# Local star-forming galaxies build up central mass concentration most
actively near $M_{\ast}=10^{10}M_{\sun}$

Zhizheng Pan<EMAIL_ADDRESS>Purple Mountain Observatory, Chinese Academy of
Sciences, 10 Yuan Hua Road, Nanjing, Jiangsu 210033, China  School of
Astronomy and Space Sciences, University of Science and Technology of China,
Hefei, 230026, China  Xianzhong Zheng  Purple Mountain Observatory, Chinese
Academy of Sciences, 10 Yuan Hua Road, Nanjing, Jiangsu 210033, China  School
of Astronomy and Space Sciences, University of Science and Technology of
China, Hefei, 230026, China  Xu Kong  School of Astronomy and Space Sciences,
University of Science and Technology of China, Hefei, 230026, China  CAS Key
Laboratory for Research in Galaxies and Cosmology, Department of Astronomy,
University of Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

To understand in what mass regime star-forming galaxies (SFGs) build up
central mass concentration most actively, we present a study on the
luminosity-weighted stellar age radial gradient ($\nabla_{\rm age}$)
distribution of $\sim 3600$ low-redshift SFGs using the MaNGA Pipe3D data
available in the SDSS DR17. The mean age gradient is negative, with
$\nabla_{\rm age}=-0.14$ log Gyr/$R_{\rm e}$, consistent with the inside-out
disk formation scenario. Specifically, SFGs with positive $\nabla_{\rm age}$
consist of $\sim 28\%$ at log$(M_{\ast}/M_{\sun})<9.5$, while this fraction
rises up to its peak ($\sim 40\%$) near log$(M_{\ast}/M_{\sun})=10$ and then
decreases to $\sim 15\%$ at log$(M_{\ast}/M_{\sun})=11$. At fixed $M_{\ast}$,
SFGs with positive $\nabla_{\rm age}$ typically have more compact sizes and
more centrally concentrated star formation than their counterparts, indicative
of recent central mass build-up events. These results suggest that the build-
up of central stellar mass concentration in local SFGs is mostly active near
$M_{\ast}=10^{10}M_{\sun}$. Our findings provide new insights on the origin of
morphological differences between low-mass and high-mass SFGs.

galaxies: evolution

††software: astropy (Astropy Collaboration et al., 2013, 2018), matplotlib
(Hunter, 2007)

##  1 Introduction

The origin of galaxy morphology is a fundamental puzzle of galaxy formation
and evolution. Bulges, characterized by dense spheroidal swarm of stars, are
commonly found in the centers of galaxies. The origin of bulge stars could be
quit complicated. Various physical processes such as mergers (Hopkins et al.,
2009, 2010), giant clump migration (Dekel et al., 2022), disk instability
induced central star formation (Kormendy & Kennicutt, 2004; Athanassoula,
2005) and rapid early-on star formation (Okamoto, 2013) are considered to play
a role in bulge formation, although their relative contributions are still
unclear (Tacchella et al., 2019; Huško et al., 2023; Boecker et al., 2023).

In the past two decades, much efforts are made to understand the physics
behind the empirical relationship linking the star formation histories of
galaxies and their morphologies. Observational studies show that star
formation quenching is associated with the presence of a bulge component,
while galaxies with bulges are not necessarily to be quenched (Bell et al.,
2012; Fang et al., 2013; Barro et al., 2017). Multiple lines of evidence
suggest that a dissipative core-building event (”compaction”) that feeds an
active galactic nucleus (AGN) can quench star formation (e.g., Bluck et al.,
2014; Wang et al., 2018; Woo & Ellison, 2019; Xu & Peng, 2021). Although the
relation between star formation quenching and the presence of a bulge is still
a matter of debate, the coupling of the two phenomena implies that the build-
up of central mass concentration is a critical step in galaxy evolution.

In this work, we aim to explore in what mass regime SFGs build up central mass
concentration most actively. To answer this question, one needs to investigate
the spatially resolved star formation properties of a statistically large
galaxy sample. MaNGA is the largest integral filed unit (IFU) spectroscopic
survey to date, providing an excellent dataset suited for such studies. In
this work, we present a study on the luminosity-weighted stellar age radial
gradient ($\nabla_{\rm age}$) distribution of $\sim 3600$ local SFGs
($0.01<z<0.15$) using data from the MaNGA survey available in the SDSS DR17
release. Throughout this paper, we adopt a concordance $\Lambda$CDM cosmology
with $\Omega_{\rm m}=0.3$, $\Omega_{\rm\Lambda}=0.7$, $H_{\rm 0}=70$ $\rm
km~{}s^{-1}$ Mpc-1 and a Chabrier (2003) initial mass function (IMF).

##  2 data

MaNGA is an IFU spectroscopic survey for obtaining two-dimensional spectral
mapping of $\sim 10,000$ nearby galaxies (Bundy et al., 2015). The sizes of
the IFUs vary from 19 to 127 fibers, and the effective spatial resolution is
$2.^{\arcsec}5$ (Law et al., 2015). The data products of the MaNGA project
have been released in SDSS DR17.

We use the reduced data from the MaNGA Pipe3D value-added catalog (Sánchez et
al., 2022). The stellar mass ($M_{\ast}$) and photometric measurements are
drawn from the NASA-Sloan Atlas catalog111nsatlas.org, including SDSS $r-$band
effective radius $R_{\rm e}$, Sérsic index $n$, minor-to-major axis ratio
$b/a$. Star formation rate (SFR) is derived from the dust-corrected H$\alpha$
luminosity. This SFR is an upper limit to the real one, since all H$\alpha$
flux is integrated irrespective of the nature of ionization (Cano-Díaz et al.,
2016). To remove edge-on objects, galaxies with $b/a>0.4$ are selected. We
further restrict our analysis to galaxies with redshift $0.01<z<0.15$,
$10^{9.0}M_{\sun}<M_{\ast}<M^{12}M_{\sun}$, and $\rm
10^{-5}M_{\sun}yr^{-1}<SFR<10^{3}M_{\sun}yr^{-1}$. We also use the quality
control flag QCFLAG to select datacubes that have good-quality analysis by the
Pipe3D pipeline (with $\rm QCFLAG=0$). Galaxies observed with the 19-fiber
IFUs are excluded from analysis because the low spatial-resolution IFUs may
result in large uncertainties in the radial gradient measurements. The final
sample contains 6916 galaxies.

The Pipe3D pipeline uses a library which compromises 273 simple stellar
population (SSP) spectra, sampling 39 ages from 1 Myr to 13.5 Gyr and seven
metallicities ($Z/Z_{\sun}$=0.006, 0.029, 0.118, 0.471, 1, 1.764 and 2.353),
to model the observed spectra. This library is a subset of the MaStar SSP
library, which is generated from an updated set of Bruzual and Charlot (BC03)
stellar population synthesis model (Bruzual & Charlot, 2003; Plat et al.,
2019). Pipe3D derived the luminosity- and mass-weighted parameters of the
stellar population using the coefficients of stellar decomposition, with the
equation:

$\begin{split}{\rm log}P_{\rm LW}=\sum\nolimits_{\rm ssp}w_{\rm ssp,L}{\rm
log}P_{\rm ssp}\\\ ~{}{\rm log}P_{\rm MW}=\frac{\sum_{\rm ssp}w_{\rm
ssp,L}\Upsilon_{\rm ssp,\bigstar}{\rm log}P_{\rm ssp}}{\sum_{\rm ssp}w_{\rm
ssp,L}\Upsilon_{\rm ssp,\bigstar}},\end{split}$ (1)

where $P_{\rm ssp}$ is the value of a particular parameter for each SSP,
$w_{\rm ssp,L}$ is the weight contributed by each SSP, and $\Upsilon_{\rm
ssp,\bigstar}$ is the mass-to-light ratio. The weight $w_{\rm ssp,L}$ is
determined at a fixed spectral range of $5450<\lambda<5550$ Å, similar to the
$V$-band central wavelength. In the spectral modeling, the dust extinction law
of Cardelli et al. (1989) is adopted.

In Pipe3D, a radial gradient of a quantity of interest is computed as the
slope of the linear fit of the quantity versus the galactic centric distance
of the spaxel in unit of $R_{\rm e}$. First, the position angle, ellipticity,
and $R_{\rm e}$ provided by the NSA catalog for each galaxy are used to create
elliptical apertures of 0.15 $R_{\rm e}$ width, covering the galactic centric
distance from 0 to 3.5 $R_{\rm e}$. The average value for each parameter at
each distance is then estimated. For the radial distribution of each
parameter, the slope of the average gradient is derived based on a linear
regression of the considered parameter along the radius. The fitting is
restricted between $0.5-2.0R_{\rm e}$. When the field of view (FOV) dose not
reach 2.0$R_{\rm e}$, the regression is restricted to the largest distance
covered by the FOV. In this work, we focus our analysis on the radial gradient
of luminosity-weighted stellar age, $\nabla_{\rm age}$. The typical
uncertainty of $\nabla_{\rm age}$ is around $\rm{0.05~{}logGyr}/R_{e}$. More
details of the MaNGA Pipe3D dataset please refer to Sánchez et al. (2022).

The morphological classification catalog of the MaNGA galaxies released by
Domínguez Sánchez et al. (2022) (DS22) is also used. This catalog provides a
T-Type parameter (ranging from $-4$ to 9) for each MaNGA galaxy, which is
obtained by training deep-leaning models based on the T-Types from Nair &
Abraham (2010). In general, T-Type¡0 corresponds to early-type galaxies, while
T-Type $\geq$ 0 correspond to late-type galaxies. For late-type galaxies,
galaxies with 0$\leq$T-Type$\leq$3 tend to have larger bulge-to-disk ratios
than those of T-Type ¿ 3 (see Figure 22 of Fischer et al. (2019)).

Figure 1: left: the log${\rm SFR}-\rm logM_{*}$ plane. Symbols are color coded
by the luminosity-weighted stellar age gradient. The dashed line indicates log
sSFR$=10^{-11}yr^{-1}$, and SFGs are selected as those with SFRs above the
line. Right: a LOESS smoothed version of the left panel.

##  3 Results

###  3.1 the distribution of luminosity-weighted age gradient of SFGs

The luminosity-weighted stellar age radial gradient $\nabla_{\rm age}$
describes the spatial distribution of stellar age within a galaxy. In this
work, we consider SFGs with $\nabla_{\rm age}>0$ (younger central stellar age
than the outer region) are undergoing/or recently underwent central mass
build-up events. This method should be valid for detecting central mass build-
up events involving recent central star formation. As mentioned above, bulge
stars could also be originated from stellar radial migration or _ex situ_ star
formation (accreted from other galaxies). Such central mass build-up events
are out of the scope of this work.

In the left panel of Figure 1, we show the SFR$-M_{\ast}$ plane for the 6916
galaxies selected in Section 2. Data points are color coded by $\nabla_{\rm
age}$. In the SFR$-M_{\ast}$ plane, SFGs distribute along the star-formation
main sequence (SFMS). To explore the $\nabla_{\rm age}$ distribution across
the SFR$-M_{\ast}$ plane, we smooth the data using locally weighted regression
method LOESS developed by Cleveland & Devlin (1988), with the `PYTHON` code
released by Cappellari et al. (2013). LOESS is developed to uncover underlying
mean trends by reducing observational errors and intrinsic scatters. We adopt
a smoothing factor of $frac=0.1$, and a linear local approximation.

The right panel of Figure 1 shows the LOESS-smoothed version of the
SFR$-M_{\ast}$ plane. As can be seen, at fixed $M_{\ast}$, galaxies with the
highest SFRs tend to have positive $\nabla_{\rm age}$. Such galaxies are
likely undergoing the compaction processes (Tacchella et al., 2016; Ellison et
al., 2018), thus having enhanced SFRs compared to the SFMS ridge line. Along
the ridge line of the SFMS, galaxies tend to have negative $\nabla_{\rm age}$.
At log$(M_{\ast}/M_{\sun})>10.5$, SFGs have steep and negative $\nabla_{\rm
age}$. This is consistent with the observation that massive SFGs typically
harbor old bulge components (Pérez et al., 2013; Pan et al., 2015, 2016;
Belfiore et al., 2017). Interestingly, SFGs near log$(M_{\ast}/M_{\sun})=10$
show shallower $\nabla_{\rm age}$ compared to those in the low-mass and high-
mass regimes. In the low-SFR regime, galaxies generally have very shallow
$\nabla_{\rm age}$. This is because the low-SFR regime is dominated by early-
type galaxies, which typically exhibit very shallow age gradients (Parikh et
al., 2021).

In this work, we focus our analysis on SFGs. SFGs are selected as those with
log sSFR$>{-11.0}~{}\rm yr^{-1}$, where sSFR=${\rm SFR}/M_{\ast}$. 3590 SFGs
are finally selected. In Figure 2, we show $\nabla_{\rm age}$ as a function of
$M_{\ast}$ for the selected SFG sample. The mean value of $\nabla_{\rm age}$
is negative, which is around $-0.136~{}\rm log~{}Gyr{/R_{e}}$. This indicates
that SFGs generally have older central stellar population than their
outskirts, consistent with the ”inside-out” galaxy formation scenario (Wang et
al., 2011; González Delgado et al., 2015; Pan et al., 2015).

To verify our SFG selection, we cross-match the MaNGA sample with the
GALEX$-$SDSS$-$WISE Legacy Catalog (GSWLC, see Salim et al. 2016), which
derive SFRs based on broad-band ultraviolet+infrared photometry. Of the 3590
MaNGA selected SFGs, 2579 ones have GSWLC counterparts. We find that 2471 ones
($\sim 95$%) also have log $\rm{sSFR_{GSWLC}>{-11.0}~{}yr^{-1}}$. This
indicates that the SFG sample selected by $\rm{sSFR_{H_{\alpha}}}$ is almost
all overlapped with that selected by $\rm{sSFR_{\rm GSWLC}}$.

Figure 2: Left: The luminosity weighted age gradient as a function of
$M_{\ast}$ for SFGs. The mean age gradient is marked by the blue dashed line.
Large symbols show the median $\nabla_{\rm age}$ values of each stellar mass
bin, with a bin size of $\Delta M_{\ast}=0.2~{}$dex. The shaded region
indicates the $16-84$ percentile region. Right: violin plot showing the
distribution of $F_{\rm positive}$ at each stellar mass bin. Figure 3:
Left:The $R_{\rm e}-M_{\ast}$ relation, color coded by the luminosity weighted
stellar age gradient (with LOESS smoothing). Large symbols show the median
$R_{\rm e}$ values of each stellar mass bin, with a bin size of $\Delta
M_{\ast}=0.2$ dex. The two dashed lines indicate the $16-84$ percentile
region. Right: Sérsic index $n$ as a function of $M_{\ast}$, with LOESS
smoothing. Symbols are color coded by luminosity weighted stellar age
gradient.

We further divide the sample into stellar mass bins and investigate their
$\nabla_{\rm age}$ distributions more specifically, with a bin size of $\Delta
M_{\ast}=0.2~{}$dex. The result is shown in Figure 2. It is clear that the
$\nabla_{\rm age}$ distribution exhibits a ”bump structure” near
log$(M_{\ast}/M_{\sun})=10$. In this work, we consider SFGs with positive
$\nabla_{\rm age}$ are undergoing/or have recently underwent central mass
build-up processes, thus exhibiting younger central stellar ages than the
outer regions. We then investigate the fraction of galaxies with positive
$\nabla_{\rm age}$ (which we denote as $F_{\rm positive}$) in each stellar
mass bin. To do this, the uncertainty of $\nabla_{\rm age}$
($\sigma_{\nabla_{\rm age}}$) needs to take into account. We first construct
the $\nabla_{\rm age}$ probability distribution function (PDF) for each SFG
assuming a Gaussian form distribution centered at $\nabla_{\rm age}$, with a
dispersion of $\sigma=\sigma_{\nabla_{\rm age}}$. With the constructed PDFs,
we then perform bootstrapped resamplings to calculate $F_{\rm positive}$. We
generate 500 realizations and derive the $F_{\rm positive}$ distribution at
each stellar mass bin. The violin plot of the right panel of Figure 2 shows
the result. As can be seen, the median $F_{\rm positive}$ is around 27% at
log$(M_{\ast}/M_{\sun})=9.1$, while it rises up to $\sim$ 40% at
log$(M_{\ast}/M_{\sun})\sim 10$ and then rapidly decreases, reaching its
minimum value of $\sim 15\%$ near log$(M_{\ast}/M_{\sun})=11.0$. The peculiar
distribution at log$(M_{\ast}/M_{\sun})=11.3$ may be of less statistical
significance due to the small number of galaxies in this mass bin. The violin
plot suggests that the excess of $F_{\rm positive}$ at
log$(M_{\ast}/M_{\sun})\sim 10$ is of high significance.

The low fraction of SFGs with positive age gradients exhibits at
log$(M_{\ast}/M_{\sun})>10.5$ is expected, since massive SFGs typically
contain old bulge components (Pan et al., 2016; Belfiore et al., 2017). An
interesting feature revealed in Figure 2 is that, low-mass SFGs seem to have a
relatively low central mass build-up efficiency compared to those near
log$(M_{\ast}/M_{\sun})=10$. We will come back to this issue in the discussion
section.

Figure 4: The T-Type distribution for $\nabla_{\rm age}>0$ and $\nabla_{\rm
age}<-0.25$ subsamples. We show the results in two stellar mass bins, with
log$(M_{\ast}/M_{\sun})<10.3$ (left panel) and log$(M_{\ast}/M_{\sun})>10.3$
(right panel). The p-value of Kolmogorov-Smirnov test is also marked.

###  3.2 The dependence of luminosity-weighted age gradient distribution on
galaxy morphology

Previous works reported that the radial gradients of many galactic properties
are dependent upon galaxy morphologies (González Delgado et al., 2015; Li et
al., 2018; Parikh et al., 2021). In this section, we investigate the
dependence of $\nabla_{\rm age}$ on the morphologies of SFGs. In the left
panel of Figure 3, we explore the distribution of $\nabla_{\rm age}$ across
the $R_{\rm e}-M_{\ast}$ plane. At fixed $M_{\ast}$, SFGs with small $R_{\rm
e}$ tend to have shallow or positive $\nabla_{\rm age}$. A similar trend is
also exhibited across the $\Sigma_{*,1kpc}-M_{\ast}$ plane of SFGs (Woo &
Ellison, 2019), where $\Sigma_{*,1kpc}$ is the stellar mass density in the
central 1kpc region of a galaxy. The large symbols show the median sizes at
fixed $M_{\ast}$, i.e, the size$-$mass relation. As can be seen, the slope of
the size$-$mass relation is shallow at log$(M_{\ast}/M_{\sun})<10.0$ and
becomes steepened and tighter toward the high mass end. In the right panel, we
investigate the $\nabla_{\rm age}$ distribution across the $n-M_{\ast}$ plane.
Data points with $n=6$ correspond to the largest value allowed in the Sérsic
profile fitting (Blanton et al., 2005). Overall, SFGs with large $n$ tend to
have shallow or positive $\nabla_{\rm age}$. At fixed $M_{\ast}$, $\nabla_{\rm
age}$ is more strongly correlated with $R_{\rm e}$ than $n$. This may be due
to the fact that a single Sérsic function is not a good parameterization of
galaxy morphology (Tasca & White, 2011).

In Figure 4, we investigate the T-Type distributions for SFGs within two
$\nabla_{\rm age}$ bins, $\nabla_{\rm age}>0$ and $\nabla_{\rm age}<-0.25$. As
can be seen, SFGs with $\nabla_{\rm age}>0$ tend to have a broader T-Type
distribution than those with $\nabla_{\rm age}<-0.25$. Also, SFGs with
$\nabla_{\rm age}>0$ contain more fractions of galaxies with T-Type $\leq$ 3\.
This trend is particularly evident at log$(M_{\ast}/M_{\sun})<10.3$, which is
consistent with the trend shown in the left panel of Figure 3. We perform a
Kolmogorov-Smirnov (K-S) test to the T-Type distributions of the two samples.
The small p-values of the K-S test suggest that these two distributions are
statistically different. To conclude, at fixed $M_{\ast}$, SFGs with positive
$\nabla_{\rm age}$ tend to have more centrally concentrated morphologies,
consistent with the central mass build-up picture.

Figure 5: Left: The $R_{\rm e}-M_{\ast}$ relation for the NSA SFG sample
selected at $0.02<z<0.05$. Large symbols show the median $R_{\rm e}$ values of
each stellar mass bin, with a bin size of $\Delta M_{\ast}=0.2$ dex. The two
blue dashed lines indicate the $16-84$ percentile region. The red region
indicates the $16-84$ region of MaNGA SFGs as shown in Figure 3. Right: the
fraction of SFGs with $n>2$ in the MaNGA and NSA-sloan sample. The shaded
region indicates the $1\sigma$ region. Figure 6: Left: The luminosity weighted
age gradient as a function of $M_{\ast}$ for SFGs with $R_{\rm e}>R_{\rm
e,median}$. The mean age gradient is marked by the red dashed line. Large
symbols show the median $\nabla_{\rm age}$ values of each stellar mass bin,
with a bin size of $\Delta M_{\ast}=0.2~{}$dex. The shaded region indicates
the region of $16-84$ percentile. Right: similar to the left panel, but for
SFGs with $R_{\rm e}<R_{\rm e,median}$. The red region indicates the region of
$16-84$ percentile for the $R_{\rm e}>R_{\rm e,median}$ subsample.

##  4 Discussion

###  4.1 Is the MaNGA SFG sample representative?

One may ask whether the MaNGA SFG sample is representative for the local SFGs
in terms morphologies. To answer this question, we compare the $R_{\rm
e}-M_{\ast}$ relation and the $f_{\rm n>2}-M_{\ast}$ relation of MaNGA SFGs
with that of a local SFG comparison sample, where $f_{\rm n>2}$ is the
fraction of SFGs with Sérsic index $n>2$. In the literature, galaxies with
$n>2.5$ are considered to harbor prominent bulges, while those with $n<1.5$
are considered as bulgeless galaxies (Blanton et al., 2003; Bell et al.,
2012). Inspecting the $n-M_{\ast}$ relation, we consider galaxies with $n>2$
have built a bulge component. The comparison SFG sample is selected from the
NSA-Sloan catalog, which is the master photometric catalog of MaNGA Pipe3D. We
first cross-match the NSA catalog with the GSWLC catalog to drive SFR
measurements. Then we select SFGs with $b/a>0.4$ at $0.02<z<0.05$ as the
comparison sample, with $\rm log~{}sSFR_{\rm GSWLC}>-11.0~{}yr^{-1}$. In this
redshift range, the SDSS spectroscopic main galaxy sample is completed to
$M_{*}\sim 10^{9}M_{\sun}$ (Schawinski et al., 2010). As mentioned before, the
$\rm sSFR_{\rm GSWLC}$ selected SFGs are almost all overlapped with those
selected by $\rm sSFR_{\rm H\alpha}$, thus this comparison should be
physically meaningful. The comparison SFG sample contains $\sim 37000$
galaxies.

We show the result in Figure 5. As can be seen, the $R_{\rm e}-M_{\ast}$
relation of MaNGA SFGs is consistent with that of the NSA catalog at
log$(M_{\ast}/M_{\sun})>10.0$. At log$(M_{\ast}/M_{\sun})<10.0$, MaNGA SFGs
are biased against compact galaxies. On the one hand, this is partially due to
the fact that we have removed the 19-fiber IFUs in the sample selection. We
confirmed that when the 19-fiber IFUs are included, the discrepancy between
the two samples in the $R_{\rm e}-M_{\ast}$ relation is reduced at
log$(M_{\ast}/M_{\sun})<10.0$. On the other hand, this should be mainly due to
the selection effect of MaNGA, because low-mass compact galaxies are difficult
to spatially resolve in the MaNGA observation. Such a trend is confirmed by
the $f_{\rm n>2}-M_{\ast}$ relation, where the $f_{\rm n>2}$ of MaNGA SFGs is
lower than that of the NSA sample at log$(M_{\ast}/M_{\sun})<10.0$.

Figure 5 suggests that at fixed $M_{\ast}$, galaxies located in the high
$R_{\rm e}$ region should be less affected by the selection effect of MaNGA.
We thus divide the MaNGA SFG sample into two subsamples according to their
$R_{\rm e}$, one with $R_{\rm e}>R_{\rm e,~{}median}$ and the other with
$R_{\rm e}<R_{\rm e,~{}median}$. We show the $\nabla_{\rm age}-M_{\ast}$
relation for these two samples in Figure 6. As can be seen, the bump structure
at log$(M_{\ast}/M_{\sun})\sim 10.0$ exhibits in both subsamples.
Interestingly, it seems that the bump is more prominent in the $R_{\rm
e}>R_{\rm e,~{}median}$ subsample, for which the result should be less
affected by the MaNGA selection effect. To conclude, although the MaNGA SFG
sample is biased against low-mass compact galaxies, our findings should not be
significantly affected by this sample selection effect.

Figure 7: Top left: the mass-weighted age gradient as a function of
$M_{\ast}$. The stellar mass bin size is $\Delta M_{\ast}=0.2$ dex. Large
symbols show the median values of each stellar mass bin. Top right: D4000
gradient as a function of $M_{\ast}$. Bottom left: the H$\alpha$ flux gradient
as a function of $M_{\ast}$. Bottom right: the H$\alpha$ flux gradient as a
function of luminosity-weighted age gradient. large symbols show the median
values of each $\nabla_{\rm age}$ bin. In each panel, the shaded region
indicates the $16-84$ percentile region.

###  4.2 Using other stellar age indicators

In this section, we first check the robustness of our analysis that based on
luminosity-weighted age gradient. In the top left panel of Figure 7, we show
the mass-weighted age gradient as a function of $M_{\ast}$. As can be seen,
the mass-weighted age gradient exhibits no ”bump structure” near
log$(M_{\ast}/M_{\sun})=10$, in stark contrast to Figure 2. The mean mass-
weighted age gradient is also much shallower compared to the mean luminosity-
weighted age gradient. This is because the mass-weighted quantities typically
have narrower dynamical range than the luminosity-weighted quantities. In the
top right panel, we show the 4000 Å break $D_{n}(4000)$ gradient as a function
of $M_{\ast}$. $D_{n}(4000)$ is a commonly used stellar age indicator, which
is sensitive to star formation activity on timescales of $\sim 1-2$ Gyr
(Kauffmann et al., 2003; Wang et al., 2018). The $D_{n}(4000)$ gradient is
flat at log$(M_{\ast}/M_{\sun})<10.2$ and turns negative toward the massive
end, consistent with previous findings (Wang et al., 2018). Overall, the
behavior of $D_{n}(4000)$ gradient follows that of the mass-weighted stellar
age gradient, showing no excess of central mass build-up events near
$M_{\ast}=10^{10}M_{\sun}$.

In the bottom left panel of Figure 7, we show the gradient of H$\alpha$ flux
as a function of $M_{\ast}$. H$\alpha$ emission is sensitive to star formation
activity on timescales of several Myr and is a traditional tracer of SFR.
Thus, a very steep H$\alpha$ flux gradient indicates a high star formation
concentration. Interestingly, the H$\alpha$ flux gradient steepens near
log$(M_{\ast}/M_{\sun})=10.0$, which echoes the ”bump structure” we identify
in Figure 2. In the bottom right panel of Figure 7, we show that SFGs with
positive $\nabla_{\rm age}$ indeed have steepened H$\alpha$ flux gradient,
indicative of more centrally concentrated star formation. To conclude, Figure
7 suggests that signs of central mass build-up events can be better detected
using the radial gradients of star formation tracers on short timescales. In
stead, mass-weighted stellar age and $D_{n}(4000)$ gradients are less
sensitive to such events, possibly due to their narrow dynamical range.

###  4.3 Bulge formation in different mass regime

At log$(M_{\ast}/M_{\sun})<9.5$, $\sim 80$% SFGs have $n<2$, i.e., the
majority of low-mass SFGs have not built a central bulge component. In
observation, low-mass SFGs are rich in neutral hydrogen (Hi) gas (Saintonge &
Catinella, 2022). As shown in previous works, Hi gas plays an important role
in the build-up of a disk component (Wang et al., 2011; Chen et al., 2020; Pan
et al., 2021). With a high Hi gas fraction, disk rebuilding could be quit
common for low-mass compact SFGs. In addition, major merger events, which are
considered to be an important channel for bulge formation, are rare in this
mass regime (Casteels et al., 2014; Rodriguez-Gomez et al., 2015). Finally,
the bulge formation efficiency of low-mass galaxies can be suppressed by their
high gas richness during merger events (Hopkins et al., 2009, 2010). These
effects all potentially contribute to a low central mass build-up efficiency
as seen in the low-mass regime.

At log$(M_{\ast}/M_{\sun})>10.5$, the majority of SFGs have built a bulge
component (with $n>2$). For massive galaxies, major mergers are considered to
play an important role in bulge formation (Hopkins et al., 2010). Recent
studies showed that stellar migration may also contribute significantly to the
build-up of central mass for Milky Way-mass galaxies (Boecker et al., 2023).
The relative contribution of different mechanisms to the bulge formation of
massive galaxies is a matter of intense debate. Regardless the detail
formation mechanism, many bulges of massive SFGs show very low star formation
activity, indicating that the central mass build-up process is largely
completed, and the galaxies are on the path of star formation quenching
(Belfiore et al., 2017).

SFGs with intermediate mass stand in a unique position in bulge formation. In
the mass range of $9.5<{\rm log}(M_{\ast}/M_{\sun})<10.5$, neither ”mass
quenching” is operated effectively (Peng et al., 2010), nor the SFGs are too
rich in Hi gas. We indeed find evidence that SFGs near
log$(M_{\ast}/M_{\sun})=10.0$ build up their central mass concentration most
actively. Such a phenomenon is helpful to interpret the systematic
morphological differences between low-mass and high-mass SFGs, as shown in
Figure 5. We also note that our results are based on the low-redshift SFG
sample. The bulge formation picture could be different at high redshift, as
high-z galaxies exhibit more intense and clumpy star formation. In future
work, it would be interesting to extend our study to higher redshifts to
explore bulge formation across cosmic time.

##  5 Conclusion

In this paper, we investigate the luminosity-weighted age gradient
distribution of $\sim 3600$ local SFGs based on the MaNGA Pipe3D dataset. The
mean age gradient is negative, with $\nabla_{\rm age}=-0.14$ log Gyr/$R_{e}$,
consistent with the inside-out disk formation scenario. Specifically, SFGs
with positive $\nabla_{\rm age}$ consist of $\sim 28\%$ at
log$(M_{\ast}/M_{\sun})<9.5$, while this fraction rises up to its peak ($\sim
40\%$) near log$(M_{\ast}/M_{\sun})=10$ and then rapidly decreases, reaching
its minimum value of $\sim 15\%$ near log$(M_{\ast}/M_{\sun})=11.0$. We find
that galaxies with positive $\nabla_{\rm age}$ typically have more compact
sizes and more centrally concentrated star formation than their counterparts,
indicating that they are undergoing/or recently underwent the compaction
process to build up central mass concentration. We conclude that the build-up
of central stellar mass concentration in local SFGs is mostly active near
$M_{\ast}=10^{10}M_{\sun}$. These findings are helpful to interpret the
systematic morphological difference across the SFG population from low-mass to
high-mass regime.

We are grateful to the anonymous referee for very useful comments that have
improved the clarity of this paper. This work was partially supported by the
National Natural Science Foundation of China (NSFC, Nos. 12173088, 12233005
and 12073078), the science research grants from the China Manned Space Project
with NO. CMS-CSST-2021-A02, CMS-CSST-2021-A04 and CMS-CSST-2021-A07. Funding
for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan
Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.SDSS-IV acknowledges support and resources from the
Center for High Performance Computing at the University of Utah. The SDSS
website is www.sdss4.org. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution for
Science, Carnegie Mellon University, Center for Astrophysics — Harvard &
Smithsonian, the Chilean Participation Group, the French Participation Group,
Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli
Institute for the Physics and Mathematics of the Universe (IPMU) / University
of Tokyo, the Korean Participation Group, Lawrence Berkeley National
Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik
(MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University, New
York University, University of Notre Dame, Observatário Nacional / MCTI, The
Ohio State University, Pennsylvania State University, Shanghai Astronomical
Observatory, United Kingdom Participation Group,Universidad Nacional Autónoma
de México, University of Arizona, University of Colorado Boulder, University
of Oxford, University of Portsmouth, University of Utah, University of
Virginia, University of Washington, University of Wisconsin, Vanderbilt
University, and Yale University. In the main body of this work, we select
galaxy sample with a redshift range of $0.01<z<0.15$, which corresponds to a
cosmic time interval of $\sim 1.8$ Gyr. To investigate whether our results are
biased due to the redshift range adopted, we select SFGs within a narrower
redshift range ($0.02<z<0.05$) and repeat our analysis. In Figure 8, we show
the age gradient as a function of stellar mass with $\sim 2600$ SFGs at
$0.02<z<0.05$. As can be seen, the main feature is unchanged compared to
Figure 2. Thus the results presented in the main body of the article should
not be affected by the adopted redshift range in the sample selection.

Figure 8: Similar to Figure 2, with the SFG sample of $0.02<z<0.05$.

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# On the analyticity of static solutions of a field equation in Finsler
gravity

Erasmo Caponio  Department of Mechanics, Mathematics and Management,
Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy<EMAIL_ADDRESS>and
Antonio Masiello  Department of Mechanics, Mathematics and Management,
Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy
<EMAIL_ADDRESS>
###### Abstract.

It is well-known that static vacuum solutions of Einstein equations are
analytic in suitable coordinates. We ask here for an extension of this result
in the context of Finsler gravity. We consider Finsler spacetimes that retain
several properties of static Lorentzian spacetimes, are Berwald and have
vanishing Ricci scalar.

##  1\. Introduction

In a couple of papers appeared in 1970 [1, 2], H. Müller zum Hagen proved that
on any $C^{3}$ static or stationary spacetime which is a vacuum solution of
the Einstein equations there exists an appropriate analytic atlas such that
the metric coefficients of the solution are also analytic. Our aim in this
paper is to investigate if this result can be extended to static Finsler
spacetimes of Berwald type. This goal forces us to analyse at least three
questions:

> What is the convenient definition of a Finsler spacetime?
>
> What the one of a static Finsler spacetime?
>
> What the field equations extending Einstein equations?

We will then consider each of the above questions in the next three sections.
Section 5 will be devoted to the extension of Müller zum Hagen’s result to a
static Berwald  spacetime.

##  2\. On the Definition of a Finsler Spacetime

The idea of replacing the Lorentzian norm of a spacetime by a function
positively homogeneous on the velocities goes back to G. Randers [3]. He
introduced a complex-valued norm $F(x,y)=\sqrt{h_{x}(y,y)}+\omega_{x}(y)$,
where $h$ is a Lorentzian metric and $\omega$ a one-form on a four-dimensional
manifold $\tilde{M}$, that could take more into account the asymmetries of the
physical world, in particular the “uni-direction of timelike intervals”. After
decades, Lorentz–Finsler norms, eventually defined only on a cone sub-bundle
$\mathcal{A}$ of $T\tilde{M}$ in order to avoid complex and/or negative
values, appeared again in the work of G. S. Asanov (see [4] and the references
therein) about general relativity and gauge field theory. Afterwards, they
have been considered in the study of multirefringence models in optics [5], in
the classical limit of modified dispersion relations encompassing Lorentz
violation in quantum gravity and in the Standard Model Extension (see, e.g.,
[6, 7, 8, 9, 10]), in studies about causality and superluminal signals (see,
for example, [11, 12]). Actually, Lorentz–Finsler norms had already emerged
some years before the work of Asanov in a paper by H. Busemann [13], in
relation to the local description of the spacetime according to an axiomatic
definition called by the author timelike $G$-space. Inspired by [13], J.K.
Beem in [14] introduced the notion of an indefinite Finsler metric as a non-
reversible fiberwise positively homogeneous of degree 2 function $L=L(x,y)$,
defined on the tangent bundle of $\tilde{M}$, whose fiberwise Hessian

$\displaystyle\tilde{g}(x,y)[u,v]:=\frac{1}{2}\frac{\partial^{2}}{\partial
s\partial t}L(x,y+su+tv)|_{(s,t)=(0,0)}$ (1)

$u,v\in T_{x}\tilde{M}$, has index $1$ for all $x\in\tilde{M}$ and $y\in
T_{x}\tilde{M}\setminus\\{0\\}$. This definition widely extends Lorentzian
geometry, with the fundamental tensor $g$ replacing the Lorentzian metric and
the function

$F(x,\cdot):=\sqrt{|L(x,\cdot)|},$

giving a positively homogeneous Lorentz–Finsler norm at each tangent space
$T_{x}\tilde{M}$. Notice that $F(x.\cdot)$ becomes absolutely homogeneous if
$L$ is reversible, i.e., $L(x,y)=L(x,-y)$. Geodesics of $(\tilde{M},L)$
connecting two points $p,q\in\tilde{M}$ can be defined as extremal curves of
the energy functional
$\gamma\in\mathcal{C}_{pq}(\tilde{M})\mapsto\int_{0}^{1}L(\gamma,\dot{\gamma})ds$,
where $\mathcal{C}_{pq}(\tilde{M})$ is the set of all the piecewise smooth
curves $\gamma:[a,b]\to\tilde{M}$ such that $\gamma(a)=p$ and $\gamma(b)=q$.
It is soon realized that geodesics must satisfy the conservation law
$L(\gamma,\dot{\gamma})=\mathrm{const.}$ and, as a consequence, world-lines of
freely falling particles are introduced kinematically as those timelike
geodesics ($L(\gamma,\dot{\gamma})<0$) parametrized with
$L(\gamma,\dot{\gamma})=-1$. More recently, V. Perlick [15] used Beem’s
definition of the Finsler spacetime in order to extend Fermat’s principle for
light rays (i.e., geodesics satisfying $L(\gamma,\dot{\gamma})=0$) between a
point and a light source modelled as a timelike curve. His Finslerian Fermat’s
principle recovers also some results that had already appeared in the study of
optics in an anisotropic medium and also of sound rays in an anisotropic
elastic medium (see references in [15]).

In [14], some $2$-dimensional examples of indefinite Finsler metrics $L$,
reversible and not, are given where the set of lightlike vectors has more than
two connected components. E. Minguzzi [16] showed later that multiple light
cones do not occur if $L$ is smooth on $T\tilde{M}\setminus 0$, reversible and
$\mathrm{dim}\,\tilde{M}\geq 3$. Anyway, for non reversible and in particular
for functions $L$ that are not smooth on the whole slit tangent bundle,
multiple connected components are to be expected. This fact had led several
authors to assume that only one of these connected components should be
considered as a privileged one by the point of view of causality. The choice
can be done, e.g., by prescribing a timelike, globally defined, vector field
$Y$ and taking at each $x\in\tilde{M}$ the connected component which is the
boundary of the set of timelike vectors containing $Y(x)$ (such as, for
example, in [17, 16]) or by a priori restricting $L$ to a cone sub-bundle
$\mathcal{A}$ of $T\tilde{M}$, like in Asanov’s definition of a Finsler norm
$F$ (such as, for example, in [18, 19, 20, 21]) or by looking only at the cone
structure, without considering as fundamental the function $L$ (see [22, 23,
24, 25]). In some physical models, anyway, indefinite Finsler metrics $L$
arise as the metrics invariant under the action of the symmetry group
considered and, in general, they are defined and smooth only on an open cone
sub-bundle of $T\tilde{M}$. In particular, this is the case of the Bogoslovsky
metric (see, for example, [26, 27]). It was observed that this is the metric
that is preserved under the action of the group of transformations of the so-
called Very Special Relativity [28].

Recently, a definition of a Finsler spacetime has been proposed [29] that
encompasses definitions which generalize Beem’s one as those in [30, 18, 19,
20]. The authors declare in [29] that their definition does not include some
classes of Finsler spacetimes studied in [31, 32] which can be seen as
generalizations of standard static and stationary Lorentzian spacetimes and
that have already appeared in other papers [33, 34, 35, 36]. Thus, it is worth
to relax slightly the definition in [29] in order to include them.

######  Definition 1 (Open cone sub-bundle of $T\tilde{M}$).

Let $\tilde{M}$ be a smooth connected manifold of dimension $n+1$ and
$\tilde{\pi}:T\tilde{M}\to\tilde{M}$ its tangent bundle. A subset $C\subset
T\tilde{M}$ will be said an open cone sub-bundle of $T\tilde{M}$, if

  * (i)

$\tilde{\pi}(C)=\tilde{M}$;

  * (ii)

for all $x\in\tilde{M}$, $C_{x}:=T_{x}\tilde{M}\cap C$ is a pointed open cone,
i.e., $0\in C_{x}$, $C_{x}\setminus\\{0\\}$ is an open subset of
$T_{x}\tilde{M}$ and if $y\in C_{x}$ then
$\lambda{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}y}\in
C_{x}$ for each $\lambda>0$;

  * (iii)

$C_{x}$ varies smoothly with $x\in\tilde{M}$ meaning that
$C_{x}\setminus\\{0\\}$ is defined by the union of the solutions of a finite
number of systems of inequalities in the variable $y$

$\begin{cases}E_{1,k}(x,y)>0\\\ \ldots\\\ E_{m_{k},k}(x,y)>0\end{cases}$

where, for each $k\in\\{1,\ldots,l\\}$, $E_{1,k},\ldots,E_{m_{k},k}\colon
T\tilde{M}\to\mathbb{R}$ are $m_{k}$ smooth functions on $T\tilde{M}\setminus
0$, positively homogeneous of degree $\alpha_{1,k},\ldots,\alpha_{m_{k},k}$ in
$y$.

An open cone sub-bundle such that for all $x\in\tilde{M}$, $C_{x}$ is convex
will be said a convex open cone sub-bundle. Moreover, an open cone sub-bundle
without the zero section will be called a slit cone sub-bundle.

######  Remark 1.

Notice that we do not assume that $C_{x}$ is convex nor that it is salient,
i.e., that if $y\in C_{x}$ then $-y\not\in C_{x}$ (indeed the open cone sub-
bundle $C$ can be equal to $T\tilde{M}$). Being salient is instead certainly
true for a convex slit cone sub-bundle. Finally, notice that (iii) implies
that the boundary of a fibre of an open cone sub-bundle is the union of a
finite number of piecewise smooth hypersurfaces in $T_{x}\tilde{M}$.

######  Definition 2 (Lorentz–Finsler metric and Finsler spacetimes).

Let $C$ be an open cone sub-bundle of $T\tilde{M}$. A Lorentz–Finsler metric
on $\tilde{M}$ is a continuous function $L\colon C\to\mathbb{R}$ which
satisfies:

  * (i)

$L=L(x,y)$ is fiberwise positively homogeneous of degree two, i.e.
$L(x,\lambda y)$ $=\lambda^{2}L(x,y)$, for all $x\in\tilde{M}$, $y\in C_{x}$
and all $\lambda\geq 0$;

  * (ii)

there exist a slit cone sub-bundle $A$ and an open cone sub-bundle $B$ such
that $A\subset B\subset C$ and $L$ is at least $C^{1}$ on $B$ and at least
$C^{4}$ on $A$ with its fundamental tensor $\tilde{g}(x,y)$, defined as in
(1), being non-degenerate for all $(x,y)\in A$;

  * (iii)

there exists a slit cone sub-bundle $T\subset A\cap L^{-1}((-\infty,0))$ such
that its closure in $L^{-1}((-\infty,0))$, denoted by $T^{A}$, is a convex
connected component of $L^{-1}((-\infty,0))$ contained in $B$ and, for all
$(x,y)\in T$, $\tilde{g}(x,y)$ has index $1$.

A Finsler spacetime is a smooth finite dimensional manifold $\tilde{M}$
endowed with a Lorentz–Finsler metric.

This definition differs from the one in [29] essentially because we relax the
condition that there exists a connected component of $L^{-1}((-\infty,0))$,
the slit cone sub-bundle which represents all the future-pointing timelike
directions physically admissible, where $L$ is smooth (and $g$ has index $1$
on it). As done in [34], a quick and elegant definition of a Lorentz–Finsler
metric might consist in requiring that $L$ is $C^{2}$ a.e. on $T\tilde{M}$
with fundamental tensor having index $1$ a.e. on $T\tilde{M}$. Anyway, it is
preferable to control the lack of smoothness of $L$, hence we allow the
possibility that $\tilde{g}$ is not defined along some relevant future-
pointing timelike direction where $L$ remains differentiable at least once.
This requirement allows us to get geodesics at least as weak extremal
contained in $B$ of the energy functional. Let us recall that a piecewise
$C^{1}$ curve $\gamma:[a,b]\to\tilde{M}$ is a continuous curve admitting a
partition $\\{t_{i}\\}_{i\in\\{0,\ldots,m\\}}$, $m\in\mathbb{N}$, of $[a,b]$
such that $\gamma|_{[t_{i-1},t_{i}]}$, for all $i\in\\{1,\ldots,m\\}$, is
$C^{1}$. Let us denote by $\partial_{x}L:B\to T^{*}\tilde{M}$ and
$\partial_{y}L:B\to T^{*}\tilde{M}$ the partial differentials of $L$ w.r.t.
the first and the second variable respectively.

######  Definition 3 (Geodesics contained in $B$).

Let $\gamma\colon[a,b]\to\tilde{M}$ be a piecewise $C^{1}$ curve such that
$(\gamma,\dot{\gamma})\subset B$ then we say that $\gamma$ is a geodesic of
$(\tilde{M},L)$ if for any piecewise $C^{1}$ vector field $\zeta$ along
$\gamma$ with $\zeta(a)=\zeta(b)=0$ it holds

$\int_{a}^{b}(\partial_{x}L(\gamma,\dot{\gamma})[\zeta]+\partial_{y}L(\gamma,\dot{\gamma})[\dot{\zeta}])ds=0.$

Arguing as in [37], Prop. 2.51, it can be proved that the Legendre map of $L$
on $T^{A}$, i.e., $(x,y)\in T^{A}\mapsto\partial_{y}L(x,y)[\cdot]\in
T_{x}^{*}\tilde{M}$ is injective on $T^{A}$. Thus, if
$\gamma:[a,b]\to\tilde{M}$ is a geodesic such that
$(\gamma,\dot{\gamma})\subset T$ then, by a standard argument about regularity
of weak extremal and classical Finslerian computations (see, e.g., [38]), we
get that $\gamma$ must be a $C^{3}$ curve satisfying the equation

$D^{\dot{\gamma}}_{\dot{\gamma}}\dot{\gamma}=0,$ (2)

where $D_{\dot{\gamma}}^{\dot{\gamma}}$ is the covariant derivative along
$\gamma$ with reference vector $\dot{\gamma}$ defined by the Chern connection
of $L$ which is well-defined on the open subset $A$ of $T\tilde{M}\setminus 0$
by (ii) of Definition 2. In local natural coordinates on $T\tilde{M}$,
Equation (2) corresponds to
$\ddot{\gamma}^{i}+\Gamma^{i}_{jl}(\gamma,\dot{\gamma})\dot{\gamma}^{j}\dot{\gamma}^{l}=0$,
where the components of the Chern connection are defined in Equation (9)
below. If we know that the Legendre map is injective on $A$, the same result
holds for all weak extremals $\gamma$ such that $(\gamma,\dot{\gamma})\subset
A$. In any case, if we know that a weak extremal $\gamma$ is $C^{1}$ and
$(\gamma,\dot{\gamma})\subset A$ then it satisfies (2) (and therefore it is
actually $C^{3}$).111In some cases, smoothness or at least $C^{1}$-regularity
of weak extremals hold; for example, this is true for some stationary
splitting Finsler spacetimes and for standard static Finsler spacetimes in
next section, see, respectively, [32], Prop. A2 and [31], Th. 2.13. In
particular, from (2) it follows that there exists a unique geodesic for each
initial condition in $T$.

As Randers spacetime metrics show, in general $L$ will be not differentiable
along null directions, i.e., along non-zero tangent vectors $(x,y)$ such that
$L(x,y)=0$. In order to have a definition for lightlike geodesics of a non
smooth $L$, a possible way is to require that, for every initial null
conditions $(x,y)$, there exists an open maximal interval
$(-\epsilon,\epsilon)$ and a $C^{1}$ curve
$\gamma:(-\epsilon,\epsilon)\to\tilde{M}$, with $\gamma(0)=x$ and
$\dot{\gamma}(0)=y$, such that for every sequence $(x_{k},y_{k})\subset
T\tilde{M}$ of initial conditions of solutions $\gamma_{k}$ of (2), $\gamma$
is the limit in the $C^{1}$ topology of $\gamma_{k}$ (see [29], Def. 1-(iv),
[34], Def. 1-(d)). A more general way of defining them (see [24, 25]) is
inspired by a well-known local property of lightlike geodesics in a spacetime:

######  Definition 4 (Lightlike pregeodesics).

Let $N\subset TM\setminus 0$ be the set of null directions, i.e.,
$N:=\\{(x,y)\in
T\tilde{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\setminus
0}:L(x,y)=0\\}$. Let also $T^{N}$ be the closure of $T$ in
$L^{-1}((-\infty,0])$. A Lipschitz curve $\gamma\colon[a,b]\to\tilde{M}$, such
that $(\gamma,\dot{\gamma})\subset T^{N}\cap N$ a.e., is a lightlike
pregeodesic if for any $s_{0}\in[a,b]$ there exists a neighbourhood $U$ of
$\gamma(s_{0})$ such that any two points in $\gamma([a,b])\cap U$ are not
connected by any Lipschitz curve $\alpha$ such that
$(\alpha,\dot{\alpha})\subset T^{A}$.

As a consequence of [25], Theorem 6.6, we have that if $T^{N}\cap N\subset A$
(i.e., $L$ is smooth on a neighbourhood of the null directions in the boundary
of $T^{A}$) then any lightlike pregeodesics in the sense of Definition 4, is
actually a geodesic, up to reparametrization, i.e., it satisfies Equation (2).

##  3\. About the Notion of Stationary and Static Finsler Spacetimes

Let us recall the notion of a Killing vector field for a Finsler metric. We
refer to [32] for details. Since $L$ is $C^{1}$ on the open cone sub-bundle
$B$, we take $B\subset T\tilde{M}\setminus 0$ as the base space instead of the
slit tangent bundle which is usual in Finsler geometry (compare with [39]). A
vector field $K$ on $\tilde{M}$ is a Killing vector field for
$(\tilde{M},L,B)$ if $K^{c}|_{B}(L)=0$, where $K^{c}$ denotes the complete
lift of $K$ to $T\tilde{M}$ (restricted to the open subset $B$). This is the
vector on $T\tilde{M}$ whose local flow $\tilde{\psi}$ is given by
$\tilde{\psi}_{t}(v)=(\psi_{t}(p),d\psi_{t}(p)[v])$, where $\psi$ is the flow
of $K$, $p=\tilde{\pi}(v)$, $v\in TM$. Thus, if $K$ is a Killing vector field
then $L$ is invariant under the flow of $K^{c}$. In natural local coordinates
of $T\tilde{M}$, $K^{c}(L)$ is given by:

$K^{c}(L)(x,y)=K^{h}(x)\frac{\partial L}{\partial x^{h}}(x,y)+\frac{\partial
K^{h}}{\partial x^{i}}(x)y^{i}\frac{\partial L}{\partial y^{h}}(x,y),$

for all $(x,y)\in B$ (the Einstein’s sum convention is used in the above and
in the following equations). It is not difficult to prove also that $K$ is a
Killing field iff $K^{c}|_{A}$ is an infinitesimal generator of local
$\tilde{g}$-isometries, i.e., for each $v\in A$ and for all $v_{1},v_{2}\in
T_{\pi(v)}\tilde{M}$, we have

$\tilde{g}(\tilde{\psi}_{t}(v))\big{[}d\psi_{t}(p)[v_{1}],d\psi_{t}(p)[v_{2}]\big{]}=\tilde{g}(v)[v_{1},v_{2}].$

for all $t\in I_{p}$, where $I_{p}\subset\mathbb{R}$ is an interval containing
$0$ such that the stages $\psi_{t}$ are well-defined in a neighbourhood
$U\subset\tilde{M}$ of $p=\pi(v)$ and $d\psi_{t}(p)[v]\in A$, for each $t\in
I_{p}$. Thus, the Lie derivative of $\mathcal{L}_{K}\tilde{g}$ in $A$
vanishes. In local natural coordinates on $T\tilde{M}$, this amount to say
that

$K^{c}(\tilde{g}_{lj})+\frac{\partial K^{h}}{\partial
x^{l}}\tilde{g}_{hj}+\frac{\partial K^{h}}{\partial x^{j}}\tilde{g}_{lh}=0.$

######  Definition 5 (Stationary Finsler spacetime).

A Finsler spacetime $(\tilde{M},L)$ is said stationary if it is endowed with a
Killing vector field $K$ which is timelike, i.e., $L(x,K(x))<0$ for all
$x\in\tilde{M}$.

In a Lorentzian manifold $(M,h)$, a timelike Killing vector field $K$ is said
static if $\mathrm{curl}K|_{\mathcal{D}}=0$, where $\mathcal{D}$ is the
orthogonal distribution to $K$. Equivalently, $K$ is static iff $\mathcal{D}$
is locally integrable; thus, for each $p\in M$ there exist a spacelike
hypersurface $S$, through $p$, orthogonal to $K$, and an open interval $I$
such that the pullback of the metric $h$ by the flow of $K$, defined in
$I\times S$, is given by $-\Lambda dt^{2}+h_{0}$, where $t\in I$,
$\partial_{t}$ is the pullback of $K$, $\Lambda=-h(K,K)$ and $h_{0}$ is the
Riemannian metric induced on $S$ by $h$ (see [40], Proposition 12.38). In
order to generalize this notion to Finsler spacetimes, requiring minimal
regularity assumptions on the Lorentz–Finsler metric $L$, we consider
$\mathcal{D}$ as the distribution of dimension $n$ on $\tilde{M}$ defined
pointwise by the kernel of the one-form $\partial_{y}L(x,K(x))[\cdot]$.

######  Definition 6 (Static Finsler spacetime).

Let $(\tilde{M},L)$ be a stationary Finsler spacetime endowed with a timelike
Killing vector field $K$, such that $(x,K(x))\subset B$. We say that $K$ is
static if
$\mathcal{D}:=\mathrm{ker}\big{(}\partial_{y}L(x,K(x))[\cdot]\big{)}$ is
locally integrable.

######  Remark 2.

Let $U$ be a vector field in $\tilde{M}$ such that $\big{(}x,U(x)\big{)}\in
T^{A}$, for all $x\in\tilde{M}$ and $L\big{(}x,U(x)\big{)}=-1$. If the
distribution
$\mathcal{D}=\mathrm{ker}\big{(}\partial_{y}L(x,U(x))[\cdot]\big{)}$ is
integrable then its integral manifold can be used to define the rest spaces of
the observer field $U$ (see the question posed in the final paragraph of
[41]). From [32], Theorem 4.8, if $K:=\sigma U$ is also a Killing vector
field, for some positive function $\sigma$ on $\tilde{M}$, $B=T\tilde{M}$ and
$L$ satisfies $L\big{(}x,K(x)\big{)}=L\big{(}x,-K(x)\big{)}$ and
$L\big{(}x,w\pm K(x)\big{)}=L(x,w)+L\big{(}x,K(x)\big{)}$ for all
$(x,w)\in\mathcal{D}$ then $(\tilde{M},L)$ is locally is isometric to a
standard static Finsler spacetime (see the definition below).

Recall that we have assumed that in the open cone sub-bundle $B$, $L$ is at
least $C^{1}$, thus $\mathcal{D}$ above is well-defined. In some Finsler
spacetimes, this is the best possible regularity level of $L$. Consider, for
example, a type of stationary splitting Finsler spacetime introduced in [32]:
assume that $\tilde{M}=\mathbb{R}\times M$ and denote with $(t,x)$ points in
$\tilde{M}$ and by $(\tau,y)$ tangent vectors of $T\tilde{M}$. Let

$L\big{(}(t,x),(\tau,y)\big{)}:=-\Lambda(x)\tau^{2}+2{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}(x,y)\tau+F^{2}(x,y),$
(3)

where $\Lambda$ is a smooth positive function on $M$,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}:TM\to\mathbb{R}$
a fiberwise positively homogeneous function and $F$ a Finsler metric on $M$.
Both
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$
and $F$ are assumed to be smooth on $TM\setminus 0$; moreover, the fundamental
tensor $g$ of $F$ (defined as in (1) with $F^{2}$ replacing $L$) is positive
definite for any $(x,y)\in TM\setminus 0$ while the fiberwise Hessian of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$
(defined analogously with
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$
in place of $L$) is positive semi-definite for all $(x,y)\in TM\setminus 0$.
Let us denote by $\mathcal{T}$ the trivial line sub-bundle of $T\tilde{M}$
defined by the vector field $\partial_{t}$. In this case $C=T\tilde{M}$,
$B=T\tilde{M}\setminus\mathcal{T}$,
$A=\big{(}T\tilde{M}\setminus\mathcal{T}\big{)}\cap\big{\\{}\big{(}(t,x),(\tau,y)\big{)}\in
T\tilde{M}:\tau>0\big{\\}}$ (see [32], Prop. 3.3),

$\displaystyle T=$ $\displaystyle\left\\{\\!\\!\Big{(}(t,x),(\tau,y)\Big{)}\in
T\tilde{M}:y\in T_{x}M\setminus\\{0\\},\
\tau>\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}(x,y)}{\Lambda(x)}+\sqrt{\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}^{2}(x,y)}{\Lambda^{2}(x)}+\frac{F^{2}(x,y)}{\Lambda(x)}}\right\\}\\!\subset\\!A$

and $\partial_{t}$ is timelike and Killing. A Finsler spacetime of the type
(3) has been considered in [42], where it has been shown to be a solution of
the field equation $R=0$ (see next section). In that paper, $L$ is a Finsler
perturbation of the Schwarzschild metric, indeed $F$ is the norm of the
Riemannian metric in the spacelike base and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$
is a function conformal to the norm of the standard Riemannian metric on
$S^{2}$, see Equation (40) in [42]:

$\displaystyle
L\big{(}(t,r,\theta,\varphi),(\tau,y_{r},y_{\theta},y_{\varphi})\big{)}:=$
$\displaystyle\quad-\left(1-\frac{2m}{r}\right)\tau^{2}+\epsilon\left(1-\frac{2m}{r}\right)\tau\sqrt{y_{\theta}^{2}+\sin^{2}\theta
y_{\varphi}^{2}}+\frac{y_{r}^{2}}{1-\frac{2m}{r}}+r^{2}(y_{\theta}^{2}+\sin^{2}\theta
y_{\varphi}^{2}),$

where $\epsilon$ is a perturbation parameter. A particular case in type (3),
is when
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$
is equal to a one-form $\omega$ on $M$. In such a case, $C=B=T\tilde{M}$
(i.e., $L$ is of class $C^{1}$ on $T\tilde{M}$) and
$A=T\tilde{M}\setminus\mathcal{T}$. The slit cone sub-bundle $T$ of timelike
future-pointing vector is defined as above with $\omega$ replacing
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}$;
now there is also another slit cone sub-bundle associated to $L$ which is

$\displaystyle T^{-}=$ $\displaystyle\left\\{\Big{(}(t,x),(\tau,y)\Big{)}\in
T\tilde{M}:y\in T_{x}M\setminus\\{0\\},\
\tau<\frac{\omega_{x}(y)}{\Lambda(x)}-\sqrt{\frac{\omega^{2}_{x}(y)}{\Lambda^{2}(x)}+\frac{F^{2}(x,y)}{\Lambda(x)}}\right\\}\subset
A.$

In particular, in this case both $T^{A}$ and $T^{-,A}$ are convex and $L$ is
smooth on $N=\big{\\{}\big{(}(t,x),(\tau,y)\big{)}\in T\tilde{M}\setminus
0:L\big{(}(t,x),(\tau,y)\big{)}=0\big{\\}}=\big{\\{}\big{(}(t,x),(\tau,y)\big{)}\in
T\tilde{M}\setminus
0:\tau=\frac{\omega_{x}(y)}{\Lambda(x)}\pm\sqrt{\frac{\omega^{2}_{x}(y)}{\Lambda^{2}(x)}+\frac{F^{2}(x,y)}{\Lambda(x)}}\big{\\}}$.
The vector field $\partial_{t}$ is a timelike Killing vector field of
$(\tilde{M},L)$ which is static if $\omega=0$ (with integral manifolds
$\\{t\\}\times M$, $t\in\mathbb{R}$). Finsler spacetimes $(\mathbb{R}\times
M,L)$, with $L$ of the type (3) and $\omega=0$ have been called in [31],
standard static Finsler spacetime.

######  Remark 3.

We observe that the slit cone sub-bundle $T$ is defined also as the set of
timelike vectors with positive component $\tau$ of the standard static Finsler
spacetime $(\mathbb{R}\times M,L_{\omega})$, where

$L_{\omega}\big{(}(t,x),(\tau,y)\big{)}:=-\tau^{2}+F_{\omega}^{2}(x,y),$ (4)

and $F_{\omega}$ is given by

$F_{\omega}(x,y)=\frac{\omega_{x}(y)}{\Lambda(x)}+\sqrt{\frac{\omega^{2}_{x}(y)}{\Lambda^{2}(x)}+\frac{F^{2}(x,y)}{\Lambda(x)}}.$
(5)

In fact, from [32], Th. 5.1, $F_{\omega}$ is a Finsler metric on $M$.

##  4\. Vacuum Field Equations

In general relativity, geodesics deviation equation is used to describe the
relative acceleration of a congruence of point particles. In particular, in
vacuum, the absence of tidal forces implies that $R^{i}_{jil}y^{j}y^{l}=0$,
where $R^{i}_{jkl}$ are the components of the Riemann curvature tensor; as a
consequence Einstein field equations $R_{jl}=R^{i}_{jil}=0$ are satisfied and,
vice versa, if $R_{jl}=0$, then $R^{i}_{jil}y^{j}y^{l}=0$. In [42], S. Rutz
used this equivalence to generalize Einstein vacuum field equations to the
Finsler setting as a single scalar equation $R(x,y)=R^{i}_{i}(x,y)=0$ on the
slit tangent bundle. Here $R=R(x,y)$ is the Finsler Ricci scalar defined as
follows. Let $\tilde{g}^{ij}(x,y)$ the components of the inverse of the matrix
representing the fundamental tensor $\tilde{g}$ at the point $(x,y)\in A$ and
let $G^{i}(x,y)$, $(x,y)\in A$, be the spray coefficients of $L$:

$G^{i}(x,y):=\frac{1}{4}\tilde{g}^{ij}(x,y)\left(\frac{\partial^{2}L}{\partial
x^{k}\partial y^{j}}(x,y)y^{k}-\frac{\partial L}{\partial x^{j}}(x,y)\right),$
(6)

so that a smooth curve $\gamma$, such that $(\gamma,\dot{\gamma})\subset A$,
is a geodesic of $L$ if and only if, in natural local coordinate on
$T\tilde{M}$, $\ddot{\gamma}^{i}+2G^{i}(\gamma,\dot{\gamma})=0$. Let

$R^{i}_{k}(x,y):=2\frac{\partial G^{i}}{\partial
x^{k}}(x,y)-y^{m}\frac{\partial^{2}G^{i}}{\partial x^{m}\partial
y^{k}}(x,y)+2G^{m}(x,y)\frac{\partial^{2}G^{i}}{\partial y^{m}\partial
y^{k}}(x,y)\\\ -\frac{\partial G^{i}}{\partial y^{m}}(x,y)\frac{\partial
G^{m}}{\partial y^{k}}(x,y).$ (7)

The Riemann curvature of $L$ at $(x,y)\in A$ is the linear map
$\mathbf{R}_{y}:T_{x}M\to T_{x}M$ given by
$\mathbf{R}_{y}:=R^{i}_{k}(x,y)\partial_{x^{i}}\otimes dx^{k}$. It can be
shown (see [43], Equations (8.11)–(8.12)) that
$R^{i}_{k}(x,y)=R^{i}_{jkl}(x,y)y^{j}y^{l}$ where $R^{i}_{jkl}$ are the
components of the $hh$ part of the curvature $2$-forms of the Chern connection
which are equal, for any $(x,y)\in A$, to

$R^{i}_{jkl}(x,y):=\frac{\delta\Gamma^{i}_{jl}}{\delta
x^{k}}(x,y)-\frac{\delta\Gamma^{i}_{jk}}{\delta
x^{l}}(x,y)+\Gamma^{m}_{jl}(x,y)\Gamma^{i}_{mk}(x,y)-\Gamma^{m}_{jk}(x,y)\Gamma^{i}_{ml}(x,y),$
(8)

being $\frac{\delta}{\delta x^{i}}$ the vector field on $A$ defined by
$\frac{\delta}{\delta x^{i}}:=\frac{\partial}{\partial
x^{i}}-N^{m}_{i}(x,y)\frac{\partial}{\partial y^{m}}$, where
$N^{m}_{i}(x,y):=\frac{\partial G^{m}}{\partial y^{i}}(x,y)$, and
$\Gamma^{i}_{jk}$ are the components of the Chern connection,

$\Gamma^{i}_{jk}(x,y):=\frac{1}{2}\tilde{g}^{il}(x,y)\left(\frac{\delta\tilde{g}_{lk}}{\delta
x^{j}}(x,y)-\frac{\delta\tilde{g}_{jk}}{\delta
x^{l}}(x,y)+\frac{\delta\tilde{g}_{lj}}{\delta x^{k}}(x,y)\right),$ (9)

for all $(x,y)\in A$. The Finsler Ricci scalar is then the contraction of the
Riemann curvature $R(x,y):=R^{i}_{i}(x,y)$, $(x,y)\in A$. It has been observed
in [29] that Rutz’s equation is not variational but can be completed, in a
suitable sense, to a variational equation on $A\setminus N$ (which coincides
with the field equation in [44] on the set $\\{(x,y)\in A:L(x,y)=-1\\}$):

$\frac{3R}{L}-\frac{1}{2}\tilde{g}^{ij}\frac{\partial^{2}R}{\partial
y^{i}\partial y^{j}}-\tilde{g}^{ij}\left(\frac{\delta P_{i}}{\delta
x^{j}}-P_{h}\Gamma^{h}_{ij}-P_{i}P_{j}+\frac{\partial}{\partial
y^{j}}\Big{(}y^{k}\big{(}\frac{\delta P_{i}}{\delta
x^{k}}-P_{h}\Gamma^{h}_{ik}\big{)}\Big{)}\right)=0$ (10)

where

$P^{i}_{jk}:=\frac{\partial^{2}G^{i}}{\partial y^{j}\partial
y^{k}}-\Gamma^{i}_{jk}$

are the components of the Landsberg tensor and $P_{i}=P^{l}_{li}$. We stress
that both equations $R=0$ and (10) are equivalent to Einstein vacuum equation
$\mathrm{Ric}(h)=0$ if $L$ comes from a Lorentzian metric $h$,
$L(x,y)=h_{x}(y,y)$ (see, respectively, [42], §3 and [29], §VII).

##  5\. On the Analyticity of the Average Metric of a Static Berwald Solution

We consider now a static Finsler spacetime $\tilde{M}=\mathbb{R}\times M$ with
$L$ of the type (4), but $F_{\omega}$ will be any Finsler metric $F$ on $M$,
not necessarily the one in (5). Let us assume also that $F$ is a Berwald
metric. This means that the components of the Chern connection of $F$ (defined
as in (9) with the fundamental tensor $g$ of $F$ replacing $\tilde{g}$) do not
depend on
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(x,y)}\in
TM$ or equivalently the components
$N^{i}_{j}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(x,y)}$
are linear in $y$ (precisely, it holds
$N^{i}_{j}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(x,y)}=\Gamma^{i}_{jk}(x)y^{k}$,
see [45], prop. 10.2.1). From (8), the components of the Riemannian curvature
tensor $R^{i}_{jkl}$ of $F$ are independent of $y$ too and the Finsler Ricci
scalar is equal to
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R(x,y)}=R^{i}_{jil}(x)y^{j}y^{l}$
for any $(x,y)\in TM\setminus 0$.

Let us use the index $0$ for the components corresponding to the coordinate
$t\in\mathbb{R}$ and by
$\alpha,\beta,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}$
the ones corresponding to coordinate systems in $M$; moreover let us
distinguish Finslerian quantities of $(\tilde{M},L)$ from the ones of $(M,F)$
by a tilde. It can be soon realized that $L$ is Berwald as well; indeed as
$\tilde{g}^{00}=-1$ and $\tilde{g}^{0\alpha}=0$, from (6), taking also into
account that $L$ does not depend on $t$ ($:=x^{0}$) and $\frac{\partial
L}{\partial x^{\alpha}}=\frac{\partial F^{2}}{\partial x^{\alpha}}$, we get
$\tilde{G}^{0}=0$ and $\tilde{G}^{\alpha}((t,x),(\tau,y))=G^{\alpha}(x,y)$.
Thus, $\tilde{N}^{0}_{i}=0$, $\tilde{N}^{\alpha}_{0}=0$ and
$\tilde{N}^{\alpha}_{\beta}((t,x),(\tau,y))=N^{\alpha}_{\beta}(x,y)$, i.e.,
they are all linear in $(\tau,y)$.

Since $L$ is Berwald, its non-vanishing spray coefficients

$\tilde{G}^{\alpha}\big{(}(t,x),(\tau,y)\big{)}=\frac{1}{2}\left(\tilde{N}^{\alpha}_{0}\big{(}(t,x),(\tau,y)\big{)}\tau+\tilde{N}^{\alpha}_{\beta}\big{(}(t,x),(\tau,y)\big{)}y^{\beta}\right)=\\\
\frac{1}{2}N^{\alpha}_{\beta}(x,y)\big{)}y^{\beta}=G^{\alpha}(x,y)y^{\beta}$

are quadratic in $y$ and then, as in [43], Prop. 7.2.2, we get
$\frac{\partial^{2}\tilde{G}^{\alpha}}{\partial y^{i}\beta\partial
y^{k}}=\tilde{\Gamma}^{\alpha}_{ik}$. Since
$\tilde{g}^{00}=\tilde{g}_{00}=-1$,
$\tilde{g}^{0,\alpha}=\tilde{g}_{0\alpha}=0$ and
$\frac{\delta\tilde{g}_{jk}}{\delta x^{0}}=0$, for all $j,k\in\\{0,\ldots
n\\}$, we also have that $\tilde{\Gamma}^{0}_{jk}=0$, for all
$j,k\in\\{0,\ldots n\\}$. Thus, the Landsberg tensor $\tilde{P}^{i}_{jk}$
vanishes. Hence, for Berwald $L$, (10) reduces to

$\frac{3\tilde{R}}{L}-\frac{1}{2}\tilde{g}^{ij}\frac{\partial^{2}\tilde{R}}{\partial
y^{i}\partial y^{j}}=0.$

Taking into account that $\frac{\partial^{2}\tilde{G}^{\alpha}}{\partial
y^{0}\partial y^{k}}=\frac{\partial^{2}G^{\alpha}}{\partial y^{0}\partial
y^{k}}=0$, for all $k\in\\{0,\ldots,n\\}$, this also implies that
$\tilde{\Gamma}^{\alpha}_{0k}=0$ and
$\tilde{\Gamma}^{\alpha}_{\beta\gamma}(t,x)=\Gamma^{\alpha}_{\beta\gamma}(x)$,
that could be also proved directly by (9). Thus, $\tilde{R}^{0}_{j0l}=0$,
$\tilde{R}^{\alpha}_{0\alpha l}=0=\tilde{R}^{\alpha}_{j\alpha 0}$ and
$\tilde{R}^{\alpha}_{\beta\alpha\gamma}\big{(}(t,x),(\tau,y)\big{)}=\tilde{R}^{\alpha}_{\beta\alpha\gamma}\big{(}t,x)=R^{\alpha}_{\beta\alpha\gamma}(x)$,
which imply that the Finsler Ricci scalar of $F$ and $L$ coincide. Thus, if
$F$, or equivalently $L$, has vanishing Ricci scalar,
$0=R(x,y)=\tilde{R}\big{(}(t,x),(\tau,y))$ for all
$\big{(}(t,x),(\tau,y)\big{)}\in T\tilde{M}\setminus 0$ then $L$ satisfies
Equation (10) on $T\tilde{M}\setminus(\mathcal{T}\cup N)$.222The converse is
true only in some cases, for example for some Bogoslowski-Berwald metric, see
[46].

It is well-known that the components (9) of the Chern connection of a Berwald
metric can be obtained from different Riemannian metrics as their Christoffel
symbols, [47]. In particular, see [48], this is true for the Riemannian metric

$h_{x}(V_{1},V_{2}):=\frac{\int_{S_{x}}g(x,y)[V_{1},V_{2}]d\lambda(y)}{\int_{S_{x}}d\lambda(y)},$
(11)

where $S_{x}:=\\{y\in T_{x}M:F(y)=1\\}$, $x\in M$ and $d\lambda(y)$ is the
measure induced on $S_{x}$ by the Lebesgue measure on $\mathbb{R}^{n}$.

Let us assume that $F$ is of class $C^{4}$ on $TM\setminus 0$; then $g(x,y)$
is of class $C^{2}$ on $TM\setminus 0$. Notice that the indicatrix bundle
$\\{(x,y)\in TM:F(x,y)=1\\}$ is a $C^{4}$ embedded hypersurface in $TM$. Thus,
both the area of $S_{x}$ and the numerator in (11) are $C^{2}$ in $x$ and then
$h$ is a $C^{2}$ Riemannian metric on $M$. From (8) and the fact that $F$ is
Berwald, the components $R^{i}_{jkl}$ are equal to the ones of the Riemannian
curvature tensor of $h$ and then we have

$\mathrm{Ric}(h)_{\alpha\beta}(x)=R^{m}_{\alpha
m\beta}(x)=\frac{1}{2}\frac{\partial^{2}}{\partial y^{\alpha}\partial
y^{\beta}}\left(R^{m}_{jml}(x)y^{j}y^{l}\right)=\frac{1}{2}\frac{\partial^{2}R}{\partial
y^{\alpha}\partial y^{\beta}}(x,y)=0.$

From [49], Theorem 4.5, it follows that in an atlas of $M$ of harmonic
coordinates of $h$, $h$ itself is analytic. We can summarize the above
reasoning in the following result:

######  Theorem 1.

Let $(\tilde{M},L)$ be a standard static Finsler spacetime,
$\tilde{M}=\mathbb{R}\times M$,
$L\big{(}(t,x),(\tau,y)\big{)}=-t^{2}+F^{2}(x,y)$. Assume that $F\in
C^{4}(TM\setminus 0)$ is Berwald with vanishing Ricci Finsler scalar $R$, then
$L$ is Berwald, satisfies the field equation (10) with and the metric
$\frac{\int_{S_{x}}g(x,y)[\cdot,\cdot]d\lambda(y)}{\int_{S_{x}}d\lambda(y)}$
on $M$, where $g$ is the fundamental tensor of the Finsler metric $F$, is
analytic in its harmonic coordinates.

Let us now consider the case when $F$ is equal to $F_{\omega}$ in (5); in
light of Theorem 1 we would like to have conditions ensuring that $F_{\omega}$
is Berwald. It is well-known that for a Randers metric $F=\alpha+\beta$, where
$\alpha$ is the norm of a Riemannian metric and $\beta$ a one-form, this holds
if and only if
$\nabla{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta}=0$,
where $\nabla$ is the Levi–Civita connection of the Riemannian metric (see
[45, Th. 11.5.1]). Let us see that a sufficient condition of this type holds
for $F_{\omega}$ as well. Let us denote by $\beta$ the one-form on $M$ defined
as $\beta:=\omega/\Lambda$ and by $G$ the Finsler metric given by
$G:=\big{(}F^{2}/\Lambda+\beta^{2}\big{)}^{1/2}$.

######  Proposition 1.

Assume that the Finsler metric $F/\sqrt{\Lambda}$ on $M$ is Berwald with
vanishing Ricci scalar and that $D\beta=0$, where $D$ is the linear covariant
derivative on $M$ induced by the Chern connection of $F/\sqrt{\Lambda}$. Then
the Finsler metric $G+\beta$ is Berwald with vanishing Ricci scalar as well.

###### Proof.

Let us show firstly that $G$ is Berwald. In order to evaluate the spray
coefficients of $G$, we compute the geodesics equation of the Finsler manifold
$(M,G)$ as the Euler–Lagrange equation of the energy functional $E_{G}$ of
$G$. Without loosing generality, we can assume for this purpose that
$\gamma\colon[a,b]\to M$, $\gamma=\gamma(s)$, is a smooth regular curve (i.e.,
$\dot{\gamma}(s)\neq 0$, for all $s\in[a,b]$) and that
$\sigma\colon[-\epsilon,\epsilon]\times[a,b]\to M$, $\sigma=\sigma(r,s)$, is a
smooth variation of $\gamma$ (i.e., $\sigma(0,\cdot)=\gamma$) such that for
all $r\in[-\epsilon,\epsilon]$, $\sigma(r,a)=\gamma(a)$,
$\sigma(r,b)=\gamma(b)$ and $\sigma(r,\cdot)$ is regular as well. Let us
denote respectively by $T$ and $U$ the vector field along $\sigma$ defined by
$\partial_{t}\sigma$ and $\partial_{r}\sigma$. Now, in order to compute the
variation of $E_{G}$ associated to $\sigma$, we can consider separately the
terms coming from the variation of the energy functional of $F/\sqrt{\Lambda}$
and the ones coming from the variation of
$\frac{1}{2}\int_{a}^{b}\beta^{2}(\dot{\gamma})ds$. The variation of the
latter functional is equal to

$\frac{1}{2}\int_{a}^{b}\partial_{r}\Big{(}\beta^{2}(T)\Big{)}ds=\int_{a}^{b}\beta(T)\Big{(}\big{(}D_{U}\beta\big{)}(T)+\beta(D_{U}T)\Big{)}ds=\int_{a}^{b}\beta(T)\beta(D_{T}U)ds$
(12)

where we have used the fact that the connection $D$ is torsion free (see [45,
p. 262]) and hence $D_{U}T=D_{T}U$. Evaluating (12) at $r=0$ gives

$\int_{a}^{b}\beta(\dot{\gamma})\beta\big{(}D_{\dot{\gamma}}V\big{)}ds=\int_{a}^{b}\beta(\dot{\gamma})\frac{d}{ds}\beta(V)ds=\\\
-\int_{a}^{b}\beta(V)\frac{d}{ds}\beta(\dot{\gamma})ds=-\int_{a}^{b}g(\gamma,\dot{\gamma})[B,V]\frac{d}{ds}\beta(\dot{\gamma})ds,$

where $V$ is the variational vector field associated to $\sigma$, i.e.,
$V:=U\big{(}\sigma(0,\cdot)\big{)}=\partial_{r}\sigma(r,\cdot)|_{r=0}$ and $B$
is the vector field along $\gamma$ representing $\beta$ with respect to the
Riemannian metric, over $\gamma$, $g(\gamma,\dot{\gamma})[\cdot,\cdot]$, $g$
being the fundamental tensor of $F/\sqrt{\Lambda}$. As the variation of the
energy functional of $F/\sqrt{\Lambda}$ at $r=0$ gives
$\int_{a}^{b}g(\gamma,\dot{\gamma})[\dot{\gamma},D_{\dot{\gamma}}V]ds$, we get
that a a smooth critical point $\gamma$ of $E_{G}$ satisfies the equation

$D_{\dot{\gamma}}\dot{\gamma}+B\frac{d}{ds}\beta(\dot{\gamma})=0$ (13)

Hence

$0=\beta(D_{\dot{\gamma}}\dot{\gamma})+\beta(B)\frac{d}{ds}\beta(\dot{\gamma})=\frac{d}{ds}\beta(\dot{\gamma})+\beta(B)\frac{d}{ds}\beta(\dot{\gamma})=\frac{d}{ds}\beta(\dot{\gamma})\big{(}1+\beta(B)\big{)}$

As $\beta(B)=g(\gamma,\dot{\gamma})[B,B]\geq 0$, we get
$\frac{d}{ds}\beta(\dot{\gamma})=0$ and hence $\gamma$ satisfy the equation
$D_{\dot{\gamma}}\dot{\gamma}=0$. This implies that the spray coefficients of
$G$ are quadratic in the velocities and then $G$ is Berwald. Let us now prove
that also $G+\beta$ is Berwald. To this end, let us compute the variation of
the length functional $\ell$ of $G+\beta$. As above, let us consider a smooth
regular curve $\gamma$. Since $\ell$ is invariant under orienting preserving
reparametrization, we can assume that $\gamma$ is parametrized w.r.t the arch
length of $G$, i.e., $G(\gamma,\dot{\gamma})=1$. Let $l$ be the length of
$\gamma$ w.r.t. $G$. Arguing as above, the first variation of $\ell$ at $r=0$
is equal to

$\int_{0}^{l}\beta(D_{\gamma}V)ds+\int_{0}^{l}\Big{(}g(\gamma,\dot{\gamma})[\dot{\gamma},D_{\dot{\gamma}}V]-g(\gamma,\dot{\gamma})[B,V]\frac{d}{ds}\beta(\dot{\gamma})\Big{)}ds.$

The first integral above is equal to $\int_{0}^{l}\frac{d}{ds}\beta(V)ds$ and
hence it vanishes for all variational vector fields $V$. Therefore, the
critical points of $\ell$ parametrized w.r.t. to the arc length of $G$
satisfies (13) and then, as above, they do satisfy equation
$D_{\dot{\gamma}}\dot{\gamma}=0$. Since $\frac{d}{ds}\beta(\dot{\gamma})=0$,
$\beta(\dot{\gamma})=\mathrm{const.}$, i.e., $\gamma$ is also affinely
parametrized for the metric $G+\beta$, hence it is a geodesic of this metric.
This implies that $D_{\dot{\gamma}}\dot{\gamma}=0$ is also the geodesics
equation of $G+\beta$ and the spray coefficients of this metric are equal to
the ones of $F/\sqrt{\Lambda}$. As a consequence, $G+\beta$ is Berwald and its
Finsler Ricci scalar vanishes because it is equal to the Finsler Ricci scalar
of $F/\sqrt{\Lambda}$ (recall (7)). ∎

######  Remark 4.

A similar proof shows that $D\beta=0$ is also a sufficient condition for a
Randers variation of a Finsler metric $F$ of Berwald type (i.e., for a Finsler
metric of the type $F+\beta$ with $F(x,y)+\beta_{x}(y)>0$ for all $(x,y)\in
TM\setminus 0$) to be Berwald as well. This extends beyond the case that $F$
is Riemannian the sufficient condition in [45], Th. 11.5.1.

##  6\. Conclusions

We have reviewed the mathematical definitions of a Finsler spacetime and of a
static timelike Killing vector field on it, based on a fundamental function
$L$ with low regularity assumptions. In particular, we have relaxed the
requirement in [29] about smoothness of $L$ on the open cone sub-bundle
defining admissible timelike future-oriented vectors, in order to include
static and stationary Finsler spacetimes that split as a product
$\mathbb{R}\times M$. We have then considered Berwald static Finsler spacetime
under the point of view of the analyticity of the solutions of the Rutz’s
equation $\tilde{R}(x,y)=0$ (and then satisfying also the field equation (10)
proposed in [29]). We have obtained a partial result in this direction stating
analyticity (in its harmonic coordinates) of any Riemannian metric whose
Levi–Civita connection coincide with the Chern connection of the Finsler
metric on the base $M$. In particular, this holds for the metric (11) obtained
as an average of the fundamental tensor of the Finsler metric on the base $M$.

The existence of analytic solutions (in a fixed coordinate system) of the
Rutz’s equation has been recently obtained for Berwald Finsler pp-waves in
[50] introduced there (see also [51], §4). The Berwald static case that we
have considered is, on the other hand, dynamical equivalent to the classical
Lorentzian static case, at least when the dynamic is governed by the Rutz’s
equation. Nevertheless, extending Theorem 1 to more general classes of Finsler
function $F$ seems difficult due to the lack of ellipticity and quasi-
diagonality of the system of equations
$R_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha\beta}}=R^{l}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\alpha}l{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta}}=0$
(that could be considered instead of the scalar equation $R=0$, see [42], §3),
even writing it in harmonic coordinates w.r.t. the horizontal Laplacian of
$F$, see [52], Remark 5. From this point of view, it might be interesting to
analyse a generalization of the Einstein field equations on the whole tangent
bundle of the spacetime, obtained recently [53], based on Sasaki type metrics
and nonlinear connections on it.

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