e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>11institutetext:
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles
University in Prague, V Holešovičkách 2, 180 00 Prague 8, Czech Republic

# Robinson–Trautman solution with nonlinear electrodynamics

T. Tahamtane1,addr1  O. Svíteke2,addr1

(August 27, 2024)

###### Abstract

Explicit Robinson–Trautman solutions with electromagnetic field satisfying
nonlinear field equations are derived and analyzed. The solutions are
generated from the spherically symmetric ones. In all studied cases the
electromagnetic field singularity is removed while the gravitational one
persists. The models resolving curvature singularity in the spherically
symmetric spacetimes could not be generalized to Robinson–Trautman geometry
using the generating method developed in this paper which indicates that the
removal of a singularity in the associated spherically symmetric case might be
just a consequence of high symmetry. We show that the obtained solutions are
generally of algebraic type II and reduce to type D in spherical symmetry.
Asymptotically they tend to the spherically symmetric case as well.

###### Keywords:

exact solution black hole nonlinear electrodynamics

###### pacs:

04.20.Jb 04.70.Bw 04.40.Nr

##  1 Introduction

Robinson–Trautman spacetimes represent an important class of geometries
defined by the requirement of possessing an expanding, nontwisting and
nonshearing null geodesic congruence RobinsonTrautman:1960 ;
RobinsonTrautman:1962 ; Stephanietal:book ; Griffiths:book . This class
contains non-spherical generalizations of black holes without rotation
(Schwarzschild solution is a special case in this class, unlike the Kerr black
hole). In general, Robinson–Trautman spacetimes do not posses any Killing
vectors thus providing solutions devoid of symmetry. Another important aspect
of this family is the presence of gravitational radiation connected with the
dynamical nature of general solutions within this class. Many global
properties of this class in four dimensions have been studied analytically,
especially in the last 25 years. In particular, based on the only nontrivial
Einstein equation (so-called Robinson–Trautman equation) the asymptotic
evolution and global structure of vacuum Robinson–Trautman spacetimes of type
II with spherical topology were investigated by Chruściel and Singleton Chru1
; Chru2 ; ChruSin . They showed that characteristic initial value problem for
generic, arbitrarily strong smooth initial data converges asymptotically in
retarded time to corresponding Schwarzschild metric near its future horizon.
Extensions across such “Schwarzschild-like” future event horizon are only of a
finite order of smoothness. These results were later extended to cover the
presence of a cosmological constant which naturally modifies the asymptotic
behavior and the solutions tend to Schwarzschild–(anti-)de Sitter podbic95 ;
podbic97 . Finally, the Chruściel–Singleton analysis was used for
Robinson–Trautman spacetimes admitting additionally pure radiation BicakPerjes
; PodSvi:2005 , where the asymptotic state is described by spherically
symmetric Vaidya–(anti-)de Sitter metric. The location of past quasilocal
horizon (which cannot be determined as an event horizon due to the
impossibility to extend the solutions to past null infinity) together with the
proof of its existence and uniqueness for the vacuum Robinson–Trautman
solutions have been studied by Tod tod . Later, Chow and Lun chow-lun  further
analyzed properties of this horizon and made numerical study of both the
horizon equation and the Robinson–Trautman equation. The analytic results were
generalized to nonvanishing cosmological constant PodSvi:2009 . The relation
between asymptotic momentum and local horizon curvature in Robinson–Trautman
class was used in the analytic explanation of an "antikick" appearing in
numerical studies of asymmetric binary black hole merger rezzolla . Recently,
solution with minimally coupled free scalar field was derived in Tahamtan-
Svitek-PRD  and shown to posses singularity which is initially naked and only
later gets covered by horizon.

Type D vacuum solutions of Robinson-Trautman family contain, apart from
Schwarzschild solution, also the C-metric Krtous  representing uniformly
accelerated pair of black holes. C-metric is as well a natural future
asymptotics in case of non-smooth initial data for certain subclass of
Robinson-Trautman spacetime Hoenselaers . By leaving out the usual spherical
topology assumption one can obtain a special case of Kasner metric
Stephanietal:book . Including null radiation source into type D leads to, e.g.
Vaidya solution or Kinnersley’s rocket Kinnersley  which is interpreted as an
object propelled by emitting directional null radiation (these "rocket"
solutions can be generalized to type II). All type D null radiation metrics
are known Frolov .

Vacuum type N solutions which correspond to spacetimes containing just
gravitational radiation have singularity at each wave surface which combine
into singular lines Stephanietal:book ; Griffiths:book . The general solution
was given by Foster and Newman Foster  and they are frequently used to form
various sandwich-type waves Griffiths . There are no pure radiation solutions
of type N.

There is also a higher-dimensional generalization of Robinson–Trautman
spacetimes (containing aligned pure radiation and a cosmological constant)
which, however, lacks the rich dynamics present in four dimensions podolsky-
ortaggio . The existence of horizons (which have generally richer topology
than in four dimensions) in this case was subsequently analyzed in Svitek2011
. These higher-dimensional solutions were as well generalized to admit a
source in the form of p-form fields ortaggio1 .

Robinson–Trautman spacetimes with Maxwell field were already derived in the
founding paper of this family of solutions RobinsonTrautman:1962 . Later they
were studied more extensively Lind ; Newman ; Kozameh1 . Among the special
cases of these solutions belong Reissner–Nordström black hole and charged
C-metric but it was also shown that this subfamily suffers from non well-
posedness Kozameh2 . The higher-dimensional generalization of
Robinson–Trautman spacetimes coupled to Maxwell electrodynamics was derived in
ortaggio2 .

Nonlinear Electrodynamics (NE) was founded almost a century ago and used
mainly as a solution to the problem of divergent field of a point charge in
the vicinity of its position (see e.g. Dirac ) also giving satisfactory self-
energy of charged particle. The best-known and frequently used form of the
theory was introduced already in 1934 by Born and Infeld BornInfeld . Nice
overview with a lot of useful information was given in a book by Plebański
Plebanski . More recently, the attention to Nonlinear Electrodynamics was
increased thanks to the discovery that string-generated corrections to Maxwell
field have the form of original Born–Infeld theory Wiltshire . However, it was
also noted that the electric displacement vector in Born–Infeld model has two
possible values for single value of electric field. This non-uniqueness was
soon solved by adding the so called Hoffmann term Hoffmann  in Born–Infeld
Lagrangian. Additionally, this new model was also used to resolve the
spacetime singularity in spherically symmetric case TT-PhysLettA .

Later, other NE models were considered for both solving the point charge
singularity and resolving the spacetime singularity Ayon-Beato1 ; Ayon-Beato2
. Note, however, that these results were obtained in the Hamiltonian framework
and their Legendre transform is far from trivial Bronnikov . One important
example is the so-called Bardeen black hole Bardeen  which was originally
discovered as a regular solution with horizon generated by certain stress
energy tensor. Only later this source was interpreted as being created by a
specific model of NE Ayon-Beato3  which, however, does not have a Maxwell
limit (in weak field regime).

Many spherically symmetric solutions of Einstein equations with NE were
studied, mostly with the Born–Infeld Lagrangian BI , logarithmic Lagrangians
Soleng ; Hendi-Rad , square root lagrangian square root , power Maxwell models
power Maxwell  and other forms Hendi ; Oliveira ; Habib . These solutions are
mostly thought of as a model of a charged particle. However, since General
Relativity is a nonlinear theory, one should study the stability of such
models nonperturbatively.

Our aim here is to derive Robinson–Trautman solutions coupled to several forms
of NE Lagrangians in order to investigate the influence of nonsphericity in
this family on the results gained previously in highly symmetric situations.
Since spherically symmetric solutions are a special sub case of
Robinson–Trautman spacetime it is a natural family to consider for nonlinear
stability investigation.

##  2 Vacuum Robinson–Trautman solutions and field equations

The general form of a vacuum Robinson–Trautman spacetime can be represented by
the following line element RobinsonTrautman:1960 ; RobinsonTrautman:1962 ;
Stephanietal:book ; Griffiths:book

${\rm d}s^{2}_{RT}=-2H\,{\rm d}u^{2}-\,2\,{\rm d}u\,{\rm
d}r+\frac{r^{2}}{{P}^{2}}\,({\rm d}x^{2}+{\rm d}y^{2}),$ (1)

with ${2H=\Delta(\,\ln{P})-2r(\,\ln{P})_{,u}-{2m/r}-(\Lambda/3)r^{2}}$,

$\Delta\equiv{P}^{2}(\partial_{xx}+\partial_{yy}),$ (2)

and where $\Lambda$ is the cosmological constant. The metric depends on two
functions, ${\,{P}(u,x,y)\,}$ and ${\,m(u)\,}$, which satisfy the nonlinear
fourth-order PDE (so called Robinson–Trautman equation)

$\Delta\Delta(\,\ln{P})+12\,m(\,\ln{P})_{,u}-4\,m_{,u}=0\,.$ (3)

The function $m(u)$ might be set to a constant by a suitable coordinate
transformation for vacuum solution. However, for a null radiation field source
which is aligned with the principal null direction the solution represents
generalization of Vaydia spacetime with time-dependent $m(u)$.

The spacetime is defined by geodesic, shearfree, twistfree and expanding null
congruence generated by ${\mbox{\boldmath$k$}=\partial_{r}}$. The coordinate
$r\in(0,\infty)$ is an affine parameter along this congruence,
$u\in(u_{0},\infty)$ is a retarded time coordinate (the initial data for this
class of spacetimes are specified on null hypersurface $u=u_{0}$), and $x,y$
are spatial coordinates spanning transversal 2-space which have Gaussian
curvature (for ${r=1}$)

${K}(u,x,y)\equiv\Delta(\,\ln{P})\,.$ (4)

If we select the transversal 2-spaces to be topological spheres (standard
assumption in this class) then $x,y$ are real stereographic-type coordinates
on a deformed spheres $r=const,u=const$. For general fixed values of $r$ and
$u$, the Gaussian curvature is ${{K}/r^{2}}$ so that, as ${r\to\infty}$, they
become locally flat.

##  3 Robinson–Trautman and static spherically symmetric solutions

In this section we will develop a generating method to obtain a
Robinson–Trautman solution coupled to NE based on a static spherically
symmetric spacetime solution with NE. In order to have a straightforward
generalization, we use the Kerr–Schild-type modification of the vacuum
Robinson–Trautman metric. First, we find the Einstein equations for
Robinson–Trautman spacetime coupled to NE and then compare them to SSS ones.
Based on the similarities we formulate the theorem summarizing the generating
method.

We consider the following action, describing nonlinear electrodynamics coupled
to gravity,

$S=\int d^{4}x\sqrt{-g}[\mathcal{R}-2\Lambda+\mathcal{L}(F)]$ (5)

where $\mathcal{R}$ is the Ricci scalar for the metric $g_{\mu\nu}$ and
Lagrangian of the nonlinear electromagnetic field $\mathcal{L}(F)$ is an
arbitrary function of the electromagnetic field invariant
$F=F_{\mu\nu}F^{\mu\nu}$ constructed from closed Maxwell 2-form $F_{\mu\nu}$.
We use units in which $c=\hbar=8\pi G=1$. By applying the variation with
respect to the metric for the action (5), we get Einstein equations

$G^{\mu}{}_{\nu}=T^{\mu}{}_{\nu}-\Lambda\delta^{\mu}{}_{\nu}.$ (6)

The energy momentum tensor generated by NE Lagrangian is given by

$T^{\mu}{}_{\nu}=\frac{1}{2}\\{\delta^{\mu}{}_{\nu}\mathcal{L}-4\,(F_{\nu\lambda}F^{\mu\lambda})\mathcal{L}_{F}\\}$
(7)

and the modified Maxwell (nonlinear electrodynamics) field equations are then
given in the following form

$\partial_{\mu}(\sqrt{-g}\mathcal{L}_{F}F^{\mu\nu})=0$ (8)

in which $\mathcal{L}_{F}=\frac{d\mathcal{L}(F)}{dF}$.

Our metric ansatz is

${\rm d}s^{2}={\rm d}s^{2}_{RT}-Q(u,r)\,{\rm d}u^{2}\ ,$ (9)

where we modify the vacuum Robinson–Trautman metric, i.e. (1), by adding
function $Q(u,r)$ to the $g_{uu}$ component. This corresponds to general Kerr-
Schild metric form given by background vacuum Robinson-Trautman geometry
$g^{RT}_{\mu\nu}$ and null, shearfree, twistfree and geodetic vector $k$

$g_{\mu\nu}=g^{RT}_{\mu\nu}-Q(u,r)k_{\mu}k_{\nu}$ (10)

Einstein tensor has now more nontrivial components compared to the original
Robinson–Trautman metric (1)

$\displaystyle{G^{u}}_{u}$ $\displaystyle=$
$\displaystyle{G^{r}}_{r}=-\Lambda+\frac{Q_{,r}}{r}+\frac{Q}{r^{2}}$ (11)
$\displaystyle{G^{x}}_{x}$ $\displaystyle=$
$\displaystyle{G^{y}}_{y}=-\Lambda+\frac{1}{2}{Q_{,rr}}+\frac{Q_{,r}}{r}$ (12)
$\displaystyle{G^{r}}_{u}$ $\displaystyle=$
$\displaystyle{}^{\mathrm{RT}}{G^{r}}_{u}-\frac{1}{r}[(\,\ln{P})_{,u}\\{r{Q_{,r}}-2Q\\}+{Q_{,u}}]$
(13)

where

${}^{\mathrm{RT}}{G^{r}}_{u}=-\frac{1}{2r^{2}}\\{\Delta\Delta(\,\ln{P})+12\,m(\,\ln{P})_{,u}-4\,m_{,u}\\}$

We assume the following specific Maxwell 2-form in the coordinates of (9)

$\mathbf{F}=F_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=E(u,r){\rm d}u\wedge{\rm
d}r$ (14)

where the electromagnetic invariant $F=F_{\mu\nu}F^{\mu\nu}$ for the above
Maxwell 2-form simplifies to $-2E^{2}$. Then from (8) and the metric (9) one
can find dynamical equation for electromagnetic field

$\frac{r^{2}}{P(u,x,y)^{2}}\mathcal{L}_{F}F_{ur}=F_{0}(x,y),$ (15)

which can be solved by

$\mathcal{L}_{F}F_{ur}=\frac{q(u)^{2}}{r^{2}},$ (16)

in which $P(u,x,y)^{2}F_{0}(x,y)=q(u)^{2}$ in order to satisfy the assumed
form of $\mathbf{F}$ (14). The energy momentum tensor given by (7) can then be
expressed in the diagonal form

$T^{\mu}{}_{\nu}=diag\left\\{\frac{\mathcal{L}}{2}-F\mathcal{L}_{F},\frac{\mathcal{L}}{2}-F\mathcal{L}_{F},\frac{\mathcal{L}}{2},\frac{\mathcal{L}}{2}\right\\}$
(17)

for our form of Maxwell field (14) and with respect to the coordinates
$(u,r,x,y)$ of metric (9).

Let us note that the assumption (14) necessarily leads to
$F_{\mu\nu}F^{\mu\nu}\neq 0$ and $F^{*}_{\mu\nu}F^{\mu\nu}=0$ (where $*$ means
a Hodge dual) as in the static spherically symmetric cases considered below.
This also means that our field is non-null and purely electric. At the same
time we do not need to consider generalization of Lagrangian that would
additionally contain terms dependent on the second invariant.

Now, we will turn our attention to the general form of previously derived
static spherically symmetric (SSS) solutions with NE. The metric has the
following form encompassing all the models that we wish to generalize

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-\left[1-{\textstyle{\frac{\Lambda}{3}}}r^{2}+f(r)\right]\,{\rm
d}t^{2}+\,\frac{{\rm
d}r^{2}}{\left[1-{\textstyle{\frac{\Lambda}{3}}}r^{2}+f(r)\right]}$ (18)
$\displaystyle+{r^{2}}\,({\rm d}{\theta}^{2}+{{\sin}^{2}{\theta}}\,{\rm
d}{\phi}^{2}),$

Here, we prefer not to introduce the null coordinate in order to keep the
correspondence with published spherically symmetric models. In this case one
can of course as well use the Kerr-Schild form to introduce function $f$ into
the background Minkowski/(anti-)de Sitter metric.

We obtain these nonzero components of the Einstein tensor for line element
(18)

$\displaystyle{G^{t}}_{t}$ $\displaystyle=$
$\displaystyle{G^{r}}_{r}=-\Lambda+\frac{f_{,r}}{r}+\frac{f}{r^{2}}$ (19)
$\displaystyle{G^{\theta}}_{\theta}$ $\displaystyle=$
$\displaystyle{G^{\phi}}_{\phi}=-\Lambda+\frac{1}{2}{f_{,rr}}+\frac{f_{,r}}{r}$
(20)

Maxwell 2-form has the following form resembling (14),

$\mathbf{F}=\tilde{E}(r){\rm d}t\wedge{\rm d}r$ (21)

although the electromagnetic field is static now (like before, field invariant
simplifies considerably $F=F_{\mu\nu}F^{\mu\nu}=-2\tilde{E}^{2}$). From the
modified Maxwell equation (8) and the metric (18), we find

$\mathcal{L}_{F}F_{tr}=\frac{q_{0}^{2}}{r^{2}},$ (22)

where $q_{0}$ is a constant. One can easily calculate the energy momentum
tensor in this case and recovers the diagonal form (17) but now with respect
to the coordinates $(t,r,\theta,\phi)$ of metric (18).

We can see the structural similarity between set of equations (11,12) and
(19,20). This suggests trying to generalize SSS solutions with NE source to
Robinson–Trautman geometry via this similarity. Note, that this similarity is
not analyzed based on some coordinate transformation but purely from the
perspective of coincidence between the form of differential equations. Of
course this similarity has underlying reason in the fact that
Robinson–Trautman metrics are generalization of spherically symmetric models
but this is not important for the following observations. Namely, the solution
$f(r)$ of equations (19,20) can be evidently transformed into a particular
solution $Q(u,r)$ of equations (11,12) by promoting integration constants in
$f$ into functions of coordinate $u$. However, one has to ensure that the
newly constructed function $Q(u,r)$ satisfies additional equation
${G^{r}}_{u}=0$ coming from combining corresponding components of Einstein
tensor (13) and energy momentum tensor (17) into field equations (6). This
equation is the basic constraint on generalizing SSS solutions with NE into
Robinson–Trautman case.

We summarize the method in the following theorem:

###### Theorem 3.1

The SSS solution of Einstein equations coupled to arbitrary NE model which is
given by line element in the form of (18) and Maxwell field (21) with specific
function $f(r)$ can be generalized into Robinson–Trautman solution coupled to
the same NE model with metric (9) and Maxwell field (14) where the function
$Q(u,r)$ is obtained from $f(r)$ by promoting the integration constants
(appearing in $f$) into functions of $u$ provided the additional constraint
equation

$(\,\ln{P})_{,u}\\{r{Q_{,r}}-2Q\\}+{Q_{,u}}=0$ (23)

is satisfied for such $Q(u,r)$.

##  4 Explicit examples

Now, we will present several important forms of NE Lagrangians and the
associated solutions for both electromagnetic field and Robinson–Trautman
metric (giving the form of metric function $Q(u,r)$). For each model we first
solved the spherically symmetric equations (19) and (20) to obtain the most
general function $f(r)$, then used the Theorem 3.1 to generate $Q(u,r)$ with
which we try to solve the constraint (23). We do not present the original
function $f(r)$ since it can be easily read of from $Q(u,r)$ by putting all
functions of $u$ to constants. Finally, we specify in which references one can
find the spherically symmetric case.

We plotted the profile of Lagrangian (Figure 1) for all the models considered
below and electromagnetic field invariant $F$ (Figure 2) in case of static
spherically symmetric solutions (alternatively it can be viewed as a profile
of invariant on $u=const$ hypersurface). Evolution of the invariant for
Robinson–Trautman solutions is presented in three-dimensional plots in
Appendix.

###  4.1 Maxwell

As a starting point we briefly show the linear Maxwell case which generalizes
the standard Reissner-Nordström solution

$\mathcal{L}(F)=-F$ (24)

Applying the method described in section 3 and summarized in Theorem 3.1 we
obtain this solution on Robinson–Trautman background

$\displaystyle Q(u,r)$ $\displaystyle=$
$\displaystyle\frac{q(u)^{4}}{r^{2}}+\frac{C_{1}(u)}{r}$ (25)

If we put this form into the field equation constraint (23) we find

$\displaystyle\Delta\Delta(\,\ln{P})+12\,M(\,\ln{P})_{,u}-4\,M_{,u}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle q\,P_{,u}-q_{,u}\,P$
$\displaystyle=$ $\displaystyle 0$ (26)

where

$M=m(u)-\frac{C_{1}(u)}{2}.$ (27)

This result is a special case of algebraic type II Einstein–Maxwell spacetimes
analyzed in Stephanietal:book  where a theorem (Theorem 28.3 in the reference)
for generating these solutions from the vacuum Robinson–Trautman spacetime is
presented.

###  4.2 Born-Infeld

As was mentioned before, NE started with this model of dynamics and it has
been studied widely in three, four and higher dimensions, also their physical
properties were analyzed in detail. The famous Born–Infeld Lagrangian has the
following form BornInfeld

$\mathcal{L}_{BI}=4\beta^{2}\left(1-\sqrt{1+\frac{F}{2\beta^{2}}}\right),$
(28)

$\beta$ is a critical field length. When $\beta\rightarrow\infty$ this
Lagrangian goes to Maxwell form which is as well recovered in the weak field
regime.

Applying the method described in Section 3 we obtain this solution on
Robinson–Trautman background

$F=-\frac{2q(u)^{4}}{r^{4}+{q(u)^{4}}/{\beta^{2}}}.$ (29)

As it was expected the electric field remains regular at $r=0$. Lagrangian
(28) is only defined for field invariant $F\geq-2\beta^{2}$ (see as well
Figure 1) which corresponds to $r\geq 0$ according to (29) so it covers the
whole range of coordinates. The metric function corresponding to NE source
becomes

$\displaystyle
Q(u,r)=\frac{C_{1}(u)}{r}+\frac{2\beta^{2}}{3}r^{2}-\frac{2\beta^{2}}{r}\int{\sqrt{r^{4}+{q(u)^{4}}/{\beta^{2}}}}\,dr$

If we put these forms in field equations we will find the same restrictions as
for Maxwell theory (4.1). However, in this case the constraint (23) enforces
$C_{1}(u)=0$ and so $M=m$.

Such solution is a straightforward generalization of the SSS solutions given
for example in Fernando ; Oliveira . This shows that there is indeed a
solution in Robinson–Trautman class that settles down smoothly to known
spherically symmetric one when function $P$ attains its special form
describing the geometry on a sphere (this is elaborated in Section 5). Then,
from (4.1) one immediately concludes that $q=const$.

###  4.3 Logarithmic form of nonlinear electrodynamics theory

The logarithmic form of Lagrangian can be given by

$\mathcal{L}_{LNE}=-8\beta^{2}\ln{\left({1+\frac{F}{8\beta^{2}}}\right)}.$
(31)

In Soleng  it was used to present a model of point particle without divergence
in electromagnetic field. However, even with vanishing "Schwarzschild mass"
and finite electromagnetic field the curvature singularity at the origin is
present. For special value of $\beta$ the horizon radius shrinks to zero and
so called black point is created (see, e.g., black-point  for their previous
occurrences).

Applying the method described in section 3 we obtain

$F=-\frac{8\beta^{2}}{q(u)^{4}}\,{\left(\beta
r^{2}-\sqrt{\beta^{2}r^{4}+q(u)^{4}}\right)}^{2}\ .$ (32)

The presence of curvature singularity can be understood from Lagrangian (31)
since it has singularity for field invariant value $F=-8\beta^{2}$ which
corresponds to $r=0$ according to (32) and so this Lagrangian provides
singular source for Einstein equations. The Robinson–Trautman geometry is
specified by the following metric function

$\displaystyle
Q(u,r)=\frac{C_{1}(u)}{r}+\frac{20}{9}\beta^{2}r^{2}-4\beta\sqrt{r^{4}\beta^{2}+q(u)^{4}}$
$\displaystyle-\frac{4}{3}\beta^{2}r^{2}\ln\left({\frac{-2r^{4}\beta^{2}+2\beta
r^{2}\sqrt{r^{4}\beta^{2}+q(u)^{4}}}{q(u)^{4}}}\right)$
$\displaystyle+\frac{16\beta^{3}}{r}\int{\frac{r^{4}}{\sqrt{r^{4}\beta^{2}+q(u)^{4}}}}\,dr$
(33)

If we put these $Q$ and $F$ into the constraint (23) we recover conditions
(4.1) and (27).

This solution is again a straightforward generalization of the SSS solutions
given in Soleng . The results concerning spherically symmetric limit are the
same as in the previous analysis of Born–Infeld example.

Figure 1: The plot of all considered Lagrangians for NE: Maxwell (dotted),
Born–Infeld (solid line - it is defined only for restricted values of $F$),
Logarithmic model (dashed), New Lagrangian 1 (dash-dotted), New Lagrangian 2
(long-dashed). The free parameter was fixed to following value: $\beta=1$.
Evidently, all the nonlinear Lagrangians have steeper growth with increasing
absolute value of $F$ than Maxwell model and as well either diverge for
certain finite value of $F$ or fail to be defined (Born–Infeld). Except for
the New Lagrangian 2 model the value at $F=0$ is the same for all models.

###  4.4 New Lagrangian 1

Now we will consider another form of dynamics for NE given by the following
Lagrangian Habib3

$\mathcal{L}_{New}=-\frac{2}{\alpha^{4}}\ln{\left(1-{\alpha^{2}}\sqrt{-{F}}\right)}-\frac{2\sqrt{-{F}}}{\alpha^{2}},$
(34)

which has correct Maxwell limit in both weak field regime and for $\alpha\to
0$. The generating method from section 3 yields the solution

$F=-\frac{2q(u)^{4}}{\left(r^{2}+q(u)^{2}\beta^{2}\right)^{2}}$ (35)

in which $\beta^{2}=\sqrt{2}\alpha^{2}$. Again, the field is regular however
the Lagrangian has singularity for $F=-\alpha^{-4}$ (see Figure 1) which is
attained at $r=0$. Note that in this case one needs to select negative root
when solving for $E(r,u)$ in order to satisfy the field equations. Metric
function becomes

$\displaystyle Q(u,r)$ $\displaystyle=$
$\displaystyle\frac{C_{1}(u)}{r}-\frac{2}{3}\frac{q(u)^{2}}{\beta^{2}}-\frac{4}{3}\frac{q(u)^{3}}{r\beta}\arctan{\left(\frac{r}{q(u)\beta}\right)}$
(36)
$\displaystyle-\frac{2r^{2}}{3\beta^{4}}\ln{\left(\frac{r^{2}}{r^{2}+q(u)^{2}\beta^{2}}\right)}$

If we put these forms in field equations we will find the already known set of
restrictions (4.1) and (27). This solution generalizes the SSS solutions given
in Habib3  where this type of Lagrangian is used for the first time.

Figure 2: The plot of electromagnetic field invariant $F$ for Maxwell
(dotted), Born–Infeld (solid line), Logarithmic(dashed), New Lagrangian 1
(dash-dotted), New Lagrangian 2 (long-dashed) models. The free parameter was
fixed to following value: $\beta=1$. We have also fixed the function $q$ to a
constant $q(u)=1$ which can be considered as $u=const$ slice of the complete
three-dimensional plots given in Appendix. The graph corresponding to New
Lagrangian 2 is limited only to the coordinate range where the model satisfies
energy conditions. Only in the Maxwell case the invariant $F$ diverges at
origin.

###  4.5 New Lagrangian 2

Finally, we consider the following NE Lagrangian Habib2

$\mathcal{L}_{New2}=\frac{1}{\beta-\sqrt{-{F}}},$ (37)

which does not have a Maxwell limit in the weak field regime. However, models
containing square root (or arbitrary powers) and generally devoid of Maxwell
limit were extensively discussed before square root ; power Maxwell . And
these examples can serve as an approximation of true dynamics in strong field
regime. Additional motivation to consider a model without Maxwell limit is to
test whether the generating method used here is not limited (via the
constraint (23)) only to those NE models with correct Maxwell limit (which are
all those considered so far).

The generating method from section 3 yields the solution

$F=-\frac{\left(\beta\,\,q(u)-r\right)^{2}}{q(u)^{2}}$ (38)

and we need $q(u)>0,\ \beta q(u)-2r>0$ to fulfill the weak energy condition so
this solution cannot cover the whole coordinate range but can serve as an
inner solution for small $r$. Evidently, without introducing the parameter
$\beta$ we cannot satisfy the energy conditions at all (for more discussion
and interpretation see Habib2 ).

The geometry is in this case defined by

$\displaystyle Q(u,r)$ $\displaystyle=$
$\displaystyle\frac{C_{1}(u)}{r}-\frac{\beta q(u)^{2}}{2}+\frac{q(u)}{2}r\ .$
(39)

If we put these forms in field equations we will find the already known set of
restrictions (4.1) and (27). This solution is a straightforward generalization
of the SSS solutions given for example in Habib2 .

The Lagrangian (37) has a singularity at $\sqrt{-F}=\beta$ (see as well the
Figure 1) which corresponds to $r=0$ according to (38). This means that
although the field invariant $F$ is regular the curvature singularity is
created via singular behavior of the Lagrangian.

##  5 Asymptotic behavior

The metric (1) admits coordinate freedom already noted by Robinson and
Trautman

$u^{\prime}=U(u),\,\,r^{\prime}=\frac{r}{U_{,u}},\,\,m^{\prime}=\frac{m}{U_{,u}^{3}},\,\,P^{\prime}=\frac{P}{U_{,u}},$
(40)

which can be used to set mass $M^{\prime}$ to a positive constant by proper
function $U(u)$. In the case of nonlinear electrodynamics, when considering
metric (9) and modified Maxwell equations (8), one has to supplement (40) (now
with $M=m-\frac{C_{1}}{2}$ instead of $m$) by transforming $q(u)$ as well

$q^{\prime}=\frac{q}{U_{,u}},$ (41)

Note that one can not set $q(u)$ and $M(u)$ to constant simultaneously since
one has only single function in hand. This means that $q(u)$, although looking
like physical charge based on (16), shares the same interpretation problems as
$m(u)$. Only now these difficulties are combined together.

Now we are ready to investigate asymptotic behavior for our solutions,
separately for the retarded time $u\to\infty$ and $r\to\infty$:

###  5.1 Asymptotics $u\to\infty$

One can use (40) to put $M^{\prime}$ to constant in (27) for all models
exactly the form considered in Chruściel and Singleton Chru1 ; Chru2 ; ChruSin
analysis of asymptotic behavior to recover the spherically symmetric final
state and also exponentially fast decay of dependence of function $P^{\prime}$
on new coordinate $u^{\prime}$. Due to the last equation in (4.1), the
asymptotic behavior of function $q^{\prime}$ is the same as $P^{\prime}$.
Namely, it tends to a constant which is completely consistent with the final
state approaching the corresponding spherically symmetric solution.

###  5.2 Asymptotics $r\to\infty$

In this limit the electromagnetic field vanishes for all cases and for
Born–Infeld, Logarithmic and New Lagrangian 1 models the asymptotic behavior
is identical to the Maxwellian case. Analyzing behavior of function $Q(u,r)$
(after moving $C_{1}$ term into redefinition of "mass" $m\to M$) we
immediately see that in all cases the metric is locally asymptotically flat
(or (anti–)de Sitter) as for the vacuum Robinson–Trautman solution. We have
neglected the New Lagrangian 2 model due to its restricted coordinate range.

##  6 Horizons

Since in all cases the Robinson–Trautman spacetimes with arbitrary form of
electrodynamics still posses curvature singularity at $r=0$, as one can
confirm by computing Kretschmann scalar (singularity is milder for NE models
with regular EM field at the origin), we would like to know if it is covered
by a horizon. Due to the dynamical nature of our spacetimes we will look for
quasilocal horizon. So we need to find a marginally trapped surfaces and one
can select any of the most popular horizon definitions — apparent hawking-
ellis , trapping hayward  or dynamical horizon krishnan . We will be looking
for a horizon hypersurface given by the equation $r=\mathcal{N}(u,x,y)$ with
the $u=u_{0}=const$ slices

$r=\mathcal{N}(u_{0},x,y)=N(x,y)$ (42)

being marginally trapped surfaces. We shall investigate the expansions of both
null normals to this surface

$\displaystyle\mathbf{{k}}$ $\displaystyle=$ $\displaystyle\partial_{r}$ (43)
$\displaystyle\mathbf{{l}}$ $\displaystyle=$
$\displaystyle\partial_{u}+\left[\frac{P^{2}}{r^{2}}(N_{,x}^{2}+N_{,y}^{2})-H-\frac{Q}{2}\right]\partial_{r}+$
$\displaystyle\frac{P^{2}}{r^{2}}(N_{,x}\partial_{x}+N_{,y}\partial_{y})$

that are normalized using $g(\mathbf{{l}},\mathbf{{k}})=-1$. The congruence
generated by $\mathbf{{k}}$ is the one defining Robinson–Trautman family and
thus apart from being shearfree and twistfree it has positive expansion
everywhere $\Theta_{\mathbf{{k}}}>0$. So by demanding the other expansion to
vanish $\Theta_{\mathbf{{l}}}=0$ we are looking for the past horizon according
to the definition by Hayward hayward . One can express this expansion in the
following form

$\displaystyle\Theta_{\mathbf{{l}}}$ $\displaystyle=$
$\displaystyle\frac{-1}{r}\left[K-\frac{2m}{r}-\frac{\Lambda}{3}r^{2}+Q\right.$
$\displaystyle\qquad\quad\left.-\frac{r\Delta
N-P^{2}(N_{,x}^{2}+N_{,y}^{2})}{r^{2}}\right]$

which leads upon evaluation on the horizon surface to equation

$K-\frac{2m}{N}-\frac{\Lambda}{3}N^{2}-\Delta\ln N+Q(u_{0},N)=0$ (45)

where only the last term represents generalization of the horizon equation
derived in PodSvi:2009 . We can check using expansion at origin and infinity
that $Q(u,r)$ (after moving the $C_{1}$ term into redefinition of "mass") is
regular everywhere for all NE models considered above while it naturally
diverges at origin for Maxwell theory. For the last NE model this regularity
is in fact caused by the restricted range of coordinate for $r$.

First, let us use the following redefinition of the function describing
horizon $N(x,y)=2mce^{-\Phi(x,y)}$ given in PodSvi:2009  to obtain an equation
(45) in better form

$\Delta\Phi=\frac{1}{c}\,e^{\Phi}+\frac{4}{3}\Lambda
m^{2}c^{2}\,e^{-2\Phi}-K-Q(u_{0},2mce^{-\Phi})\,,$ (46)

where we assume $\Phi>0$.

In the case of NE models the regularity of $Q$ at origin and infinity (or
maximum allowed value of $r$) means that it has finite supremum and infimum

$Q_{\sup}=\sup_{r\in(0,\infty)}Q(u_{0},r)\,;\
Q_{\inf}=\inf_{r\in(0,\infty)}Q(u_{0},r)$ (47)

These, together with minimum $K_{\min}$ and maximum $K_{\max}$ of Gaussian
curvature of compact surface spanned by $x$ and $y$, can be used to
straightforwardly generalize the results given in PodSvi:2009  which use
theorem (Theorem 1 therein) relying on the existence of sub- and super-
solutions footnote  $\Phi_{\pm}$ for (46) satisfying $0<\Phi_{-}\leq\Phi_{+}$.
In our case the constant sub- and super-solutions can be given depending on
the value of cosmological constant:

  * $\bullet$

$\Lambda\leq 0$

$\displaystyle\Phi_{-}$ $\displaystyle=$
$\displaystyle\ln\left(\,c\,[K_{\min}+Q_{\inf}]\,\right)\,$
$\displaystyle\Phi_{+}$ $\displaystyle=$
$\displaystyle\ln\Big{(}\,c\,[K_{\max}+Q_{\sup}]-\frac{4}{3}\Lambda
m^{2}c^{3}\,\Big{)}\,$ (48)

provided $c[K_{\min}+Q_{\inf}]>1$,

  * $\bullet$

$\Lambda>0$

$\displaystyle\Phi_{-}$ $\displaystyle=$
$\displaystyle\ln\Big{(}\,c\,[K_{min}+Q_{\inf}]-\frac{4}{3}\Lambda
m^{2}c^{3}\,\Big{)}\,$ $\displaystyle\Phi_{+}$ $\displaystyle=$
$\displaystyle\ln\left(\,c\,[K_{max}+Q_{\sup}]\,\right)\,$ (49)

if ${\,c\,[K_{min}+Q_{\inf}]-\frac{4}{3}\Lambda m^{2}c^{3}>1}$. By using the
optimal choice of constant $c$ this constraint reduces to condition on
"physical" quantities

$9\Lambda m^{2}<(K_{min}+Q_{\inf})^{3}.$ (50)

So the restrictions on existence of horizon are stronger for positive
cosmological constant which is natural since as a special case for $Q=0$ we
have asymptotically (there we have $K_{min}=1$) Schwarzschild–de Sitter
solution which can have naked singularity (see PodSvi:2009  for extended
discussion).

For the Maxwell theory one has to be more careful when constructing sub- and
super-solutions. The sub-solutions for both cases of $\Lambda$ can be used
here straightforwardly by setting $Q_{\inf}=0$. For super-solution one uses
the explicit form $Q=q(u_{0})^{4}/r^{2}$ to derive quadratic equation for
$z=e^{\Phi}$

$\frac{1}{c}\,z+\frac{4}{3}\Lambda
m^{2}c^{2}-K_{\max}-\frac{q(u_{0})^{4}}{4m^{2}c^{2}}z^{2}=0$ (51)

based on the right-hand side of (46) after using upper bound for the second
and third term. Upon finding positive solution of (51) for an optimal choice
of free constant $c$ one can give the following super-solutions (notice
slightly different division of cases according to $\Lambda$):

  * $\bullet$

$\Lambda<0$

$\Phi_{+}=\ln\left(\frac{2m^{2}c_{1}}{q(u_{0})^{4}}\right)\,\\\ $

where $c_{1}=-\frac{\sqrt{3\Lambda(K_{\max}q(u_{0})^{4}-m^{2})}}{2\Lambda
q(u_{0})^{2}m}$, so the necessary condition is

$m^{2}>K_{\max}q(u_{0})^{4}$ (52)

  * $\bullet$

$\Lambda\geq 0$ In this case one can neglect the cosmological constant term
since it has the preferable sign anyway and directly obtain

$\Phi_{+}=\ln\left(\frac{2mc}{q(u_{0})^{4}}(m+\sqrt{m^{2}-K_{\max}q(u_{0})^{4}})\right)\,\\\
$

and the condition is same as in the previous case (52).

Asymptotically $K_{\max}\to 1$ since the solution tends to spherically
symmetric case and $q(u)$ approaches constant $q_{0}$. If one would select
traditional notation $q_{0}^{4}=Q^{2}$ (here and only here $Q$ denotes charge
of Reisner–Nordström solution) one recovers natural condition $m^{2}>Q^{2}$.

##  7 Algebraic type of the solution

Now, we would like to see if the geometry of our spacetime is sufficiently
general. Since vacuum Robinson–Trautman spacetime is generally of algebraic
type II we would like our solution to be at least of the same type and not
more special. Our preferred tetrad for determining the Weyl scalars of our
solution is given by different null vectors compared to (43)

$\displaystyle\mathbf{\tilde{k}}$ $\displaystyle=$ $\displaystyle\partial_{r}$
$\displaystyle\mathbf{\tilde{l}}$ $\displaystyle=$
$\displaystyle\partial_{u}-(H+Q/2)\partial_{r}$ (53)
$\displaystyle\mathbf{\tilde{m}}$ $\displaystyle=$
$\displaystyle\frac{P}{\sqrt{2}r}(\partial_{x}+i\partial{y})$

where $i$ is a complex unit. The Weyl spinor computed from this tetrad has
only the following nonzero components

$\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\left[Q_{,rr}-\frac{2}{r}Q_{,r}+\frac{2}{r^{2}}Q\right]-\frac{m}{r^{3}}$
$\displaystyle\Psi_{3}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}P}{4r^{2}}(K_{,x}-iK_{,y})$ (54)
$\displaystyle\Psi_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{4r^{2}}\left[\\{P^{2}(K_{,x}-iK_{,y})\\}_{,x}-i\\{P^{2}(K_{,x}-iK_{,y})\\}_{,y}\right]$

Now, we can easily determine the type irrespective of possible non-optimal
choice of tetrad by using the review of explicit methods for determining the
algebraic type in Zakhary  that are based on Penrose . Namely, when we use
invariants

$I=\Psi_{0}\Psi_{4}-4\Psi_{1}\Psi_{3}+3\Psi_{2}^{2},\ J={\rm
det}\left(\begin{array}[]{ccc}\Psi_{4}&\Psi_{3}&\Psi_{2}\\\
\Psi_{3}&\Psi_{2}&\Psi_{1}\\\ \Psi_{2}&\Psi_{1}&\Psi_{0}\end{array}\right)$

we can immediately confirm that $I^{3}=27J^{2}$ is satisfied so that we are
dealing with type II or more special. At the same time generally $IJ\neq 0$ so
it cannot be just type III. Moreover, the spinor covariant $R_{ABCDEF}$ has
nonzero components

$\displaystyle R_{011111}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Psi_{2}(2\Psi_{3}^{2}-3\Psi_{2}\Psi_{4})$ (55)
$\displaystyle R_{111111}$ $\displaystyle=$
$\displaystyle\Psi_{3}(2\Psi_{3}^{2}-3\Psi_{2}\Psi_{4})$ (56)

which means that generally the spacetime cannot be of type D. So indeed our NE
solution is of the most general type possible for the Robinson–Trautman vacuum
class. Which does not mean that there cannot be a NE solution of type I when
one considers completely general Maxwell tensor ${\mathbf{F}}$. Moreover,
inspecting the components of Weyl spinor (7) one concludes that in the special
case of $K(x,y)=const>0$ (constant positive Gaussian curvature of compact two-
space spanned by $x,y$) the algebraic type becomes D consistent with spherical
symmetry. Finally, since $\Psi_{3}=0$ implies $\Psi_{4}=0$ we cannot have all
components of spinor covariant $Q_{ABCD}$ (see Penrose  and Zakhary )
vanishing while having nonvanishing Weyl spinor. This means that our family of
solutions does not contain type N geometries.

##  8 Conclusion and final remarks

We have derived Robinson–Trautman solutions with source given by nonlinear
electrodynamics for several specific models of NE Lagrangian (both with
Maxwell limit and without). The solutions were derived based on known
spherically symmetric ones by a method described in Section 3. The Maxwell
case was included for comparison as well. In all cases of NE the singularity
of electromagnetic field is resolved as in the static spherically symmetric
cases. However, it was not possible to satisfy additional constraint for
having Robinson–Trautman solution with Hoffmann–Born–Infeld model or with NE
model which provides source of Bardeen black hole. Both these models can be
used to construct spherically symmetric solutions without curvature
singularity. The impossibility to generalize these models in the absence of
this symmetry suggests that this kind of resolution of curvature singularity
might not be stable under nonlinear perturbations (at least within the
Robinson–Trautman class). However, the Robinson–Trautman class does not
contain rotating black holes (due to twist-free condition) and therefore our
results do not need to be universally valid. Unfortunately, the twisting class
of solutions does not permit analysis on the level presented here (there are
no asymptotic behavior studies in the dynamical regime).

Since in all models the curvature singularity is present we analyzed the
existence of horizons using the quasilocal concepts. All solutions are
generally of algebraic type II and asymptotically in retarded time approach
their spherically symmetric versions. All models with unrestricted coordinate
range also remain locally asymptotically flat (or (anti–)de Sitter) as their
vacuum counterparts.

The interpretation of "charge" $q(u)$ suffers from the same difficulties as
that of "mass" in vacuum Robinson–Trautman solutions. The asymptotic behavior
of $q$ in all our models is identical to that of function $P$ which describes
geometry of two spaces of constant $u$ and $r$, namely in preferred coordinate
$u$ it settles exponentially fast to a constant.

## Appendix

Here we plot graphs of electromagnetic field invariant $F$ for all the
considered models. We have put $C_{1}(u)=1,\beta=1,\alpha=2^{-1/4}$ and we
have selected exponential decay behavior for function $q(u)=1+e^{-u}$ which
corresponds to the asymptotic behavior derived in Section 5. The plots show
that, except for the Maxwell case, $F$ is finite at $r=0$ for all models. The
difference is only in the rate of approach to this finite value. All the
models give evidently asymptotically ($r\to\infty$) vanishing $F$ and also
fast exponential decay to SSS form of $F$ is clear.

Figure 3: The plot for Maxwell model Figure 4: The plot for Born–Infeld model
Figure 5: The plot for Logarithmic model Figure 6: The plot for New Lagrangian
1 model Figure 7: The plot for New Lagrangian 2 model

###### Acknowledgements.

We are grateful to prof. Jiří Bičák for discussion and valuable comments. This
work was supported by grant GAČR 14-37086G.

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11institutetext: King’s College London

# A Measure of Explanatory Effectiveness ††thanks: Presented at the 1st
International Workshop on Trusted Automated Decision-Making (TADM) co-located
with ETAPS 2021. Work done by DC is thanks to the UKRI Centre for Doctoral
Training in Safe and Trusted AI (EPSRC Project EP/S023356/1). PM wishes to
thank Simon Parsons and Elizabeth Sonenberg for discussions on these topics.
DC would like to thank Alex Jackson and Nandi Schoots for helping to
understand understanding.

Towards a Formal Model of Explanation

Dylan R. Cope Correspondence<EMAIL_ADDRESS>Peter McBurney

(Received: date / Accepted: date)

###### Abstract

In most conversations about explanation and AI, the recipient of the
explanation (the explainee) is suspiciously absent, despite the problem being
ultimately communicative in nature. We pose the problem ‘explaining AI
systems’ in terms of a two-player cooperative game in which each agent seeks
to maximise our proposed measure of explanatory effectiveness. This measure
serves as a foundation for the automated assessment of explanations, in terms
of the effects that any given action in the game has on the internal state of
the explainee.

###### Keywords:

Explanation XAI Explainee-centric Artificial Intelligence Algorithmic
Information Theory Dialogues

##  1 Introduction

The term _explanation_ in artificial intelligence (AI) is often conflated with
the concepts of _interpretability_ and _explainable AI_ (XAI), but there are
important distinctions to be made. Miller (2019) defines interpretability and
XAI as the process of building AI systems that humans can understand. In other
words, by design, the AI’s decision-making process is inherently transparent
to a human. In contrast, explicitly explaining the decision-making to an
arbitrary human is explanation generation. The latter is the subject of this
paper. More specifically, we are working towards developing a formal framework
for the automated generation and assessment of explanations.

Firstly, some key terminology: an explanation is generated through a
dialectical interaction whereby one agent, the explainer, seeks to ‘explain’
some phenomenon, called the explanandum, to another agent, the explainee. In
this article, we propose a novel measure of _explanatory effectiveness_ that
can be used to motivate artificial agents to generate good explanations (e.g.
in the form of a reward signal), or to analyse the behaviours of existing
communicating agents. We then define _explanation games_ as cooperative games
where two (or more) agents seek to maximise the effectiveness measure.

##  2 Related Literature

Intepretability and XAI have received an abundance of recent attention (see
Adadi & Berrada (2018) for a review). This is largely due to two factors;
regulatory demands (UK Information Commissioner’s Office 2019) and the
emergence of highly-performant black-box models, such as deep neural networks,
that are naturally inscrutable. However, the central crux of interpretability
techniques is the need to define a fixed interpretable domain from which we
can derive explanations. This presents two challenges: there are no formal
procedures for determining if a given domain is interpretable; and a domain
may be interpretable to some agents, but not others, or only within some
contexts. Moving away from interpretability, the problem of explanation
generation has a long history in AI (Mueller et al. 2019). To some, there is a
sense in which generating explanations is the hallmark of intelligence itself
(Schank 1984). To others, explanation is simply about building models – a
process which is seen as merely instrumental to intelligent behaviour (Russell
& Norvig 2010, Hutter 2005, Chaitin 2006).

In the philosophy of science the concept of explanation is posed in terms of
generating descriptions of, or hypotheses regarding, latent phenomena. This
has led to investigations of formal measures of _explanatory power_ , with an
early example being Popper’s (1959) notion of the ‘degree of corroboration’.
This developed into a line of philosophers devising subjectivist definitions
for capturing aspects of the ‘goodness’ of explanations or hypotheses (Lipton
2003, Glass 2002, Okasha 2000, Schupbach & Sprenger 2011). However, by the
subjectivity of these measures they may only assess the degree to which one
believes (or simply likes) an explanation, which is not necessarily correlated
with the degree to which an explanation is actually true (or representative of
the world).

Recently, calls have been made for the need for human-centred explanation
(Kirsch 2017, Abdul et al. 2018). Yet, the framing of explanation generation
as a cooperative problem between a human and machine dates back to the era of
expert systems (Karsenty & Brezillon 1995, Johnson & Johnson 1993, Graesser et
al. 1996). By articulating explanation as a formal dialogue, a related
direction of investigation is dialogue games (McBurney & Parsons 2002). In
particular, _information-seeking_ (Walton & Krabbe 1995) and _education_
(Sklar & Parsons 2004) dialogues are especially relevant. Sklar & Azhar (2018)
conducted empirical research with a human-machine collaboration task where the
agents participated in a dialogue and explanations were provided to a human
based of an argumentation framework (Dung 1995).

##  3 What is Explanation?

In this work we treat explanatory processes as involving two agents — an
explainer and an explainee — and the result is that the explainee understands
the explanandum better by the end than they did at the start. We define ‘an
explanation’ as any sequence of observations made by the explainee that leads
to this result. Thus an explanation could be a piece of text or spoken
language, but it could also be a diagram or a piece of interactive media.

With this we shift the problem onto formally defining a measure of an agent’s
‘understanding’ of some arbitrary phenomenon. We approach the question in
terms of four stances111These stances do not represent arguments defended by
anyone in particular, but rather we are constructing them here as rhetorical
tools to help decompose the problem. towards comprehension, understanding as:
(1) a sensation (Hume 1751); (2) information compression (Chaitin 2006, Zenil
2019, Maguire et al. 2016); (3) performance capacity (Turing 1950, Perkins
1993); or (4) organised information (Lakoff & Johnson 1980, Hofstadter &
Sander 2012).

The sensation stance states that comprehension is a conscious experience — you
understand something if you feel that you apprehend it. The compression stance
says that understanding is the formulation of concise and accurate
descriptions of phenomena. The performance stance argues that having
information is not enough; you must also know how to use the information. The
organised-information stance tells us that utilisation and compression are a
byproduct of something more important; namely that the agent represents
information in relation to their own conceptual framework. While each of the
stances has issues of their own, combined they provide a persuasive account.
In other words, if someone claims they understand something, they can use
their information to do things, and their description of the phenomenon is
concise, accurate, and grounded in other concepts that they understand, then
it is hard to argue that they do not grasp the phenomenon.

##  4 Technical Background

###  4.1 Algorithmic Information Theory

Algorithmic Information Theory (AIT) is a view of information that takes a
fundamentally computational approach (Solomonoff 1964, Kolmogorov 1968,
Chaitin 1975). Formally, AIT is built on the notion of _Kolmogorov complexity_
, denoted $K(x)$. $K(x)$ is defined as the length of the shortest program,
$p$, on a Universal Turing Machine (UTM), $U$, that outputs $x$.

$\displaystyle K(x)=\min_{p}\left\\{|p|\leavevmode\nobreak\
:\leavevmode\nobreak\ U(p)=x\right\\},\leavevmode\nobreak\ \text{where $|p|$
measures the length of $p$}$ (1)

The conditional Kolmogorov complexity, $K(x|y)$, is similarly defined by the
length of the shortest program that produces $x$ when given input $y$.

$\displaystyle K(x|y)=\min_{p}\left\\{|p|\leavevmode\nobreak\
:\leavevmode\nobreak\ U(yp)=x\right\\}$ (2)

Thus we can define a measure of mutual information:

$\displaystyle I(x;y)=K(y)-K(y|x)$ (3)

Unless otherwise specified, when we talk of the mutual information between two
objects we will be referring to an application of Equation 3.

###  4.2 Agents

In its most basic conception, ‘an agent’ is any system that makes observations
and takes actions. For any agent $X^{t}\in\mathcal{X}$ at time $t$, we will
denote that they make observations $o_{X}^{t}\in\mathcal{O}_{X}$ and take
actions $a_{X}^{t}\in\mathcal{A}_{X}$. Another important factor in describing
agents is their internal state. This phrase can refer to various aspects of an
agent’s cognition, but we are mostly interested in this object insofar as it
stores information. Firstly, we will assume that an agent’s internal state may
fall into a variety of configurations, i.e. there exists a set of possible
internal states for an agent, which we will denote $\mathcal{Z}_{X}$.
Secondly, we will talk of information being ‘encoded’ in an agent’s internal
state. Given an object $o$, we will denote $X$’s encoding of $o$ as $\langle
o\rangle_{X}$, where $\langle o\rangle_{X}\in\left\\{p\leavevmode\nobreak\
:\leavevmode\nobreak\ U(p)=o\right\\}$ for some UTM $U$. We will speak of the
agent ‘having’ this encoding, or its internal state ‘containing’ this
encoding. This is independent to how this is achieved, e.g. the agent’s
internal state may simply store a list of encodings, or multiple encodings may
overlap in a distributed storage medium such as a neural network.

###  4.3 Universal Intelligence Theory

Universal Intelligence Theory (UIT), proposed by Legg & Hutter (2007),
establishes a definition of machine intelligence based on algorithmic
information theory and reinforcement learning. In order to meaningfully
compare different performances over a potentially infinite number of time
steps, the scope of possible environments is limited such that the sum of
rewards (the return) is always less than one. We will refer this as the set of
bounded-test environments. With this, the universal intelligence of an agent
$\pi$ is given by:

$\displaystyle\Upsilon(\pi)=\sum_{\mu\in E}2^{-K(\mu)}V_{\mu}^{\pi}$ (4)

Where $V_{\mu}^{\pi}$ is the return that $\pi$ achieves in environment $\mu$.

###  4.4 Universal Artificial Intelligence

Consider a stochastic environment with dynamics described by a probability
distribution $\mu(e_{k}|\text{\ae}_{<k})$, where $e_{k}$ is the percept
(observation-reward tuple) given at time $k$, and $\text{\ae}_{<k}$ is the
action-percept history. In order to perform optimally, the agent in this
environment must infer $\mu$. This is known as the problem of induction. By
combining Solomonoff induction (Solomonoff 1964) with von Neumann-Morgenstern
rational decision-making (Morgenstern & von Neumann 1953), Hutter (2005)
defines AIXI; an agent that chooses the best possible action at every time
step given perfect inductive inference.

##  5 Formalising Understanding

###  5.1 Partitioning the Internal State

In order to devise a measure of understanding, we will start by defining
partitions of the information in the internal state. These partitions are
constructed with respect to a given phenomenon $p\in\mathcal{P}$. There are
four: the $p$-relevant information (all information related to $p$), the
$p$-irrelevant (all information completely unrelated to $p$), the $p$-specific
(the information that only relates to $p$), and the $p$-background information
(all information that is not specifically related to $p$). In the following
formal definitions we are using a particular notation that warrants
explanation. As we have already established, $z_{X}$ denotes the internal
state of agent $X$. We denote $p$-relevant notation with a comma after the $X$
followed by $*p$, $z_{X,*p}$. The star indicates that we are ‘selecting’ all
of the information relevant to $p$, rather than only the information specific
to $p$. When the star is omitted we are referring to to specific information
regarding whatever follows the comma, e.g. $z_{X,p}$ is the $p$-specific
information and $z_{X,\neg p}$ is the information specific to everything that
is not $p$ (the $p$-irrelevant information).

######  Definition 1 ($p$-Relevant Information)

Given a phenomenon $p\in\mathcal{P}$ and an agent $X$ with internal state
$z_{X}$, the $p$-relevant information $z_{X,*p}\in\mathcal{Z}_{X}$ is the
object where $I(z_{X,*p};p)=I(z_{X};p)$ and $I(z_{X,*p};z_{X})$ is minimised,
i.e. there exists no $z_{X,*p}^{\prime}$ such that
$I(z_{X,*p}^{\prime};p)=I(z_{X};p)$ and
$I(z_{X,*p}^{\prime};z_{X})<I(z_{X,*p};z_{X})$.

######  Definition 2 ($p$-Irrelevant Information)

Given a phenomenon $p\in\mathcal{P}$ and an agent $X$ with internal state
$z_{X}\in\mathcal{Z}_{X}$, the $p$-irrelevant information $z_{X,\neg p}$ is
the object where $I(z_{X,\neg p};p)=0$ and $I(z_{X,\neg p};z_{X})$ is
maximised, i.e. there exists no $z_{X,\neg p}^{\prime}$ such that $I(z_{X,\neg
p}^{\prime};p)=0$ and $I(z_{X,\neg p}^{\prime};z_{X})>I(z_{X,\neg p};z_{X})$.

######  Definition 3 ($p$-Specific Information)

Given a phenomenon $p\in\mathcal{P}$ and an agent $X$ with internal state
$z_{X}\in\mathcal{Z}_{X}$, the $p$-specific information $z_{X,p}$ is the
object where $I(z_{X,p};p)>0,\leavevmode\nobreak\
I(z_{X,p};p^{\prime})=0\leavevmode\nobreak\ \forall
p^{\prime}\in\mathcal{P},p^{\prime}\neq p$ and the mutual information
$I(z_{X,p};z_{X})$ is maximised, i.e. there exists no $z_{X,p}^{\prime}$ such
that $I(z_{X,p}^{\prime};p)>0,\leavevmode\nobreak\
I(z_{X,p}^{\prime},p^{\prime})=0\leavevmode\nobreak\ \forall
p^{\prime}\in\mathcal{P},p^{\prime}\neq p$ and
$I(z_{X,p}^{\prime};z_{X})>I(z_{X,p};z_{X})$.

######  Definition 4 ($p$-Background Information)

Given a phenomenon $p\in\mathcal{P}$ and an agent $X$ with internal state
$z_{X}\in\mathcal{Z}_{X}$ and $p$-specific information $z_{X,p}$, the
$p$-background information222By the notation $*\neg p$ we can see that this is
‘everything relevant’ to ‘not $p$’. $z_{X,*\neg p}$ is the object where
$I(z_{X,p};z_{X,*\neg p})=0$ and $I(z_{X,*\neg p};z_{X})$ is maximised, i.e.
there exists no $z_{X,*\neg p}^{\prime}$ such that $I(z_{X,p};z_{X,*\neg
p}^{\prime})=0$ and $I(z_{X,*\neg p}^{\prime};z_{X})>I(z_{X,*\neg p};z_{X})$.

###  5.2 Information Compression

With these partitions we can define how compressed the $p$-relevant
information is:

######  Definition 5 ($p$-Compression Factor)

Suppose a phenomenon $p\in\mathcal{P}$ and an agent $X$. The $p$-compression
factor $c:\mathcal{X}\times\mathcal{P}\rightarrow(0,1]$ is given as the ratio
of the Kolmogorov complexity of the $p$-relevant information object to the
size of the agent’s encoding of that information:

$\displaystyle c(X,p)=\frac{K(z_{X,*p})}{|\langle z_{X,*p}\rangle_{X}|}$ (5)

###  5.3 Information Utilisation

Next, we will attempt to formalise the performance stance on understanding,
i.e. we will try to define $X$’s information utilisation of $p$. To do this,
we will need to construct a set of ‘fair tests of $p$’ for $X$. We will start
by noting: (1) A fair test for $X$ should require $X$’s background
information; (2) a test of $p$ should require information about $p$. We will
use the formalisation of rational decision-making, AIXI, to ‘benchmark’ how
information is utilised in a given environment. Unlike a typical test-taker,
AIXI enters into an environment with no prior knowledge, and thus we must
present any priors to AIXI as a part of its percept sequence. Therefore, to
decide whether or not a given task meets the criteria outlined above we will
construct a ‘meta-task’ for AIXI where relevant observations are prepended to
the task.

######  Definition 6 ($(X,p)$-tests)

Given a phenomenon $p\in\mathcal{P}$ and an agent $X$ with internal state
$z_{X}\in\mathcal{Z}_{X}$, we start with the set of bounded-test environments
$E$, we define the set of $(X,p)$-tests, $E_{X,p}$, as follows:

$\displaystyle E_{X,p}=\left\\{\mu\in E\leavevmode\nobreak\
:\leavevmode\nobreak\
V_{(p,b)\circ\mu}^{AIXI}=V_{\mu}^{*}>0,\leavevmode\nobreak\
V_{(p)\circ\mu}^{AIXI}=V_{(b)\circ\mu}^{AIXI}=V_{\mu}^{AIXI}=0\right\\}$ (6)

Where $b$ is a shorthand for the $p$-background information $b=z_{X,*\neg p}$,
and $\mathbf{x}\circ\mu$ denotes the construction of a new environment
$\mu^{\prime}$ such that:

$\displaystyle\forall x_{i}\in\mathbf{x},\leavevmode\nobreak\ \forall
a_{<i},\leavevmode\nobreak\ \mu^{\prime}\left((x_{i},0)\leavevmode\nobreak\
|\leavevmode\nobreak\ a_{<i}\right)=1$ (7) $\displaystyle\forall
k>|\mathbf{x}|,\leavevmode\nobreak\ \mu^{\prime}(e_{k}\leavevmode\nobreak\
|\leavevmode\nobreak\ a_{<k})=\mu(e_{k}\leavevmode\nobreak\
|\leavevmode\nobreak\ a_{j\ldots k}),\leavevmode\nobreak\
\text{where}\leavevmode\nobreak\ j=|\mathbf{x}|$ (8)

It is worth noting why we are using only the $p$-background information and
not the agent’s entire internal state as required prior knowledge. This is
because if the agent knows anything about $p$ then AIXI would be able to use
the information encoded in the internal state to pass the test when only given
$b$. We want AIXI to only get information about $p$ from $p$ itself so that we
can strictly outline the criteria above.

Using the set of fair tests for $X$, we can define a measure of information
utilisation by measuring the agent’s intelligence across these environments.
This is an adaptation of Hutter’s (2005) measure of intelligence (Equation 4).

######  Definition 7 ($p$-Utilisation)

Given an agent $X$ and phenomenon $p$, the $p$-utilisation
$\Upsilon_{p}:\mathcal{X}\rightarrow[0,1]$ is defined:

$\displaystyle\Upsilon_{p}(X)=\sum_{\mu\in E_{X,p}}2^{-K(\mu)}V_{\mu}^{X}$ (9)

###  5.4 Information Integration

With the definitions we have constructed here, we can also introduce a measure
of how ‘integrated’ the $p$-relevant information is.

######  Definition 8 ($p$-Integration)

Suppose we have a phenomenon $p\in\mathcal{P}$, and an agent $X\in\mathcal{X}$
with $p$-relevant information $z_{X,*p}$ and $p$-specific information
$z_{X,p}$. The $p$-integration,
$\phi:\mathcal{X}\times\mathcal{P}\rightarrow[0,1)$, is defined,

$\displaystyle\phi(X,p)=\tanh\left(\frac{|\langle
z_{X,*p}\rangle_{X}|}{|\langle z_{X,p}\rangle_{X}|}-1\right)$ (10)

As the $p$-relevant information will always be larger-than or equal to the
$p$-specific information ($|\langle z_{X,*p}\rangle_{X}|\geq|\langle
z_{X,p}\rangle_{X}|$), the ratio in this measure will equal 1 when all
relevant information is specific. In this case, there is no relevant
information that is used for anything else, i.e. the $p$-relevant information
is not at all integrated into the rest of the internal state (or nothing else
exists to integrate with). Conversely, the smaller the specific information
gets, the more the relevant information must be sharing with encodings for
other phenomena.

###  5.5 The Measure of Understanding

Finally we bring these ideas together to define our measure of understanding.
The resulting measure is bounded by 0 and 1.

######  Definition 9 (Understanding)

Given an agent $X\in\mathcal{X}$ with internal state $z_{X}$ and phenomenon
$p\in\mathcal{P}$, the measure of $X$’s understanding of phenomenon $p$,
$\kappa:\mathcal{X}\times\mathcal{P}\rightarrow[0,1)$, is defined as:

$\displaystyle\kappa(X,p)$ $\displaystyle=\frac{\hat{\kappa}(X,p)\cdot
c(X,p)\cdot\phi(X,p)\cdot\Upsilon_{p}(X)\cdot I(z_{X};p)}{K(p)}$ (11)

Where $\hat{\kappa}(X,p)\in\\{0,1\\}$ is $X$’s self-reported understanding of
$p$.

##  6 Explanation Games

With our measure of understanding, we are ready to define explanatory
effectiveness:

######  Definition 10 (Explanatory Effectiveness)

The effectiveness of an explanation is the change in explainee’s understanding
of the explanandum $p\in\mathcal{P}$ over the course of the explanatory
process. Formally, given an explainer agent $A$ and an explainee agent $B$
that interact over $\tau$ time steps, the explanatory effectiveness is a
function $\xi:\mathcal{O}_{B}^{*}\times\mathcal{P}\rightarrow(-1,1)$ defined
as:

$\xi(\mathbf{o}_{B},\leavevmode\nobreak\
p)=\kappa(B^{\tau},\leavevmode\nobreak\ p)-\kappa(B^{1},\leavevmode\nobreak\
p)$ (12)

Where $B^{t}$ denotes $B$ at time $t$ and $\mathbf{o}_{B}$ is the sequence of
observations that $B$ made during the interaction.

######  Definition 11 (Explanation Game)

Suppose an explainer agent $A$, explainee agent $B$, and explanandum
$p\in\mathcal{P}$. An explanation game $G=(A,B,p,\tau)$ is a cooperative
finite sequential game with asymmetric information in which the participants
seek to maximise $\xi(\mathbf{o}_{B},p)$ over the course of $\tau$ time steps,
where $\mathbf{o}_{B}$ is the sequence of all observations made by $B$.

From these definitions, there are a few observations that we can make.
Firstly, there is nothing to stop a game from having negative effectiveness,
i.e. the explainee understands the phenomenon less after the ‘explanation’. As
$\kappa$ is bounded by 0 and 1, $\xi$ is bounded by -1 and 1. Secondly, there
is no necessary link between effectiveness and the explainee’s beliefs
regarding their own understanding. It is possible for the explainee to believe
that the explanation was more effective than it was (e.g.
$\hat{\kappa}(X,p)=1$, but $I(z_{X};p)=0$). Thirdly, we can use this notion to
discuss the motivation of the explainer. It makes sense to consider an agent
as an explainer, rather than a deceiver, only if they expect the sign of the
$\xi$ to be positive. Finally, it is worth noting that this measure changes
according to time in which we choose to record it. The explainer may start out
strong and increase the explainee’s understanding of the explanandum, but then
say something that leads to confusion.

##  7 Discussion

In this paper we have presented a formal model for assessing the the
‘explanatory effectiveness’ $\xi$ of a dialectical process between two agents.
We used this to define explanation games in which participants seek to
maximise $\xi$. Along the way we used AIT and UIT to develop a measure of an
agent’s ‘understanding’ of a given phenomenon $p$. This involved partitioning
the information in the agent’s mental state into four objects relative to $p$;
the $p$-relevant, $p$-irrelevant, $p$-specific, and $p$-background
information. We used these to define the $p$-compression factor (how
compressed the agent’s representation of $p$ is), $p$-integration (what
proportion of the representation is only encoding for $p$), and the
$p$-utilisation. For the last of these we needed to construct a set of ‘fair
tests’, i.e. a set of environments that would rely on both knowledge of $p$
and the agent’s background knowledge to solve. We find these environments by
asking: “Could AIXI solve this environment when given this information?”.
However, it should not be taken for granted that this is the right question to
ask, and thus we should study this space of environments more precisely to see
if it includes unfair tests or leaves out potential fair tests.

Future work should investigate the trustworthiness of explanations generated
in our framework, as we have made the implicit assumption that if an agent
understands something they can assess whether or not they trust it. One
direction to look in is the implications of explainees with limited
capacities, i.e. either time/space complexity constraints, or explainees who
are biased in particular ways. Additionally, the assumption that explanation
games are always cooperative should be challenged, as in many real situations
participants may have conflicting or ulterior agendas. For both the
cooperative and non-cooperative case a useful research project will be to
articulate rules for the dialogue game between explainer and explainee
(McBurney & Parsons 2002) and to develop strategies for each player, given
their goals. Finally, as $K$ and AIXI are not computable, alternatives for
these components (for the purposes of this framework) should be devised and
studied.

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