USTC-ICTS/PCFT-22-01

# Ghost instability in the teleparallel gravity model with parity violations

Mingzhe Li  Zhihao Li  Haomin Rao  Interdisciplinary Center for Theoretical
Study, University of Science and Technology of China, Hefei, Anhui 230026,
China  Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

###### Abstract

In this paper we consider the parity violating gravity model within the
framework of teleparallel gravity. The parity violations are caused by the
couplings of a scalar field to the scalar invariants which are parity-odd and
quadratic in the torsion tensor. Totally there are two such type independent
invariants, and one of them is the Nieh-Yan density. Through investigations on
the cosmological perturbations of this model, we find that in general it
suffers from the difficulties of ghost instability in the scalar and vector
perturbations. But in the special case only the coupling to the Nieh-Yan
density exists, this model is ghost free and reduces to the Nieh-Yan modified
Teleparallel Gravity model. We also analyze the severity of the ghost
instability by studying the perturbations around the Minkowski background.

##  I introduction

In recent years, there are many interests in investigating possible parity
violations in gravity theories in the literature, partly stimulated by the
experimental detections of gravitational waves (GWs) ligo1 ; ligo2  and the
developments in the cosmic microwave background radiation (CMB) experiments
CMB1 ; CMB2 . A famous and frequently studied parity violating (PV) gravity
model is the so-called Chern-Simons (CS) modified gravity CSgravity1 ;
CSgravity2 , which within the framework of Riemannian geometry modifies
general relativity (GR) by a gravitational CS term $\theta R\tilde{R}$, where
$R\tilde{R}\equiv\varepsilon^{\mu\nu\rho\sigma}R_{\mu\nu}^{~{}~{}~{}\alpha\beta}R_{\rho\sigma\alpha\beta}$,
$\theta$ is scalar field, $R_{\mu\nu\rho\sigma}$ is the Riemann tensor
constructed from Levi-Civita connection,
$\varepsilon^{\mu\nu\rho\sigma}=\epsilon^{\mu\nu\rho\sigma}/\sqrt{-g}$ is
Levi-Civita tensor, $\epsilon^{\mu\nu\rho\sigma}$ is antisymmetry symbol and
$g$ is determinant of the metric. The CS modified gravity makes a difference
between the amplitudes of the left- and right-handed polarized components of
gravitational waves (GWs), but no difference between their velocities. This is
the so-called amplitude birefringence phenomenon. However, the CS modified
gravity suffers from the problem of vacuum instability because one of the
circularly polarized components of GWs becomes a ghost at high frequencies
CSgravity3 , i.e., its kinetic term has a wrong sign. Further extensions to
the CS modified gravity were made in Refs. Crisostomi:2017ugk ; Gao:2019liu ;
Zhao:2019xmm , but these did not stop the ghost mode at high frequencies, as
shown explicitly by Eqs. (3.5)-(3.6) and (3.39)-(3.40) in Ref. Bartolo:2020gsh
. It is very difficult to have a ghost-free PV gravity model within the
framework of Riemannian geometry.

To seek the possibilities we may go beyond the Riemannian geometry. Along this
way, the Nieh-Yan modified Teleparallel Gravity (NYTG) model PVtele1 ; PVtele2
was recently proposed. The NYTG model is based on the teleparallel gravity
(TG) Tele ; tele2021  which may be considered as a constrained metric-affine
theory and is formulated in a spacetime endowed with a metric compatible but
curvature free connection, the gravity is identified with the spacetime
torsion. One may have a GR equivalent model within the framework of TG (we may
call it TGR). The NYTG model PVtele1 ; PVtele2  modifies TGR slightly by the
anomalous coupling $\theta\mathcal{T}\widetilde{\mathcal{T}}$ between an
axion-like field (it is a pseudo scalar field) $\theta(x)$ and the Nieh-Yan
density Nieh:1981ww :
$\mathcal{T}\widetilde{\mathcal{T}}=(1/2)\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}^{\lambda}_{~{}\mu\nu}\mathcal{T}_{\lambda\rho\sigma}$
with $\mathcal{T}^{\lambda}_{~{}\mu\nu}$ being the torsion tensor. The Nieh-
Yan density is parity-odd, so at the background with $\partial_{\mu}\theta\neq
0$, the Nieh-Yan coupling term $\theta\mathcal{T}\widetilde{\mathcal{T}}$
violates parity spontaneously. Also in Refs. PVtele1 ; PVtele2 , the NYTG
model has been applied to cosmology. It was found that this model makes a
difference between the propagating velocities of the left- and right-handed
polarized components of GWs, but makes no difference between their amplitudes.
This is the so-called velocity birefringence phenomenon. More importantly,
through detailed studies on the cosmological perturbations, it was shown in
Refs. PVtele1 ; PVtele2  that the NYTG model is ghost-free. More recently,
this model was found to be compatible with the results of most local tests in
the Solar System at the post-Newtonian order Rao:2021azn ; Qiao:2021fwi , the
upper limit on its model parameters by the GWs data of LIGO/Virgo
Collaboration was obtained in Ref. Wu:2021ndf , and the enhancement of
primordial GWs during inflation due to the velocity birefringence of NYTG
model and its implications in the air-based GWs experiments were studied in
Ref. Cai:2021uup . The application of Nieh-Yan term on the big-bounce
cosmology was considered in Ref. Bombacigno:2021bpk . More recent constraints
on the parity violations in gravities can be found in Ref. Gong:2021jgg .

Within the TG framework, the Nieh-Yan density
$\mathcal{T}\widetilde{\mathcal{T}}$ is not the only parity-odd invariant one
can construct from the terms quadratic in the torsion tensor. A more general
model including all the parity-odd quadratic invariants (coupled with the
scalar field $\theta(x)$) in the action was considered in Ref. PVtele3  and
was applied to cosmology. However, in Ref. PVtele3 , only the background
evolution of the universe was studied. In this paper, we will investigate the
cosmological perturbations of this general model 111In Ref. PVtele3 , the
modifications to TGR by parity-even terms were also considered. We will not
consider these further extensions in this paper.. We will show that, in
general this model is pathological due to the ghost modes appeared in the
scalar and vector perturbations, unless all the PV terms except the Nieh-Yan
coupling drop out from the action, if so then this model reduces to the NYTG
model of Refs. PVtele1 ; PVtele2 . Furthermore, by studying the properties of
the ghost modes in Minkowski spacetime, we will find that it is almost
impossible to avoid the ghost instability by lowering the energy scale of the
effective field theory.

It deserves mentioning that there is also consideration on PV gravity model
STGPV1  based on the symmetric teleparallel gravity Nester:1998mp . However,
as it was pointed out in Ref. STGPV2 , such kind of models also suffer from
the ghost instability in the vector perturbations.

This paper is organized as follows. In Sec. II we present the basics of the TG
theory and the model which extends NYTG by including extra PV terms in the
action. The cosmological perturbations are studied in Sec. III, we will show
with the quadratic actions of the linear perturbations that in general both
the scalar and vector perturbations suffer from the problem of ghost
instability, unless this model reduces to the NYTG model. To further explore
the severity of the ghost instability in this model, we study the
perturbations around the Minkowski spacetime in Sec. IV. The conclusion is
presented in Sec. V.

In this paper, we adopt the unit $8\pi G=1$, and use the signature $(+,-,-,-)$
for the metric. The local space tensor indices are denoted by
$A,B,C,...=0,1,2,3$ and by $a,b,c,...=1,2,3$ when limiting to spatial
components. They are lowered and raised by the Minkowski metric $\eta_{AB}$
and its inverse $\eta^{AB}$. The spacetime tensor indices are denoted by Greek
$\mu,\nu,\rho,...=0,1,2,3$ and by Latin $i,j,k,...=1,2,3$ when limiting to
spatial components. They are lowered and raised by the spacetime metric
$g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. In addition, we distinguish the
spacetime affine connection $\hat{\Gamma}^{\rho}_{~{}\mu\nu}$ and its
associated covariant derivative $\hat{\nabla}$ from the Levi-Civita connection
${\Gamma}^{\rho}_{~{}\mu\nu}$ and its associated covariant derivative
${\nabla}$ respectively.

##  II The Parity Violating gravity model as an extension of NYTG model

The TG theory can be considered as a constrained metric-affine theory. It is
formulated in a spacetime endowed with a metric $g_{\mu\nu}$ and an affine
connection $\hat{\Gamma}^{\rho}_{~{}\mu\nu}$, which is curvature free and
metric compatible,

$\hat{R}^{\sigma}_{~{}\rho\mu\nu}=\partial_{\mu}\hat{\Gamma}^{\sigma}_{~{}\nu\rho}-\partial_{\nu}\hat{\Gamma}^{\sigma}_{~{}\mu\rho}+\hat{\Gamma}^{\sigma}_{~{}\mu\lambda}\hat{\Gamma}^{\lambda}_{~{}\nu\rho}-\hat{\Gamma}^{\sigma}_{~{}\nu\lambda}\hat{\Gamma}^{\lambda}_{~{}\mu\rho}=0~{},~{}\hat{\nabla}_{\rho}g_{\mu\nu}=\partial_{\rho}g_{\mu\nu}-\hat{\Gamma}^{\lambda}_{~{}\rho\mu}g_{\lambda\nu}-\hat{\Gamma}^{\lambda}_{~{}\rho\nu}g_{\mu\lambda}=0~{}.$
(1)

Without curvature and nonmetricity in the TG theory, the gravity is identified
with spacetime torsion
$\mathcal{T}^{\rho}_{~{}\mu\nu}=2\hat{\Gamma}_{\,[\mu\nu]}^{\rho}$. In terms
of the tetrad $e^{A}_{~{}\mu}$ and spin connection $\omega^{A}_{~{}B\mu}$,
which relates the metric and affine connection through
$g_{\mu\nu}=\eta_{AB}e^{A}_{~{}\mu}e^{B}_{~{}\nu}$ and
$\hat{\Gamma}^{\rho}_{~{}\mu\nu}=e_{A}^{~{}\,\rho}(\partial_{\mu}e^{A}_{~{}\nu}+\omega^{A}_{~{}B\mu}e^{B}_{~{}\nu})$,
the torsion tensor can be expressed as

$\mathcal{T}^{\rho}_{~{}\mu\nu}=2e_{A}^{~{}\,\rho}(\partial_{[\mu}e^{A}_{~{}\nu]}+\omega^{A}_{~{}B[\mu}e^{B}_{~{}\nu]})~{}.$
(2)

The teleparallel constraints (1) dictate that the spin connection can be in
general expressed as,

$\omega_{~{}B\mu}^{A}=(\Lambda^{-1})^{A}_{~{}C}\partial_{\mu}\Lambda_{~{}B}^{C}~{},$
(3)

where $\Lambda^{A}_{~{}B}$ is arbitrary element of Lorentz transformation
matrix which is position dependent and satisfies the relation
$\eta_{AB}\Lambda^{A}_{~{}C}\Lambda^{B}_{~{}D}=\eta_{CD}$ at any spacetime
point. Therefore, the tetrad $e^{A}_{~{}\mu}$ and the Lorentz matrix
$\Lambda^{A}_{~{}B}$ can be regarded as the basic variables of the TG theory.

The TGR model, as the GR equivalent TG model, has the following action,

$\displaystyle S_{TGR}=\frac{1}{2}\int d^{4}x~{}{\|e\|}\mathbb{T}\equiv\int
d^{4}x~{}{\|e\|}\left(-\frac{1}{2}\mathcal{T}_{\mu}\mathcal{T}^{\mu}+\frac{1}{8}\mathcal{T}_{\alpha\beta\mu}\mathcal{T}^{\alpha\beta\mu}+\frac{1}{4}\mathcal{T}_{\alpha\beta\mu}\mathcal{T}^{\beta\alpha\mu}\right)~{},$
(4)

where ${\|e\|}=\sqrt{-g}$ is the determinant of the tetrad, $\mathbb{T}$ is
the torsion scalar, and
$\mathcal{T}_{\mu}=\mathcal{T}^{\alpha}_{~{}~{}\mu\alpha}$ is the torsion
vector. Since we have the identity
$-{R}(e)=\mathbb{T}+2{\nabla}_{\mu}\mathcal{T}^{\mu}$, the action (4) is
identical to the Einstein-Hilbert action up to a surface term, where the
curvature scalar $R(e)$ is defined by the Levi-Civita connection and
considered as being fully constructed from the metric, and in turn from the
tetrad.

As mentioned before, the NYTG model PVtele1 ; PVtele2  modifies the TGR model
by a coupling Lagrangian between an axion-like field and the Nieh-Yan density.
Generally we should also consider its own dynamics of the axion-like field and
take other matter into account, so the full action of the NYTG model is

$\displaystyle S_{NYTG}=\int
d^{4}x~{}{\|e\|}\left[\frac{1}{2}\mathbb{T}+\frac{c}{4}\,\theta\,\mathcal{T}_{\lambda\mu\nu}\widetilde{\mathcal{T}}^{\lambda\mu\nu}+\frac{1}{2}\nabla_{\mu}\theta\nabla^{\mu}\theta-V(\theta)\right]+S_{m}~{},$
(5)

where $c$ is the coupling constant and
$\widetilde{\mathcal{T}}^{\lambda\mu\nu}=(1/2)\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}^{\lambda}_{~{}~{}\rho\sigma}$.
Other matter with the action $S_{m}$ is assumed to be coupled to spacetime
minimally through the tetrad. At the background in which the axion-like field
has non-zero spacetime derivatives, the Nieh-Yan coupling term breaks parity
spontaneously. Because only the first-order derivatives of the basic variables
appears in the action, the NYTG model can avoid the Ostrogradski ghost mode
originated from higher-order derivatives. Detailed analyses on the
cosmological perturbations PVtele1 ; PVtele2  showed that this model is ghost
free and exhibits a velocity birefringence phenomenon of GWs, which is an
explicit signal of parity violation.

Besides the Nieh-Yan density, there are other scalar invariants which are
parity-odd and quadratic in the torsion tensor within the TG framework.
Totally we have the following four invariants of this type:

$\mathcal{P}_{1}=\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}^{\lambda}_{~{}\mu\nu}\mathcal{T}_{\lambda\rho\sigma}~{},~{}\mathcal{P}_{2}=\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}_{\mu}\mathcal{T}_{\nu\rho\sigma}~{},~{}\mathcal{P}_{3}=\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}^{\lambda}_{~{}\mu\nu}\mathcal{T}_{\rho\sigma\lambda}~{},~{}\mathcal{P}_{4}=\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}_{\mu\nu}^{~{}~{}\lambda}\mathcal{T}_{\rho\sigma\lambda}~{},$
(6)

where $\mathcal{P}_{1}$ is just the Nieh-Yan density. None of these terms hide
higher derivatives. However, these invariants are not fully independent. There
exist two identities: $\mathcal{P}_{3}=-\mathcal{P}_{1}+\mathcal{P}_{2}$ and
$\mathcal{P}_{4}=(1/2)\mathcal{P}_{1}-\mathcal{P}_{2}$, so only two of them
are independent, this is consistent with the result of Ref. PVtele3 . For
convenience we need only to consider $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$
as independent parity-odd invariants. As a generalization of the NYTG model,
we will in this paper consider the couplings of $\theta(x)$ to both
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ as in Ref. PVtele3 . In another word,
the PV gravity model we will study in more detail in this paper has the full
action:

$\displaystyle S=\int
d^{4}x~{}{\|e\|}\left[\frac{1}{2}\mathbb{T}+\frac{1}{4}\,\theta\left(c_{1}\mathcal{P}_{1}+c_{2}\mathcal{P}_{2}\right)+\frac{1}{2}\nabla_{\mu}\theta\nabla^{\mu}\theta-V(\theta)\right]+S_{m}~{},$
(7)

where both $c_{1}$ and $c_{2}$ are coupling constants, if $c_{2}=0$ the model
returns to the NYTG model. This action is our starting point in this paper.

Like the NYTG model, only the first-order derivatives of the basic variables
appear in the action (7), this general model can avoid the Ostrogradski ghost
mode from higher-order derivatives. But this does not mean this model is ghost
free, it only means that if ghost modes appear, they are not the Ostrogradski
type. The equations of motion follow from the variations with respect to
$e^{A}_{~{}\mu}$ and $\Lambda^{A}_{~{}B}$ separately:

$\displaystyle G^{\mu\nu}+N^{\mu\nu}$ $\displaystyle=$ $\displaystyle
T^{\mu\nu}+T^{\mu\nu}_{\theta}~{},$ (8) $\displaystyle N^{[\mu\nu]}$
$\displaystyle=$ $\displaystyle 0~{},$ (9)

where $G^{\mu\nu}$ is the Einstein tensor, $T^{\mu\nu}=-(2/\sqrt{-g})(\delta
S_{m}/\delta g_{\mu\nu})$ and
$T^{\mu\nu}_{\theta}=[V(\theta)-\nabla_{\alpha}\theta\nabla^{\alpha}\theta/2]g^{\mu\nu}+\nabla^{\mu}\theta\nabla^{\nu}\theta$
are the energy-momentum tensors for the matter and the scalar field $\theta$
respectively, and

$N^{\mu\nu}=\frac{c_{2}}{2}\left[\hat{\nabla}^{\mu}(\theta\tilde{\mathcal{T}}^{\nu})-\varepsilon^{\mu\nu\rho\sigma}\hat{\nabla}_{\rho}(\theta\mathcal{T}_{\sigma})-g^{\mu\nu}\tilde{\mathcal{T}}^{\rho}\partial_{\rho}\theta-\theta\left(2\tilde{\mathcal{T}}^{(\mu\nu)\rho}\mathcal{T}_{\rho}+\frac{1}{2}g^{\mu\nu}\mathcal{P}_{1}-\mathcal{T}^{\mu}\tilde{\mathcal{T}}^{\nu}\right)\right]+c_{1}\tilde{\mathcal{T}}^{\mu\nu\rho}\partial_{\rho}\theta~{},$
(10)

where $\tilde{\mathcal{T}}^{\mu}=g_{\rho\nu}\tilde{\mathcal{T}}^{\rho\mu\nu}$.
Similar to most modified TG models, the equation of motion (9) from the
variation of $\Lambda^{A}_{~{}B}$ is not independent of Eq. (8), it is just
the antisymmetric part of the latter. As explained in Ref. PVtele2 , this is
due to the local Lorentz invariance of the action, any change caused by
$\delta\Lambda^{A}_{~{}B}$ can always be equivalent to the change caused by
$\delta e^{A}_{~{}\mu}$, so requiring the action to take the extremum under
$\delta e^{A}_{~{}\mu}$ already includes the case where the action takes the
extremum under $\delta\Lambda^{A}_{~{}B}$. There is another equation following
from the variation of the action (7) with respect to $\theta$,

${\square}\theta+V^{(1)}-\frac{c_{1}}{4}\mathcal{P}_{1}-\frac{c_{2}}{4}\mathcal{P}_{2}=0~{},$
(11)

where ${\square}=g^{\mu\nu}{\nabla}_{\mu}{\nabla}_{\nu}$ and $V_{\theta}$ is
the first derivative of the potential to the scalar field $\theta$. All these
equations of motion are consistent with the Bianchi identity
$\nabla_{\mu}G^{\mu\nu}=0$ and the covariant conservation law
$\nabla_{\mu}T^{\mu\nu}=0$.

Again, like most modified TG models, this model (7) has two kinds of gauge
symmetries: the diffeomorphism invariance and the local Lorentz invariance,
the local Lorentz transformation makes the following change:

$e^{A}_{~{}\mu}\rightarrow(L^{-1})^{A}_{~{}B}e^{B}_{~{}\mu}~{},~{}\Lambda^{A}_{~{}B}\rightarrow\Lambda^{A}_{~{}C}L^{C}_{~{}B}~{},$
(12)

where $L^{A}_{~{}B}$ is also the element of Lorentz matrix. It’s easy to prove
that the metric $g_{\mu\nu}$ and the torsion tensor
$\mathcal{T}_{~{}\mu\nu}^{\rho}$ are invariant under the local Lorentz
transformation (12), so is the action (7). Due to the local Lorentz
invariance, we can always choose the gauge
$\Lambda^{A}_{~{}B}=\delta^{A}_{~{}B}$, i.e., $\omega^{A}_{~{}B\mu}=0$. This
is the Weitzenböck connection which had been frequently adopted in the
literature. This gauge is also called the Weitzenböck gauge. Note that once
the Weitzenböck gauge $\omega^{A}_{~{}B\mu}=0$ is adopted, the model will no
longer be local Lorentz invariant.

Even taking the Weitzenböck gauge, the model still has sixteen basic variables
in the gravity sector, while GR has only ten. At the same time, the model also
has sixteen gravitational field equations implied in Eqs. (8) and (9). One can
find from the expression (10) that the extra six equations in (9) are second
order in time derivatives unless $c_{2}=0$. So naively, it can be expected
that the general model (7) presents six more dynamical degrees of freedom than
the case of GR with a minimally coupled scalar field and other matter. We will
confirm this judgement in Sec. III. It is some of these extra degrees of
freedom bring ghost instabilities.

##  III Cosmological perturbations and quadratic actions

From now on, we apply the model (7) to cosmology. This model corresponds to a
special case of a more general model considered in Ref. PVtele3  where the
dynamics of the cosmological background has been studied in detail. Hence, we
will not consider the evolution of the background any more. We will focus on
the linear cosmological perturbations around the Friedmann-Robertson-Walker
(FRW) background. For simplicity, we take the Weitzenböck gauge
$\omega^{A}_{~{}B\mu}=0$, and we only consider the spatially-flat FRW universe
as the background because this is enough for us to see the difficulties of
this model.

We use the following parametrization for perturbed tetrad PVtele2 ; tetradper1
:

$\displaystyle e^{0}_{\ 0}=a(1+A)~{},~{}e^{0}_{\
i}=a(\beta_{,i}+\beta_{i}^{V})~{},~{}e^{c}_{\
0}=a\delta_{ci}(\chi_{,i}+\chi_{i}^{V})~{},$ $\displaystyle e^{c}_{\
i}=a\delta_{cj}[(1-\psi)\delta_{ij}+\alpha_{,ij}+\alpha_{j,i}^{V}-\epsilon_{ijk}(\lambda_{,k}+\lambda_{k}^{V})+\frac{1}{2}h^{T}_{ij}]~{},$
(13)

where $a$ is the scale factor of the universe and the subscript $``,i"$ means
$\partial_{i}$. So the perturbed metric components have the familiar forms:

$\displaystyle g_{00}=a^{2}(1+2A)~{},~{}g_{0i}=-a^{2}(B_{,i}+B_{i}^{V})~{},$
$\displaystyle
g_{ij}=-a^{2}[(1-2\psi)\delta_{ij}+2\alpha_{,ij}+\alpha_{i,j}^{V}+\alpha_{j,i}^{V}+h^{T}_{ij}]~{},$
(14)

where $B=\chi-\beta$ and $B^{V}_{i}=\chi^{V}_{i}-\beta^{V}_{i}$. Besides the
familiar scalar perturbations ($A,B,\psi,\alpha$), vector perturbations
($B_{i}^{V},\alpha^{V}_{i}$), and tensor perturbations $h^{T}_{ij}$ in the
metric, the parametrization of tetrad brings six extra variables, which are
scalar perturbation $\lambda,\chi+\beta$ and vector perturbation
$\lambda_{i}^{V},\chi_{i}^{V}+\beta_{i}^{V}$. All the vector perturbations are
transverse and denoted by the superscript $V$, both the tensor perturbations
are transverse and traceless and denoted by the superscript $T$. In addition,
the scalar field $\theta$ is decomposed as
$\theta(\eta,\vec{x})=\bar{\theta}(\eta)+\delta\theta(\eta,\vec{x})$ with
$\eta$ being the conformal time. In addition, we should also consider the
perturbations to the matter contained in $S_{m}$.

With this scheme one can obtain the perturbation equations directly for the
model (7), but in order to see whether there are dangerous propagating modes
it is better to consider the quadratic actions for the linear perturbations.
When applying the model to the early universe, such as the inflationary epoch,
in that case the scalar field $\theta$ may be considered as the inflaton, we
need to quantize these perturbations to have a mechanism for generating the
primordial perturbations which seed the large scale structure at later time.
For this purpose, the quadratic actions are indispensable. For simplicity in
the following discussions, we ignore other matter so that $S_{m}=0$ in Eq. (7)
from now on.

Even we have taken the Weitzenböck connection, the diffeomorphism invariance
is preserved, it is safe to take the unitary gauge
$\delta\theta=0,~{}\alpha=0,~{}\alpha_{i}^{V}=0$. This simplifies the
calculations, for example, the gauge invariant scalar perturbation
$\zeta=-(\psi+\mathcal{H}\delta\theta/\theta^{\prime})$ representing the
curvature perturbation of the hypersurfaces of constant $\theta$ field reduces
to $-\psi$ under the unitary gauge, where $\mathcal{H}=a^{\prime}/a$ is the
conformal Hubble rate. With these we may choose $A$, $\zeta$, $B$, $\beta$,
$\lambda$, $B_{i}^{V}$, $\beta_{i}^{V}$, $\lambda_{i}^{V}$ and $h^{T}_{ij}$ as
independent variables.

As mentioned before, there is no higher-order derivative in the action of the
extended model (7), so it avoids the Ostrogradski ghost mode. But the
couplings $\theta\left(c_{1}\mathcal{P}_{1}+c_{2}\mathcal{P}_{2}\right)$ are
possible to cause mixings among perturbation variables, such as
$x^{\prime}y^{\prime}=(1/2)[(x^{\prime}+y^{\prime})^{2}-(x^{\prime}-y^{\prime})^{2}]\equiv(1/2)(z_{1}^{\prime
2}-z_{2}^{\prime 2})$, here the prime represents the derivative with respect
to the conformal time. Such mixings have the potential to introduce ghost
modes into the quadratic actions. We will investigate the quadratic actions
for the scalar, vector, and tensor perturbations of the model (7) separately
in the following subsections.

###  III.1 Quadratic action for scalar perturbations

For the model (7) with $S_{m}=0$, one can obtain directly the quadratic action
for the scalar perturbations:

$\displaystyle S^{(2)}_{S}$ $\displaystyle=$ $\displaystyle-\int d\eta
d^{3}k~{}a^{2}\Big{\\{}3{\zeta^{\prime}}^{2}-6\mathcal{H}\zeta^{\prime}A-k^{2}(2A+\zeta)\zeta+a^{2}V(\theta)A^{2}+2k^{2}(\zeta^{\prime}-\mathcal{H}A)B$
(15) $\displaystyle-(2c_{1}+3c_{2})\theta^{\prime}k^{2}\zeta\lambda-
c_{2}\theta
k^{2}\left[\beta^{\prime}\lambda^{\prime}-k^{2}(B+\beta)\lambda+(\zeta-A)\lambda^{\prime}\right]\Big{\\}}~{},$

we have changed to the momentum space in terms of Fourier transformations. We
also simply mark $A^{*}B$ as $AB$, $A^{*}A$ as $A^{2}$, and so on. No matter
what the values the coupling constants $c_{1}$ and $c_{2}$ have, the variables
$A$ and $B$ are non-dynamical fields and the variations of the quadratic
action with them lead to the following two constraints:

$\displaystyle 2(\mathcal{H}A-\zeta^{\prime})-c_{2}\theta k^{2}\lambda=0~{},$
(16) $\displaystyle
6\mathcal{H}\zeta^{\prime}+2k^{2}\zeta-2a^{2}V(\theta)A+2\mathcal{H}k^{2}B-c_{2}\theta
k^{2}\lambda^{\prime}=0~{}.$ (17)

These constraint equations are just used to solve the non-dynamical variables
$A$ and $B$. One can eliminate these two non-dynamical variables from the
action (15) by substituting the constraints (16) and (17) back into it. After
that, the quadratic action for scalar perturbations can be expressed as

$\displaystyle S^{(2)}_{S}=\int d\eta
d^{3}k\,\Bigg{\\{}\frac{a^{2}{\theta^{\prime}}^{2}}{2\mathcal{H}^{2}}\left({\zeta^{\prime}}^{2}-k^{2}\zeta^{2}\right)+2c_{1}a^{2}\theta^{\prime}k^{2}\zeta\lambda+c_{2}a^{2}\theta
k^{2}\bigg{[}\lambda^{\prime}\beta^{\prime}-\frac{1}{\mathcal{H}}\lambda^{\prime}\zeta^{\prime}-\Big{(}1-\frac{{\theta^{\prime}}^{2}}{2\mathcal{H}^{2}}\Big{)}\lambda\zeta^{\prime}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-k^{2}\lambda\beta+\Big{(}\frac{k^{2}}{\mathcal{H}}+2\frac{\theta^{\prime}}{\theta}-2\mathcal{H}\Big{)}\lambda\zeta+\frac{c_{2}(\theta\theta^{\prime
2}+2\mathcal{H}\theta^{\prime}-2\mathcal{H}^{2}\theta)}{4\mathcal{H}^{2}}k^{2}\lambda^{2}\bigg{]}\Bigg{\\}}~{}.$
(18)

From this action, one will see that whether the coefficient $c_{2}$ vanishes
or not determines whether the scalar perturbations of the model (7) contain
ghost modes.

In the case where $c_{2}=0$, the model (7) reduces to the NYTG model, the
quadratic action (18) reduces to the following one,

$S^{(2)}_{S}=\int d\eta
d^{3}k\,\left[\frac{a^{2}{\theta^{\prime}}^{2}}{2\mathcal{H}^{2}}\left({\zeta^{\prime}}^{2}-k^{2}\zeta^{2}\right)+2c_{1}a^{2}\theta^{\prime}k^{2}\zeta\lambda\right]~{}.$
(19)

Evidently there is no ghost mode in this action. In fact $\lambda$ in this
case is a Lagrange multiplier and leads to $\zeta=0$, all the scalar
perturbations are non-dynamical. This is a special feature of the linear
perturbation theory of the NYTG model when applied to spatially flat universe
PVtele1 . If the background of the universe has nonzero spatial curvature,
there is one dynamical scalar mode which is normal and healthy PVtele2 .

In the more general case with $c_{2}\neq 0$, the kinetic terms
$\lambda^{\prime}\beta^{\prime}$ and $\lambda^{\prime}\zeta^{\prime}$ led by
$c_{2}$ in the action (18) generally contain ghost modes, as mentioned in the
paragraph just before this subsection. In order to explicitly see how many
dynamical degrees of freedom and how many ghost modes there are, we define new
perturbation variables $\gamma_{i}$ with $i=1,2,3$ in terms of the old
variables $\zeta$, $\beta$, and $\lambda$:

$\displaystyle\gamma_{1}=\frac{a\sqrt{|3+3\tau_{1}-\tau_{1}^{2}}|}{\sqrt{2}\mathcal{H}\theta^{\prime}(\tau_{1}-\tau_{2})(\tau_{1}-\tau_{3})}\left[(1-\tau_{2}-\tau_{3})\theta^{\prime
2}\zeta+\mathcal{H}\theta^{\prime 2}(\tau_{2}+\tau_{3})\beta-
c_{2}\mathcal{H}(2+\tau_{2}\tau_{3})k^{2}\lambda\right]~{},$
$\displaystyle\gamma_{2}=\frac{a\sqrt{|3+3\tau_{2}-\tau_{2}^{2}}|}{\sqrt{2}\mathcal{H}\theta^{\prime}(\tau_{2}-\tau_{1})(\tau_{2}-\tau_{3})}\left[(1-\tau_{1}-\tau_{3})\theta^{\prime
2}\zeta+\mathcal{H}\theta^{\prime 2}(\tau_{1}+\tau_{3})\beta-
c_{2}\mathcal{H}(2+\tau_{1}\tau_{3})k^{2}\lambda\right]~{},$
$\displaystyle\gamma_{3}=\frac{a\sqrt{|3+3\tau_{3}-\tau_{3}^{2}}|}{\sqrt{2}\mathcal{H}\theta^{\prime}(\tau_{3}-\tau_{1})(\tau_{3}-\tau_{2})}\left[(1-\tau_{1}-\tau_{2})\theta^{\prime
2}\zeta+\mathcal{H}\theta^{\prime 2}(\tau_{1}+\tau_{2})\beta-
c_{2}\mathcal{H}(2+\tau_{1}\tau_{2})k^{2}\lambda\right]~{},$ (20)

where $\tau_{1}$, $\tau_{2}$ and $\tau_{3}$ are the three solutions of the
equation $1-2\tau-\tau^{2}+\tau^{3}=0$, and numerically $\tau_{1}\approx
0.445$, $\tau_{2}\approx 1.802$ and $\tau_{3}\approx-1.247$. It can be
verified that the transformation (20) is linearly reversible. Then we can
express the action (18) in terms of the new perturbation variables as,

$S^{(2)}_{S}=\int d\eta
d^{3}k\,\Big{\\{}{\gamma_{1}^{\prime}}^{2}+{\gamma_{2}^{\prime}}^{2}-{\gamma_{3}^{\prime}}^{2}+\mathring{\mathcal{C}}_{12}\gamma^{\prime}_{1}\gamma_{2}+\mathring{\mathcal{C}}_{13}\gamma^{\prime}_{1}\gamma_{3}+\mathring{\mathcal{C}}_{23}\gamma^{\prime}_{2}\gamma_{3}+\hat{\mathcal{C}}_{1}\gamma_{1}^{2}+\hat{\mathcal{C}}_{2}\gamma_{2}^{2}+\hat{\mathcal{C}}_{3}\gamma_{3}^{2}+\tilde{\mathcal{C}}_{12}\gamma_{1}\gamma_{2}+\tilde{\mathcal{C}}_{13}\gamma_{1}\gamma_{3}+\tilde{\mathcal{C}}_{23}\gamma_{2}\gamma_{3}\Big{\\}}~{},$
(21)

where

$\displaystyle\mathring{\mathcal{C}}_{ij}=\frac{2(\tau_{i}-\tau_{j})}{\sqrt{|3+3\tau_{i}-\tau_{i}^{2}||3+3\tau_{j}-\tau_{j}^{2}|}}\left(2\frac{\theta^{\prime\prime}}{\theta^{\prime}}-\frac{\theta^{\prime}}{\theta}\right)~{},$
$\displaystyle\hat{\mathcal{C}}_{i}=(i^{2}-3i+1)k^{2}+\frac{1}{|3+3\tau_{i}-\tau_{i}^{2}|}\Bigg{\\{}4\tau_{i}\mathcal{H}^{2}+2\left[6+\tau_{i}-3\tau_{i}^{2}+2\frac{c_{1}}{c_{2}}(2-\tau_{i}^{2})\right]\mathcal{H}\frac{\theta^{\prime}}{\theta}+\frac{1}{2}\left(\tau_{i}^{2}-3\tau_{i}-5-\frac{2\tau_{i}}{\theta^{2}}\right)\theta^{\prime
2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{\tau_{i}(\tau_{i}+1)}{2}\frac{\theta^{\prime
4}}{\mathcal{H}^{2}}+3\tau_{i}\frac{\theta^{\prime\prime}}{\theta}-2\tau_{i}\left(\frac{\theta^{\prime\prime}}{\theta^{\prime}}\right)^{2}+(\tau_{i}^{2}-1)\frac{\theta^{\prime
3}}{\mathcal{H}\theta}+2(\tau_{i}+1)\frac{\theta^{\prime}\theta^{\prime\prime}}{\mathcal{H}}+(3+\tau_{i}-\tau_{i}^{2})\frac{\theta^{\prime\prime\prime}}{\theta^{\prime}}\Bigg{\\}}~{},$
$\displaystyle\tilde{\mathcal{C}}_{ij}=\frac{1}{\sqrt{|3+3\tau_{i}-\tau_{i}^{2}||3+3\tau_{j}-\tau_{j}^{2}|}}\Bigg{\\{}4(\tau_{i}+\tau_{j})\mathcal{H}^{2}-2\theta^{\prime
2}+2\left[12+\tau_{i}+\tau_{j}-3\tau_{i}^{2}-3\tau_{j}^{2}+2\frac{c_{1}}{c_{2}}(4-\tau_{i}^{2}-\tau_{j}^{2})\right]\mathcal{H}\frac{\theta^{\prime}}{\theta}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-2\tau_{j}\left(\frac{\theta^{\prime}}{\theta}\right)^{2}+(\tau_{i}^{2}+\tau_{j}^{2}-\tau_{i}-\tau_{j}-3)\frac{\theta^{\prime
4}}{\mathcal{H}^{2}}+2(\tau_{i}+2\tau_{j})\frac{\theta^{\prime\prime}}{\theta}-4\tau_{i}\left(\frac{\theta^{\prime\prime}}{\theta^{\prime}}\right)^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+(\tau_{i}^{2}+\tau_{j}^{2}-2)\frac{\theta^{\prime
3}}{\mathcal{H}\theta}+2(\tau_{i}^{2}+\tau_{j}^{2}-2\tau_{i}-2\tau_{j}-4)\frac{\theta^{\prime}\theta^{\prime\prime}}{\mathcal{H}}-4\tau_{j}\frac{\theta^{\prime\prime\prime}}{\theta^{\prime}}\Bigg{\\}}~{}.$

This quadratic action (21) for the scalar perturbations shows that all three
$\gamma_{i}$s represent dynamical degrees of freedom. That means the model (7)
with $c_{2}\neq 0$ has two more dynamical degrees of freedom than the model of
GR with a minimally coupled scalar field. At the same time, the quadratic
action (21) clearly shows that $\gamma_{3}$ is a ghost mode because its
kinetic term has a wrong sign. This will cause the vacuum instability. This
ghost mode in the scalar perturbations is completely brought by the PV
coupling term $c_{2}\theta\mathcal{P}_{2}$ in the action (7) and independent
of the scale.

###  III.2 Quadratic action for vector perturbations

For vector perturbations, direct calculation shows that the quadratic action
of the model (7) is

$\displaystyle S^{(2)}_{V}=\int
d^{4}x\Bigg{\\{}-\frac{a^{2}}{4}B^{V}_{i}\nabla^{2}B^{V}_{i}+\frac{1}{2}c_{1}a^{2}\theta^{\prime}\epsilon_{ijk}\left(\beta^{V}_{i}\beta^{V}_{j,k}-\lambda^{V}_{i}\lambda^{V}_{j,k}\right)$
$\displaystyle\quad\quad\quad\quad\quad+c_{2}a^{2}\theta\bigg{[}\beta_{i}^{V\prime}\lambda_{i}^{V\prime}+\lambda_{i}^{V}\nabla^{2}\beta_{i}^{V}+\frac{1}{2}B_{i}^{V}(\nabla^{2}\lambda^{V}_{i}-\epsilon_{ijk}\beta^{V\prime}_{j,k})+\frac{1}{2}\Big{(}\frac{\theta^{\prime}}{\theta}-\mathcal{H}\Big{)}\epsilon_{ijk}\left(\beta^{V}_{i}\beta^{V}_{j,k}-\lambda^{V}_{i}\lambda^{V}_{j,k}\right)\bigg{]}\Bigg{\\}}~{}.$
(22)

where $\nabla^{2}=\partial_{i}\partial_{i}$ is Laplacian. Whatever the values
the coefficients $c_{1}$ and $c_{2}$ take, the variable $B^{V}_{i}$ is a non-
dynamical field. It induces the constraint,

$\nabla^{2}B^{V}_{i}+c_{2}\theta\left(\epsilon_{ijk}\beta^{V\prime}_{j,k}-\nabla^{2}\lambda^{V}_{i}\right)=0~{}.$
(23)

This constraint equation just solves the variable $B_{i}^{V}$. For further
simplification, we use the circular polarization bases $\hat{e}^{L}_{i}$,
$\hat{e}^{R}_{i}$, so that $B_{i}^{V}$ is expanded as,

$B^{V}_{i}(\eta,\vec{x})=\sum_{A}\int\frac{d^{3}k}{(2\pi)^{3/2}}B_{A}(\eta,\vec{k})\hat{e}^{A}_{i}(\vec{k})e^{i\vec{k}\cdot\vec{x}}~{}.$
(24)

The bases satisfy the relation:
$\epsilon_{ijk}n^{j}\hat{e}_{k}^{A}=i\kappa_{A}\hat{e}_{ij}^{A}$, here $A=L,R$
and $\kappa_{L}=-1$, $\kappa_{R}=1$ for the left- and right-handed polarized
components respectively, $\vec{n}$ is the unit vector of $\vec{k}$. We also
expand the variables ${\lambda}^{V}_{i}$ and ${\beta}^{V}_{i}$ in the same
way. Substituting the constraint (23) back into the action (22), the quadratic
action for vector perturbations can be rewritten as

$\displaystyle S^{(2)}_{V}=\sum_{A}\int d\eta
d^{3}ka^{2}\,\Bigg{\\{}\frac{1}{2}c_{1}\kappa_{A}\theta^{\prime}k\left(\beta^{*}_{A}\beta_{A}-\lambda^{*}_{A}\lambda_{A}\right)+$
$\displaystyle\quad\quad
c_{2}\theta\bigg{[}{\beta^{*}_{A}}^{\prime}\lambda_{A}^{\prime}-\frac{1}{4}c_{2}\theta{\beta^{*}_{A}}^{\prime}\beta_{A}^{\prime}-\frac{1}{2}c_{2}\kappa_{A}\theta
k{\beta^{*}_{A}}^{\prime}\lambda_{A}-k^{2}{\beta^{*}_{A}}\lambda_{A}+\frac{1}{2}\kappa_{A}k\Big{(}\frac{\theta^{\prime}}{\theta}-\mathcal{H}\Big{)}\left(\beta^{*}_{A}\beta_{A}-\lambda^{*}_{A}\lambda_{A}\right)-\frac{1}{4}c_{2}\theta
k^{2}\lambda^{*}_{A}\lambda_{A}\bigg{]}\Bigg{\\}}~{}.$

Again, from this action, one will see that whether the coefficient $c_{2}$
vanishes or not determines whether the vector perturbations of the model (7)
contain ghost modes.

In the case where $c_{2}=0$, the model (7) reduces to the NYTG model, the
quadratic action (III.2) reduces to the following one,

$S^{(2)}_{V}=\sum_{A}\int d\eta
d^{3}k\,\frac{1}{2}c_{1}\kappa_{A}a^{2}\theta^{\prime}k(\beta^{*}_{A}\beta_{A}-\lambda^{*}_{A}\lambda_{A})~{}.$
(26)

Evidently, the vector perturbations $\beta_{A}^{V}$ and $\lambda_{A}^{V}$ are
non-dynamical fields. There is no propagating ghost mode in this action. In
fact, without source to the vector perturbations, all of them vanish in the
expanding universe. This is what happens in the NYTG model PVtele1 ; PVtele2 .

In the more general case with $c_{2}\neq 0$, the mixing terms
${\beta^{*}_{A}}^{\prime}\lambda_{A}^{\prime}$ and
$-{\beta^{*}_{A}}^{\prime}\beta_{A}^{\prime}$ led by $c_{2}$ in the action
(III.2) generally indicate the existences of ghost modes. In order to
explicitly see how many dynamical degrees of freedom and how many ghost modes
there are, we redefine the following independent vector perturbation variables
through $\beta^{V}_{A}$ and $\lambda^{V}_{A}$:

$\hat{\beta}_{A}=\frac{1}{2}c_{2}a\theta\beta_{A}-a\lambda_{A}~{}~{},~{}~{}\hat{\lambda}_{A}=a\lambda_{A}~{}.$
(27)

The above transformation from $(\beta_{A},\lambda_{A})$ to
$(\hat{\beta}_{A},\hat{\lambda}_{A})$ is reversible. Then we can rewrite the
quadratic action (III.2) as

$S^{(2)}_{V}=\sum_{A}\int d\eta
d^{3}k\left(\hat{\lambda}_{A}^{*\prime}\hat{\lambda}_{A}^{\prime}-\hat{\beta}_{A}^{*\prime}\hat{\beta}_{A}^{\prime}+\mathcal{C}_{1}\hat{\beta}_{A}^{*}\hat{\lambda}_{A}^{\prime}+\mathcal{C}_{2}\hat{\beta}_{A}^{*}\hat{\lambda}_{A}+\mathcal{C}_{3}\hat{\lambda}_{A}^{*}\hat{\lambda}_{A}+\mathcal{C}_{4}\hat{\beta}_{A}^{*}\hat{\beta}_{A}\right)~{},$
(28)

where

$\displaystyle\mathcal{C}_{1}=\kappa_{A}c_{2}\theta
k-2\frac{\theta^{\prime}}{\theta}~{},$
$\displaystyle\mathcal{C}_{2}=-2k^{2}-2\frac{\theta^{\prime\prime}}{\theta}-2\mathcal{H}\frac{\theta^{\prime}}{\theta}+\kappa_{A}c_{2}k(2\theta^{\prime}+\mathcal{H}\theta)+\frac{4\kappa_{A}k}{(c_{2}\theta)^{2}}\left[(c_{1}+c_{2})\theta^{\prime}-c_{2}\mathcal{H}\theta\right]~{},$
$\displaystyle\mathcal{C}_{3}=-2k^{2}+2\mathcal{H}^{2}-\frac{1}{2}\theta^{\prime
2}-\frac{\theta^{\prime
2}}{\theta^{2}}-\frac{1}{4}c_{2}^{2}\theta^{2}k^{2}+\frac{\kappa_{A}k}{2}\left(3c_{2}\mathcal{H}\theta+2c_{2}\theta^{\prime}-c_{1}\theta^{\prime}\right)+\frac{2\kappa_{A}k}{(c_{2}\theta)^{2}}\left[(c_{1}+c_{2})\theta^{\prime}-c_{2}\mathcal{H}\theta\right]~{},$
$\displaystyle\mathcal{C}_{4}=-2\mathcal{H}^{2}+\frac{1}{2}\theta^{\prime
2}-\frac{\theta^{\prime\prime}}{\theta}-2\mathcal{H}\frac{\theta^{\prime}}{\theta}+\frac{2\kappa_{A}k}{(c_{2}\theta)^{2}}\left[(c_{1}+c_{2})\theta^{\prime}-c_{2}\mathcal{H}\theta\right]~{}.$

The quadratic action (28) shows that all the four components of vector
perturbations, $\hat{\beta}_{A}$ and $\hat{\lambda}_{A}$ with $A=L~{}{\rm
and}~{}R$, are dynamical modes. It also clearly shows that both components of
$\hat{\beta}_{A}$ are ghost modes because their kinetic terms have wrong
signs. Again, this will cause the vacuum instability and such instability in
the vector perturbations is completely brought by the PV coupling term
$c_{2}\theta\mathcal{P}_{2}$ in the action (7) and is scale independent.

As a comparison, the model of GR with a scalar field or the NYTG model have no
propagating dynamical degrees of freedom in the vector perturbations.
Combining with the analysis on the scalar perturbations in the above
subsection, the model (7) with $c_{2}\neq 0$ brings totally six more dynamical
degrees of freedom than the model of GR with a minimally coupled scalar field.
This confirms the judgement in Sec. II. Furthermore, three of these extra
dynamical degrees of freedom represent ghost modes.

###  III.3 Quadratic action for tensor perturbations

Finally, we have the quadratic action for tensor perturbations in the model
(7),

$S^{(2)}_{T}=\int d^{4}x\
\frac{a^{2}}{8}\left[{h^{T}_{ij}}^{\prime}{h^{T}_{ij}}^{\prime}-h^{T}_{ij}\nabla^{2}h^{T}_{ij}-(c_{1}\theta^{\prime}-3c_{2}\mathcal{H}\theta)\epsilon_{ijk}h^{T}_{il}h^{T}_{lj,k}\right]~{}.$
(29)

In terms of the circular polarization bases $\hat{e}^{L}_{ij}$ and
$\hat{e}^{R}_{ij}$, where
$n^{l}\epsilon_{lik}\hat{e}_{jk}^{A}=i\kappa_{A}\hat{e}_{ij}^{A}$ with $A=L,R$
and $\kappa_{L}=-1$, $\kappa_{R}=1$, the tensor perturbations are expanded as

$h_{ij}^{T}(\eta,\vec{x})=\sum_{A}\int\frac{d^{3}k}{(2\pi)^{3/2}}\,h_{A}(\eta,\vec{k})\,\hat{e}_{ij}^{A}(\vec{k})\,e^{-i\vec{k}\cdot\vec{x}}~{}.$
(30)

The final form of the quadratic action for tensor perturbations is

$S^{(2)}_{T}=\sum_{A}\int d\eta
d^{3}k~{}\frac{a^{2}}{4}\left[h_{A}^{*\prime}h_{A}^{\prime}-k^{2}\left(1+\kappa_{A}\frac{c_{1}\theta^{\prime}-3c_{2}\mathcal{H}\theta}{k}\right)h_{A}^{*}h_{A}\right]~{}.$
(31)

First, we see that the tensor perturbations are ghost free. Second, the
modified dispersion relation
$\omega_{A}^{2}=k^{2}\left[1+\kappa_{A}(c_{1}\theta^{\prime}-3c_{2}\mathcal{H}\theta)/k\right]$
is helicity dependent. Consider small coupling and slow evolution of $\theta$,
one can find that GWs with different helicities will have different phase
velocities $v_{p}^{A}=\omega_{A}/k\simeq
1+\kappa_{A}(c_{1}\theta^{\prime}-3c_{2}\mathcal{H}\theta)/(2k)$, i.e., the
velocity birefringence phenomenon. This is the explicit signal of parity
violation in this model. We can also see that the phase velocity difference
become important only at the region of small $k$ (large scales), so this is an
infrared effect. Besides the velocity birefringence, the amplitudes for both
components of GWs are the same. When $c_{2}=0$, these results go back to those
obtained in the NYTG model PVtele1 ; PVtele2 .

##  IV Perturbations around the Minkowski background

To further explore the severity of the ghost instability in the model (7), we
study the perturbations around the Minkowski background in this section. We
still take the Weitzenböck gauge $\omega^{A}_{~{}B\mu}=0$, and still use Eq.
(III) to parameterize the perturbations of the tetrad, but the scale factor
$a=1$ in the Minkowski background. In addition, the scalar field $\theta$ is
decomposed as $\theta=\theta_{0}+\delta\theta$, where the constant
$\theta_{0}$ is the lowest point of potential $V(\theta)$, satisfying
$V(\theta_{0})=0$ and $V_{\theta}(\theta_{0})=0$. Note that $\delta\theta$ is
gauge invariant in this case, so we can no longer take the unitary gauge in
this section. Instead we take the conformal Newton gauge:
$B=0,~{}\alpha=0,~{}\alpha_{i}^{V}=0$, and we choose $A$, $\psi$,
$\delta\theta$, $\beta$, $\lambda$, $B_{i}^{V}$, $\beta_{i}^{V}$,
$\lambda_{i}^{V}$ and $h^{T}_{ij}$ as independent variables. Since the purpose
of this section is to explore the severity of the ghost instability in the
model (7), for simplicity, we only consider the case of $S_{m}=0$, $c_{1}\neq
0$, $c_{2}\neq 0$ and $\theta_{0}\neq 0$ in the following.

For scalar perturbations, direct calculation shows that the quadratic action
of the model (7) is

$\displaystyle S^{(2)}_{S}=\int
dtd^{3}k~{}\bigg{\\{}\frac{1}{2}\Big{[}\delta{\theta^{\prime}}^{2}-(k^{2}+m_{\theta}^{2})\delta\theta^{2}\Big{]}-3{\psi^{\prime}}^{2}+k^{2}\psi^{2}-2k^{2}A\psi+c_{2}\theta_{0}\Big{[}k^{2}(\beta^{\prime}-A-\psi)\lambda^{\prime}-k^{4}\beta\lambda\Big{]}\bigg{\\}}~{},$
(32)

where $m_{\theta}^{2}=d^{2}V(\theta)/d\theta^{2}|_{\theta=\theta_{0}}$. We
have changed to the momentum space in terms of Fourier transformations and we
also simply mark $A^{*}\psi$ as $A\psi$, $\psi^{*}\psi$ as $\psi^{2}$, and so
on. The variable $A$ is non-dynamical field and it induces the constraint

$2\psi+c_{2}\theta_{0}\lambda^{\prime}=0~{}.$ (33)

Substituting the constraint (33) back into the action (32), the quadratic
action for scalar perturbations can be rewritten as

$\displaystyle S^{(2)}_{S}=\int
dtd^{3}k~{}\bigg{\\{}\frac{1}{2}\Big{[}\delta{\theta^{\prime}}^{2}-(k^{2}+m_{\theta}^{2})\delta\theta^{2}\Big{]}+\frac{1}{4}\Big{[}3c_{2}^{2}\theta_{0}^{2}(-{\lambda^{\prime\prime}}^{2}+k^{2}{\lambda^{\prime}}^{2})-4c_{2}\theta_{0}k^{2}(\beta\lambda^{\prime\prime}+k^{2}\beta\lambda)\Big{]}\bigg{\\}}~{}.$
(34)

The action (34) is equivalent to the following action,

$\displaystyle S^{(2)}_{S}=\int
dtd^{3}k~{}\bigg{\\{}\frac{1}{2}\Big{[}\delta{\theta^{\prime}}^{2}-(k^{2}+m_{\theta}^{2})\delta\theta^{2}\Big{]}+\frac{1}{4}\Big{[}c_{2}\theta_{0}\phi\lambda^{\prime\prime}+\frac{1}{12}\phi^{2}+\frac{2}{3}k^{2}\beta\phi+\frac{4}{3}k^{4}\beta^{2}+3c_{2}^{2}\theta_{0}^{2}k^{2}{\lambda^{\prime}}^{2}-4c_{2}\theta_{0}k^{4}\beta\lambda\Big{]}\bigg{\\}}~{}.$
(35)

The variation of the action (35) on $\phi$ leads to constraint

$\phi=-2(2k^{2}\beta+3c_{2}\theta_{0}\lambda^{\prime\prime})~{}.$ (36)

It can be verified that the action (34) can be obtained by substituting the
constraint (36) back into the action (35), so the action (34) and the action
(35) are equivalent. On the other hand, the variation of the action (35) on
$\beta$ leads to the constraint

$\beta=\frac{3}{2}c_{2}\theta_{0}\lambda-\frac{1}{4k^{2}}\phi~{}.$ (37)

Substituting the constraint (37) back into the action (35), the quadratic
action for scalar perturbations can be rewritten as

$\displaystyle S^{(2)}_{S}=\int
dtd^{3}k~{}\bigg{\\{}\frac{1}{2}\Big{[}\delta{\theta^{\prime}}^{2}-(k^{2}+m_{\theta}^{2})\delta\theta^{2}\Big{]}+\frac{c_{2}\theta_{0}}{4}\Big{[}3c_{2}\theta_{0}k^{2}{\lambda^{\prime}}^{2}-\phi^{\prime}\lambda^{\prime}+k^{2}\lambda\phi-3c_{2}\theta_{0}k^{4}\lambda^{2}\Big{]}\bigg{\\}}~{}.$
(38)

In order to further simplify the quadratic action, we define new perturbation
variables $q_{1},q_{2}$ in terms of the old variables $\lambda,\phi$:

$\displaystyle
q_{1}=2^{-\frac{3}{2}}\left(1+9c_{2}^{2}\theta_{0}^{2}k^{4}\right)^{-\frac{1}{4}}\left[\phi-\left(3c_{2}\theta_{0}k^{2}+\sqrt{1+9c_{2}^{2}\theta_{0}^{2}k^{4}}\right)\lambda\right]~{},$
$\displaystyle
q_{2}=2^{-\frac{3}{2}}\left(1+9c_{2}^{2}\theta_{0}^{2}k^{4}\right)^{-\frac{1}{4}}\left[\phi-\left(3c_{2}\theta_{0}k^{2}-\sqrt{1+9c_{2}^{2}\theta_{0}^{2}k^{4}}\right)\lambda\right]~{}.$
(39)

Then we can express the action (38) in terms of the new perturbation variables
as,

$\displaystyle S^{(2)}_{S}=\int
dtd^{3}k~{}\bigg{\\{}\frac{1}{2}\Big{[}\delta{\theta^{\prime}}^{2}-(k^{2}+m_{\theta}^{2})\delta\theta^{2}\Big{]}+c_{2}\theta_{0}\Big{[}\frac{1}{2}({q_{1}^{\prime}}^{2}-k^{2}q_{1}^{2})-\frac{1}{2}({q_{2}^{\prime}}^{2}-k^{2}q_{2}^{2})\Big{]}\bigg{\\}}~{}.$
(40)

The quadratic action (40) for the scalar perturbations shows that
$\delta\theta,q_{1},q_{2}$ are all dynamical degrees of freedom and $q_{2}$ is
a ghost mode because its kinetic term has a wrong sign. Such a result is
consistent with the results in Sec. III.1. Furthermore, the action (40) shows
that the mass of the ghost mode $q_{2}$ is zero. This means that no matter how
low the energy scale of the effective field theory is, the ghost mode $q_{2}$
can always be excited. Therefore, even from the point of view of effective
field theory, the model (7) still suffers from the difficulty of ghost
instability.

For vector perturbations, its quadratic action can be obtained through the
same process as in Sec. III.2, and the result is still the action (28), but
here $\mathcal{C}_{1}=\kappa_{A}c_{2}\theta_{0}k$, $\mathcal{C}_{2}=-2k^{2}$,
$\mathcal{C}_{3}=-(2+c_{2}^{2}\theta_{0}^{2}/4)k^{2}$ and $\mathcal{C}_{4}=0$.
The action (28) shows that $\hat{\beta}_{A}$ and $\hat{\lambda}_{A}$ are all
dynamical modes but $\hat{\beta}_{A}$ are ghost modes. The fact that the mass
term coefficient $\mathcal{C}_{4}$ of the ghost modes $\hat{\beta}_{A}$ vanish
implies that the ghost modes $\hat{\beta}_{A}$ can always be excited no matter
how low the energy scale of the effective field theory is. Such results again
illustrate that even from the point of view of effective field theory, the
model (7) suffers from the difficulty of ghost instability.

For tensor perturbations, its quadratic action is exactly the same as that of
GR, so we won’t discuss it anymore.

##  V Conclusion

We studied in this paper a parity violating gravity model within the framework
of teleparallel gravity. This model modifies the gravity by the couplings of a
scalar field to the scalar invariants which are parity-odd and quadratic in
the torsion tensor. Totally there are two such type independent invariants,
and one of them is the Nieh-Yan density. Through investigations on its
cosmological perturbations, we found that in general this model has six more
dynamical degrees of freedom compared with model of GR with a minimally
coupled scalar field. Among these extra dynamical degrees of freedom, two of
them are scalar perturbations and the rest four are vector perturbations.
Furthermore, half of them are ghost modes and cause instabilities at all
scales. However, in the special case only the coupling to the Nieh-Yan density
exists, this model is ghost free and reduces to the NYTG model. Finally, by
investigating the properties of the ghost modes in Minkowski spacetime, we
find that it is almost impossible to avoid ghost instability in the model (7)
by adjusting the energy scale of the effective field theory.

Acknowledgement: This work is supported in part by NSFC under Grant No.
12075231 and 12047502, and by National Key R&D Program of China Grant No.
2021YFC2203100.

## References

  * (1) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 116, no.6, 061102 (2016) doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].
  * (2) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 119, no.16, 161101 (2017) doi:10.1103/PhysRevLett.119.161101 [arXiv:1710.05832 [gr-qc]].
  * (3) H. Li, S. Y. Li, Y. Liu, Y. P. Li, Y. Cai, M. Li, G. B. Zhao, C. Z. Liu, Z. W. Li and H. Xu, et al. “Probing Primordial Gravitational Waves: Ali CMB Polarization Telescope,” Natl. Sci. Rev. 6, no.1, 145-154 (2019) doi:10.1093/nsr/nwy019 [arXiv:1710.03047 [astro-ph.CO]].
  * (4) K. Abazajian et al. [CMB-S4], “CMB-S4: Forecasting Constraints on Primordial Gravitational Waves,” [arXiv:2008.12619 [astro-ph.CO]].
  * (5) R. Jackiw and S. Y. Pi, Phys. Rev. D 68, 104012 (2003) doi:10.1103/PhysRevD.68.104012 [arXiv:gr-qc/0308071 [gr-qc]].
  * (6) S. Alexander and N. Yunes, Phys. Rept. 480, 1-55 (2009) doi:10.1016/j.physrep.2009.07.002 [arXiv:0907.2562 [hep-th]].
  * (7) S. Dyda, E. E. Flanagan and M. Kamionkowski, Phys. Rev. D 86, 124031 (2012) doi:10.1103/PhysRevD.86.124031 [arXiv:1208.4871 [gr-qc]].
  * (8) M. Crisostomi, K. Noui, C. Charmousis and D. Langlois, Phys. Rev. D 97, no.4, 044034 (2018) doi:10.1103/PhysRevD.97.044034 [arXiv:1710.04531 [hep-th]].
  * (9) X. Gao and X. Y. Hong, Phys. Rev. D 101, no.6, 064057 (2020) doi:10.1103/PhysRevD.101.064057 [arXiv:1906.07131 [gr-qc]].
  * (10) W. Zhao, T. Zhu, J. Qiao and A. Wang, “Waveform of gravitational waves in the general parity-violating gravities,” Phys. Rev. D 101, no.2, 024002 (2020) doi:10.1103/PhysRevD.101.024002 [arXiv:1909.10887 [gr-qc]].
  * (11) N. Bartolo, L. Caloni, G. Orlando and A. Ricciardone, JCAP 03, 073 (2021) doi:10.1088/1475-7516/2021/03/073 [arXiv:2008.01715 [astro-ph.CO]].
  * (12) M. Li, H. Rao and D. Zhao, JCAP 11, 023 (2020) doi:10.1088/1475-7516/2020/11/023 [arXiv:2007.08038 [gr-qc]].
  * (13) M. Li, H. Rao and Y. Tong, Phys. Rev. D 104, no.8, 084077 (2021) doi:10.1103/PhysRevD.104.084077 [arXiv:2104.05917 [gr-qc]].
  * (14) R. Aldrovandi and J. G. Pereira, Teleparallel Gravity, Vol. 173. Springer, 23 Dordrecht, (2013).
  * (15) S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. L. Said, J. Mifsud and E. Di Valentino, [arXiv:2106.13793 [gr-qc]].
  * (16) H. T. Nieh and M. L. Yan, J. Math. Phys. 23, 373 (1982) doi:10.1063/1.525379
  * (17) H. Rao, Phys. Rev. D 104, no.12, 124084 (2021) doi:10.1103/PhysRevD.104.124084 [arXiv:2107.08597 [gr-qc]].
  * (18) J. Qiao, T. Zhu, G. Li and W. Zhao, [arXiv:2110.09033 [gr-qc]].
  * (19) Q. Wu, T. Zhu, R. Niu, W. Zhao and A. Wang, [arXiv:2110.13870 [gr-qc]].
  * (20) R. G. Cai, C. Fu and W. W. Yu, [arXiv:2112.04794 [astro-ph.CO]].
  * (21) F. Bombacigno, S. Boudet, G. J. Olmo and G. Montani, Phys. Rev. D 103, no.12, 124031 (2021) doi:10.1103/PhysRevD.103.124031 [arXiv:2105.06870 [gr-qc]].
  * (22) C. Gong, T. Zhu, R. Niu, Q. Wu, J. L. Cui, X. Zhang, W. Zhao and A. Wang, [arXiv:2112.06446 [gr-qc]].
  * (23) M. Hohmann and C. Pfeifer, Eur. Phys. J. C 81, no.4, 376 (2021) doi:10.1140/epjc/s10052-021-09165-x [arXiv:2012.14423 [gr-qc]].
  * (24) A. Conroy and T. Koivisto, JCAP 12, 016 (2019) doi:10.1088/1475-7516/2019/12/016 [arXiv:1908.04313 [gr-qc]].
  * (25) J. M. Nester and H. J. Yo, Chin. J. Phys. 37, 113 (1999) [arXiv:gr-qc/9809049 [gr-qc]].
  * (26) M. Li and D. Zhao, [arXiv:2108.01337 [gr-qc]].
  * (27) K. Izumi and Y. C. Ong, JCAP 06, 029 (2013) doi:10.1088/1475-7516/2013/06/029 [arXiv:1212.5774 [gr-qc]].


# Gromov–Hausdorff Distance Between Segment and Circle

Yibo Ji, Alexey A. Tuzhilin

###### Abstract

We calculate the Gromov–Hausdorff distance between a line segment and a circle
in the Euclidean plane. To do that, we introduced a few new notions like round
spaces and nonlinearity degree of a metric space.

##  1 Introduction

The Gromov–Hausdorff distance measures the difference between any two metric
spaces. There are a few possibilities to do that. One of them, to embed
isometrically the both spaces into all possible other metric spaces, then the
least possible Hausdorff distance between the images will be just that
characteristic. Another way is to establish correspondences between these
spaces and to measure the least possible distortion of the spaces metrics
produced by these correspondences.

The Gromov–Hausdorff distance has many applications, for example, it helps to
investigate the growth of groups, or it can be used in image recognition.
However, to get concrete values of the distance, even in the cases of “simple
spaces” like line segment and the standard circle, is a very non-trivial task.
In this paper we discuss our original technique enabled to obtain the exact
values in the latter case.

##  2 Preliminaries

In what follows, we work with various non-empty metric spaces, and the
distance between points $p$ and $q$ of a metric space we denote by $|pq|$,
independently on the choice of the space. For a metric space $X$, $x\in X$,
non-empty $A,B\subset X$, $r>0$, and $s\geq 0$, we use the following notions
and notations:

  * •

$U_{r}(x)=\bigl{\\{}y\in X:|xy|<r\bigr{\\}}$ (_open ball with center $x$ and
radius $r$_);

  * •

$B_{s}(x)=\bigl{\\{}y\in X:|xy|\leq s\bigr{\\}}$ (_closed ball with center $x$
and radius $s$_);

  * •

$|xA|=\inf_{a\in A}|xa|$ (_distance between $x$ and $A$_);

  * •

$U_{r}(A)=\bigl{\\{}y\in X:|yA|<r\bigr{\\}}$ (_open $r$-neighborhood of $A$_);

  * •

$B_{s}(A)=\bigl{\\{}y\in X:|yA|\leq s\bigr{\\}}$ (_closed $s$-neighborhood of
$A$_);

  * •

$d_{H}(A,B)=\inf\bigl{\\{}t:A\subset U_{t}(B)\text{ and }U_{t}(A)\supset
B\bigr{\\}}$ (_Hausdorff distance between $A$ and $B$_).

For any non-empty set $X$, we denote by ${\cal P}_{0}(X)$ the set of all non-
empty subsets of $X$. If $X$ is a metric space, then the Hausdorff distance on
${\cal P}_{0}(X)$ satisfies the triangle inequality, but can be equal
$\infty$, and can vanish for different subsets. However, if we restrict
$d_{H}$ to the set ${\cal H}(X)$ of all non-empty closed bounded subsets of
$X$, then $d_{H}$ becomes a metric, see [1].

######  Theorem 2.1 ([1]).

Given a metric space $X$, the following properties are simultaneously
presented or not in the both $X$ and ${\cal H}(X)$_:_ completeness, total
boundness, compactness.

Given $r>0$, a subset of a metric space is called _$r$ -separated_ if the
distance between any its different points is at least $r$.

Given two sets $X$ and $Y$, _a correspondence between $X$ and $Y$_ is each
subset $R\subset X\times Y$ such that for any $x\in X$ there exists $y\in Y$
with $(x,y)\in R$ and, vise versa, for any $y\in Y$ there exists $x\in X$ with
$(x,y)\in R$. Let ${\cal R}(X,Y)$ denote the set of all correspondences
between $X$ and $Y$. If $X$ and $Y$ are metric spaces, and $R\in{\cal
R}(X,Y)$, then we define _the distortion $\operatorname{dis}R$ of $R$_ as
follows:

$\operatorname{dis}R=\sup\Bigl{\\{}\bigl{|}|x_{1}x_{2}|-|y_{1}y_{2}|\bigr{|}:(x_{1},y_{1}),\,(x_{2},y_{2})\in
R\Bigr{\\}}.$

Further, _the Gromov–Hausdorff distance $d_{GH}(X,Y)$ between metric spaces
$X$ and $Y$_, or for short _GH-distance_ , is the value

$d_{GH}(X,Y)=\frac{1}{2}\inf\bigl{\\{}\operatorname{dis}R:R\in{\cal
R}(X,Y)\bigr{\\}}.$

An equivalent definition: it is the infimum of Hausdorff distances
$d_{H}(X^{\prime},Y^{\prime})$ between all possible subsets $X^{\prime}$ and
$Y^{\prime}$ of the metric spaces $Z$, provided $X^{\prime}$ is isometric to
$X$, and $Y^{\prime}$ is isometric to $Y$. It is well-known [1] that $d_{GH}$
is a metric on the set of isometry classes of compact metric spaces, in
particular, two compact metric spaces are isometric if and only if the GH-
distance between them vanishes. In general situation, $d_{GH}$ satisfies the
triangle inequality, is bounded for bounded spaces (for non-bounded spaces it
can be infinite), and can vanish for non-isometric spaces. Also, for any
$A,B\in{\cal P}_{0}(X)$, it holds $d_{GH}(A,B)\leq d_{H}(A,B)$. In particular,
if ${\bar{A}}$ is the closure of $A$, then
$d_{GH}(A,{\bar{A}})=d_{H}(A,{\bar{A}})=0$.

If $X_{1},X_{2},\ldots$ and $X$ are some metric spaces such that
$d_{GH}(X_{i},X)\to 0$, then we say that the sequence $X_{i}$ is
_Gromov–Hausdorff convergent_ to $X$ and write
$X_{i}\xrightarrow{\operatorname{GH}}X$.

It is easy to see that for any correspondence $R\in{\cal R}(X,Y)$, its closure
in $X\times Y$ has the same distortion, thus, to achieve $d_{GH}(X,Y)$, it
suffices to consider only closed correspondences. In other words, if ${\cal
R}_{c}(X,Y)$ is the set of all closed correspondences between $X$ and $Y$,
then we have

$d_{GH}(X,Y)=\frac{1}{2}\inf\bigl{\\{}\operatorname{dis}R:R\in{\cal
R}_{c}(X,Y)\bigr{\\}}.$

For any metric space $X$ and a real number $\lambda>0$, we denote by $\lambda
X$ the metric space which differs from $X$ by multiplication of all its
distances by $\lambda$. For $\lambda=0$, we define $\lambda X$ as a single
point space.

######  Theorem 2.2 ([1]).

Let $X$ and $Y$ be metric spaces. Then

  1. (1)

if $X$ is a single-point metric space, then
$d_{GH}(X,Y)=\frac{1}{2}\operatorname{diam}Y$_;_

  2. (2)

if $\operatorname{diam}X<\infty$, then

$d_{GH}(X,Y)\geq\frac{1}{2}|\operatorname{diam}X-\operatorname{diam}Y|;$

  3. (3)

$d_{GH}(X,Y)\leq\frac{1}{2}\max\\{\operatorname{diam}X,\operatorname{diam}Y\\}$,
in particular, $d_{GH}(X,Y)<\infty$ for bounded $X$ and $Y$_;_

  4. (4)

for any metric spaces $X$, $Y$ and any $\lambda>0$, we have $d_{GH}(\lambda
X,\lambda Y)=\lambda d_{GH}(X,Y)$.

##  3 Homogeneous and round metric spaces

Fix a real $0<b\leq\operatorname{diam}X$ and an integer $n\geq 2$. A metric
space $X$ is called _$(b,n)$ -homogeneous_ if for any point $x\in X$ there
exists a $b$-separated $n$-point subset $S\subset X$ such that $x\in S$.

######  Theorem 3.1.

Let $X$ be a $(b,n)$-homogeneous metric space. Suppose that $Y$ is not
$(a,n)$-homogeneous for some $0<a<b$. Then $2d_{GH}(X,Y)\geq b-a$.

###### Proof.

Put $c=b-a$ and suppose to the contrary that $2d_{GH}(X,Y)<c$, then there
exists $R\in{\cal R}(X,Y)$ such that $\operatorname{dis}R<c$, therefore, for
any $(x_{1},y_{1}),\,(x_{2},y_{2})\in R$ with $|x_{1}x_{2}|\geq b$, it holds
$|y_{1}y_{2}|>|x_{1}x_{2}|-c\geq b-c=a$. Since $Y$ is not $a$-homogeneous,
there exists $y\in Y$ such that there is no $(a,n)$-separated $n$-point subset
$T\subset Y$ with $y\in T$. Since $R$ is a correspondence, there exists $x\in
X$ such that $(x,y)\in R$. Since $X$ is $(b,n)$-homogeneous, there exists
$b$-separated $S=\\{x_{1},\ldots,x_{n}\\}\subset X$ such that $x_{1}=x$. Let
$T=\\{y_{1},\ldots,y_{n}\\}$ with $y_{1}=y$ and $(x_{i},y_{i})\in R$ for any
$i$. Then $T$ is an $a$-separated $n$-point subset $Y$ containing $y$, a
contradiction. ∎

A metric space $X$ is called _round_ if it is $(b,2)$-homogeneous for any
$0<b<\operatorname{diam}X$.

######  Example 3.2.

Let $X\subset\mathbb{R}^{n}$ be a sphere of radius $r>0$ w.r.t. some norm.
Then $X$ is round, but not $(a,2)$-homogeneous for each $a>2r$.

######  Corollary 3.3.

Let $X$ be a round metric space, $0<a<\operatorname{diam}X$, and suppose that
a metric space $Y$ is not $(a,2)$-homogeneous. Then
$2d_{GH}(X,Y)\geq\operatorname{diam}X-a$.

###### Proof.

Theorem 3.1 implies that $2d_{GH}(X,Y)\geq b-a$ for each $b$,
$0<a<b<\operatorname{diam}X$, and the result follows from the arbitrariness of
$b$. ∎

######  Example 3.4.

Let $X\subset\mathbb{R}^{m}$ and $Y\subset\mathbb{R}^{n}$ be spheres of radii
$b$ and $a$ w.r.t. some norms, then

$2d_{GH}(X,Y)\geq 2b-2a.$

##  4 Nonlinearity degree of a metric space

In this section, we introduce the notion of nonlinearity degree and
investigate several its properties. To start with, we denote by
$\operatorname{Lip}_{a}(X)$ the set of all real-valued $a$-Lipschitz functions
defined on a metric space $X$.

######  Definition 4.1.

_The nonlinearity degree of a metric space $X$_ is defined as follows:

$c(X):=\inf_{f\in\operatorname{Lip}_{1}(X)}\sup\bigl{\\{}|xy|-|f(x)-f(y)|:x,y\in
X\bigr{\\}}.$

######  Remark 4.2.

Clearly, $c(X)\geq 0$, and for any $X\subset\mathbb{R}$ we have $c(X)=0$
(indeed, to achieve the value $0$, we can take the inclusion mapping as $f$).
Also, since $c(X)\leq\operatorname{diam}X$, then for each bounded space $X$,
the value $c(X)$ is finite.

If $f\colon X\to\mathbb{R}$ is $a$-Lipschitz, then for any $b\in\mathbb{R}$
the function $f+b$ is $a$-Lipschitz as well, and for any $x,y\in X$ the value
$|xy|-|f(x)-f(y)|$ remains the same. In addition, for a bounded metric space,
each $a$-Lipschitz function is bounded, thus, to calculate the value $c(X)$ in
this case, we can restrict ourselves by those $f\in\operatorname{Lip}_{1}(X)$
that satisfy $\inf f(X)=0$. The set of all such functions we denote by
$\operatorname{Lip}_{1}^{0}(X)$. Hence, we got the following

######  Theorem 4.3.

For any bounded metric space $X$, we have

$c(X):=\inf_{f\in\operatorname{Lip}_{1}^{0}(X)}\sup\bigl{\\{}|xy|-|f(x)-f(y)|:x,y\in
X\bigr{\\}}.$

The nonlinearity degree can be used to estimate the minimal Gromov–Hausdorff
distance between metric space $X$ and non-empty subsets of $\mathbb{R}$.

######  Theorem 4.4.

For any metric space $X$, we have

$\bigl{|}X\,{\cal P}_{0}(\mathbb{R})\bigr{|}=\inf_{Z\in{\cal
P}_{0}(\mathbb{R})}d_{GH}(X,Z)\leq c(X)/2.$

If $X$ is bounded, then there exists $Z\in{\cal H}(\mathbb{R})$ such that
$d_{GH}(X,Z)\leq c(X)/2$.

###### Proof.

If $c(X)=\infty$ then the result holds.

Now, suppose that $c(X)<\infty$. Take $f_{k}\in\operatorname{Lip}_{1}(X)$ such
that

$\sup\Bigl{\\{}|xy|-\bigl{|}f_{k}(x)-f_{k}(y)\bigr{|}:x,y\in X\Bigr{\\}}\leq
c(X)+\frac{1}{k},$

and put $X_{k}=f_{k}(X)$. Let $R_{k}\in{\cal R}(X,X_{k})$ be the graph of
$f_{k}$, namely,

$R_{k}=\Bigl{\\{}\bigl{(}x,f_{k}(x)\bigr{)}:x\in X\Bigr{\\}}.$

Since $f_{k}$ is $1$-Lipschitz, then for any
$\bigl{(}x,f_{k}(x)\bigr{)},\,\bigl{(}y,f_{k}(y)\bigr{)}\in R_{k}$ we have
$|f_{k}(x)-f_{k}(y)|\leq|xy|$, therefore,

$\operatorname{dis}R_{k}=\sup_{x,y\in
X}\Bigl{\\{}|xy|-\bigl{|}f_{k}(x)-f_{k}(y)\bigr{|}\Bigr{\\}}\leq
c(X)+\frac{1}{k}.$

Due to the arbitrariness of $k\in{\mathbb{N}}$, we get

$\inf_{Z\in{\cal P}_{0}(\mathbb{R})}d_{GH}(X,Z)\leq\inf_{k}d_{GH}(X,X_{k})\leq
c(X)/2.$

Now we prove the existence of $Z$ for bounded $X$. By Theorem 4.3, we can take
$f_{k}$ from $\operatorname{Lip}_{1}^{0}(X)$ instead of from
$\operatorname{Lip}_{1}(X)$. Let $\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu_{k}$ be the closure of
$X_{k}=f_{k}(X)$. Since $d_{GH}(X_{k},\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu_{k})=0$, then we have
$\inf_{k}d_{GH}(X,\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern
1.5mu_{k})\leq c(X)/2$ by the triangle inequality. The boundedness of $X$
implies $d=\operatorname{diam}X<\infty$, and by Theorem 2.2, we have

$\operatorname{diam}\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern
1.5mu_{k}=\operatorname{diam}X_{k}\leq\operatorname{diam}X+2d_{GH}(X,\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu_{k})\leq d+c(X)<\infty.$

Since $X_{k}\geq 0$ and $0\in\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu_{k}$ for each
$k\in{\mathbb{N}}$, then $X_{k}\subset\bigl{[}0,d+c(X)\bigr{]}=:I$ for all
$k\in{\mathbb{N}}$, and, by compactness of ${\cal H}(I)$ due to Theorem 2.1,
there exists a convergent subsequence $\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu_{k_{i}}\in{\cal H}(I)$.
Put $Z=\lim_{i\to\infty}\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern
1.5mu_{k_{i}}$. Since $d_{GH}(\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu_{k_{i}},Z)\leq
d_{H}(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern
1.5mu_{k_{i}},Z)$, we obtain $\mkern
1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern
1.5mu_{k_{i}}\xrightarrow{\operatorname{GH}}Z$, therefore $d_{GH}(X,Z)\leq
c(X)/2$. ∎

######  Example 4.5.

Now we show that the inequality in Theorem 4.4 cannot be changed to equality,
even in the case of compact $X$. Let $X$ be the standard unit circle in the
Euclidean plane, endowed with the intrinsic metric. We shall find a compact
$Z\subset\mathbb{R}$ such that $d_{GH}(X,Z)<c(X)/2$. Let us take as $Z$ the
segment $[0,\frac{2}{3}\pi]$. Below, in Lemma 6.1, we shall prove that
$d_{GH}(X,Z)=\frac{1}{3}\pi$. On the other hand, given any
$f\in\operatorname{Lip}_{1}(S^{1},\mathbb{R})$, the function $g(x)=f(x)-f(-x)$
has at least one zero point $x_{0}$. Then

$\sup\Bigl{\\{}|xy|-\bigl{|}f(x)-f(y)\bigr{|}:x,y\in
X\Bigr{\\}}\geq|x_{0}(-x_{0})|-\bigl{|}f(x_{0})-f(-x_{0})\bigr{|}=\pi.$

Since the choice of $f$ is arbitrary, we know that $c(X)\geq\pi>2d_{GH}(X,Z)$.

The next result characterizes all compact metric spaces with $c(X)=0$.

######  Corollary 4.6.

Each compact metric space $X$ with $c(X)=0$ is isometric to a compact subset
of $\mathbb{R}$.

###### Proof.

By Theorem 4.4, there exists a compact subset $Z\subset\mathbb{R}$ with
$d_{GH}(X,Z)=0$, thus $X$ and $Z$ are isometric. ∎

######  Proposition 4.7.

For a compact metric space $X$, there exists a function
$f\in\operatorname{Lip}_{1}(X)$ such that

$\sup\bigl{\\{}|xy|-|f(x)-f(y)|:x,y\in X\bigr{\\}}=c(X).$

###### Proof.

By Theorem 4.3, we can choose a sequence
$f_{n}\in\operatorname{Lip}_{1}^{0}(X)$ such that for each $n\in{\mathbb{N}}$
it holds

$\sup\bigl{\\{}|xy|-|f_{n}(x)-f_{n}(y)|:x,y\in X\bigr{\\}}<c(X)+\frac{1}{n}.$

The next lemma is a direct consequence of Arzelà-Ascoli theorem [1].

######  Lemma 4.8.

Each uniformly bounded sequence of $L$-Lipschitz functions on a compact metric
space contains a subsequence uniformly convergent to an $L$-Lipschitz
function.

Since any collection of functions from $\operatorname{Lip}_{1}(X)$ is
uniformly bounded, we can apply Lemma 4.8, thus, w.l.o.g. we can suppose that
the sequence $f_{n}$ itself uniformly converges to some
$f\in\operatorname{Lip}_{1}(X)$. Take arbitrary $\varepsilon>0$ and choose
$N\in{\mathbb{N}}$ such that for any $n\geq N$ and any $x\in X$ we have

$\bigl{|}f(x)-f_{n}(x)\bigr{|}<\varepsilon/3\ \ \text{and}\ \
1/n<\varepsilon/3,$

hence for any $x,y\in X$ and any $n\geq N$ it holds

$\sup\bigl{\\{}|xy|-|f(x)-f(y)|:x,y\in
X\bigr{\\}}\leq\sup\bigl{\\{}|xy|-|f_{n}(x)-f_{n}(y)|:x,y\in
X\bigr{\\}}+2\varepsilon/3<c(X)+\varepsilon.$

Due to the arbitrariness of $\varepsilon$, we have
$\sup\bigl{\\{}|xy|-|f(x)-f(y)|:x,y\in X\bigr{\\}}\leq c(X)$, that concludes
the proof. ∎

######  Remark 4.9.

Proposition 4.7 gives another proof of Theorem 4.4 when $X$ is compact.

######  Theorem 4.10.

Suppose we have a connected compact metric space $X$ equipped with a
homeomorphism $\alpha$ of order $2$ such that
$\bigl{|}\alpha(x)\,x\bigr{|}=\operatorname{diam}X$ for each $x\in X$, and a
compact metric space $Y$ such that $c(Y)<\operatorname{diam}X$. Then we have
$2d_{GH}(X,Y)\geq m$, where
$m=\frac{2}{3}\bigl{(}\operatorname{diam}X-c(Y)\bigr{)}$.

###### Proof.

Suppose to the contrary that $2d_{GH}(X,Y)<m$, thus there is a closed
correspondence $R\in{\cal R}_{c}(X,Y)$ such that $\operatorname{dis}R<m$.

By Proposition 4.7, there exists $v\in\operatorname{Lip}_{1}(Y)$ such that

$\sup\bigl{\\{}|y_{1}y_{2}|-|v(y_{1})-v(y_{2})|:y_{1},y_{2}\in
Y\bigr{\\}}=c(Y).$

In the following text, we will abbreviate $c(Y)$ as $c$ and $\alpha(x)$ as
$\alpha_{x}$.

Since $X\times Y$ is compact and $R$ is its closed subset, then $R$ is
compact, together with $R(x)=R\cap\bigl{(}\\{x\\}\times Y\bigr{)}$ and
$R(\alpha_{x})=R\cap\bigl{(}\\{\alpha_{x}\\}\times Y\bigr{)}$. Since the
function $v$ is continuous, then the restrictions of $v$ to $R(x)$ and
$R(\alpha_{x})$ are bounded and attain their maximal and minimal values, thus
the following function $f\colon X\to\mathbb{R}$ is correctly defined:

$f(x)=\max v\bigl{(}R(x)\bigr{)}-\min v\bigl{(}R(\alpha_{x})\bigr{)}.$

######  Lemma 4.11.

For any sequence $x_{i}\in X$ such that $x_{i}\to{\tilde{x}}\in X$ it holds

$\limsup_{i\to+\infty}f(x_{i})\leq f({\tilde{x}}).$

###### Proof.

Let us put $R^{v}:=\bigl{\\{}\bigl{(}x,v(y)\bigr{)}:(x,y)\in
R\bigr{\\}}\subset X\times\mathbb{R}$, then we can rewrite $f$ as follows:

$f(x)=\max R^{v}(x)-\min R^{v}(\alpha_{x}).$

Since $R$ is compact, then $R^{v}$ is compact as the image of $R$ under the
continuous mapping $\operatorname{id}\times v$. Thus $R^{v}$ is closed and
bounded.

We put $z_{i}=\max R^{v}(x_{i})$, $w_{i}=\min R^{v}(\alpha_{x_{i}})$, then
$f(x_{i})=z_{i}-w_{i}$, both the sequences $z_{i}$, $w_{i}$ are bounded, and
both the $z:=\limsup_{i\to+\infty}z_{i}$, $w:=\liminf_{i\to+\infty}w_{i}$ are
finite. Now we choose subsequences $z_{k_{i}}$ and $w_{l_{i}}$ such that

$z=\lim_{i\to+\infty}z_{k_{i}}\ \ \text{and}\ \
w=\lim_{i\to+\infty}w_{l_{i}}.$

Since $R^{v}$ is closed,
$(x_{k_{i}},z_{k_{i}}),(\alpha_{x_{l_{i}}},w_{l_{i}})\in R^{v}$, and $z$, $w$
are finite, then

$({\tilde{x}},z)=\lim_{i\to+\infty}(x_{k_{i}},z_{k_{i}})\in R^{v}\ \
\text{and}\ \
(\alpha_{\tilde{x}},w)=\lim_{i\to+\infty}(\alpha_{x_{l_{i}}},w_{l_{i}})\in
R^{v},$

therefore, $z\in R^{v}({\tilde{x}})$ and $w\in R^{v}(\alpha_{\tilde{x}})$.
Thus, we get

$\max R^{v}({\tilde{x}})\geq z=\limsup_{i\to+\infty}z_{i}\ \ \text{and}\ \
\min R^{v}(\alpha_{\tilde{x}})\leq w=\liminf_{i\to+\infty}w_{i}.$

In account,

$f({\tilde{x}})\geq
z-w=\limsup_{i\to+\infty}z_{i}-\liminf_{i\to+\infty}w_{i}=\\\
=\limsup_{i\to+\infty}z_{i}+\limsup_{i\to+\infty}(-w_{i})\geq\limsup_{i\to+\infty}(z_{i}-w_{i})=\limsup_{i\to+\infty}f(x_{i}),$

and the proof is completed. ∎

Since $\operatorname{dis}R<m$ and $|x\,\alpha_{x}|=\operatorname{diam}X$, then
for any $y_{1}\in R(x)$ and $y_{2}\in R(\alpha_{x})$ we have
$|y_{1}y_{2}|>\operatorname{diam}X-m$, thus

$\bigl{|}v(y_{1})-v(y_{2})\bigr{|}\geq|y_{1}y_{2}|-c>\operatorname{diam}X-m-c=:d.$

So, either $v(y_{1})-v(y_{2})>d$ or $v(y_{2})-v(y_{1})>d$. Since the values
$\max v\bigl{(}R(x)\bigr{)}$ and $\min v\bigl{(}R(\alpha_{x})\bigr{)}$ are
attained, then for each $x\in X$ it holds either $f(x)>d$ or $-f(x)>d$, and
the latter is equivalent to $f(x)<-d$.

We put $A:=\bigl{\\{}x\in X:f(x)>d\bigr{\\}}$ and $B:=\bigl{\\{}x\in
X:f(x)<-d\bigr{\\}}$. Since $c\leq\operatorname{diam}X$ and
$m=\frac{2}{3}(\operatorname{diam}X-c)$, we get

$d=\operatorname{diam}X-m-c=\frac{1}{3}(\operatorname{diam}X-c)\geq
0\geq-\frac{1}{3}(\operatorname{diam}X-c)=c+m-\operatorname{diam}X=-d,$

hence $A$ is the complement of $B$. Moreover, since $f(x)\neq\pm d$, we have

$A=\bigl{\\{}x\in X:f(x)\geq d\bigr{\\}}\ \ \text{and}\ \ B=\bigl{\\{}x\in
X:f(x)\leq-d\bigr{\\}}.$

Now we show that $A$ is closed and, thus, $B$ is open. Indeed, consider an
arbitrary sequence $x_{i}\in A$ converging to some $x\in X$. Then, by Lemma
4.11, we have $f(x)\geq\limsup\limits_{x_{i}\to x}f(x_{i})\geq d$, thus $x\in
A$ and, therefore, $A$ is closed.

######  Lemma 4.12.

We have $\alpha(B)\subset A$.

###### Proof.

Take arbitrary $x\in B$, then, by definition, $\max v\bigl{(}R(x)\bigr{)}-\min
v\bigl{(}R(\alpha_{x})\bigr{)}\leq-d$, thus

$-f(x)=\min v\bigl{(}R(x)\bigr{)}-\max v\bigl{(}R(\alpha_{x})\bigr{)}\leq\max
v\bigl{(}R(x)\bigr{)}-\min v\bigl{(}R(\alpha_{x})\bigr{)}\leq-d,$

so $f(x)\geq d$ and, hence, $\alpha_{x}\in A$. ∎

######  Lemma 4.13.

It holds that $\alpha(A)\subset B$.

###### Proof.

If not, there exists $x\in A$ such that $\alpha_{x}\in A$. Without loss of
generality, suppose that $\max v\bigl{(}R(\alpha_{x})\bigr{)}\geq\max
v\bigl{(}R(x)\bigr{)}$. Since $x\in A$, we have, by definition, $\max
v\bigl{(}R(x)\bigr{)}\geq d+\min v\bigl{(}R(\alpha_{x})\bigr{)}$.

Recall that for any $y\in R(x)$ and $y^{\prime}\in R(\alpha_{x})$ it holds
$\bigl{|}v(y)-v(y^{\prime})\bigr{|}>d$, thus we get

$\max v\bigl{(}R(\alpha_{x})\bigr{)}\geq d+\max v\bigl{(}R(x)\bigr{)}\geq
2d+\min v\bigl{(}R(\alpha_{x})\bigr{)}.$

As we mentioned above, $R(\alpha_{x})$ is compact and $v$ is continuous, hence
there exist $y_{1},y_{2}\in R(\alpha_{x})$ such that $v(y_{1})=\max
v\bigl{(}R(\alpha_{x})\bigr{)}$ and $v(y_{2})=\min
v\bigl{(}R(\alpha_{x})\bigr{)}$. Since $v$ is $1$-Lipschitz, we have
$|y_{1}y_{2}|\geq v(y_{1})-v(y_{2})$, therefore,

$m>\operatorname{dis}R\geq\operatorname{diam}R(\alpha_{x})\geq|y_{1}y_{2}|\geq
v(y_{1})-v(y_{2})=\\\ =\max v\bigl{(}R(\alpha_{x})\bigr{)}-\min
v\bigl{(}R(\alpha_{x})\bigr{)}\geq
2d=2\bigl{(}\operatorname{diam}X-m-c\bigr{)}.$

Thus $m>\frac{2}{3}(\operatorname{diam}X-c)$ which achieves contradiction and,
hence, we have $\alpha(A)\subset B$. ∎

Since $X=A\sqcup B$ and $\alpha\colon X\to X$ is a homeomorphism, Lemmas 4.12
and 4.13 imply that the restriction of $\alpha$ to $A$ is a homeomorphism onto
$B$, hence both $A$ and $B$ are proper closed-open, therefore, $X$ is not
connected, a contradiction which completes the proof of the theorem. ∎

##  5 Geometric calculation of distortion

Consider the special case we are interesting in, namely, let
$X=I_{\lambda}\subset\mathbb{R}$ be a segment of the length $\lambda$, and
$Y=S^{1}=\bigl{\\{}(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1\bigr{\\}}$ be the
standard circle endowed with the intrinsic metric. Since we are interesting in
the Gromov–Hausdorff distance between $I_{\lambda}$ and $S^{1}$, it does not
matter where we place $I_{\lambda}$ on the line $\mathbb{R}$. We will use two
possible placements: $[0,\lambda]$ and $[-\lambda/2,\,\lambda/2]$. In this
section we will use the latter one.

Take a correspondence $R\in{\cal R}(I_{\lambda},S^{1})$. Our goal is to
calculate geometrically the distortion of $R$. To do that, we parameterize the
circle $S^{1}$ by the standard angular coordinate $-\pi\leq\varphi\leq\pi$,
thus the interior distance between points $\varphi_{1},\varphi_{2}\in S^{1}$
equals
$\min\bigl{\\{}|\varphi_{1}-\varphi_{2}|,2\pi-|\varphi_{1}-\varphi_{2}|\bigr{\\}}$.
Let $-\lambda/2\leq t\leq\lambda/2$ be the standard coordinate on the segment
$I_{\lambda}$. Now we represent the correspondence $R$ as a subset of
rectangle $Q:=[-\lambda/2,\,\lambda/2]\times[-\pi,\pi]\subset\mathbb{R}^{2}$.

To calculate the distortion, we need to maximize the function

$f\bigl{(}(t_{0},\varphi_{0}),(t,\varphi)\bigr{)}=\begin{cases}\bigl{|}\,|t-t_{0}|-|\varphi-\varphi_{0}|\,\bigr{|}&\text{if
$|\varphi-\varphi_{0}|\leq\pi$},\\\
\bigl{|}\,|t-t_{0}|-2\pi+|\varphi-\varphi_{0}|\,\bigr{|}&\text{if
$|\varphi-\varphi_{0}|\geq\pi$}\\\ \end{cases}$

over all $(t_{0},\varphi_{0}),\,(t,\varphi)\in R$. In other words, we need to
find the least possible $a$ such that
$f\bigl{(}(t_{0},\varphi_{0}),(t,\varphi)\bigr{)}\leq a$ for all
$(t_{0},\varphi_{0}),\,(t,\varphi)\in R$.

Take arbitrary $a>0$, and denote by $D_{a}(t_{0},\varphi_{0})$ the subset of
the plane $\mathbb{R}^{2}$ consisting of all $(t,\varphi)\in\mathbb{R}^{2}$
such that $f\bigl{(}(t_{0},\varphi_{0}),(t,\varphi)\bigr{)}\leq a$. Clearly
that $D_{a}(t_{0},\varphi_{0})$ can be obtained from $D_{a}:=D_{a}(0,0)$ by
shifting by $(t_{0},\varphi_{0})$, i.e.,
$D_{a}(t_{0},\varphi_{0})=(t_{0},\varphi_{0})+D_{a}$. Evident observation
shows that $D_{a}$ looks like the blue domain in Figure 1, left-hand side.

Figure 1: Geometric calculation of distortion.

It is bounded by the segments parallel to bisectors of quadrants, and its
sizes are given by the labels in the figure. Thus, we get the following
result.

######  Theorem 5.1.

Under the notations introduced above, $\operatorname{dis}R\leq a$ if and only
if for any $(t_{0},\varphi_{0})\in R$ it holds $R\subset
D_{a}(t_{0},\varphi_{0})$. In particular, the distortion of $R\subset Q$
equals the least possible $a$ satisfying the previous condition.

######  Remark 5.2.

If a graph $P$ satisfies the above property with respect to $a$, then any
subgraph of it also satisfies the above property.

Let us note that the domain $Q\cap D_{a}(t_{0},\varphi_{0})$ is depicted on
the right-hand side of Figure 1 in green, as the intersection of blue and
yellow domains.

##  6 Calculating GH-distance between circle and segment

Now we apply the technique described above to calculate the Gromov–Hausdorff
distance between the standard circle
$S^{1}=\bigl{\\{}(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1\bigr{\\}}$ with
intrinsic metric and the segment $I_{\lambda}=[0,\lambda]\subset\mathbb{R}$.
We start from the following

######  Lemma 6.1.

For each $0\leq\lambda<\pi$ we have

$d_{GH}(I_{\lambda},S^{1})\geq\frac{\pi}{2}-\frac{\lambda}{4}.$

###### Proof.

Notice that the metric space $S^{1}$ is round, $\operatorname{diam}S^{1}=\pi$,
and for any $\lambda^{\prime}>\lambda$ the segment $I_{\lambda}$ is not
$(\lambda^{\prime}/2,2)$-homogeneous. Thus, for any
$\pi>\lambda^{\prime}>\lambda$ we can use Corollary 3.3 and get

$d_{GH}(I_{\lambda},S^{1})\geq\frac{\pi}{2}-\frac{\lambda^{\prime}}{4}.$

The result follows from arbitrariness of $\lambda^{\prime}$. ∎

######  Proposition 6.2.

For each $0\leq\lambda\leq 2\pi/3$ we have

$d_{GH}(I_{\lambda},S^{1})=\frac{\pi}{2}-\frac{\lambda}{4}.$

###### Proof.

By Lemma 6.1, we get the necessary lower bound for the
$d_{GH}(I_{\lambda},S^{1})$. It remains to get the same upper bound. To do
that, we will construct a special correspondence $R_{\lambda}\in{\cal
R}(I_{\lambda},S^{1})$ with $\operatorname{dis}R_{\lambda}=\pi-\lambda/2$,
which completes the proof.

For $\lambda=0$ we put $R_{\lambda}=I_{\lambda}\times S^{1}$. Since the space
$I_{\lambda}$ consists of a single point, we get

$\operatorname{dis}R_{\lambda}=\operatorname{diam}S^{1}=\pi=\pi-\lambda/2.$

Now, suppose that $\lambda>0$. Identify $\mathbb{R}^{2}$ with ${\mathbb{C}}$,
and let $R_{\lambda}$ be the graph of the mapping $\gamma_{\lambda}\colon
I_{\lambda}\to S^{1}$ defined as $\gamma\colon t\mapsto e^{2\pi
i\,t/\lambda}$. Notice that the image of the mapping $t\mapsto 2\pi
t/\lambda$, $t\in I_{\lambda}$, equals $\\{e^{it}:t\in[0,2\pi]\\}$, thus
$R_{\lambda}$ is a correspondence, and for any $t,t^{\prime}\in I_{\lambda}$,
we have

$\bigl{|}\gamma(t)\gamma(t^{\prime})\bigr{|}=\min\Bigl{\\{}\frac{2\pi}{\lambda}|t-t^{\prime}|,\,2\pi-\frac{2\pi}{\lambda}|t-t^{\prime}|\Bigr{\\}}.$

Since the value $\bigl{|}\gamma(t)\gamma(t^{\prime})\bigr{|}$ depends only on
$s:=|t-t^{\prime}|$, we put
$f(s)=\bigl{|}\gamma(t)\gamma(t^{\prime})\bigr{|}-|t-t^{\prime}|$, then

$\operatorname{dis}R_{\lambda}=\max_{0\leq s\leq\lambda}\bigl{|}f(s)\bigr{|}.$

Since the function

$f(s)=\begin{cases}\frac{2\pi}{\lambda}s-s&\text{for
$0\leq\frac{2\pi}{\lambda}s\leq\pi$},\\\
2\pi-\frac{2\pi}{\lambda}s-s&\text{for $\pi\leq\frac{2\pi}{\lambda}s\leq
2\pi$},\end{cases}$

is linear on $[0,\lambda/2]$ and on $[\lambda/2,\lambda]$, we get

$\max_{0\leq
s\leq\lambda}\bigl{|}f(s)\bigr{|}=\max\Bigl{\\{}\bigl{|}f(0)\bigr{|},\,\bigl{|}f(\lambda/2)\bigr{|},\,\bigl{|}f(\lambda)\bigr{|}\Bigr{\\}}=\max\bigl{\\{}0,\,|\pi-\lambda/2|,\,\lambda\bigr{\\}}.$

Since $0\leq\lambda\leq 2\pi/3$, then
$|\pi-\lambda/2|=\pi-\lambda/2\geq\lambda>0$, therefore
$\operatorname{dis}R_{\lambda}=\pi-\lambda/2$. ∎

######  Proposition 6.3.

If $\frac{2}{3}\pi\leq\lambda\leq\frac{7}{6}\pi$, then we have
$d_{GH}(I_{\lambda},S^{1})=\frac{\pi}{3}$.

###### Proof.

Applying Lemma 6.5, it suffices to construct a correspondence
$R_{\lambda}\in{\cal R}(I_{\lambda},S^{1})$ with
$\operatorname{dis}R_{\lambda}=2\pi/3$, which completes the proof.

Again, we identify $\mathbb{R}^{2}$ with ${\mathbb{C}}$, and let $R_{\lambda}$
be the graph of the mapping $\gamma_{\lambda}\colon I_{\lambda}\to S^{1}$
defined as $\gamma\colon t\mapsto e^{3\pi i\,t}$. Notice that the image of the
mapping $t\mapsto 3\pi t$, $t\in I_{\lambda}$, contains
$\\{e^{it}:t\in[0,2\pi]\\}$, thus $R_{\lambda}$ is a correspondence, and for
any $t,t^{\prime}\in I_{\lambda}$, we have

$\bigl{|}\gamma(t)\gamma(t^{\prime})\bigr{|}=\min\Bigl{\\{}3|t-t^{\prime}|,\,2\pi-3|t-t^{\prime}|,\,3|t-t^{\prime}|-2\pi,\,4\pi-3|t-t^{\prime}|\Bigr{\\}}.$

Since the value $\bigl{|}\gamma(t)\gamma(t^{\prime})\bigr{|}$ depends only on
$s:=|t-t^{\prime}|$, we put
$f(s)=\bigl{|}\gamma(t)\gamma(t^{\prime})\bigr{|}-|t-t^{\prime}|$, then

$\operatorname{dis}R_{\lambda}=\max_{0\leq s\leq\lambda}\bigl{|}f(s)\bigr{|}.$

Since the function

$f(s)=\begin{cases}2s&\text{for $0\leq s\leq\frac{\pi}{3}$},\\\
2\pi-4s&\text{for $\frac{\pi}{3}\leq s\leq\frac{2}{3}\pi$},\\\
2s-2\pi&\text{for $\frac{2}{3}\pi\leq s\leq\pi$},\\\ 4\pi-4s&\text{for
$\pi\leq s\leq\frac{7}{6}\pi$},\\\ \end{cases}$

is linear on $[0,\pi/3]$, $[\pi/3,2\pi/3]$, $[2\pi/3,\pi]$, $[\pi,7\pi/6]$, we
get

$\max_{0\leq s\leq
7\pi/6}\bigl{|}f(s)\bigr{|}=\max\Bigl{\\{}\bigl{|}f(0)\bigr{|},\,\bigl{|}f(\pi/3)\bigr{|},\,\bigl{|}f(2\pi/3)\bigr{|},\,\bigl{|}f(\pi)\bigr{|},\,\bigl{|}f(7\pi/6)\bigr{|}\Bigr{\\}}=\max\\{0,\,2\pi/3\\}=2\pi/3,$

therefore, $\operatorname{dis}R_{\lambda}=2\pi/3$. ∎

######  Proposition 6.4.

If $5\pi/3\leq\lambda\leq 2\pi$, then we have
$d_{GH}(I_{\lambda},S^{1})=\frac{\lambda-\pi}{2}$.

###### Proof.

By Theorem 2.2, we have

$d_{GH}(I_{\lambda},S^{1})\geq\frac{\operatorname{diam}I_{\lambda}-\operatorname{diam}S^{1}}{2}=\frac{\lambda-\pi}{2}.$

Thus, it suffices to construct a correspondence $R\in{\cal
R}(I_{\lambda},S^{1})$ with $\operatorname{dis}R\leq\lambda-\pi=:a$. We will
use notations from Section 5 and apply Theorem 5.1.

Consider the correspondence depicted in Figure 2.

Figure 2: A correspondence $R$ to prove Proposition 6.4.

We have to show that for any $(t_{0},\varphi_{0})\in
R\subset[-\lambda/2,\lambda/2]\times[-\pi,\pi]$ the domain
$D_{a}(t_{0},\varphi_{0})$ covers $R$. In Figure 3 we show the corresponding
domain $D_{a}$ endowed with its special sizes; also we write down the
corresponding sizes of segments forming $R$.

Figure 3: A correspondence $R$ to prove Proposition 6.4.

Due to symmetry reasons, it suffices to verify the cases of
$(t_{0},\varphi_{0})$ lying on blue, green or cyan segments of $R$. We show
that in these cases $R$ is covered by orange cross and rose rectangle from the
left-hand side of Figure 3.

To simplify verification of the formulas below, let us explicitly write down
the coordinates of the points $A$, $B$, $C$, and $D$ from Figure 2 and 3
(points with the same names supplied by primes are symmetric w.r.t. the
origin):

$A=\Bigl{(}\frac{\lambda}{2}-\frac{\pi}{2},\pi\Bigr{)},\ \
B=\Bigl{(}\frac{\pi}{2},\frac{\pi}{2}\Bigr{)},\ \
C=\Bigl{(}\frac{\lambda}{4}+\frac{\pi}{4},\frac{\lambda}{4}+\frac{\pi}{4}\Bigr{)},\
\ D=\Bigl{(}\frac{\lambda}{2},\frac{\pi}{2}\Bigr{)}.$

It is easy to calculate that

$|AC|=\frac{3\pi-\lambda}{2\sqrt{2}}\leq\frac{\lambda-\pi}{\sqrt{2}},\ \
|CD|=\frac{\lambda-\pi}{2\sqrt{2}}<\frac{\lambda-\pi}{\sqrt{2}},\ \
|B^{\prime}C|=|BC^{\prime}|=\frac{\lambda+3\pi}{2\sqrt{2}}<\frac{\lambda+\pi}{\sqrt{2}}.$

Since the length and width of each of the two rectangles forming the orange
cross are $\sqrt{2}(\lambda+\pi)$ and $\sqrt{2}(\lambda-\pi)$, respectively,
then, once the point $(t_{0},\phi_{0})$ lies on the blue segment of the
relation $R$, the orange cross covers $R$.

Now, consider the case when $(t_{0},\phi_{0})$ lies on the green segment.
Still, the whole correspondence $R$ is covered by the orange cross. To see
that, it suffices to notice the following evident formulas:

$\displaystyle|CC^{\prime}|=\frac{\lambda+\pi}{\sqrt{2}},\ \
|CD|+|C^{\prime}D^{\prime}|=\frac{\lambda-\pi}{\sqrt{2}},\ \
|A^{\prime}C^{\prime}|-|CD|=\frac{2\pi-\lambda}{\sqrt{2}}<\frac{\lambda-\pi}{\sqrt{2}},$
$\displaystyle|BC|=\frac{\lambda-\pi}{2\sqrt{2}}<\frac{\lambda-\pi}{\sqrt{2}},\
\ |AD|=\frac{\pi}{\sqrt{2}}<\frac{\lambda+\pi}{\sqrt{2}}.$

At last, let $(t_{0},\phi_{0})$ lie on the cyan segment. Then, similarly, the
whole correspondence $R$ except the magenta segment is covered by the orange
cross. Let us show that the magenta segment is covered by the both orange
cross and the rose rectangle. This can be extracted from the following
inequalities:

$|AC|+|A^{\prime}C^{\prime}|=\frac{3\pi-\lambda}{\sqrt{2}}<\frac{\lambda+\pi}{\sqrt{2}},\
\ |BB^{\prime}|=\sqrt{2}\pi<\sqrt{2}\pi-\frac{\lambda-\pi}{\sqrt{2}},\ \
|CC^{\prime}|=\frac{\lambda+\pi}{\sqrt{2}}.$

The proof is completed. ∎

######  Lemma 6.5.

For each $\pi\leq\lambda\leq\frac{5\pi}{3}$ we have
$d_{GH}(I_{\lambda},S^{1})\geq\frac{\pi}{3}$.

###### Proof.

Since $I_{\lambda}$ is a subset of $\mathbb{R}$, we have $c(I_{\lambda})=0$ by
Remark 4.2.

For $x\in S^{1}$, denote the antipodal point of $x$ by $\alpha(x)$, then
$\alpha\colon S^{1}\to S^{1}$ is a homeomorphism of order $2$ and
$\bigl{|}x\alpha(x)\bigr{|}=\operatorname{diam}S^{1}$. Since $S^{1}$ is
connected and compact, and $I_{\lambda}$ is compact, and
$c(I_{\lambda})=0<\pi=\operatorname{diam}S^{1}$, we can apply Theorem 4.10 to
the pair $(S^{1},I_{\lambda})$ and, thus, we get

$2d_{GH}(I_{\lambda},S^{1})\geq\frac{2\pi}{3}.$

∎

######  Proposition 6.6.

If $\pi\leq\lambda\leq\frac{5\pi}{3}$, then we have
$d_{GH}(I_{\lambda},S^{1})=\frac{\pi}{3}$.

###### Proof.

Lemma 6.5 implies that it suffices to prove the inequality
$d_{GH}(I_{\lambda},S^{1})\leq\frac{\pi}{3}$. To do that, we construct a
similar correspondence $R^{\prime}_{\lambda}\in{\cal R}(I_{\lambda},S^{1})$ as
in the proof of Proposition 6.4 as follows. Compared with the previous
correspondence, the endpoints of the cyan (respectively, magenta) segment are
$(\frac{\pi}{3},\pi)$ and $(\frac{\pi}{2},\frac{\pi}{2})$ (respectively,
$(-\frac{\pi}{3},-\pi)$ and $(-\frac{\pi}{2},-\frac{\pi}{2})$).

Let us put $P=R^{\prime}_{\frac{5\pi}{3}}$. Note that $P$ is exactly the same
construction for $\lambda=\frac{5\pi}{3}$ as in Proposition 6.4. Thus, $P$
satisfies the condition of Theorem 5.1 with respect to $a=\frac{2\pi}{3}$.

Note that for $\lambda\in[\pi,\frac{5\pi}{3}]$, the correspondence
$R^{\prime}_{\lambda}$ is a subgraph of $P$ since $R^{\prime}_{\lambda}$ only
has shorter green and red segments compared with $P$. Since $P$ satisfies
Theorem 5.1 for $a=\frac{2\pi}{3}$, we conclude that $R^{\prime}_{\lambda}$
also satisfies Theorem 5.1 for $a=\frac{2\pi}{3}$. The proof is completed. ∎

######  Proposition 6.7.

If $\lambda\geq 2\pi$, then we have
$d_{GH}(I_{\lambda},S^{1})=\frac{\lambda-\pi}{2}$.

###### Proof.

Again, by Theorem 2.2, we have

$d_{GH}(I_{\lambda},S^{1})\geq\frac{\operatorname{diam}I_{\lambda}-\operatorname{diam}S^{1}}{2}=\frac{\lambda-\pi}{2}.$

Let $Y=\\{e^{it}:0<t<2\\}\subset{\mathbb{C}}$ be the standard unit circle, and

$X=\Bigl{[}-1-\frac{\lambda-\pi}{2},-1\Bigr{]}\cup\
Y\cup\Bigl{[}1,1+\frac{\lambda-\pi}{2}\Bigr{]}\subset{\mathbb{C}}$

is equipped with intrinsic metric. Then $S^{1}$ is isometric to $Y$, and
$I_{\lambda}$ is isometric to

$Z=\Bigl{[}-1-\frac{\lambda-\pi}{2},-1\Bigr{]}\cup\ \\{e^{it}:\pi\leq t\leq
2\pi\\}\cup\Bigl{[}1,1+\frac{\lambda-\pi}{2}\Bigr{]}\subset X.$

Clearly that $d_{H}(Y,Z)=\frac{\lambda-\pi}{2}$, thus

$d_{GH}(I_{\lambda},S^{1})\leq\frac{\lambda-\pi}{2}.$

∎

Gathering together Propositions 6.2–6.7, we get

######  Theorem 6.8.

Let $S^{1}=\bigl{\\{}(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1\bigr{\\}}$ be the
standard circle with intrinsic metric, and
$I_{\lambda}=[0,\lambda]\subset\mathbb{R}$ the segment of the length
$\lambda$. Then

$d_{GH}(I_{\lambda},S^{1})=\begin{cases}\dfrac{\pi}{2}-\dfrac{\lambda}{4}&for\
\ 0\leq\lambda\leq\dfrac{2}{3}\pi,\\\\[10.0pt] \dfrac{\pi}{3}&for\ \
\dfrac{2}{3}\pi\leq\lambda\leq\dfrac{5}{3}\pi,\\\\[10.0pt]
\dfrac{\lambda-\pi}{2}&for\ \ \lambda\geq\frac{5}{3}\pi,\end{cases}$

see Figure 4.

Figure 4: Gromov–Hausdorff distance between circle and segment.

## References

  * [1] D.Burago, Yu.Burago, S.Ivanov, _A Course in Metric Geometry_. Graduate Studies in Mathematics, vol.33, A.M.S., Providence, RI, 2001.