# Black to white transition of a charged black hole

Antoine Rignon-Bret  École Normale Supérieure, 45 rue d’Ulm, F-75230 Paris,
France  Carlo Rovelli  Aix-Marseille University, Université de Toulon, CPT-
CNRS, F-13288 Marseille, France.  Department of Philosophy and the Rotman
Institute of Philosophy, 1151 Richmond St. N London N6A5B7, Canada  Perimeter
Institute, 31 Caroline Street N, Waterloo ON, N2L2Y5, Canada

###### Abstract

We present an exact solution of the Maxwell-Einstein equations, which
describes the exterior of a charged spherical mass collapsing into its own
trapping horizon and then bouncing back from an anti-trapping horizon at the
same space location of the same asymptotic region. The solution is locally but
not globally isometric to the maximally extended Reissner-Nordström metric and
depends on seven parameters. It is regular, and defined everywhere except for
a small region, where quantum tunnelling is expected. This region lies
_outside_ the mass: the mass-bounce and its near exterior are governed by
classical general relativity. We discuss the relevance of this result for the
fate of realistic black holes. We comment on possible effects of the classical
instabilities and the Hawking radiation.

## Introduction

Contrary to what is sometimes assumed, the long term evolution of the
_exterior_ of a black hole is likely to be affected by quantum gravitational
effects. This is because the back-reaction of the Hawking radiation makes the
curvature grow to Planckian values _outside_ the horizon too. At this point,
and its immediate future, the Einstein equations are likely to be violated by
quantum gravity effects, in particular by conventional quantum tunnelling.
Elsewhere, it is reasonable to expect the evolution of spacetime to be
governed by the classical Einstein equations. What do _these_ permit for the
long term evolution of a black hole?

A surprisingly result obtained in [1] is that the Einstein equations admit a
spherically symmetric vacuum solution that describes a black hole that tunnels
into a white hole. This spacetime does not contradict Birkhoff’s theorem,
because it is locally –but not globally– isometric to the Kruskal spacetime.
The result reveals an interesting scenario: (i) black hole horizons are
_event_ horizons only in the classical limit; (ii) before the end of the
evaporation, the trapping horizon tunnels into an antitrapping horizon; (iii)
the black hole interior tunnels into a white hole interior; (iv) the collapsed
matter bounces out. This scenario [2, 3, 4, 5, 6, 7, 8, 9, 10], its possible
astrophysical implications [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22],
and its relevance for the black hole information paradox [23, 24] are
currently under intense investigation.

So far the literature has focused on the non-rotating and non-charged case.
Realistic black holes rotate and are approximated by the Kerr-Newman metric,
whose maximal extension has a markedly different structure than the Kruskal
spacetime. Is the intriguing black to white transition a peculiarity of the
Schwarzschild metric, or is it possible in general?

The lack of spherical symmetry makes the analysis of the Kerr-Newman case
harder, but there is an interesting intermediate case that has the same global
structure as extended Kerr (the Carter-Penrose diagrams of their maximal
extension is similar) and yet has spherical symmetry: the Reissner-Nordström
metric. This is the solution of the Maxwell-Einstein equations around a
spherical symmetric _charged_ mass. In this paper, we extend the result of [1]
to this case. We show that there is an exact solution of the Einstein-Maxwell
equations that describes the exterior of a charged spherical mass that
collapses into its own trapping horizon and then bounces back from an anti-
trapping horizon at the same spacial location of the same asymptotic region.
The solution is locally but not globally isometric to the maximally extended
Reissner-Nordström metric, is regular, and is everywhere defined except for a
compact finite region, where a quantum gravitational tunnelling transition can
be expected.

What we find is surprising. Unlike the non-charged case, the quantum region
does not continue inside the hole all the way to the collapsing and bouncing
matter. The bounce of the collapsing matter and its surrounding evolve
classically without ever entering the quantum region. This is comprehensible,
as the global structure of Reissner-Nordström (and Kerr) allows a time-like
geodesic to enter a black hole, traverse it and exit from a white hole without
encountering singularities or high curvature regions. In the maximally
extended metric, the white hole is in a different asymptotic region; here we
show (following [1]) that the white hole can be in the same asymptotic region
as the black hole: in its immediate future.

This result makes the charged case transition (and presumably the rotating
case as well) easier to understand and treat than the Schwarzschild case. In a
sense, the spacetime region which needs to be described by quantum gravity is
smaller than in the Schwarzschild case: some part of the mechanism of the
black to white transition is already contained in the classical solution.
Quantum effects are not needed for the charged mass to bounce, nor for the
black hole interior to evolve into a white hole exterior. Only the horizon
area undergoes a quantum transition, when it reaches Planckian curvature.

We recall the main features of the black to white transition in the
Schwarzschild case in Section I and the causal structure of the maximally
extended Reissner-Nordström metric in Section II. The solution of the Maxwell-
Einstein equations that describes the black to white transition of the charged
black hole is built in Section III and the reason a quantum tunnelling is to
be expected is discussed in Section IV. Then we comment the possible effects
of the classical instabilities [25, 26, 27] and the Hawking evaporation in
Sections V and VI. We use units where $G$ = $c$ = $\hbar$ = 1.

##  I Schwarzschild

Figure 1: The Carter-Penrose diagram of a Schwarzshild black hole until the
onset of quantum gravity. The light grey region is the collapsing star. The
dark grey region is where quantum gravity becomes relevant.

The Carter-Penrose diagram of a (classical) Schwarzschild black hole created
from a gravitational collapse is depicted on Figure 1. The dark grey region is
where the classical theory becomes unreliable, due to quantum gravitational
effects. We expect this to happen when the curvature becomes Planckian, for
instance when the Kretschmann scalar

$K^{2}=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}=48\frac{M^{2}}{r^{6}}$
(1)

becomes of order 1. Here $M$ is the black hole mass and $r$ is the
Schwarzschild radius. This happens before the $r=0$ singularity inside the
black hole. Just outside the horizon, $K\sim\frac{1}{M^{2}}$. The Hawking
evaporation steadily decreases the mass $M$ of an isolated black hole,
bringing it down to Planckian values, hence the quantum region extends outside
the horizon. General arguments and some specific calculation [5, 28] indicate
that the transition probability $P$ from black holes to white holes is
proportional to

$P\sim e^{-M^{2}}.$ (2)

Thus becoming dominant at the end of the evaporation, where $M\sim 1$. It is
also possible that quantum effects could appear earlier [1, 29], at a time of
order $M^{2}$ after the collapse.

Here we are not directly concerned with these estimates. We only retain the
fact that the quantum region extends outside the horizon. This region can be
organized into three sub regions [9] (see Figures 1 and 2):

  * •

Region B : The horizon region.

  * •

Region C : The collapsing star region.

  * •

Region A : The region which is neither directly causally connected to the
horizon nor to the collapsing star.

It is shown in [9] that the phenomena in these three regions can be considered
causally disconnected, as they are separated by a large spacelike distance,
which at the end of the evaporation grows to

$L\simeq{M}^{\frac{10}{3}},$ (3)

which is huge for a macroscopic black hole. To understand the physics of the
end of a black hole, we have to understand the quantum evolution of these
three regions of spacetime.

The Carter-Penrose diagram of the classical metric describing the black to
white bounce [1] is depicted in Figure 2. The white region is locally but not
globally isomorphic to the Kruskal metric, the maximal extension of the
Schwarzshild metric. There is a trapped as well as a later anti-trapped region
(see also Figure 8 in [30]), separated by a small compact region where the
Einstein equations are violated by quantum gravity effects. The horizon of the
black hole is not an event horizon.

Figure 2: The Carter-Penrose diagram of the black to white transition. The
dark grey region is the quantum gravity region. The black hole (trapped
region) is below the quantum gravity region while the white hole is above. The
trapping horizons are the dashes lines.

##  II The maximally extended Reissner-Nordström metric

Figure 3: Maximally extended Reissner-Nordström metric

The Reissner-Nordström metric describing a massive charged star (or black
hole) is

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega^{2}$ (4)

with

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}},$ (5)

where $M$ is the mass of the star and $Q$ its charge. Together with the
Maxwell potential

$A_{a}=\left(\frac{Q}{r},0,0,0\right),$ (6)

equations (4–5) solve the Maxwell-Einstein equations.

Notice that the $Q\to 0$ limit that reduces Reissner-Nordström to
Schwarzschild is subtle at small radius, as

$\lim_{Q\to 0}\lim_{r\to 0}f(r)=\infty,\ \ \ {\rm while}\ \ \ \lim_{r\to
0}\lim_{Q\to 0}f(r)=-\infty,$ (7)

which shows that even a small charge changes the inner geometry radically.

As Schwarzschild, the Reissner-Nordström metric is static outside the horizon,
and spherically symmetric. The main difference with Schwarzschild is that the
equation $f(r)=0$ that gives the position of the trapping horizons has two
solutions rather than one:

$r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}.$ (8)

The two solutions $r_{+}$ and $r_{-}$ define the outer and inner horizon,
respectively. They separate trapped, non-trapped and anti-trapped regions: the
surfaces of constant $r$-coordinate are timelike for $r>r_{+}$, becomes
spacelike for $r_{-}<r<r_{+}$, and are timelike again for $r<r_{-}$. The
difference with Schwarzschild is therefore the presence of the inner horizon
$r_{-}$, a feature in common with the Kerr metric. As for Schwarzschild, the
metric has four Killing fields, the three associated to the rotations, and the
Killing field $\frac{\partial}{\partial t}=\xi$ associated to the invariance
with respect to the $t$ coordinate. Notice that $\xi$ is timelike for
$r>r_{+}$ and $r<r_{-}$, null at the two horizons and spacelike for
$r_{-}<r<r_{+}$.

The Penrose diagram of the Maximally extended Reissner-Nordström spacetime is
given in Figure 3. The interior region continues into an anti-trapped region,
namely a white hole region, that in turns exits via another outer horizon into
a different asymptotic region.

Consider a neutral particle with mass $m$ and energy momentum
$p_{\alpha}=mu_{\alpha}$ falling into a Reissner-Nordström black hole. The
quantity $E=-g_{\alpha\beta}p^{\alpha}\xi^{\beta}$ is a constant of motion. A
straightforward calculation gives

$\dot{r}^{2}+f(r)=E^{2},$ (9)

where $\dot{r}=u^{r}$. Notice that $\dot{r}$ vanishes at the radius $r_{b}$
determined by

$f(r_{b})=E^{2}.$ (10)

If the initial radial velocity of the particle vanishes at large $r$, then
$E=1$ and

$r_{b}=\frac{Q^{2}}{2M}<r_{-}.$ (11)

Hence, the particle enters the black hole, crosses the outer horizon at
$r_{+}$, then crosses the inner horizon at $r_{-}$, and reaches $r=r_{b}$
where its velocity goes to zero. The particle bounces and starts moving
outward. This is permitted because the interior region is not trapped. By time
inversion symmetry, its geodesics exit then into the next asymptotic region
through the anti-trapped region. See Figure 4.

Figure 4: The portion of the maximally extended Reissner-Nordstrom spacetime
relative for the black to white bounce. The grey line is a geodesic that
enters the black hole from a asymptotic region, and exit in a different
asymptotic region.

The conclusion is also valid for a charged particle. Due to (6), the
electromagnetic effect on the energy of a charged particle is simply to shift
it to $p_{0}=mu_{0}+qA_{0}$ where $A_{0}=\frac{Q}{r}$ (we use units where
$\frac{1}{4\pi\epsilon_{0}}=1$). The conserved quantity is
$E=-g^{\alpha\beta}p_{\alpha}\xi_{\beta}=-p_{\alpha}\xi^{\alpha}$, giving

$\bigg{(}{E}-\frac{Q^{2}}{Mr}\bigg{)}^{2}=f(r)+\dot{r}^{2}.$ (12)

Charged particles do not follow geodesics, but (12) is similar to (9) up to a
shift in the energy. If the initial radial velocity vanishes, it implies from
(12) that ${E}\simeq 1$ by assuming that the initial radius is sufficiently
large. Hence, we have to solve Eq. (12) with the turning point condition
($\dot{r}^{2}=0$), which gives

$\bigg{(}1-\frac{Q^{2}}{Mr}\bigg{)}^{2}=f(r)$ (13)

Interestingly, the solution of this equation is still (9) where we didn’t take
the electrostatic repulsion into account.

Figure 5: The surface of the star (grey line), the boundary of the quantum
region (continuous blue line) and the constant time surface that can be
identified.

Thus, a collapsing massive spherical charged mass that enters its own outer
horizon can cross the inner horizon and bounce out at $r_{b}$.

In the Schwarzschild case the bounce of the star was an hypothesis based on
quantum gravity. In the charged case, the bounce of the star is predicted by
_classical_ general relativity. The physical problem of the C region defined
above is solved without the need of quantum mechanics.

In other words, the presence of a charge opens up a _classical_ throat for the
passage of the star and its surrounding spacetime from the black to the white
hole region.

If the charge is small this passage is narrow. Its size can be estimated from
the spacelike distance of the boundary of the star to the outer boundary of
the classical region on a constant $t$ surface. This is

$\displaystyle d$ $\displaystyle=$
$\displaystyle\int_{r_{b}}^{r_{-}}\frac{dr}{\sqrt{f(r)}}=\int_{r_{b}}^{r_{-}}\frac{r}{\sqrt{(r-r_{+})(r-r_{-})}}dr$
(14)
$\displaystyle\leq\int_{0}^{r_{-}}\frac{r_{-}}{\sqrt{(r_{-}-r_{+})(r-r_{-})}}dr$
$\displaystyle=\sqrt{2}M\frac{\bigg{(}1-\sqrt{1-\eta^{2}}\bigg{)}^{\frac{3}{2}}}{(1-\eta^{2})^{\frac{1}{4}}}$

where $\eta=\frac{Q}{M}$. If $\eta$ is small, the throat is small. Only matter
falling just after the star collapses reaches it, rather than the Cauchy
horizon, But if $\eta$ is close to 1

$ds=\frac{r}{\sqrt{(r-r_{+})(r-r_{-})}}dr\simeq\frac{r}{r-r_{-}}dr$ (15)

which is not integrable in $r_{-}$. Hence the throat can be arbitrarily large.

Figure 6: The spacetime of the black to white charged transition outside the
star.

The result suggests that the Reissner-Nordström black to white transition may
be more natural than the Schwarzschild one: allready in the classical theory
the white hole is in the future of the black hole.

##  III The Reissner-Nordström black to white transition

Here we are not interested in an hypothetical emergence in a different
universe. We are interested in the compatibility of classical general
relativity with a bounce within the _same_ universe, generated by a quantum
tunnelling in a small compact spacetime region.

To show that this is possible, we can construct a solution of the Maxwell-
Einstein equations, following [1], by cutting and gluing relevant portions of
the maximally extended Reissner-Nordström metrics. The way this can be done is
sketched in Figure 5.

Figure 7: The full black to white transition, for a charged black hole.

To construct the spacetime we are interested in, we proceeded as follows. (i)
We cut away from the maximally extended Reissner-Nordström spacetime all the
regions on the left of the grey line in Figure 5, and replace it with the
interior of a classical bouncing star. (ii) We cut away all the region to the
right of the blue line in Figure 5. (iii) We glue the two dotted portions of
the blue line. These are both constant-Schwarzschild-time surfaces, and
therefore the gluing gives a smooth junction, where the Einstein equations are
satisfied.

More precisely, to glue the spacetime smoothly both metric and extrinsic
curvature must match. The metric of two surfaces of constant time coordinate
$t$ is clearly the same. By construction, the extrinsic curvature $k$ changes
sign (because of the time reversal symmetry exploited in cutting the
spacetime). But the curvature vanishes on these surfaces as the normal to the
spacelike surfaces of constant $t$ is the vector
$N_{\alpha}=\partial_{\alpha}t=\xi$ where $\xi$ is the Killing vector
associated to the $t$-coordinate invariance and

$k=\frac{1}{2}\mathcal{L}_{N}q=\frac{1}{2}\mathcal{L}_{\xi}q=0$ (16)

where $q$ is the induced metric on the spacelike hypersurface.

The resulting spacetime is depicted in Figure 6, and in 7 by including the
star. The central grey area represents the quantum tunnelling region, where
the classical evolution is violated. Let us do so more explicitly.

Let us break the spacetime of Figure 5 into two overlapping regions, as
depicted in Figure 8, and introduce distinct coordinates in the two regions.

The left panel includes the exterior $r>r_{+}$, the black hole
$r_{-}<r<r_{+}$, and the interior $r<r_{-}$. A set of coordinates covering
this entire left region is given by $r$ and the advanced time null coordinate
$v$ defined by

$v=t+r^{*}(r)$ (17)

where

$r^{*}(r)=\int\frac{dr}{f(r)}.$ (18)

The integration must be done separately in the regions separated by $r_{+}$
and $r_{-}$, because the integral is ill behaved at the horizons. (Remember
that the $t$ coordinates in the three regions are actually unrelated to one
another, each going from $-\infty$ to $+\infty$ in its own region. Properly
speaking, they should have different names.) In the interior region, it is
convenient to choose the integration constant (or the lowest boundary of the
integral) so that the $t$ coordinate of the bounce point $b$ is zero. In the
external region, it is convenient to choose it so that the $t$ coordinate of
the outermost point of the quantum region is zero.

Figure 8: The part of the maximally extended Reissner-Nordström spacetime
surrounding to the black hole region on the left, that can be labeled by
$(r,v)$ coordinates, and the part corresponding to the white hole region on
the right, by $(r,u)$ patch. Picture taken from [31].

In these coordinates the metric reads

$ds^{2}=-f(r)dv^{2}+2drdv+r^{2}d\Omega^{2}.$ (19)

The right panel of Figure 8 includes the exterior, the white hole and again
the interior region, separated by $r_{+}$ and $r_{-}$. A set of coordinates
covering the left region is given by $r$ and the retarded time

$u=t-r^{*}(r).$ (20)

In these coordinates, the metric reads

$ds^{2}=-f(r)du^{2}-2drdu+r^{2}d\Omega^{2}.$ (21)

The two coordinate patches overlap in the interior ($r<r_{-}$) and exterior
($r>r_{+}$) regions, where their relation is easily deduced by equating the
$t$ coordinate of the two:

$u+r^{*}(r)=v-r^{*}(r).$ (22)

Notice that identifying the coordinates in the two overlapping regions
(exterior and interior) adds a parameter to the definition of the spacetime
(irrespectively from the detailed location of the quantum region). This can be
seen as follows. Consider two points $p_{1}$ and $p_{2}$ both on $t=0$ (hence
$u_{1}=-v_{1}$ and $u_{2}=-v_{2}$), and let $r_{1}$ and $r_{2}$ be their
radius. If the two points are both in the exterior or both in the interior
region, the difference of their retarded (or advanced) time is

$u_{2}-u_{1}=r^{*}(r_{2})-r^{*}(r_{1})=\int_{r_{1}}^{r_{2}}\frac{dr}{f(r)}.$
(23)

But if $p_{2}$ is in the interior and $p_{1}$ in the exterior, the above
integral must be broken into two and therefore depends on an arbitrary
integration constant, not determined by the radius of the points. This
additional parameter is a global topological parameter which distinguished
spacetimes with the same $M$ and $Q$, obtained in this manner.

To specify the metric entirely (including the location of the quantum region)
is it convenient to focus on three special events: the event $b$ where the
radius of the star reaches its mininal value $r_{b}$ given in (11), the event
$P$ where the quantum tunnelling starts and the point $P^{\prime}$ where it
ends. We take the quantum region to be the causal diamond defined by $P$ and
$P^{\prime}$, namely the intersection between the causal future of $P$ and the
causal past of $P^{\prime}$: see Figure 9.

Figure 9: The parameters characterising the bouncing geometry.

Let the coordinates of the points $b,P$ and $P^{\prime}$ be
$(v_{b},u_{b}),(v_{P},u_{P})$ and $(v_{P^{\prime}},u_{P^{\prime}})$,
respectively. The two points of the quantum region $min$ and $max$ with the
minimal and maximal radius have coordinates $(v_{P},u_{P^{\prime}})$ and
$(v_{P^{\prime}},u_{P})$ respectively.

As a first step, consider a time reversal symmetric bounce geometry. In this
case we can assume $u_{b}=-u_{v},u_{P^{\prime}}=-v_{P}$ and
$u_{P}=-v_{P^{\prime}}$ (see Figure 9). The maximal radius reached by the
quantum region is $r_{max}=(v_{P^{\prime}}-u_{P})/2>r_{+}$, at $max$. The
minimal radius reached by the quantum region is
$r_{min}=v_{P}-u_{P^{\prime}}<r_{-}$, at $min$. We can chose the zero of the
Schwarzschild time coordinate $t$ so that $b$, $min$ and $max$ are all on
$t=0$.

Let us discuss the interpretation of these quantities. The quantities $v_{0}$
and $u_{0}$ are the advanced time of the collapse of the star into the black
hole and the retarded time of exit from the white hole. The difference
$v_{b}-v_{o}$ is the (short) time determined by the details of the dynamics of
the collapsing star. For simplicity, we can take the limit of a star
collapsing at very high relativistic speed so that $v_{b}-v_{o}\sim 0$ and
neglect their difference.

The advanced time $\tau_{t}=v_{P^{\prime}}-v_{P}$ is the time of the
tunnelling transition (likely to be short). The crucial physical parameter of
the process is the difference $\tau_{BH}=v_{P}-v_{b}$. Notice that we must
have $r_{min}<r_{-}$, but as $r_{min}$ approaches $r_{-}$, the advanced time
$v_{P}$ goes to infinity. Hence $\tau_{BH}$ can be arbitrary long.

Disregarding the time $v_{b}-v_{o}$, the symmetric bounce is then fully
characterized by five parameters: $M,Q,r_{P},\tau_{BH}$ and $\tau_{t}$. A
theory of quantum gravity must give the transition probability
$W(M,Q,r_{P},\tau_{BH},\tau_{t})$ between the black hole state and the white
hole state. For the formulation of this computation in Loop Quantum Gravity,
see [5, 6, 28, 10]

A non-time-reversal-symmetric bounce can be obtained by having $b$ and
$(v_{P},u_{P^{\prime}})$ on different constant $t$ surfaces and $P$ and
$P^{\prime}$ at different radius. This gives $u_{P^{\prime}}\neq-v_{P}$ and
$\tau_{WH}=u_{b}-u_{P^{\prime}}$ different from $\tau_{BH}$. The two long
times $\tau_{BH}$ and $\tau_{WH}$ can be interpreted as the lifetimes of the
black hole and the white hole. The general case is therefore characterized by
seven parameters: $M,Q,r_{P},r_{P^{\prime}},\tau_{BH}$ and $\tau_{WH}$, plus
the tunnelling time $\tau_{t}$.

##  IV The point $P$ and the onset of the tunnelling

The spacetime described above is characterized by the region
$(u_{P}<u<u_{P^{\prime}},v_{P}<v<v_{P^{\prime}})$ where the Einstein equations
are not satisfied. Following [10], we call this region the B region. As
mentioned in the introduction, this is the region where we assume quantum
mechanics alters the classical spacetime dynamics. Why quantum theory should
be relevant here? There are three answers to this question:

  1. (i)

Quantum mechanics allows tunnelling transition with a given finite
probability. A rough estimate of a the tunnelling probability may include a
factor $e^{-S}$, where $S$ is the action of the process. Since $Q$ is of
little relevance near the outer horizon, we may expect this factor to be
proportional to $e^{-M^{2}}$. In the classical theory, the lifetime of a black
hole is infinite and during an infinite time, an event may happen even if its
probability per unit of time is very small. This would give an exponentially
large factor $e^{M^{2}}$ in the black hole lifetime, but still a finite
lifetime.

  2. (ii)

However, we expect an isolated black hole to evaporate by Hawking radiation.
During the evaporation, the mass of the hole decreases, reaching a Planckian
value in a time of the order $M_{0}^{3}$, where $M_{0}$ is the initial mass of
the hole. When $M$ approaches the Planck mass, the suppressing factor
$e^{-M^{2}}$ approaches unit, and the tunnelling becomes likely to happen.
Hence we may expect a transition to happen shortly before the end of the
Hawking evaporation, hence in a time of order $M_{0}^{3}$ after the collapse.
It is important to recall again that when $M$ is near the Planck mass, the
curvature becomes Planckian _outside_ the horizon. We can view the point $P$
as a point where the curvature is close enough to a Planckian curvature to
trigger the quantum tunnelling.

  3. (iii)

Finally, we also recall the hypothesis considered in [1]: quantum corrections
to the Einstein equations can be small but are never exactly zero, so they can
pile up in proper time $\tau$. Assuming a linear deviation to first order in
$\hbar$, a non negligible quantum phenomenon may start when $q=K\tau$ is
Planckian, where $K$ is a curvature scalar such as the Kretchman invariant

$K^{2}=C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}=48\frac{\left(M-\frac{Q^{2}}{r}\right)^{2}}{r^{6}}$
(24)

and $\tau$ is the proper time along a stationary time-like geodesic. Since the
curvature near the horizon is of order $M^{-2}$, this has suggested the
(speculative) possibility of a transition triggered already after a black hole
lifetime of order $M_{0}^{2}$. This consideration suggests also what is the
possible radius of $P$. The parameter $q$ is maximized at a finite distance
from the horizon, of the order of $M$. This is because the curvature increases
with smaller radius, but the proper time at a stationary point is red shifted
near the outer horizon as $\tau=\sqrt{f(r)}\ \tau_{BH}$, so that

$\displaystyle q$ $\displaystyle=$ $\displaystyle
K\tau\sim\frac{M-\frac{Q^{2}}{r}}{r^{3}}\sqrt{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}\
\tau_{BH},$ (25)

which has a minimum in $r$ just outside the outer horizon. The minimum sets
the radius of $P$, while $q\sim 1$ at this minimum sets $\tau_{BH}$ [1].

One way or the other, at some point $P$ outside the horizon the dynamics of
the gravitational field enters the quantum tunnelling region. By causality,
this region must then be in the causal future of $P$. In the future of this
region, we can assume spacetime to be described again by a solution of the
Einstein equations. Depending on the interpretation of quantum theory one
prefers, this can be seen as a de-cohered many world branch, or the result of
a (“measuring”) interaction with other degrees of freedom (the “observer”)
after the quantum process.

Notice that the entire time-like singularity and the non-causal region of the
Reissner-Nordström spacetime past the Cauchy horizon (the boundary of the
causal future of the singularity) disappear, replaced by the quantum region: a
part from the quantum transition, the spacetime is regular.

This completes the construction of the Reissner-Nordström black to white hole
transition spacetime.

Before concluding this section, we observe that the above result can also be
taken as a possible form of the effective metric in the C and A regions in the
case of vanishing electrical charge. (Another simple guess for the effective
metric in the $A$ region can be obtained by replacing $r$ with
$\sqrt{r^{2}+L^{2}}$ in the Schwarzschild metric [32]: see also [33] for the
same idea). Loop quantum cosmology [34] suggests that the dominant quantum
effect at high curvature is a repulsive force, precisely as in the Reissner-
Nordström geometry (where repulsion does not act on charged matter only). This
is the short scale quantum pressure which is also responsible for the quantum
bounce of a Planck star [2, 14]. Hence the Reissner-Nordström geometry can
also be taken as a guess for the quantum corrections to the metric at short
radius, for an effective value of a “charge” $Q_{eff}^{2}$ determined by $M$
(and the Planck mass $M_{Planck}$). This can be estimated assuming that the
corrections becomes relevant when the curvature is Planckian, which gives

$Q_{eff}\sim M^{\frac{2}{3}}.$ (26)

Hence, restoring physical units

$\eta_{eff}\sim\left(\frac{M}{M_{Planck}}\right)^{-\frac{1}{3}}.$ (27)

For a macroscopic mass $M$, $\eta_{eff}\ll 1$ and so the resulting metric is
very close to Schwarzschild, but since, as pointed out above, the $Q\to 0$ and
$r\to 0$ limits do not commute, the global structure of the metric is
radically changed nevertheless. If this is correct, the actual effect of
quantum gravity makes the Reissner-Nordström geometry studied here more
realistic than the Schwarzschid geometry.

##  V Cauchy horizon instabilities

There are three important physical phenomena that the model defined in this
paper disregards: rotation, instabilities and the Hawking radiation. In this
last part of the paper we briefly comment on their possible effect.

To account for rotation, we must study the Kerr-Newman metric. As already
noticed, the Carter-Penrose maximal extension of the Kerr-Newman metric is
similar to the one of the maximal extension of the Reissner-Nordström metric.
It is therefore reasonable to expect that some qualitative aspects of the
black to white bounce extend to the rotating case. This will be done
elsewhere.

Instabilities are expected before the Cauchy horizon [35, 36]. There is a
simple argument that illustrates why. Consider a sequence of pulses emitted
radially at regular time intervals from past infinity. They all reach an
observer moving towards the Cauchy horizon in a finite proper time. Thus, if
perturbations enter the black hole for an arbitrary long external time, they
all pile up in the finite time of the observer before the Cauchy horizon. In
turn, this is likely to cause a concentration of energy near the Cauchy
horizon. This is going to generate a strong curvature, which at some point may
become Planckian.

Poisson and Israel [26, 27] have in fact shown that small perturbations of the
metric outside the black hole during or after the star collapsing become
unstable once they penetrate inside the black hole and lead the Cauchy horizon
to become a null curvature singularity. This phenomenon is known as mass
inflation and is confirmed by numerical investigations [37, 38, 39, 40, 41,
42]. Furthermore, physical and numerical indications point to the expectation
that the perturbation grows into a _spacelike_ singularity in the classical
theory.

Before diverging the curvature must become Planckian. Hence, the region along
the Cauchy horizon is situated inside a quantum gravity region. This result
strengthen the hypothesis that the spacetime dynamics enters a quantum region
before the development of the Cauchy horizon, as in the hypothesis of this
paper, and no Cauchy horizon develops in reality. Quantum gravity should
correct not only the singularities but also the lack of global hyperbolicity
of classical general relativity (see also [41, 42].).

A quick estimate of where the instability drives the spacetime dynamics into
the quantum region can be obtained as follows. Near the inner horizon, the
Reissner-Nordström metric can be written as (see [31])

$ds^{2}\simeq-2e^{-\kappa_{-}v}e^{\kappa_{-}u}dudv+r^{2}d\Omega^{2}$ (28)

where $\kappa_{-}=\frac{1}{2}f^{\prime}(r_{-})$ is the surface gravity on the
inner horizon. The proper time interval $d\tau$ between the reception of
signals by an observer approaching the Cauchy horizon along a (timelike)
constant $u+v=2t$ trajectory inside the black hole is related to the proper
time interval $dv$ of emission of these signals from infinity by

$d\tau\simeq\sqrt{2}\ e^{\kappa_{-}t}e^{-\kappa_{-}v}dv$ (29)

approaching the Cauchy horizon ($v\to\infty$), giving a blue shift growing
exponentially as

$\frac{d\tau}{dv}\sim e^{-\kappa_{-}v}$ (30)

This blue shift is expected to hold whatever is the observer motion across the
inner horizon [38]. According to [37, 38], the perturbation generated by this
instability gives rise, for $v\to\infty$, to a curvature of the order of

$K^{2}=g(u)\frac{e^{2\kappa_{-}v}}{v^{2q}}$ (31)

where $g(u)$ is a function that vanishes at the outer horizon and increases
monotonically and $q$ is a positive integer depending on the characteristics
of the in-falling perturbations. Since $v$ is large, the term $v^{2q}$ can be
neglected and we have

$v\simeq-\frac{\log{g(u)}}{2\kappa_{-}}$ (32)

So we may expect that the spacetime enters the quantum region just because of
the instabilities. Since this formula is only reliable in the large $v$ limit,
is not clear to us whether this can happen before the onset of the quantum
regime studied above.

But notice that in any case these instabilities happens only in the interior
of the black hole as $g(u)$ vanishes on the outer boundary. The black to white
transition requires that the region affected by quantum gravity leaks outside
the horizon. Without this, the apparent horizon becomes an event horizon and
the black hole remains such forever, if observed from the exterior.

As Reissner-Nordström and Kerr share the same causal diagram, the existence of
the infinite blue shift and the consequences discussed above, i.e the
instability the (Kerr) metric, the null weak singularity along the Cauchy
horizon, and its shrinking up to a spacelike singularity at $r=0$, are also
present for the rotating black hole and the charged and rotating Kerr-Newman
black hole.

Notice that there is also the possibility that the quantum gravity region
generated by the instabilities extend all the way to the trajectory of the
bouncing star, in which case the physics would be more similar to the one
studied in [1]. See Figure 10.

Figure 10: The Carter-Penrose diagram for the black to white transition of the
charged black hole when the the quantum gravity region (in green) extends up
to the entire internal region. The difference with Figure 7 is that here the
star encounters a quantum gravity region, and the white hole region is not
classically connected to the black hole region anymore.

##  VI Hawking evaporation

In the Schwarzschild case, the mass of the hole decreases because of the
Hawking Radiation (HR) falling into the hole. What about its charge ?

One may speculate that the HR is dominated by massless particles that can
escape to infinity more easily. Massless particles in the standard model are
not charged, so they may not carry charge away and diminish the black hole
charge $Q$. The same is not true for the rotating case, because photons can
carry angular momentum, although the HR might still be dominated by $s$-waves.
According to [43] charged black holes do not necessarily evolve towards the
Schwarzschild limit contrary to Kerr black holes. If they are massive enough
and if their charge is large enough ($\frac{3}{4}<\eta^{2}<1$ according to
[43]), they evolve toward the extremal limit ($\eta$ = 1). If so, the ratio
$\eta=Q/M$ increases at the end of the Hawking evaporation, making the
Reissner-Nordsrtöm model more realistic than the Schwarzschild model for the
transition.

In the absence of Hawking evaporation, the black to white transition might
take exponentially long times to happen. The Hawking evaporation shrinks the
mass, making it increasingly more probable, hence likely to happen within a
time $M_{0}^{3}$. But part of the HR falls into the black hole, affecting the
internal metric. Some consequences of the backreaction of the HR on the bounce
have been explored in [8]. In the rest of this section we present some sketchy
considerations on the possible effect of this back-reaction in the charged
case. These are speculative, because not much is solidly known about the
energy-momentum tensor of the HR falling inside the hole and its back-
reaction. For simplicity, we assume here that the ratio $\eta=\frac{Q}{M}$
remains constant. If so, both the outer $r_{+}=M(1+\sqrt{1-\eta^{2}})$ and
inner $r_{-}=M(1-\sqrt{1-\eta^{2}})$ horizons decrease.

Figure 11: A pair of particles is created just outside the horizon (red
lines). One, with mass $+\delta m$ and charge $+\delta q$ escapes to null
infinity. The other, with mass $-\delta m$ and charge $-\delta q$ falls into
the black hole along a constant $v$ null line. The infalling particle modifies
the metric inside the black hole in its future, producing a Reissner-Nordström
metric of a black hole of mass $M-\delta m$ and charge $Q-\delta q$. Both
inner and outer horizons shrink. (Here and in the next figure depicted without
the gluing of the asymptotic regions.

There are two simple models of the HR falling inside the hole. The first is
that it is made by negative energy quanta moving along ingoing null geodesics.
The second is that it is made by negative energy quanta moving along outgoing
null geodesics (the radius of which is still decreasing, since it is a trapped
region). Reality is probably between the two [44, 8, 45].

Consider an Hawking quantum of mass $-\delta m$ and charge $-\delta q$ in-
falling along a constant $v$ null line. A simple estimate of its backreaction
is to modify the geometry by shifting the Reissner-Nordström mass and charges
by these amounts in its future. Notice that this shifts the two horizons
inward. Let $r_{-}^{\prime}$ be the radius of the new inner horizon. See
Figure 11.

Call $c$ the point where the in-falling quantum meets the bouncing star and
$r_{c}$ the radius of this point. There are two possibilities. If
$r_{-}^{\prime}>r_{c}$, the negative energy quanta crosses the new inner
horizon $r_{-}^{\prime}$ before reaching the star. This does not change the
overall picture much.

However, it might also happen that $r_{-}^{\prime}<r_{c}$, namely the negative
energy in-falling quanta meets the bouncing star _before_ entering the new
inner horizon. If so, $c$ is in a trapped region hence the star has to fall
again, until it crosses new inner horizon and be allowed to bounce at a point
$b^{\prime}$ with radius $r_{b^{\prime}}<r_{b}$. This process might disrupt
the classical throat. In particular, notice that if $r_{P}$ is near Planckian,
so must be the outer and inner horizons at $v_{P}$, and hence so must be the
corresponding $r_{c}$, which implies that the star itself must have entered a
quantum regime. See Figure 12. While in the non charged case the infalling HR
always encounters the star in the antitrapped region, in the charged case it
is the HR that may make its charge and mass decrease to Planckian values
before the final bounce. Our control of the energy momentum tensor of the HR
and its back-reaction is still insufficient to judge if this is truly the
case.

Figure 12: The Hawking radiation shifts the inner horizon for each quanta. A
shell (grey line) falls again and bounces at increasingly smaller radius. The
vertically hatched area is trapped, the horizontally hatched area anti-
trapped. Radiation continues to be emitted outside until the point $P$ outside
the trapped region becomes planckian

An argument can be given, suggesting that the onset of the quantum gravity
region may be spacelike, especially for small $\eta$ (on this, see also [46]).
As ingoing HR falls into the hole, it carries negative energy and therefore
decreases the local mass of the geometry.

We may expect the metric to be a Reissner-Nordström metric locally but with a
decreasing mass function $M(v)$. On the inner horizon, the curvature of a
given slice (from eq. (24)) is

$K^{2}=48\frac{\bigg{(}1-\frac{\eta^{2}}{1-\sqrt{1-\eta^{2}}}\bigg{)}^{2}}{M(v)^{4}(1-\sqrt{1-\eta^{2}})^{6}}\simeq\frac{3072}{M(v)^{4}\eta^{12}}$
(33)

where the last approximation holds if $\eta$ is small. This indicates when the
inner horizon enters a region of planckian curvature. Let $S_{in}$ be the
point where the curvature becomes planckian on the inner horizon and $v_{in}$
its advanced time. This is the innermost point included in the quantum region.
The curvature at the _outer_ horizon $r_{+}=M(v)(1+\sqrt{1-\eta^{2}})$ is

$K^{2}=48\frac{\bigg{(}1-\frac{\eta^{2}}{1+\sqrt{1-\eta^{2}}}\bigg{)}^{2}}{M(v)^{4}(1+\sqrt{1-\eta^{2}})^{6}}\simeq\frac{3}{4}\frac{1}{M(v)^{4}}$
(34)

where again the last approximation holds if $\eta$ is small. Comparing (33)
and (34) for $K^{2}\sim 1$,

$\frac{M(v_{in})}{M(v_{P})}=\frac{8}{\eta^{3}}>1$ (35)

Hence, $v_{{in}}<v_{P}$ as $M$ is a decreasing function of $v$ and so the line
joining the points $P$ and $S_{in}$ is spacelike. The line joining $P$ and
$S_{in}$ can be taken as the boundary of the quantum region. This picture has
similarities with the Schwarzschild black to white transition.

Finally, outgoing HR quanta inside the hole, on the other hand, fall into the
quantum region. The negative energy they carry can emerge into the white hole
and decrease the mass and charge of the star inside the white hole and the
classical throat is not destroyed. See [8, 47]. The same may happen if the
Hawking quanta are massive (see Figure 13.)

Figure 13: Massive Hawking radiation quanta enter the quantum region, and make
the inner (and outer) horizon shrinks. (Depicted without the gluing of the two
asymptotic regions.)

##  VII Conclusion

The presence of charge renders the black to white transition more interesting
than the non charged case. The bounce of the star and the region immediately
surrounding it evolve into a white hole simply by following the classical
dynamics. Only the horizon region tunnels. The white hole is in the same
location of the same asymptotic region as the black hole that originates it.

The extension of the spacetime region surrounding the star that evolves into a
white hole classically depends on the charge: in the limit of vanishing
charge, the situation is similar to the Schwarzschild tunnelling.

We have also tentatively explored the effects of the classical instabilities
and the back-reaction of the HR on the process. These may alter the picture of
the interior, creating a spacelike onset of the quantum region (as in the
Schwarzschild case) and decreasing the range of the classical region. But they
do not seem to alter the basic possibility of black to white hole quantum
tunnelling.

The qualitative similarities of the Reissner-Nordström and Kerr-Newman metrics
suggest that the entire black to white hole quantum tunnelling may be a
general possibility, and therefore represent a likely scenario for the fate of
all real black holes.

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$\hat{\nabla}$, in the basis $\mathfrak{B}_{T\widetilde{Q}}$ and
$\mathfrak{B}_{\mathfrak{g}_{S}}$, are given by

$\left(\\!\\!\begin{array}[]{c}\hat{\Gamma}_{01}^{1}\\\
\hat{\Gamma}_{01}^{2}\end{array}\\!\\!\right)=\frac{1}{2}[\kappa|_{S}]^{-1}\\!\\!\left(\\!\\!\\!\\!\begin{array}[]{c}\kappa^{\prime}_{11}(1-\gamma_{3}^{2})\\\
(\kappa^{\prime}_{12}+m\varrho
B)(1-\gamma_{3}^{2})-H_{21}\end{array}\\!\\!\\!\\!\right)\mbox{ \ and \
}\left(\\!\\!\begin{array}[]{c}\hat{\Gamma}_{02}^{1}\\\
\hat{\Gamma}_{02}^{2}\end{array}\\!\\!\right)=\frac{1}{2}[\kappa|_{S}]^{-1}\\!\\!\left(\\!\\!\\!\\!\begin{array}[]{c}(\kappa^{\prime}_{12}+mAL)(1-\gamma_{3}^{2})-H_{12}\\\
\kappa^{\prime}_{22}(1-\gamma_{3}^{2})\end{array}\\!\\!\\!\\!\right),$

and $\hat{\Gamma}_{ij}^{1}=\hat{\Gamma}_{ij}^{2}=0$. Following Def. 3.20, the
bilinear form $\Sigma=\Sigma^{1}\otimes\xi_{1}+\Sigma^{2}\otimes\xi_{2}$ is
given by

$\Sigma^{1}=-(\hat{\Gamma}_{0j}^{1}+R_{1j})X^{0}\wedge
Y^{j}\quad\mbox{and}\quad\Sigma^{2}=-(\hat{\Gamma}_{0j}^{2}+R_{2j})X^{0}\wedge
Y^{j},$

where the functions $R_{ij}$ are given in (5.43). Then, the horizontal gauge
symmetries can be seen as parallel sections along the dynamics with respect to
the $\Sigma$-connection:

$\overset{\textit{\tiny{$\Sigma$}}}{\nabla}_{\dot{\gamma}}\zeta=0.$

###  5.3 A homogeneous ball on a surface of revolution

Let us consider the holonomic system formed by a homogeneous sphere of mass
${\bf m}$ and radius $r>0$, which center $C$ is constrained to belong to a
convex surface of revolution $\Sigma$ (i.e., the ball rolls on the surface
$\tilde{\Sigma}$, see Figure 3). The surface $\Sigma$ is obtained by rotating
about the $z$-axis the graph of a convex and smooth function
$\phi:\mathbb{R}_{+}\longrightarrow\mathbb{R}$. Thus, $\Sigma$ is described by
the equation $z=\phi(x^{2}+y^{2})$. To guarantee smoothness and convexity of
the surface, we assume that $\phi$ verifies that $\phi^{\prime}(0^{+})=0$,
$\phi^{\prime}(s)>0$ and $\phi^{\prime\prime}(s)>0$, when $s>0$. To ensure
that the ball has only one contact point with the surface we ask the curvature
of $\phi(s)$ to be at most 1/r. The configuration manifold $Q$ is
$\mathbb{R}^{2}\times SO(3)$ with coordinates $(x,y,g)$ where $G$ is the
orthogonal matrix fixing the attitude of the sphere and $(x,y)$ are the
coordinates of $C$ with respect to a reference frame with origin $O$ and
$z$-axis coinciding with the figure axis of $\Sigma$.

$\widetilde{\Sigma}$$\Sigma$$x$$y$$z$$O$$C$$\vec{n}$$P$$\bf g$

Figure 3: The homogeneous ball on a convex surface of revolution.

Let us denote by $n=n(x,y)$ the outward normal unit vector to $\Sigma$ with
components $(n_{1},n_{2},n_{3})$ given by

$\frac{n_{1}}{n_{3}}=2x\phi^{\prime},\quad\frac{n_{2}}{n_{3}}=2y\phi^{\prime}\quad\mbox{and}\quad
n_{3}=-\frac{1}{\sqrt{1+4(x^{2}+y^{2})(\phi^{\prime})^{2}}}.$

If $\omega=(\omega_{1},\omega_{2},\omega_{3})$ is the angular velocity of the
ball in the space frame, then the Lagrangian of the holonomic system on $TQ$
is

$L(x,y,g,\dot{x},\dot{y},\omega)=\frac{{\bf
m}}{2n_{3}^{2}}\left((1-n_{2}^{2})\dot{x}^{2}+2n_{1}n_{2}\,\dot{x}\dot{y}+\dot{y}^{2}(1-n_{1}^{2})\right)+\frac{1}{2}\langle\mathbb{I}\omega,\omega\rangle-{\bf
m}{\bf g}\phi\,,$ (5.46)

where ${\bf g}$ denotes the gravity acceleration and $\mathbb{I}$ the moment
of inertia of the sphere with respect to its center of mass.

Geometry of the constrained system. The ball rotates without sliding on the
surface $\widetilde{\Sigma}$, and hence the nonholonomic constraints equations
are

$\dot{x}=-r\left(\omega_{2}n_{3}-\omega_{3}n_{2}\right)\,,\qquad\dot{y}=-r\left(\omega_{3}n_{1}-\omega_{1}n_{3}\right).$

We denote by $\\{X_{1}^{R},X_{2}^{R},X_{3}^{R}\\}$ the right invariant vector
fields on $SO(3)$ and by $\\{\rho_{1},\rho_{2},\rho_{3}\\}$ the right Maurer-
Cartan 1-forms, that form a basis of $T^{*}SO(3)$ dual to
$\\{X_{1}^{R},X_{2}^{R},X_{3}^{R}\\}$. Then the constraint 1-forms are given
by

$\epsilon^{1}:=dx-r\left(n_{2}\rho_{3}-n_{3}\rho_{2}\right)\,,\qquad\epsilon^{2}:=dy-r\left(n_{3}\rho_{1}-n_{1}\rho_{3}\right)\,.$

The constraint distribution $D$ defined by the annihilator of $\epsilon^{1}$
and $\epsilon^{2}$ has fiber, at $q=(x,y,g)$, given by

$D_{q}=\textrm{span}\left\\{Y_{x}:=\partial_{x}-\frac{1}{rn_{3}}(n_{2}X_{n}-X_{2}^{R}),\
Y_{y}:=\partial_{y}+\frac{1}{rn_{3}}(n_{1}X_{n}-X_{1}^{R}),\
X_{n}\right\\}\,,$ (5.47)

where $X_{n}:=n_{1}\,X_{1}^{R}+n_{2}\,X_{2}^{R}+n_{3}\,X_{3}^{R}$. Consider
the basis of $TQ$

${\bf B}_{TQ}=\left\\{Y_{x},Y_{y},X_{n},Z_{1},Z_{2}\right\\}\,,$ (5.48)

where $Z_{1}:=\frac{1}{rn_{3}}X_{2}^{R}-\frac{n_{2}}{rn_{3}}X_{n}$ and
$Z_{2}:=-\frac{1}{rn_{3}}X_{1}^{R}+\frac{n_{1}}{rn_{2}}X_{n}$ with associated
coordinates $(\dot{x},\dot{y},\omega_{n},w^{1},w^{2})$, for
$\omega_{n}=n\cdot\omega=n_{i}\omega_{i}$, the normal component of the angular
velocity $\omega$. The dual frame of (5.48) is

${\bf
B}_{T^{*}Q}=\left\\{dx,dy,\rho_{n},\epsilon^{1},\epsilon^{2},\right\\}\,,$
(5.49)

where $\rho_{n}=n_{i}\rho_{i}$, with associated coordinates
$(p_{x},p_{y},p_{n},M_{1},M_{2})$ on $T^{*}Q$. The manifold
$\mathcal{M}=\kappa^{\sharp}(D)$ is given by

$\mathcal{M}=\left\\{(x,y,g;p_{x},p_{y},p_{n},M_{1},M_{2})\ :\
M_{1}=\tfrac{-I}{I+mr^{2}}p_{x},\ M_{2}=\tfrac{-I}{I+mr^{2}}p_{y}\right\\}.$

The symmetries. Consider the action $\Psi$ of the Lie group $G=SO(2)\times
SO(3)$ on the manifold $Q$ given, at each $(x,y,g)\in Q$ and
$(h_{\theta},h)\in SO(2)\times SO(3)$, by

$\Psi_{(h_{\theta},h)}(x,y,g)=(h_{\theta}(x,y)^{t},\tilde{h}_{\theta}gh),$

where $\tilde{h}_{\theta}$ is the $3\times 3$ rotational matrix of angle
$\theta$ with respect to the $z$-axis. In other words, $SO(3)$ acts on the
right on itself and $SO(2)$ acts by rotations about the figure axis of the
surface $\Sigma$. The Lagrangian (5.46) and the constraints (5.47) are
invariant with respect to the lift of this action to $TQ$ given by
$\Psi_{(h_{\theta},h)}(x,y,g,\dot{x},\dot{y},\omega)=(h_{\theta}(x,y)^{t},\tilde{h}_{\theta}gh,h_{\theta}(\dot{x},\dot{y})^{t},\omega)$.
The invariance of the kinetic energy and the constraints $D$ ensures that
$\Psi$ restricts to an action on $\mathcal{M}$, that leaves the equations of
motion invariant.

The Lie algebra $\mathfrak{g}$ of $G$ is isomorphic to
$\mathbb{R}\times\mathbb{R}^{3}$ with the infinitesimal generators

$(1;{\bf
0})_{Q}=-y\partial_{x}+x\partial_{y}+X^{R}_{3}\quad\mbox{and}\quad(0;{\bf
e}_{i})_{Q}=\alpha_{i}\,X_{1}^{R}+\beta_{i}\,X_{2}^{R}+\gamma_{i}\,X_{3}^{R},\textrm{
for }i=1,2,3,$

where ${\bf e}_{i}$ denotes the $i$-th element of the canonical basis of
$\mathbb{R}^{3}$ and, $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$,
$\beta=(\beta_{1},\beta_{2},\beta_{3})$
$\gamma=(\gamma_{1},\gamma_{2},\gamma_{3})$ the rows of the matrix $g\in
SO(3)$. Observe that $(1;{\bf 0})_{Q}$ is an infinitesimal generator of the
$SO(2)$-action and the others are infinitesimal generators of the
$SO(3)$-action. We then underline that the $G$-symmetry satisfies the
dimension assumption and it is proper and free whenever $(x,y)\neq(0,0)$ (note
that the rank of $V$ is 3 for $(x,y)=(0,0)$ and it is 4 elsewhere, showing
that the action is not even locally free).

Let us denote by $\widetilde{Q}\subset Q$ and
$\widetilde{\mathcal{M}}\subset\mathcal{M}$ the manifolds where the $G$-action
is free, i.e. $(x,y)\neq(0,0)$. The vertical distribution $S=D\cap V$ on
$\widetilde{Q}$ has rank 2 with fibers

$S_{q}=\textrm{span}\\{Y_{1}:=-yY_{x}+xY_{y},Y_{2}:=X_{n}\\}.$

The bundle $\mathfrak{g}_{S}\rightarrow Q$ has a global basis
$\mathfrak{B}_{\mathfrak{g}_{S}}$ of sections given by

$\mathfrak{B}_{\mathfrak{g}_{S}}=\left\\{\xi_{1}:=\left(1;\frac{x}{r\,n_{3}},\frac{y}{r\,n_{3}},0\right),\xi_{2}:=(0;n\,g)\right\\}$

and we check that $(\xi_{1})_{Q}=Y_{1}$ and $(\xi_{2})_{Q}=Y_{2}$. Finally we
observe that $\widetilde{Q}/G$ has dimension 1
($\rho_{\tilde{Q}}:\tilde{Q}\to\tilde{Q}/G$ is given by
$\rho_{Q}(x,y,g)=x^{2}+y^{2}$) and hence the $G$-symmetry satisfies Conditions
$(\mathcal{A}1)$-$(\mathcal{A}4)$ on $\widetilde{Q}$.

The existence of horizontal gauge momenta. Using the basis (5.48) and the
definition of $S$, we consider the decomposition

$T\widetilde{Q}=H\oplus S\oplus W,$

where $W$ is a vertical complement of the constraints given by
$W:=\textrm{span}\\{Z_{1},Z_{2}\\}$ and $H:=S^{\perp}\cap D$ is generated by
$X_{0}:=xY_{x}+yY_{y}$. As in Example 5.2, in this case, it is enough (and
straightforward using that $n_{3}(x,y)$ is rotational invariant and that
$[X_{1}^{R},X_{2}^{R}]=-X_{3}^{R}$ for all cyclic permutations) to check that
$\kappa([Y_{1},Y_{2}],Y_{1})=0$ and $\kappa([Y_{1},Y_{2}],Y_{2})=0$ to
guarantee that the kinetic energy is strong invariant on $S$. Finally, we also
see that $\kappa(X_{0},[Y_{i},X_{0}])=0$, for $i=1,2$ . Therefore, following
Theorem 3.15, the system admits two $G$-invariant (functionally independent)
horizontal gauge momenta ${\mathcal{J}}_{1}$ and ${\mathcal{J}}_{2}$, showing
that the first integrals obtained in [52, 38, 59, 16, 27] can be obtained from
the symmetry of the system as horizontal gauge momenta.

The computation of the 2 horizontal gauge momenta. We now characterize the
coordinate functions of the horizontal gauge symmetries written in the basis
$\mathfrak{B}_{\mathfrak{g}_{S}}$ on $\widetilde{Q}$. That is, let us denote
by

$J_{1}:={\bf i}_{Y_{1}}\Theta=-yp_{x}+xp_{y}\qquad\mbox{and}\qquad J_{2}:={\bf
i}_{Y_{2}}\Theta=p_{n}.$

Using the orbit projection $\rho_{\tilde{Q}}:\tilde{Q}\to\tilde{Q}/G$, a
$G$-invariant function $f$ on $Q$ can be thought as depending on the variable
$\tau=x^{2}+y^{2}$, i.e., $f=f(\tau)$. Following Theorem 3.15$(ii)$, a
function $\mathcal{J}=f_{1}J_{1}+f_{2}J_{2}$ for $f_{1},f_{2}\in
C^{\infty}(Q)^{G}$ is a horizontal gauge momenta if and only if
$(f_{1},f_{2})$ is a solution of the linear system of ordinary differential
equations on $\widetilde{Q}/G$,

$R\left(\\!\\!\begin{array}[]{c}f_{1}\\\
f_{2}\end{array}\\!\\!\right)=\left(\\!\\!\begin{array}[]{c}\bar{X}_{0}(f_{1})\\\
\bar{X}_{0}(f_{2})\end{array}\\!\\!\right)\quad\mbox{where}\quad
R=2\tau\left(\begin{array}[]{cc}0&-2\frac{rI}{E}n_{3}^{2}(2(\phi^{\prime})^{3}-\phi^{\prime\prime})\\\
\tfrac{A}{r}n_{3}^{2}&0\end{array}\\!\\!\right)$ (5.50)

for $A=\phi^{\prime}+2\tau\phi^{\prime\prime}$ and
$\bar{X}_{0}=T\rho_{\widetilde{Q}}(X_{0})=2\tau\tfrac{\partial}{\partial\tau}$.
The matrix $R$ is computed using that $R=[\kappa|_{S}]^{-1}[N]$ where

$[N]=\frac{2I}{r}\tau\left(\\!\\!\begin{array}[]{cc}0&-2\tau
n_{3}^{2}(2(\phi^{\prime})^{3}-\phi^{\prime\prime})\\\
An_{3}^{2}&0\end{array}\\!\\!\right)\quad\mbox{and}\quad[\kappa|_{S}]=\left(\\!\\!\begin{array}[]{cc}\tfrac{E}{r^{2}}\tau&0\\\
0&I\end{array}\\!\\!\right).$

Since this system admits two independent solutions
$f^{1}=(f_{1}^{1},f_{2}^{2})$ and $f^{2}=(f_{1}^{2},f_{2}^{2})$ on
$\widetilde{Q}/G$, then the nonholonomic system admits two $G$-invariant
horizontal gauge momenta $\mathcal{J}_{1}$, $\mathcal{J}_{2}$ defined on
$\widetilde{\mathcal{M}}$ of the form

$\mathcal{J}_{1}=f_{1}^{1}J_{1}+f_{1}^{2}J_{2}\qquad\mbox{and}\qquad\mathcal{J}_{2}=f_{1}^{2}J_{1}+f_{2}^{2}J_{2}.$
(5.51)

recalling that $J_{1}=-yp_{x}+xp_{y}$ and $J_{2}=p_{n}$

######  Remark 5.5.

Let us denote by $\bar{\mathcal{J}}_{1}$, $\bar{\mathcal{J}}_{2}$ the
functions on $\widetilde{\mathcal{M}}/G$ associated to (5.51).

  * $(i)$

The (reduced) first integrals $\bar{\mathcal{J}}_{1}$, $\bar{\mathcal{J}}_{2}$
can be extended by continuity to the differential space $\mathcal{M}/G$ and
thus $\mathcal{J}_{1}$, $\mathcal{J}_{2}$ are $G$-invariant functions on
$\mathcal{M}$ (see [27] for details) and in this case we say that the system
admits $2=\textup{rank}(\mathfrak{g}_{S})$ horizontal gauge momenta.

  * $(ii)$

The system of differential equations (5.50) can be written as

$R_{1}f_{2}=f_{1}^{\prime}\qquad\mbox{and}\qquad R_{2}f_{1}=f_{2}^{\prime},$

where
$R_{1}=R_{1}(\tau)=-2\tfrac{rI}{E}n_{3}^{2}(2(\phi^{\prime})^{3}-\phi^{\prime\prime})$
and $R_{2}=R_{2}(\tau)=\tfrac{A}{r}n_{3}^{2}$. Hence $\bar{\mathcal{J}}_{1}$,
$\bar{\mathcal{J}}_{2}$ are first integrals of Routh type found in [38] (see
also [23, 59, 16, 54]) and shown to be horizontal gauge momenta in [27, 29].

$\diamond$

Integrability and reconstruction. The reduced integrability of this system was
established in [52] and its complete broad integrability has been extensively
studied in [38, 59, 16, 27, 26], using the existence of first integrals
$\mathcal{J}_{1}$ and $\mathcal{J}_{2}$, without relating their existence to
the symmetry group. The symmetry origin of $\mathcal{J}_{1}$ and
$\mathcal{J}_{2}$ was announced in [9], and then proved in [54, 29]. Here we
want to stress how Theorem 3.15 can be applied and therefore the reduced
integrability of the system is ensured. That is, $\bar{\mathcal{J}}_{1}$,
$\bar{\mathcal{J}}_{2}$, $H_{\mbox{\tiny{red}}}$ are first integrals of the
reduced dynamics $X_{\mbox{\tiny{red}}}$ defined on the manifold
$\tilde{\mathcal{M}}/G$ of dimension 4. Moreover, as proved in [38, 59] the
reduced dynamics is made of periodic motions or of equilibria, and hence,
since the symmetry group is compact, the complete dynamics is generically
quasi-periodic on tori of dimension 3 (see Theorem 4.12 and [38, 26]). Indeed
one could can say more on the geometric structure of the phase space
$\widetilde{\mathcal{M}}$ of the complete system, it is endowed with the
structure of a fibration on tori of dimension at most 3 (see [26] for a
detailed study of the geometry of the complete system on
$\widetilde{\mathcal{M}}$).

Hamiltonization. Even though the hamiltonization of this example has been
studied in [8], in this section we see the hamiltonization as a consequence of
Theorem 3.15 and how the resulting Poisson bracket on
$\widetilde{\mathcal{M}}/G$ depends on the linear system of ordinary
differential equations (5.50).

By Theorem 4.5, the nonholonomic system is hamiltonizable by a gauge
transformation; that is, on $\widetilde{\mathcal{M}}/G$ the reduced
nonholonomic system is described by a Poisson bracket with 2-dimensional
leaves given by the common level sets of the horizontal gauge momenta
$\bar{\mathcal{J}}_{1}$, $\bar{\mathcal{J}}_{2}$, induced by (5.51), (recall
that $\bar{\mathcal{J}}_{i}$ are the functions on $\widetilde{\mathcal{M}}/G$,
such that $\rho^{*}(\bar{\mathcal{J}}_{i})={\mathcal{J}}_{i}$).

Following Theorem 4.10, we compute the 2-form $B_{\mbox{\tiny{HGS}}}$,
defining the dynamical gauge transformation, using the momentum equation
(5.50). Since $dY^{1}|_{D}=0$, then

$B_{\mbox{\tiny{HGS}}}:=\langle
J,\mathcal{K}_{\mbox{\tiny{$\mathcal{W}$}}}\rangle-
p_{1}R_{12}{\mathcal{X}}^{0}\wedge{\mathcal{Y}}^{2}+p_{2}R_{21}X^{0}\wedge{\mathcal{Y}}^{1}+p_{2}d\mathcal{Y}^{2},$
(5.52)

where $\mathcal{X}^{0}=\tau_{\mbox{\tiny{$\mathcal{M}$}}}^{*}X^{0}$ and
$\mathcal{Y}^{i}=\tau_{\mbox{\tiny{$\mathcal{M}$}}}^{*}Y^{i}$. That is,

$\begin{split}\langle
J,\mathcal{K}_{\mbox{\tiny{$\mathcal{W}$}}}\rangle|_{\mathcal{C}}&=M_{1}\,d\epsilon^{1}|_{\mathcal{C}}+M_{2}\,d\epsilon^{2}|_{\mathcal{C}},\\\
&=-\frac{Ir}{E(x^{2}+y^{2})}\left(p_{1}(\tfrac{1}{rn_{3}^{2}}+2n_{3}^{2}A){\mathcal{X}}^{0}\wedge{\mathcal{Y}}^{2}+p_{0}n_{3}(2\phi^{\prime}n_{3}+\tfrac{1}{r}{\mathcal{Y}}^{1}\wedge{\mathcal{Y}}^{2}\right)|_{\mathcal{C}},\end{split}$

and using that
$d{\mathcal{Y}}^{2}|_{\mathcal{C}}=\frac{(x^{2}+y^{2})}{n_{3}}p_{2}X^{0}\wedge{\mathcal{Y}}^{1}|_{\mathcal{C}}$
we obtain

$B_{\mbox{\tiny{HGS}}}=(x^{2}+y^{2})p_{2}(\tfrac{1}{n_{3}}+2\tfrac{A}{r}n_{3}^{2}){\mathcal{X}}^{0}\wedge{\mathcal{Y}}^{1}+\tfrac{rI}{E}(\tfrac{1}{rn_{3}}+2\phi^{\prime})(p_{1}{\mathcal{X}}^{0}\wedge{\mathcal{Y}}^{2}-p_{0}n_{3}^{2}{\mathcal{Y}}^{1}\wedge{\mathcal{Y}}^{2}).$
(5.53)

######  Remark 5.6.

Since the action is not free, $\mathcal{M}/G$ is a semialgebraic variety that
consists in two strata: a singular 1-dimensional stratum corresponding to the
points in which the action is not free; and the four dimensional regular
stratum $\widetilde{\mathcal{M}}/G$ (where the action is free). Moreover,
analyzing the change of coordinates between ${\bf B}_{T^{*}Q}$ and
${\mathfrak{B}}_{T^{*}Q}$ we get

$\tau=x^{2}+y^{2},\ p_{0}=xp_{x}+yp_{y},\ p_{1}=-yp_{x}+xp_{y},\ p_{2}=p_{n},$

and adding $p_{3}=p_{x}^{2}+p_{y}^{2}$ we recover the coordinates used in [38,
27] on $\widetilde{\mathcal{M}}/G$. $\diamond$

######  Remark 5.7.

Since the convexity of the function $\phi$ that parametrizes the surface
$\Sigma$ is not strictly used, this example also describes the geometry and
dynamics of a homogeneous ball rolling on surface of revolution such that its
normal vector fields has $n_{3}\neq 0$. $\diamond$

###  5.4 Comments on the hypothesis of Theorem 3.15: examples and
counterexamples

Theorem 3.15 shows that a nonholonomic system with symmetries satisfying
certain hypotheses admits the existence of $k$ functionally independent
$G$-invariant horizontal gauge momenta. Next, assuming Conditions
$(\mathcal{A}1)$-$(\mathcal{A}3)$, we study what may happen if the other
hypotheses of Theorem 3.15 are not satisfied. In particular we study three
cases: when the metric is not strong invariant, when $\kappa(X_{0},[X_{0},Y])$
is different from zero, and finally when Condition $(\mathcal{A}4)$ is not
verified (i.e., $\textup{dim}(Q/G)\neq 1$). For each case we give examples and
counterexamples to illustrate our conclusions.

#### Analyzing the strong invariance condition and $\kappa(X_{0},[X_{0},Y])=0$

Consider a nonholonomic system
$(\mathcal{M},\Omega_{\mbox{\tiny{$\mathcal{M}$}}}|_{\mathcal{C}},H_{\mbox{\tiny{$\mathcal{M}$}}})$
with a $G$-symmetry satisfying Conditions $(\mathcal{A}1)$-$(\mathcal{A}4)$.
Suppose that $(f_{1},...,f_{k})$ is a solution of the system of differential
equations (3.22), then, from (3.20), we observe that $\mathcal{J}=f_{j}J_{i}$
is a horizontal gauge momentum if and only if

$f_{i}\kappa(X_{0},[Y_{i},X_{0}])=0\quad\mbox{and}\quad
f_{i}(\kappa(Y_{j},[Y_{i},Y_{l}])+\kappa(Y_{l},[Y_{i},Y_{j}))=0,\mbox{ for
each }j,l.$

for a $S$-orthogonal horizontal space $H$. That is, in some cases, even if
$\kappa(X_{0},[X_{0},Y_{i_{0}}])\neq 0$ for some $Y_{i_{0}}\in\Gamma(S)$ or
the metric is not strong invariant, we may still have a horizontal gauge
momentum.

We now present two examples that show the main features of these phenomenon.

The metric is not strong invariant on $S$. The following is a mathematical
example, that has the property that the metric is not strong invariant, and it
admits only 1 horizontal gauge momenta even though the rank of the
distribution $S$ is 3. Precisely, consider the nonholonomic system on the
manifold $Q=\mathbb{R}^{3}\times SE(2)$ with coordinates
$(u,v,x)\in\mathbb{R}^{3}$ and $(y,z,\theta)\in SE(2)$ with Lagrangian given
by

$L(q,\dot{q})=\frac{1}{2}\left(u^{2}+v^{2}+\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}+\dot{\theta}^{2}+4(\sin\theta\,\dot{z}+\cos\theta\,\dot{y})\dot{\theta}\right),$

and constraints 1-forms given by

$\epsilon^{u}=du-(1+\cos x)d\theta\qquad\mbox{and}\qquad\epsilon^{v}=dv-\sin
xd\theta.$

The symmetry is given by the action of the Lie group $G=\mathbb{R}^{2}\times
SE(2)$ defined, at each $(a,b;c,d,\beta)\in G$, by

$\Psi((a,b;c,d,\beta),(u,v,x,y,z,\theta))=(u+a,v+b,x,h_{\beta}\left(\\!\begin{matrix}y\\\
z\end{matrix}\\!\right)+\left(\\!\begin{matrix}c\\\
d\end{matrix}\\!\right),\theta+\beta),$

where $h_{\beta}$ is the $2\times 2$ rotational matrix of angle $\beta$. The
distribution $S=D\cap V$ is generated by the $G$-invariant vector fields
$\\{Y_{\theta},Y_{1},Y_{2}\\}$ given by

$Y_{\theta}:=\partial_{\theta}+(1+\cos x)\partial_{u}+\sin
x\partial_{v},\,Y_{1}:=\cos\theta\partial_{y}+\sin\theta\partial_{z},\,Y_{2}:=-\sin\theta\partial_{y}+\cos\theta\partial_{z},$

and $X_{0}=\partial_{x}$ generates $H=S^{\perp}\cap D$. It is straightforward
to check that Conditions $(\mathcal{A}1)$-$(\mathcal{A}4)$ are satisfied and
that $\kappa(X_{0},[X_{0},Y])=0$ for all $Y\in\Gamma(S)$. However, the metric
is not strong invariant on $S$: $\kappa(Y_{2},[Y_{\theta},Y_{1}])=1$ and
$\kappa(Y_{\theta},[Y_{1},Y_{2}])=0$. From (3.20), we can observe that
$\mathcal{J}=2p_{1}+p_{\theta}$ is the only horizontal gauge momentum of the
system in spite of the rank of $S$ being 3 (where, as usual, $p_{1}={\bf
i}_{Y_{1}}\Theta_{\mbox{\tiny{$\mathcal{M}$}}}$ and $p_{\theta}={\bf
i}_{Y_{\theta}}\Theta_{\mbox{\tiny{$\mathcal{M}$}}}$).

Dropping condition $\kappa(X_{0},[X_{0},Y])=0$. We illustrate with a
multidimensional nonholonomic particle the different scenarios obtained when
$\kappa(X_{0},[X_{0},Y])\neq 0$ for a section $Y\in\Gamma(S)$ (see Table 5.4).

Consider the nonholonomic system on $\mathbb{R}^{5}$ with Lagrangian
$L(q,\dot{q})=\frac{1}{2}\dot{q}\cdot\kappa\,\dot{q}-V(x_{1})$, where $\kappa$
is the kinetic energy metric

$\kappa=\begin{pmatrix}1&0&1&0&1\\\ 0&1&0&0&0\\\ 1&0&1&0&0\\\ 0&0&0&1&1\\\
1&0&0&1&1\\\ \end{pmatrix},$

and with the nonintegrable distribution $D$ given, at each
$q=(x_{1},\ldots,x_{5})\in\mathbb{R}^{5}$, by

$\displaystyle D_{q}=\textrm{span}\\{$ $\displaystyle
D_{1}=f(x_{1})\,\partial_{x_{1}}+b(x_{1})\,\partial_{x_{3}}+c(x_{1})\,\partial_{x_{4}}\,,D_{2}=h(x_{1})\,\partial_{x_{1}}+g(x_{1})\,\partial_{x_{2}}\,,$
$\displaystyle
D_{3}=d(x_{1})\,\partial_{x_{1}}+j(x_{1})\,\partial_{x_{4}}+l(x_{1})\,\partial_{x_{5}}\\}\,,$

where
$b(x_{1}),c(x_{1}),d(x_{1}),f(x_{1}),g(x_{1}),h(x_{1}),j(x_{1}),l(x_{1})$ are
functions on $\mathbb{R}^{5}$ depending only on the coordinate $x_{1}$. The
group $\mathbb{R}^{4}$ of translations along the $x_{2}$, $x_{3}$, $x_{4}$ and
$x_{5}$ directions acts on the system and leaves both the Lagrangian and the
nonholonomic constraints invariant. It is straightforward to see that this
$G$-symmetry satisfies Conditions $(\mathcal{A}1)$-$(\mathcal{A}4)$. The fiber
of the distribution $S$ over $q\in Q$ is
$S_{q}=\textrm{span}\\{Y_{1}:=f(x_{1})D_{2}-h(x_{1})D_{1}\,,Y_{2}:=h(x_{1})D_{3}-d(x_{1})D_{2}\\}$.
Since the translational Lie group $\mathbb{R}^{4}$ is abelian then the kinetic
energy is strong invariant on $V$ (see Example 3.10). The distribution
$H=S^{\perp}\cap D$ is generated by the vector field
$X_{0}=\beta_{1}(x_{1})\,D_{1}+\beta_{2}(x_{1})\,D_{2}+\beta_{3}(x_{1})\,D_{3}$,
for $\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ suitable functions (defined on
$\mathbb{R}^{5}$ but depending only on the coordinate $x_{1}$).

For particular choices of the functions
$b(x_{1}),c(x_{1}),d(x_{1}),f(x_{1}),g(x_{1}),h(x_{1}),j(x_{1}),l(x_{1})$ the
two terms $\kappa(X_{0},[Y_{1},X_{0}])$ and $\kappa(X_{0},[Y_{2},X_{0}])$ may
not vanish. The computations and their expression are rather long and were
implemented with Mathematica. The next table shows different situations that
we obtain:

multidimensional nonholonomic particle ($\textup{rank}(S)=2$)
---
behaviour of $\kappa(X_{0},[X_{0},Y])$ |  $\sharp$ horizontal gauge momenta
$\kappa(X_{0},[Y_{1},X_{0}])=0$ and $\kappa(X_{0},[Y_{2},X_{0}])\neq 0$ | 0
$\kappa(X_{0},[Y_{1},X_{0}])=0$ and $\kappa(X_{0},[Y_{2},X_{0}])\neq 0$ | 1
$\kappa(X_{0},[Y_{1},X_{0}])\neq 0$ and $\kappa(X_{0},[Y_{2},X_{0}])\neq 0$ | 0

#### Cases when Condition $(\mathcal{A}4)$ is not satisfied (or
$\textup{rank}(H)\neq 1$)

When Condition $(\mathcal{A}4)$ is not verified, it is still possible to work
with the momentum equation stated in Proposition 3.3. Basically, for the case
when $\textup{rank}(H)=0$ we still have $\textup{rank}(S)$ horizontal gauge
momenta, while if $\textup{rank}(H)>1$ we cannot say anything.

If $\textup{rank}(H)=0$. In this case, $TQ=V$ which means that $Q\simeq G$.
That is, consider a nonholonomic system $(L,D)$ on a Lie group $G$ for which
the left action is a symmetry of the system. Since the only $G$-invariant
functions are constant, we need to check that, for a basis
$\mathfrak{B}_{\mathfrak{g}_{S}}=\\{\xi_{1},...,\xi_{k}\\}$ of
$\Gamma(\mathfrak{g}_{S})$, the momentum equation (3.17) is satisfied only for
constant functions $f_{i}=c_{i}$. In this case, since
$X_{\mbox{\tiny{nh}}}\in\Gamma(\mathcal{V})$, the coordinate momentum equation
(3.20), for $f\in C^{\infty}(Q)^{G}$, remains

$f_{i}v^{l}v^{j}\kappa(Y_{j},[Y_{i},Y_{l}])=0.$

The constant functions $f_{i}=c_{i}$ are $k$ (independent) solutions of the
momentum equation if and only if the kinetic energy is strong invariant on
$S$, and hence the sections of the basis $\mathfrak{B}_{\mathfrak{g}_{S}}$ are
horizontal gauge symmetries.

As illustrative examples, see the vertical disk and the Chaplygin sleigh in
[28] and [11] respectively.

If $\textup{rank}(H)>1$. In this case we cannot assert the existence of a
global basis of $H$. However, in some examples the horizontal space $H$ may
admit a global basis which we denoted by $\\{X_{1},...,X_{n}\\}$ for
$n=\textup{rank}(H)$. In this case, we observe that the second summand of the
momentum equation (3.20) gives the condition

$\kappa(X_{\alpha},[Y_{i},X_{\beta}])-\kappa(X_{\beta},[Y_{i},X_{\alpha}])=0\quad\mbox{for
all }\alpha,\beta=1,...,n$

and the third summand gives a system of partial differential equations whose
solutions induce the horizontal gauge momenta. As an illustrative example, we
can work out the Chaplygin ball [19, 24]: this example has a $G$-symmetry so
that $\textup{rank}(S)=1$ and $\textup{rank}(H)=2$ with a global basis (see
e.g. [36, 3]). However, working with the momentum equation (3.17), it is
possible to show that the system admits 1 horizontal gauge momentum,
recovering the known result in [19, 24, 15].

##  Appendix A Appendix: Almost Poisson brackets and gauge transformations

Almost Poisson brackets. An almost Poisson bracket on a manifold $M$ is a
bilinear bracket $\\{\cdot,\cdot\\}:C^{\infty}(M)\times C^{\infty}(M)\to
C^{\infty}(M)$ that is skew-symmetric and satisfies Leibniz identity (but does
not necessarily satisfy Jacobi identity). Due to the bilinear property, an
almost Poisson bracket induces a bivector field $\pi$ on $M$ defined, for each
$f,g\in C^{\infty}(M)$ by

$\pi(df,dg)=\\{f,g\\}.$

The vector field $X_{f}:=\\{\cdot,f\\}$ is the hamiltonian vector field of
$f$. Equivalently, $X_{f}=-\pi^{\sharp}(df)$, where $\pi^{\sharp}:T^{*}M\to
TM$ is the map such that for $\alpha,\beta\in T^{*}M$,
$\beta(\pi^{\sharp}(\alpha))=\pi(\alpha,\beta)$. The characteristic
distribution of the bracket $\\{\cdot,\cdot\\}$ is the distribution on $M$
generated by the hamitonian vector fields.

An almost Poisson bracket $\\{\cdot,\cdot\\}$ is Poisson when the Jacobi
identity is satisfied, i.e.,

$\\{f,\\{g,h\\}\\}+\\{g,\\{h,f\\}\\}+\\{h,\\{f,g\\}\\}=0,\qquad\mbox{for
}f,g,h\in C^{\infty}(M).$

Equivalently, a bivector field $\pi$ is Poisson if and only if $[\pi,\pi]=0$
where $[\cdot,\cdot]$ is the Schouten bracket, see e.g. [46]. The
characteristic distribution of a Poisson bracket is integrable and foliated by
symplectic leaves.

######  Definition A.1.

[56] An almost Poisson bracket $\\{\cdot,\cdot\\}$ on $M$ is twisted Poisson
if there exists a closed 3-form $\Phi$ on $M$ such that, for each $f,g,h\in
C^{\infty}(M)$

$\\{f,\\{g,h\\}\\}+\\{g,\\{h,f\\}\\}+\\{h,\\{f,g\\}\\}=\Phi(X_{f},X_{g},X_{h}),$

where $X_{f},X_{g},X_{h}$ are the hamiltonian vector fields of $f,g,h,$ with
respect to $\\{\cdot,\cdot\\}$. In other words, a bivector field $\pi$ on $M$
is twisted Poisson if $[\pi,\pi]=\frac{1}{2}\pi^{\sharp}(\Phi)$.

######  Remark A.2.

The characteristic distribution of a twisted Poisson bracket is integrable and
it is foliated by almost symplectic leaves. Conversely, it was shown in [6],
that any regular almost Poisson bracket with integrable characteristic
distribution is a twisted Poisson bracket. $\diamond$

A regular almost Poisson bracket $\\{\cdot,\cdot\\}$ on $M$ is determined by a
2-form $\Omega$ and a distribution $F$ defined on $M$ so that $\Omega|_{F}$ is
nondegenerate. In fact, for $f\in C^{\infty}(M)$,

$X_{f}=\\{\cdot,f\\}\quad\mbox{if and only if}\quad{\bf
i}_{X_{f}}\Omega|_{F}=df|_{F},$ (A.54)

(actually, the bracket is determined by the nondegenerate 2-section
$\Omega|_{F}$ on $M$). The distribution $F$ is the characteristic distribution
of the bracket. If $F$ is integrable, then $\\{\cdot,\cdot\\}$ is a (regular)
twisted Poisson bracket by the 3-form $\Phi=d\Omega$ ($\Omega$ is not
necessarily closed). A Poisson bracket has $F$ integrable and $\Omega$ closed.

Gauge transformations of a (regular) bracket by a 2-form.

######  Definition A.3.

[56] Consider a (regular) bracket $\\{\cdot,\cdot\\}$ on the manifold $M$ as
in (A.54) and a 2-form $B$ satisfying that $(\Omega+B)|_{F}$ is nondegenerate.
A gauge transformation of $\\{\cdot,\cdot\\}$ by the 2-form $B$ defines a
bracket $\\{\cdot,\cdot\\}_{\mbox{\tiny{$B$}}}$ on $M$ given, at each $f\in
C^{\infty}(M)$, by

${\bf i}_{X_{f}}(\Omega+B)|_{F}=df|_{F}\quad\mbox{if and only if}\quad
X_{f}=\\{\cdot,f\\}_{\mbox{\tiny{$B$}}}.$

In this case, we say that the brackets $\\{\cdot,\cdot\\}$ and
$\\{\cdot,\cdot\\}_{\mbox{\tiny{$B$}}}$ are gauge related .

######  Remark A.4.

  1. $(i)$

The brackets $\\{\cdot,\cdot\\}$ and $\\{\cdot,\cdot\\}_{\mbox{\tiny{$B$}}}$
have the same characteristic distribution $F$. Therefore, if an almost Poisson
bracket has a nonintegrable characteristic distribution, all gauge related
brackets will be almost Poisson with a nonintegrable characteristic
distribution.

  2. $(ii)$

If the bracket $\\{\cdot,\cdot\\}$ is twisted Poisson by a 3-form $\Phi$, then
the gauge related bracket $\\{\cdot,\cdot\\}_{\mbox{\tiny{$B$}}}$ is twisted
Poisson by the 3-form $(\Phi+dB)$. Moreover, they share the characteristic
foliation but the 2-form on each leaf $F_{\mu}$ changes by the term
$B_{\mu}=\iota_{\mu}B$ for $\iota_{\mu}:F_{\mu}\to M$ the inclusion.

  3. $(iii$)

The original definition of a gauge transformation in [56] was given on Dirac
structures and then the 2-form $B$ does not need to satisfy the nondegenerate
condition $(\Omega+B)|_{F}$.

$\diamond$

######  Definition A.5.

Let $\tau:M\to P$ be a vector bundle and let $\alpha$ be a $k$-form on the
manifold $M$. We say that $\alpha$ is semi-basic with respect to the bundle
$M\to P$ if

${\bf i}_{X}\alpha=0\quad\mbox{for all $X\in TM$ such that $T\tau(X)=0$}.$

The $k$-form $\alpha$ is basic if there exists a $k$-form $\bar{\alpha}$ on
$P$ such that $\tau^{*}\bar{\alpha}=\alpha$.

######  Remark A.6.

Consider the canonical symplectic 2-form $\Omega_{\mbox{\tiny{$Q$}}}$ on
$T^{*}Q$. If $B$ is a semi-basic 2-form with respect to the bundle $T^{*}Q\to
Q$, then $\Omega_{\mbox{\tiny{$Q$}}}+B$ is a nondegenerate 2-form on $T^{*}Q$.
$\diamond$

Symmetries. Let us consider an almost Poisson manifold $(M,\\{\cdot,\cdot\\})$
given as in (A.54) and a Lie group $G$ acting on $M$ and leaving
$\\{\cdot,\cdot\\}$ invariant. Then on the reduced manifold $M/G$ there is an
almost Poisson bracket $\\{\cdot,\cdot\\}_{\mbox{\tiny{red}}}$ defined, at
each $f,g\in C^{\infty}(M/G)$ by

$\\{f,g\\}_{\mbox{\tiny{red}}}\circ\rho=\\{\rho^{*}f,\rho^{*}g\\},$

where $\rho:M\to M/G$ is the orbit projection.

If a $G$-invariant 2-form $B$ satisfies that $(\Omega+B)|_{F}$ is
nondegenerate, then the gauge related bracket
$\\{\cdot,\cdot\\}_{\mbox{\tiny{$B$}}}$ is $G$-invariant as well. Both
brackets $\\{\cdot,\cdot\\}$ and $\\{\cdot,\cdot\\}_{\mbox{\tiny{$B$}}}$ can
be reduced to obtain the corresponding reduced brackets
$\\{\cdot,\cdot\\}_{\mbox{\tiny{red}}}$ and
$\\{\cdot,\cdot\\}_{\mbox{\tiny{red}}}^{\mbox{\tiny{$B$}}}$ on the quotient
manifold $M/G$ as the diagram shows:

(A.55)

As it was observe in [36, 6], the brackets
$\\{\cdot,\cdot\\}_{\mbox{\tiny{red}}}$ and
$\\{\cdot,\cdot\\}_{\mbox{\tiny{red}}}^{\mbox{\tiny{$B$}}}$ can have different
properties. More precisely, they are not necessarily gauge related and hence
one can be Poisson while the other not. In fact,
$\\{\cdot,\cdot\\}_{\mbox{\tiny{red}}}$ and
$\\{\cdot,\cdot\\}_{\mbox{\tiny{red}}}^{\mbox{\tiny{$B$}}}$ are gauge related
if and only if the 2-form $B$ is basic with respect to the principal bundle
$M\to M/G$.

##  Appendix B Appendix: Some facts on reconstruction theory

The reconstruction of the dynamics from reduced equilibria and reduced
periodic orbits has been well studied in [35, 42], when the symmetry group is
compact and in [2] in the non–compact case. In this subsection we shortly
review the basic results of reconstruction theory in the simplest framework,
of free and proper group actions. We consider a Lie group $G$ that acts freely
and properly on a manifold $M$. The freeness and properness of the action
guarantee that the quotient space $M/G$ has the structure of a manifold and
$\tau:M\longrightarrow M/G$ is a principal bundle with structural group $G$.
Let $X$ be a $G$–invariant vector field on $M$, then there exists a vector
field $\bar{X}$ on $M/G$, which is $\tau$–related to $X$. We recall that a
$G$–orbit $\mathcal{O}_{m_{0}}=G\cdot m_{0}$, with $m_{0}\in M$, is a relative
equilibrium for $X$, if it is invariant with respect to the flow of $X$ and
its projection to the reduced space $M/G$ is an equilibrium of the reduced
dynamics $\bar{X}$. Moreover a $G$–invariant subset $\mathcal{P}$ of $M$ is
called a relative periodic orbit for $X$, if it is invariant by the flow and
its projection to the quotient manifold $M/G$ is a periodic orbit of
$\bar{X}$.

Let $\mathcal{P}$ be a relative periodic orbit and $\gamma$ a curve in
$\mathcal{P}$. By the periodicity of the reduced dynamics, the integral curves
of the complete system, that pass through $\gamma(0)$, returns periodically,
with period $T>0$, to the $G$–orbit through $\gamma(0)$. The freeness of the
action of $G$ on $M$ guarantees that $\forall\gamma$ in $\mathcal{P}$ there
exists a unique $p(\hat{\gamma})$ in $G$ such that

$\phi^{X}_{T}(\gamma)=\psi_{p(\hat{\gamma})}(\gamma)\,,$

where $\phi^{X}_{T}$ is the flow of $X$ at time $T$, $\psi_{g}$ is the action
of $G$ on $M$, $\hat{\gamma}$ is the projection of $\gamma$ on $M/G$ with
respect to $\tau$, and the map $p:\mathcal{P}\rightarrow G$, $\gamma\mapsto
p=p(\hat{\gamma})$ is the so–called _phase_ [26]. The phase $p$ is a piecewise
smooth map, constant along the orbits of $X$ (i.e. $p\circ\phi^{X}_{t}=p$,
$\forall t$) and it is equivariant with respect to conjugation, that is $\quad
p(h\cdot\gamma)=h\,p(\hat{\gamma})h^{-1},\quad\forall h\in
G,\forall\gamma\in\mathcal{P}$. Then the following Theorem holds.

######  Proposition B.1.

[35, 42, 2] Let $\mathcal{P}$ be a relative periodic orbit of $X$ on $M$. Then

  * i)

if the group $G$ is compact, the flow of $X$ in $\mathcal{P}$ is
quasi–periodic with at most $rank\,G+1$ frequencies;

  * ii)

if $G$ is non–compact, the flow of $X$ in $\mathcal{P}$ is either
quasi–periodic, or escaping.

The non–compact case is the most frequent and also the most interesting, for
example one could say more on which of the two behaviours of the dynamics,
namely quasi-periodicity or escaping, is “generic” by studying the group $G$
(but this goes beyond our scopes, for more details see [2]).

######  Remark B.2.

In [35, 42, 2] reconstructions results are given from the point of view of Lie
Algebras, while [33] develops a theory in terms of groups. Moreover [33]
investigates the structure of the copies of $\mathbb{R}$ and shows that one
can define an intrinsic notion of a certain number of frequencies that gives
rise to the idea that, in this case, the reconstructed dynamics ‘spirals’
toward a certain direction. $\diamond$

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