# Local Metric with Parameterized Evolution

Martin Land  Department of Computer Science, Hadassah College, Jerusalem
<EMAIL_ADDRESS>
(0/00/2019; 0/00/2019; 0/00/2019)

###### Abstract

We present a canonical Hamiltonian formulation of GR in which $\tau$, the
parameter of system evolution, is external to spacetime, playing a role
similar to what we call time in nonrelativistic mechanics. This approach,
known as Stueckelberg-Horwitz-Piron (SHP) theory, inherits the full
computational power of classical analytical mechanics while maintaining
manifest covariance throughout and eliminating possible conflict with general
diffeomorphism invariance. In particular, SHP simplifies the initial value
problem with potential applications in highly dynamical interactions, such as
black hole collisions. By allowing the energy-momentum tensor and metric to
depend explicitly on $\tau$, we may describe particle motion in geodesic form
with respect to a dynamically evolving background metric. As a toy model, we
consider a $\tau$-dependent mass $M(\tau)$, first as a perturbation in the
Newtonian approximation and then for a Schwarzschild-like metric. As expected,
the extended Einstein equations imply a non-zero energy-momentum tensor,
proportional to $dM/d\tau$, representing a flow of mass and energy into
spacetime that corresponds to the changing source mass. In $\tau$-equilibrium,
this system becomes a generalized Schwarzschild solution for which the
extended Ricci tensor and energy-momentum tensor vanish.

###### keywords:

general relativity, evolution theories, initial value problem

††articletype: Article Type

##  1 Introduction

In general relativity, matter tells spacetime how to curve and spacetime
curvature tells matter how to move. More precisely, a mass distribution given
by $T^{\alpha\beta}\left(x\right)$ induces a local metric
$g_{\alpha\beta}\left(x\right)$ through the Einstein equations, and the metric
enters the equations of motion for particles. One may, in principle, construct
the $T^{\alpha\beta}\left(x\right)$ associated with particles
$x^{\mu}_{i}\left(\tau\right)$, find the induced metric and obtain equations
of motion for matter in the resulting gravitational field. But calculations of
this type may be intractable — even for linearized field equations. As Fock
observed [Fock (1937)], it may be impractical to replace $\tau$ by $t$ as
parameter of particle motion, in order to obtain $T^{\alpha\beta}$ as a
function of $x^{\mu}$ alone. Later, Stueckelberg [Stueckelberg (19411),
Stueckelberg (19412)] argued that to fully characterize all possible spacetime
trajectories, the 8D phase space
$x^{\mu}\left(\tau\right),\dot{x}^{\mu}\left(\tau\right)$ must be
unconstrained, and $\tau$ cannot be identified with the proper time of the
motion, which may not be well-defined. Horwitz and Piron [Horwitz Piron
(1973)] developed this idea into a manifestly covariant canonical mechanics
for the many-body problem [Horwitz (2015), Horwitz Arshansky (2018)], in which
gauge fields may depend explicitly on $\tau$ so that the Stueckelberg-Horwitz-
Piron (SHP) theory remains well-posed [Saad . (1989)].

The classical and quantum mechanics of particles in a spacetime with local
metric $g_{\mu\nu}(x)$ has been studied extensively by Horwitz [Horwitz
(2019)] and will not be discussed at length here. Our goal is to find a
consistent prescription for extending general relativity to accommodate an
energy-momentum tensor $T^{\alpha\beta}(x,\tau)$ and a metric
$g_{\alpha\beta}(x,\tau)$ satisfying $\tau$-dependent Einstein equations. As a
guide we recall the construction of the SHP particle action as
$S_{\text{Maxwell}}\longrightarrow S_{\text{SHP}}$, where

$S_{\text{Maxwell}}=\int
d\tau~{}\frac{1}{2}M\dot{x}^{\mu}\dot{x}_{\mu}+\frac{e}{c}\dot{x}^{\mu}A_{\mu}\left(x^{\lambda}\right)$
(1)

for $\lambda,\mu,\nu=0,1,2,3$, and

$\displaystyle S_{\text{SHP}}$ $\displaystyle=$ $\displaystyle\int
d\tau~{}\frac{1}{2}M\dot{x}^{\mu}\dot{x}_{\mu}+\frac{e}{c}\dot{x}^{\mu}a_{\mu}\left(x^{\lambda},\tau\right)+\frac{e}{c}c_{5}a_{5}\left(x^{\lambda},\tau\right)$
(2) $\displaystyle=$ $\displaystyle\int
d\tau~{}\frac{1}{2}M\dot{x}^{\mu}\dot{x}_{\mu}+\frac{e}{c}\dot{x}^{\beta}a_{\beta}\left(x^{\lambda},\tau\right)$

where $\alpha,\beta,\gamma=0,1,2,3,5$, and in analogy to $x^{0}=ct$, we write
$x^{5}=c_{5}\tau$. Variation with respect to $x^{\mu}$ leads to the Lorentz
force [Land Horwitz (1991)]

$\displaystyle M\ddot{x}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{e}{c}\left(\dot{x}^{\nu}f_{\mu\nu}+c_{5}f_{\mu
5}\right)=\frac{e}{c}\dot{x}^{\beta}f_{\mu\beta}$ (3)
$\displaystyle\frac{d}{d\tau}\left(-\frac{1}{2}M\dot{x}^{\mu}\dot{x}_{\mu}\right)$
$\displaystyle=$ $\displaystyle c_{5}\frac{e}{c}\dot{x}^{\beta}f_{5\beta}$ (4)

where (4) permits the exchange of mass between particles and fields. The
$\beta=5$ index is understood to be a formal convenience, indicating O(3,1)
scalar quantities, and not elements of 5D tensors. Still, terms of the type
$\dot{x}^{\beta}a_{\beta}\left(x,\tau\right)$ suggest that (3) and (4) exhibit
$5D\longrightarrow 4D$ symmetry breaking, in which we first specify
$\dot{x}^{5}=c_{5}$ as a scalar constant, restricting the dynamical degrees of
freedom to the 4D manifold, before applying a dynamical principle.

##  2 Canonical mechanics

###  2.1 An approach to a $\tau$-dependent metric

Standard general relativity considers the invariant interval

$\delta x^{2}=g_{\mu\nu}\delta x^{\mu}\delta
x^{\nu}=\left(x_{2}-x_{1}\right)^{2}$ (5)

between two neighboring spacetime points, viewed as an instantaneous
displacement. This invariance is a geometrical statement expressing a wide
freedom in assigning coordinates to the manifold. To transform geometry into
dynamics, a spacetime path maps an arbitrary parameter $\zeta$ to a continuous
sequence of events $x^{\mu}(\zeta)$ such that any two points on the path have
timelike separation, and so the proper time can be taken as the parameter.
Consistent with the notion of a 4D block universe, the path consists of
instantaneous displacements, observed as “motion” through changes in
$x^{0}(\zeta)$ with $\zeta$. Parameterizing by proper time $s$, the line
element

$\delta x^{2}=g_{\mu\nu}\delta x^{\mu}\delta
x^{\nu}=g_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}\delta
s^{2}=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\delta s^{2}$ (6)

suggests a dynamical description of the path by the action

$S=\int dx=\int ds\ \sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}$ (7)

and leading to geodesic equations of motion as an expression of the
equivalence principle. The geodesic equations can also be derived from the
action

$S=\int ds\ \frac{1}{2}\ g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$ (8)

which removes the constraint $\dot{x}^{2}=-c^{2}$ associated with (7).

In SHP a physical event $x^{\mu}(\tau)$ theory occurs at time $\tau$ and
chronologically precedes events occurring at subsequent times. The physical
picture describes the evolution of one 4D block universe defined at time
$\tau$ to another, infinitesimally close, at $\tau+d\tau$. Evolution slows to
zero in equilibrium. The $\tau$-dependence of $a^{\alpha}(x,\tau)$ reminds us
that while geometric relations on spacetime are defined within a given block
universe, the dynamics are defined in the transition from one 4D block
manifold to another. We therefore consider the interval

$dx^{\mu}=\bar{x}^{\mu}(\tau+\delta\tau)-x^{\mu}(\tau)$ (9)

between an event $x^{\mu}$ occurring at time $\tau$ and an event
$\bar{x}^{\mu}$ occurring at a different spacetime location at a subsequent
time $\tau+\delta\tau$. With $\delta x^{5}=c_{5}\delta\tau$ we may write

$dx^{2}=g_{\alpha\beta}\left(x,\tau\right)\delta x^{\alpha}\delta x^{\beta}$
(10)

which contains both the geometrical distance $\delta x^{\mu}$ between two
neighboring points in one manifold, and the dynamical distance $\delta x^{5}$
between events occurring at two sequential times.

This line element suggests the free particle Lagrangian

$L=\frac{1}{2}Mg_{\alpha\beta}\left(x,\tau\right)\dot{x}^{\alpha}\dot{x}^{\beta}$
(11)

containing both geometry and dynamics. In a formal and restricted sense, this
system is defined on 5D coordinates $\left(x^{\mu},x^{5}\right)$, as was done
for the fields in (2) to (4). Nevertheless, we maintain the distinction
between geometry and evolution. The parameter $x^{5}$ does not become a
dynamical degree of freedom, and no five dimensional symmetries are ascribed
to this 5D structure. While the terms $c_{5}g_{\mu
5}\left(x,\tau\right)\dot{x}^{\mu}$ and $c_{5}^{2}g_{55}\left(x,\tau\right)$
appear as interactions, we view them as features of an extended equivalence
principle, in which $g_{\mu\nu}$ expresses geodesic motion within the curved
geometry of a 4D block universe at any fixed $\tau$, while $g_{\alpha 5}$
express geodesic motion along the evolving structure of one 4D block universe
to another.111Horwitz and Gershon [Gershon Horwitz (2009)] have shown that a
covariant action with scalar potential is equivalent to a free action with
local conformal metric. In their work, metric component $g_{5\alpha}$ and
related components of the connection similarly appear.

###  2.2 General 5D spacetime

If (11) were a 5D free particle Lagrangian for $x^{\alpha}(\xi)$ then the
Euler-Lagrange equations would provide geodesic equations

$0=\frac{D\dot{x}^{\gamma}}{D\xi}=\ddot{x}^{\gamma}+\Gamma_{\alpha\beta}^{\gamma}\dot{x}^{\alpha}\dot{x}^{\beta}$
(12)

where $D/D\xi$ is the absolute derivative (in the notation of Weinberg
[Weinberg (1972)]) and

$\Gamma_{\alpha\beta}^{\gamma}=g^{\gamma\delta}\Gamma_{\delta\alpha\beta}=\frac{1}{2}g^{\gamma\delta}\left(\partial_{\alpha}g_{\delta\beta}+\partial_{\beta}g_{\delta\alpha}-\partial_{\delta}g_{\beta\alpha}\right)$
(13)

is the standard Christoffel symbol in 5D.

Writing the canonical momentum

$p_{\alpha}=\frac{\partial
L}{\partial\dot{x}^{\alpha}}=Mg_{\alpha\beta}\dot{x}^{\beta}\qquad\longrightarrow\qquad\dot{x}^{\alpha}=\frac{1}{M}g^{\alpha\beta}p_{\beta}$
(14)

the scalar Hamiltonian, representing particle mass,

$K=\dot{x}^{\alpha}p_{\alpha}-L=\frac{1}{2M}g^{\alpha\beta}p_{\alpha}p_{\beta}=L$
(15)

is conserved, as seen directly through

$\frac{d}{d\tau}\left(\frac{1}{2}Mg_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}\right)=Mg_{\alpha\beta}\dot{x}^{\alpha}\frac{D\dot{x}^{\beta}}{D\tau}=0$
(16)

where we used metric compatibility $Dg_{\alpha\beta}/D\xi=0$. This result may
also be seen from the canonical equations of motion

$\dot{x}^{\alpha}=\frac{dx^{\alpha}}{d\xi}=\frac{\partial K}{\partial
p_{\alpha}}\qquad\qquad\dot{p}_{\alpha}=\frac{dp_{\alpha}}{d\xi}=-\frac{\partial
K}{\partial x^{\alpha}}$ (17)

and the Poisson bracket

$\left\\{F,G\right\\}=\frac{\partial F}{\partial x^{\alpha}}\frac{\partial
G}{\partial p_{\alpha}}-\frac{\partial F}{\partial p_{\alpha}}\frac{\partial
G}{\partial x^{\alpha}}$ (18)

so that

$\frac{dK}{d\xi}=\left\\{K,K\right\\}+\frac{\partial
K}{\partial\xi}=\frac{1}{2M}p_{\alpha}p_{\beta}\ \frac{\partial
g^{\alpha\beta}}{\partial\xi}=0$ (19)

using the $\xi$-independence of the metric.

###  2.3 Breaking the 5D symmetry to 4D+1

We break the 5D symmetry by putting $x^{5}(\tau)=c_{5}\tau$ in (11), which
becomes

$L=\frac{1}{2}Mg_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\frac{1}{2}Mc_{5}g_{5\mu}\dot{x}^{\mu}+\frac{1}{2}Mc^{2}_{5}g_{55}\
.$ (20)

Comparing (12) with the equations of motion

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\ddot{x}^{\mu}+\Gamma_{\lambda\sigma}^{\mu}\dot{x}^{\lambda}\dot{x}^{\sigma}+2c_{5}\Gamma_{5\sigma}^{\mu}\dot{x}^{\sigma}+c^{2}_{5}\Gamma_{55}^{\mu}$
(21) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{D\dot{x}^{5}}{D\tau}=\frac{d\dot{x}^{5}}{d\tau}=\frac{dc_{5}}{d\tau}$
(22)

the prescription for symmetry breaking connection is

$\Gamma_{5\alpha}^{\mu}=\frac{1}{2}g^{\mu\nu}\left(\partial_{5}g_{\nu\alpha}+\partial_{\alpha}g_{\nu
5}-\partial_{\nu}g_{\alpha 5}\right)\qquad\Gamma_{\alpha\beta}^{5}\equiv 0\ .$
(23)

We notice in particular the violation of the 5D symmetry

$\Gamma_{5\mu\nu}=-\Gamma_{\mu 5\nu}+\partial_{\nu}g_{\mu 5}$ (24)

which no longer applies because the label 5 indicates a scalar quantity, and
is not a tensor index.

Because $x^{5}$ is not a dynamical quantity and has no conjugate momentum, the
Hamiltonian can be written

$K=\frac{1}{2}Mg_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}-\frac{1}{2}Mc^{2}_{5}g_{55}$
(25)

which is not conserved for a $\tau$-dependent metric, as seen from

$\frac{dK}{d\tau}=\frac{\partial
K}{\partial\tau}=-\frac{1}{2}M\dot{x}^{\mu}\dot{x}^{\nu}\frac{\partial
g_{\mu\nu}}{\partial\tau}-\frac{1}{2}Mc_{5}^{2}\frac{\partial
g_{55}}{\partial\tau}.$ (26)

Expression (26) can also be found by taking the total $\tau$-derivative of
(25) and inserting the equations of motion (21).

###  2.4 Einstein equations

We define $n(x,\tau)$ to be the number of events per spacetime volume, so that
the total number of events is

$N(\tau)=\int d\Omega\ n(x,\tau)$ (27)

where $d\Omega$ is a scalar spacetime volume element, and

$j^{\alpha}\left(x,\tau\right)=\rho(x,\tau)\dot{x}^{\alpha}(\tau)=Mn(x,\tau)\dot{x}^{\alpha}(\tau)$
(28)

is the 5-component event current. The continuity equation in flat space

$\partial_{\alpha}j^{\alpha}=\partial_{\mu}j^{\mu}+\partial_{5}j^{5}=\partial_{\mu}j^{\mu}+\frac{\partial\rho}{\partial\tau}=0$
(29)

leads to conservation of total event number through

$\frac{d}{d\tau}\int d\Omega\ n(x,\tau)+\frac{1}{M}\int d\Omega\
\partial_{\mu}j^{\mu}=\frac{dN}{d\tau}=0\ .$ (30)

With a local metric (29) is generalized (in the notation of Wald [Wald
(1984)]) to

${\boldsymbol{\nabla}}_{\alpha}j^{\alpha}={\boldsymbol{\nabla}}_{\alpha}X^{\alpha}=\partial_{\alpha}j^{\alpha}+j^{\gamma}\Gamma^{\alpha}_{\gamma\alpha}=0$
(31)

and since $j^{5}$ is a scalar on physical grounds, prescription (23) requires
$\Gamma_{\alpha\beta}^{5}=0$, so that the continuity equation becomes

${\boldsymbol{\nabla}}_{\alpha}j^{\alpha}=\frac{\partial\rho}{\partial\tau}+{\boldsymbol{\nabla}}_{\mu}j^{\mu}=0\
.$ (32)

Generalizing the 4D stress-energy-momentum tensor to 5D, we write the mass-
energy-momentum tensor [Land (2019)] as

$T^{\alpha\beta}=Mn\dot{x}^{\alpha}\dot{x}^{\beta}=\rho\dot{x}^{\alpha}\dot{x}^{\beta}$
(33)

where in addition to the standard 4D components $T^{\mu\nu}$, we have the
current density $T^{5\beta}=\dot{x}^{5}\dot{x}^{\beta}\rho=c_{5}j^{\beta}$.
Conservation of mass-energy-momentum
${\boldsymbol{\nabla}}_{\beta}T^{\alpha\beta}=0$ follows from the geodesic
equations
$\dot{x}^{\beta}{\boldsymbol{\nabla}}_{\beta}\dot{x}^{\alpha}=D\dot{x}^{\alpha}/D\tau=0$
and continuity ${\boldsymbol{\nabla}}_{\alpha}j^{\alpha}=0$, when the
equations of motion (21) and (22) are evaluated under the prescription (23).

The Einstein equations are similarly extended to

$G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}Rg_{\alpha\beta}=\frac{8\pi
G}{c^{4}}T_{\alpha\beta}$ (34)

where the Ricci tensor $R_{\alpha\beta}$ and scalar $R$ are found by
contracting indices of the curvature tensor $R_{\gamma\alpha\beta}^{\delta}$
found from the Christoffel symbols. Conservation of $T_{\alpha\beta}$ depends
on the prescription (23), and so in order to insure

${\boldsymbol{\nabla}}_{\beta}G^{\alpha\beta}=0$ (35)

we must similarly suppress $\Gamma_{\alpha\beta}^{5}$ when constructing the
Ricci tensor. Writing
$R_{\alpha\beta}=R_{\alpha\beta\gamma}^{\gamma}=R_{\alpha\beta\lambda}^{\lambda}+R_{\alpha\beta
5}^{5}$, we proceed by separating out terms containing the 5 index, leading to

$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(R_{\mu\nu}\right)^{4D}$ (36) $\displaystyle R_{\mu 5}$
$\displaystyle=$
$\displaystyle\frac{1}{c_{5}}\partial_{\tau}\Gamma_{\mu\lambda}^{\lambda}-\partial_{\lambda}\Gamma_{\mu
5}^{\lambda}+\Gamma_{\sigma
5}^{\lambda}\Gamma_{\mu\lambda}^{\sigma}-\Gamma_{\sigma\lambda}^{\lambda}\Gamma_{\mu
5}^{\sigma}$ (37) $\displaystyle R_{55}$ $\displaystyle=$
$\displaystyle\frac{1}{c_{5}}\partial_{\tau}\Gamma_{5\lambda}^{\lambda}-\partial_{\lambda}\Gamma_{55}^{\lambda}+\Gamma_{\sigma
5}^{\lambda}\Gamma_{5\lambda}^{\sigma}-\Gamma_{\sigma\lambda}^{\lambda}\Gamma_{55}^{\sigma}$
(38)

with new 5-terms from $g_{5\mu}$ and the $\tau$-dependence of $g_{\mu\nu}$.

##  3 Newtonian approximation

At low energy the velocity can be approximated as

$\dot{x}=\left(c\frac{dt}{d\tau},\frac{d\mathbf{x}}{d\tau},c_{5}\right)\simeq\left(c\frac{dt}{d\tau}\left(1,\mathbf{0}\right),c_{5}\right)$
(39)

so the equations of motion reduce to

$0\simeq\ddot{x}^{\mu}+c^{2}\Gamma_{00}^{\mu}\dot{t}^{2}+2cc_{5}\Gamma_{50}^{\mu}\dot{t}+c_{5}^{2}\Gamma_{55}^{\mu}\
.$ (40)

Taking $\partial_{0}g_{\mu\nu}=0$ and $g_{0k}=0$ in the weak field
approximation

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ (41)

we have the new Christoffel symbols

$\Gamma_{50}^{\mu}=\frac{1}{2c_{5}}g^{\mu
0}\partial_{\tau}h_{00}\qquad\Gamma_{55}^{\mu}=-\frac{1}{2}g^{\mu\nu}\partial_{\nu}h_{55}\
.$ (42)

The equations of motion

$\ddot{x}^{\mu}=\frac{1}{2}c^{2}g^{\mu\nu}\partial_{\nu}h_{00}\dot{t}^{2}-c\left(g^{\mu
0}\partial_{\tau}h_{00}\right)\dot{t}+\frac{1}{2}c_{5}^{2}g^{\mu\nu}\partial_{\nu}h_{55}$
(43)

split into

$\ddot{t}=\left(\partial_{\tau}h_{00}\right)\dot{t}\qquad\qquad\mathbf{\ddot{x}}=\frac{1}{2}c^{2}\dot{t}^{2}\nabla
h_{00}+\frac{1}{2}c_{5}^{2}\nabla h_{55}$ (44)

and writing

$\frac{d^{2}\mathbf{x}}{d\tau^{2}}=\frac{d}{d\tau}\left(\frac{dt}{d\tau}\frac{d\mathbf{x}}{dt}\right)=\dot{t}^{2}\frac{d^{2}\mathbf{x}}{dt^{2}}+\left(\partial_{\tau}h_{00}\right)\dot{t}\frac{d\mathbf{x}}{dt}$
(45)

we obtain

$\frac{d^{2}\mathbf{x}}{dt^{2}}=\frac{1}{2}c^{2}\nabla
h_{00}+\frac{1}{\dot{t}^{2}}\left[\frac{1}{2}c_{5}^{2}\nabla
h_{55}-\left(\partial_{\tau}h_{00}\right)\dot{t}\frac{d\mathbf{x}}{dt}\right]\
.$ (46)

The unperturbed motion ($\partial_{\tau}h_{\mu\nu}=0$ and $\nabla h_{55}=0$)
recovers Newtonian gravitation

$\frac{d^{2}\mathbf{x}}{dt^{2}}=-\frac{Gm}{r^{2}}\mathbf{\hat{r}}=-\nabla\phi\mbox{\qquad}\qquad\phi=-\frac{Gm}{r}$
(47)

in the usual manner, with

$g_{00}=-1-\frac{2}{c^{2}}\phi=-\left(1-\frac{2GM}{rc^{2}}\right)\ .$ (48)

As a perturbation at the Newtonian level, we treat the mass $M=M(\tau)$ as
$\tau$-dependent. As was shown in [Land (2019)], under certain circumstances,
an accelerated charged particle in SHP electrodynamics may radiate in such a
way that its energy and momentum vary independently, leading to a change in
the particle’s mass. This mass is transferred to the electromagnetic field,
producing mass radiation terms in the mass-energy-momentum tensor. Now the $t$
equation

$\ddot{t}=\left(\partial_{\tau}h_{00}\right)\dot{t}=\frac{2G\dot{M}}{rc^{2}}\dot{t}$
(49)

can be solved as

$\dot{t}=\exp\left(\frac{2G\ \Delta M(\tau)}{rc^{2}}\right)$ (50)

where $\Delta M(\tau)=M(\tau)-M(\tau_{0})$ is the change in mass from its
initial value. Neglecting $\left(\Delta M\right)^{2}$ for small changes in
mass and putting $h_{55}=0$, the equations for Newtonian gravity in plane
polar coordinates (with $\dot{\theta}=d\theta/dt$) become

$\displaystyle\frac{d^{2}r}{dt^{2}}-r\dot{\theta}^{2}+\frac{2G}{rc^{2}}\frac{dM}{dt}\frac{dr}{dt}$
$\displaystyle=$ $\displaystyle-\frac{GM}{r^{2}}$ (51)
$\displaystyle\frac{1}{r}\frac{d}{dt}\left(r^{2}\dot{\theta}\right)+\frac{2G}{c^{2}}\frac{dM}{dt}\dot{\theta}$
$\displaystyle=$ $\displaystyle 0$ (52)

after separating the radial and angular parts. We see that the nonrelativistic
angular momentum $Mr^{2}\dot{\theta}$ is not generally conserved in this
limit, because the transition to $t$ as evolution parameter in (45) introduced
a non-radial velocity-dependent force component. Neglecting deviation from
radial descent (taking $\dot{\theta}^{2}\approx 0$), the radial equation
reduces to

$\frac{d^{2}r}{dt^{2}}+\frac{2G}{c^{2}}\frac{dM}{dt}\frac{d}{dt}\ln
r=-\frac{GM}{r^{2}}$ (53)

describing an object accelerating in a Newtonian gravitational field with an
additional dissipative term.

##  4 Extended Schwarzschild solution

We extend the spherically symmetric metric by permitting $g_{00}=B$ and
$g_{11}=A$ (in the notion of Weinberg [Weinberg (1972)]) to be
$\tau$-dependent (but still $t$-independent) and introducing a fifth metric
component $g_{55}=Q$. Using the spherical symmetry to put $\theta=\pi/2$, the
Lagrangian becomes

$L=\frac{1}{2}\left[-c^{2}B\left(r,\tau\right)\dot{t}^{2}+A\left(r,\tau\right)\dot{r}^{2}+r^{2}\dot{\phi}^{2}+c_{5}^{2}Q\left(r,\tau\right)\right]$
(54)

from which we find new non-zero Christoffel symbols

$\Gamma_{50}^{0}=\dfrac{1}{2c_{5}}\dfrac{\partial_{\tau}B}{B}\qquad\Gamma_{55}^{1}=-\dfrac{1}{2}\dfrac{\partial_{r}Q}{A}\mbox{\qquad}\Gamma_{15}^{1}=\dfrac{1}{2c_{5}}\dfrac{\partial_{\tau}A}{A}$
(55)

and new non-zero Ricci tensor components

$\begin{array}[]{l}R_{05}=-\dfrac{1}{rc_{5}}\dfrac{\partial_{\tau}B}{B}\qquad\qquad
R_{15}=\dfrac{1}{2c_{5}}\left[\dfrac{\partial_{r}\partial_{\tau}B}{B}+2\dfrac{1}{r}\dfrac{\partial_{\tau}B}{B}\right]\rule[-18.0pt]{0.0pt}{18.0pt}\\\
R_{55}=\dfrac{1}{2}\left(\dfrac{\partial_{r}^{2}Q}{B}-\dfrac{\left(\partial_{r}Q\right)\left(\partial_{r}B\right)}{B^{2}}\right)+\dfrac{1}{2c_{5}^{2}}\left(\dfrac{\partial_{\tau}B}{B}\right)^{2}+\dfrac{\partial_{r}Q}{rB}\end{array}$
(56)

suggesting a highly dynamical system. Taking $M=M(\tau)$ to generalize the
Schwarzschild solution

$B(r,\tau)=A^{-1}(r,\tau)=1-\frac{2GM(\tau)}{rc^{2}}$ (57)

we still have $R_{\mu\nu}=\left(R_{\mu\nu}\right)^{4D}=0$, but
$R_{05},R_{15}$, and $R_{55}$ will contain non-vanishing terms proportional to
$dM/d\tau$. Setting the terms in $R_{55}$ containing $Q$ to zero we find

$Q=\int_{r}^{\infty}dr\
\frac{B}{r^{2}}=\frac{1}{r}\left(1-\frac{M\left(\tau\right)G}{c^{2}r}\right)=\frac{1}{2r}\left(3-B\right)$
(58)

leading to the Einstein tensor (34) with components

$\begin{array}[]{l}G_{\mu\nu}=-\gamma_{r}^{2}\left(1-\dfrac{MG}{c^{2}r}\right)^{-1}g_{\mu\nu}\qquad
G_{55}=\dfrac{1}{r^{2}}\gamma_{r}^{2}\\\
G_{50}=\dfrac{2\gamma_{r}}{r}\left(1-\dfrac{MG}{c^{2}r}\right)^{-1}\quad
G_{51}=-\dfrac{\gamma_{r}}{r}\left(1-\dfrac{MG}{c^{2}r}\right)^{-1}\end{array}$
(59)

where
$\gamma_{r}=\dfrac{G}{c_{5}^{2}c^{2}}\dfrac{dM}{d\tau}\left(1-\dfrac{2MG}{c^{2}r}\right)^{-1}$.

The equations of motion found from Lagrangian (54) admit solutions

$\dot{t}=\frac{1}{B}=\left(1-\frac{2GM(\tau)}{rc^{2}}\right)^{-1}\qquad\dot{\phi}=\frac{J}{r^{2}}$
(60)

for constant $J$ and the radial equation

$\frac{d}{d\tau}\frac{1}{2}\left(-c^{2}\frac{1}{B}+A\dot{r}^{2}+\frac{J^{2}}{r^{2}}-c_{5}^{2}Q\right)=-\frac{1}{2}\dot{r}^{2}\partial_{\tau}A-\frac{1}{2}c_{5}^{2}\partial_{\tau}Q\
.$ (61)

We recognize the LHS as the Hamiltonian

$K=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}-\frac{1}{2}c_{5}^{2}g_{55}$
(62)

in this parameterization, which as we saw in (26) is not conserved when the
metric is $\tau$-dependent. In equilibrium, when $dM/d\tau=0$, we recover

$R_{\alpha\beta}=0\qquad G_{\alpha\beta}=0\qquad
j_{\alpha}=0\qquad\frac{dK}{d\tau}=0$ (63)

describing a massless system (away from the origin), and the radial equation
admits a first integral solution.

##  5 Discussion

Interactions in SHP electrodynamics form an integrable system in which event
evolution generates an instantaneous current defined over spacetime at $\tau$,
and in turn, these currents induce $\tau$-dependent fields that act on other
events at $\tau$. We expect that in a similar way, a fully-developed SHP
formulation of general relativity will describe how the instantaneous
distribution of mass and energy at $\tau$ expressed through
$T_{\alpha\beta}(x,\tau)$ induces the local metric $g_{\alpha\beta}(x,\tau)$,
which in turn, determines geodesic equations of motion for any particular
event at $x^{\mu}(\tau)$.

Through the simplified example of a $\tau$-dependent mass point, we found that
the effect of the mass shift on the metric leads, by way of the Einstein
equations, to mass and energy radiation into the radial and time directions of
spacetime. We also saw that this metric induces non-conservation of mass in a
particle in extended geodesic motion. In this sense, as in SHP
electrodynamics, matter may exchange mass through the medium of local
spacetime. We therefore expect that SHP general relativity will be a useful
computational tool in relativistic dynamics.

## Acknowledgments

The author wishes to thank Lawrence P. Horwitz for useful comments on this
work, and to acknowledge his guiding wisdom over many decades. The author also
wishes to thank Aurora Perez Martinez for providing the impetus for this work.

## References

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# Free unitary groups are (almost) simple

Alexandru Chirvasitu111University of California at Berkeley,
<EMAIL_ADDRESS>
###### Abstract

We show that the quotients of Wang and Van Daele’s universal quantum groups by
their centers are simple in the sense that they have no normal quantum
subgroups, thus providing the first examples of simple compact quantum groups
with non-commutative fusion rings.

Keywords: free unitary group, simple compact quantum group, CQG algebra

###### Contents

  1. 1 Preliminaries
    1. 1.1 Free unitary groups
    2. 1.2 Normal quantum subgroups
  2. 2 Statements and proofs

## Introduction

A short note such as this one can hardly do justice to the richness of the
subject of quantum groups, so we will simply refer the reader to the papers
mentioned below, as well as to the vast literature cited by those papers, for
a comprehensive view of the history and intricacies of the subject. Let us
just mention that the quantum groups in this paper are of function algebra
type, rather than of the quantized universal enveloping algebra kind
introduced in [13, 9]. The former are non-commutative analogues of algebras of
continuous functions on a (here compact) group, and were formalized in
essentially their present form in [22]. To be more specific, we only work with
the algebraic counterparts of the objects described by Woronowicz. These are
the so-called CQG algebras of [8], and mimic the algebras of representative
functions on a compact group, minus the commutativity. They are complex Hopf
$*$-algebras satisfying an additional condition whih ensures, among other
things, the semisimplicity of their categories of comodules.

It makes sense, in view of the importance and ubiquity of simple compact Lie
groups, to study analogous notions in the quantum setting. This program was
initiated in [21], where simple compact quantum groups are defined (along with
the notion of normal compact quantum subgroup) and examples are provided. It
is observed there that these examples are all almost classical, in the sense
that their so-called fusion algebras (meaning Grothendieck rings of their
categories of finite-dimensional comodules) are isomorphic to fusion algebras
of compact Lie groups.

One source of compact quantum groups is provided by ordinary compact Lie
groups deformed in some sense ([15, 18] and references therein), but more
exotic examples can be obtained as quantum automorphism groups of various
structures, such as, say, bilinear forms [10], finite-dimensional
$C^{*}$-algebras endowed with a trace [20, 3], finite graphs [5], finite
metric spaces [4], etc. In the context of simplicity, it turns out [21, $\S$5]
that deforming simple compact Lie groups produces, as expected, simple compact
quantum groups. On the other hand, it is shown in the same paper ($\S$4) that
quantum automorphism groups of traced finite-dimensional $C^{*}$-algebras are
simple, while quantum automorphism groups of non-degenerate bilinear forms
have a single non-trivial normal compact quantum subgroup of order $2$.

The universal quantum groups $A_{u}(Q)$ (the free unitary groups of the title)
parametrized by invertible positive matrices $Q$ were introduced in [19].
Since they jointly play the same role as that of the family of unitary groups
in the theory of compact Lie groups, they are arguably the first examples one
should test compact quantum group notions or tentative results against. Wang
notes in [21, Proposition 4.5] that $A_{u}(Q)$ always has a central one-
dimensional torus (just like classical unitary groups), and hence cannot
possibly be simple. The main result of this paper is that the quotient by this
central circle group is nevertheless simple, again as in the classical case
(see Section 1 for an explanation of the terminology):

######  Theorem.

Let $Q\in GL_{n}$ be a positive invertible matrix. The quotient of the quantum
group $A_{u}(Q)$ by its central circle subgroup $A_{u}(Q)\to
C({\mathbb{S}}^{1})$ is simple.

The outline of the paper: We recall the relevant terminology, conventions and
results in Section 1, and prove the main results (one of which is stated
above) in Section 2.

### Acknowledgement

I am grateful to Shuzhou Wang for fruitful conversations on the contents of
[21], and for his many suggestions on the improvement of this paper.

##  1 Preliminaries

We assume the basics of Hopf algebra and coalgebra theory, as covered in, say,
[16, 7]. Comultiplications, counits, and antipodes are denoted as usual by
$\Delta,\varepsilon$ and $S$ perhaps adorned with the name of the coalgebra or
Hopf algebra as in $\Delta_{H}$, and we use Sweedler notation with suppressed
summation symbol for comultiplication, as in
$\displaystyle\Delta(x)=x_{1}\otimes x_{2}$. Comodules are always right and
finite-dimensional, and everything in sight is complex.

For CQG algebras, unitary and unitarizable comodules, and all other notions
that go into formalizing compact quantum groups in a purely algebraic setting
we refer to [8] or [14, $\S$11]. Here, we only remind the reader that a CQG
algebra is a Hopf $*$-algebra (Hopf algebra endowed with a conjugate linear,
involutive algebra anti-automorphism ‘$*$’ making both $\Delta$ and
$\varepsilon$ morphisms of $*$-algebras) satisfying an additional condition
which ensures that all finite-dimensional comodules admit an inner product
invariant under the coaction in some sense ([14, 11.1.5]). This accords with
the point of view that representations of the compact quantum group are
comodules over the Hopf algebra which is supposed to behave like the algebra
of representative functions on the otherwise fictitious group. Denote by
$\operatorname{\mathrm{CQG}}$ the category of CQG algebras and Hopf
$*$-algebra morphisms.

We also take for granted the correspondence between comodules over a coalgebra
$C$ and subcoalgebras of $C$, sending a $C$-comodule $V$ to the smallest
subcoalgebra $C_{V}\subseteq C$ such that the structure map $V\mapsto V\otimes
C$ factors through $V\otimes C_{V}$. This correspondence induces a bijection
between isomorphism classes of simple comodules and simple subcoalgebras of
$C$, and in a Hopf algebra $H$ it behaves well with respect to multiplication
and the antipode: The coalgebra associated to the tensor product $V\otimes W$
is exactly the product $C_{V}C_{W}$ in $H$ (that is, the linear span of
products of elements from $C_{V}$ and $C_{W}$), while the coalgebra
corresponding to the dual $V^{*}$ is $S(C_{V})$.

The fusion semiring of a Hopf algebra is the Grothendieck semiring of its
category of (finite-dimensional, right) comodules. Note that it admits a
natural anti-endomorphism $\alpha\mapsto\alpha^{*}$, sending (the class of) a
comodule to (the class of) its dual. When the Hopf algebra is cosemisimple
(e.g. a CQG algebra), this anti-endomorphism is involutive. By a slight abuse
of notation, if $x$ is the class of $V$ in the fusion semiring, we write
$C_{x}$ for $C_{V}$.

###  1.1 Free unitary groups

The main characters in this paper are the objects $A_{u}(Q)$ mentioned in the
introduction. Here, $Q$ is a positive invertible $n\times n$ matrix, and by
definition, $A=A_{u}(Q)$ is the $*$-algebra freely generated by $n^{2}$
elements $u_{ij}$, $i,j=\overline{1,n}$ subject to the conditions that both
$u=(u_{ij})_{i,j}$ and $Q^{\frac{1}{2}}\overline{u}Q^{-\frac{1}{2}}$ be
unitary $n\times n$ matrices in $A$ (where $\overline{u}=(u_{ij}^{*})_{i,j}$);
cf. [14, 11.3.1, Example 6], or [19], where these objects were first
introduced in their $C^{*}$-algebraic versions.

It turns out that $A$ can be made into a CQG algebra by demanding that
$u_{ij}$ be the $n^{2}$ “matrix units” of an $n\times n$ matrix coalgebra:

$\Delta(u_{ij})=\sum_{k}u_{ik}\otimes
u_{kj},\quad\varepsilon(u_{ij})=\delta_{ij}.$

The antipode is then defined by $S(u)=u^{*}=(u_{ji}^{*})_{i,j}$, and one
checks the CQG condition by observing that both the comodule $V$ with basis
$e_{i}$, $i=\overline{1,n}$ whose comodule structure is defined by
$e_{j}\mapsto\sum_{i}e_{i}\otimes u_{ij}$ and its dual are unitarizable; we
refer to the cited sources for details on how this works.

The reason why these are called ‘free unitary’ is that every CQG algebra
finitely generated as an algebra is a quotient of some $A_{u}(Q)$; keeping in
mind the arrow reversal implicit in passing from groups to functions on them,
this means that every “compact quantum Lie group” embeds in the compact
quantum group associated to some $A_{u}(Q)$. In other words, collectively, the
$A_{u}(Q)$’s play the same role in the world of compact quantum groups as the
family of all unitary groups does in the theory of ordinary compact groups.

It will be important to recall the structure of the fusion semiring $R_{+}$ of
$A=A_{u}(Q)$, as worked out in [2]. One of the main results of that paper is
that there is a bijection ${\mathbb{N}}*{\mathbb{N}}\ni x\mapsto r_{x}\in
R_{+}$ between the free monoid on two elements and the set of simple
comodules, with multiplication in $R_{+}$ given by

$r_{x}r_{y}=\sum_{x=ag,y=g^{*}b}r_{ab}.$ (1)

Here, ‘$*$’ is the involutive anti-automorphim of ${\mathbb{N}}*{\mathbb{N}}$
interchanging the two copies of ${\mathbb{N}}$, and the generators $\alpha$
and $\alpha^{*}$ of the two ${\mathbb{N}}$’s correspond respectively to the
fundamental comodule $V$ from the next-to-last paragraph and its dual.

###  1.2 Normal quantum subgroups

Always keeping in mind arrow reversal, a (closed) quantum subgroup of the
(compact quantum group with) CQG algebra $A$ should be a quotient CQG algebra
$A\to B$. This is indeed the standard definition in the literature, and the
one we employ here. We regard the arrow itself as the quantum subgroup, and
identify $A\to B$ and $A\to B^{\prime}$ as quantum subgroups provided the two
are isomorphic in the category of arrows in $\operatorname{\mathrm{CQG}}$
sourced at $A$ (in the terminology of [21, 2.7], we identify quantum subgroups
whenever they have the same imbedding).

Following [1, 1.1.5] and [21, 2.2], a quantum subgroup $\pi:A\to B$ of a
compact quantum group is said to be normal if the corresponding right and left
quantum coset spaces

$\operatorname{\textsc{LKer}}(\pi)=\\{a\in A\ |\ \pi(a_{1})\otimes
a_{2}=1_{B}\otimes a\\}$

and respectively

$\operatorname{\textsc{RKer}}(\pi)=\\{a\in A\ |\
a_{1}\otimes\pi(a_{2})=a\otimes 1_{B}\\}$

coincide. The resulting linear subspace
$\operatorname{\textsc{LKer}}=\operatorname{\textsc{RKer}}=C\leq A$ then turns
out to be a CQG subalgebra of $A$ and can be interpreted as functions on the
quotient of the compact quantum group corresponding to $A$ by the normal
quantum subgroup $\pi:A\to B$.

It is perhaps worth pointing out that $\pi$ and $\iota$ determine each other:
We have just seen how to get $\iota$ from $\pi$, and conversely, it can be
shown (e.g. [21, 4.4]) that $\pi:A\to B$ is precisely the quotient of $A$ by
the ideal $AC^{+}=C^{+}A$, where $C^{+}=\ker(\varepsilon_{C})$. What this
means, in other words, is that the inclusion $iota:C\to A$ and the surjection
$\pi:A\to B$ fit into an exact sequence $0\to C\to A\to B\to 0$ of Hopf (in
our case also $*$-) algebras in the sense of [1, 1.2.3].

######  1.2.1 Remark.

If $\iota:C\to P$ is to be half of an exact sequence, then $C$ must be
invariant under the right (and also left) adjoint action of $P$ on itself
given by $q\triangleleft p=S(p_{1})qp_{2}$ (respectively $p\triangleright
q=p_{1}qS(p_{2})$). Indeed, it is observed in the proof of [1, 1.1.12] that
for any Hopf algebra morphism $f$, $\operatorname{\textsc{LKer}}(f)$ as
defined in § 1.2 is invariant under the right adjoint action. It is for this
reason that we refer to the CQG subalgebras $\iota:C\to P$ of interest, i.e.
those giving rise to exact sequences, as ad-invariant.

In fact, invariance of a Hopf $*$-subalgebra $C\to P$ under either the left or
right adjoint action is also sufficient in order that it be part of an exact
sequence. This is proven in [1, 1.2.5] provided $P$ is faithfuly flat over
$C$; the latter condition always holds (that is, a CQG algebra is always
faithfully flat over a CQG subalgebra) by [6]. $\blacklozenge$

In view of the above discussion, the following is very reasonable:

######  1.2.2 Definition.

A compact quantum group $A\in\operatorname{\mathrm{CQG}}$ is simple if there
are no normal quantum subgroups $\pi:A\to B$ apart from $\varepsilon_{A}$ and
$\operatorname{id}_{A}$. $\blacklozenge$

######  1.2.3 Example.

As shown in [21, 4.5], sending $u_{ij}$ to $\delta_{ij}t$ implements a normal
embedding of the one-dimensional torus ${\mathbb{S}}^{1}$ with algebra of
representative functions $C({\mathbb{S}}^{1})=\mathbb{C}[t,t^{-1}]$ into any
of the free unitary groups $A_{u}(Q)$. The aim of this paper is to prove that
the resulting quotient is simple in the sense of 1.2.2.

We denote the CQG algebra associated to this quotient by $P_{u}(Q)$, standing
for ‘projective’. This is motivated by the fact that $P_{u}(Q)$ is a kind of
“projectivized” version of the free unitary group. $\blacklozenge$

######  1.2.4 Remark.

The terminology conflicts slightly with that of [21, 3.3]: On the one hand,
Wang’s definition of simplicity only demands that there be no non-trivial
connected normal quantum subgroups, and refers to the stronger form of
simplicity from 1.2.2 as absolute. I prefer the shorter term because there is
no need for that distinction in this paper. In other ways though, 1.2.2 might
seem weaker than [21, 3.3], because it makes no mention of the other three
conditions of the latter (numbered as in that paper):

(1) In order to be simple, Wang requires that a CQG algebra be finitely
generated. This is indeed not the case for $P_{u}(Q)$ (2.0.3), but we remedy
the problem in Proposition 2.0.5, where we provide a smaller, “less canonical”
example of a simple CQG satisfying this additional condition.

(2) The simple CQG’s of [21] are required to be connected, in the sense that
the $*$-subalgebra generated by any simple subcoalgebra is infinite-
dimensional. $A_{u}(Q)$ and hence its CQG subalgebras are easily seen to
satisfy this condition, so we need not worry about it any longer.

(4) Simple CQG’s are not supposed to have any non-trivial group-like elements,
or equivalently, one-dimensional comodules, or again, one-dimensional
subcoalgebras. Once more, this is automatically satisfied by $A_{u}(Q)$ and
its CQG subalgebras, for example because the fusion ring is freely generated
(as a ring) by the fundamental representation and its dual ([2, Théorème 1
(ii)]), which immediately shows that its only invertible elements are $\pm 1$.
$\blacklozenge$

##  2 Statements and proofs

Let us restate the result announced in the introduction:

######  2.0.1 Theorem.

For any positive invertible matrix $Q$, the compact quantum group $P_{u}(Q)$
is simple in the sense of 1.2.2.

The proof consists of showing that whenever a CQG subalgebra $\iota:C\to
P=P_{u}(Q)$ fits into an exact sequence as explained in § 1.2, $C$ is either
the scalars or the entire $P$. Since inclusions of CQG subalgebras induce
inclusions of fusion semirings, we can attack the problem by limiting the
possibilities for a fusion semiring of $R_{+}=R_{+}(P)$ if it is to correspond
to such an embedding $\iota$.

In working with fusion semirings, all of which are embedded in that of
$A=A_{u}(Q)$ described in § 1.1, we will identify the free monoid
${\mathbb{N}}*{\mathbb{N}}$ (and hence the set of simple comodules of $A$)
with words in the generators $\alpha$ and $\alpha^{*}$. It will be less
cumbersome, notationally, to substitute $0$ and $1$ respectively for $\alpha$
and $\alpha^{*}$, and write the elements of the free monoid as binary words;
note that the ‘$*$’ involution changes $0$’s into $1$’s and vice versa. First,
we determine precisely which binary words correspond to simple comodules of
$P_{u}(Q)$.

######  2.0.2 Lemma.

The simple comodules of $P$ are parametrized by the binary words with equal
numbers of $0$’s and $1$’s.

######  Proof.

Since by definition $P$ is the quotient of the free unitary group by the
circle subgroup $A\to C({\mathbb{S}}^{1})$, its simple comodules are precisely
those which, when regarded as comodules over $C({\mathbb{S}}^{1})$, break up
as direct sums of the trivial comodule. The conclusion follows from the
observations that (a) simple representations over the circle group are
parametrized by ${\mathbb{Z}}$, and (b) under this identification, scalar
corestriction via $A\to C({\mathbb{S}}^{1})$ turns the simple corresponding to
a binary word $w$ into a direct sum of copies of the simple
$C({\mathbb{S}}^{1})$-comodule corresponding to the integer $(\sharp\text{ of
}0\text{'s in }w)-(\sharp\text{ of }1\text{'s in }w)$. $\blacksquare$

We refer to binary words with equal numbers of $0$’s and $1$’s as balanced.

######  2.0.3 Remark.

Using the fusion rules 1, it can be shown that given any finite set $S$ of
binary words, no simple in the semiring generated by $S$ can start with a
longer contiguous segment of $0$’s than the longer such segment in a member of
$S$. Together with Lemma 2.0.2, this shows that as noted in 1.2.4, $P$ is not
finitely generated. $\blacklozenge$

We now start working towards proving Theorem 2.0.1.

######  2.0.4 Lemma.

If $\iota:C\to P$ is ad-invariant and strictly larger than the scalars, then
both $r_{01}$ and $r_{10}$ are in the fusion semiring $R_{+}(C)\subseteq
R_{+}$ of $C$.

######  Proof.

By assumption, $C$ has some non-trivial simple comodule, whose corresponding
binary word we may as well assume starts with a zero: $r_{0x}$. But then its
dual will be $r_{x^{*}1}$, and it follows from the fusion rules 1 that
$r_{01}$ is a summand in the product $r_{0x}r_{x^{*}1}$. In conclusion, the
simple $r_{01}$ must be in $R_{+}(C)$.

We now have to prove the same about $r_{10}$, and this is where ad-invariance
comes into the picture. First, the argument in the previous paragraph solves
the problem as soon as we can show that some $r_{1\cdots}$ is in $R_{+}(C)$
(then multiply it with its dual $r_{\cdots 0}$, etc.). To this end, fix some
non-zero $p\in P$ in the coalgebra $C(r_{10})$, and let it act on a non-zero
$c\in C$ in the coalgebra $C(r_{01})$ via the left adjoint action. On the one
hand, the result must be in $C$ by ad-invariance. On the other, the fusion
rules 1 say that $r_{10}r_{01}r_{10}$ equals $r_{100110}$, and hence the
multiplication map $C_{r_{10}}\otimes C_{r_{01}}\otimes C_{r_{10}}\to
C_{r_{100110}}$ is an isomorphism. It follows from this that $p\triangleright
c=p_{1}cS(p_{2})$ is a non-zero element of $C_{r_{100110}}$ (note that
$C_{r_{10}}$ is fixed by the antipode, so it contains both $p_{1}$ and
$S(p_{2})$). All in all, we get $r_{100110}\in R_{+}(C)$, and as observed at
the beginning of this paragraph, this will do to finish the proof.
$\blacksquare$

In the above proof and elsewhere we are tacitly using the correspondence
between CQG subalgebras of a CQG algebra $A$ and sub-semirings of its fusion
semiring $R_{+}(A)$. In one direction, an inclusion $\iota:C\to A$ of a CQG
subalgebra induces a fully faithful monoidal functor between categories of
comodules, and hence a fusion semiring inclusion. Moreover, $R_{+}(C)$ is the
${\mathbb{N}}$-span of precisely those simples in $R_{+}(A)$ which correspond
to simple subcoalgebras of $C$. In the other direction, given a sub-semiring
$R_{+}$ of $R_{+}(A)$ which is an ${\mathbb{N}}$-span of simples and is closed
under the involution, the direct sum of simple subcoalgebras of $A$
corresponding to the simples in $R_{+}$ is a CQG subalgebra. These two
constructions are inverse to one another, and implement the correspondence.

Let us stop for a moment to record the fact that we now already have an
example of simple compact quantum group with non-commutative fusion semiring:
The proof of Lemma 2.0.4 only uses $r_{01}$ and $r_{10}$ and the fusion rules
of $A_{u}(Q)$, so it provides a proof for

######  2.0.5 Proposition.

For any positive invertible matrix $Q$, the compact quantum group whose
underlying CQG algebra is the subalgebra of $A_{u}(Q)$ generated by
$u_{ij}u_{kl}^{*}$ and $u_{ij}^{*}u_{kl}$ is simple.

As promised in 1.2.4, this example, unlike $P_{u}(Q)$, is finitely generated
as an algebra and hence conforms to all of the conditions for simplicity in
[21, 3.3]. Apart from commutativity of the fusion semiring, this example
felicitously lacks another property. It is observed in [21, $\S$5] that the
examples of simple compact quantum groups mentioned in the introduction have
what in that paper is called property F. It means that every embedding of a
CQG subalgebra is part of an exact sequence, or, in view of 1.2.1, every CQG
subalgebra is invariant under the left adjoint action. The CQG algebra of
Proposition 2.0.5 clearly does not have property F, as its CQG subalgebra
generated by $u_{ij}u_{kl}^{*}$, whose fusion semiring is generated by
$r_{01}$, is not ad-invariant by the proof of Lemma 2.0.4.

######  2.0.6 Lemma.

Keeping the hypotheses and notation of Lemma 2.0.4, all simples of the form
$r_{0\cdots 01\cdots 1}$ and $r_{1\cdots 10\cdots 0}$ belong to $R_{+}(C)$.

######  Proof.

Let us prove the statement for $r_{0\cdots 01\cdots 1}$ ($n$ $0$’s and also
$n$ $1$’s, since the binary word must be balanced). First, note that it is
enough to show that some simple of the form $r_{0\cdots 0x}$ ($n$ $0$’s) is in
$R_{+}(C)$. Indeed, its dual would correspond to the word $x^{*}$ followed by
$n$ $1$’s, and their product would contain the desired simple as a summand. We
know from Lemma 2.0.4 that $r_{10}$ is in $R_{+}(C)$, and now we can argue as
in the proof of that lemma, acting on a non-zero element of $C_{r_{10}}$ by a
non-zero element of $C_{r_{0\cdots 01\cdots 1}}$ via the left adjoint action
to conclude. $\blacksquare$

######  Proof of Theorem 2.0.1.

Keeping the notations and assumptions of Lemma 2.0.4, we show by induction on
the length of a balanced word $x\in{\mathbb{N}}*{\mathbb{N}}$ that $r_{x}$
belongs to $R_{+}(C)$. Having taken care of the base step of the induction in
Lemma 2.0.4, we can assume $x$ is at least four symbols long. There are two
possibilities:

(1) $x$ consists of only two contiguous blocks of symbols, one of $0$’s and
one of $1$’s, as in, say, $x=0\cdots 01\cdots 1$. This case is covered by
Lemma 2.0.6.

(2) $x$ consists of more than two contiguous blocks of symbols. Write it
without loss of generality as, say, $x=0\cdots 01y$, starting with $n\geq 1$
$0$’s. Then, by the fusion rules 1, $x$ is a summand in the product of
$r_{0\cdots 01\cdots 1}$ ($n$ $0$’s) and $r_{0\cdots 0y}$ ($n-1$ $0$’s). These
two words are both strictly shorter than $x$, so the induction hypothesis
implies that both simples are in $R_{+}(C)$. This finishes the proof.
$\blacksquare$

While these results partially address [21, Problem 4.6 (2)], which asks for
examples of simple compact quantum groups with non-commutative fusion
semirings, it will still be interesting to investigate how much further
Proposition 2.0.5 can be pushed, and hence make some progress towards the
classification of simple quotients of free unitary groups. Note that here
‘quotient quantum group’ is used in the weak sense, meaning simply ‘CQG
subalgebra’. Quotients in the stronger sense of left hand halves of exact
sequences are taken care of by the following consequence of Theorem 2.0.1 (or
rather of its proof):

######  2.0.7 Corollary.

Any proper normal quantum subgroup of $A=A_{u}(Q)$ is contained in the circle
subgroup $A\to C({\mathbb{S}}^{1})$ from 1.2.3.

######  Proof.

The previous results go through practically verbatim for $A$ (instead of
$P=P_{u}(Q)$), and show that any non-trivial ad-invariant $\iota:C\to A$
contains the CQG subalgebra $P\subset A$. $\blacksquare$

We saw in 1.2.1 that CQG subalgebras $\iota:C\to H$ corresponding to quotients
by normal quantum subgroups have a simple characterization as precisely those
which are ad-invariant. It would be interesting though, as well as convenient,
to have a purely combinatorial characterization in terms of fusion semirings:

######  2.0.8 Question.

Let $\iota:C\to H$ be an inclusion of CQG algebras. Can the ad-invariance of
$C$ be characterized solely in terms of the fusion ring inclusion
$R_{+}(C)\subseteq R_{+}(H)$?

Alternatively, and also probably more tractably,

######  2.0.9 Question.

Does simplicity for a compact quantum group depend only on its fusion
semiring?

Positive answers would provide an alternative approach to the invariance of
simplicity under deformation proved in [21, $\S$5], and would be natural
companions to such results as the possibility of lifting isomorphisms of
fusion semirings to honest isomorphisms for compact connected Lie groups
([12]) and the fusion semiring characterization of the center for a compact
group (as in [17] or [11, 3.9]).

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