Collapsing Wormholes Sustained by Dustlike Matter

Pavel E. Kashargin

Sergey V. Sushkov

Institute of Physics, Kazan Federal University, Kremliovskaya str. 16a, Kazan 420008, Russia

It is well known that static wormhole configurations in general relativity (GR) are possible only if matter threading the wormhole throat is “exotic”—i.e., violates a number of energy conditions. For this reason, it is impossible to construct static wormholes supported only by dust-like matter which satisfies all usual energy conditions. However, this is not the case for non-static configurations.
In 1934, Tolman found a general solution describing the evolution of a spherical dust shell in GR. In this particular case, Tolman's solution describes the collapsing dust ball; the inner space–time structure of the ball corresponds to the Friedmann universe filled by a dust.
In the present work we use the general Tolman's solution in order to construct a dynamic spherically symmetric wormhole solution in GR with dust-like matter.
The solution constructed represents the collapsing dust ball with the inner wormhole space–time structure. It is worth noting that, with the dust-like matter, the ball is made of satisfies the usual energy conditions and cannot prevent the collapse. We discuss in detail the properties of the collapsing dust wormhole.

§ INTRODUCTION

Wormholes are solutions of general relativity which possess a throat.
We define a wormhole throat as a closed 2-dimensional hypersurface of minimal area.
Such solutions were first mentioned in the works [1, 2, 3, 4], but the greatest interest in these objects arose after the work of Michael Morris and Kip Thorne in 1988 [5].
As noted by the authors, one of the “exotic” properties of these objects is that the presence of a throat leads to a violation of the energy conditions for the energy-momentum tensor of matter.
An overview of research on wormholes can be found, for example, in [6, 7].
Wormholes have been considered in various theories of gravity, including the Brans-Dicke theory of gravity [8], in the Einstein-Born-Infeld theory of gravity [11], in the Einstein-Gauss-Bonnet theory [12], in $f(R)$ gravity [14], in Rastall's theory of gravity [15] and many other theories.
Various models of matter were considered as a source of wormholes:
scalar and electromagnetic fields [11, 16, 17], Chaplygin gas [19], various models of phantom energy and quintessence, including models of matter with the structure of the energy–momentum tensor of a perfect fluid [20, 21, 22, 23, 24].

Models of wormholes were also built using the thin-shell formalism, the works of [25, 26] being among the first.

Most of the literature is devoted to the study of static and spherically symmetric wormholes.
An important generalization of these studies is the study of dynamic solutions.
Dynamic wormholes have been considered in various aspects. One of the ways for building dynamic models  [27, 28, 29, 30, 31] is to add a time-dependent scaling factor to the metric.
Dynamic models of wormholes were constructed using the thin shell formalism [32].
General properties of an arbitrary dynamic wormhole are considered in the papers [33, 34].
In recent papers [35, 36, 37], dynamic solutions have been constructed on the basis of a family of solutions describing regular black holes.

It is worth noting that the throat of a wormhole is defined differently by different authors.
In the static case, the different definitions agree with each other; in the dynamic case contradictions may arise.
The general defenition of the throat, including for time-dependent metrics, is considered, for example, in the following
works [34, 38, 39]. We will use the definition that was used in the works [27, 29, 30].

The aim of this work is to construct a solution describing a dynamic wormhole in the theory of gravity with dust-like matter.
The general solution of Einstein's equations in the theory of gravity with dust-like matter for a spherically symmetric metric was obtained by
Tolman in 1934  [40, 41, 42, 43, 44]; the solution will be summarized in Section <ref>.
This solution contains three arbitrary functions.
In Section <ref> we construct a solution that describes a collapsing wormhole, choosing these functions in a certain way.
In Section <ref> we investigate the properties of the obtained solution.

§ GRAVITATIONAL COLLAPSE OF A DUST-LIKE SPHERICAL SHELL

The spherically symmetric solution of the Einstein equations in the theory of gravity with dust-like matter was obtained by Tolman in 1934 [40, 41, 42, 43, 44].  Briefly, the solution will be presented in this section.
Dust-like matter allows the choice of a frame that is both synchronous and co-moving.
In this case, the spherically symmetric metric has the form[Henceforth, we denote the speed of light as
$c =1$.]:

\begin{equation}\label{metric-4}
ds^2=d\tau^2-e^{\lambda(\tau,R)}dR^2-r^2(\tau,R)\big[d\theta^2+\sin^2\theta d\varphi^2\big],
\end{equation}

where $\lambda(\tau,r)$ and $r(\tau,R)$ are functions of $\tau$ and $R$.
The energy–momentum tensor of dust-like matter in the co-moving frame has the form

\begin{equation}
T_{i}^{j}=\varepsilon u_i u^j, \label{fluid}
\end{equation}

where $(u^{i})=(1,0,0,0)$ is a velocity four vector, and $\varepsilon$ is the energy density.
Below, we present the solution of the Einstein equations for the metric (<ref>) and the energy–momentum tensor (<ref>).
The function $\lambda(\tau,R)$ has the form

\begin{equation}
\end{equation}

where the prime means differentiation with respect to the coordinate $R$, the dot means differentiation with respect to the coordinate $\tau$,  $f(R)$ is an arbitrary function, and $f(R)$ satisfies the condition $1+f>0$. Function $r(\tau,R)$  can be represented in parametric form.
In case $f>0$, the function has the form:

\begin{eqnarray}
&r=\dfrac{F(R)}{2f(R)}\left[ \cosh\eta-1 \right],
\quad
{\tau_0}(R)-\tau=\dfrac{F(R)}{2f(R)^{3/2}}\left[ \sinh\eta -\eta\right];&\label{Tolman1}
\end{eqnarray}

in the case of $f<0$, the function has the form:

\begin{eqnarray}
&r=\dfrac{F(R)}{2|f(R)|}\left[1- \cos\eta \right],\quad
{\tau_0}(R)-\tau=\dfrac{F(R)}{2|f(R)|^{3/2}}\left[\eta- \sin\eta\right],\label{Tolman2}&
\end{eqnarray}

where $F(R)$ and ${\tau_0}(R)$ are an arbitrary functions.
In the case of $f=0$, the function has the form:

\begin{eqnarray}
\end{eqnarray}

The energy density $\varepsilon$ is:

\begin{equation}
8\pi k \varepsilon=\dfrac{F'}{r'r^2 }.\label{varepsilon}
\end{equation}

The scalar curvature $\mathcal{R}$ and the Kretschman scalar $\mathcal{K}$ for the Tolman solution (<ref>–<ref>) read

\begin{eqnarray}
& \mathcal{R} = \dfrac{F'}{r^2r'}, \label{R} &\\
& \mathcal{K} = \dfrac{3F'^2}{r^4r'^2}+ \dfrac{16 F^2}{r^6} - \dfrac{8FF'}{r^5r'}. \label{K}&
\end{eqnarray}

The scalar curvature and the Kretschmann scalar diverge at $r=0$ or $r'=0$ (in the case $F'\neq 0$), the solution in this case has a singularity.

The general solution (<ref>)–(<ref>)
depend on three arbitrary functions $f(R)$, $F(R)$ and ${\tau_0}(R)$.
The solution allows an arbitrary transformation of the radial coordinate $\tilde{R}=\tilde{R}(R)$, where $\tilde{R}(R)$ is a differentiable function.
Using this, we can give any of the three functions $f(R)$, $F(R)$ or ${\tau_0}(R)$ some specific form.
Thus, only two of the three functions can be considered arbitrary.
The particular cases of the solution are the Schwarzschild solution and the Friedmann model with dust.

§ THROAT CONDITIONS

In this section, we will obtain the conditions under which the general solution (<ref>)–(<ref>) will describe the wormhole geometry.
A characteristic feature of wormholes is the presence of a throat—i.e., the presence of a space-like closed two-dimensional surface of the minimum area.
Various definitions of the mouth of a wormhole are considered in the literature [34, 38, 39, 27, 29, 30],
we will follow the approach used, for example, in works [27, 29, 30].

In order to determine the throat conditions, following the approach used in [30], we constructed an embedding diagram for the metric (<ref>).
For convenience, we rewrote the function $f(R)$ in the form $f(R)=-b(R)/R$,  where $ b (R) $ is an arbitrary function.
Taking into account the relation (<ref>), the metric (<ref>) can be represented as

\begin{equation}\label{metric-41}
ds^2=d\tau^2-\dfrac{r'^2 dR^2}{1-b/R}-r^2\big[d\theta^2+\sin^2\theta d\varphi^2\big].
\end{equation}

Metric (<ref>) is defined for $1-b(R)/R>0$, and for $b(R)=R$ has a singularity.
Consider the metric (<ref>) on the spatial slice $\tau=const$ and $\theta=\pi/2$

\begin{equation}\label{3_d}
dl^2=\dfrac{r'^2 dR^2}{1-b/R}+r^2d\varphi^2,
\end{equation}

as a metric on a surface of revolution $\rho=\rho(z)$ embedded in a three-dimensional space with an Euclidean metric

\begin{equation}\label{R3d}
\end{equation}

where $z$, $\rho$ and $\varphi$ are cylindrical coordinates (see Figure <ref>).
Comparing (<ref>) and (<ref>), we get

\begin{eqnarray}
\end{eqnarray}

Taking into account that for constant $\tau=const$

\begin{eqnarray}
&d\rho|_{\tau_0}=r'_R dR,&
\end{eqnarray}

we have

\begin{eqnarray}
&\dfrac{d\rho}{dz}=\left(\dfrac{R}{b}-1\right)^{1/2},\quad \dfrac{d^2\rho}{dz^2}=\dfrac{b-R b'}{2 r' b^2}\label{ee2}.&
\end{eqnarray}

axis y line = left,
axis x line = middle,
xlabel     = $\rho$,

xtick=, ytick=,
samples     = 160,
domain      = 0.5:3,
xmin = 0.5, xmax = 3,
ymin = -1.8, ymax = 1.8,
[blue,thick]  sqrt(x-1);
[blue,thick]  -sqrt(x-1);
[label=right:$z$] (A) at (0cm, 5cm);

[view=6030,axis lines=none,colormap/cool, fill opacity=.9]
domain=-2:2,y domain=0:2*pi,
z buffer=sort]
((x^2+1) * cos(deg(y)),
(x^2+1) * sin(deg(y)),

The embedding diagram. The function $\rho(z)$ is shown on the left panel, and the surface obtained by rotating the curve $\rho(z)$ about the axis $Oz$ is shown on the right panel.

The throat of the wormhole will have the shape of a sphere, which is located at a certain value of the radial coordinates $R=R_{th}$.
On the embedding diagram, the sphere $R=R_{th}$ corresponds to a circle of radius $\rho$ on the surface of revolution;
at the throat the radius of the circle $\rho(z)$ has a minimum.
Conditions for the minimum of the function $\rho(z)$ at $R=R_{th}$ have the form:

\begin{eqnarray}
&\dfrac{d\rho}{dz}\Big|_{R_{th}}=0,\quad \dfrac{d^2\rho}{dz^2}\Big|_{R_{th}}>0.&\label{ee4}
\end{eqnarray}

Comparing (<ref>) and (<ref>), we get throat conditions for the metric (<ref>):

\begin{eqnarray}
\end{eqnarray}

We will assume that the radial coordinate takes the following positive range of values:
$0<R_{th}<R<+\infty$, where the value $R=R_{th}$ corresponds to the throat.
The condition (<ref>) implies that the function $b(R)$ is positive at the throat: $b(R_{th})>0$.
Therefore,  $b(R)$ is positive in some neighborhood of the throat by continuity.
Therefore, the function $f=-b/R$ is negative in the vicinity of the throat,
and to construct a wormhole model  we will consider the solution (<ref>) with a negative value of $f$.

§ COLLAPSING WORMHOLE MODEL WITH DUST-LIKE MATTER

The general solution (<ref>)–(<ref>) of the Einstein equations for a spherically symmetric metric in the theory of gravity with dust-like matter depend on three arbitrary functions $f(R)$, $F(R)$ and ${\tau_0}(R)$. In this section, we will consider a special case describing a wormhole.
We choose arbitrary functions as follows[This case is not the only one possible. Further we will restrict ourselves to considering this example as one of the simplest.]:

\begin{eqnarray}
&F=R,\quad {\tau_0}(R)={\tau_0}, \quad f(R)=-\dfrac{b}{R},\label{FT}&
\end{eqnarray}

where ${\tau_0}$, $b$ are constants, $b>0$ and $f<0$. The solution takes the form:

\begin{eqnarray}
&ds^2=d\tau^2-\dfrac{r'^2dR^2}{1-b/R}-r^2(t,R)\big[d\theta^2+\sin^2\theta d\varphi^2\big],\label{Twh1}&\\
&r=\dfrac{R^2}{2b}\left[1- \cos\eta \right],\quad {\tau_0}-\tau=\dfrac{R^{5/2}}{2b^{3/2}}\left[\eta- \sin\eta\right],\label{Twh3}&
\end{eqnarray}

where $R\in[b,+\infty)$. In this case, the first throat condition (<ref>) is satisfied for $R=b$.
The second condition (<ref>) will be satisfied when $\left.r'\right|_{R = b}>0$.
We calculate the derivative of the function $r(\tau,R)$ (<ref>):

\begin{eqnarray}
(1-\cos\eta)^2-\dfrac{5}{4}\sin\eta(\eta-\sin\eta) \right].\label{r'}
\end{eqnarray}

For  $\eta\in(0,2\pi)$ and $R\in[b,+\infty)$ the expression  (<ref>)  is positive: $r'>0$.
Therefore, for the solution (<ref>, <ref>), the throat conditions (<ref>, <ref>) are satisfied for $R=b$.

To build a model of a wormhole, two identical sets $M_{+}$ and $M_{-}$ are required, each of which has a metric (<ref>, <ref>) with the same parameter $b$

M_{\pm}=\left\{ (\tau,R,\theta,\varphi)| \; R\in [b,+\infty) \right\}.

$M_{+}$ and  $M_{-}$ are manifolds with boundaries. The boundary is a timelike hypersurface

\Sigma_{\pm}=\left\{ (\tau,R,\theta,\varphi)| \; R=b \right\}.

We identify $M_{+}$ and $M_{-}$ by boundaries. We get a new manifold $M=M_{+}\cup M_{-}$.
The resulting space $M$ has a wormhole geometry that connects two space–times, in each of which the radial coordinate takes the values $R\in[b,+\infty)$.
The throat of the wormhole corresponds to the value of the radial coordinate $R=b$.

As $R\to b$, one of the metric coefficients in (<ref>), approaches infinity.
However, the values of $r'$ (<ref>) and $r$ (<ref>) are nonzero for $\eta\in(0,2\pi)$, and the scalar curvature (<ref>) and the Kretschmann scalar (<ref>) are regular in this case. Thus, the singularity of the metric at the throat has a purely coordinate character.

It is known [25, 26, 45, 46, 47, 48, 49],
that the energy–momentum tensor on the shell $S_{ij}$ is proportinal to the Dirac delta function:

where $T_{ij}^{\pm}$ is the energy–momentum tensor in the corresponding region $M_{\pm}$.
$S_{ij}$ is calculated as follows:

S_{ij}=\dfrac{1}{8\pi G}\left( [k_{ij}]-[k]h_{ij} \right)

where $k_{ij}$ is the second fundamental form of $\Sigma$,
$[k_{ij}]$ is the difference of limiting values of the second fundamental form on both sides of the surface $\Sigma$ in $M^{\pm}$

$h_{\alpha\beta}$ is a projection tensor on $\Sigma$,

and $n_{\alpha}$ is the normal vector to $\Sigma$.
In our case, the unit normal vector to the surface has the form
(n^i)=\left( 0, \pm e^{-\lambda/2},0,0 \right).
The tensor $[k_{ij}]$ on the surface $R=b$

\left(
\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 \\
0 & 0 & 2r\sqrt{1-b/R} & 0 \\
0 & 0 & 0 & 2r\sqrt{1-b/R}\sin^2\theta
\end{array}\right)

equals zero. Consequently, the shell momentum energy tensor
is zero. Thus, there is no material shell on the surface $\Sigma$, and the energy–momentum tensor of the model contains only dust-like matter.

§ ANALYSIS OF THE MODEL

In this section, we will consider the dynamics of the model from the point of view of an external observer.
We will assume that the dusty matter is distributed inside some spherical layer.
Outside this layer, the geometry is described by the spherically symmetric Schwarzschild solution in the vacuum [43]:

\begin{eqnarray}
\end{eqnarray}

where  $r_g=2M$ is a gravitational radius, $M$ is a mass that creates a gravitational field.
In the absence of pressure, each element of matter moves along a geodesic path. All geodesics are radial due to spherical symmetry.
From the point of view of an external observer, the radial coordinate of the surface of the star is a function of time $r=r(\tau)$, where $r(\tau)$ is the radial geodesic in the Schwarzschild space–time.
It is convenient to represent the geodesic equation in parametric form [50]:

\begin{eqnarray}
\end{eqnarray}

where the parameter $\chi\in[0,\pi]$, $\tau$ is a time coordinate in the co-moving coordinate system, $r_0$ is the value of the radial coordinate at the initial moment of time $\tau=0$ ($\chi=0$).
Expressions (<ref>–<ref>) are valid provided that at the initial moment of time $\tau=0$ the shell was at rest $dr/d\tau=0$.
Substituting expressions (<ref>–<ref>) into the metric (<ref>), we find the 3-geometry of the surface of the star in the external metric:

\begin{eqnarray}
\end{eqnarray}

Now we will consider the internal geometry of the dusty layer (<ref>),   (<ref>).
Let us assume that dust-like  matter is enclosed in a spherical layer $R\in[b,R_{0}]$, where $R_0$ is its outer radius.
The internal 3-geometry of the surface of the star is found by substituting expression (<ref>) into the metric (<ref>) at $R=R_0$:

\begin{eqnarray}
\end{eqnarray}

On the surface of the star the Schwarzschild metric should be smoothly matched with the intrinsic metric.
Comparing (<ref>) and (<ref>), we see that these 3-geometries are smoothly sewn together if we identify the following parameters

\begin{eqnarray}
&\eta=\pi-\chi,\quad r_0=\dfrac{R_0^2}{b},\quad M=\dfrac{R_0}{2}. \label{junctioncons}&
%\quad \tau_0=\dfrac{\pi r_0^{9/2}}{2b^{3/2}}.&
\end{eqnarray}

At the initial moment of time $\tau=0$ ($\chi=0$), the dust-like layer with the outer radius $r_0$ begins to collapse from the rest position towards the center.
As follows from formulas (<ref>, <ref>), at the moment of collapse, the outer radius vanishes $r=0$,
and the watch of the co-moving observer shows  $\tau_0=\dfrac{\pi r_0^{5/2}}{2b^{3/2}}$ ($\chi=\pi$).
The time $\tau_0 $ is the lifetime of the wormhole.
At the moment of collapse, solutions (<ref>, <ref>) are singular, since the scalar curvature and the Kretschmann scalar (<ref>, <ref>) approach infinity.
By construction, $R_0>b$, which means that due to the relations (<ref>) the initial outer radius of the shell will be greater than the gravitational radius—i.e., $r_0>r_g$. From the point of view of an infinitely distant observer, gravitational collapse will last an infinitely long time.

The figure shows the graphs of the function $r(R,\tau)/b$ depending on $\tau/b$ for different values of $R$: $R=b$, $R=1.1b$ and $R=R_0=1.2b$.
Red-blue lines are the null radial geodesics. (See the text for more details.)

It is very important to answer the question: Is a photon able to cross the collapsing wormhole? To solve this problem we need to find null geodesics inside the collapsing dust ball. An equation for radial geodesics can be directly found from the condition $ds^2=0$. The metric (<ref>) yields

\begin{equation}\label{geod_R}
\frac{dR}{d\tau}=\mp \frac{1}{r'}\,\sqrt{1-\frac{b}{R}},
\end{equation}

where $r'$ is given by Eq. (<ref>). Using the relations (<ref>), one can rewrite Eq. (<ref>) as follows

\begin{equation}\label{geod_r}
\frac{dr}{d\tau}=\left(\frac{b}{R}\right)^{1/2}\,
\left[-\frac{\sin\eta}{1-\cos\eta} \mp\sqrt{\frac{R}{b}-1}\right].
\end{equation}

Integrating this equation gives null geodesics in the coordinates $(r,\tau)$. Note that the sign $"-"$ corresponds to ingoing geodesics moving towards the wormhole throat, while $"+"$ corresponds to outgoing geodesics moving away from the throat. At the throat $R=b$ Eq. (<ref>) reduces to

\begin{equation}
\left.\frac{dr}{d\tau}\right|_{R=b}=-\frac{\sin\eta}{1-\cos\eta},
\end{equation}

therefore null geodesics are tangent to the cycloid $R=b$.

In the figure <ref> we illustrate the main features of the obtained solution. In particular, the dependence of the radius $r(R,t)$ of various layers of the dust-like shell $R\in[b,R_0]$ on time $\tau$ for $R_0=1.2b$ (see Eq. (<ref>)) is shown.
Region I: $ R <b $ is not included in space–time and is not considered.
Region II: $R\in[b,R_0]$ is filled with dust-like matter.
Region III is the empty area outside the dust-like shell.
The three solid curves show the dynamics of the dust layers over time: curve at $R=R_0$ is the outer shell;
curve at $R=b$ is the wormhole throat; intermediate curve for the inner layer $b<R<R_0$, $R=1.1b$.
The dash-dotted line corresponds to the gravitational radius $r=r_g$.
In a finite proper time $\tau_0$, the shell reaches the gravitational radius and collapses towards the singularity $r=0$.
Red-blue lines are the null radial geodesics. The red curves are ingoing geodesic lines of photons who started their travel towards the wormhole throat from the outer shell. These photons are reaching the throat in a finite time; at the point where the photon is crossing the throat, its geodesic line becomes tangent to the cycloid $R=b$. After crossing the throat, the photon continues its movement on the other side of the wormhole space–time. In the figure <ref> the blue curves correspond to outgoing geodesic lines of photons who started their travel towards the outer shell from the throat.

The figure shows the graphs of the energy density $8\pi k\varepsilon b^2$ depending on $\tau/b$ for different values of $R$: $R=b$, $R=1.1b$ and $R=R_0$, $R_0=1.2b$.  As $\tau\to\tau_0$ the energy tends to infinity, the vertical asymptote is shown by dash-dot on the right in the figure.

Figure <ref> shows the dependence (<ref>) of the energy density $\varepsilon$ on the time $\tau$ for different values of $R\in[b,R_0]$.
The energy density is positive, and at the moment of collapse $\tau\to\tau_0$, the energy tends to infinity.
The energy–momentum tensor of “ordinary” matter satisfies certain physical requirements, one of which is the null energy condition:

T_{\mu\nu}V^{\mu}V^{\nu}\leqslant 0

for any null vector $V^{\mu}$. It is known [6], that the null energy condition can be violated in wormhole space–time.
Consider a region in the vicinity of the wormhole throat $R\in[b,R_0]$, where the model is described by solutions (<ref>, <ref>).
We choose a null vector $V^{\ mu}=\left( 1,e^{-\lambda/2},0,0 \right)$ and calculate the following contraction:

\begin{equation}
\end{equation}

which is positive for $R\in[b,R_0]$, $\tau\in [0,\tau_0]$. Thus, in this model, the null energy condition is fulfilled.

We also present this metric in the limiting case $\tau\to\tau_0$ ($\eta\to+0$, $\chi\to\pi-0$):

\begin{eqnarray}
ds^2=d\tau^2-\left(\dfrac{3}{2}\dfrac{\tau_0-\tau}{R}\right)^{4/3}\left[\dfrac{1}{12^{2/3}}\dfrac{dR^2}{1-b/R}+R^2\left(d\theta^2+\sin^2\theta d\varphi^2\right)\right]
\end{eqnarray}

§ CONCLUSION

It is well known that static wormhole configurations in general relativity (GR) are possible only if matter threading the wormhole throat is “exotic”—i.e., violates a number of energy conditions. For this reason, it is impossible to construct static wormholes supported only by dust-like matter which satisfies all usual energy conditions. However, this is not the case for non-static configurations.
In 1934, Tolman found a general solution describing an evolution of a spherical dust shell in GR. In this particular case, Tolman's solution describes the collapsing dust ball; the inner space–time structure of the ball corresponds to the Friedmann universe filled by a dust.
In the present work we have used Tolman's solution in order to construct a dynamic spherically symmetric wormhole solution in GR with dust-like matter.
For this aim, we have considered the specific subclass of arbitrary functions $f(R)$, $F(R)$ and $t_0(R)$ constituting Tolman's solution,
and written solutions (<ref>)–(<ref>), which satisfy the throat conditions.
The solution constructed represents the collapsing dust ball with the inner wormhole space–time structure.

To construct a wormhole model we considered two identical solutions $M_{+}$ and $M_{-}$ with metrics (<ref>), (<ref>), in each of which the radial coordinate $R$ takes the range of values $R\in[b,+\infty)$. The sets $M_{+}$ and $M_{-}$ are identified by their boundaries $\Sigma$. The result is a new manifold $M=M_{+}\cup M_{-}$ that has a wormhole geometry.
The throat of the wormhole corresponds to the junction surface $\Sigma$: $R=b$.
Due to the smoothness of the functions, the identity of $M_{+}$ and $M_{-}$ and to the throat conditions, the energy–momentum tensor $S_{ij}$ of the shell $\Sigma$ is equal to zero. Thus, the energy–momentum tensor of the constructed model contains only dust-like matter, and does not contain a contribution of the Dirac delta function on the junction surface.
For $\eta\in(0,2\pi)$, the scalar curvature (<ref>) and the Kretschmann scalar (<ref>) are regular, and the solution does not have singularities.

We assumed that the dusty matter is enclosed inside some spherical layer and considered the dynamics of a wormhole from the point of view of an external observer.
Gluing solutions (<ref>), (<ref>) and the Schwarzschild solution, we obtained the condition (<ref>) that connects the model parameters, the gravitational radius and the mass of the dust ball.
The dynamics of dust layers over time is depicted in Figure <ref>.
At the initial moment of time, a dusty layer begins to collapse towards the center, at the moment of collapse the outer radius of the shell vanishes, and the clock of the co-moving observer shows $\tau_0={\pi r_0^{5/2}}/{2b^{3/2}}$. The $\tau_0$ is the wormhole's lifetime.
From the point of view of an infinitely distant observer, gravitational collapse will last indefinitely.
At the moment of collapse, the scalar curvature and the Kretschmann scalar (<ref>), (<ref>) become singular and the solution has a singularity.
Figure <ref> shows the dependence of the energy density of the dust layer on time, the energy density is positive, and tends to infinity at the moment of collapse. It is shown that the null energy condition for this model is satisfied (<ref>). Therefore, the dust-like matter that the ball is made of satisfies the usual energy conditions and cannot prevent the collapse.

We acknowledge the contribution of Evdokim Isanaev on the very beginning of this work. P.E.K. and S.V.S. are supported by RSF grant No. 16-12-10401. Partially, this work was done in the framework of the Russian Government Program of Competitive Growth of the Kazan Federal University.

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# Detecting correlated errors in twin-field quantum key distribution

B. Panchumarthi  A. Stephens and M. Beck  Department of Physics, Reed College,
3203 SE Woodstock Blvd., Portland, OR 97202<EMAIL_ADDRESS>
††journal: osajournal††articletype: Research Article

We experimentally demonstrate that we can detect correlated errors in a twin-
field quantum key distribution (TFQKD) system by using a technique that is
related to self-consistent tomography. We implement a TFQKD system based on a
fiber-Sagnac loop, in which Alice and Bob encode information in the phase of
weak coherent states that propagate in opposite directions around the loop.
These states interfere as they exit the loop and are detected by a third
party, Charlie, who reports the results of their measurements to Alice and
Bob. We find that it is possible for Alice and Bob to detect correlated state-
preparation and measurement errors while trusting only their own individual
states, and without trusting Charlie’s measurements.

##  1 Introduction

Quantum-state tomography (QST) estimates the density operator of an unknown
quantum state by performing a series of measurements with perfectly calibrated
detectors [1, 2, 3]. Quantum-detector tomography (QDT) estimates the positive-
operator-valued measure (POVM) that describes a detector by probing it with a
series of perfectly characterized quantum states [4, 5, 6]. However, it is now
appreciated that in real experiments there is no such thing as perfectly
calibrated detectors or perfectly characterized states. There are invariably
errors in both of these, and we refer to them as state-preparation and
measurement (SPAM) errors. As such, techniques have been developed for
estimating the quantum operators that characterize both the state and the
measurements, in a self-consistent manner [7, 8, 9, 10].

Recently it has been shown that there exist measurement techniques related to
self-consistent tomography that are capable of detecting certain types of SPAM
errors, without needing to explicitly reconstruct the state or measurement
operators. Loop (or holonomic) SPAM tomography has been described
theoretically for N-qubit systems [11, 12, 13], and demonstrated
experimentally for 1- and 2-qubit systems [14, 15]. It is capable of detecting
correlated errors between state preparations and measurements, or between
measurements performed in different locations. Loop SPAM tomography is
powerful because the only assumption made about the states and the
measurements is that their underlying Hilbert-space dimensions are known. It
works by looking for self-consistency (or a lack thereof in the case of
errors) in an over-complete set of measurements.

Consider a quantum key distribution (QKD) system in which Alice and Bob wish
to share a secret key. The expectation values of the measurements performed
with their system can be described by

$S^{ij}=\text{Tr}\left[\left({\hat{a}^{i}}\otimes{\hat{b}^{j}}\right){\hat{\xi}}\right].$
(1)

One possible protocol for such a QKD system is one that is based on Ekert’s
protocol, or something similar [16, 17]. In this case ${\hat{\xi}}$ describes
the entangled state of a pair of particles that travel to Alice and Bob,
${\hat{a}^{i}}$ refers to Alice’s $i$’th measurement operator, and
${\hat{b}^{j}}$ refers to Bob’s $j$’th measurement operator. In this protocol
there is a joint state, but the measurements are independent. Correlated
errors can lead Alice and Bob to $think$ their generated key is secure, when
in fact it is not. However, it has been demonstrated that loop SPAM tomography
can detect these correlated errors, demonstrating to Alice and Bob the
vulnerability of their system [15].

A different QKD protocol is one in which Alice and Bob prepare independent
states, and then a joint measurement is performed on them. Equation (1) still
applies, but in this case ${\hat{\xi}}$ describes the joint measurement,
${\hat{a}^{i}}$ refers to Alice’s $i$’th state preparation, and
${\hat{b}^{j}}$ refers to Bob’s $j$’th state preparation. This form of QKD is
known as measurement-device-independent (MDI) QKD [18]. The joint measurement
can be performed by a third party, Charlie. It is assumed that Alice and Bob
know their own states, but it can be shown that they do not need to know
${\hat{\xi}}$ (they do not need to trust Charlie) in order for their key to be
secure. Recently, Moore and van Enk have proposed a method, related to self-
consistent tomography, to determine if there are correlated errors present in
an MDIQKD system [19]. Such errors could be due to an eavesdropper, or they
could be due to more mundane, but hard to detect, hardware or software errors.
Here we describe an experimental implementation of their method, using a form
of MDIQKD known as twin-field (TF) QKD.

TFQKD was introduced as a way of extending the range of QKD systems [20, 21].
Traditional QKD schemes have a maximum key rate that scales as the channel
transmittance $\eta$, whereas the key rate for TFQKD scales as $\sqrt{\eta}$.
There have been several experimental demonstrations of TFQKD, which have
demonstrated its ability to transmit keys over ever larger distances, and in
some cases have exceeded the repeaterless secret key capacity (the Pirandola-
Laurenza-Ottaviani-Banchi, PLOB, bound) [22, 23, 24, 25, 26, 27, 28, 29]. Here
we implement a TFQKD system based on that of Zhong _et al._ [25]. Our goal is
not to implement a long-distance TFQKD system, but rather to show that we can
use the technique proposed by Moore and van Enk to detect correlated errors in
this system.

##  2 Theory

###  2.1 Correlated error detection

Here we provide an overview of the theory behind our error detection
technique; further details can be found in Ref. [19]. The operators describing
the states produced by Alice and Bob can be written in the Pauli basis as

${\hat{a}^{i}}=\sum_{k}a_{k}^{i}{\hat{\sigma}_{k}}\>,\>\>\>{\hat{b}^{j}}=\sum_{\ell}b_{\ell}^{j}{\hat{\sigma}_{\ell}},$
(2)

where the $a_{k}^{i}$’s can be considered to be elements of a 4-dimensional
vector $a^{i}$ that describes Alice’s $i$’th state preparation, and the
$b_{\ell}^{j}$’s similarly describe Bob’s $j$’th state. The operator that
describes the joint measurement $\hat{\xi}$ can be written as

${\hat{\xi}}=\frac{1}{4}\sum_{m,n}x_{mn}\>({\hat{\sigma}_{m}}\otimes{\hat{\sigma}_{n}}),$
(3)

where the $x_{mn}$’s are the elements of a $4\times 4$ matrix $X$.
Substituting Eqs. (2) and (3) into Eq. (1) and taking the trace yields

$S^{ij}=\sum_{k,\ell}x_{k\ell}a_{k}^{i}b_{\ell}^{j}.$ (4)

Here the expectation values $S^{ij}$ are the elements of a $4\times 4$ matrix
$S$. From Eq. (4) we see that $S$ can be expressed as a matrix product:

$S=A^{T}{X}{B},$ (5)

where the 4 columns of ${A}$ and ${B}$ are the vectors that describe 4
different preparations of Alice’s and Bob’s states. If Alice ensures that the
vectors that describe her 4 state preparations are linearly independent, we
can multiply by the inverse $\left(A^{T}\right)^{-1}$ to get

$\left({A}^{T}\right)^{-1}{S}={X}{B}.$ (6)

Suppose that Bob prepares 4 states that are described by the matrix $B$, and
Alice prepares 4 states that are described by the matrix $A_{1}$. By Eq. (5)
they will measure the set of expectation values described by $S_{1}$. Now
suppose that they perform a different set of measurements in which Bob keeps
his state preparations the same, but Alice changes at least one of her state
preparations. Alice’s states are now described by the matrix $A_{2}$, and they
measure expectation values $S_{2}$. Since $B$ is unchanged, the condition that
there is a unique $X$ that is independent of Alice’s state preparations is
then, from Eq. (6),

$\left({A_{1}}^{T}\right)^{-1}{S_{1}}=\left({A_{2}}^{T}\right)^{-1}{S_{2}}.$
(7)

We define the matrix $M$ as

$M\equiv\left({A_{1}}^{T}\right)^{-1}{S_{1}}-\left({A_{2}}^{T}\right)^{-1}{S_{2}}.$
(8)

From Eqs. (6)-(8) it is apparent that the condition $M=0$ ensures that the
measurements, described by $X$, are uncorrelated with Alice’s state
preparations.

Alice can test whether or not there are any correlated errors between her
state preparations and the measurements that Charlie is performing in the
following way. Alice prepares five different states–four are used to construct
the matrix $A_{1}$, and the fifth is combined with three of the first set to
construct the matrix $A_{2}$. Bob prepares four different states, and Alice
receives the measurement results for each trial from Charlie. From the
measurements she can construct the matrices of expectation values $S_{1}$ and
$S_{2}$. Note that Alice does not need to know what states Bob produces (she
doesn’t need to know $B$), but on each trial she does need to know which of
the four states Bob is sending, so that she can properly arrange the
measurements into $S_{1}$ and $S_{2}$. She does not need to trust the
measurements that Charlie is performing (she doesn’t need to know $X$). Alice
constructs the matrix $M$ and if $M=0$, to within the uncertainty in the
measurements, Alice can be confident that Charlie’s measurements are
independent of her state preparations. Furthermore, note that if Bob were also
to prepare five different states, he could use the same process to test if
Charlie’s measurements are correlated with his state preparations.

Note that Alice may use as few as 5 state preparations. But if she and Bob
decide on a QKD protocol that uses more states she can use as many as 8
preparations. The constraint is that each of the matrices $A_{1}$ and $A_{2}$
must contain independent columns so that they can be inverted.

###  2.2 Twin-field quantum key distribution

For our TFQKD system we use the encoding scheme described by Yin and Fu, which
is essentially the same as that of Zhong _et al._ with the addition of
$Y$-basis states [30, 25]. We use weak coherent states, and post-select all
measurements on the presence of a single photon, so Alice and Bob’s states
consist of vacuum $|{0\rangle}$ and single-photon $|{1\rangle}$ contributions.
The Bloch vectors that describe the states are distinguished by the phases of
the corresponding coherent states. The $X$-basis states correspond to the
phases $\theta=0,\pi$,

$|{+X\rangle}=|{\alpha\rangle}\>,\>\>\>|{-X\rangle}=|{\alpha
e^{i\pi}\rangle}=|{-\alpha\rangle},$ (9)

while the $Y$-basis states correspond to the phases $\theta=\pi/2,3\pi/2$,

$|{+Y\rangle}=|{\alpha e^{i\pi/2}\rangle}\>,\>\>\>|{-Y\rangle}=|{\alpha
e^{i3\pi/2}\rangle}.$ (10)

The $Z$-basis states in this encoding scheme correspond to the vacuum and a
phase-randomized weak coherent state. We do not need
$|{+Z\rangle}=|{0\rangle}$ for our purposes. To keep the phase randomization
practical, for $|{-Z\rangle}$ the phase is chosen from one of the four values
$\theta=n\pi/2\>(n=0,1,2,3)$. Note that due to limitations of our hardware, we
do not randomly choose the phase on every state preparation. Rather, we step
through the phases in sequence. After averaging, the measured expectation
value will be the same as if the phase was chosen randomly. While this would
not work for a truly secure QKD system, it is sufficient for our purpose of
demonstrating an error detection technique.

##  3 Experiments

###  3.1 Experimental design

Our experimental apparatus, based on that of Zhong _et al._ , is shown in Fig.
1 [25]. Charlie generates 1 ns pulses at a rate of 1 MHz that he sends to
Alice and Bob through single-mode optical fibers. Alice and Bob use phase
modulators to select which of the basis states they transmit back to Charlie.
To maintain temporal overlap and phase stability between Alice and Bob’s
states, the geometry is chosen to be that of a Sagnac interferometer, in which
Alice’s and Bob’s photons propagate in opposite directions around a loop.
Charlie uses single-photon counting modules (SPCMs) to detect the photons that
return to him (the detected signal level is kept to < 0.01 photons per pulse).
To eliminate dark and background counts the photon arrival times are recorded
with time-to-digital converters, and only photons that arrive within 2.4 ns of
the expected time are counted.

Figure 1: The experimental apparatus. An amplitude modulator (AM) produces 1
ns pulses from a 1550 nm laser diode. These pulses pass through a circulator
(C), and are split by a 2x2 fiber coupler (FC) before entering a Sagnac
interferometer. Pulses travel in opposite directions around the Sagnac loop.
Alice encodes information on the phase of the clockwise traveling pulse using
a waveguide phase modulator (PM) synchronized with the arrival of her pulse,
while Bob similarly encodes information in the phase of the counter-clockwise
pulse. Polarization controllers (PC) and linear polarizers (LP) control the
polarization in the interferometer; FS is a 40 m spool of optical fiber, which
is used to introduce a time delay that ensures Alice’s and Bob’s state
preparations are independent of each other. The pulses interfere at the
coupler, and depending on their relative phases are detected with single-
photon-counting modules at D+ or D-. By knowing which detector fires, Alice
and Bob can generate a quantum key, or they calculate expectation values in
order to perform error detection measurements.

The expectation values for a given pair of Alice and Bob’s state preparations
are given by the number of detected photons $N_{+}$ and $N_{-}$ at detectors
$D_{+}$ and $D_{-}$ as

$S=\frac{N_{+}-N_{-}}{N_{+}+N_{-}}.$ (11)

Shown in Fig. 2 are plots of the expectation values as a function of the
voltage that Alice applies to her phase modulator, for four different phase
shifts applied by Bob. We see that the expectation values oscillate
sinusoidally between $\pm 1$, as expected. Careful adjustment of the
polarizations of the counter-propagating pulses is necessary to obtain the
high-visibility interference that leads to this result. To calibrate the
voltage that we need to apply to Alice’s modulator to obtain a given phase
shift, we fix Bob’s phase shift, then we obtain 10 trials of $S$ as a function
of voltage and fit each one to a sinusoidal function. We acquire these 10
trials during a time interval that is comparable to the time it takes for us
to acquire the data we use to calculate $M$, in order to assure that any drift
will be approximately the same for these different experiments. We then
calculate the mean and standard deviation of the 10 sets of fit parameters. We
find that the uncertainty in Alice’s ability to produce a particular phase
shift is $\pm~{}.029\>\text{rad}\cong\pm~{}\pi/100$. This uncertainty
determines the uncertainties in the elements of the matrix $A$ that describes
Alice’s state preparations.

Figure 2: The expectation value $S$ as a function of the voltage applied
Alice’s phase modulator. Markers are measured, while the solid lines are
sinusoidal fits to the data. Each curve corresponds to a different phase shift
applied by Bob: 0 (red •), $\pi/2$ (orange $\blacksquare$), $\pi$ (green
$\blacktriangle$), $3\pi/2$ (blue $\blacktriangledown$).

###  3.2 Results

On each shot Charlie reports to Alice and Bob which of his detectors have
fired, and Bob reports to Alice a label that refers to which state he has
produced. From this information, Alice can calculate expectation values of the
measurements, and arrange them into the $S$ matrices as described above.
Recall that Alice does not need to know Bob’s states, but she does need enough
information from Bob to distinguish between different states so that she can
can properly construct the $S$ matrices. For each pair of Alice and Bob’s
state preparations Alice collects data from approximately $10^{6}$ shots.
Alice also needs to know the uncertainty in $S$, so she repeats the above
process 10 times, and calculates the mean and standard deviation of the
elements of $S$. By knowing which states she produces on each shot, Alice can
construct the $A$ matrices, including the uncertainty due to her ability to
control the phase, and finally she constructs the matrix $M$ and its
uncertainty $\Delta M$.

Figure 3:  Results corresponding to measurements when no errors have been
introduced. The theoretically expected values for the mean $M_{th}$ and its
standard deviation $\Delta M_{th}$ are shown in (a) and (b), while the
corresponding experimentally measured values are sown in (c) and (d). A plot
of the individual matrix elements $M_{ij}$ for the experimentally measured
data is given in (e); here the error bars represent the 95% confidence
interval [Eq. (12)]. It can be seen that the confidence interval for each
matrix element contains 0.

Shown in Fig. 3 are both theoretical and experimental matrices when we expect
there to be no correlated errors present. The theoretical matrix $M_{th}$ is 0
for no errors [Fig. 3(a)], while the corresponding experimental matrix $M$ is
nonzero due the the experimental uncertainties [Fig. 3(c)]. The theoretically
expected uncertainties $\Delta M_{th}$ [Fig. 3(b)] are calculated by using the
uncertainty in Alice’s phase shift, and by assuming that the photocount
statistics are Poissonian with a mean number of photons $N_{pho}$ equal to the
experimental mean, and an uncertainty of $\sqrt{N_{pho}}$. We can see that the
theoretically expected uncertainties are comparable to the experimental
uncertainties [Fig. 3(d)].

In order to determine whether or not the data shown in Fig. 3 are consistent
with $M=0$, we compute 95% confidence intervals ($CI$) for all the elements in
the experimentally determined $M$. These are given by

$CI_{ij}=t^{*}_{N-1}\;\frac{M_{ij}}{\Delta M_{ij}/\sqrt{N}}.$ (12)

Here $N=10$ is the number of trials and $t^{*}_{N-1}=2.262$ is the cutoff
corresponding to the 95% confidence interval of a $t$-distribution with $N-1$
degrees of freedom. As seen in Fig. 3(e), 0 is contained within the confidence
interval for every matrix element. As such, our measurements are consistent
with $M=0$, and hence no correlated errors have been detected.

Figure 4: Results corresponding to phase errors of $\pi/30$ introduced
whenever Alice sets her state to $|{-X\rangle}$. The theoretically expected
values for the mean $M_{th}$ and its standard deviation $\Delta M_{th}$ are
shown in (a) and (b), while the corresponding experimentally measured values
are sown in (c) and (d). A plot of the individual matrix elements $M_{ij}$ for
the experimentally measured data is given in (e); here the error bars
represent the 95% confidence interval [Eq. (12)]. It can be seen that the
confidence intervals for several matrix elements do not include 0.

Shown in Fig. 4 are results in which errors corresponding to a phase shift of
$\pi/30$ on Alice’s state preparations are introduced every time Alice
prepares state $|{-X\rangle}$. The theoretical and experimental matrices and
uncertainties are seen to be in good agreement. Figure 4(e) shows the
individual matrix elements and their corresponding 95% confidence intervals
for the experimental measurements. Note that 0 is not contained within the
confidence interval for several of the matrix elements. The probability that
we would find a particular value for $M_{ij}$ that is as at least as far from
0 as the measured value is given by the $p$-value of that matrix element, and
we compute the $p$-value corresponding to each matrix-element. Even a single
non-zero matrix element would mean $M\neq 0$. As such, we use the minimum
$p$-value of the matrix elements (corresponding to the most extreme matrix
element) to represent the $p$-value of the matrix. For the data of Fig. 4 we
find the $p$-value to be $2\times 10^{-8}$. The probability that the data of
Fig. 4 is consistent with no error is quite low, and we have thus detected the
$\pi/30$ errors on Alice’s preparations of the state $|{-X\rangle}$ with high
confidence.

In other experiments we have determined that we can also detect $\pi/30$ phase
errors in the preparation of the states $|{+X\rangle}$ ($p=5\times 10^{-8}$),
$|{+Y\rangle}$ ($p=5\times 10^{-5}$) and $|{-Y\rangle}$ ($p=7\times 10^{-5}$)
with similarly high confidence. We have also found that we can detect larger
phase errors in all of these states with even higher confidence. Note that we
cannot detect phase errors in the state $|{-Z\rangle}$, as this corresponds to
a state of random phase. Adding a constant phase shift to a random phase state
has no effect on the state.

##  4 Conclusions

We have demonstrated that we can detect correlated SPAM errors in TFQKD using
a technique related to self-consistent tomography. Specifically, Alice and Bob
can share a secret key by preparing single-photon level coherent states, and
using Charlie as an intermediary to report the results of joint measurements
performed on their states. Alice can detect the presence of correlated errors
without having to trust Charlie. All she needs is the information that Charlie
makes publicly available, enough information from Bob to label his different
state preparations (she does not need to know what the states actually are),
and knowledge of her own state preparations. We find that we can detect phase
shift errors as small as $\pi/30$ in the state preparations with high
confidence. It is reasonable that we should be able to detect phase errors of
this size, given that our uncertainty in controlling the phase is
$\pm\pi/100$.

While the experimental apparatus we have used was not intended to be a fully
working TFQKD system, it does incorporate all of the major pieces of such a
system. As such, we believe that our technique could be a useful tool for
detecting correlated errors in TFQKD.

Funding National Science Foundation (NSF) (1855174).

Acknowledgments We thank S. J. van Enk for helpful discussions, and for
sharing a pre-publication version of Ref. [19]. We thank A. Jones for
assistance with the statistical analysis of our data, L. Illing for the loan
of some equipment, and J. Ewing for help with fabricating equipment. B. P.
acknowledges support from the Gordon and Betty Moore Foundation.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time, but may be obtained from the authors upon
reasonable request.

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