# Particle motion under the conservative piece of the self-force is
Hamiltonian

Francisco M. Blanco  Department of Physics, Cornell University, Ithaca, NY
14853, USA.  Éanna É. Flanagan  Department of Physics, Cornell University,
Ithaca, NY 14853, USA.

###### Abstract

We consider the motion of a point particle in a stationary spacetime under the
influence of a scalar, electromagnetic or gravitational self-force. We show
that the conservative piece of the first-order self-force gives rise to
Hamiltonian dynamics, and we derive an explicit expression for the Hamiltonian
on phase space. Specialized to the Kerr spacetime, our result generalizes the
Hamiltonian function previously obtained by Fujita et. al., which is valid
only for non-resonant orbits. We discuss implications for the first law of
binary black hole mechanics.

Introduction: The two body problem in general relativity has been the focus of
intense observational and theoretical interest in recent years. On the
observational side, LIGO and VIRGO have detected several dozen coalescences of
binary systems containing black holes and neutron stars [1, 2, 3] via the
gravitational waves that they emit. The near future should bring many more
detections from upgraded instruments, from the next generation ground based
detectors Cosmic Explorer [4] and Einstein Telescope [5], from the space based
detector LISA [6], and potentially from pulsar timing arrays [7]. On the
theoretical side a wide variety of approaches valid in different regimes have
been used to understand the dynamics of black hole binaries with ever greater
precision: numerical relativity [8], the post-Newtonian approximation [9, 10,
11], the post-Minkowskian approximation [12] for which amplitude methods from
quantum field theory have been fruitfully brought to bear [13], the small mass
ratio approximation [14, 15], and the effective one-body framework which
synthesizes information from the other approaches [16, 17].

An issue that arises in this field is whether one can define dissipative and
conservative sectors of the dynamics for which the conservative sector admits
a Hamiltonian description. While this is not possible in the fully nonlinear,
dynamical regime, it has been achieved in the post-Newtonian and post-
Minkowskian approximations to various orders, and it is a foundational
assumption of the effective one body framework. Its status within the small
mass ratio regime, however, has been an open question beyond the leading order
of geodesic motion. In that regime the small body is treated as a point
particle, and the leading order self-force acting on that body is computed by
taking a gradient of a suitably regularized version of the body’s self field
[14, 15], computed as a perturbation of the large black hole spacetime. That
force can be split into time-even conservative and time-odd dissipative
pieces. Hamiltonian descriptions of the conservative motion have been derived
in special cases (orbits in the Schwarzschild spacetime [18] and non-resonant
orbits in Kerr [19]). General orbits in Kerr however have been an open
question.

In this Letter we show that the leading order self-forced motion of a
nonspinning body in any stationary spacetime admits a Hamiltonian description,
and derive an explicit expression for the Hamiltonian. We then discuss a
number of applications in the context of black holes: implications for our
understanding of the integrability of the motion, a clarification of the
limited domain of validity of the first law of binary black hole mechanics
[20], and the identification of a new class of gauge invariant observables
that may be useful for comparing different computational methods.

General result in Hamiltonian dynamics: We start by deriving a general result
in the theory of Hamiltonian systems. We define a pseudo-Hamiltonian dynamical
system to consist of a phase space $\Gamma$, a closed, non-degenerate two form
$\Omega_{AB}$ and a smooth pseudo-Hamiltonian function ${\cal
H}:\Gamma\times\Gamma\to{\bf R}$, for which the dynamics are given by integral
curves of the vector field

$v^{A}=\Omega^{AB}\frac{\partial}{\partial Q^{B}}\left.{\cal
H}(Q,Q^{\prime})\right|_{Q^{\prime}=Q},$ (1)

where $\Omega^{AB}\Omega_{BC}=\delta^{A}_{C}$ and $Q^{A}$ are coordinates on
$\Gamma$. Pseudo-Hamiltonian systems need not be Hamiltonian, and can be used
to describe dissipation [21].

We now specialize to a pseudo-Hamiltonian system which is a perturbation of a
Hamiltonian system, with symplectic form and pseudo-Hamiltonian

$\displaystyle\Omega_{AB}$ $\displaystyle=$ $\displaystyle\Omega_{0\,AB},$
(2a) $\displaystyle{\cal H}(Q,Q^{\prime})$ $\displaystyle=$ $\displaystyle
H_{0}(Q)+\varepsilon{\cal H}_{1}(Q,Q^{\prime})+O(\varepsilon^{2}).$ (2b)

Here $\varepsilon$ is a formal expansion parameter. We denote by
$Q\to\varphi_{\tau}(Q)$ the zeroth order Hamiltonian flow, defined by the
condition

$\left.{d\over
d\tau}\right|_{\tau=0}\varphi^{A}_{\tau}(Q)=\Omega_{0}^{AB}\partial_{B}H_{0},$
(3)

which satisfies the group composition law

$\varphi_{\tau}[\varphi_{\tau^{\prime}}(Q)]=\varphi_{\tau+\tau^{\prime}}(Q).$
(4)

The pseudo-Hamiltonian perturbation ${\cal H}_{1}$ is defined in terms of a
function $G:\Gamma\times\Gamma\to{\bf R}$ via

${\cal
H}_{1}(Q,Q^{\prime})=\int_{-\infty}^{\infty}d\tau^{\prime}{\tilde{G}}(0,Q,\tau^{\prime},Q^{\prime}),$
(5)

where we have defined

${\tilde{G}}(\tau,Q,\tau^{\prime},Q^{\prime})=G\left[\varphi_{\tau}(Q),\varphi_{\tau^{\prime}}(Q^{\prime})\right].$
(6)

The function $G$ is assumed to satisfy the conditions

$\displaystyle G(Q,Q^{\prime})$ $\displaystyle=$ $\displaystyle
G(Q^{\prime},Q),$ (7a)
$\displaystyle{\tilde{G}}(\tau,Q,\tau^{\prime},Q^{\prime})$ $\displaystyle\to$
$\displaystyle 0\ \ {\rm as}\ \tau\ {\rm or}\ \tau^{\prime}\to\pm\infty.$ (7b)

We now show that with these assumptions, the pseudo-Hamiltonian system (2) is
Hamiltonian to linear order in $\varepsilon$. To do so we need to find a
perturbed Hamiltonian
${\tilde{H}}=H_{0}+\varepsilon{\tilde{H}}_{1}+O(\varepsilon^{2})$, and a
perturbed symplectic form
${\tilde{\Omega}}_{AB}=\Omega_{0\,AB}+\varepsilon{\tilde{\Omega}}_{1\,AB}+O(\varepsilon^{2})$,
for which the equation of motion
$dQ^{A}/d\tau={\tilde{\Omega}}^{AB}\partial_{B}{\tilde{H}}$ coincides with
that given by Eqs. (1) and (2) to $O(\varepsilon)$. This yields the
requirement

$\partial_{B}{\tilde{H}}_{1}-{\tilde{\Omega}}_{1\,BC}\Omega_{0}^{CD}\partial_{D}H_{0}=\frac{\partial}{\partial
Q^{B}}\left.{\cal H}_{1}(Q,Q^{\prime})\right|_{Q^{\prime}=Q}.$ (8)

We choose the perturbation to the symplectic form to be111We originally
arrived at this obscure formula by applying the prescription of Llosa and
Vives [22] for obtaining Hamiltonians from non-local in time Lagrangians to
the non-local in time action principle for the conservative self-force of
Refs. [23, 24].

${\tilde{\Omega}}_{1\,BC}=\left[{\partial\over\partial
Q^{B}}{\partial\over\partial Q^{C^{\prime}}}\int d\tau\int
d\tau^{\prime}\chi(\tau,\tau^{\prime}){\tilde{G}}(\tau,Q,\tau^{\prime},Q^{\prime})\right]_{Q^{\prime}=Q}$
(9)

where

$\chi(\tau,\tau^{\prime})={1\over 2}{\rm sgn}(\tau)-{1\over 2}{\rm
sgn}(\tau^{\prime}).$ (10)

Because of the antisymmetry property
$\chi(\tau^{\prime},\tau)=-\chi(\tau,\tau^{\prime})$ and the symmetry property
(7a) of $G$, the expression (9) defines a closed two form on phase space.
Using the symplectic form perturbation (9) and the pseudo-Hamiltonian
perturbation (5) we find that the requirement (8) reduces to

$\displaystyle\partial$ ${}_{B}{\tilde{H}}_{1}=\left[{\partial\over\partial
Q^{B}}\int
d\tau^{\prime}{\tilde{G}}(0,Q,\tau^{\prime},Q^{\prime})\right]_{Q^{\prime}=Q}$
(11)
$\displaystyle+\Omega_{0}^{CD}\partial_{D}H_{0}\left[{\partial\over\partial
Q^{B}}{\partial\over\partial Q^{C^{\prime}}}\int d\tau\int
d\tau^{\prime}\chi\,{\tilde{G}}(\tau,Q,\tau^{\prime},Q^{\prime})\right]_{Q^{\prime}=Q}.$

We now proceed to simplify the second term in Eq. (11), in several stages.
First, we bring the factor $\Omega_{0}^{CD}\partial_{D}H_{0}$ inside the
square brackets and replace it with the tensor
$\Omega_{0}^{C^{\prime}D^{\prime}}\partial_{D}^{\prime}H_{0}$ at $Q^{\prime}$.
This replacement is valid because of the subsequent evaluation at
$Q^{\prime}=Q$. Second, we can replace the differential operator
$\Omega_{0}^{C^{\prime}D^{\prime}}\partial_{D}^{\prime}H_{0}\partial_{C^{\prime}}$
using the zeroth order Hamiltonian flow (3). The second term becomes

$\displaystyle\left\\{\frac{\partial}{\partial Q^{B}}\right.$
$\displaystyle\left.\frac{d}{d\Delta\tau^{\prime}}\right|_{\Delta\tau^{\prime}=0}\int
d\tau\left.\int
d\tau^{\prime}\chi\,\tilde{G}[\tau,Q,\tau^{\prime},\varphi_{\Delta\tau^{\prime}}(Q^{\prime})]\right\\}_{Q^{\prime}=Q}.$
(12)

Third, using the definition (6) of ${\tilde{G}}$ together with the group
property (4) of the Hamiltonian flow we have

${\tilde{G}}[\tau,Q,\tau^{\prime},\varphi_{\Delta\tau^{\prime}}(Q^{\prime})]={\tilde{G}}(\tau,Q,\tau^{\prime}+\Delta\tau^{\prime},Q^{\prime}).$
(13)

Hence the term (12) can be rewritten as

$\left[{\partial\over\partial Q^{B}}\int d\tau\int d\tau^{\prime}\chi\,{d\over
d\tau^{\prime}}{\tilde{G}}(\tau,Q,\tau^{\prime},Q^{\prime})\right]_{Q^{\prime}=Q}.$
(14)

Fourth, we integrate by parts with respect to $\tau^{\prime}$ and make use of
the condition (7b) to eliminate the boundary terms. The derivative of the
expression (10) for the function $\chi$ gives a delta function,
$d\chi/d\tau^{\prime}=-\delta(\tau^{\prime})$. The final result is

$\left[{\partial\over\partial Q^{B}}\int
d\tau{\tilde{G}}(\tau,Q,0,Q^{\prime})\right]_{Q^{\prime}=Q}.$ (15)

Using the definition (6), the symmetry property (7a) and relabeling
$\tau\to\tau^{\prime}$ this can be written as

$\left[{\partial\over\partial Q^{B^{\prime}}}\int
d\tau^{\prime}{\tilde{G}}(0,Q,\tau^{\prime},Q^{\prime})\right]_{Q^{\prime}=Q}.$
(16)

Finally inserting this expression as a replacement for the second term in the
condition (11), we see that the right hand side is now a total derivative, as
desired, and the resulting expression for the perturbation to the Hamiltonian
is

${\tilde{H}}_{1}(Q)=\int d\tau^{\prime}{\tilde{G}}(0,Q,\tau^{\prime},Q).$ (17)

This completes the proof that the system (2) is Hamiltonian.

We can obtain a more convenient representation of this Hamiltonian system by
making a linearized phase space diffeomorphism parameterized by the vector
field $\varepsilon\xi^{A}$, under which we have

$\displaystyle{\tilde{H}}_{1}$ $\displaystyle\to$ $\displaystyle
H_{1}={\tilde{H}}_{1}+\mathcal{L}_{\xi}H_{0},$ (18a)
$\displaystyle{\tilde{\Omega}}_{1\,AB}$ $\displaystyle\to$
$\displaystyle\Omega_{1\,AB}={\tilde{\Omega}}_{1\,AB}+(\mathcal{L}_{\xi}\Omega_{0})_{AB}.$
(18b)

If we choose $\xi^{A}=\Omega_{0}^{AB}\eta_{B}$ then we find
$\Omega_{1\,AB}={\tilde{\Omega}}_{1\,AB}-\partial_{A}\eta_{B}+\partial_{B}\eta_{A}$.
We now choose

$\eta_{A}={1\over 2}\left[{\partial\over\partial Q^{A^{\prime}}}\int d\tau\int
d\tau^{\prime}\chi\,{\tilde{G}}(\tau,Q,\tau^{\prime},Q^{\prime})\right]_{Q^{\prime}=Q},$
(19)

which yields from Eq. (9) that $\Omega_{1\,AB}=0$. Hence the new symplectic
form coincides with the unperturbed symplectic form:

$\Omega_{AB}=\Omega_{0\,AB}+O(\varepsilon^{2}).$ (20)

Similarly by inserting Eq. (19) into Eq. (18a) and simplifying using the same
techniques as for Eq. (11) yields

$H_{1}(Q)={1\over 2}\int d\tau^{\prime}{\tilde{G}}(0,Q,\tau^{\prime},Q),$ (21)

which differs from the original result (17) by a factor of $2$.

Application to motion under the conservative self-force: We now explain how
the motion of a particle under the action of its conservative first order
gravitational self-force in a stationary spacetime $(M,g_{ab})$ can be cast as
a pseudo-Hamiltonian system of the form (2), by modifying slightly the pseudo-
Hamiltonian construction of Fujita et. al. [19]. This will allow us to apply
our Hamiltonian result (20) and (21).

For the zeroth order geodesic motion we use phase space coordinates
$(x^{\mu},p_{\mu})$ with symplectic form $\Omega_{0}=dp_{\mu}\wedge dx^{\mu}$
and Hamiltonian222This differs from the Hamiltonian of [19] in that it
includes a square root, which is necessary to make $G(Q,Q^{\prime})$ symmetric
in Eq. (23) below.

$H_{0}=-\sqrt{-g^{\mu\nu}(x)p_{\mu}p_{\nu}}.$ (22)

The time parameter $\tau$ associated with this Hamiltonian is then proper time
normalized with respect to $g_{ab}$, while the conserved value of $-H_{0}$ is
the mass of the particle.

For the first order motion, consider a particle at location $x^{\mu^{\prime}}$
with initial 4-momentum $p_{\mu^{\prime}}$. Writing
$Q^{\prime}=(x^{\prime},p^{\prime})$, we denote by333Our index conventions are
unadorned indices for the point $Q=(x,p)$, primed indices for the point
$Q^{\prime}=(x^{\prime},p^{\prime})$, and barred indices for
$\varphi_{\tau^{\prime}}(Q^{\prime})$.
$\varphi_{\tau^{\prime}}(Q^{\prime})=[x^{{\bar{\mu}}}(\tau^{\prime}),p_{{\bar{\mu}}}(\tau^{\prime})]$
the geodesic with initial data $Q^{\prime}$. From this geodesic we can compute
the Lorenz gauge metric perturbation

$h^{\mu\nu}(x;Q^{\prime})=\frac{1}{\sqrt{-g^{\mu^{\prime}\nu^{\prime}}p_{\mu^{\prime}}p_{\nu^{\prime}}}}\int
d\tau^{\prime}G^{\mu\nu\,{\bar{\mu}}{\bar{\nu}}}[x,x^{\prime}(\tau^{\prime})]p_{{\bar{\mu}}}(\tau^{\prime})p_{{\bar{\nu}}}(\tau^{\prime}).$

Here the symmetric Green’s function $G^{\mu\nu\,{\bar{\mu}}{\bar{\nu}}}$ is
the average of the retarded and advanced Green’s functions, regularized
according to the Detweiler-Whiting prescription [25, 14]. The conservative
forced motion of the particle is then equivalent at linear order to geodesic
motion in the metric $g_{\mu\nu}+h_{\mu\nu}$, where $Q^{\prime}$ is held fixed
when evaluating the geodesic equation and then evaluated at $Q^{\prime}=Q$
[25, 15].

We can therefore obtain a pseudo-Hamiltonian description of the dynamics by
replacing the metric $g_{\mu\nu}(x)$ in Eq. (22) with
$g_{\mu\nu}(x)+h_{\mu\nu}(x,Q^{\prime})$. Expanding to linear order in
$h_{\mu\nu}$, comparing with Eqs. (2b), (5) and (6), and setting to unity the
formal expansion parameter $\varepsilon$ we can read off the function
$G(Q,Q^{\prime})$ on phase space to be444Similar constructions work for scalar
and electromagnetic self-forces. For a particle endowed with a scalar charge
$q$ and electromagnetic charge $e$ we replace the initial Hamiltonian
expression (22) with
$-\sqrt{-g^{\mu\nu}(p_{\mu}-eA_{\mu})(p_{\nu}-eA_{\nu})}-q\Phi$. The
expression (23) gets replaced by $-q^{2}G_{\rm sc}(x,x^{\prime})$ in the
scalar case, where $G_{\rm sc}$ is the scalar Green’s function, and with
$-\frac{e^{2}G^{\mu\mu^{\prime}}(x,x^{\prime})p_{\mu}p_{\mu^{\prime}}}{\sqrt{-g^{\lambda\sigma}p_{\lambda}p_{\sigma}}\sqrt{-g^{\lambda^{\prime}\sigma^{\prime}}p_{\lambda^{\prime}}p_{\sigma^{\prime}}}}$
in the electromagnetic case, where $G^{\mu\mu^{\prime}}$ is the Lorenz gauge
electromagnetic Green’s function.

$G(Q,Q^{\prime})=-\frac{G^{\mu\nu\,\mu^{\prime}\nu^{\prime}}(x,x^{\prime})p_{\mu}p_{\nu}p_{\mu^{\prime}}p_{\nu^{\prime}}}{2\sqrt{-g^{\lambda\sigma}p_{\lambda}p_{\sigma}}\sqrt{-g^{\lambda^{\prime}\sigma^{\prime}}p_{\lambda^{\prime}}p_{\sigma^{\prime}}}}.$
(23)

This function satisfies the symmetry property (7a). It will also satisfy the
decay property (7b) if the retarded555The singular Green’s function that is
subtracted off in the Detweiler-Whiting regularization prescription does not
contribute here since it vanishes at timelike separations. Green’s function
falls off at late times at fixed spatial position. This is known to be true
for scalar fields in a class of stationary spacetimes [26], while for black
holes it is a lore of the field that perturbations decay at late times as a
power law [27]. This decay was shown for the Weyl scalars in black hole
spacetimes by Barack [28], and it is also generally believed to be true for
tensor perturbations, although it has not yet been established rigorously; see
Refs. [29, 30] for recent developments.

From this pseudo-Hamiltonian formulation it follows that the motion under the
conservative self-force is described by the Hamiltonian (21), in any
stationary spacetime for which the retarded Green’s function goes to zero at
late times.

Specialization to motion near a black hole: Specialize now to the motion of a
particle orbiting a Kerr black hole. In this context it is useful to derive an
explicit form for the Hamiltonian in action angle variables.

We use the variables
$(q^{\alpha},j_{\alpha})=(q^{t},q^{r},q^{\theta},q^{\phi},j_{t},j_{r},j_{\theta},j_{\phi})$
defined in Refs. [31, 32], deformed via Eq. (19). In these variables the
symplectic form is $\Omega=dj_{\alpha}\wedge dq^{\alpha}$ and the full
Hamiltonian from Eqs. (21) and (22) is

$H=H_{0}(j_{\alpha})+H_{1}(q^{\alpha},j_{\alpha}).$ (24)

The zeroth order geodesic motion is given by
$q^{\alpha}(\tau)=q^{\alpha}_{0}+\Omega^{\alpha}_{0}(j)\tau$, $j_{\alpha}=$
const, where $\Omega^{\alpha}_{0}=\partial H_{0}/\partial j_{\alpha}$ are the
zeroth order frequencies.

We now fix a value $m$ of the conserved quantity $-H$, which is the mass of
the particle to leading order. For describing motion on the mass shell $H=-m$
it will be convenient to define rescaled versions of the symplectic form and
Hamiltonian,

${\hat{\Omega}}_{AB}=\Omega_{AB}/m,\ \ \ {\hat{H}}=H/m.$ (25)

This rescaling preserves Hamilton’s equations. Using the fact that under the
transformation $(x^{\mu},p_{\mu})\to(x^{\mu},sp_{\mu})$ with $s>0$ we have
$(q^{\alpha},j_{\alpha})\to(q^{\alpha},sj_{\alpha})$ [32], $H_{0}\to sH_{0}$
and $H_{1}\to s^{2}H_{1}$ [cf. Eq. (23)], the dynamical system can be written
as

${\hat{\Omega}}=dJ_{\alpha}\wedge dq^{\alpha},\ \ \
{\hat{H}}={\hat{H}}_{0}(J)+m{\hat{H}}_{1}(q,J),$ (26)

where $J_{\alpha}=j_{\alpha}/m$.

Motion on this mass shell can be described in terms a 6 dimensional
Hamiltonian system, which can be derived from the 8 dimensional system (26) as
follows [33]. Because of the symmetries of the Kerr background the Hamiltonian
is independent of $q^{t}$, ${\hat{H}}={\hat{H}}(q^{i},J)$ where
$q^{i}=(q^{r},q^{\theta},q^{\phi})$. Consider paths in the 9-dimensional
extended phase space $(q,J,\tau)$ that join an initial point
$(q_{1},J_{1},\tau_{1})$ to a final point $(q_{2},J_{2},\tau_{2})$. Paths that
extremize the line integral of the Poincaré-Cartan one form
$\int[J_{\alpha}dq^{\alpha}-{\hat{H}}(q^{i},J)d\tau],$ with $\delta
q^{\alpha}=\delta\tau=0$ at the endpoints, satisfy the 8-dimensional Hamilton
equations of motion [33]. We now restrict to paths lying within the surface
${\hat{H}}=-1$. Within this surface we can solve for $J_{t}=-h(q^{i},J_{i})$
in terms of the other parameters from the equation

${\hat{H}}[q^{i},-h(q^{i},J_{i}),J_{i}]=-1,$ (27)

where $J_{i}=(J_{r},J_{\theta},J_{\phi})$. The line integral now reduces to

$\int\Big{[}J_{i}dq^{i}-hdq^{t}\Big{]}+(\tau_{2}-\tau_{1}).$ (28)

The second term is a constant and the first term is an extremum under the
variation of paths $[q^{i}(q^{t}),J_{i}(q^{t})]$ that connect the two
endpoints for which $\delta q^{i}=0$. Hence we obtain a 6-dimensional
Hamiltonian system with Hamiltonian $h(q^{i},J_{i})$, time parameter $q^{t}$
and symplectic form $dJ_{i}\wedge dq^{i}$. By combining Eqs. (25) and (27) it
follows that the Hamiltonian can be expanded as

$h(q^{i},J_{i})=h_{0}(J_{i})+mh_{1}(q^{i},J_{i})+O(m^{2}).$ (29)

where $h_{0}$ and $h_{1}$ are given by ${\hat{H}}_{0}(-h_{0},J_{i})=-1$ and
$h_{1}={\hat{H}}_{1}(q^{i},-h_{0},J_{i})/\Omega^{t}_{0}$. The zeroth order
frequencies are now $\omega^{i}_{0}=\partial h_{0}/\partial
J_{i}=\Omega^{i}_{0}/\Omega^{t}_{0}$.

The Hamiltonian perturbation $h_{1}$ is independent of $q^{\phi}$ due to the
symmetry of the Kerr background, and can be expanded in Fourier modes666It is
possible to obtain an explicit formula for the coefficients $h_{1\,{\bf k}}$
starting from a Fourier expansion of the function (23) in action angle
variables  $\displaystyle G(q,J,q^{\prime},J^{\prime})$ $\displaystyle=$
$\displaystyle\int d\omega\sum_{m,{\bf k},{\bf
k}^{\prime}}e^{-i\omega(q^{t}-q^{t^{\prime}})-im(q^{\phi}-q^{\phi^{\prime}})}e^{i{\bf
k}\cdot{\bf q}+i{\bf k}^{\prime}\cdot{\bf q}^{\prime}}G_{\omega m{\bf k}{\bf
k}^{\prime}}(J,J^{\prime}).$ Combining this with Eqs. (21), (27), (29) and
(31) gives  $h_{1\,{\bf k}}=\frac{\pi}{(\Omega^{t}_{0})^{2}}\sum_{m,{\bf
l}}G_{\omega m({\bf k}/2+{\bf l}/2)({\bf k}/2-{\bf
l}/2)}(J_{t},J_{i},J_{t},J_{i}),$ (30) where we sum over all pairs of integers
${\bf l}=(l_{r},l_{\theta})$ for which $k_{r}+l_{r}$ and
$k_{\theta}+l_{\theta}$ are even, and we evaluate at $J_{t}=-h_{0}(J_{i})$ and
at $\omega=m\omega^{\phi}_{0}+({\bf k}-{\bf
l})\cdot{\mbox{\boldmath$\omega$}}_{0}/2$. on the torus parameterized by ${\bf
q}=(q^{r},q^{\theta})$:

$h_{1}({\bf
q},J_{i})=\sum_{k_{r}=-\infty}^{\infty}\sum_{k_{\theta}=-\infty}^{\infty}e^{i{\bf
k}\cdot{\bf q}}h_{1\,{\bf k}}(J_{i}).$ (31)

Application: Integrability of dynamics: We now turn to discussing some
applications. Since the motion is Hamiltonian one can ask whether it is also
integrable. It will be integrable to linear order if and only if all the
resonant mode amplitudes vanish, that is,

$h_{1\,{\bf k}}(J_{i})=0\ \ \ {\rm whenever}\ \ \ {\bf
k}\cdot\mbox{\boldmath$\omega$}_{0}(J_{i})=0,\ {\bf k}\neq{\bf 0}.$ (32)

This is easy to see, since under a linearized canonical transformation with
generating function $G({\bf q},J_{i})=\sum_{\bf k}\exp[i{\bf k}\cdot{\bf
q}]G_{\bf k}(J_{i})$ we have $h_{1\,{\bf k}}\to h_{1\,{\bf k}}+i({\bf
k}\cdot\mbox{\boldmath$\omega$}_{0})G_{\bf k}.$ Thus choosing $G_{\bf
k}(J_{i})=-ih_{1\,{\bf k}}(J_{i})/{\bf k}\cdot\mbox{\boldmath$\omega$}_{0}$
yields $h_{1\,{\bf k}}=0$ for all nonzero ${\bf k}$ and thus an integrable
system777The resulting Hamiltonian coincides with that found by Ref. [19], who
excluded resonances., and this choice is possible without divergences only
when the condition (32) is satisfied. Conversely, if the system is integrable
there must exist perturbed versions $J_{i}+m\delta J_{i}$ of the action
variables which have vanishing Poisson brackets with the Hamiltonian
$h_{0}+mh_{1}$, which yields at linear order the relation

$k_{i}h_{1\,{\bf k}}=({\bf k}\cdot\mbox{\boldmath$\omega$}_{0})\delta
J_{i\,{\bf k}}$ (33)

between Fourier components, enforcing the condition (32).

An alternative version of the integrability condition (32) is that the average
of the conservative time derivative of the Carter constant $Q(J_{i})$ over any
orbit on any resonant torus should vanish. Computing a time derivative using
Eqs. (29) and (31) gives $dQ/d\tau=\Omega^{t}_{0}(\partial Q/\partial
J_{i})dJ_{i}/dq^{t}=-i\Omega^{t}_{0}(\partial Q/\partial J_{i})\sum_{\bf
k}k_{i}h_{1\,{\bf k}}e^{i{\bf k}\cdot{\bf q}}$. Now using ${\bf q}(\tau)={\bf
q}_{0}+\mbox{\boldmath$\Omega$}_{0}\tau$, writing the resonant vectors as
${\bf k}=N{\bf k}_{0}=N(n,-p,0)$ for integers $N$ and taking an orbit average
gives888We neglect in this calculation the coordinate transformation (19),
because under $J_{i}\to J_{i}+\delta J_{i}$ we have
${\dot{J}}_{i}\to{\dot{J}}_{i}+\omega_{0}^{j}\partial\delta J_{i}/\partial
q^{j}$ and the resonant Fourier components of the correction evaluated on a
resonant torus vanish.

$\left<{dQ\over d\tau}\right>=-i\Omega^{t}_{0}\left(n{\partial Q\over\partial
J_{r}}-p{\partial Q\over\partial
J_{\theta}}\right)\sum_{N=-\infty}^{\infty}N\,h_{1\,N{\bf k}_{0}}\,e^{iNq_{\rm
res}},$ (34)

where $q_{\rm res}={\bf k}_{0}\cdot{\bf q}_{0}=nq^{r}_{0}-pq^{\theta}_{0}$ is
the resonant combination of the phases. The left hand side vanishing for all
$q_{\rm res}$ is equivalent to all the resonant amplitudes $h_{1\,N{\bf
k}_{0}}$ vanishing.

One of us conjectured in Ref. [34] that the linear integrability condition
(32) is satisfied in Kerr, based on the fact that enhanced symmetries present
in the post-Newtonian limit enforce this condition. However, this was a weak
argument, since it is possible for symmetries to be present only near the
boundary of phase space that corresponds to the post-Newtonian limit, and not
in the interior (just as for asymptotic spacetime symmetries). Indeed,
recently Nasipak and Evans have shown numerically that
$\left<dQ/d\tau\right>=0$ fails for conservative scalar self-forces in Kerr on
resonances [35, 36]. The gravitational self-force case is presumably similar,
although this will need to be confirmed numerically (see Ref. [37]).

If the gravitational case is indeed non-integrable, the qualitative
consequences for the conservative dynamics are well understood in general
contexts from the theory of weakly perturbed Hamiltonian systems [33, 38].
They have been explored in the contexts of tidal and other perturbations to
extreme mass ratio inspirals in Refs. [39, 40, 41, 42]. Suppose we focus
attention on one resonant torus $J_{i}=J^{*}_{i}$ and neglect the effect of
other resonances. First, away from this torus the invariant tori $J_{i}=$
constant are deformed [cf. Eq. (33)] but preserved (as predicted by the KAM
theorem [33]). Second, within a shell of width $J_{i}-J^{*}_{i}\sim\sqrt{m}$
the dynamics is altered: In the $m\to 0$ limit the resonant torus is destroyed
and replaced by a number of islands of size $\sim\sqrt{m}$ in phase space
within which the motion is integrable999This can be seen explicitly in the
description of the near-resonance dynamics derived by van de Meent, Eq. (18)
of Ref. [43], dropping the dissipative terms (the first term on the right hand
side and half of the oscillatory terms); the solutions consist of rotational
or librational (islands) motions, depending on the energy. [42]. One can
define action angle variables within each island, but they do not join
continuously onto the global action angle variables. At finite $m$ chaotic
regions develop within the shell. Third, motion that starts within the shell
is confined to remain within it by the surrounding surviving invariant tori,
since the system is effectively two dimensional ($J_{\phi}$ is conserved)
[38]. There are no large excursions to $J_{i}-J^{*}_{i}\sim O(1)$, unlike in
higher dimensions.

When one considers the full $O(m)$ dynamics with the dissipative component of
the self force included, the non-integrable mode coefficients $h_{1\,{\bf k}}$
can drive transient resonances which give $O(\sqrt{m})$ kicks to the action
variables $J_{i}$ [34], and also sustained resonances in which the orbit
evolves along a non-adiabatic path in the space of parameters $J_{i}$
maintaining the condition ${\bf k}\cdot\mbox{\boldmath$\omega$}_{0}(J_{i})=0$
[43]. However neither of these are smoking gun signatures of the breakdown of
integrability, since both can be produced when $h_{1\,{\bf k}}=0$ by the
oscillatory dissipative components of the self force [43].

Non-integrability would also complicate the dynamics away from the resonant
islands in phase space. If one computes the dynamics using the linear
prescription described after Eq. (32) for eliminating the oscillatory terms in
the Hamiltonian (29), ignoring the divergences, the resulting fractional
errors caused by the nearest strong resonance scale as $\sim m^{2}|h_{1{\bf
k}}|^{2}(J-J^{*})^{-4}$. It is possible to achieve smaller errors $\sim
m^{3}|h_{1{\bf k}}|^{3}(J-J^{*})^{-6}$ by using a second order canonical
transformation to eliminate the oscillatory terms in (29) from $h_{1}$ through
$O(m^{2})$, at the price of a more complicated description of the dynamics. In
either case the errors become of order unity in the vicinity of the resonant
islands.

Application: First law of binary black hole mechanics: In the absence of
resonances, our Hamiltonian (24) directly yields a version of the first law,
as in Ref. [19]. We eliminate all $q$ dependent terms in (24) using a
canonical transformation as described after Eq. (32). We regard $H$ as a
function $H=H(j_{\alpha},M_{\rm irr},S_{\rm bh})$ of the action variables
$j_{\alpha}$ and of the irreducible mass $M_{\rm irr}$ and spin $S_{\rm bh}$
of the large black hole. Taking a variation and using $H=-m$ gives

$-\delta m=\Omega^{t}\delta j_{t}+\Omega^{i}\delta j_{i}+\frac{\partial
H}{\partial M_{\rm irr}}\delta M_{\rm irr}+\frac{\partial H}{\partial S_{\rm
bh}}\delta S_{\rm bh},$ (35)

where $\Omega^{\alpha}=\partial H/\partial j_{\alpha}$ are the frequencies
accurate to subleading order in $m$. Identifying $-j_{t}$ as the orbital
energy $E$, dividing by $\Omega^{t}$, and adding the variation of the
background black hole mass $M_{\rm bh}(M_{\rm irr},S_{\rm bh})$ gives

$\delta(M_{\rm bh}+E)=z\delta m+\omega^{i}\delta j_{i}+z_{\rm bh}\delta M_{\rm
irr}+\Omega_{\rm bh}\delta S_{\rm bh}.$ (36)

Here $z=1/\Omega^{t}$ is the redshift invariant,
$\omega^{i}=\Omega^{i}/\Omega^{t}$, $z_{\rm bh}=\partial M_{\rm bh}/\partial
M_{\rm irr}+z\partial H/\partial M_{\rm irr}$ and $\Omega_{\rm bh}=\partial
M_{\rm bh}/\partial S_{\rm bh}+z\partial H/\partial S_{\rm bh}$. Equation (36)
yields a form of the first law for binaries101010Equation (36) is not quite
the conventional form of the first law beyond the leading order in $m$. The
conventional form would require the quantity $M_{\rm bh}+E$ on the left hand
side to coincide with the Bondi mass to $O(m^{2})$, whereas it is known to
coincide only to $O(m)$ [44]. Additionally we have identified the on-shell
value of the Hamiltonian with minus the particle mass $m$, and this relation
can have a correction to subleading order in $m$. Nevertheless, our form is
sufficient to illustrate the difficulties caused by non-integrability, which
are generic for all forms of the first law. to subleading order in $m$.

This derivation of the first law required the integrability assumption (32).
We now explain how the first law would break down if that assumption is
violated as discussed above. The first law requires a labeling of time-
averaged orbits by some smooth set of parameters. However, when (32) is
violated integrable motions near a resonance fall into different types that
are disconnected from one another. Within an island one can define new action
angle variables ${\tilde{q}}^{i},{\tilde{j}}_{i}$, but these cannot join
smoothly onto the deformed action-angle variables outside the islands. Thus,
the best one can hope for is set of distinct first laws, one for each
disconnected component of integrable motion. Also the number of such
components is formally infinite, since the resonances are dense in phase
space. In practice only the few resonances for which the order
$|k_{r}|+|k_{\theta}|$ is not large will be significant: the width of an
island scales as $\sim\sqrt{m}|h_{1{\bf k}}|$ [43] which will go exponentially
to zero as the order increases, assuming the Hamiltonian is a smooth function
on the torus.

Application: Gauge invariant observables: Gauge invariant observables such as
invariant redshifts, frequencies of innermost stable circular orbits, etc.
have proven enormously useful for cross checks between different computational
methods [15]. The simple form (21) of our Hamiltonian may be helpful for
computing such observables, since one expects the complicated phase space
coordinate transformation (19) not to be relevant for gauge invariant
observables.

For generic orbits, non-integrability of the dynamics would impede the
definition of such observables. For example one can no longer label orbits by
their three fundamental frequencies of motion. However new gauge invariant
observables do arise in this context, the resonant amplitudes $h_{1{\bf k}}$
themselves, for which the action-angle variables are defined geometrically at
zeroth order [32] and which at first order are invariant under linearized
phase space coordinate transformations. These observables are not accessible
from within post-Newtonian or post-Minkowski theory, but could be useful for
comparisons between self-force theory and numerical relativity.

Conclusions: We have shown that the conservative dynamics of two body problem
in general relativity in the small mass ratio limit is Hamiltonian to the
first subleading order, when the small body is nonspinning. It would be
interesting to extend this result to include the spin of the small body, and
to second-order conservative self-forces.

Acknowledgments: We thank Adam Pound, Leo Stein, Justin Vines and Neils
Warburton for helpful discussions. This research was supported in part by NSF
grant PHY-2110463.

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$[U,V]=\sum_{i,j=1}^{s}\left\\{a^{i}b^{j}\left[X_{i},X_{j}\right]+\left(a^{i}X_{i}\left(b^{j}\right)X_{j}-b^{j}X_{j}\left(a^{i}\right)X_{i}\right)\right\\}.$

So the value of $[U,V]$ in $v\tau=TM/\tau$ at a point $x\in M$ depends only on
the value of $U$ and $V$ at $x$. Hence, $Q_{\tau}(U,V)=\pi\left([U,V]\right)$
is a skew-symmetric bilinear map $Q_{\tau}:\tau\times\tau\rightarrow\nu\tau$
where $\pi:TM\rightarrow\nu\tau$ is the canonical projection. In other words,
$Q_{\tau}$ is a global section of the vector bundle $\Lambda^{2}(\tau,v\tau)$
of skew-symmetric bilinear forms on the bundle $\tau$. The integrability
condition of $\tau$ is equivalent to $Q_{\tau}$ identically equals to $0$. So
we get a map

$Q:C^{\infty}(\mathcal{G}(M,s))\longrightarrow\mathscr{Z},$

where $\mathscr{Z}$ is a fibre bundle over $\mathcal{G}(M,s)$ whose fibre over
a point $\sigma\in\mathcal{G}(M,s)$ is the infinite-dimensional space
$\Omega^{2}(\sigma,v\sigma)$ of global sections of the bundle
$\Lambda^{2}(\tau,v\tau)$. The space

$\mathcal{FOL}(M,s)$

of codimension $s$ foliations on $M$ is exactly the set $\\{Q=0\\}.$ It will
be equipped with the $C^{\infty}$-topology induced by the topology of the
Fréchet manifold $C^{\infty}(\mathcal{G}(M,s)).$ Let $\mathcal{D}(M)$ be the
diffeomorphism group of $M$. Then $\mathcal{D}(M)$ acts on
$C^{\infty}(\mathcal{G}(M,s))$ and the action preserves
$\mathcal{F}\mathcal{O}\mathcal{L}(M,s)$. We denote by
$\mathcal{O}_{\mathcal{F}}$ the orbit of
$\mathcal{F}\in\mathcal{F}\mathcal{O}\mathcal{L}(M,s)$.

######  Definition 6.21.

A deformation of $\mathcal{F}$ parametrized by an open neighborhood $U$ of 0
in $\mathbb{R}^{d}$ is a continuous map
$\rho:U\rightarrow\mathcal{FOL}(M,s),t\mapsto\mathcal{F}_{t}$ such that
$\rho(0)=\mathcal{F}_{0}=\mathcal{F}$.

######  Definition 6.22.

A deformation $\rho:U\rightarrow\mathcal{FOL}(M,s),t\mapsto\mathcal{F}_{t}$ of
$\mathcal{F}$ parametrized by an open neighborhood $U$ of 0 in
$\mathbb{R}^{d}$ is called with fixed differentiable type if for all $t\in
U,\,\rho(t)\in\mathcal{O}_{\mathcal{F}}$, i.e., there exists a differentiable
family $\left\\{h_{t}\right\\}_{t\in U}$ of diffeomorphisms on $M$ such that
$h_{t}^{*}\mathcal{F}_{t}=\mathcal{F}$.

One has the following nice properties when the initial foliation is
transversely holomorphic.

######  Proposition 6.23 ([EKAG97, Proposition 4.3]).

Let $\mathcal{F}_{t}$ be the deformation of an $r$-complex codimensional
$(s=2r)$ transversely holomorphic foliation $\mathcal{F}$ with fixed
differentiable type.

  1. $(i)$

Suppose that $\mathcal{F}$ is Riemannian. Then for any $t\in U$,
$\mathcal{F}_{t}$ is also Riemannian.

  2. $(ii)$

The family $\left\\{\mathcal{F}_{t}\right\\}_{t\in U}$ is lifted to $M^{\\#}$
in a new family of TP foliations $\left\\{\mathcal{F}_{t}^{\\#}\right\\}_{t\in
U}$ invariant under $SO(s)$ with fixed differentiable type.

  3. $(iii)$

Assume that the foliations $\mathcal{F}_{t}$ are transversely holomorphic and
$\mathcal{F}_{0}=\mathcal{F}$ possesses a transversely Hermitian metric
$\Omega_{0}.$ Then $\left\\{\mathcal{F}_{t}\right\\}_{t\in U}$ are provided
with transversely Hermitian metrics $\Omega_{t}$ varying differentiably
depending on t.

###  6.5. Proof of Theorem 1.7

We split the proof of Theorem 1.7 in four steps.

######  Step $($I$)$.

Integrable Beltrami differentials, Kuranishi family: foliated version.

We first briefly introduce the main results obtained in [EKAN89], which is of
great significance in our proof.

With the same setting as in Theorem 1.7, El Kacimi Alaoui–M. Nicolau proved
the existence of _versal space_ (also called _Kuranishi space_) for
deformations mimicking the construction of the Kuranishi space presented in
[Ku64], and the universal property of the versal space was strengthened in
[Gb92]. As a result, one can get a family, denoted by

$\phi(t)\in A^{0,1}(M/\mathcal{F},\mathcal{N}\mathcal{F}^{1,0})$

for the Kuranishi family, satisfying the equation

$\overline{\partial}\phi(t)=\frac{1}{2}\left[\phi(t),\phi(t)\right]$

on some analytic set and determining the transversely holomorphic structure on
$(M,\mathcal{F}_{t})$. Hence, this construction and $\phi(t)$ will play the
same roles as the Kuranishi’s completeness theorem (introduced in Section 3.2)
and the integrable Beltrami differentials $\varphi(t)$, respectively. Let
$z^{1},\cdots,z^{r}$ and $\zeta^{1},\cdots,\zeta^{r}$ be the transverse
holomorphic coordinates on $\mathcal{F}_{0}$ and $\mathcal{F}_{t}$,
respectively. Then $\phi(t)$ can be defined by

(6.12) $\phi(t):=\left(\left(\frac{\partial\zeta}{\partial
z}\right)^{\\!\\!\\!-1}\left(\frac{\partial\zeta}{\partial\bar{z}}\right)\right)^{i}_{\overline{j}}d\bar{z}^{j}\otimes\frac{\partial}{\partial\bar{z}^{i}}.$

It will be often written as $\phi$ briefly. Also, we can define the extension
map

$e^{\iota_{\phi}|\iota_{\overline{\phi}}}:A^{p,q}(M/\mathcal{F}_{0})\longrightarrow
A^{p,q}(M/\mathcal{F}_{t}),$

analogous to (3.5) which can preserve all $(p,q)$-basic forms.

######  Step $($II$)$.

Transverse positive $(p,p)$-basic forms: deformation openness.

To prove Theorem 1.7, we need the following key observation, which will be the
deformation openness property of the transverse positivity for $(p,p)$-basic
forms. Here we follow the idea as that of [RWZ21, the second proof of Theorem
4.9]. Note that to make the proposition holds, the new extension map
$e^{\iota_{\phi}|\iota_{\overline{\phi}}}$ is a key player. We will use its
two properties in the following proof: one is that it depends smoothly on $t$;
the other one is that it can send each kind of (strictly) positive
$(p,p)$-basic forms on $(M,\mathcal{F}_{0})$ bijectively onto the
corresponding one on $(M,\mathcal{F}_{t})$, which can be showed by an argument
similar to the one in [RWZ21, Proposition 4.11].

######  Proposition 6.24.

Let $\left\\{\mathcal{F}_{t}\right\\}_{t\in U}$ be a smooth family of
transversely Hermitian structures on a compact foliated manifold
$(M,\mathcal{F})$ with fixed differentiable type, and $\Omega_{t}$ a family of
real $(p,p)$-basic forms, depending smoothly on $t$. Assume that $\Omega_{0}$
is a transverse positive $(p,p)$-basic form on $(M,\mathcal{F}_{0})$. Then
$\Omega_{t}$ is also transverse positive on $(M,\mathcal{F}_{t})$ for
sufficiently small $t$.

###### Proof.

Let $\Omega_{0}$ be a transverse positive $(p,p)$-basic form on
$(M,\mathcal{F}_{0})$ and $\Omega_{t}$ its real extension on
$(M,\mathcal{F}_{t})$. It is sufficient to show that: at any given point $x\in
M$, there exists a uniform small constant $\epsilon$ such that for any $t\in
U_{\epsilon}$, where $U_{\epsilon}=\\{t\in\mathbb{R}^{d}\ \big{|}\
|t|<\epsilon\\}$, and any nonzero decomposable $(q,0)$-basic form $\tau$ with
$p+q=r$, we have

(6.13) $\Omega_{t}(x)\wedge\sigma_{q}e^{{\iota_{\phi}}}(\tau)\wedge
e^{\iota_{\bar{\phi}}}(\bar{\tau})>0.$

Indeed, let $\omega_{0}$ be a transversely Hermitian metric on
$(M,\mathcal{F}_{0})$. For any fixed point $x\in M$, we define a continuous
function $f_{x}(t,[\tau])$ on $U_{\epsilon}\times Y_{x}$ by

(6.14)
$f_{x}(t,[\tau]):=\frac{\Omega_{t}(x)\wedge\sigma_{q}e^{\iota_{\phi}}(\tau)\wedge
e^{\iota_{\bar{\phi}}}(\bar{\tau})}{|\tau|_{\omega_{0}(x)}^{2}\cdot\omega_{0}(x)^{r}},$

where
$Y_{x}=\left.\rho(G(q,r))\right|_{x}\subset\mathbb{P}\left(A^{q,0}(M/\mathcal{F}_{0})|_{x}\right)$
is compact. Here we view

$\Omega_{t}\wedge\sigma_{q}e^{\iota_{\phi}}(\tau)\wedge
e^{\iota_{\bar{\phi}}}(\bar{\tau})$

as an $(r,r)$-basic form on $(M,\mathcal{F}_{0})$. By assumption,

$f_{x}(0,[\tau])=\frac{\Omega_{0}(x)\wedge\sigma_{q}\tau\wedge\overline{\tau}}{|\tau|_{\omega_{0}(x)}^{2}\cdot\omega_{0}(x)^{r}}>0.$

Hence, by the continuity of $f_{x}(0,[\tau])$ on $t$ and $[\tau]$ plus the
compactness of $Y_{x}$, there exists a constant $\epsilon_{x}>0$ depending
only on $x$, such that

$f(x,\overline{U}_{\epsilon_{x}/2},Y_{x}):=f_{x}(\overline{U}_{\epsilon_{x}/2},Y_{x})>0.$

Let $\\{T_{j}\\}_{j\in J}$ be trivializing adapted covering of $M$, and choose
any $x\in T_{j}$ for some $j$. So we can identify $Y_{x}$ and $Y_{y}$ for any
point $y\subset T_{j}$, and $f_{y}$ is defined on $Y_{x}$. Similarly, by the
continuity of $f$ on $\overline{U}_{\epsilon_{x}/2}\times Y_{x}$, there exists
an open neighbourhood $R_{x}\subset T_{j}$ of $x$ such that

$f(R_{x},\overline{U}_{\epsilon_{x}/2},Y_{x})>0.$

Since $M$ is compact, one gets a finite open covering $R_{x_{i}}$,
$i=1,\cdots,m$, $M=\mathop{\bigcup}_{i=1}^{m}R_{x_{i}}$. Set

$\varepsilon:=\min_{1\leq i\leq m}\epsilon_{x_{i}}/2>0.$

Then

$f(x,\overline{U}_{\varepsilon},Y_{x})=f_{x}(\overline{U}_{\varepsilon},Y_{x})>0$

for any $x\in M$.

Therefore, by virtue of the definition (6.14), for any $|t|\leq\varepsilon$,
we have

$\Omega_{t}(x)\wedge\sigma_{q}e^{\iota_{\phi}}(\tau)\wedge
e^{\iota_{\bar{\phi}}}(\bar{\tau})=f_{x}(t,[\tau])|\tau|^{2}_{\omega_{0}(x)}\cdot\omega_{0}(x)^{r}>0$

for any nonzero decomposable $(q,0)$-basic form $\tau$ at $x$. This is the
desired inequality (6.13).

∎

With these preparations, to prove Theorem 1.7, it suffices to prove the
special case $p=q$ of the following theorem.

######  Theorem 6.25.

Let $\left\\{\mathcal{F}_{t}\right\\}_{t\in\Delta_{\epsilon}}$ be a smooth
family of transversely Hermitian structures on a compact foliated manifold
$(M,\mathcal{F})$ with fixed differentiable type, parametrized by a small
complex disc. If $\mathcal{F}_{0}$ satisfies the $(p,q+1)$-th and $(q,p+1)$-th
mild $\partial\bar{\partial}$-lemmata, then there is a $d$-closed
$(p,q)$-basic form $\Omega(t)$ on $(M,\mathcal{F}_{t})$ depending smoothly on
$t$ with $\Omega(0)=\Omega_{0}$ for any $d$-closed $\Omega_{0}\in
A^{p,q}(M/\mathcal{F})$.

Next we will sketch the proof of Theorem 6.25 by adopting the power series
idea used in [RWZ19, Section 4]. Here we just indicate the main procedures
emphasizing how we adapt the method to our new setting.

######  Step $($III$)$.

Obstruction equation and construction of power series.

As both $e^{{\iota}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}$ and
$e^{{\iota}_{\phi}}$ are invertible operators when $t$ is sufficiently small
by (6.12), it follows that for any $\Omega\in A^{p,q}(M/\mathcal{F})$,

(6.15) $\displaystyle
e^{{\iota}_{\phi}|{\iota}_{\bar{\phi}}}(\Omega)=e^{{\iota}_{\phi}}\circ
e^{{\iota}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}\circ
e^{-{\iota}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}\circ
e^{-{\iota}_{\phi}}\circ e^{{\iota}_{\phi}|{\iota}_{\bar{\phi}}}(\Omega).$

Here we follow the notations:
$\overline{\phi}\phi=\phi\lrcorner\overline{\phi}$, $\mathds{1}$ is the
identity operator defined as:

$\mathds{1}=\frac{1}{p+q}\left(\sum_{1}^{r}dz^{i}\otimes\frac{\partial\
}{\partial z^{i}}+\sum_{1}^{r}d\bar{z}^{i}\otimes\frac{\partial\
}{\partial\bar{z}^{i}}\right)$

when it acts on $(p,q)$-basic forms, and likewise for others. Set

(6.16)
$\displaystyle\widetilde{\Omega}=e^{-{\iota}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}\circ
e^{-{\iota}_{\phi}}\circ
e^{{\iota}_{\phi}|{\iota}_{\bar{\phi}}}(\Omega)=(\mathds{1}-\bar{\phi}\phi)\Finv\Omega,$

where $\Omega$ and $\widetilde{\Omega}$ are apparently one-to-one
correspondence. Here the notation $\Finv$ denotes the simultaneous
contraction. By observing the local expression for any element in
$A^{0,1}(M/\mathcal{F},\mathcal{N}\mathcal{F}^{1,0})$ (see (6.1)), we have

######  Proposition 6.26.

Let $\phi\in A^{0,1}(M/\mathcal{F},\mathcal{N}\mathcal{F}^{1,0})$ on
$(M,\mathcal{F})$. Then on the space $A^{\bullet,\bullet}(M/\mathcal{F})$,

$e^{-\iota_{\phi}}\circ d\circ
e^{\iota_{\phi}}=d-{\mathcal{L}}_{\phi}^{1,0}+\iota_{\overline{\partial}\phi-\frac{1}{2}[\phi,\phi]},$

where ${\mathcal{L}}_{\phi}^{1,0}:=\iota_{\phi}\partial-\partial\iota_{\phi}$
is the Lie derivative.

Together with (6.15) and (6.16), Proposition 6.26 implies that

$\displaystyle\begin{split}d(e^{{\iota}_{\phi}|{\iota}_{\bar{\phi}}}(\Omega))&=d\circ
e^{{\iota}_{\phi}}\circ
e^{{\iota}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}(\widetilde{\Omega})\\\
&=e^{{\iota}_{\phi}}\circ(\bar{\partial}+[\partial,{\iota}_{\phi}]+\partial)\circ
e^{{\iota}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}(\widetilde{\Omega})\\\
&=e^{{\iota}_{\phi}}(\bar{\partial}_{\phi}+\partial)\sum_{k=0}^{+\infty}A_{k}\\\
&=e^{{\iota}_{\phi}}\left(\bar{\partial}_{\phi}A_{0}+\sum_{k=0}^{+\infty}(\partial
A_{k}+\bar{\partial}_{\phi}A_{k+1})\right),\end{split}$

where

$A_{k}:=\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})$

is a $(p+k,q-k)$-basic form and

$\bar{\partial}_{\phi}:=\bar{\partial}+[\partial,{\iota}_{\phi}].$

Thus, $d(e^{{\iota}_{\phi}|{\iota}_{\bar{\phi}}}(\Omega))=0$ amounts to

(6.17) $\bar{\partial}_{\phi}A_{0}=0,\quad\partial
A_{k}+\bar{\partial}_{\phi}A_{k+1}=0,\quad k=0,1,2,\ldots.$

Following the proof given in [RWZ21, Propositions 4.2 and 4.5], one sees that
(6.17) is equivalent to

(6.18)
$\displaystyle\sum_{k=0}^{\infty}\left(\bar{\partial}\circ\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}+\partial\circ\frac{{\iota}^{k}_{\phi}}{k!}\right)A_{r-p-(l-k)}=0,$

where $\max\\{1,r-p-q\\}\leq l\leq\min\\{2r-p-q,r+1\\}$,
${\iota}^{k}_{\phi}=0$ for $k<0$ and $0!=1$. And the system (6.18) of
obstruction equations can be reduced to the following one with only two
equations:

(6.19)
$\begin{cases}\sum_{k=0}^{\infty}\left(\bar{\partial}\circ\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}+\partial\circ\frac{{\iota}^{k}_{\phi}}{k!}\right)\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})=0,\\\
\sum_{k=0}^{\infty}\left(\bar{\partial}\circ\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}+\partial\circ\frac{{\iota}^{k}_{\phi}}{k!}\right)\circ\frac{{\iota}^{k-1}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{(k-1)!}(\widetilde{\Omega})=0.\end{cases}$

Rewrite (6.19) as

$\begin{cases}\partial\widetilde{\Omega}^{\prime}=-\bar{\partial}\sum_{k=1}^{\infty}\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega}),\\\
\bar{\partial}\widetilde{\Omega}^{\prime}=-\partial\sum_{k=0}^{\infty}\frac{{\iota}^{k+1}_{\phi}}{(k+1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega}),\end{cases}$

where

$\widetilde{\Omega}^{\prime}:=\widetilde{\Omega}+\sum_{k=1}^{\infty}\frac{{\iota}^{k}_{\phi}}{k!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega}).$

By Lemma 6.20, we need to prove

(6.20)
$\displaystyle\begin{cases}\partial\bar{\partial}\sum_{k=1}^{\infty}\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})=0,\\\
\partial\bar{\partial}\sum_{k=0}^{\infty}\frac{{\iota}^{k+1}_{\phi}}{(k+1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})=0,\end{cases}$

at the $(N+1)$-th order if it has been solved for the orders $\leq N$.
Analogous to [RWZ19, p. 471], we write the power series $\alpha(t)$

$\alpha(t)=\sum_{k=0}^{\infty}\sum_{i+j=k}\alpha_{i,j}t^{i}\bar{t}^{j},$

of (bundle-valued) $(p,q)$-basic forms as

$\begin{cases}\alpha(t)=\sum^{\infty}_{k=0}\alpha_{k},\\\\[4.0pt]
\alpha_{k}=\sum_{i+j=k}\alpha_{i,j}t^{i}\overline{t}^{j},\end{cases}$

where $\alpha_{k}=\sum_{i+j=k}\alpha_{i,j}t^{i}\overline{t}^{j}$ is the
$k$-order homogeneous part in the expansion of $\alpha(t)$ and all
$\alpha_{i,j}$ are smooth (bundle-valued) $(p,q)$-basic forms on
$\mathcal{F}_{0}$ with $\alpha(0)=\alpha_{0,0}$.

Indeed, one can prove (6.20) by repeating the procedures presented in [RWZ19,
p. 476]. Then by Lemmata 6.15 and 6.20, one has:

######  Proposition 6.27.

With the notations of Lemmata 6.15 and 6.20, the system of equations (6.11)
has a canonical solution

$x=\overline{\partial}(\partial\overline{\partial})^{*}\mathbb{G}^{b}_{BC}\overline{\partial}\zeta-\partial(\partial\overline{\partial})^{*}\mathbb{G}^{b}_{BC}\partial\bar{\xi}.$

Using Proposition 6.27, one obtains a formal solution of (6.19) by induction,

(6.21) $\displaystyle\widetilde{\Omega}_{l}$
$\displaystyle=-\Big{(}\sum_{k=1}^{\infty}\frac{{\iota}^{k}_{\phi}}{k!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})\Big{)}_{l}-\overline{\partial}(\partial\overline{\partial})^{*}\mathbb{G}^{b}_{BC}\overline{\partial}\Big{(}\sum_{k=1}^{\infty}\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})\Big{)}_{l}$
$\displaystyle\quad+\partial(\partial\overline{\partial})^{*}\mathbb{G}^{b}_{BC}\partial\Big{(}\sum_{k=0}^{\infty}\frac{{\iota}^{k+1}_{\phi}}{(k+1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})\Big{)}_{l}$
$\displaystyle=-\Big{(}\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k}_{\phi}}{k!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})\Big{)}_{l}-\overline{\partial}(\partial\overline{\partial})^{*}\mathbb{G}^{b}_{BC}\overline{\partial}\Big{(}\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})\Big{)}_{l}$
$\displaystyle\quad+\partial(\partial\overline{\partial})^{*}\mathbb{G}^{b}_{BC}\partial\Big{(}\sum_{k=0}^{\min\\{q,r-p\\}}\frac{{\iota}^{k+1}_{\phi}}{(k+1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})\Big{)}_{l}.$

######  Step $($IV$)$.

Regularity argument.

From the induction expression (6.21), one obtains the formal expression of
$\widetilde{\Omega}:$

(6.22) $\displaystyle\widetilde{\Omega}$
$\displaystyle=-\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k}_{\phi}}{k!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})-\overline{\partial}(\partial\overline{\partial})^{*}\mathbb{G}_{BC}^{b}\overline{\partial}\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})$
$\displaystyle\quad+\partial(\partial\overline{\partial})^{*}\mathbb{G}_{BC}^{b}\partial\sum_{k=0}^{\min\\{q,r-p\\}}\frac{{\iota}^{k+1}_{\phi}}{(k+1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})+\Omega_{0}.$

Set

$\displaystyle F=$
$\displaystyle\partial(\partial\overline{\partial})^{*}\mathbb{G}_{BC}^{b}\partial\sum_{k=0}^{\min\\{q,r-p\\}}\frac{{\iota}^{k+1}_{\phi}}{(i+1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}$
$\displaystyle-\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k}_{\phi}}{k!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}$
$\displaystyle-\overline{\partial}(\partial\overline{\partial})^{*}\mathbb{G}_{BC}^{b}\overline{\partial}\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}$

and write

(6.23) $\Omega_{0}=(\mathds{1}-F)\widetilde{\Omega}.$

We claim that $\widetilde{\Omega}(t)$ converges in Hölder norm (6.5) as
$t\rightarrow 0$. Here, we resort to Construction 6.11 and Remark 6.14 to get
two transversely elliptic estimates for a basic form $\psi$,

$\|\overline{\partial}^{*}\psi\|_{k-1,\alpha}\leq C_{1}\|\psi\|_{k,\alpha}$

and

$\|\mathbb{G}_{BC}^{b}\psi\|_{k,\alpha}\leq
C_{k,\alpha}\|\psi\|_{k-4,\alpha},$

where $k>3$ and $C_{k,\alpha}$ depends on only on $k$ and $\alpha$, not on
$\psi$. As $\phi(t)$ converges smoothly to zero as $t\rightarrow 0$, one
estimates by (6.23),

(6.24)
$\|\Omega_{0}\|_{k,\alpha}\geq(1-\epsilon_{k,\alpha})\|\widetilde{\Omega}\|_{k,\alpha},$

where $0<\epsilon_{k,\alpha}\ll 1$ is some constant depending on $k,\alpha$.

Finally, we proceed to the regularity of $\widetilde{\Omega}(t)$ since there
is possibly no uniform lower bound for the convergence radius obtained as
above in the $C^{k,\alpha}$-norm when $k$ converges to $+\infty$. Similarly,
we can transfer the world of transversely elliptic theory to the one of
elliptic theory in the ordinary sense as aforementioned. So our argument also
lies heavily in the elliptic estimates [Ko86, Appendix 8], [DN55] and also
[RWZ21, Subsection 3.2]. Without loss of generality, we just consider the
equation:

$\displaystyle\square_{\overline{\partial}}^{b}\widetilde{\Omega}$
$\displaystyle=-\square^{b}_{\overline{\partial}}\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k}_{\phi}}{k!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})-\overline{\partial}\overline{\partial}^{*}\overline{\partial}(\partial\overline{\partial})^{*}\mathbb{G}_{BC}^{b}\overline{\partial}\sum_{k=1}^{\min\\{q,r-p\\}}\frac{{\iota}^{k-1}_{\phi}}{(k-1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})$
$\displaystyle\quad+\square_{\overline{\partial}}^{b}\partial(\partial\overline{\partial})^{*}\mathbb{G}_{BC}^{b}\partial\sum_{k=0}^{\min\\{q,r-p\\}}\frac{{\iota}^{k+1}_{\phi}}{(k+1)!}\circ\frac{{\iota}^{k}_{(\mathds{1}-\bar{\phi}\phi)^{-1}\bar{\phi}}}{k!}(\widetilde{\Omega})$

by applying the basic Dolbeault Laplacian
$\square^{b}_{\overline{\partial}}=\overline{\partial}^{*}\overline{\partial}+\overline{\partial}\overline{\partial}^{*}$
to the expression formula (6.22) and omitting the lower-order term
$\square^{b}_{\overline{\partial}}\Omega_{0}$ in this expression.

For each $l=1,2,\cdots$, choose a smooth function $\eta^{l}(t)$ with values in
$[0,1]$:

$\eta^{l}(t)\equiv\begin{cases}1,\ \text{for
$|t|\leq(\frac{1}{2}+\frac{1}{2^{l+1}})\gamma$};\\\ 0,\ \text{for
$|t|\geq(\frac{1}{2}+\frac{1}{2^{l}})\gamma$},\end{cases}$

where $\gamma$ is a positive constant to be determined. Inductively, by
Douglis–Nirenberg’s interior estimates [Ko86, Appendix.Theorem 2.3], [DN55],
for any $l=1,2,\cdots$, $\eta^{2l+1}\widetilde{\Omega}(t)$ is
$C^{k+l,\alpha}$, where $\gamma$ can be chosen independent of $l$. Since
$\eta^{2l+1}(t)$ is identically equal to $1$ on $|t|<\frac{\gamma}{2}$ which
is independent of $l$, $\widetilde{\Omega}(t)$ is $C^{\infty}$ on
$(M,\mathcal{F}_{0})$ with $|t|<\frac{\gamma}{2}$. Then
$\widetilde{\Omega}(t)$ can be considered as a real analytic family of
$(p,q)$-basic forms in $t$ and thus is smooth on $t$.

The proof of Theorem 6.25 (and thus Theorem 1.7) is completed.

######  Remark 6.28.

On the assumptions of Theorem 1.7, we should notice:

  1. (a)

One can conclude from our proof that the homological orientability assumption
used in [EKAG97, Rź21] is not necessary in our approach. In the previous
works, they follow the original cohomological argument used by
Kodaira–Spencer, and it seems that the homological orientability assumption on
$\mathcal{F}$ is necessary for the upper semi-continuity theorem for
transversely elliptic operators to work (see [Rź21, Remark 4.6] for example).
In our proof, we directly construct the desired transversely $p$-Kähler form.
We more faithfully follow the scalar product or Hölder norm on the space of
basic forms introduced in Construction 6.11 and allow the existence of a
correction term involving the mean curvature of the foliation when defining
the adjoint operators, see Remarks 6.12, 6.13.

  2. (b)

Raźny [Rź17, Theorem 7.1] showed that a homologically orientable transversely
Kähler foliation on a compact manifold satisfies the
$\partial\overline{\partial}$-lemma (6.10), and thus obviously satisfies the
$(1,2)$-th mild $\partial\overline{\partial}$-lemma. Compared with the works
of [EKAG97, Rź21], Theorem 1.7 uses weaker assumptions when $p=1$
(transversely Kähler).

###  6.6. Deformation invariance of basic Hodge/Bott–Chern numbers

Recently, Raźny showed the rigid property of basic Hodge numbers under
deformations of homologically orientable transversely Hermitian foliations on
a compact manifold when $\mathcal{F}_{0}$ satisfies
$\partial\overline{\partial}$-lemma and the foliations are fixed [Rź21,
Corollary 5.3]. Motivated by this, we study the deformation invariance of
basic Hodge/Bott–Chern numbers with ‘weak’
$\partial\overline{\partial}$-lemmata in this subsection.

The following theorem can be seen as the foliated version of Kodaira–Spencer’s
results. It seems that we cannot drop the homological orientability assumption
as already mentioned in Remark 6.28.

######  Theorem 6.29 ([EKAG97, Theorem 5.4], [Rź21, Theorem 4.5]).

Let $\left\\{\mathcal{F}_{t}\right\\}_{t\in U}$ be a smooth family of
homologically orientable transversely Hermitian structures on a compact
manifold $M$ with fixed differentiable type, parametrized by an open
neighborhood $U$ of $0$ in $\mathbb{R}^{d}$. Let
$D_{t}:A^{p,q}(M/\mathcal{F}_{t})\rightarrow A^{p,q}(M/\mathcal{F}_{t})$ be a
family of transversely elliptic operators of even order. Denote
$h(t):=\dim\ker(D_{t})$. Then $h(t)$ is upper semi-continuous. Furthermore, if
$h(t)$ is independent of $t\in U$, then the operators $\mathbb{G}_{t}$ and
$\mathbb{H}_{t}$ depend differentiably on $t\in U$. Here, $\mathbb{G}_{t}$ and
$\mathbb{H}_{t}$ are the associated Green’s operator and harmonic projection
operator, respectively.

Consequently, one can repeat the same procedures in Section 4, [RZ18, Section
3] and [RWZ19, Section 5] to get

######  Theorem 6.30.

Let $\left\\{\mathcal{F}_{t}\right\\}_{t\in U}$, $M$, $U$ be as in Theorem
6.29.

  1. $(a)$

If $\mathcal{F}_{0}$ satisfies both $\mathbb{B}^{p,q+1}$ and
$\mathcal{B}^{p+1,q}$ with the deformation invariance of
$h^{p,q-1}_{\overline{\partial}_{t}}(M/\mathcal{F}_{t})$ established, then
$h^{p,q}_{\overline{\partial}_{t}}(M/\mathcal{F}_{t})$ are independent of $t$.

  2. $(b)$

If $\mathcal{F}_{0}$ satisfies both $\mathbb{S}^{p,q+1}$ and
$\mathbb{B}^{p+1,q}$ with the deformation invariance of
$h^{p,q-1}_{\overline{\partial}_{t}}(M/\mathcal{F}_{t})$ established, then
$h^{p,q}_{\overline{\partial}_{t}}(M/\mathcal{F}_{t})$ are independent of $t$.

  3. $(c)$

If $\mathcal{F}_{0}$ satisfies both $\mathbb{B}^{p,q+1}$ and
$\mathbb{B}^{q,p+1}$ with the deformation invariance of the $(p-1,q-1)$-basic
Aeppli number $h^{p-1,q-1}_{A}(M/\mathcal{F}_{t})$ established, then the
$(p,q)$-basic Bott–Chern number are independent of $t$.

Finally, we mention an interesting question proposed by Raźny, whose main
point lies in whether one can generalize [Rź21, Corollary 5.3] to the case
when the foliations are allowed to vary.

######  Question 6.31 ([Rź21, Question 1.2]).

Are the basic Hodge numbers rigid under deformations of (homologically
orientable) transversely Kähler foliations on compact manifolds?

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