# Evidence for Planck Luminosity Bound in Quantum Gravity

Wolfgang<EMAIL_ADDRESS>Institute for Quantum Gravity,
Theoretical Physics III, Department of Physics
Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91052
Erlangen, Germany
19 February 2024

###### Abstract

Recently, we introduced a non-perturbative quantization of impulsive
gravitational null initial data. In this note, we investigate an immediate
physical implication of the model. One of the quantum numbers is the total
luminosity carried to infinity. We show that a transition happens when the
luminosity reaches the Planck power $\mathcal{L}_{\mathrm{P}}$. Below
$\mathcal{L}_{\mathrm{P}}$, the spectrum of the radiated power is discrete.
Above the Planck power, the spectrum is continuous and contains caustics that
can spoil the semi-classical interpretation of the resulting quantum states of
geometry.

##  1 Introduction

It has been often argued that the Planck power

$\mathcal{L}_{\mathrm{P}}=\frac{m_{\mathrm{P}}c^{2}}{t_{\mathrm{P}}}=c^{5}/G\approx
3,63\times 10^{52}\,\mathrm{W}$ (1)

places an upper bound on the gravitational wave luminosity, see e.g. [1, 2],
but also [3] for more critical remarks. The perhaps simplest argument in
favour of this idea can be found in Misner, Thorne, Wheeler [1]. Take the
virial theorem, i.e. $E_{\mathrm{kin}}\sim M\omega^{2}R^{2}\sim GM^{2}/R$, and
the quadrupole formula for the gravitational wave luminosity, i.e.
$\mathcal{L}_{\mathrm{GW}}\sim\mathcal{L}_{\mathrm{P}}^{-1}(MR^{2}\omega^{3})^{2}$,
where $M$ is the mass of the system, $R$ denotes its spatial extension and
$\omega$ is the frequency at which the system oscillates. Since the emission
can only happen when $R$ is greater than the Schwarzschild radius $2GM/c^{2}$,
we obtain $\mathcal{L}_{\mathrm{GW}}\lesssim\mathcal{L}_{\mathrm{P}}$. Such a
simple bound can not exist in higher dimensions. In $D>4$ spacetime
dimensions, the Planck power depends not only on $G$ and $c$, but also on
$\hbar$, which can not appear at the classical level. It is no surprise
therefore that it is in $D=4$ alone that we have an equation for the peak
luminosity of two coalescing black holes, which depends only on dimensionless
observables, such as the mass ratio $\eta=m_{1}m_{2}/(m_{1}+m_{2})^{2}$ and
dimensionless spin components [4, 5]. In higher dimensions, we need an
additional 1length scale [6]. This makes it doubtful that there is a universal
bound on the gravitational wave luminosity when $D>4$.

Recently, we introduced a non-perturbative quantization of impulsive
gravitational null initial data in $D=4$, see [7]. The proposal relies on the
geometry of light-like boundaries, which simplifies the construction of gauge-
invariant observables [8, 9, 10, 11]. In this note, we investigate the role of
the Planck power $\mathcal{L}_{\mathrm{P}}$ in the model. The analysis is
based on a combination of non-perturbative and semi-classical techniques. The
bound appears upon adding a parity-violating $\gamma$-term [12, 13, 14] to the
gravitational action in the bulk, which, by coincidence or not, only appears
in $D=4$. The resulting $\gamma$-Hilbert–Palatini action is

$\displaystyle S[A,e]=\frac{1}{16\pi G}\int_{\mathcal{M}_{4}}$

where $F=\mathrm{d}A+\tfrac{1}{2}[A,A]$ is the curvature of the
$\mathfrak{so}(1,3)$ connection, $e^{\alpha}$ is the co-tetrad and $\gamma$ is
the Barbero–Immirzi parameter [15, 16]. Since
$\mathcal{L}_{\mathrm{P}}\rightarrow\infty$ as $G\rightarrow 0$, we also note
that it seems implausible to find a bound on the gravitational wave luminosity
from a perturbative quantisation, where we have a formal perturbative
expansion with respect to $\varkappa=\sqrt{8\pi G/c^{3}}$.

##  2 Phase space of impulsive data

In the following, we consider the phase space of a pulse of gravitational null
initial data on a null boundary $\mathcal{N}_{3}\simeq[-1,1]\times S_{2}$.
Since the boundary is null, we can introduce a co-dyad
$(\tensor{e}{{}^{1}_{a}},\tensor{e}{{}^{2}_{a}})\in\Omega^{1}(\mathcal{N}_{3}:\mathbb{R}^{2})$,
intrinsic to $\mathcal{N}_{3}$, that diagonalizes the signature $(0$$+$$+)$
metric

$q_{ab}:=\varphi^{\ast}_{\mathcal{N}_{3}}g_{ab}=\delta_{ij}\tensor{e}{{}^{i}_{a}}\tensor{e}{{}^{j}_{b}},$
(2)

where $\delta_{ij}$ is the Kronecker delta and
$\varphi^{\ast}_{\mathcal{N}_{3}}g_{ab}$ is the pull-back of the spacetime
metric $g_{ab}$ in $\mathcal{M}_{4}$ to the boundary
$\mathcal{N}_{3}\subset\partial\mathcal{M}_{4}$. Any such co-dyad can be
parametrized by a conformal factor $\Omega$ and an $SL(2,\mathbb{R})$ holonomy
$S$,

$\tensor{e}{{}^{i}_{a}}=\Omega\,\tensor{S}{{}^{i}_{m}}{}^{\circ}\tensor{e}{{}^{m}_{a}},$
(3)

where ${}^{\circ}\tensor{e}{{}^{m}_{a}}$ is a fiducial dyad, which we keep
fixed once and for all. A possible choice is
${}^{\circ}\tensor{e}{{}^{1}_{a}}=\partial_{a}\vartheta$,
${}^{\circ}\tensor{e}{{}^{2}_{a}}=\sin(\vartheta)\partial_{a}\varphi$, where
$(\vartheta,\varphi)$ are standard spherical coordinates that are Lie dragged
along the null generators.

By introducing a time coordinate
$\mathcal{U}:\mathcal{N}_{3}\rightarrow[-1,1]$, we extend the angular
coordinates $(\vartheta,\varphi)$ into a three-dimensional coordinate system
of $\mathcal{N}_{3}$. The resulting vector field $\partial_{\mathcal{U}}$ is
null. We assume it to be future pointing. The boundary conditions are

$\text{for}\quad\partial\mathcal{N}_{3}=\mathcal{C}_{+}\cup\mathcal{C}_{-}^{-1}:\mathcal{U}\big{|}_{\mathcal{C}_{\pm}}=\pm
1.$ (4)

In the absence of additional structure, such as a preferred foliation,
symmetries or asymptotic boundary conditions, there is no natural torsionless
and metric compatible covariant derivative in $T\mathcal{N}_{3}$, see e.g.
[17]. A natural derivative $D_{a}$ exists on the extended vector bundle
$\bigcup_{p\in\mathcal{N}_{3}}\\{p\\}\times T_{p}\mathcal{M}_{4}$, where it is
induced from the bulk, i.e.
$D_{a}=\varphi^{\ast}_{\mathcal{N}_{3}}\nabla_{a}$, given the usual Levi-
Civita covariant derivative $\nabla_{a}$ in $(\mathcal{M}_{4},g_{ab})$. The
corresponding connection, which depends on both the extrinsic and intrinsic
geometry of $\mathcal{N}_{3}$, is the metric analogue of the self-dual
Ashtekar connection [18].

There are infinitely many clock variables $\mathcal{U}$ that satisfy the
boundary condition (4). To pick a unique representative, we impose the gauge
condition

$\partial^{a}_{\mathcal{U}}D_{a}\partial_{\mathcal{U}}=-\Omega^{-1}\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}[\Omega]\partial_{\mathcal{U}}.$
(5)

Upon choosing this gauge, the residual constraints simplify [11]. We are left
with a transport equation for the $SL(2,\mathbb{R})$ holonomy and the
Raychaudhuri equation [19] for $\Omega^{2}$. The Rachaudhuri equation becomes

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}\mathcal{U}^{2}}\Omega^{2}$
$\displaystyle=-2\sigma\bar{\sigma}\Omega^{2},$ (6)

where $\sigma$ is the shear of the null generators $\partial_{\mathcal{U}}$ of
$\mathcal{N}_{3}$. It defines a transport equation for the $SL(2,\mathbb{R})$
holonomy,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\mathcal{U}}H$
$\displaystyle=\mathopen{}\mathclose{{}\left(\sigma\bar{X}+\bar{\sigma}X}\right)H,\qquad
H(\mathcal{U}=-1,\vartheta,\varphi)=\bbvar{1},$ (7)

where we introduced a decomposition of $\mathfrak{sl}(2,\mathbb{R})$ into
translational components $X$ and $\bar{X}$ and a $U(1)$ generator $J$ with
$\mathfrak{sl}(2,\mathbb{R})$ commutation relations

$[J,X]=-2\mathrm{i}X,\;[J,\bar{X}]=+2\mathrm{i}\bar{X},\;[X,\bar{X}]=\mathrm{i}J.$
(8)

The $SL(2,\mathbb{R})$ element $S(\mathcal{U},\vartheta,\varphi)$ that
parametrizes the co-dyads in (3) is then given by

$S=\mathrm{e}^{\Delta J}HS_{-},$ (9)

where $\Delta:\mathcal{N}_{3}\rightarrow[0,2\pi)\mathrm{mod}2\pi$ is an
unspecified $U(1)$ angle with boundary condition
$\Delta(\mathcal{U}=-1,\vartheta,\varphi)=0$. In addition,
$S(\mathcal{U}=-1,\vartheta,\varphi)=S_{-}(\vartheta,\varphi)$. In the
interior of $\mathcal{N}_{3}$, the value of $\Delta$ can be gauged to zero. At
the upper boundary $\mathcal{C}_{+}$ it can not. The boundary data
$S_{-}(\vartheta,\varphi)$ and
$\Delta_{+}(\vartheta,\varphi)=\Delta(\mathcal{U}=1,\vartheta,\varphi)$ are an
example of gravitational edge modes [20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30].

The shear $\sigma:\mathcal{N}_{3}\rightarrow\mathbb{C}$ is unconstrained. It
determines the free radiative data along $\mathcal{N}_{3}$. To describe the
quantum geometry of a single pulse of radiation, we consider only those
configurations on phase space, where $\sigma$ is constant along the null
generators of $\mathcal{N}_{3}$. We can then integrate the Raychaudhuri
equation and obtain

$\displaystyle\Omega^{2}(\mathcal{U},\vartheta,\varphi)=\frac{E_{+}(\vartheta,\varphi)+E_{-}(\vartheta,\varphi)}{2}$
$\displaystyle\frac{\cos\mathopen{}\mathclose{{}\left(\sqrt{2\sigma\bar{\sigma}}\,\mathcal{U}}\right)}{\cos\mathopen{}\mathclose{{}\left(\sqrt{2\sigma\bar{\sigma}}}\right)}+\frac{E_{+}(\vartheta,\varphi)-E_{-}(\vartheta,\varphi)}{2}\frac{\sin\mathopen{}\mathclose{{}\left(\sqrt{2\sigma\bar{\sigma}}\,\mathcal{U}}\right)}{\sin\mathopen{}\mathclose{{}\left(\sqrt{2\sigma\bar{\sigma}}}\right)},$
(10)

where $E_{\pm}$ are free corner data at the initial and final cross section of
$\mathcal{N}_{3}$.

The action (LABEL:gammaactn) determines the symplectic structure for the
initial data on $\mathcal{N}_{3}$. Upon taking the pull-back to the solution
space of the transport equations, we obtain canonical commutation relations
[7]. The only non-vanishing brackets are

$\displaystyle\big{\\{}a(\boldsymbol{z}),\bar{a}(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=\mathrm{i}\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime}),$
(11a)
$\displaystyle\big{\\{}b(\boldsymbol{z}),\bar{b}(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=\mathrm{i}\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime}),$
(11b)
$\displaystyle\big{\\{}c(\boldsymbol{z}),\bar{c}(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=2\,\mathrm{i}\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime})\,L(\boldsymbol{z}),$
(11c)
$\displaystyle\big{\\{}L(\boldsymbol{z}),{c}(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=-\mathrm{i}\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime})\,c(\boldsymbol{z}),$
(11d)
$\displaystyle\big{\\{}L(\boldsymbol{z}),\bar{c}(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=+\mathrm{i}\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime})\,\bar{c}(\boldsymbol{z}),$
(11e)

with $\boldsymbol{z}=(\vartheta,\varphi)$ and

$\displaystyle\big{\\{}c(\boldsymbol{z}),U(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=XU(\boldsymbol{z})\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime}),$
(12a)
$\displaystyle\big{\\{}\bar{c}(\boldsymbol{z}),U(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=\bar{X}U(\boldsymbol{z})\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime}),$
(12b)
$\displaystyle\big{\\{}L(\boldsymbol{z}),U(\boldsymbol{z}^{\prime})\big{\\}}$
$\displaystyle=-\frac{1}{2}JU(\boldsymbol{z})\,\delta^{(2)}(\boldsymbol{z}|\boldsymbol{z}^{\prime}).$
(12c)

The relationship between the canonical variables and the geometry of the
impulsive boundary data is determined by two sets of equations. First of all,
we have

$\displaystyle U$
$\displaystyle=\mathrm{e}^{\gamma\ln\mathopen{}\mathclose{{}\left(\tan\mathopen{}\mathclose{{}\left(\sqrt{2\sigma\bar{\sigma}}}\right)/\sqrt{2\sigma\bar{\sigma}}}\right)J}S_{-},$
(13) $\displaystyle a$ $\displaystyle=\frac{\sqrt{E_{+}}}{\sqrt{8\pi\gamma
G}}\operatorname{ch}\bigl{(}2\sqrt{\sigma\bar{\sigma}}\bigr{)}\mathrm{e}^{-\mathrm{i}\mathopen{}\mathclose{{}\left[\Delta_{+}+2\gamma\ln\mathopen{}\mathclose{{}\left(\cos\mathopen{}\mathclose{{}\left(\sqrt{2\sigma\bar{\sigma}}}\right)}\right)}\right]},$
(14) $\displaystyle b$ $\displaystyle=\frac{\sqrt{E_{+}}}{\sqrt{8\pi\gamma
G}}\operatorname{sh}\bigl{(}2\sqrt{\sigma\bar{\sigma}}\bigr{)}\mathrm{e}^{\mathrm{i}\mathopen{}\mathclose{{}\left[\Delta_{+}+\phi+2\gamma\ln\mathopen{}\mathclose{{}\left(\frac{\sin\mathopen{}\mathclose{{}\left(\sqrt{2\sigma\bar{\sigma}}}\right)}{\sqrt{2\sigma\bar{\sigma}}}}\right)}\right]},$
(15)

where $\Delta_{+}(\vartheta,\varphi)=\Delta(\mathcal{U}=1,\vartheta,\varphi)$
and $\phi:\sigma=|\sigma|\mathrm{e}^{\mathrm{i}\phi}$ are $U(1)$ angles. The
$SL(2,\mathbb{R})$ element $U$ determines additional corner data that
parametrize the shape degrees of freedom of the signature $(0$$+$$+)$ metric
at the boundary. The overall scale of the boundary metric is set by the $U(1)$
generator $L$ and the norm of the oscillator variables $a$ and $b$. We obtain

$\displaystyle E_{-}+E_{+}$ $\displaystyle=16\pi\gamma
G\mathopen{}\mathclose{{}\left(L+a\bar{a}}\right),$ (16) $\displaystyle
E_{-}-E_{+}$ $\displaystyle=16\pi\gamma
G\mathopen{}\mathclose{{}\left(L+b\bar{b}}\right).$ (17)

The quotient of the two oscillators determines the shear

$\operatorname{th}\bigl{(}2\sqrt{\sigma\bar{\sigma}}\bigr{)}=\sqrt{\frac{\bar{b}b}{\bar{a}a}}.$
(18)

Finally, there is one residual pair of second-class constraints, imposing
recurrence relations for physical states,

$\displaystyle c\bar{a}\bar{b}$
$\displaystyle=-\sqrt{2}\gamma\mathopen{}\mathclose{{}\left(L+\bar{a}a}\right)\,\sqrt{\bar{a}a\bar{b}b}\,\tan\bigl{(}\sqrt{2\sigma\bar{\sigma}}\bigr{)}-\mathrm{i}\bar{a}a\bar{b}b,$
(19) $\displaystyle\bar{c}{a}{b}$
$\displaystyle=-\sqrt{2}\gamma\mathopen{}\mathclose{{}\left(L+\bar{a}a}\right)\,\sqrt{\bar{a}a\bar{b}b}\,\tan\bigl{(}\sqrt{2\sigma\bar{\sigma}}\bigr{)}+\mathrm{i}\bar{a}a\bar{b}b.$
(20)

The constraints are second-class. At the quantum level, only one of them can
be imposed strongly. The other maps the physical Hilbert space into its
orthogonal complement.

##  3 Critical luminosity

Upon quantizing the oscillators $a,\bar{a}$, and $b,\bar{b}$ and the
$SL(2,\mathbb{R})\times\mathfrak{sl}(2,\mathbb{R})$ variables
$(U,L,c,\bar{c})$, we obtain a kinematical Hilbert space. Physical states lie
in the kernel of one of the constraints, e.g. (19). The kinematical Hilbert
space carries a unitary representation of $SL(2,\mathbb{R})$. The
representations are characterized by the value of the Casimir. At the
classical level,

$\displaystyle L^{2}-c\bar{c}$ $\displaystyle=\frac{1}{8\pi
G}\biggl{[}\frac{1}{4\gamma^{2}}(E_{-}-E_{+})^{2}-\frac{1}{\gamma^{2}}\operatorname{sh}^{2}\bigl{(}2\sqrt{\sigma\bar{\sigma}}\bigr{)}E_{+}E_{-}-\frac{1}{2}(E_{+}+E_{-})^{2}\tan^{2}\bigl{(}\sqrt{2\sigma\bar{\sigma}}\bigr{)}\biggr{]}.$
(21)

When $L^{2}>c\bar{c}$, the spectrum of the Casimir is discrete. When
$L^{2}<c\bar{c}$, it is continuous. For $\sigma=0$ and $E_{+}\neq E_{-}$, the
Casimir is positive, i.e. $L^{2}>c\bar{c}$. As we increase $\sigma$, we will
reach a critical value $\sigma_{\text{\it crit}}$, where the sign will flip.
Expanding the Casimir (21) for small shear, we obtain

$|\sigma_{\text{\it
crit}.}|^{2}=\frac{1}{4}\frac{(E_{-}-E_{+})^{2}}{\gamma^{2}(E_{+}+E_{-})^{2}+4E_{+}E_{-}}+\mathcal{O}(|\sigma_{\text{\it
crit}}|^{3}).$ (22)

In a neighbourhood of future null infinity $\mathcal{I}^{+}$, we have an
asymptotic expansion with respect to an affine radial Bondi coordinate $r$
[31, 32, 33]. The shear of the ingoing null generators vanishes as
$\mathcal{O}(r^{-1})$. The area density $\Omega^{2}$ blows up as
$\mathcal{O}(r^{2})$. We can thus use (22) to evaluate $\sigma_{\text{\it
crit}}$ in the asymptotic limit, in which we take $\mathcal{N}_{3}$ to future
null infinity $\mathcal{I}^{+}$. Below, we have a pictorial representation of
the resulting geometry.

Impulsive data at future null infinity. In the shaded region
$\mathcal{N}_{3}$, the time derivative of the asymptotic shear
${\sigma}^{(0)}(u,\vartheta,\varphi)$ is constant in $u$, everywhere else
$\dot{\sigma}^{(0)}=0$.

If we consider such boundary data at future null infinity $\mathcal{I}^{+}$,
we can embed them into Bondi coordinates $(u,r,\vartheta,\phi)$. The pulse
starts at an asymptotic Bondi time $u_{-}(\vartheta,\varphi)$ and terminates
at $u_{+}(\vartheta,\varphi)$ with total duration $(\Delta
u)(\vartheta,\varphi)=u_{+}(\vartheta,\varphi)-u_{-}(\vartheta,\varphi)$. To
leading order in the $1/r$-expansion, the map between the boundary intrinsic
time coordinate $\mathcal{U}:\mathcal{N}_{3}\rightarrow[-1,1]$ and the
asymptotic Bondi time $u$ is a mere angle-dependent dilation,

$\partial^{a}_{\mathcal{U}}=\frac{(\Delta
u)(\vartheta,\varphi)}{2}\partial^{a}_{u}+\mathcal{O}(r^{-1}).$ (23)

Upon introducing an adapted Newman–Penrose tetrad
$(k^{a},\ell^{a},m^{a},\bar{m}^{a})$, see [34], where $k^{a}=\partial^{a}_{r}$
and $\ell^{a}=\partial^{a}_{u}+\mathcal{O}(r^{-1})$, and both
$k_{a}=-\nabla_{a}u$ and $\ell_{a}$ are surface orthogonal, we obtain the
asymptotic expansion

$\displaystyle\sigma_{(\ell)}$
$\displaystyle=m^{a}m^{b}\nabla_{a}\ell_{b}=-\frac{\dot{\sigma}^{(0)}(u,\vartheta,\varphi)}{r}+\mathcal{O}(r^{-2}),$
(24) $\displaystyle\sigma_{(k)}$
$\displaystyle={m}^{a}{m}^{b}\nabla_{a}k_{b}=\frac{\sigma^{(0)}(u,\vartheta,\varphi)}{r^{2}}+\mathcal{O}(r^{-3}).$
(25)

The map (23) between the two clock variables implies a relationship between
the asymptotic Bondi shear $\sigma^{(0)}$ and a family of radiative data
$\\{\sigma_{r}\\}$ at the abstract boundary $\mathcal{N}_{3}$,

$\sigma_{r}(\vartheta,\varphi)=\frac{(\Delta
u)(\vartheta,\varphi)}{2}\frac{\dot{\sigma}^{(0)}(\vartheta,\varphi)}{r}+\mathcal{O}(r^{-2}).$
(26)

This equation allows us to relate the asymptotic Bondi shear to the critical
shear (22), where the Casimir changes its sign. Using the standard round
metric at future null infinity $\mathcal{I}^{+}$, we then also have

$\displaystyle
E_{+}(\vartheta,\varphi)=E_{-}(\vartheta,\varphi)+\mathcal{O}(r^{1})=r^{2}+\mathcal{O}(r^{1}),$
(27) $\displaystyle
E_{-}(\vartheta,\varphi)-E_{+}(\vartheta,\varphi)=-4r(\Delta
u)(\vartheta,\varphi)+\mathcal{O}(r^{0}).$ (28)

We insert the expansion back into (22) and obtain the critical shear of the
null generators $\partial_{\mathcal{U}}^{a}$

$|\sigma_{\text{\it crit.}}|^{2}=\frac{1}{4}\frac{(\Delta
u)^{2}(\vartheta,\varphi)}{\gamma^{2}+1}\frac{1}{r^{2}}+\mathcal{O}(r^{-3}).$
(29)

We translate this value back into the asymptotic Bondi frame. Equation (26)
implies that $\sigma_{\text{\it crit.}}$ corresponds to a critical value for
the asymptotic shear given by

$|\dot{\sigma}^{(0)}_{\text{\it
crit.}}|=\frac{1}{\sqrt{\gamma^{2}+1}}+\mathcal{O}(r^{-1}).$ (30)

Using the Bondi mass loss formula, we infer a critical value for the
luminosity of the gravitational wave pulse,

$\mathcal{L}_{\text{\it crit.}}=\frac{c^{5}}{4\pi
G}\oint_{S_{2}}d^{2}\Omega\,|\dot{\sigma}^{(0)}_{\text{\it
crit.}}|^{2}=\frac{\mathcal{L}_{\mathrm{P}}}{\gamma^{2}+1}.$ (31)

If $L^{2}-c\bar{c}>0$, the impulsive wave will have a luminosity smaller than
$\mathcal{L}_{\text{\it crit.}}$ In this regime, each light ray carries a
discrete unitary representation of $SL(2,\mathbb{R})$. The fundamental
operators are

$\displaystyle
c^{\dagger}|N,n_{a},n_{b},n_{c}\rangle=\mathrm{i}\sqrt{(N+n_{c})(n_{c}-N+1)}|N,n_{a},n_{b},n_{c}+1\rangle,$
(32)
$\displaystyle\mathopen{}\mathclose{{}\left(L^{2}-\tfrac{1}{2}(cc^{\dagger}+c^{\dagger}c)}\right)|N,n_{a},n_{b},n_{c}\rangle=N(N-1)|N,n_{a},n_{b},n_{c}\rangle,$
(33) $\displaystyle
a^{\dagger}|N,n_{a},n_{b},n_{c}\rangle=\sqrt{n_{a}+1}|N,n_{a}+1,n_{b},n_{c}\rangle,$
(34) $\displaystyle
b^{\dagger}|N,n_{a},n_{b},n_{c}\rangle=\sqrt{n_{b}+1}|N,n_{a},n_{b}+1,n_{c}\rangle,$
(35)

where $n_{a}$ and $n_{b}$ are integers, $n_{c}=N,N+1,\dots$ is integer or
half-integer and $N=1,\tfrac{3}{2},\dots$. Physical states are annihilated by
the constraint (19). For the discrete series representations of
$SL(2,\mathbb{R})$, a unique solution can be found by a linear combination of
states

$\displaystyle\bigl{\\{}|N,n+r,r,N+m-r\rangle:r=0,\dots,m\bigr{\\}}.$ (36)

For the continuous series representations of $SL(2,\mathbb{R})$,
$L^{2}-c\bar{c}<0$. The spectrum of the Casimir is continuous, for any
$s\in\mathbb{R}$, we have

$\displaystyle
c^{\dagger}|s,n_{a},n_{b},n_{c}\rangle=\frac{1}{2\mathrm{i}}(\mathrm{i}s-2n_{c}-1)|s,n_{a},n_{b},n_{c}+1\rangle,$
$\displaystyle\mathopen{}\mathclose{{}\left(L^{2}-\tfrac{1}{2}(cc^{\dagger}+c^{\dagger}c)}\right)|s,n_{a},n_{b},n_{c}\rangle=-\frac{1}{4}(s^{2}+1)|s,n_{a},n_{b},n_{c}\rangle.$
(37)

In this regime, the impulsive wave will have a luminosity bigger than
$\mathcal{L}_{\text{\it crit.}}$ The spectrum of the Casimir is continuous,
and the operator $L$ is no longer bounded from below. This has important
consequences. The recurrence relations (19) will not terminate, physical
states will be superpositions of kinematical states, where the quantum numbers
$n_{a}$ and $n_{b}$ will become arbitrarily large. Therefore, the shear
$\sigma\bar{\sigma}$ will be unbounded from above, see (18). If we take the
Born rule and compute the probabilities for $\sigma\bar{\sigma}$ to take a
certain value, there will always be a chance that an observer obtains
$\sqrt{2\sigma\bar{\sigma}}>\pi/2$. In this case, the profile of the area
density (10) will pass through a caustic, where $\Omega^{2}=0$. When there is
a caustic, we violate the implicit assumption that we are in a smooth
asymptotic region, in which $\Omega^{2}=r^{2}+\mathcal{O}(r)$, as
$r\rightarrow\infty$. This can be avoided only when $L^{2}>c\bar{c}$, i.e.
when the luminosity of the gravitational wave pulse is below
$\mathcal{L}_{\text{\it crit.}}$. Then, the physical states are built from
superpositions of the discrete series unitary representations of
$SL(2,\mathbb{R})$ and for any physical state the quantum numbers $n_{a}$ and
$n_{b}$ will be bounded from above.

##  4 Outlook and Conclusion

The critical luminosity (31) separates the continuous spectrum from the
discrete eigenvalues of the $SL(2,\mathbb{R})$ Casimir. The bound depends on
the Barbero–Immirzi parameter $\gamma$. This is a common feature in $D=4$.
When $\gamma\neq 0$, the boundary charges are a mixture of electric and
magnetic contributions that otherwise vanish in the $\gamma\rightarrow\infty$
limit [35, 11, 7]. The spectrum of the charges is determined by their
algebraic properties alone, but the map between the charges and the physical
observables depends on $\gamma$. In this way, the spectrum of physical
observables can depend on $\gamma$. This is analogous to how the
$\theta$-angle in quantum electrodynamics enters the Dirac quantization
condition between magnetic and electric charges [36]. In loop quantum gravity,
this effect is responsible for the quantization of geometric observables, such
as area, angles, volumes and length [37, 38, 39, 40, 41]. Such a fundamental
quantum discreteness of geometry affects other physical observables. It
creates a fundamental bound on the energy density of matter [42, 43] and
perhaps also acceleration [44]. Here, we found a similar bound on the
gravitational wave luminosity (31). The analysis is based on a non-
perturbative quantization of radiative data at finite distance. The result is
only partial, because there is an implicit assumption: the validity of the
classical asymptotic $1/r$-expansion when applied to the spectrum of
gravitational observables at finite null boundaries. If the luminosity exceeds
the critical luminosity (31), this assumption may no longer be valid due to
the possible creation of caustics. What we have shown so far is only a first
step. A more refined investigation will follow to understand the significance
of the Planck power for the spectrum of the gravitational wave luminosity in
non-perturbative quantum gravity.

## References

  * [1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation.  W. H. Freeman, San Francisco, 1973.
  * [2] V. Cardoso, T. Ikeda, C. J. Moore, and C.-M. Yoo, “Remarks on the maximum luminosity,” Phys. Rev. D 97 (2018), no. 8, 084013, arXiv:1803.03271.
  * [3] A. Jowsey and M. Visser, “Reconsidering maximum luminosity,” Int. J. Mod. Phys. D 30 (2021), no. 14, 2142026, arXiv:2105.06650.
  * [4] D. Keitel et al., “The most powerful astrophysical events: Gravitational-wave peak luminosity of binary black holes as predicted by numerical relativity,” Phys. Rev. D 96 (2017), no. 2, 024006, arXiv:1612.09566.
  * [5] F. Zappa, S. Bernuzzi, D. Radice, A. Perego, and T. Dietrich, “Gravitational-Wave Luminosity of Binary Neutron Stars Mergers,” Phys. Rev. Lett. 120 (Mar, 2018) 111101, arXiv:1712.04267.
  * [6] V. Cardoso, O. J. C. Dias, and J. P. S. Lemos, “Gravitational radiation in D-dimensional space-times,” Phys. Rev. D 67 (2003) 064026, arXiv:hep-th/0212168.
  * [7] W. Wieland, “Quantum geometry of the null cone,” arXiv:2401.17491.
  * [8] M. P. Reisenberger, “The symplectic 2-form for gravity in terms of free null initial data,” Class. Quant. Grav. 30 (2013) 155022, arXiv:1211.3880.
  * [9] M. P. Reisenberger, “The Poisson brackets of free null initial data for vacuum general relativity,” Class. Quant. Grav. 35 (2018), no. 18, 185012, arXiv:1804.10284.
  * [10] L. Ciambelli, L. Freidel, and R. G. Leigh, “Null Raychaudhuri: Canonical Structure and the Dressing Time,” arXiv:2309.03932.
  * [11] W. Wieland, “Gravitational SL(2, $\mathbb{R}$) algebra on the light cone,” JHEP 07 (2021) 057, arXiv:2104.05803.
  * [12] J. Samuel, “A Lagrangian basis for Ashtekar’s formulation of canonical gravity,” Pramana 28 (1987) L429–L432, doi:10.1007/BF02847105.
  * [13] T. Jacobson and L. Smolin, “The Left-Handed Spin Connection as a Variable for Canonical Gravity,” Phys. Lett. B 196 (1987) 39–42, doi:10.1016/0370-2693(87)91672-8.
  * [14] S. Holst, “Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action,” Phys. Rev. D 53 (November, 1996) 5966–5969, arXiv:gr-qc/9511026.
  * [15] J. F. Barbero G., “Real Ashtekar variables for Lorentzian signature space times,” Phys. Rev. D 51 (1995) 5507–5510, arXiv:gr-qc/9410014.
  * [16] G. Immirzi, “Real and complex connections for canonical gravity,” Class. Quant. Grav. 14 (1997) L177–L181, arXiv:gr-qc/9612030.
  * [17] A. Ashtekar and S. Speziale, “Horizons and Null Infinity: A Fugue in 4 voices,” arXiv:2401.15618.
  * [18] A. Ashtekar, “New Variables for Classical and Quantum Gravity,” Phys. Rev. Lett. 57 (1986) 2244–2247, doi:10.1103/PhysRevLett.57.2244.
  * [19] A. Raychaudhuri, “Relativistic Cosmology. I,” Phys. Rev. 98 (1955) 1123–1126, doi:10.1103/PhysRev.98.1123.
  * [20] A. P. Balachandran, L. Chandar, and A. Momen, “Edge states in gravity and black hole physics,” Nucl. Phys. B 461 (1996) 581–596, arXiv:gr-qc/9412019.
  * [21] S. Carlip, “Statistical mechanics of the (2+1)-dimensional black hole,” Phys. Rev. D 51 (1995) 632–637, arXiv:gr-qc/9409052.
  * [22] G. Barnich and C. Troessaert, “BMS charge algebra,” JHEP 12 (2011) 105, arXiv:1106.0213.
  * [23] W. Donnelly and L. Freidel, “Local subsystems in gauge theory and gravity,” JHEP 09 (2016) 102, arXiv:1601.04744.
  * [24] S. Carrozza and P. A. Hoehn, “Edge modes as reference frames and boundary actions from post-selection,” JHEP 02 (2022) 172, arXiv:2109.06184.
  * [25] L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale, “Extended corner symmetry, charge bracket and Einstein’s equations,” JHEP 09 (2021) 083, arXiv:2104.12881.
  * [26] L. Freidel, M. Geiller, and D. Pranzetti, “Edge modes of gravity. Part I. Corner potentials and charges,” JHEP 11 (2020) 026, arXiv:2006.12527.
  * [27] L. Freidel, M. Geiller, and D. Pranzetti, “Edge modes of gravity. Part II. Corner metric and Lorentz charges,” JHEP 11 (2020) 027, arXiv:2007.03563.
  * [28] L. Freidel, M. Geiller, and W. Wieland, “Corner symmetry and quantum geometry,” in Handbook of Quantum Gravity, L. M. Cosimo Bambi and I. Shapiro, eds.  Springer, 2023.  arXiv:2302.12799.
  * [29] W. Wieland, “Null infinity as an open Hamiltonian system,” JHEP 04 (2021) 095, arXiv:2012.01889.
  * [30] W. Wieland, “Discrete gravity as a topological field theory with light-like curvature defects,” JHEP 05 (2017) 142, arXiv:1611.02784.
  * [31] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, “Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system,” Proc. of the Royal Soc. Lond. A: Mathematical, Physical and Engineering Sciences 269 (1962), no. 1336, 21–52, doi:10.1098/rspa.1962.0161.
  * [32] R. Sachs, “Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time,” Proceedings of the Royal Society London A 270 (1962), no. 1340, 103–126, doi:10.1098/rspa.1962.0206.
  * [33] H. Bondi, “Gravitational Waves in General Relativity,” Nature 186 (1960), no. 4724, 535–535, doi:10.1038/186535a0.
  * [34] E. Newman and R. Penrose, “An Approach to Gravitational Radiation by a Method of Spin Coefficients,” Journal of Mathematical Physics 3 (1962), no. 3, 566–578, doi:10.1063/1.1724257.
  * [35] W. Wieland, “Fock representation of gravitational boundary modes and the discreteness of the area spectrum,” Ann. Henri Poincaré 18 (2017) 3695–3717, arXiv:1706.00479.
  * [36] E. Witten, “Dyons of Charge $e\theta/2\pi$,” Phys. Lett. B 86 (1979) 283–287, doi:10.1016/0370-2693(79)90838-4.
  * [37] C. Rovelli and L. Smolin, “Discreteness of area and volume in quantum gravity,” Nuclear Physics B 442 (1995), no. 3, 593–619, arXiv:gr-qc/9411005.
  * [38] A. Ashtekar and J. Lewandowski, “Quantum theory of geometry I.: Area operators,” Class. Quant. Grav. 14 (1997) A55–A82, arXiv:gr-qc/9602046.
  * [39] A. Ashtekar and J. Lewandowski, “Quantum Theory of Geometry II: Volume operators,” Advances in Mathematical and Theoretical Physics 1 (1997) 388–429, arXiv:gr-qc/9711031.
  * [40] E. Bianchi, “The Length operator in Loop Quantum Gravity,” Nucl. Phys. B 807 (2009) 591–624, arXiv:0806.4710.
  * [41] E. Bianchi and H. M. Haggard, “Discreteness of the volume of space from Bohr-Sommerfeld quantization,” Phys. Rev. Lett. 107 (2011) 011301, arXiv:1102.5439.
  * [42] A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum nature of the big bang,” Phys. Rev. Lett. 96 (2006) 141301, arXiv:gr-qc/0602086.
  * [43] A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum Nature of the Big Bang: Improved dynamics,” Phys. Rev. D 74 (2006) 084003, arXiv:gr-qc/0607039.
  * [44] C. Rovelli and F. Vidotto, “Evidence for Maximal Acceleration and Singularity Resolution in Covariant Loop Quantum Gravity,” Phys. Rev. Lett. 111 (2013) 091303, arXiv:1307.3228.


# Podcast Metadata and Content: Episode Relevance and Attractiveness in Ad Hoc
Search

Ben Carterette1, Rosie Jones1, Gareth F. Jones2, Maria Eskevich3, Sravana
Reddy1, Ann Clifton1, Yongze Yu1, Jussi Karlgren1, Ian Soboroff4 1Spotify,
2Dublin City University, 3CLARIN ERIC, 4NIST1United States and Sweden,
2Ireland, 3Netherlands, 4United States

(2021; 2021)

###### Abstract.

Rapidly growing online podcast archives contain diverse content on a wide
range of topics. These archives form an important resource for entertainment
and professional use, but their value can only be realized if users can
rapidly and reliably locate content of interest. Search for relevant content
can be based on metadata provided by content creators, but also on transcripts
of the spoken content itself. Excavating relevant content from deep within
these audio streams for diverse types of information needs requires varying
the approach to systems prototyping. We describe a set of diverse podcast
information needs and different approaches to assessing retrieved content for
relevance. We use these information needs in an investigation of the utility
and effectiveness of these information sources. Based on our analysis, we
recommend approaches for indexing and retrieving podcast content for ad hoc
search.

datasets, search, information retrieval, spoken content retrieval, podcasts,
broadcast media

††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the
44th International ACM SIGIR Conference on Research and Development in
Information Retrieval; July 11–15, 2021; Virtual Event, Canada††booktitle:
Proceedings of the 44th International ACM SIGIR Conference on Research and
Development in Information Retrieval (SIGIR ’21), July 11–15, 2021, Virtual
Event, Canada††journalyear: 2021††copyright: acmlicensed††conference:
Proceedings of the 44th International ACM SIGIR Conference on Research and
Development in Information Retrieval; July 11–15, 2021; Virtual Event,
Canada.††booktitle: Proceedings of the 44th International ACM SIGIR Conference
on Research and Development in Information Retrieval (SIGIR ’21), July 11–15,
2021, Virtual Event, Canada††price: 15.00††isbn: 978-1-4503-8037-9/21/07††doi:
10.1145/3404835.3463101††ccs: Information systems Test collections††ccs:
Information systems Relevance assessment††ccs: Information systems
Presentation of retrieval results

##  1\. Introduction

Podcasts are a rapidly expanding as a popular medium for delivery of spoken
audio content. As of 2021, more than 38 million podcast episodes are available
online (Whitner, 2021). Given the amount of podcast material available, we
believe it is increasingly important that it be fully searchable if it is to
be fully exploited by users.

At present, most podcast search is done via catalog match, using show titles,
episode titles, and sometimes metadata provided by podcast creators. This
metadata is of highly varied quality and hence usefulness in supporting search
operations. Recommendation from friends and family (Research, 2019) remains in
the top-three ways people find podcasts, while non-podcast-listeners in the
same study say that they do not know how to find a podcast, or that they do
not really know where to start.

In this paper we report experiments using the test collection from the TREC
2020 Podcasts Track which show that the automatically transcribed content of
podcast episodes is more reliably useful for search than metadata provided by
podcast creators.

##  2\. Podcast Format

Podcasts are distributed as audio streams or files, through RSS feeds
containing multiple metadata fields (Apple, 2020). A podcast show has a title,
description, language, consumption order (episodic or sequential), and a list
of categories (e.g., News, Sports, Comedy) selected by the creator from a
predefined taxonomy. A show typically has multiple episodes, which are the
distinct audio files. Each episode has its own title, description, and other
information. All this metadata may be noisy or inadequate (Sharpe, 2020).

Podcasting is typically a spoken-word medium. However, the relative ease and
low cost of recording and publishing means there is great variability in the
specifics. Podcast episodes have a wide range of lengths, from just a few
minutes to hours, although they tend to be between half an hour and an hour
long. Podcast content can vary in form, e.g. being scripted or informal
dialogues. These can pose problems for search systems built for text archives,
and also for the transcription of the content.

##  3\. Speech retrieval

At first glance, podcasts are spoken documents, which have been well-studied
in the TREC Spoken Document Retrieval Track which ran from 1997-2000 (Garofolo
et al., 2000). However, this work was based on news corpora, which are
relatively homogenous in genre, style, and speaking professionalism, while
podcasts come in many disparate forms, increasing the difficulty of the task.
Retrieval from an archive of oral history has also been well-studied (Pecina
et al., 2008), but lacks the multi-speaker, multi-genre aspect of podcasts.
The NTCIR Spoken Query and Document tasks (Akiba et al., 2011) used a document
collection of spontaneous speech, but the 600 hours of speech is tiny compared
to the 50,000 hours of speech in the Spotify podcasts corpus (Clifton et al.,
2020). The identification of “jump-in” points in multimedia content based on
the spoken soundtrack at Mediaeval from 2011-2015 (Larson et al., 2011;
Eskevich et al., 2012, 2015) motivates an approach to podcast search in which
we identify the best place to start listening. We refer to Jones (2019) for a
more complete overview of research in spoken content retrieval from its
beginnings in the early 1990s to today.

Because of the availability of metadata fields, podcasts can be represented as
semi-structured documents, and models like BM25F (Robertson et al., 2004),
Field Relevance Models (Kim and Croft, 2012), and NRMF (Zamani et al., 2018),
can be adopted for podcast search tasks. Evaluation campaigns such as the INEX
XML retrieval initiative (Fuhr et al., 2006; Lalmas and Tombros, 2007) have
studied such models. As argued by Besser et al. (2008), the goals of podcast
search may be similar to those for blog search, if podcasts are viewed as
audio blogs.

In addition, the publication format of podcasts, as series of episodes
typically consumed in sequence, and the prominence of hosts and certain
popular guests, act as a filter on top of the topical search. Tsagkias et al.
(2008) argued that the quality and credibility of podcasts, which are
sometimes considered during the relevance assessment process, can be
characterized using four types of indicators pertaining to the podcast
content, the podcast creator, the podcast context, or the technical execution
of the podcast. These facts distinguish podcast search from most well-
established search tasks including adhoc, web, personal, and enterprise
search.

##  4\. TREC 2020 Podcasts Track

The Podcasts Track ran for the first time at TREC 2020 (Jones et al., 2020).
It featured two tasks: a search task and a summarization task, the former is
the focus of this paper.

Podcast search focused on a segment retrieval task defined as the problem of
finding relevant segments of podcast episodes given a query representing a
specific information need. The corpus for the task comprised 100,000 podcast
episodes released in 2019, including metadata, audio, and full transcripts
produced by automatic speech recognition (ASR) (Clifton et al., 2020).
Participants were asked to retrieve unique episode identifiers (episode URIs)
along with a time offset to the start of a two-minute segment starting on the
minute within that episode. The segment corpus comprises 3.4M segments with an
average word count of $340\pm 70$ per segment.

Seven participating groups submitted a total of 24 runs to the search task.
Each run ranked up to 1,000 segments for each of the 50 queries. Runs used
retrieval techniques such as relevance feedback, query expansion, word2vec,
BERT reranking, and fusion.

##  5\. Podcast search

We now take a step back to ask: why are we interested in this kind of podcast
search? What are the user needs that can be addressed by retrieving segments
of episodes? What other information could be retrieved that might improve the
user experience?

###  5.1. Information needs for podcast search

User needs pertaining to podcasts include education, entertainment, and
information. The format and variability of the podcast medium – series of
episodes, the importance of the specific host or guest in the episode, range
of presentation styles from monologues to interviews and banter, production
quality – are additional aspects that may determine whether a user is
interested in listening or not, entirely separate from topical relevance.
These factors make podcast search different from traditional search tasks.

###  5.2. Topic development for TREC

Since there is currently no large-scale content-based podcast search engine,
there is no simple source of sample topics. Thus we developed topics by
introspection, considering what we might use a podcast search engine for, what
we could imagine others using it for, what types of information needs make it
more attractive than web search, etc. Topic development used several sources:
lists of events in 2019, topic creators interests, and browsing metadata for
potentially interesting content. To determine whether the topic would be
interesting, we roughly compared metadata matches, web search results, and the
results from our in-house transcript search engine. Topics were finally
selected based on whether they retrieved interesting content from a simple
search index of the podcast transcripts.

We also experimented with “known item” and “refinding” information needs. In
total, eight development and 50 test topics were developed, with 13 topics
within the test set being labeled as refinding or known-item.

    <topic>
    <num>34</num>
    <query>halloween stories and chat</query>
    <type>topical</type>
    <description>I love Halloween and I want to hear stories and conversations
                 about things people have done to celebrate it.  I am not looking
                 for information about the history of Halloween or generalities
                 about how it is celebrated, I want specific stories from
                 individuals.
    </description>
    </topic>

Figure 1. Example search topics

###  5.3. Assessing relevance

####  5.3.1. TREC

Participant submissions were pooled to depth 20 (minimum 128 segments per
topic, maximum 306) and reviewed by NIST assessors. The assessors primarily
reviewed the ASR text of the segment, but could listen to audio if the need
arose. The assessor’s view showed each retrieved segment within the context of
the transcript of the full episode, so that they could explore the context of
the segment to better understand it. All segments from the same episode were
judged in sequence together.

The relevance judgments were on a four-point scale of bad (0), fair (1), good
(2), or excellent (3). For known-item and refinding topics, an additional
level of “perfect” (4) was added and meant that the segment was precisely what
the user was looking for. Excellent segments were completely on-topic,
provided highly relevant information, and represented an ideal entry point
into the episode.

####  5.3.2. Additional Assessments

source | no. topics | type | no. assessments
---|---|---|---
TREC | 16 | title | 1,381
|  | title+description | 1,381
|  | transcript-segment | 1,863
|  | transcript-full | 114
metadata | 50 | title | 1,049
+transcript |  | title+description | 1,049
indexes |  | transcript-segment | 514
|  | transcript-full | 254

Table 1. New assessments collected for this work. Assessments were collected
from documents retrieved for a subsample of topics from TREC runs, and from
all documents retrieved from indexes described in Section 6.

The authors of this work independently assessed podcast metadata and full
transcripts. From the metadata, we extracted episode titles and descriptions.
Episode titles and descriptions are frequently too short or unspecified to
accurately reflect episode relevance, so instead we assess how attractive they
might be to a user with the stated information need.

We produced the following additional assessments:

  * •

Title attractiveness. Would a user with the given information need find the
episode a good candidate to stream based solely on the title?

  * •

Title and description attractiveness. Would the user find the episode a good
candidate to stream based on the title and description together?

  * •

Full transcript relevance. Is the episode relevant to the information need
(based on the full transcript)?

We also used TREC segment-level judgments to obtain transcript-level judgments
by taking the maximum relevance of any segment of an episode as the relevance
of the transcript. We refer to this as the transcript-segment judgment. We
acknowledge that these may be unreliable; it is possible that there is a
segment of the episode more relevant than any seen by a TREC assessor and
therefore our transcript-level judgment is low.

We selected documents to judge as follows: First, we built five new indexes
using different combinations of metadata and transcripts; see Section 6 for
details. We retrieved the top 10 results for all TREC topics from each of
these five indexes, pooled them, and judged all titles and descriptions for
attractiveness, plus a select subset of transcripts for relevance. This
provided about 1,000 episode title and description attractiveness judgments,
plus about 250 transcripts that had not previously been assessed for TREC.

We also pooled the top 10 results from all 24 TREC submitted runs. We selected
a random sample of 16 topics for judging title and description attractiveness
in this pool. We judged another 1,400 episode titles and descriptions in this
set, as well as another 100 transcripts. Episodes were ordered by a function
of rank position in runs, which may produce ordering effects (Eisenberg and
Barry, 1988). We did not investigate this. Table 1 summarizes the new
assessments.

There is very little overlap between episodes from our new indexes and
episodes from TREC submitted runs. Across all 50 topics, there are only 79
episodes that occur in the top 10 of both our new runs and the 24 TREC
submissions—only $7.7\%$ of all episodes retrieved by our five new runs. By
focusing on metadata we retrieve many more potentially relevant episodes.

##  6\. Indexing and Searching Podcasts

An RSS feed with podcast episode titles, descriptions, and other metadata, in
addition to the audio file, includes a lot of information that could be
indexed for retrieval. In order to understand the relative utility of various
free-text fields, we constructed Lucene indexes of episode titles, episode
descriptions, titles and descriptions concatenated, full ASR transcripts, and
transcripts concatenated with titles and descriptions. We retrieved rankings
of episode URIs for all 50 TREC topics from each of these five indexes.

We also obtained the TREC submitted runs. In order to make the segment
rankings comparable to URI rankings from our indexes, we compressed all
retrieved segments from one URI to a single result for that URI, ranked at the
position of the top-ranked segment, and then removed all subsequent mentions
of that URI.

Table 2 demonstrates how the choice of assessment and index impact retrieval
results. Rows correspond to retrieval from the indexes described above.
Columns correspond to the four different judgment types summarized in Table 1.
Conclusions from this table:

| assessment type
---|---
|  | title + | transcript- | transcript-
index | title | description | segment | full
title | 0.43 | 0.33 | 0.19 | 0.21
description | 0.40 | 0.51 | 0.24 | 0.45
title+description | 0.49 | 0.56 | 0.27 | 0.46
full transcript | 0.37 | 0.42 | 0.41 | 0.52
transcript+title+description | 0.43 | 0.51 | 0.45 | 0.61

Table 2. NDCG@10 results for five different indexes evaluated with four
assessment types.

| assessment type
---|---
|  | title + | transcript- | transcript-
run | title | description | segment | full
oudalab1 | 0.03 | 0.05 | 0.02 | 0.00
hltcoe1 | 0.17 | 0.23 | 0.18 | 0.26
hltcoe5 | 0.16 | 0.18 | 0.19 | 0.22
hltcoe3 | 0.15 | 0.19 | 0.24 | 0.20
hltcoe2 | 0.25 | 0.28 | 0.31 | 0.29
LRGREtvrs-r_3 | 0.20 | 0.29 | 0.32 | 0.37
BERT-DESC-TD | 0.20 | 0.27 | 0.33 | 0.41
BM25 | 0.20 | 0.27 | 0.33 | 0.41
BERT-DESC-Q | 0.20 | 0.27 | 0.33 | 0.41
RERANK-QUERY | 0.20 | 0.27 | 0.33 | 0.41
RERANK-DESC | 0.20 | 0.27 | 0.33 | 0.41
BERT-DESC-S | 0.20 | 0.27 | 0.33 | 0.41
LRGREtvrs-r_2 | 0.24 | 0.30 | 0.35 | 0.36
QL | 0.21 | 0.26 | 0.37 | 0.38
LRGREtvrs-r_1 | 0.25 | 0.32 | 0.37 | 0.38
hltcoe4 | 0.29 | 0.29 | 0.43 | 0.32
run_dcu1 | 0.33 | 0.37 | 0.44 | 0.44
run_dcu3 | 0.33 | 0.36 | 0.46 | 0.43
UTDThesis_Run1 | 0.28 | 0.32 | 0.46 | 0.28
run_dcu5 | 0.35 | 0.36 | 0.47 | 0.45
run_dcu2 | 0.33 | 0.37 | 0.48 | 0.44
UMD_IR_run2 | 0.32 | 0.34 | 0.48 | 0.37
run_dcu4 | 0.35 | 0.37 | 0.49 | 0.45
UMD_ID_run4 | 0.32 | 0.37 | 0.54 | 0.40
UMD_IR_run1 | 0.29 | 0.37 | 0.55 | 0.35
UMD_IR_run3 | 0.31 | 0.39 | 0.58 | 0.41
UMD_IR_run5 | 0.35 | 0.42 | 0.58 | 0.40

Table 3. Evaluation of TREC submitted runs and baselines by NDCG@10 with four
new types of assessments.

| transcript | description | % transcript words
---|---|---|---
| avg. length | vocab size | avg. ratio | avg. length | vocab size | in description
True Crime | 2200 | 870 | 0.48 | 100 | 71 | 4.3
Religion and Spirituality | 2000 | 700 | 0.44 | 79 | 55 | 3.5
Government | 2500 | 860 | 0.41 | 82 | 60 | 3.1
Business and Technology | 2900 | 910 | 0.38 | 141 | 94 | 4.1
News and Politics | 2800 | 1200 | 0.46 | 100 | 72 | 2.8
History | 2300 | 1000 | 0.49 | 81 | 58 | 2.9
Fiction | 1900 | 770 | 0.52 | 78 | 53 | 2.6

Table 4. Terminological coverage of description with respect to transcript,
content words

  1. (1)

A ranking of episode titles from a title-only index is more attractive than a
ranking of episode titles from a description-only index, while a ranking of
descriptions from a description-only index is more attractive than a ranking
of titles from a description-only index, suggesting that titles and
descriptions are not always signaling the same topical relatedness.

  2. (2)

The most attractive results are achieved by retrieving against an index of
episode titles and descriptions.

  3. (3)

A ranking based on retrieval of transcripts returns many more relevant
episodes, though titles and descriptions are much less attractive.

  4. (4)

Indexing transcripts and metadata together provides the strongest relevance by
either transcript judgment type with a relatively small decrease in
attractiveness.

  5. (5)

Result attractiveness remains surprisingly low in all cases—many results will
appear not relevant.

From this we further conclude that there is a great deal of relevant material
to be excavated from the transcripts that cannot be accessed via the metadata.
But enabling users to find that information via full-document search means
that result presentation will be negatively impacted. In other words, metadata
search is inadequate for finding relevant information, but full-text search
results in unappealing rankings. Thus some form of query-biased summarization
(Spina et al., 2017) or passage retrieval is necessary for presenting results
to users in an appealing way.

Table 3 shows the TREC runs evaluated by the four new assessment types. We
observe the following:

  * •

There is stronger correlation between different assessment types among TREC
runs than among those in Table 2.

  * •

By the transcript-segment judgments, our indexes are not competitive with the
best TREC submitted runs.

  * •

By the new transcript-level relevance assessments, all but one of our runs are
better than the best TREC run.

  * •

Our runs are substantially better than TREC runs on title and description
attractiveness.

Based on these findings, it could be argued that indexing metadata and
transcripts together, then searching full transcripts rather than segments, is
likely to provide the best overall user experience.

##  7\. Assessment agreement

Transcript relevance. Transcript-segment and transcript-full assessments agree
that an episode is relevant in 71% of cases. 43% of episodes have agreement on
the exact grade of relevance. This is a high level of agreement compared to
other retrieval tasks (Voorhees, 1998).

Attractiveness and relevance. When either the title is judged attractive or
the episode is judged relevant, in about 50% of cases both are true. This is a
decent level of agreement for a retrieval task, but it means that episode
title attractiveness is often not a very good predictor of episode relevance.
The results are very similar for description attractiveness.

Interannotator agreement. URIs that overlapped between the five metadata runs
and the TREC runs were independently judged by two different assessors. Among
episode titles that at least one assessor marked attractive, assessors agreed
88% of the time. Similarly with descriptions, in 82% of cases both assessors
found the description attractive. Agreement on exact grade of attractiveness
was high as well: 72% for titles and 68% for descriptions.

##  8\. Topical focus

Table 4 shows the coverage of content words for select categories of podcasts.
Some genres have episodes with relatively high topical focus, which is
reflected in the topical coverage of the descriptions. Each transcript and
description is filtered to only contain nouns, verbs, and adjectives; the
table reports counts of these tokens. The average ratio of vocabulary size to
length, a measure of topical variation within a text, varies across the
categories. While all these scores are normal, the topical variation of
”Fiction” podcasts is considerably wider than that of ”Business and
Technology” podcasts. The last column of the table demonstrates a difference
in terminological coverage. This can be understood as a measure of the topical
representativity of the descriptions with respect to the episode. A higher
score will mean that the description represents more of the topical content.
The score variation indicates a potential for determining the topicality of
the episode or show, and thus the utility of using search technology optimised
for topical retrieval and content analysis (as opposed to usage-based
similarity measures).

##  9\. Conclusion

It is not difficult to find information needs for podcast search such that
highly relevant content is buried in episodes, its presence not indicated by
either episode title or description. Episode titles and descriptions that
appear attractive may lead to irrelevant content and frustrated users. Podcast
search engines should index both metadata and episode content, and episode
segments or query-biased summaries may be necessary to help users understand
why retrieved episodes are relevant to their need.

Acknowledgments Thanks to the reviewers for their constructive suggestions,
and to TREC participants and organizers for helping make the track a success.
Gareth Jones is partially supported by Science Foundation Ireland as part of
the ADAPT Centre (Grant 13/RC/2106) at Dublin City University.

## References

  * (1)
  * Akiba et al. (2011) Tomoyosi Akiba, Hiromitsu Nishizaki, Hiroaki Nanjo, and Gareth J. F. Jones. 2011.  Overview of the NTCIR-12 SpokenQuery & Doc-2 Task. In _Proc. NTCIR_.
  * Apple (2020) Apple. 2020.  A Podcaster’s Guide to RSS.  https://help.apple.com/itc/podcasts_connect/#/itcb54353390.  (Accessed on 01/30/2021).
  * Besser et al. (2008) Jana Besser, Katja Hofmann, and Martha A. Larson. 2008\.  An Exploratory Study of User Goals and Strategies in Podcast Search. In _LWA 2008 - Workshop-Woche: Lernen, Wissen & Adaptivität, Würzburg, Deutschland, 6.-8. Oktober 2008, Proceedings_ _(Technical Report, Vol. 448)_.
  * Clifton et al. (2020) Ann Clifton, Sravana Reddy, Yongze Yu, Aasish Pappu, Rezvaneh Rezapour, Hamed Bonab, Maria Eskevich, Gareth J. F. Jones, Jussi Karlgren, Ben Carterette, and Rosie Jones. 2020.  100,000 Podcasts: A Spoken English Document Corpus. In _Proc. COLING’20_.
  * Eisenberg and Barry (1988) Michael Eisenberg and Carol Barry. 1988.  Order effects: A study of the possible influence of presentation order on user judgments of document relevance.  _JASIST_ 39, 5 (1988).
  * Eskevich et al. (2015) Maria Eskevich, Robin Aly, Roeland Ordelman, David N. Racca, Shu Chen, and Gareth J. F. Jones. 2015\.  SAVA at MediaEval 2015: Search and Anchoring in Video Archives. In _Working Notes Proceedings of the MediaEval 2015 Multimedia Benchmark Workshop_.
  * Eskevich et al. (2012) Maria Eskevich, Gareth J. F. Jones, Shu Chen, Robin Aly, Roeland Ordelman, and Martha Larson. 2012\.  Search and Hyperlinking Task at MediaEval 2012. In _Working Notes Proceedings of the MediaEval 2012 Multimedia Benchmark Workshop_.
  * Fuhr et al. (2006) Norbert Fuhr, Mounia Lalmas, Saadia Malik, and Gabriella Kazai. 2006.  _Advances in XML Information Retrieval and Evaluation: 4th International Workshop of the Initiative for the Evaluation of XML Retrieval_.
  * Garofolo et al. (2000) John S. Garofolo, Cedric G. P. Auzanne, and Ellen M. Voorhees. 2000\.  The TREC Spoken Document Retrieval Track: A Success Story (RIAO). In _Content-Based Multimedia Information Access - Volume 1_.
  * Jones (2019) Gareth J. F. Jones. 2019\.  About Sound and Vision: CLEF Beyond Text Retrieval Tasks.  In _Information Retrieval Evaluation in a Changing World - Lessons Learned from 20 Years of CLEF_. Springer.
  * Jones et al. (2020) Rosie Jones, Ben Carterette, Ann Clifton, Maria Eskevich, Gareth Jones, Jussi Karlgren, Aasish Pappu, Sravana Reddy, and Yongze Yu. 2020.  Overview of the TREC 2020 Podcasts Track.  In _Proc. TREC_.
  * Kim and Croft (2012) Jin Young Kim and W. Bruce Croft. 2012.  A Field Relevance Model for Structured Document Retrieval. In _Proc. ECIR_.
  * Lalmas and Tombros (2007) Mounia Lalmas and Anastasios Tombros. 2007.  Evaluating XML Retrieval Effectiveness at INEX.  _SIGIR Forum_ 41, 1 (June 2007), 40–57.
  * Larson et al. (2011) Martha Larson, Maria Eskevich, Roeland Ordelman, Christoph Kofler, Sebastian Schmiedeke, and Gareth J. F. Jones. 2011.  Overview of MediaEval 2011 Rich Speech Retrieval Task and Genre Tagging Task. In _Proc. MediaEval_.
  * Pecina et al. (2008) Pavel Pecina, Petra Hoffmannová, Gareth J. F. Jones, Ying Zhang, and Douglas W. Oard. 2008\.  Overview of the CLEF-2007 Cross-Language Speech Retrieval Track. In _Proc. CLEF_.
  * Research (2019) Edison Research. 2019\.  The podcast consumer.  _available athttps://www.edisonresearch.com/the-podcast-consumer-2019/ (accessed Feb. 9, 2021)_ (2019).
  * Robertson et al. (2004) Stephen Robertson, Hugo Zaragoza, and Michael Taylor. 2004\.  Simple BM25 Extension to Multiple Weighted Fields. In _Proc. CIKM_.
  * Sharpe (2020) Matthew Sharpe. 2020\.  A review of metadata fields associated with podcast RSS feeds. In _PodRecs: Workshop on Podcast Recommendations_.
  * Spina et al. (2017) Damiano Spina, Johanne R. Trippas, Lawrence Cavedon, and Mark Sanderson. 2017.  Extracting audio summaries to support effective spoken document search.  _Journal for the Association of Information Science and Technology (JASIST)_ 68, 9 (2017).
  * Tsagkias et al. (2008) Manos Tsagkias, Martha Larson, Wouter Weerkamp, and Maarten de Rijke. 2008.  PodCred: A Framework for Analyzing Podcast Preference. In _Proceedings of the 2nd ACM Workshop on Information Credibility on the Web_ (Napa Valley, California, USA) _(WICOW ’08)_.
  * Voorhees (1998) Ellen M. Voorhees. 1998\.  Variations in relevance judgments and the measurement of retrieval effectiveness. In _Proc. SIGIR_.
  * Whitner (2021) Gavin Whitner. 2021\.  The Meteoric Rise of Podcasting.  https://musicoomph.com/podcast-statistics https://musicoomph.com/podcast-statistics (Accessed February 2021).
  * Zamani et al. (2018) Hamed Zamani, Bhaskar Mitra, Xia Song, Nick Craswell, and Saurabh Tiwary. 2018.  Neural Ranking Models with Multiple Document Fields. In _Proc. WSDM_.