred years,
the SETI Ellipsoid will be largely incomplete in the 100 pc neighborhood. We
thus expand our search to fully utilize the data available in Gaia DR3.

Furthermore, Davenport et al. (2022) primarily consider the present SETI
Ellipsoid of SN 1987A, which tells us only which category each star falls in,
as described above. To calculate which stars are currently on the SETI
Ellipsoid, we determine the distances $d_{1}$ and $d_{2}$ for each star, then
use the inequality

$\lvert d_{1}+d_{2}-2\mathscr{A}\rvert\leq\tau$

where $\tau$ is a 0.1 lyr distance tolerance of being on the Ellipsoid. The
current SETI Ellipsoid for SN1987A is presented in Figure 4, expanded to
include most of the Gaia variable star catalog. Davenport et al. (2022) also
calculate the crossing time for each star,

$T_{x}=\frac{d_{1}+d_{2}-\mathcal{D}}{c}$

which tells us the time at which the star will be on the SETI Ellipsoid, and
they search through the Gaia Science Alerts archive (Hodgkin et al., 2021) for
any variability alerts from any stars with crossing times within Gaia’s
operational phase. The crossing time diagram for SN 1987A is shown in Figure
5. We further develop this idea by selecting only stars that have crossing
times, including the error bounds, between mid-2014 and mid-2017, and then
performing a variability analysis that compares the light curve before and
after the crossing time. Our analysis is described in more detail in Section
6.

The crossing time is dependent on distances to both the source event and
stars, as well as the time of Earth observation of the source event, so
uncertainties in these directly affect uncertainties in the time of arrival of
signals to Earth. For all the SNe considered, the uncertainty in the date of
observation is negligible. For small angles, the timing uncertainty is
dominated by the distance uncertainty to the star, because the light travel
time from the source event to Earth and the candidate star is nearly
identical. This is an additional reason why Davenport et al. (2022) use SN
1987A as a source event, because its distance is large relative to nearby
stars. However, this is not generally the case for Galactic SNe; the timing
uncertainties for these are significantly affected by the distance
uncertainties to the SNe, which are listed in Table 1. For many stars, these
large SNe distance errors will translate to timing uncertainties that make
their observation impractical, so we disregard these errors and echo the
sentiment of Seto (2021) that we, along with several other fields of
astrophysics, await high-precision distances to these objects. These are
projected to become available through future projects such as the Square
Kilometre Array, which will have many small dish antennas with a wider field
of view that will improve parallax measurements of very radio-bright sources
such as the Crab pulsar (Kaplan et al., 2008), which corresponds to SN 1054.

In total, we find 465 targets with crossing times and error bounds within the
date range of available Gaia epoch photometry using the SETI Ellipsoid for the
four SNe. Their spatial distribution is shown in Figure 6.

##  4 Target Selection via the Seto Scheme

Though the data from Gaia are a vast improvement over previous astrometric
measurements, at large distances, the uncertainties in stellar distances
correspond to large uncertainties in timing, which make both scheduling
observations and searching through past data nonviable. To remove this error,
and to complement our search of Gaia DR3 with the SETI Ellipsoid, we also use
the signaling framework described by Seto (2021), which does not require
precise stellar distances. The basic geometry of the Seto scheme is shown in
Figure 1 and summarized here.

The Seto scheme relies on finding a unique datum point in space-time that is
common between Earth and an extraterrestrial civilization; in particular, the
point $x$ in Figure 1, which is the closest point to the source event on the
line Ex connecting Earth and a technosignature candidate system, is such a
point in space. And, the time interval
$t(\theta)=l\cdot\textrm{sin}(\theta)/c$ plus the time epoch of the source
event, $t_{0}$, is a unique temporal point. We should then search for signals
that are synchronized with a transmission from the datum point
$(x,t(\theta)+t_{0})$. The line segment Ex in Figure 1, with length
$d=l\cdot\textrm{cos}(\theta)$ represents the depth we can search to, because
any extraterrestrial civilizations on the line beyond $x$ would need to be
able to predict the source event in order to send a signal that will reach
Earth in synchronization with a signal from the datum point.

The time on Earth to observe each angle is given by the time of arrival of a
signal sent from the datum point,

$t_{E}(\theta)=(t_{0}+t(\theta))+\frac{l\cdot\textrm{cos}(\theta)}{c}-(t_{0}+\frac{l}{c})$
$=\frac{l}{c}(\textrm{sin}(\theta)+\textrm{cos}(\theta)-1)$

where $t_{E}$ is the time on Earth since the observation of the source event
(which occurs at time $t_{0}+\frac{l}{c}$). To simplify, Seto (2021)
introduces the normalized time,

$\tau_{\text{E}}\triangleq\frac{c}{l}t_{E}$

in which case the normalized times of observation are

$\tau_{\text{E}}(\theta)=\textrm{sin}(\theta)+\textrm{cos}(\theta)-1$

Figure 7: Seto scheme crossing time diagram with SN 1054 as the source event.
This scheme covers half the sky, centered at SN 1054, with two rings, one
expanding with time and one contracting. The contour lines show the
progression of these two rings. For SN 1054, the search window closes at
around the year 3620, corresponding to a normal time of $\sqrt{2}-1$. Figure
8: Stars with Seto scheme crossing times within dates of available Gaia light-
curve data for SNe 1604, 1572, and 1054. Each supernova has two rings of
candidates, corresponding to the $\theta_{+}$ and $\theta_{-}$ rings. There
are 403 candidates total.

The above equation tells us the time to observe a specific angle in the sky;
this can give us the crossing times of stars, directly analogous to the SETI
Ellipsoid crossing times, because we can trivially calculate the angle between
two sets of celestial coordinates (i.e., that of the source event and that of
each star). The crossing time diagram via the Seto scheme for SN 1054 is shown
in Figure 7. To find what angles to observe at given times, which is useful
for forecasting targets of interest, we can invert the above equation. For
$0\leq\tau_{\text{E}}\leq\sqrt{2}-1$, there exist two solutions:

$\theta_{\pm}=\frac{\pi}{4}\pm\textrm{cos}^{-1}\bigr{(}\frac{1+\tau_{\text{E}}}{\sqrt{2}}\bigr{)}$

So, for normalized times $\tau_{\text{E}}<\sqrt{2}-1$, the directions to
observe form two concentric rings on the celestial sphere centered at the
direction of the source event. The $\theta_{-}$ ring begins as a point in the
event direction and expands over time, while the $\theta_{+}$ ring begins as a
great circle perpendicular to the event direction and shrinks over time; at
$\tau_{\text{E}}=\sqrt{2}-1$, the rings merge, and beyond that, the search
window of the source event has closed, and half the sky has been covered.

As noted previously, this search framework only requires that the distance to
the star be less than the search depth of the respective angle,
$d=l\cdot\textrm{cos}(\theta)$. At $\tau_{\text{E}}=0$, the $\theta_{-}$ ring
has a search depth of $l$ and the $\theta_{+}$ ring has a search depth of $0$;
these values decrease and increase, respectively, to a final value of
$\frac{\sqrt{2}l}{2}$.

Though not essential for a civilization that is only searching, the signaling
schematic is similarly time-dependent. The directions that an extraterrestrial
civilization following the Seto framework must transmit in are identical to
the searching directions, but reflected across the plane perpendicular to the
source event direction. In other words, the two signaling rings have the same
angles as the receiving rings, but are concentric about the direction
antipodal to the event direction.

A strong connection exists between the SETI Ellipsoid and the Seto scheme. In
particular, the two search directions in the Seto scheme correspond to the
angles at which the lines $d_{1}$ and $d_{2}$ in Figure 3 are perpendicular,
and the search depth is the length $d_{2}$. This can be seen through the
equation

$t_{E}(\theta)=\frac{l}{c}(\textrm{sin}(\theta)+\textrm{cos}(\theta)-1)=\frac{d_{1}+d_{2}-l}{c}$

which is identical to the crossing time equation for the SETI Ellipsoid. As
seen in Figure 1, two such angles exist, corresponding to $\theta_{+}$ and
$\theta_{-}$, which form two rings in the celestial sphere when rotated about
the Earth-source event axis. The normalized time $\tau_{\text{E}}=\sqrt{2}-1$
corresponds to the point at which no such perpendicular lines exist in the
SETI Ellipsoid.

Although stellar distance uncertainties do not factor into the timing
uncertainties using the Seto scheme, distance errors to the SNe will have an
effect. The timing uncertainty is

$dt=\partial t_{\theta}+\partial t_{l}$

This is strongly dominated by the $\partial t_{l}$ term, because the angular
positions of the stars and SNe have extremely precise values, so $d\theta$ is
small. So,

$dt=\frac{dt_{E}}{dl}dl=\frac{dl(\textrm{sin}(\theta)+\textrm{cos}(\theta)-1)}{c}$

which has a maximum value of $dl\frac{\sqrt{2}-1}{c}$ in the domain of
$\theta$. Hence, precise measurements of SNe distances are necessary for
constraining crossing times to within a reasonable interval.

However, as noted in Section 3, the distances to supernova remnants have large
uncertainties corresponding to unfeasibly large timing uncertainties for most
angles. Thus, we again disregard these errors.

Figure 9: The top panel shows the spatial distribution of all 868 candidates
selected via the SETI Ellipsoid and Seto methods for all four SNe, with
crossing times within the date range of available Gaia light curves, combining
the targets from Figures 8 and 6. The bottom panel displays the color-
magnitude diagram of these targets, each marked with its Gaia variable type
classification.

For our Seto scheme target selection, we again choose stars with crossing
times within the dates of Gaia epoch photometry, excluding error bounds as
noted above, and with distances less than the search depths. We use SNe 1604,
1572, and 1054 as source events. SN 1987A is excluded because its normalized
time is very small, so the $\theta_{+}$ ring has a very low depth and the
$\theta_{-}$ ring is very thin; when calculated, there are zero stars for the
Seto scheme with respect to SN 1987A within the desired date range. In total,
we find 403 targets with crossing times within the date range of available
Gaia epoch photometry using the Seto method for the three SNe. Their spatial
distribution is shown in Figure 8.

##  5 Candidate Exploration

We find a total of 868 candidates with SETI Ellipsoid or Seto crossing times
falling in the time range of Gaia DR3 epoch photometry. Their astrometric
properties, crossing times, and corresponding source supernovae are listed in
Table 2, which is available in full in the machine-readable format. The
spatial distribution and color-magnitude diagram, with Gaia variability
classification, of the target stars is shown in Figure 9.

The majority of candidates are classified as solar-like variables, which have
rotational modulation, flares, and spots. These are more difficult to analyze,
because they have a less well-defined model that involves the many parameters
of the star’s magnetic active regions. However, a possible advantage is their
similarity to the Sun, which may indicate a greater chance at hosting life
compared with other variable types, many of which clearly lie above the main
sequence, but we do not eliminate the possibility of extraterrestrial
civilizations with stars of different evolutionary states or with more extreme
variability.

Although variable stars are astrophysically interesting, and we expect a form
of artificial variability when searching for extraterrestrial intelligence, it
is unclear whether or not any type of technosignature signal in a stellar
light curve would be classified as a variable star by Gaia’s machine-learning
classification algorithm (Eyer et al., 2023). Future Gaia data releases will
include epoch photometry for all sources in the catalog, allowing for a more
complete analysis. However, we note that the classifications from Gaia are
only the most likely variable type for each star, outputted from the machine-
learning algorithm, and so the stars may be classified incorrectly, may not
fall within any of Gaia’s predefined variability types, or may not even be

=2.75in

Table 2: Table of Candidate Targets

ID | RA | Dec | DistaaDistances and distance uncertainties (upper and lower bounds at the 84th and 16th percentiles, respectively), are taken from Bailer-Jones et al. (2021) | Dist84aaDistances and distance uncertainties (upper and lower bounds at the 84th and 16th percentiles, respectively), are taken from Bailer-Jones et al. (2021) | Dist16aaDistances and distance uncertainties (upper and lower bounds at the 84th and 16th percentiles, respectively), are taken from Bailer-Jones et al. (2021) | XbbGalactocentric Cartesian $X$, $Y$, and $Z$ coordinates. | YbbGalactocentric Cartesian $X$, $Y$, and $Z$ coordinates. | ZbbGalactocentric Cartesian $X$, $Y$, and $Z$ coordinates. |  $G$ccGaia magnitudes in $G$, $G_{\text{BP}}$, and $G_{\text{RP}}$ bands. |  $G_{BP}$ccGaia magnitudes in $G$, $G_{\text{BP}}$, and $G_{\text{RP}}$ bands. |  $G_{RP}$ccGaia magnitudes in $G$, $G_{\text{BP}}$, and $G_{\text{RP}}$ bands. | ClassddGaia variability type classification. | XTimeeeStar crossing time via either the SETI Ellipsoid or the Seto method. | SetoffTrue if the star is a candidate via the Seto method, and false if the star is a candidate via the SETI Ellipsoid. | SN | XTimeErreeStar crossing time via either the SETI Ellipsoid or the Seto method.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| ∘ | ∘ | pc | pc | pc | pc | pc | pc |  |  |  |  | BJD |  |  | yr
5277882523178810112 | 126.44 | -61.89 | 166.7108 | 166.95 | 166.42 | 18.46 | -161.00 | -39.15 | 12.31 | 12.81 | 11.65 | SOLAR_LIKE | 2457250.23 | False | 1987A | 0.85
5278211586393119104 | 125.85 | -60.68 | 160.97 | 161.21 | 160.70 | 14.58 | -156.07 | -36.63 | 13.35 | 14.04 | 12.54 | SOLAR_LIKE | 2457562.62 | False | 1987A | 0.83
5085376179094311680 | 55.70 | -24.46 | 26.00 | 26.01 | 25.98 | -12.62 | -10.15 | -20.32 | 14.03 | 15.83 | 12.76 | SOLAR_LIKE | 2457030.67 | False | 1987A | 0.04
5882042344287383936 | 232.72 | -58.86 | 25.15 | 25.16 | 25.14 | 19.91 | -15.37 | -0.98 | 10.20 | 11.18 | 9.21 | SOLAR_LIKE | 2457524.21 | False | 1987A | 0.04
5793290899490805888 | 220.68 | -74.31 | 54.03 | 54.08 | 54.00 | 34.09 | -40.10 | -12.31 | 8.30 | 8.64 | 7.78 | ECL | 2457762.37 | False | 1987A | 0.13
5573439907376720896 | 95.79 | -40.22 | 66.51 | 66.56 | 66.46 | -23.11 | -57.09 | -25.09 | 10.62 | 11.19 | 9.90 | SOLAR_LIKE | 2457294.68 | False | 1987A | 0.16
5575165556516524416 | 96.17 | -36.67 | 51.70 | 51.74 | 51.67 | -20.84 | -43.61 | -18.33 | 12.99 | 14.28 | 11.87 | SOLAR_LIKE | 2456961.20 | False | 1987A | 0.11
5764995891157361792 | 203.08 | -88.44 | 128.03 | 128.22 | 127.87 | 63.18 | -96.57 | -55.47 | 11.49 | 11.92 | 10.88 | SOLAR_LIKE | 2457519.76 | False | 1987A | 0.57
5765214766988828672 | 268.90 | -87.94 | 109.54 | 109.66 | 109.41 | 56.30 | -80.06 | -49.22 | 13.68 | 14.65 | 12.70 | SOLAR_LIKE | 2457048.89 | False | 1987A | 0.40
5258643234376355840 | 153.30 | -57.82 | 63.07 | 63.12 | 63.03 | 14.31 | -61.42 | -1.33 | 13.19 | 14.31 | 12.14 | SOLAR_LIKE | 2457547.73 | False | 1987A | 0.16

Note. — Table 2 is published in its entirety in the machine-readable format. A
portion is shown here, rounded to two decimals, for guidance regarding its
form and content.

variable stars at all, and so a technosignature search in the light curves of
these objects is not futile. Our light-curve analysis described in Section 6
is sensitive to both changes in variability parameters and flux, and so can be
applied to both variable and nonvariable stars, regardless of their Gaia
classification. Furthermore, it can be modified given prior information about
the sample of stars to be analyzed; for example, we perform a more detailed
analysis for the sample of eclipsing binaries, which comprise a significant
fraction of the candidates, as seen in Figure 9.

The distance errors for the 465 SETI Ellipsoid candidates must be less than
1.5 ly, because otherwise the timing uncertainties would necessarily extend
outside the Gaia data window, which spans three years. The actual maximum
error is 0.46 ly, and 68% of candidates have errors less than 0.19 ly,
indicating that for the majority of these targets, the crossing times are
accurate to approximately two months.

Predictably, the distance errors for the 403 Seto method candidates are much
greater, as they have no constraints apart from the upper bound distance being
less than the search depth. For these targets, the maximum error is 2000 ly,
and 66% have errors less than 55 ly.

##  6 Variability Analysis

Figure 10: Sample light curve for one of the selected targets, with solar-like
variable type classification, in the three Gaia bands: $G$, $BP$, and $RP$.
The solid vertical line marks the crossing time, via either the SETI Ellipsoid
or the Seto scheme. The solid horizontal lines denote the median flux,
calculated separately before and after the crossing time, and the dashed
horizontal lines are the $\pm 3\times\textrm{median absolute deviation (MAD)}$
levels. This target is a candidate from the SETI Ellipsoid method, so we
include the crossing time error due to the stellar distance uncertainty, which
is denoted by the vertical dashed lines. The data are noticeably sparse and
incomplete, consisting of only three years of Gaia observations, but have
long-term stability.

To fully utilize the available epoch photometry data from Gaia (see Figure
10), which is limited by sparsity and incompleteness, we use a novel
technosignature search approach that splits the light curves at the crossing
time and looks for variations between the two halves. We focus on periodic
variables, for which the sparsity issue can be alleviated by period folding.

In particular, eclipsing binaries are the most ideal variable type for our
analysis, because they have easily parametrized light curves as double
Gaussians. This allows for a numerical measure of the dissimilarity of the
left and right halves of the light curve using an error-weighted difference
for each parameter. Ranking these measures can then give the most anomalous
light curve, which would correspond to the light curve with the greatest
change at the crossing time and thus the highest potential to be a
technosignature system.

Figure 11: Sample candidate light curve with SN 1987A light curve overlain in
green. The red dots show the SN light-curve interpolated to match the Gaia
light curve data points; the orange dots are the same for a top-hat function
with width equal to the FWHM of the SN light curve, a baseline equal to a
normalized flux of 1, and a peak equal to the maximum normalized flux of the
stellar light curve. The vertical blue line indicates the crossing time of the
star, which is where the supernova light curve and top-hat function begin. The
timescale of the SN, on the order of hundreds of days, is close to that of the
Gaia epoch photometry sampling and is also longer than the periods of most
stars that are classified as variable by Gaia. This approach would be better
suited to a dataset that has denser time sampling and is not limited to only
variable stars.

Here we point out that rather than a traditional SETI approach, which
generally looks for direct electromagnetic emission from technology with
spectrotemporal properties consistent with known or hypothesized behaviors of
technology, our method is sensitive to civilizations that can modulate their
host star’s period, amplitude, or phase. While this may be possible for a
sufficiently advanced civilization, we ideally would like to search for
several types of potential transmissions, such as a signal mimicking the light
curve of the source event, across all stars, not just those classified as
periodic or variable. We attempted a search for such signals by cross-
correlating a normalized supernova light curve, as well as a simpler top-hat
function derived from the same parameters of the supernova light curve, with
each candidate’s light curve; however, the Gaia epoch photometry is too sparse
for this analysis, as the timescale of a supernova is on the same order as the
spacing between photometry measurements, making any correlation spurious. This
can be seen in Figure 11.

Of the 868 candidate targets, 73 are classified as eclipsing binaries. For
each of these eclipsing binaries, we first split the normalized $G$-band light
curve at the crossing time. We refer to the light curves before and after the
crossing time as the left and right light curves, respectively. Then, with a
Lomb-Scargle Periodogram, we extract the peak frequency of the full, left, and
right light curves. Each of these frequencies also carries a false alarm
probability, which is the probability that, assuming the light curve has no
periodic signal, we will observe a periodogram power at least as high; we take
these values to be the error, although they do not directly represent a
statistical uncertainty.

All three light curves are subsequently folded with the same frequency, namely
the peak frequency with the lowest false alarm probability, and in reference
to the same epoch. A fit to a double Gaussian with constant baseline is then
performed for each folded light curve, and seven relevant parameters are
extracted: the depth, phase, and width of the two Gaussians, and the baseline
flux. For these parameters, plus the median flux and the peak frequency, we
calculate the error-weighted difference,

$\frac{\lvert
X_{\textrm{right}}-X_{\textrm{left}}\rvert}{\sqrt{\sigma_{\textrm{right}}^{2}+\sigma_{\textrm{left}}^{2}}}$

where $\sigma$ is the square root of the variance for the seven double
Gaussian parameters, the MAD for the median flux, and the false alarm
probability for the peak frequency.

For many of these targets, the crossing time is close to either the start or
the end of available observations, which causes either the left or right light
curve to contain very few data points. In these cases, we are unable to
perform a double Gaussian fit at all. Thus, from the 73 eclipsing binaries, we
ultimately perform a full analysis on 45 of them. The analysis was performed
solely on $G$-band data, because we find that trying to fit all three bands
simultaneously further reduces the number of successful fits; however, if a
promising signal is found from the $G$-band data, it would be valuable to
analyze the $RP$ and $BP$ bands to confirm that the signal is present in them
as well.

For each parameter, we rank the targets in order of increasing error-weighted
difference, so the $0^{\textrm{th}}$ target has the most similar left and
right light curves with respect to a given parameter, and the
$44^{\textrm{th}}$ target has the most dissimilar light curves. We then take
the sum of the rankings for each target and divide by the maximum, leaving us
with a single interest index ranging from 0 to 1, where 1 represents the most
interesting target (i.e., the target with the greatest difference between its
left and right light curves).

Figure 12: (a) Full, left, and right light curves and their Lomb-Scargle
Periodograms for two eclipsing binaries. The left panel has the highest-
interest index and the right panel is near the middle of the interest
rankings. The analysis seems to successfully fold the data both before and
after the crossing time, and there is a small but noticeable change in the
double Gaussian eclipsing binary fit between the left and right light curves,
though this is most likely due to sparse sampling. It is difficult to
distinguish between eclipsing binaries with high interest, as we do not
normally expect the variability parameters for an eclipsing binary to change,
so the left and right light curves should be similar. (b) Lomb-Scargle
Periodograms and light curves for two eclipsing binaries with some of the
lowest-interest indices. The periodograms are unable to find strong peak
frequencies and the double Gaussian fit fails in these two panels, indicating
that the analysis and ranking system successfully assigns low interest to
targets that do not have clean data.

Figure 12: (Continued.)

The periodograms and light curves for four eclipsing binaries, including the
target with the highest-interest index, are shown in Figure 12. There is not a
large difference between targets of high interest, as seen in the top two
panels, but this is expected given the null hypothesis of no signals. In the
most interesting candidate (top left panel), the greatest difference between
the left and right light curves is due to a change in the relative depths of
the two Gaussians. However, upon inspection, we see that this difference, and
likely most other changes between the left and right light curves for these
systems, is caused by the sparsity of the light curve, even when folded.

However, our variability analysis and ranking algorithm are successful in
separating eclipsing binaries with clearly distinctive light curves from those
without. The lowest-ranked light curves, in the bottom panels of Figure 12,
clearly have much worse Lomb-Scargle Periodogram frequency extractions and
double Gaussian fits, largely because there are very few points in the light-
curve dips.

Although a full periodicity analysis cannot be implemented on the entire
sample of variable stars, because many are not periodic, we perform a similar
but simpler analysis on the remaining candidates. Rather than comparing the
double Gaussian fit parameters before and after the crossing time, we compare
the median flux, the standard deviation, and the best-fit slope for these
targets. We follow the same process of calculating the error-weighted
differences for the median flux and best-fit slope, but for the standard
deviation, we instead normalize by the mean flux and calculate a simple
difference, because there is no well-defined measure of the error of a
standard deviation value. An advantage of this simpler procedure is that we
can incorporate all three Gaia filter bands, because a linear fit will fail
only if there are fewer than two points on either side of the crossing time,
so we ultimately compute nine parameters, the same number as for the eclipsing
binary analysis, for a total of 734 out of the 795 variable stars not
classified as eclipsing binaries.

We also repeat the same target ranking process, and show the most and least
interesting candidates in Figure 13. It again appears that our analysis
successfully finds uninteresting candidates with seemingly well-understood or
periodic light curves, giving them low rankings and separating them from more
interesting candidates, which appear to have more significant changes in light
curves before and after the crossing time. However, it remains difficult to
separate any possible technosignature signal from random variation and noise
given the limits of the Gaia epoch photometry.

Figure 13: Light curves in all three Gaia bands for the most (top panel) and
least (bottom panel) interesting targets not classified as variable stars,
formatted in the same manner as in Figure 10. The most interesting candidate
displays a nonsignificant but noticeable shift in the median flux before and
after the crossing time, as well as possible increase in the scatter. The
best-fit slope also appears to be strongly negative before the crossing time
but near-horizontal afterward. On the other hand, the least interesting
candidate shows almost zero change in the median flux, and seemingly small
change in the scatter and best-fit slope.

##  7 Conclusions

We have presented both a spatiotemporal technosignature candidate search
framework, which combines the SETI Ellipsoid and Seto methods, and a novel
SETI approach that is sensitive to changes in a periodic star’s variability
parameters. With precise astrometry and distance from Gaia and Bailer-Jones et
al. (2021), we explore this framework with several nearby SNe as source events
for signal synchronization to select candidates with crossing times within the
time range of available Gaia epoch photometry, and perform our variability
analysis on the well-parametrized subset of eclipsing binary candidate
systems.

Given the lack of any statistically significant changes in the light curves of
the selected eclipsing binaries, we can place constraints on the prevalence of
extraterrestrial civilizations using either of these spatiotemporal signaling
schemes and behaving in a manner detectable by our time domain analysis, as
has been done by Price et al. (2020), Franz et al. (2022), and others for
previous SETI observations. This limit can be calculated with a one-sided 95%
Poisson confidence interval, assuming a conservative 50% probability of
detecting such a signal if present (Gehrels, 1986). We performed our analysis
on a total of 779 variable star systems, and so we place an upper limit of
0.47% on the percentage of such systems hosting civilizations that are
behaving in the hypothesized manner.

Although we use only the error associated with the distance to the stars to
calculate crossing time uncertainties, the errors in the SNe distances are
also significant, and will have a nonnegligible effect on the timing
uncertainty for most targets via both the SETI Ellipsoid and the Seto method.
We, like Seto (2021), again emphasize that while precise distances to SNe will
help us more accurately constrain our technosignature searches, the advent of
these measurements is beneficial not just to SETI but to many areas of
astrophysics. With tighter SNe distance errors, we will also be able to expand
our framework to other historic Galactic SNe.

Our geometric framework constrains the SETI parameter space by telling us
where and when to search, for both planned observations and archival data, but
the nature of the synchronized transmission is still unknown. To best utilize
the light curves from Gaia, we look for modulations in frequency, amplitude,
or phase of the candidate systems, which is reasonably feasible for an
advanced civilization. Another possible intentional signals in optical bands
is a peak in the light curve that mimics the shape and width of the source
event. Although this is not yet feasible given the limitations of Gaia data
and photometry, the situation may change in the near future. This method may
also be better suited for a telescope like TESS, which has more densely
sampled data at the cost of some long-term stability and discontinuous
observations for most stars. Furthermore, there is the possibility of using
other datasets and observatories to search for synchronized bright radio
flashes or other types of transient radio signals. In particular, the SETI
Ellipsoid and Seto schemes could be used to select targets for radio SETI
observations.

This work is also limited by the incompleteness of Gaia DR3 light curve data,
which consist of a curated set of variable objects that have passed through a
classification algorithm and may not include other potentially interesting
candidates, such as stars that may have a short, small increase in flux.
Moreover, only three years of photometric data have been released for the
stars with light curves. Gaia has made this work and other geometric
approaches to SETI possible, and we expect that the next big revolution will
arrive with Gaia Data Release 4, which will include 66 months of epoch
photometry for all 2 billion sources.

The authors thank the anonymous referee for their helpful comments, which
substantially improved this work. The authors wish to thank Sofia Sheikh and
Barbara Cabrales for helpful conversations about the SETI Ellipsoid. The
Breakthrough Prize Foundation funds the Breakthrough Initiatives which manages
Breakthrough Listen. A.N. was funded as a participant in the Berkeley SETI
Research Center Research Experience for Undergraduates Site, supported by the
National Science Foundation under grant No. 1950897. J.R.A.D. acknowledges
support from the DiRAC Institute in the Department of Astronomy at the
University of Washington. The DiRAC Institute is supported through generous
gifts from the Charles and Lisa Simonyi Fund for Arts and Sciences, and the
Washington Research Foundation.

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Null Lagrangians in Schwarzian mechanics

Pratik Majhi and Madan Mohan Panja

Department of Mathematics, Visva-Bharati University, Santiniketan 731235,
India

Pranab Sarkar

Department of Chemistry, Visva-Bharati University, Santiniketan 731235, India

and

Benoy Talukdar

Department of Physics, Visva-Bharati University, Santiniketan 731235, India

## Abstract

In addition to standard and non-standard Lagrangians of classical mechanics,
we consider, in this work, null Lagrangians that (i) identically satisfy the
Euler-Lagrange equation and at the same time can be expressed as (ii) the
total derivative of some scalar function. As an addendum to the properties in
(i) and (ii) we find that null Lagrangians are also characterized by (iii)
vanishing energy functions or Jacobi integrals. By working with higher-order
${SL(2,R)}$ invariant Schwarzian derivatives introduced recently by Krivonos
(Phys. Rev. D 109, 065029(2024)) we demonstrate that these Schwarzians,
especially the even –order ones, provide a natural basis to introduce higher-
order null Lagrangians in Schwarzian mechanics.

Physical theories are often formulated using standard Lagrangian functions [1]
represented by $L=T-V$, where $T$ is the kinetic energy of the system under
study and $V$, the corresponding potential function. There also exist
Lagrangian functions which involve neither $T$ nor $V$. Despite this, the
sought functions lead to the equations of motion via the Euler-Lagrange
equation. Consequently, these Lagrangian functions are qualified as ‘non-
standard’. The non-standard Lagrangians play a role in the study of non-linear
systems [2,3]. An additional family of Lagrangian functions often called the
null Lagrangians (NLs) is also of considerable current interest. By a null
Lagrangian, we mean one, the Euler-Lagrange equation of which vanishes
identically. The relevance of NL to ordinary differential equations has been
discussed by Olver [4] who clearly pointed out that the Euler-Lagrange
equations $E(NL)=0$ vanish identically for all values of the arguments (say
$t,x$) of $NL$ only if it is a total derivative of some function $P$ of $t,x$
and of derivatives of $x$ with respect to $t$. The function $P$ is often
called the gauge function. In the following, we deal with two well-known
examples of null Lagrangians quoted by Musielak [5] and point out that, in
addition to the well-known properties of NL quoted above, the energy functions
or Jacobi integrals of NLs also vanish identically.

The first null Lagrangian of our interest is given by

$\mathrm{NL_{1}(x,\dot{x})=c_{1}x\dot{x}}$ (1)

where $c_{1}$ is an arbitrary constant and, as before, $x$ is function of time
$t$. Here over dot denotes differentiation with respect to time $t$. The
first-order Lagrangian (1) leads to vanishing Euler-Lagrange equation with a
gauge function $P=({\frac{1}{2c_{1}}})x^{2}$. The second example of our
interest is again a first-order Lagrangian

$\mathrm{NL_{2}(x,\dot{x})=\frac{a_{1}\dot{x}}{a_{2}x+a_{4}}}$ (2)

Here, $a$’s are arbitrary constant. We have now $E(NL_{2})=0$ with
$P=\frac{a_{1}}{a_{2}}|a_{2}x+a_{4}|$.
The Jacobi integral or the energy function of any first-order Lagrangian
($L^{(1)}$) is given by

$\mathrm{J(L^{(1)})=x\frac{\delta{L^{(1)}}}{\delta{\dot{x}}}-L^{(1)}}$ (3)

Using Eqs. (1) and (2) successively in Eq. (3) we find that
$J(NL_{1})=J(NL_{2})=0$. This indicates that the essential properties of a
null Lagrangian, namely, $E(NL)=0$ and that $NL$ is the gradient of a scalar
function should be supplemented by simultaneous vanishing of its Jacobi
integral.
The properties and applications of $NLs$ have been explored in different areas
of physics and mathematics [6,7]. The construction procedure of null
Lagrangians have also been studied in the literature [8] with particular
emphasis on their use in the calculus of variations. The cases of both first
and higher-order Lagrangians were considered. The object of the present work
is to demonstrate that the null Lagrangian has a natural space in non-
relativistic conformally invariant systems.

For an arbitrary function $q(t)$the expression

$\mathrm{\\{q(t),t\\}=\frac{\dddot{q}(t)}{\dot{q}(t)}-\frac{3}{2}\left(\frac{\ddot{q}(t)}{\dot{q}(t)}\right)^{2}}$
(4)

defines the well known Schwarzian derivative [9]. It is invariant under
fractional transformation and allows one to construct the
$SL(2,{R})$-invariant action functional

$\mathrm{S=1/2\int dt\ \\{q(t),t\\}}$ (5)

such that any function of $\\{q(t),t\\}$ can be used as Lagrangians to
determine the equation of motion of a higher-derivative $SL(2,{R})$-invariant
system. One dimensional mechanics so defined is often called the Schwarzian
mechanics[10].

The Schwarzian derivative (4) can be written in an alternative form

$\mathrm{\\{q(t),t\\}=\frac{d}{dt}\left(\frac{\ddot{q}(t)}{\dot{q}(t)}\right)-\frac{1}{2}\left(\frac{\ddot{q}(t)}{\dot{q}(t)}\right)^{2}},$
(6)

where $\frac{\ddot{q}(t)}{\dot{q}(t)}$ is called the pre-Schwarzian
derivative. While the Schwarzian derivative is invariant under $SL(2,{R})$
transformation $q(t)=\frac{af(t)+b}{cf(t)+d}$, with $a,b,c,d$ real or complex
constants such that $ad-bc=0$, the pre-Schwarzian is not. Interestingly, if
one chooses to work with a Lagrangian defined by the square of the pre-
Schwarzian derivative, namely,

${L^{(2)}={\frac{1}{2}}\left(\frac{\ddot{q}(t)}{\dot{q}(t)}\right)^{2}},$ (7)

one arrives at the following results[11].
The second-order energy function
$J\left(L^{(2)}\right)=\ddot{q}(t)\frac{dL^{(2)}}{d\ddot{q}}-L^{(2)}+\dot{q}(t)\left(\frac{dL^{(2)}}{d\dot{q}(t)}-\frac{d}{dt}\left(\frac{dL^{(2)}}{d{q(t)}}\right)\right)$
becomes equal to -$\\{q(t),t\\}$ and while the equation of motion is obtained
as a fourth-order nonlinear differential equation

$\mathrm{q^{(4)}(t)\dot{q}(t)^{2}-4\dot{q}(t)\ddot{q}(t)\dddot{q}(t)+3\ddot{q}^{(3)}(t)=0}$
(8)

The result in (8) has been obtained from the higher-order Euler-Lagrange
equation[12]

$ELeq=\sum_{i=0}^{n}(-1)^{i}\frac{d^{i}}{dt^{i}}\frac{\partial L}{\partial
q^{(i)}}$ (9)

for $n=2$. In this context, it appears rather interesting to envisage a
similar study for a third-order Lagrangian defined by the Schwarzian
derivative. The energy function of our third-order Lagrangian can be obtained
from the general expression

$J(L)=\sum_{r=1}^{n}q^{(r)}(t)\sum_{k=0}^{n}r(-1)^{k}\frac{d^{k}}{dt^{k}}\frac{\partial
L}{\partial q^{(r+k)}(t)}$ (10)

for the Jacobi integral of the $n^{th}$-order Lagrangian. Interestingly, we
find $J(L^{(3)})=\\{q(t),t\\}$ and an equation of motion which is exactly the
same as that in (8). Thus it is natural to compute results for equations of
motion and Jacobi integrals using Lagrangians represented by higher-order
Schwarzian derivatives and thus gain some physical insight from them. We shall
see in the course of our study this will provide an opportunity to look for
null Lagrangians in Schwarzian mechanics.

Higher-order generalization of the Schwarzian derivative in (4) has been
considered by a number of authors. For example, more than fifty years ago
Aharonov [13] provided definitions of higher-order analogues of
$\\{q(t),t\\}$. Relatively recently, Tamanoi [14] introduced another set of
higher-order Schwarzians. In close analogy with Aharonov invariants, Schippers
[15] defined higher Schwarzians $S_{n}\\{q(t)\\}$ inductively by means of the
recurrence relation

$\mathrm{S_{n+1}(q(t))=\frac{d}{dt}S_{n}(q(t))-(n-1)\frac{\ddot{q}(t)}{\dot{q}(t)}S_{n}(q(t))}$
(11)

$n\geq 3$, with $S_{3}(q(t))=\\{q(t),t\\}$. The higher-order Schwarzians
obtained from (11) is not invariant under $SL(2,{R})$ transformation. The same
is true for other such existing results.

In a recent publication Krivonos [16] proposed a physical model for the origin
of higher Schwarzians and treated them as Goldstone fields associated with the
generators of the Virasoro algebra [17]. This provided a new set of
$SL(2,{R})$ invariant higher Schwarzians. We denote them by $\sigma,s$ rather
than $z,s$ as in ref.16 such that $\sigma_{3}=\\{q(t),t\\}$ the Schwarzian
derivative while $\sigma_{4},\sigma_{5}$ etc stand for higher Schwarzians. In
the following we choose these higher Schwarzians from ref.16 as Lagrangian
functions and compute the Euler-Lagrange equation and Jacobi integrals by
using (9) and (10) respectively. For $L^{(4)}=\sigma_{4}$ we found

${ELeq=0}\hskip 8.5359ptand\hskip 8.5359pt{J(L^{(4)})=0}$ (12)

with the Gauge function $P=\\{q(t),t\\}$. Thus the higher Schwarzian
$\sigma_{4}$ represents a null Lagrangian.

Let us now consider the next higher-order Lagrangian given by
$L^{(5)}=\sigma_{5}$. In this case we get a sixth-order nonlinear equation of
motion

$\begin{split}\mathrm{{\dot{q}(t)^{4}q^{(6)}(t)-6\dot{q}(t)^{3}\ddot{q}(t){q^{(5)}}(t)}}-10\dot{q}(t)^{3}\dddot{q}(t)\\\
{q^{(4)}}(t)+{\frac{45}{2}}\dot{q}(t)^{2}\ddot{q}(t)^{2}{q^{(4)}}(t)+30\dot{q}(t)^{2}\ddot{q}(t)\\\
{{\dddot{q}(t)^{(2)}}-60\dot{q}(t)^{3}\ddot{q}(t){q^{(5)}}(t)}+{\frac{45}{2}}\ddot{q}(t)^{5}=0\end{split}$
(13)

with the Jacobi integral

$\begin{split}\mathrm{J(L^{(5)})=\frac{45\ddot{q}(t)^{4}}{4\dot{q}(t)^{4}}-\frac{25\ddot{q}(t)^{2}\dddot{q}(t)}{4\dot{q}(t)^{3}}}\\\
{+\frac{5\dddot{q}(t)^{2}}{\dot{q}(t)^{2}}+\frac{10\ddot{q}(t)\dddot{q}^{(4)}(t)}{\dot{q}(t)^{2}}-\frac{2{q^{(5)}(t)}}{\dot{q(t)}}},\end{split}$
(14)

with $q^{(n)}(t)=\frac{d^{n}q(t)}{dt^{n}}$. Clearly’ $L^{(5)}$is not a null
Lagrangian although $\sigma_{5}$ is $SL(2,{R})$ invariant. The reason behind
is that $L^{(5)}$ does not follow from a Gauge function. But working with a
sixth-order Lagrangian defined by $L^{(6)}=\sigma_{6}$ we found

${ELeq=0}\hskip 8.5359ptand\hskip 8.5359pt{J(L^{(6)})=0}$ (15)

and with the gauge function

$\begin{split}\mathrm{P=\frac{{q^{(5)}(t)}}{\dot{q}(t)}-\frac{5\ddot{q}(t){q^{(4)}(t)}}{\dot{q}(t)^{2}}-\frac{5\dddot{q}(t)^{2}}{\dot{q}(t)^{2}}}\\\
{+\frac{20\ddot{q}(t)^{2}\dddot{q}(t)}{\dot{q}(t)^{3}}-\frac{45\ddot{q}(t)^{4}}{4\dot{q}(t)^{4}}}\end{split}$
(16)

In this way we could demonstrate that all even-order higher Schwarzians of
Krivonos[16] represent null Lagrangians while the odd-order ones are non-null.

We conclude by noting that fist-order null Lagrangians corresponding to both
standard and non-standard Lagrangians have been extensively treated in the
literature with a view to look for their usefulness in physics [18,19]. This
is , however, not the case with higher-order null Lagrangians, Thus it remains
an interesting curiosity to look for the role of higher-order null Lagrangians
and/or of higher Schwarzians of Krivonos in physical theories.

## Acknowledgement

One of the authors (PM) would like to thank University Grants Commission for a
research fellowship.

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