# Exact Geometries from Boundary Gravity

Rohit K. Gupta, Supriya Kar, R. Nitish and Monika Verma  Department of Physics
and Astrophysics
University of Delhi, New Delhi 110007, India

###### Abstract

We show that the extremal Reissner-Nordström type multi black holes in an
emergent scenario are exact in General Relativity. It is shown that an axion
in the bulk together with a geometric torsion ensure the required energy-
momentum to source the $(3$$+$$1)$ geometry in the Einstein tensor. Analysis
reveals a significant role of dark energy to the curved space-time.

††preprint: APS/123-QED

##  I Introduction

Einstein gravity elegantly ensures the deformation geometries of space-time
underlying the Riemann curvature tensor ${\cal R}_{\mu\nu\lambda\rho}$. It is
a classical theory of metric $g_{\mu\nu}$ field and delves with the continuum
space-time. Furthermore the ${\cal R}_{\mu\nu\lambda\rho}$ is reducible and is
re-expressed with three irreducible curvature tensors, $i.e.$ Conformal-Weyl
$C_{\mu\nu\lambda\rho}$, trace-less Ricci $S_{\mu\nu}$ and Ricci scalar ${\cal
R}$. Thus the curved space-time in Einstein gravity is contained in one or
more of the three irreducible curvatures. However the Einstein gravity was
elegantly described by the Einstein-Hilbert action which is expressed only in
terms of the scalar ${\cal R}$. The metric field equations turn out be non-
linear and they are beautifully expressed in terms of the Einstein tensor
$G_{\mu\nu}$=${\kappa}T_{\mu\nu}$, where $\kappa$ denotes the coupling and
$T_{\mu\nu}$ ensures the energy-momentum tensor. Importantly the $T_{\mu\nu}$
is believed to source the intrinsic curvatures of space-time. There were
numerous attempts in the past hundred years to apprehend the $T_{\mu\nu}$
which could describe the geometry in $G_{\mu\nu}$ but none of them were
completely successful Ashtekar and Magnon-Ashtekar (1979); Ó Murchadha (1986);
Sharif (2004); Gümrükçüoğlu and Namba (2019). In fact, a firm believe leading
to a conjecture seeking an appropriate $T_{\mu\nu}$ remains unanswered.

The unresolved issue in Einstein gravity provokes thought to believe in an
alternate formulation Wilczek (1998); Blagojević  _et al._ (2013); Anastasiou
_et al._ (2018); Benisty  _et al._ (2022) to go beyond General Relativity
(GR). This was with a plausible aim to find a satisfactory $T_{\mu\nu}$. Along
the line, higher dimensional theories underlying the Riemann curvature tensor
have been explored in the past Cho (1975); Troncoso and Zanelli (2000); Wesson
(2015). In this paper we attempt to address an apparently unresolved problem
in GR but with a higher dimensional gauge theoretic framework Singh  _et al._
(2013a, b, 2014). Research ensures that an order two anti-symmetric
$B_{\mu\nu}$ field underlying a $U(1)$ gauge symmetry in $(4$$+$$1)$ may serve
as an insightful tool to find the required $T_{\mu\nu}$. Interestingly a
geometric torsion (GT) theory is known to ensure an open geometry which is
unlike to the closed geometries in GR. The observed perihelion precision
Nitish  _et al._ (2020) reconfirms the open geometry in presence of an extra
dimension to GR. Interestingly the proposed bulk/boundary correspondence
resembles to an established open/closed string duality Sen (2003).

In this article we revisit a GT description underlying a $5D$ bulk gauge
theory with a renewed interest for a boundary gravity Nitish  _et al._ (2019);
Gupta  _et al._ (2020). A topological $BF$ term is argued to incorporate the
winding modes in gravity. It is believed to ensure an UV finite quantum
correction to GR. It was shown that a torsion connection modifies a covariant
derivative $\nabla_{\mu}$$\rightarrow$${\cal D}_{\mu}$ in a $U(1)$ gauge
theory Singh  _et al._ (2013a); Nitish and Kar (2021). The non-Riemann
curvature tensors were derived in the modified theory which have been checked
to ensure the space-time beyond GR. It was demonstrated that an emergent
metric underlying the known black hole geometries in Einstein gravity is a low
energy phenomenon derived from a symmetric matrix comprising the quantum
fluctuations of $B_{\mu\nu}$ field. Thus the modified gauge theory with the
Minkowski signature of space-time has been shown to incorporate intrinsic
curvatures in presence of a GT connection.

It was argued that a gravitational pair of membrane/anti-membrane across the
horizon of a background black hole can be created by the quantum of a GT in a
gauge theory. This is analogous to the pair creation process
($\gamma\rightarrow e^{+}e^{-}$) known in quantum electrodynamics. Similarly
the $B_{\mu\nu}$ field quantum is believed to vacuum create a pair of
string/anti-string Bachas and Porrati (1992). In fact a vacuum pair creation
is a powerful quantum tool and is believed to provide an enhanced vision to
explore more in the quantum gravity domain. Interestingly the idea has been
exploited to explain the Hawking radiation phenomenon at the horizon of a
black hole Hawking (1975). We may mention that the tool was also beautifully
explored to address a cosmic pair Majumdar and Davis (2002).

##  II Aspects of multi black holes in GR

Reissner-Nordström (RN) black hole is known as exact in GR. It is
characterised by two unequal deformation parameters ($M$$>$$|Q_{e}|$) and they
source two independent potentials within the RN black hole. The instability
with two horizons in RN black hole stabilizes to an extremal geometry with an
equipotential. Thus a charged, or more than one parameter, black hole is
likely to provide a clue to unfold some aspects of quantum gravity.
Interestingly a drastic change in feature from an extremal RN black hole to
multi black holes in isotropic coordinates Carroll (2019) may be perceived
with a notion of cloning in a quantum scenario. In fact, the cloning of black
hole is believed to be associated with the non-locality on the event horizon
Singh  _et al._ (2014). Intuitively a non-local horizon in an extremal RN
black hole may be viewed as a collection of closely spaced horizons which
ensure the exact multi black holes in GR.

In addition, an extremal black hole is believed to provide a clue to a quantum
tunneling phenomenon between the multiple black holes. Intuitively, the non-
zero integer values of the conserved charge $M$=$|Q_{e}|$ reconfirms the
multiple black holes. We recall that $M$ takes continuum range of values and a
typical electric charge can take non-zero integer values in a non-extremal RN
black hole. However the RN black hole shrinks ($r_{+}\leftrightarrow r_{-}$)
to ensure a stable configuration for a positive integer value to $M$. This in
turn would describe an extremal RN multi black hole as $M$ takes $M_{i}$
positive integer values. A similar analysis in an emergent quantum gravity
scenario Singh  _et al._ (2013b) would like to incorporate the multi black
hole scenario even for non-extremal RN black hole. We postpone a detailed
analysis to the section IV.

##  III Boundary Quantum Gravity

A promising theoretical idea relating a bulk $B_{\mu\nu}$ field theory to a
boundary gravity Gupta  _et al._ (2020) is believed to be insightful to a
theory of quantum gravity. The boundary action comprises a local sector GR and
a global sector which turns out to be topological. In the case the holographic
correspondence naturally ensures that an emergent gravity is described by a
closed theory and the bulk is an open theory. At this point we may recall the
established $AdS_{5}$ bulk/boundary $CFT_{4}$ correspondence Maldacena (1999),
where the gravity in bulk describes an isolated system due to the $AdS$
radius. However this feature is not feasible with an arbitrary geometry in
bulk.

Now we begin with a $B_{\mu\nu}$ field dynamics ensuring a $U(1)$ flux
$H_{\mu\nu\lambda}$=$3\nabla_{[\mu}B_{\nu\lambda]}$ in a perturbation gauge
theory. A covariant constant ${\tilde{B}}_{\mu\nu}$ field is considered in
addition to the $B_{\mu\nu}$ field dynamics. Interestingly a constant
${\tilde{B}}_{\mu\nu}$ has been explored as a perturbation parameter.
Remarkably ${\tilde{B}}_{\mu\nu}$ theory has been shown to incorporate higher
order perturbation corrections in ${\tilde{B}}_{\mu\nu}$ field. In fact the
corrections were vital to envisage the gravitational interactions in a
plausible quantum theory. An iteration replaces
$H_{\mu\nu\lambda}\rightarrow{\cal H}_{\mu\nu\lambda}$. Then a term in GT is
given by

${\cal
D}_{\lambda}{\tilde{B}}_{\mu\nu}=\nabla_{\lambda}{\tilde{B}}_{\mu\nu}+\frac{1}{2}{\cal
H}_{\lambda\mu}{}^{\rho}{\tilde{B}}_{\rho\nu}-\frac{1}{2}{\cal
H}_{\lambda\nu}{}^{\rho}{\tilde{B}}_{\rho\mu}$ (1)

Then the GT flux is given by

$\displaystyle{\cal H}_{\mu\nu\lambda}$ $\displaystyle=$ $\displaystyle
H_{\mu\nu\lambda}+H_{\mu\nu}{}^{\alpha}{\tilde{B}}_{\alpha\lambda}+H_{\nu\lambda}{}^{\alpha}{\tilde{B}}_{\alpha\mu}$
(2)
$\displaystyle\qquad\qquad\qquad\quad+H_{\lambda\mu}{}^{\alpha}{\tilde{B}}_{\alpha\nu}+{\cal
O}({\tilde{B}}^{2})$

With an appropriate coupling or length scale $l$ the $U(1)$ gauge
transformation becomes
$\delta{\tilde{B}}_{\mu\nu}$=$l(\partial_{\mu}\Lambda_{\nu}$$-$$\partial_{\nu}\Lambda_{\mu})$.
Explicitly a variation of the action yields

$\displaystyle\delta{\cal H}_{\mu\nu\lambda}^{2}=6l\left({\cal
H}^{\mu\nu\lambda}H_{\mu\nu}{}^{\alpha}-{\cal
H}^{\mu\nu\alpha}H_{\mu\nu}{}^{\lambda}\right)\nabla_{\alpha}\Lambda_{\lambda}$
(3)

The broken $U(1)$ gauge symmetry may be restored Singh  _et al._ (2013a) with
a symmetric matrix $f_{\mu\nu}$$=$$\pm l^{2}({\cal
H}_{\mu\alpha\beta}H^{\alpha\beta}{}_{\nu})$. It shows that a non-linear GT
and a linear gauge theoretic torsion together ensure the matrix fluctuations
and they have been argued to source an emergent quantum gravity potential
$V_{q}$ in ref.Nitish  _et al._ (2020). Analysis reveals that $V_{q}$ should
be a non-Newtonian potential and hence is believed to be defined with a lower
cutoff on the radial coordinate. Alternately, the required UV cutoff can also
be governed by a non-point (or extended) charge. It was argued that a non-
point charge, naturally modifies a point notion, and incorporates winding
modes into the theory. In a large (length) scale limit, the symmetric matrix
reduces to a non-trivial metric correction to the otherwise flat metric
$g_{\mu\nu}$. It is given by

${\tilde{g}}_{\mu\nu}=g_{\mu\nu}\pm f_{\mu\nu}\ |_{\rm large\ scale}$ (4)

With the Minkowski signature of space-time and for on-shell
${\tilde{B}}_{\mu\nu}$, the generic curvature tensor has been shown to
describe a $4$-flux ${\cal F}_{\mu\nu\lambda\rho}$=$4l\nabla_{[\mu}{\cal
H}_{\nu\lambda\rho]}$ in addition to ${\cal K}_{\mu\nu\lambda\rho}$. The
reduced effective curvatures in the case are given by

$\displaystyle\left[{\cal D}_{\mu},{\cal
D}_{\nu}\right]A_{\lambda}=\left({\cal K}_{\mu\nu\lambda}{}^{\rho}+{\cal
F}_{\mu\nu\lambda}{}^{\rho}\right)A_{\rho}$
$\displaystyle\qquad\qquad\qquad\quad+\left({\cal F}_{\nu\mu}+{\cal
H}_{\mu\nu}{}^{\rho}{\cal D}_{\rho}\right)A_{\lambda}\ ,$ $\displaystyle{\rm
where}\quad$ $\displaystyle{{\cal
K}}_{\mu\nu\lambda}{}^{\rho}=\frac{1}{4}\left({\cal
H}_{\mu\lambda}{}^{\sigma}{\cal H}_{\nu\sigma}{}^{\rho}-{\cal
H}_{\nu\lambda}{}^{\sigma}{\cal H}_{\mu\sigma}{}^{\rho}\right)$ (5)

Then the effective action Gupta  _et al._ (2020) underlying an open geometry
(say $R\times S_{4}$ topology) becomes

$S_{\rm V}=\frac{-1}{48l^{3}}\int d^{5}x\sqrt{-g}\ \left(l^{2}{\cal
F}_{\mu\nu\lambda\rho}^{2}+4{\cal
H}_{\mu\nu\lambda}^{2}+4H_{\mu\nu\lambda}^{2}\right)$ (6)

A priori, the first term reconfirms a non-interacting GT in bulk with one
propagating degree of freedom (PDF). It can be checked to confirm a dynamical
axion in the quantum theory. An axion is known to source an instanton and is
believed to ensure the quantum tunneling among the multiple vacua. The second
term in eq(6) ensures a non-canonical interaction in the perturbation theory.
The third term describes a free field dynamics. Altogether the generic bulk
dynamics is governed by two interacting fields (GT and ${\tilde{B}}_{\mu\nu}$)
and a non-interacting $B_{\mu\nu}$ in the perturbation theory. The background
${\tilde{B}}_{\mu\nu}$ independence of the flux $H_{\mu\nu\lambda}$ allows the
exterior calculus to re-express the torsion as a $3$-form $H_{3}$$=$$dB_{2}$.
The third term in eq(6) is a total derivative and hence ensures a boundary
term $B_{2}$$\wedge$$F_{2}$. In the case the open geometry with a bulk GT
dynamics associates a boundary $S_{4}$ with a propagating axion along $R$. The
vacuum expectation $<$$\chi$$>$$=$$\chi_{0}$$=$ (cosmological) constant
$\Lambda$ ensures that the boundary space-time curvature is primarily sourced
by an interacting second term in eq(6).

Remarkably the precise curvature on the boundary theory has been argued to map
to ${\cal R}$ under a proposed correspondence between the bulk GT and boundary
GR Nitish  _et al._ (2019); Gupta  _et al._ (2020). In fact the bulk/boundary
gravity correspondence was primarily based on two key evidences and they are:
(i) ${\cal K}_{\mu\nu\lambda\rho}$ shares both the symmetry properties of
${\cal R}_{\mu\nu\lambda\rho}$ under the interchange of Lorentz indices
independently and in pairs though they are respectively sourced by
$B_{\mu\nu}$ field and metric field Singh  _et al._ (2013a) and (ii)
generically the PDF of a $B_{\mu\nu}$ field in bulk is always one degree
higher than that of a metric field on the boundary Gupta  _et al._ (2020).
Thus with a bulk/boundary correspondence, the second term ensures a Ricci like
scalar ${\cal K}$ and may be identified with ${\cal R}$. With a subtlety the
boundary action becomes

$S_{\partial V}=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}\ \left({\cal
R}-2\Lambda\right)-\int B_{2}\wedge F_{2}$ (7)

It reconfirms that the boundary (quantum) theory is generically described by
GR with a topological correction. In this article we show that the boundary
dynamics in the emergent scenario is an exact. A nonzero topological invariant
in GR such as Euler characteristics $\chi$ or signature $\tau$ is believed to
unfold the presence of winding modes. However an extremal RN ensures
$\chi$=$0$=$\tau$ and the quantum correction is apparently decoded with a non-
local horizon. This in turn describes multi black holes as an exact in GR. At
a first sight, these two alternate descriptions are believed to be in
agreement with the non-unique nature of quantum gravity.

##  IV RN type multi black holes

We begin with the gauge ansatz in $(3$$+$$1)$ dimensions Singh  _et al._
(2013b) for $(b,P)>0$. For simplicity we consider $l$$=$$1$ and then the
ansatz becomes

$\displaystyle B_{t\theta}=B_{r\theta}=b\;,\quad{\tilde{B}}_{\theta\phi}=p\
\sin^{2}\theta\ ,$ $\displaystyle A_{t}={{Q_{e}}\over{r}}\;,\quad
A_{\phi}=-Q_{m}\cos\theta$ (8)

They have been used to obtain the line-element for a RN type geometry in a
large length scale limit. Then the line element becomes

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{b^{2}}{r^{2}}+\frac{Q_{e}^{2}}{r^{4}}\right)dt^{2}+\left(1-\frac{b^{2}}{r^{2}}+\frac{Q_{e}^{2}}{r^{4}}\right)^{-1}dr^{2}$
(9) $\displaystyle\qquad\qquad\qquad\qquad\quad\ \ +\
r^{2}\left(1-\frac{Q^{2}_{m}}{r^{4}}\right)d\Omega^{2}\ ,$

where the integers $(b,Q_{e},Q_{m})$ are the conserved charges. The emergent
line-element empirically confirms a $(3$$+$$1)$ dimensional dyonic black hole,
with a mass $b^{2}$, embedded in $(4$$+$$1)$. Since the contribution to
geometry from the mass, electric and magnetic charges take on positive numbers
$(1,4,9,16\dots)$, their spectrum describes bound states of non-extremal black
holes for $b^{2}$$>$$\pm 2Q_{e}^{2}$ and with horizon radii $r_{\pm}$
satisfying $2r_{\pm}^{2}$$=$$b^{2}$$\pm$$\sqrt{b^{4}-4Q^{2}_{e}}$. However
each bound state describes a continuum space-time within and is separated from
all other bound states by classically forbidden regimes. Nevertheless the
quantum gravity scenario allows tunneling between the black holes. It is a new
feature associated with the non-extremal RN type black hole which is unlike to
that in GR.

In addition the multi RN type black holes are defined with a reduced radius of
$S_{2}$ due to the magnetic charge. The $S_{2}$ area becomes singular at
$r^{4}$$=$$Q_{m}^{2}$. With an imposed self duality and in a limit
$r^{4}$$\rightarrow$$Q_{e}^{2}$, the emergent RN identifies with the causal
patch of a Schwarzschild black hole with a reduced effective mass $(b^{2}/2)$
embedded in a higher dimension. This is due to a fact that the action, for a
$U(1)$ gauge field, vanishes for a self dual electromagnetic field which
ensures a vacuum solution. Interestingly some of the emerging features share
with that of a $(3$$+$$1)$ dimensional charged black hole in string theory
Garfinkle  _et al._ (1991).

Since the causal patch is independent of the magnetic charge, one may set
$Q_{m}$$=$$0$ in the extremal RN type black hole without any loss of
generality. With a shift in the square of the radial coordinate
$\rho^{2}$=$(r^{2}$$-$$Q_{e})$ the line element in isotropic coordinates
become

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-H^{-2}dt^{2}+H\left(d{\rho}^{2}+{\rho}^{2}d\Omega^{2}\right),$
$\displaystyle{\rm where}\quad H$ $\displaystyle=$
$\displaystyle\left(1+{{Q_{e}}\over{\rho^{2}}}\right)$ (10)

It is straight-forward to check that a time independent $H$ satisfies the
Laplace’s equation $\nabla^{2}H$=$0$ in $(4$$+$$1)$ dimensions. Its solution
is given by

$\displaystyle H=1+\sum_{i=1}^{N}\frac{m_{i}}{\left|y-y_{i}\right|^{2}}$ (11)

The harmonic function $H$ is well behaved at the spatial infinity. Each value
of $i$ defines a horizon and hence the solution (10) describes an extremal RN
multi black hole in near horizon geometry even in an emergent gravity
scenario. It provokes one to believe in a nonlocal horizon which in turn is a
collection of a multi horizons along a radial coordinate $\rho$. In fact an
extremal RN type is not derived as an exact solution in GR. The multi black
holes are obtained in the limit for a large scale structure of space-time in a
quantum gravity theory.

##  V Boundary GR from the Bulk $T_{\mu\nu}$

In this section we would like to review the validity of the conjectured GT in
bulk/boundary gravity correspondence. In particular we check if the Einstein
tensor type in an emergent scenario is an exact in GR? The non-vanishing
components of the Einstein type tensor ${\tilde{G}}_{\mu\nu}$$=$$({\tilde{\cal
R}}_{\mu\nu}$$-$$\frac{1}{2}{\tilde{g}}_{\mu\nu}{\tilde{\cal R}})$ are worked
out with near horizon coordinate to yield

$\displaystyle{\tilde{G}}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\frac{2Q_{e}\left(2Q_{e}-{\rho}^{2}\right)}{\left({\rho}^{2}+Q_{e}\right)^{2}}\
,\;\
{\tilde{G}}_{\phi\phi}=\frac{2Q_{e}\left(2Q_{e}-{\rho}^{2}\right)}{\left({\rho}^{2}+Q_{e}\right)^{2}}\sin^{2}\theta$
$\displaystyle{\tilde{G}}_{tt}$ $\displaystyle=$
$\displaystyle\frac{{\rho}^{4}Q_{e}\left(Q_{e}-2{\rho}^{2}\right)}{\left({\rho}^{2}+Q_{e}\right)^{5}}\;,\;{\tilde{G}}_{rr}=-\frac{Q_{e}\left(Q_{e}-2{\rho}^{2}\right)}{{\rho}^{2}\left({\rho}^{2}+Q_{e}\right)^{2}}$
(12)

The components are re-expressed in term of the radial coordinate and they
become

$\displaystyle{\tilde{G}}_{tt}$ $\displaystyle=$
$\displaystyle\frac{3Q_{e}^{4}}{r^{10}}-\frac{8Q_{e}^{3}}{r^{8}}+\frac{7Q_{e}^{2}}{r^{6}}-\frac{2Q_{e}}{r^{4}}$
$\displaystyle{\tilde{G}}_{rr}$ $\displaystyle=$
$\displaystyle\frac{-Q_{e}^{2}}{\left(r^{2}-Q_{e}\right)r^{4}}+\frac{2Q_{e}}{r^{4}}$
$\displaystyle{\tilde{G}}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\frac{6Q_{e}^{2}}{r^{4}}-\frac{2Q_{e}}{r^{2}}$
$\displaystyle{\tilde{G}}_{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{6Q_{e}^{2}\sin^{2}\theta}{r^{4}}-\frac{2Q_{e}\sin^{2}\theta}{r^{2}}$
(13)

It is important to note that the conserved charges on a brane is opposite in
sign to that on an anti-brane Singh  _et al._ (2013a). The components of
Einstein type tensor on a vacuum created gravitational pair of
$(3{\bar{3}})$-brane are worked out to yield

$\displaystyle{\tilde{G}}_{tt}$ $\displaystyle=$
$\displaystyle\frac{6Q_{e}^{4}}{r^{10}}+\frac{14Q_{e}^{2}}{r^{6}}\
,\quad{\tilde{G}}_{rr}=-\frac{2Q_{e}^{4}}{r^{10}}-\frac{2Q_{e}^{2}}{r^{6}}$
$\displaystyle{\tilde{G}}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\frac{12Q_{e}^{2}}{r^{4}}\qquad{\rm and}\quad\
{\tilde{G}}_{\phi\phi}=\frac{12Q_{e}^{2}\sin^{2}\theta}{r^{4}}$ (14)

Now it is interesting to check the validity of Einstein like tensor for the
exactness $i.e.\ {\tilde{G}}_{\mu\nu}$=${\tilde{\kappa}}T_{\mu\nu}$, where
${\tilde{\kappa}}$ is believed to be an effective gravitational coupling.
Alternately, at least the ${\tilde{G}}_{\mu\nu}$ should be proportional to
$T_{\mu\nu}$ if the emergent gravity scenario corresponds to the Einstein
gravity. In fact we would like to examine the above mentioned conjecture by
explicitly computing the components of $T_{\mu\nu}$ derived from a bulk gauge
theory in the case. It is important to realize that the perturbation gauge
theory in the case incorporates the dynamics of ${\tilde{B}}_{\mu\nu}$ as well
as the three index GT which respectively sources $T_{\mu\nu}(1)$ and
$T_{\mu\nu}(2)$. Together they define the total
$T_{\mu\nu}=T_{\mu\nu}(1)+T_{\mu\nu}(2)$. Explicitly they are given by

$\displaystyle T_{\mu\nu}(1)$ $\displaystyle=$ $\displaystyle{\cal
H}_{\mu\alpha\beta}{\cal H}^{\alpha\beta}{}_{\nu}-\frac{g_{\mu\nu}}{6}{\cal
H}_{\alpha\beta\gamma}{\cal H}^{\alpha\beta\gamma}$
$\displaystyle\qquad{T}_{\mu\nu}(2)$ $\displaystyle=$ $\displaystyle{\cal
F}_{\mu\alpha\beta\gamma}{\cal
F}^{\alpha\beta\gamma}{}_{\nu}-\frac{g_{\mu\nu}}{8}{\cal
F}_{\alpha\beta\gamma\delta}{\cal F}^{\alpha\beta\gamma\delta}$ (15)

Invoking the gauge ansatz Singh  _et al._ (2013a) we write

$\displaystyle B_{t\psi}$ $\displaystyle=$ $\displaystyle B_{r\psi}=b\;,\quad
B_{\theta\psi}=\tilde{P}^{3}\sin^{2}\psi\cot\theta$ $\displaystyle{\rm
and}\quad{\tilde{B}}_{\psi\phi}$ $\displaystyle=$
$\displaystyle{P^{3}}\sin^{2}\psi\cos\theta$ (16)

The non-vanishing components of GT are

$\displaystyle{\cal H}_{\theta\phi}{}^{\psi}$ $\displaystyle=$
$\displaystyle\frac{P^{3}}{r^{2}}\sin^{2}\psi\sin\theta$ (17)
$\displaystyle{\rm and}\quad{\cal H}_{\theta\phi}{}^{t}$ $\displaystyle=$
$\displaystyle-{\cal
H}_{\theta\phi}{}^{r}=\frac{bP^{3}}{r^{2}}\sin^{2}\psi\sin\theta$ (18)

Subsequently the nontrivial components of the $4$-index flux Gupta  _et al._
(2020) become

$\displaystyle{\cal F}_{t\psi\theta\phi}$ $\displaystyle=$
$\displaystyle-{\cal
F}_{r\psi\theta\phi}=\frac{-2bP^{3}}{r^{2}}\sin{2}\psi\sin\phi$
$\displaystyle{\cal F}_{tr\theta\phi}$ $\displaystyle=$
$\displaystyle\frac{2bP^{3}}{r^{3}}\sin^{2}\psi\sin\theta$ (19)

Explicitly the components of both the energy-momentum tensors are given by

$\displaystyle T_{tt}(1)$ $\displaystyle=$
$\displaystyle\left(1+\frac{2b^{2}}{r^{2}}\right)\frac{P^{6}}{r^{6}}\;,\quad
T_{rr}(1)=\left(-1+\frac{2b^{2}}{r^{2}}\right)\frac{P^{6}}{r^{6}}$
$\displaystyle T_{\psi\psi}(1)$ $\displaystyle=$
$\displaystyle\frac{P^{6}}{r^{4}}\ ,\
T_{\theta\theta}(1)=\frac{P^{6}}{r^{4}}\sin^{2}\psi$ $\displaystyle
T_{\phi\phi}(1)$ $\displaystyle=$
$\displaystyle\frac{P^{6}}{r^{4}}\sin^{2}\psi\sin^{2}\theta$
$\displaystyle{T}_{tt}(2)$ $\displaystyle=$
$\displaystyle-\frac{12{b}^{2}{P}^{6}}{r^{10}}\left(8\cot^{2}\psi+3\right)$
$\displaystyle{T}_{rr}(2)$ $\displaystyle=$
$\displaystyle\frac{12{b}^{2}{P}^{6}}{r^{10}}\left(-8\cot^{2}\psi+3\right)$
$\displaystyle{T}_{\psi\psi}(2)$ $\displaystyle=$
$\displaystyle\frac{12{b}^{2}{P}^{6}}{r^{8}}\
,\;{T}_{\theta\theta}(2)=\frac{36{b}^{2}{P}^{6}}{r^{8}}\sin^{2}\psi$
$\displaystyle{T}_{\phi\phi}(2)$ $\displaystyle=$
$\displaystyle\frac{36{b}^{2}{P}^{6}}{r^{8}}\sin^{2}\psi\sin^{2}\theta$ (20)

We consistently fix an angular coordinate $\psi$=${{\pi}\over{2}}$ to arrive
at a $(3$$+$$1)$-dimensional embedded regime within a $(4$$+$$1)$ bulk. Then
the energy-momentum tensors in a specific combination of their components have
been checked to satisfy the field equations. A priori they are defined with
eight constants $({\kappa}_{i},{\kappa}_{i})$ for $i$=$(0,1,2,3)$ and the
field equations are given by

$\displaystyle{\tilde{G}}_{tt}$ $\displaystyle=$
$\displaystyle\kappa_{0}T_{tt}(1)-{\tilde{\kappa}}_{0}T_{tt}(2)-{\tilde{\kappa}}_{3}T_{\phi\phi}(2)$
$\displaystyle{\tilde{G}}_{rr}$ $\displaystyle=$
$\displaystyle\kappa_{1}T_{rr}(1)-{\tilde{\kappa}}_{1}T_{rr}(2)-{\tilde{\kappa}}_{2}T_{\theta\theta}(2)$
$\displaystyle{\tilde{G}}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\kappa_{2}T_{\theta\theta}(1)\quad{\rm
and}\quad{\tilde{G}}_{\phi\phi}=\kappa_{3}T_{\phi\phi}(1)$ (21)

The emergent equations take the usual form of Einstein field equations. The
conditions are worked out to yield

$\displaystyle\kappa_{0}$ $\displaystyle=$ $\displaystyle
7\kappa_{1}=\frac{7}{6}\kappa_{2}=\frac{7}{6}\kappa_{3}=126{\tilde{\kappa}}_{2}=18{\tilde{\kappa}}_{3}\sin^{2}\theta$
$\displaystyle{\tilde{\kappa}}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\quad{\rm and}\quad{\tilde{\kappa}}_{1}=\frac{1}{18}$
(22)

They ensure only one independent constant $\kappa$ among the eight
$(\kappa_{i},{\tilde{\kappa}}_{i})$. All the resulting field equations (21)
can be checked to be defined with a nontrivial coupling as is believed in GR.
The result reconfirms our conjecture that the RN type in an emergent gravity
is an exact in GR. Analysis presumably identifies an emergent gravity scenario
with that of GR at least for the RN black hole.

In addition the complete $T_{\mu\nu}$ has been shown to be sourced by two
independent fields in a perturbation theory. Importantly a combination of
energy-momentum tensor components for both the fields were shown to source the
Riemann curvature for the extremal RN black hole. The result is remarkable and
is believed to strengthen our proposed correspondence Nitish  _et al._ (2019)
between a gauge theory in bulk and a boundary quantum gravity underlying the
Einstein gravity.

##  VI Concluding remarks

In this article we have revisited a correspondence between the $5D$ bulk
$U(1)$ gauge theory and a boundary gravityGupta  _et al._ (2020) with a
renewed interest. It was shown that the boundary gravity dynamics comprises GR
and a topological $BF$ term which in turn describes a mass dipole correction
to GR. It is argued that the topological correction incorporates winding
mode(s) into the otherwise point particle notion in GR. Arguably the winding
modes ensure an UV finite (non-perturbation) quantum correction to GR.

We have shown that the extremal RN type multi black holes are exact in GR. In
particular the energy-momentum tensor in the boundary gravity was shown to be
a special case of the bulk dynamics ensured by two completely anti-symmetric
fluxes ${\cal H}_{\mu\nu\lambda}$ and ${\cal F}_{\mu\nu\lambda\rho}$ in a
perturbation theory. Analysis reveals that the dark energy, in the disguise of
a bulk axion, and a GT together ensure the required $T_{\mu\nu}$ to source the
geometry in GR.

Acknowledgements: Author (SK) gratefully acknowledges the research grant-in-
aid 2021-22 from the Institute of Eminence at the University of Delhi.

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# ThumbTrak: Recognizing Micro-finger Poses Using a Ring with Proximity
Sensing

Wei Sun  Cornell UniversityUnited States Institute of Software, Chinese
Academy of SciencesChina School of Computer Science and Technology, University
of Chinese Academy of Sciences, China ,  Franklin Mingzhe Li  Carnegie Mellon
UniversityUnited States ,  Congshu Huang  Cornell UniversityUnited States
Peking UniversityChina ,  Zhenyu Lei  Cornell UniversityUnited States Huazhong
University of Science and TechnologyChina ,  Benjamin Steeper  Cornell
UniversityUnited States ,  Songyun Tao  Cornell UniversityUnited States ,
Feng Tian  School of Artificial Intelligence, University of Chinese Academy of
Sciences, BeijingChina State Key Laboratory of Computer Science, Institute of
Software Chinese Academy of Sciences, China and  Cheng Zhang  Cornell
UniversityUnited States

(2021)

###### Abstract.

ThumbTrak is a novel wearable input device that recognizes 12 micro-finger
poses in real-time. Poses are characterized by the thumb touching each of the
12 phalanges on the hand. It uses a thumb-ring, built with a flexible printed
circuit board, which hosts nine proximity sensors. Each sensor measures the
distance from the thumb to various parts of the palm or other fingers.
ThumbTrak uses a support-vector-machine (SVM) model to classify finger poses
based on distance measurements in real-time. A user study with ten
participants showed that ThumbTrak could recognize 12 micro finger poses with
an average accuracy of 93.6%. We also discuss potential opportunities and
challenges in applying ThumbTrak in real-world applications.

sensing, wearable sensing, gesture recognition, micro-finger poses, proximity
sensing, machine learning

††copyright: acmcopyright††journalyear: 2021††copyright:
acmcopyright††conference: 23rd International Conference on Mobile Human-
Computer Interaction; September 27-October 1, 2021; Toulouse & Virtual,
France††booktitle: 23rd International Conference on Mobile Human-Computer
Interaction (MobileHCI ’21), September 27-October 1, 2021, Toulouse & Virtual,
France††price: 15.00††doi: 10.1145/3447526.3472060††isbn:
978-1-4503-8328-8/21/09††ccs: Human-centered computing Pointing devices

##  1\. Introduction

The state-of-art interaction techniques on wearable computers (e.g.,
smartwatch, glass) are dominated by speech-based and touchscreen-based
interaction, which have not yet provided the optimal interaction experience
for certain scenarios. Speech interaction may not work well in public spaces,
where environmental noise may be high or talking out loud might be socially
inappropriate (Li et al., 2017). Touch-based interfaces on wearables are often
limited by the screen sizes (Oney et al., 2013). Therefore, the expressiveness
of the touch interaction on wearables is relatively limited, compared to the
touch interaction on tablets or phones. Furthermore, touch-based interaction
demands exclusive eye-contact, which may not always be convenient. For
instance, if a phone call comes in during a meeting, the user may want to
immediately dismiss the phone call, instead of locating the ’reject’ button on
a screen or issue voice commands (Sun et al., 2021). Therefore, there is a
need for a discrete and eyes-free input technique to promote wearable
interaction experience in these scenarios.

Figure 1. ThumbTrak System

This figure contains two sub-figures. The left one shows the sketch of a hand
wearing a Thumbtrak system and nine red arrows that show the position and the
orientation of each sensor. The right one shows a photo that a hand wearing a
Thumbtrak system.

Existing work has explored various approaches to enable hand or finger
gestures for wearable devices (e.g., (Mujibiya et al., 2013; Kim et al.,
2012)). Many of them rely on wearing sensor units on the wrist or forearm to
recognize finger or hand gestures (e.g., (Kim et al., 2012; Gong et al., 2017;
Zhang and Harrison, 2015)). However, such methods either require heavy
instrumentation, which might be obtrusive to the user, or the recognition
performance from sensor units on the wrist or forearm might be affected in
certain situations, such as the occlusions from clothing or certain hand
angles. Therefore, we are interested in exploring ring-based approaches
because many people are used to wearing rings in everyday activities and it
has potentials to detect various hand/finger gestures with less constraints
(e.g., hand angles or clothing conditions) than prior approaches.

In our work, we present ThumbTrak (as shown in Figure 1), a thumb-ring based
eyes-free pose input technique using proximity sensing. The thumb-ring is
built with a flexible printed circuit board hosting nine proximity sensors,
which measure distances from the thumb to various parts of the hand. Because
these distances vary depending on the user’s hand position, ThumbTrak can
distinguish which finger phalanges of the 12 phalanges the thumb touches in
real-time using a support-vector-machine (SVM) model based on measured
distances. The layout of the 12 finger poses (three phalanges per finger
$\times$ four fingers) is similar to a T9 keyboard, which can be naturally
extended to input numbers. A user study with ten participants showed that
ThumbTrak could recognize these 12 micro finger poses with an average accuracy
of 93.6%.

The contributions of our paper are:

  * •

A novel thumb-ring-based input method which recognizes 12 micro-finger poses
in real-time using proximity sensing and machine learning.

  * •

A user study with 10 participants which evaluated the performance of the
system (over 93%) and presented analysis on sensor numbers and positions.

  * •

A discussion of opportunities and challenges about how to apply this
technology in a wider range of real-world applications.

##  2\. Related Work

Gesture recognition has always been a classic research topic in the HCI
community. In this section, we introduce and discuss prior work in wearable
gesture input techniques involving one-handed input methods and ring-based
approaches.

###  2.1. One-handed Gesture Input

Naturally, one-handed input allows users to input information with just one
hand - often a more convenient option compared to interactions requiring both
hands. A variety of one-handed input technologies have been developed. Since
armbands and wristbands are commonly worn accessories in society, many prior
works use these two form factors to detect discrete hand or finger gestures. A
variety of different sensing modalities have been employed to detect these
gestures, including electric impedance tomography (Zhang and Harrison, 2015;
Zhang et al., 2016a), pressure sensing (Dementyev and Paradiso, 2014),
acoustic sensing (Zhang et al., 2018; Deyle et al., 2007; Laput et al., 2016;
Harrison et al., 2010; Kubo et al., 2019), Electromyography (EMG) (Lu et al.,
2014; Saponas et al., 2009), magnetic sensing (Huang et al., 2016), motion
sensing (Lu et al., 2014; Wen et al., 2016; Xu et al., 2015), electrical
sensing (Zhang et al., 2016b), cameras (Loclair et al., 2010; Way and
Paradiso, 2014; Hu et al., 2020) and proximity sensing (Fukui et al., 2011;
Gong et al., 2016). For example, Lu et al. (Lu et al., 2014) leveraged four
surface electromyography sensors to recognize hand gestures such as grasping
or opening the hand. However, prior research either required heavy
instrumentation or mostly focused on recognizing mid- or large-scale one-hand
gestures (e.g., hand movements, hand grasping), which lacks explorations on
micro-finger one-handed poses (e.g., distinguishing which finger phalanges of
the 12 phalanges the thumb touches).

###  2.2. Gesture Input with a Ring

Rings are another accessory that people are accustomed to wearing. Since they
sit on the fingers (or thumb) themselves, they are arguably a more convenient
method for sensing micro finger poses compared to wrist- or arm-related form
factors. Consequently, many previous ring-based input techniques have been
invented, recognizing discrete finger or hand gestures using a wide variety of
sensing modalities. Previously explored modalities have included visual
sensing (cameras) (Chan et al., 2015), motion sensing (Zhang et al., 2017a, b;
Gupta et al., 2019), acoustic sensing (Zhang et al., 2018), magnetic sensing
(Ashbrook et al., 2011; Chen et al., 2013), and infrared reflection sensing
(Ogata et al., 2012). Ring-based input projects have also been invented to
track continuous finger input on 2D surfaces (Kienzle and Hinckley, 2014) and
thumb input in 3D space (Chen et al., 2013).

Figure 2. ThumbTrak Pose Family

This figure contains twelve hand poses with corresponding pictures in three
rows. The first row: thumb touching little finger distal, ring finger distal,
middle finger distal, and index finger distal. The second row: thumb touching
little finger middle, ring finger middle, middle finger middle, and index
finger middle. The third row: thumb touching little finger proximal, ring
finger proximal, middle finger proximal, and index finger proximal.

The most similar previous work to ours is FingerPing (Zhang et al., 2018),
which also recognizes thumb touches on the 12 finger phalanges. The system
recognizes these poses via active acoustic sensing with a thumb-ring, and a
wristband (Zhang et al., 2018). Compared to FingerPing, ThumbTrak only
requires one piece of form factor (the thumb ring), which is arguably more
practical, since it requires less instrumentation. Furthermore, ThumbTrak is
potentially more comfortable comparing to FingerPing (Zhang et al., 2018),
given there is no vibration on the thumb. More importantly, FingerPing has a
low cross-section performance, due to inconsistencies in ring and wristband
placement, which results in inconsistent signals causing acoustic impedance to
vary. Alternatively, ThumbTrak uses proximity sensing - a relatively more
robust cross-session solution. We present a promising cross-session
recognition performance report later on in the paper.

##  3\. Design and Implementation of ThumbTrak

###  3.1. Pose Design

Inspired by the natural similarity of the layouts between the finger phalanges
and the T9 keyboard (Kubo et al., 2019), we designed 12 micro-finger poses (as
shown by Figure 2), involving thumb touching on each of the three phalanges of
the index, middle, ring, and little fingers. These micro-gestures have fewer
movement distinctions comparing with other hand or finger gestures (e.g., fist
and thumbs-up) and they allow input in a more discreet fashion for privacy.
Furthermore, adopting an existing input layout is more straightforward to map
the input poses to different functionalities in applications. For instance,
the 12 poses construct a three (phalanges) by four (fingers) matrix, which can
be used to input numbers. A subset of these 12 micro-finger poses can also be
used as a directional pad.

###  3.2. Hardware Design

Figure 3. ThumbTrak Hardware

This figure shows the thumbtrak from an overview. It combines a 3D printed
case, velcro, and a flexible printed circuit board.

Figure 3 illustrates ThumbTrak’s ring, which comprises three major components:
1) a flexible printed circuit board (FPCB), 2) a 3D printed case to provide
support, and 3) a Velcro band. The FPCB board hosts nine proximity sensors
(VL6180X)111https://www.st.com/resource/en/datasheet/vl6180x.pdf, and they
operate at around 10Hz. We choose FPCB over a traditional printed circuit
board (PCB) because FPCB is bendable and flexible, allowing it to wrap around
the thumb to better measure distances in multiple directions. The more
directions we can measure distances from, the more informative the sensor
readings will provide. The FPCB is glued on a 3D printed case for more robust
support. We use a Velcro band to fit the ring tightly and comfortably around
thumbs of different sizes. The sensor readings are communicated from FPCB to a
Teensy 3.2 using wired $\mathrm{I^{2}C}$ communication, which eventually sends
the data to a python program running on a MacBook Pro for online
classification via serial communication. We sequentially activate and change
each proximity sensor’s address to enable $\mathrm{I^{2}C}$ communication. In
the device testing, we found that ThumbTrak consumes around 120mW power.

###  3.3. Data Processing Pipeline

We developed a Python program to continuously capture and analyze the stream
of sensor data for real-time pose recognition. From our internal testings
between researchers, we found that the distance between the thumb and any
given part of the palm did not exceed 150mm when performing candidate poses.
If a distance value does exceed 150mm, we set its value to 150mm to prevent
erroneous sensing results due to the out of range sensing distances, which
signifies the sensor is out of range. Many out of range sensors are an
indicator of “no-pose” status.

Figure 4. Data Processing Pipeline

This figure shows the data processing pipeline from raw data to extract
features to determine whether it is a pose or not. Then it goes to use pose
features to classified pose, and then use a filter to output pose results.

Every sample iteration, we gather each sensor’s distance value for a total of
nine initial features. We then use these nine values to derive two extra
features: the number (quantity) of out of range sensors and the average value
of all in-range sensors. This makes for a total of 11 features. We then feed
all 11 features into a support vector machine model (SVM classifier with a
linear kernel) to predict whether or not a pose is being performed. If the SVM
model, which is responsible for segmentation, predicts a pose is being
performed, we pass that segment values on to a second SVM model (with linear
kernel) for classification. This second SVM classifier is responsible for
predicting which finger pose of the 12 poses was performed.

To prevent potential system noises, we leveraged majority voting to detect the
poses in real-time. For each data sample, we extract features, which are sent
to the first SVM model to estimate whether it is a target pose or noise (non-
pose). If the first SVM recognizes a data sample as a gesture, we activate the
second SVM and classify the pose as one of the 12 micro-finger poses. At a
specific time, if the previous 15 data points (1.5 seconds) are classified as
poses, we then perform the majority voting. As the thumb may not be stabilized
in the first half second, we only consider the classification result from the
later one second for majority voting. Then we output the labels that represent
the majority of the later 10 data points of the entire 15 data points as the
gesture recognition result. Furthermore, if the SVM model outputs five
consecutive results as noise (non-pose), ThumbTrak recognizes this as a reset
for micro-finger pose recognition.

###  3.4. Calibration

Remounting the ring can introduce displacements on the sensor positions, which
can impact the performance of the system. To address this issue, we introduce
a calibration method in our system to help the users to remount the ring to
its original position with minimal effort. We determined proximity to its
original position by outputting the absolute value of the difference between
its current distance and its original distance for two poses: little-proximal
and little-distal (the furthest fingers from the thumb). In the calibration
process, we first asked participants to perform the two poses (i.e., little-
proximal and little-distal). The researcher then empirically and iteratively
adjust the ring position based on the output value. In our user evaluation, we
set the average offset threshold value per sensor as 0.7mm.

##  4\. User Study

To evaluate ThumbTrak, we conducted a user study with ten participants with an
average age of 22 (three female). All participants were right-handed. The user
study comprises three main phases: 1) practicing phase—participants get to
know the micro-finger poses and our prototype; 2) training phase—researcher
collects training data with participants to generate our recognition model; 3)
testing phase—participants are asked to perform different micro-finger poses
to evaluate the performance of our system. Before conducting the user study
with ten participants, we piloted our system and study process with two other
participants. The study took around 60 minutes on average. The whole study
procedure was approved by the institutional review board (IRB).

###  4.1. Practicing Phase

At the beginning of the user study, a researcher introduced our project and
outlined the study’s procedure. The researcher then helped the participant to
fasten the ring, ensuring it be fixed tightly yet comfortably around the
user’s thumb to avoid unwanted rotation. Next, the researcher demonstrated how
to perform each pose, instructing them to follow along with their dominant
hand. After this, the participant familiarizes herself with different micro-
finger poses. Participants were seated for the entirety of the experiment.

###  4.2. Training Phase

To collect training data, participants were asked to participate in three
training sessions. In each training session, the participant was asked to
perform five instances of each of the 13 poses (including “no-pose”). Poses
were prompted by a screen displaying an image of an open hand with a red dot
resting over the region of the finger to be touched. Images appeared for a
total of four seconds: three seconds for participants to react and perform the
pose and one second for the system to record the sensor data. All in all, each
participant performed 65 randomly ordered poses continuously for each of the
three training sessions. Between each training session, participants got a
short break.

###  4.3. Testing Phase

Each testing session consisted of 60 poses (i.e., 12 micro-finger poses x 5).
Additionally, after each completed pose, the real-time result was presented on
the screen. If the result matched the requested pose, the interface turned
green. Otherwise, the interface turned blue and displayed the falsely
predicted pose. Every time when participants received the results, they were
asked to reset the micro-finger pose to no-pose for segmentation purposes. The
testing phase contains three sessions: we called the first two sessions as in-
session tests, which focus on the overall performance of the system, and the
final one as the cross-session test, which mainly tests for the effects of
remounting of the device. Between each testing session, participants got a
short break. After finishing the last in-session test, a researcher helped the
user to remove the ring. After a short rest, the researcher affixed the ring
back onto the user’s thumb and proceeded to undergo the calibration process.
After calibration, the participant continued with the cross-session study,
which was identical to in-session tests.

After completing all sessions, each participant was asked to fill out a
questionnaire collecting some personal information regarding age, gender, hand
dominance, wearable-device use, and thumb size. Participants also provided
subjective feedback on the device, the comfort level, and perceived social
appropriateness. Lastly, each participant received $10 for participating in
the study.

##  5\. Results

Throughout the entire user study, all participants correctly performed every
requested hand pose as the stimuli indicated on the screen, both in training
and testing sessions. In total, across all 10 participants, we collected 1950
(13 poses x 150) instances in training sessions and 1800 (12 poses x 150)
instances in testing sessions. Of the 1800 total testing instances, 1200 were
in-session, and 600 were cross-session. If the recognition results in real-
time matched the stimuli poses on the screen, we recorded it as a correct
instance. Otherwise, it was a classification. The results reported in this
section were the real-time performance recorded during the user study.

###  5.1. In-session Testing

Figure 5. Confusion Matrix for recognizing poses for in-session testing

This figure shows a confusion matrix with rows are true labels and columns are
predicted labels. For each value in the confusion matrix, the background color
close to black if the value is close to 100 and the background color close to
white if the value is close to 0. It shows a black diagonal pattern from top
left to bottom right.

Among the performance of ten participants, the average accuracy across all
poses was 93.6%. The confusion matrix in Figure 5 shows the average accuracy
results of all participants for each hand pose. Recognition was most accurate
for poses involving the little finger and least accurate at the index proximal
position (Figure 5). There is a common confusion among the proximal positions
of the index, middle, and ring fingers. Specifically, we uncovered that index
proximal (Accuracy=85.72%) was misclassified as index middle, middle middle
and middle proximal (Figure 5). From analyzing these poses and feedback from
participants, we found that some participants have a hard time performing the
pose to touch the index proximal with thumb, because index proximal is the
closest position to the thumb. Furthermore, we revealed that the “too close”
situation made sensors have a smaller variability in distance comparing with
fingertips, which results in the misclassification to index middle, middle
middle, and middle proximal (Figure 5).

Figure 6. Accuracy for 12 Phalanges (Rounded to nearest integer)

This figure shows the accuracy for 12 phalanges on a hand. For each value of
the accuracy, the background color close to red if the value is close to 100
and the background color close to white if the value is close to 0. This
figure shows the accuracy for the index (97, 97, 86, top to bottom), middle
(95, 96, 90, top to bottom), ring (90, 89, 89, top to bottom), and little
finger (97, 99, 99, top to bottom).

Figure 7. Accuracy for each participant

This figure is a bar chart of the recognition accuracy for 12 poses. X-axis:
each participant. Y-axis: the recognition accuracy in percentage. For each
participant, the figure shows two bars in blue (in-session) and yellow (cross-
session).

We then visualized the recognition performance on different finger positions
(Figure 6), which depicts the accuracy for each of the 12 phalanges across all
participants. Beyond the misclassification of the index proximal, we realized
that the recognition performance of the ring finger is relatively worse than
the middle finger and the little finger. In our study, we found that it is a
natural behavior for participants to move their middle finger when they try to
use their thumb to touch the ring finger, which may affect the recognition
performance. In Figure 7, we present the overall performance for each
participant during the in-session tests and the cross-session test. The figure
illustrates that universally, all participants were able to achieve a
relatively high average accuracy (Figure 7). Overall, we found that the
highest accuracy achieved was 97.50% while the lowest was 88.79%. The median
accuracy was 94.10%.

###  5.2. Cross-session Testing

In the cross-session testing, we found that the average accuracy across all
participants was 82.7%. Cross-session confusion error distributions were quite
similar to in-session errors. For example, we found that index proximal was
also misclassified as index middle, middle middle, and middle proximal.
Furthermore, we found middle middle was the second most erroneous pose and got
misclassified as index distal, index middle, ring middle, and index proximal.
Figure 7 shows that accuracy varies greatly among participants, from the
highest 95.00% to the lowest 67.86%, with an 82.9% median accuracy. From the
observation in the cross-session test when the researcher tried to assist
participants in remounting the device, we discovered that the existing
calibration process reduced the effort of memorizing the ring position
relative to hand or finger biometrics (e.g., ‘the ring was worn at 1cm below
the thumbnail’).

Figure 8. Left: Sensor name mapping on ThumbTrak. Right: Accuracy drop by
excluding the targeted sensor

This figure contains two sub-figures. The left shows the label of nine sensors
from S1 to S9. The right shows a bar chart. Y-axis is the accuracy of pose
detection. The X-axis is the sensor label that excluding the targeted sensor.
This right sub-figure shows that the accuracy drop start with low from
excluding the sensor 1, and it keeps increasing till sensor 4 and keeps
dropping till sensor 9.

###  5.3. Analysis of Sensor Numbers and Positions

In our study, we utilized nine proximity sensors on the ring to recognize
micro-finger poses. We then perform further analysis on how different
positions and numbers of proximity sensors may affect the recognition
performance. To understand the usefulness of different sensors at different
positions, we first applied the study segmentation results and removed one
sensor, and used the other eight sensors for training and testing, each time
we change the position of the excluded sensor. From the analysis results, we
uncovered that removing sensors in the middle caused the performance to drop
the most. For example, the accuracy dropped 15.25% if we remove Sensor 4
(Figure 8). On the contrary, the performance stayed around 90% if we remove
the sensor at the side of the ring.

Figure 9. Accuracy with different number of sensors

The figure shows a bar chart that the Y-axis is the accuracy of pose
detection, and the X-axis is the sensor label of training sets.

To further understand the effect of using fewer sensors on the performance
change, we leverage the study segmentation results and start with sensors in
the middle, because we found sensors in the middle contribute to the
performance the most. In this analysis, we start with Sensor 5; we then keep
adding sensors surrounding the sensor group. Overall, We ended with five
sensor subsets: Sensor 5, Sensor 4-6, Sensor 3-7, Sensor 2-8, and Sensor 1-9
(Figure 9). By analyzing associated performance changes (Figure 9), the
accuracy stays around 40% when we only use Sensor 5 for training and testing
(Figure 9). After we evaluate the recognition performance with using Sensor
3-7, the accuracy increases significantly to 85% (Figure 9). However, the
accuracy increases less than 10% after we add the four sensors at the side
(Figure 9). We further inspect the sensor output from the proximity sensors at
the side, and we find that they sometimes remain out-of-range for certain
micro-finger poses. Therefore, future research could perform a similar
analysis to reach a compromise between power consumption and recognition
accuracy.

###  5.4. Reduce Training session

In our in-session tests, we used data from all three training sessions to
train the classifiers. However, it is often advantageous for users to reduce
training data collection time. To investigate this, we reduced the training
data to $n$ training session(s) ($n=1,2$). This resulted in an accuracy of
83.5% for $n=1$ and 91.2% for $n=2$ (93.6% for $n=3$). Thus, future developers
should compromise between the effort of collecting training data and the
recognition accuracy.

##  6\. Discussion

In the result section, we showed the accuracy and recognition confusion in
both in-session testing and cross-session testing. We then analyzed the
recognition performance effects of sensor numbers and positions. We also
mentioned the usefulness of calibration when remounting the device and the
tradeoff of reducing training sessions. In the discussion, we will further
discuss 1) applications and opportunities for ThumbTrak, 2) potential
affordance of ThumbTrak, 3) pose subsets for cross-session, 4) calibration and
remounting process, and 5) hardware design improvements.

Figure 10. Layout of pose subsets

This figure contains three sub-figures. All sub-figures show a sketch of a
hand that labeled the layout of pose subsets the 12 phalanges(left), four
corners(middle), and navigation(right).

###  6.1. Applications and Opportunities for ThumbTrak

####  6.1.1. T9 Keyboard

ThumbTrak offers up to 12 discrete input gestures with an average accuracy
rate of 93.6%. We found that it is natural to map these 12 micro-finger poses
to a number input system due to their structural similarity (Figure 2). People
could use ThumbTrak to make phone calls to their friends, and they could also
utilize ThumbTrak as a T9 keyboard to accomplish text entry tasks in different
messaging and social media applications.

####  6.1.2. D-Pad Control

Certain micro-finger poses can also be chosen to form new pose subsets (Figure
10) to meet the demand of different applications. Many applications do not
require all 12 poses to be functional. For instance, we can use a pose set
(Figure 10) to control a D-Pad, which enables navigation through hierarchical
menus for wearable devices. The buttons of the D-Pad can be naturally mapped
to the index-middle, middle-distal, middle-proximal, and ring-middle positions
(Figure 10). If we only include these poses in the training and testing
sessions, this subset can achieve an average accuracy of 99.0% in post-
analysis (Figure 11).

####  6.1.3. Music Control

ThumbTrak can detect the “pose” and the “no-pose” gesture. Thus, we can
calculate the gesture holding time, which can provide continuous input, such
as increase or decrease the music volume. The user could hold a pose to raise
or lower volume and release the gesture when the volume has reached the
desired level. Similarly, for music control, the user could use ThumbTrak to
place commands for ‘Fast Forward’, ‘Rewind’, ‘Play’, and ‘Pause’.

####  6.1.4. Application Shortcuts

Another interesting pose subset, the four corner phalanges (index-distal,
index-proximal, little-distal, little-proximal), can also be integrated as
shortcuts with many applications. For example, the user could use the index-
distal pose to do mobile screen capturing when playing a game on a smartphone
(e.g., PUBG MOBILE). This subset presented a 97.1% accuracy in post-analysis
(Figure 11).

####  6.1.5. Virtual Reality & Augmented Reality

Much existing virtual reality and augmented reality systems utilize WIMP
(i.e., “Windows, Icons, Menus, Pointer”) for user interactions. Similar to
prior work on enabling faster control with finger or hand gestures
(Kulshreshth and LaViola Jr, 2014; Bowman and Wingrave, 2001), ThumbTrak could
also provide the opportunity of enabling fast hierarchical controls and reduce
the effort of controlling the pointer movement through button and icon
interactions, but without having to wear complex sensor instrumentation (e.g.,
sensor gloves (Bowman and Wingrave, 2001)) or instrumenting a camera in front
of the user (Fan et al., 2020; Kulshreshth and LaViola Jr, 2014).

###  6.2. Potential Affordance of ThumbTrak

In our study, we only evaluated ThumbTrak on recognizing discrete micro-finger
poses. However, in our experiment, we observed that the distance profile
between thumb to other fingers could also be very informative to predict the
thumb position in 3D space. One natural next step of ThumbTrak is to explore
continuous tracking of the thumb position in 3D space using time-series
modeling techniques, similar to the work of UTrack (Chen et al., 2013). This
would grant ThumbTrak an even richer expressiveness in input space.
Furthermore, ThumbTrak could potentially support other hand or finger
gestures, such as fist and thumbs up. Because the distance between the thumb
and the palm would change when performing these gestures. Therefore, future
applications could explore customized finger or hand gestures to naturally and
unobtrusively interact with their interfaces with ThumbTrak.

In this current research, we focused on the general proof-of-concept and
simply aligned nine sensors in a line. Beyond the existing hardware setup with
a 1D array of proximity sensors, an extension of ThumbTrak could change the
sensor layout from 1D to 2D to enable more depth detection. The angle of each
sensor, their relative position to one another, and the number of sensors all
may play a critical role in improving the ring’s performance. Other than
proximity sensing, future research could explore how to leverage the micro
depth camera on a ring-based sensing unit for hand/finger gesture recognition.
Moreover, using radar sensors or acoustic sensors could also extend the
potential applications of ring-based sensing units.

###  6.3. Pose Subsets for Cross-session

As illustrated above, some re-combined pose subsets are sufficient for
multiple tasks and applications. This also applies to cross-section subsets.
In our cross-session testing with 12 poses, the accuracy dropped from 93.6% to
82.7%. For navigation micro-finger pose subset (index-middle, middle-distal,
middle-proximal, and ring-middle), we found that the accuracy only dropped
99.0% to 94.9% (Figure 11). In terms of the four corner phalanges micro-finger
pose subset (index-distal, index-proximal, little-distal, little-proximal),
the accuracy dropped from 97.1% to 91.5% (Figure 11). Thus, we found that
using a subset of poses could enable the system to perform at high accuracy
even if the participant need to remount the device, without applying any
further calibration procedure.

Figure 11. Accuracy for different layout of pose subsets

This figure is a bar chart of the recognition accuracy for 3 pose subsets, the
Y-axis: the percentage accuracy. X-axis: each pose subset, the 12
phalanges(left), four corners(middle), and navigation(right).

###  6.4. Calibration and Remounting Process

In our calibration process, our algorithm design relies on the ring’s position
to remain stable. Once the ring shifts positions, the system may begin to
misclassify hand poses. In our cross-session results, we found that the
performance dropped around 10% after remounting the device. To better assist
the calibration procedure, we may include more training data from different
ring positions during calibration. This may compensate for the minor
displacement of the ring and provide a more adaptive system to remounting
situations.

Furthermore, we found that providing calibration feedback or adjustment
recommendations is important for users while remounting their devices. To
provide calibration feedback, we realized that it might be useful to attach
small vibration motors to different positions of the ring to provide haptic
feedback to users. Furthermore, the calibration process we used in the
study—just by reporting the absolute differences of the sensor values
comparing with the original position—may not provide enough information to a
user where she should adjust the ring position. Therefore, having a
visualization tool would be beneficial to users while they are using ThumbTrak
in real-life scenarios.

###  6.5. Hardware Design Improvement

In order to ensure that all participants could wear our ring, we could not
simply print one ring of a standard size. So we decided to create an
adjustable ring. We achieved this by 3D printing a curved frame with a hole in
each end. We then fed Velcro straps through both holes so that when the strap
is tightened, the curved 3D printed frame will flex and tighten around any
sized thumb (as Figure 3 shows). An issue with this design, however, is its
tendency to shift positions while users perform hand poses, reducing
ThumbTrak’s recognition accuracy as a consequence. In addition, this hard
plastic PLA may not be comfortable enough for everyday uses. A potential
solution could be to replace the plastic with other more elastic materials
like rubber. Increased elasticity would result in a softer, more adjustable
ring. Moreover, rubber would increase friction between the ring and the skin,
helping the ring to remain in a fixed position. Also, it is worth noting that
although it was important to make a one-size-fits-all model for the purposes
of our experiment, rings by nature are not typically designed to be one-size-
fits-all. In future form factors, ThumbTrak could therefore be integrated into
customizable ring sizes in the same way that rings are custom-fitted in the
jewelry industry today.

##  7\. Limitation and Future Work

In our paper, we conducted a quantitative study to asked participants to
evaluate ThumbTrak in relatively static manners under a lab setting. However,
people may leverage micro-finger poses while standing, walking, jogging,
holding objects in their hands, or in complex environments, all of which were
not included in the current study. Moreover, our current prototype requires
wire connections to collect data for gesture recognition. In future work, we
plan to first create a self-contained prototype that is self-powered and has
wireless communications. We will further conduct scenario-based studies (e.g.,
(Li et al., 2019; Kianpisheh et al., 2019)) to both quantitatively and
qualitatively evaluate the performance of ThumbTrack and identify potential
problems under different contexts.

Another limitation is that we only included the “no-pose” gesture as
participants’ hands stay still in static situations. In real-life scenarios,
people may accidentally move their fingers, which may result in different
sensor outputs and unintended gestures. In our evaluation, we only studied
discrete gestures under a lab setting, people might use micro-finger gestures
in a continuous manner. Future work should explore methods to reduce false
positives(e.g., activation gesture) and enable continuous gestures for real-
life scenarios.

Furthermore, our researcher helped participants to calibrate the ring position
in the cross-session testing, which limit our explorations on the behaviors of
how participants remount the ring by themselves, which they will eventually do
under real-life scenarios. In future work, we will conduct studies to
specifically understand users’ behaviors of calibration and remounting the
device. Finally, our participants are mostly younger adults (average age of
22). In future work, we will include participants from different age groups.
Although we understand our work could be strengthened with the above
modification, we think our work—a novel thumb-ring-based input method that
recognizes 12 micro-finger poses in real-time using proximity sensing and
machine learning—makes novel contributions to HCI and mobile-sensing research
community.

##  8\. Conclusion

In this paper, we present ThumbTrak, a novel Thumb-ring based input technique
able to recognize micro-finger poses in real-time with proximity sensing. A
user study with 10 participants shows that ThumbTrak can recognize thumb
touches on all 12 finger phalanges with an average accuracy of 93.6%. We
compared our in-session testing, and cross-session testing results and
uncovered design guidelines of micro-finger pose recognition system with
proximity sensing. We further discussed the potential applications for
ThumbTrak (e.g., D-Pad), the accuracy of pose subsets for cross-session
testing, calibration and remounting process, continuous tracking of thumb, and
potential improvements of hardware design. We believe our findings would shed
light on future research with micro-finger poses sensing.

###### Acknowledgements.

This project is supported by the Department of Information Science at Cornell
University. We appreciate the constructive reviews from reviewers and early
discussion and feedback from colleagues in the Smart Computer Interfaces for
Future Interactions (SciFi) Lab at Cornell University.

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