# CYLINDRICALLY SYMMETRIC FIELDS IN GENERAL RELATIVITY

N. O. Santos 1 and Anzhong Wang 2
1Sorbonne Université, UPMC Université Paris 06, LERMA, Observatoire de
Paris/Meudon,
5 place Jules Janssen, F-92195 Meudon Cedex France.
2GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(February 2023)

###### Abstract

We present a brief review of exact solutions of cylindrical symmetric fields
in General Relativity produced by different perfect fluid sources. These
sources are assumed static, stationary, translating and collapsing. Properties
of these fields are discussed and some important open questions are called the
attention for future research.

##  1 Introduction

Globally cylindrically symmetric solutions of the Einstein field equations, at
first sight, may not seem to be physically relevant since they impose
infinitely long sources. Nonetheless, under controlled circumstances, they can
provide fairly accurate descriptions of different physical phenomena.
Important features of cylindrical systems that can help relativistic
astrophysics are, for instance, exact models for rotation and the dragging of
spacetime, models of extragalactic jets, gravitational radiation, the effect
to the cosmological constant describing dark energy, translating fluids that
might model beams of light produce by stars, and many others. For references
concerning these features see [1, 2].

Cylindrically symmetric fields started to be studied in the realm of General
Relativity dating back to 1919, when Levi-Civita (LC) [3] obtained the vacuum
solution of a static cylindrical vacuum spacetime. Ever since much has been
written by researchers trying to grasp its physical and geometrical
interpretations. However, this endeavour proved to be difficult and uncertain.
Only in 1958 Marder [4] established that the LC solution has two independent
parameters usually called $\sigma$ and $a$. Understanding the origin, geometry
and physics that lies behind these two parameters is a big challenge in
understanding the solution. For small values of $\sigma$, noted by LC himself,
the corresponding Newtonian field is the external gravitational field produced
by an infinitely long homogeneous line mass with $\sigma$ representing the
mass per unit length. In this approximation the parameter $a$ is also
associated with the constant arbitrary potential that exists in the Newtonian
solution.

In 1979 Bonnor [6] observed that $a$ is also dressed with a relevant global
topological meaning, and cannot be removed by coordinate transformations.
There is a series of obstacles and apparently contradictory properties of
$\sigma$ to allow possible interpretations. Unexpectedly, the parameter
$\sigma$ is the most difficult and elusive to be interpreted and there is a
long list of articles dedicated to unveil its meaning. In spite of that, the
Newtonian limit in cylindrical models agrees well with observations (see [7],
[8] and [9]).

Research extended LC to cylindrically stationary vacuum spacetimes, obtained
independently, by Lanczos [11] in 1924 and Lewis [12] in 1932. Its physical
interpretation is taking too a long time. In its usual form it is presented by
two families, one with four real parameters and the other with complex
parameters. Only not long ago (in 1998), it was been proved by MacCallum and
Santos [35] that the four parameters can be reduced to three independent ones.
However, the general interpretation of these parameters still remain unclear,
and the constraints relating them are unknown.

The vacuum spacetime produced by a translating source has been obtained by
Griffiths and Santos in 2009 [14]. Its extension possessing a cosmological
constant has not yet considered.

In 1925 Beck was the first to study cylindrical time-dependent vacuum
spacetime. The interpretation of these solutions as representing cylindrical
gravitational waves was given by Einstein. These same solutions were later
reobtained by Einstein and Rosen [16] in 1937. It is still unknown a
collapsing source that can be matched to this solution [17].

Our aim here is to present a brief review of the exact gravitational fields
produced by different cylindrical perfect fluid sources which are static,
rotating, under translation and under gravitational collapse. Some important
unsolved questions are addressed too.

##  2 Einstein’s field equations

The world constructed by Newton to describe his theory of gravitation had much
simplicity. Space is always equal to itself where particles move and act upon
each other. Furthermore, the gravitational effects propagate with an infinite
large velocity. This concept of space sounded not plausible to Einstein and he
came with the brilliant idea that space is the gravitational field meaning the
Newtonian space itself [51]. This needed a better formulation of the field
equations. Einstein, after a long struggle, found that the Riemann geometry,
first proposed by Gauss and generalized to any dimensions by Riemann, could
properly describe the curvature of space produced by different matter
distributions. By doing this Einstein managed to express the gravitational
field equations in an arbitrary coordinate system.

In Riemann’s geometry the parallel transport of a vector is proportional to
its curvature which is described by a quantity called the Riemann curvature
tensor $R_{\alpha\beta\gamma\delta}$. The justification of calling it
curvature lies in the fact that it vanishes if and only if the space is flat.

Einstein [52] derived the field equations in the year 1915 which are given by

$R_{\alpha\beta}-\frac{1}{2}Rg_{\alpha\beta}=\kappa T_{\alpha\beta}.$ (1)

They show how the space curvature, represented by the Ricci tensor
$R_{\alpha\beta}=R^{\gamma}_{\alpha\gamma\beta}$, and its scalar
$R=g^{\alpha\beta}R_{\alpha\beta}$ are brought about by the source of
curvature, the matter distribution given by the energy momentum tensor
$T_{\alpha\beta}$. In fact not only space curves but time as well and for this
reason we call spacetime where Einstein field equations dwell. The coupling
constant $\kappa$ in normalized units, the velocity of light $c=1$ and the
Newton’s constant $G=1$, values $\kappa=8\pi G/c^{4}=8\pi$. This system of
equations, called Einstein’s field equations, constitutes a set of 10 partial
differential equations with respect to the metric $g_{\alpha\beta}$. In the
vacuum case where $T_{\alpha\beta}=0$, these equations reduce to

$R_{\alpha\beta}=0.$ (2)

When spacetime is deprived of sources producing gravitational fields, and
thereby producing curvature, the spacetime is flat and given by the metric
$\eta_{\alpha\beta}$, called the Minkowski metric,

$ds^{2}=\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}=dt^{2}-dx^{2}-dy^{2}-dz^{2},$
(3)

with $t$ being the time coordinate and $x$, $y$ and $z$ the Cartesian
coordinates. Hence, the Newtonian limit, which one expects to be obtained from
Einstein’s field equations is produced when the metric is given by

$g_{\alpha\beta}=\eta_{\alpha\beta}+f_{\alpha\beta},$ (4)

with $f_{\alpha\beta}$ being a small deviation from the Minkowski spacetime.

##  3 LC vacuum spacetime

Einstein’s field equations (1) are highly nonlinear thus imposing huge
difficulties in finding their solutions. Furthermore, after one found their
solutions, even more difficulties arise in interpreting them. In spite of
great efforts to grasp their physical meanings as is well shown in [53], the
majority or even almost the totality of these solutions are still not well
understood given their high nonlinearity. We will consider these difficulties
in the following by studying cylindrically symmetric gravitational fields.

Schwarzschild [54] in 1916 obtained the first vacuum solution to (2)
describing the spherically symmetric vacuum field with the following line
element

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
g_{\alpha\beta}dx^{\alpha}dx^{\beta}$ (5) $\displaystyle=$
$\displaystyle\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$

where the spherical coordinates are numbered $x^{0}=t$, $x^{1}=r$,
$x^{2}=\theta$ and $x^{3}=\phi$. In the Newtonian approximation (5) becomes

$g_{00}=1+2U,$ (6)

where $U$ is the Newtonian potential and comparing to (5) we have

$U=-\frac{M}{r},$ (7)

hence the only parameter stemming from the integration of the field equations
for a spherically symmetric vacuum spacetime is the Newtonian mass $M$.

The second solution of Einstein’s field equations (2) was obtained a few years
later, 1919, after Schwarzschild presented his in 1916. It corresponds to a
cylindrical vacuum spacetime, and can be cast in the form

$ds^{2}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}=\rho^{4\sigma}dt^{2}-\rho^{4\sigma(2\sigma-1)}(d\rho^{2}+dz^{2})-\frac{1}{a}\rho^{2(1-2\sigma)}d\phi^{2},$
(8)

where the cylindrical coordinates are numbered $x^{0}=t$, $x^{1}=\rho$,
$x^{2}=z$ and $x^{3}=\phi$. In the Newtonian approximation we have
corresponding to (6),

$g_{00}=e^{2U},$ (9)

and compared to (8) for this case we have the Newtonian potential

$U=2\sigma\ln\rho.$ (10)

Hence, from this expression we clearly see that for small values of $\sigma$
it is the Newtonian mass per unit length as produced by an infinitely long
homogeneous line mass, as observed by LC himself. Ever since much has been
written by researchers trying to unveil its physics and geometrical
interpretations. However, the endeavour proved to be difficult and uncertain.

Only in 1958 did Marder [4] establish that the solution (8) contains two
arbitrary independent parameters, $\sigma$ and $a$, differently from the
Newtonian fields in which there is only one independent parameter $\sigma$.

We call the attention that for spherical vacuum case, the Schwarzschild
metric, in the relativistic and Newtonian theories there appears just one
parameter. This fact already suggests some harder difficulties in
understanding cylindrically symmetric fields, which is indeed the case.

In the following we review the main properties and physics that lies behind
the LC spacetime, which so far has been grasped up to the present time in a
large number of papers. We are aware that these results sometimes appear
contradictory since some interpretations collide with others. In fact this is
one of our main motivations to deepen in Einstein’s theory.

###  3.1 Nature of the coordinates of the LC spacetime

The LC metric given by (8) can be written in the form [10]

$ds^{2}=\rho^{4\sigma}dt^{2}-\rho^{4\sigma(2\sigma-1)}\left(d\rho^{2}+\frac{1}{a_{m}}dm^{2}\right)-\frac{1}{a_{n}}\rho^{2(1-2\sigma)}dn^{2},$
(11)

where $-\infty<t<\infty$ is the time and $0\leq\rho<\infty$ the radial
coordinates, and $\sigma$, $a_{m}$ and $a_{n}$ are constants. The nature of
the coordinates $m$ and $n$, so far unspecified, depends upon the behaviour of
the metric coefficients. Either $a_{m}$ or $a_{n}$ can be transformed away by
a scale transformation depending upon the behaviour of the coordinates $m$ and
$n$, thus leaving the metric with only two independent parameters. In orderto
find that behaviour we transform the radius $\rho$ into a proper length radial
coordinate $r$ by defining

$\rho^{2\sigma(2\sigma-1)}d\rho=dr,$ (12)

thus obtaining

$\rho=R^{1/\Sigma},\;\;R=\Sigma r,\;\;\Sigma\equiv 4\sigma^{2}-2\sigma+1,$
(13)

and metric (11) becomes

$ds^{2}=f(r)dt^{2}-dr^{2}-h(r)dm^{2}-l(r)dn^{2},$ (14)

with

$f(r)=R^{4\sigma/\Sigma},\;\;h(r)=\frac{1}{a_{m}}R^{4\sigma(2\sigma-1)/\Sigma},\;\;l(r)=\frac{1}{a_{n}}R^{2(1-2\sigma)/\Sigma}.$
(15)

Consider $0<\sigma<1/2$, which implies for this range $h(r)$ diverging when
$r\rightarrow 0$ and $l(0)=0$. Then we can interpret $m$ as the axial
coordinate $-\infty<z<\infty$ with $a_{m}=1$ by rescaling $z$ and $n$ as the
angular coordinate $\phi$ with the topological identification of every $\phi$
with $\phi+2\pi$, and the metric (14) becomes

$ds^{2}=R^{4\sigma/\Sigma}dt^{2}-dr^{2}-R^{4\sigma(2\sigma-1)/\Sigma}dz^{2}-\frac{1}{a}R^{2(1-2\sigma)/\Sigma},$
(16)

where $a_{n}$ is replaced by $a$.

Now consider $1/2<\sigma<\infty$, which implies $h(0)=0$ and $l(r)$ diverging
when $r\rightarrow 0$. Then we can interpret $m$ as the angular coordinate
$\phi$ with topological identification of every $\phi$ with $\phi+2\pi$, and
$n$ as an axial coordinate $-\infty<z<\infty$ with $a_{n}=1$ by again
rescaling $z$, and metric (14) becomes

$ds^{2}=R^{4\sigma/\Sigma}-dr^{2}-R^{2(1-2\sigma)/\Sigma}dz^{2}-\frac{1}{a}R^{4\sigma(2\sigma-1)/\Sigma}d\phi^{2},$
(17)

where we replaced $a_{m}$ for $a$.

In both cases, (16) and (17), where $\sigma>0$ we have $g_{00}\rightarrow 0$
as $r\rightarrow 0$, indicating an attractive singularity. While assuming
$\sigma<0$, we obtain $g_{00}\rightarrow\infty$ as $r\rightarrow 0$,
indicating a repulsive singularity.

The invariant quantity, under coordinate transformations, built out of the
Riemann curvature tensor given by
$\mathcal{R}=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$, called
the Kretschmann scalar, is a good indicator of singularities. Calculating
$\mathcal{R}$ for metrics $(\ref{9})$ and $(\ref{10})$ we obtain

$\mathcal{R}=\frac{64\sigma^{2}(2\sigma-1)^{2}}{\Sigma^{3}r^{4}}.$ (18)

From (18) we see that $\mathcal{R}\rightarrow\infty$ as $r\rightarrow 0$, and
$\mathcal{R}\rightarrow 0$ for $\sigma=0$, $1/2$ and $\infty$.

Metric (16) for $\sigma=0$ becomes

$ds^{2}=dt^{2}-dr^{2}-dz^{2}-\frac{1}{a}r^{2}d\phi^{2},$ (19)

representing the Minkowski spacetime when $a=1$ in cylindrical polar
coordinates $(r,z,\phi)$. However if (19) has $a>0$ there is an angle deficit
of $2\pi\delta$ given by $\delta=1-1/\sqrt{a}$, producing flat spacetime
everywhere except along the axis $r=0$, which is interpreted as a cosmic
string. The deficit can represent the tension of the string with mass per unit
length $\mu=\delta/4$. If there is an angle excess $a<1$ it would represent a
cosmic string under compression with $\mu<0$. Hence the constant $a$ is
directly linked to the gravitational analogue of the Aharonov-Bohm effect
[18]. This effect shows that gravitation depends on the topological structure
of spacetime giving rise to an angular deficit $\delta$ as in the
electromagnetic Aharonov-Bohm effect, where a (classical) non-observable
quantity (the vector potential) becomes observable (part of it) through a
quantum non-local effect. Its gravitational analogue allows a (Newtonian) non-
observable quantity (the additional constant to the Newtonian potential) to
become observable in the relativistic theory through the angular deficit in
strings. For a review in cosmic strings see [19].

In the case $\sigma=1/2$ the two metric coefficients $h$ and $l$ in (15) are
constants, so both $a_{m}$ and $a_{n}$ can be set to unity. Then neither $m$
nor $n$ is entitled to be an angular coordinate, and the three coordinates
$(r,m,n)$ are better visualized as Cartesian coordinates $(x,y,z)$. Hence
metric (11) can be written as

$ds^{2}=z^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2},$ (20)

which is the static plane symmetric vacuum spacetime obtained by Rindler [20,
14, 21].

In the next section we calculate the circular geodesics for the above
spacetime to try to get further understanding of the properties of different
values that $\sigma$ can attain for $\sigma\geq 0$, since $\sigma<0$ would in
some way correspond to negative mass densities.

###  3.2 Geodesics of LC spacetime

For the metrics (16) and (17) the circular geodesics [22] have
$\dot{r}=\dot{z}=0$ and
$g_{tt,r}{\dot{t}}^{2}+g_{\phi\phi,r}{\dot{\phi}}^{2}=0$, where the dot stands
for differentiation with respect to the affine parameter $s$. The geodesic
angular velocity is defined by $\omega={\dot{\phi}}/{\dot{t}}$, and the only
non zero component of its velocity is the tangential one, given by
$W^{\phi}=\omega\sqrt{g_{tt}}$, with it modulus defined by
$W^{2}=W^{\alpha}W_{\alpha}$ [23].

In the case $0\leq\sigma<1/2$, from (16) we obtain

$\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle\frac{2\sigma}{1-2\sigma}aR^{2(4\sigma-1)/\Sigma},$ (21)
$\displaystyle W^{2}$ $\displaystyle=$
$\displaystyle\frac{2\sigma}{1-2\sigma}.$ (22)

We note that for a given $\sigma$ the velocity $W$ in (22) is the same for all
circular geodesics, in agreement with the corresponding Newtonian gravitation.
Furthermore, we see that $W$ increases monotonically with $\sigma$, that is
from $\sigma=0$ producing $W=0$, to $\sigma=1/4$ attaining $W=1$, the speed of
light, and finally $\sigma=1/2$ producing geodesics with $W=\infty$. For small
$\sigma$ and $a=1$ from (21) and (22) we obtain the Newtonian limit $W=\omega
r$.

In the case $1/2<\sigma<\infty$ from (17) we obtain

$\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sigma-1}aR^{8\sigma(1-\sigma)/\Sigma},$ (23)
$\displaystyle W^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2\sigma-1}.$
(24)

With $\sigma$ increasing beyond $1/2$, we note from (23) that $W$ diminishes,
attaining $W=1$ for $\sigma=1$ and $W=0$ for $\sigma=\infty$.

In other words, the circular geodesics are timelike when either $0<\sigma<1/4$
or $\sigma>1$, are lightlike when $\sigma=1/4$ or $1$, and are spacelike when
$1/4<\sigma<1$.

If we redefine $\sigma$ by

$\sigma=\frac{1}{4\bar{\sigma}},$ (25)

then metric (17) with a rescaling of its coordinates becomes (16), hence (22)
and (23) reduce to (20) and (21). This means that the parameter range
$1/2<\sigma<\infty$ is equivalent to the range $0<\sigma<1/2$ and the
coordinates $z$ and $\phi$ switching their nature.

Hence we might have the following picture for the different values of
$\sigma$. For small values of $\sigma$ the metric (16) with $t$ and $r$
constants describes cylindrical surfaces with $\phi$ as the periodic
coordinate. As $\sigma$ increases the cylindrical surfaces open out and become
an infinite plane for $\sigma=1/2$. For values of $\sigma$ bigger than $1/2$
the coordinate $z$ becomes periodic forming new cylindrical surfaces
perpendicular to previous ones for $0<\sigma<1/2$.

Another interesting geodesic is the one that describes the motion of the
particle along the axis of symmetry $z$. These geodesics calculated with (16)
(we restrict the calculation of this metric since (17) is equivalent) produce

${\ddot{z}}=\frac{4\sigma(1-2\sigma){\dot{r}}{\dot{z}}}{\Sigma r}.$ (26)

It means that particles increase their speed along $z$ when distancing
radially from the axis, while diminishing their axial speed when moving
radially towards the axis. This result indicates that a force parallel to the
$z$ axis appears. In the flat case $\sigma=0$ such an effect vanishes,
bringing out its non-Newtonian nature. We further discuss this weird geodesic
property in the section containing the Lewis spacetime.

For radial geodesics it has been shown [23] that there exist timelike
particles approaching $z$ that are reflected at $r=r_{min}$ and move outwards
until attaining $\dot{r}=0$ at $r=r_{m}ax$ repeating endlessly this
trajectory. This motion is called confinment of test particles.

In the next section we see some further limits that LC metric satisfies

###  3.3 LC spacetime as a limit of the $\gamma$ spacetime

In cylindrical coordinates, static axially symmetric solutions of Einstein’s
vacuum equations are given by the Weyl metric [24]

$ds^{2}=e^{2\lambda}dt^{2}-e^{-2\lambda}[e^{2\mu}(d\rho^{2}+dz^{2})+\rho^{2}d\phi^{2}],$
(27)

with $\lambda(\rho,z)$ and $\mu(\rho,z)$ satisfying

$\lambda_{,\rho\rho}+\frac{1}{\rho}\lambda_{,\rho}+\lambda_{zz}=0,$ (28)

and

$\displaystyle\mu_{,\rho}$ $\displaystyle=$
$\displaystyle\rho(\lambda^{2}_{,\rho}-\lambda^{2}_{,zz}),$ (29)
$\displaystyle\mu_{,z}$ $\displaystyle=$ $\displaystyle
2\rho\lambda_{,\rho}\lambda_{,z},$ (30)

where the comma stands for partial derivation. Synge writes [25], as the most
amazing fact, that (28) is just the Laplace equation for $\lambda$ in the
Euclidean space. Metric (27) with the general solution of (28), (29) and (30)
is usually referred to as the $\gamma$ metric, and the corresponding functions
$\lambda$ and $\mu$ are given by

$\displaystyle e^{2\lambda}$ $\displaystyle=$
$\displaystyle\left(\frac{R_{+}+R_{-}-2m}{R_{+}+R_{-}+2m}\right)^{\gamma},$
(31) $\displaystyle e^{2\mu}$ $\displaystyle=$
$\displaystyle\left[\frac{(R_{+}+R_{-}+2m)(R_{+}+R_{-}-2m)}{4R_{+}R_{-}}\right]^{\gamma^{2}},$
(32)

where

$R^{2}_{\pm}=\rho^{2}+(z\pm m)^{2},$ (33)

and $\gamma$ and $m$ are two integration constants. These solutions were first
found by Bach and Weyl in 1922 [26]. Calculating its Newtonian potential (10)
we obtain

$U=\gamma\ln\left(\frac{R_{-}+z-m}{R_{+}+z+m}\right),$ (34)

which corresponds to a potential due to a line segment of length $2l$ and mass
per unit length $\gamma/2$ symmetrically distributed along the $z$ axis. Hence
the total mass $M$ of the line segment is $M=\gamma m$. The particular case
$\gamma=1$ corresponds to the Schwarzschild metric. This can be seen by taking
Erez-Rosen coordinates [27] defined by

$\rho^{2}=(r^{2}-2mr)\sin^{2}\theta,\;\;z=(r-m)\cos\theta,$ (35)

and the $\gamma$ metric becomes

$ds^{2}=-Fdt^{2}-\frac{1}{F}[Gdr^{2}+Hd\theta^{2}+(r^{2}-2mr)\sin^{2}\theta
d\phi^{2}],$ (36)

where

$\displaystyle F$ $\displaystyle=$
$\displaystyle\left(1-\frac{2m}{r}\right)^{\gamma},$ (37) $\displaystyle G$
$\displaystyle=$
$\displaystyle\left(\frac{r^{2}-2mr}{r^{2}-2mr+m^{2}\sin^{2}\theta}\right)^{\gamma^{2}-1},$
(38) $\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{(r^{2}-2mr)^{\gamma^{2}}}{(r^{2}-2mr+m^{2}\sin^{2}\theta)^{\gamma^{2}-1}}.$
(39)

Now we can easily check that for $\gamma=1$ the metric (36) reduces to the
Schwarzschild metric.

If we want to compare the $\gamma$ metric to the LC metric, in the limit when
its length segment $m\rightarrow\infty$, one notices that by taking this limit
in (31) and (32) the metric diverges. For this reason one can use the Cartan
scalars approach to obtain a finite limit. These scalars are the components of
the Riemann tensor and its covariant derivatives calculated in a constant
frame. Two metrics are equivalent if and only if there exist coordinate and
Lorentz transformations which transform the Cartan scalars of one of the
metrics into the Cartan scalars of the other. Although the Cartan scalars
provide a local characterization of the spacetime, global properties such as
topological defects do not appear in them. By doing all this procedure one can
prove that locally in the limit $m\rightarrow\infty$ the $\gamma$ metric is
the same as the LC metric. Details of these long calculations are given in
[28]. Here we come to an interesting and, so far, weird result showing how
apparently unexpected results can stem from long known results like the
Schwarzschild and the LC solutions. When the density per unit length of the
rod $\sigma=\gamma/2$ has the value $\gamma=1$, or the mass density per unit
length $\sigma=1/2$, it becomes the Schwarzschild spherically symmetric
spacetime, and in the limit $m\rightarrow\infty$, it becomes the Rindler
static plane symmetric vacuum spacetime. This is a remarkable result.

For all the different limiting metrics that LC spacetime can undergo see [28].
The limits for the circular geodesics of the $\gamma$ spacetime to the LC
spacetime are well studied in [23].

In the next section we make a brief review of possible sources to the LC
spacetime.

###  3.4 Sources producing LC spacetime

The LC spacetime, as we saw, contains two essential constants denoted by $a$
and $\sigma$. The constant $a$ refers to an angle deficit or excess that
produces cosmological strings. However, it is $\sigma$ that presents the most
serious obtacles to its interpretation. Indeed, for small values
$0<\sigma<1/4$, LC describes the spacetime generated by an infinite line mass,
with $\sigma$ as mass per unit coordinate length. When $\sigma=0$ the
spacetime is flat. However, circular timelike geodesics only exist for
$0<\sigma<1/4$, becoming null for $\sigma=1/4$ and being spacelike for
$\sigma>1/4$. Furthermore, as the value of $\sigma$ increases from $1/4$ to
$1/2$ the corresponding Kretchmann scalar (18) diminishes monotonically and
vanishes when $\sigma=1/2$, implying that the spacetime is flat.

Thus, the question is what does the LC metric represent outside the range
$0\leq\sigma\leq 1/4$?

First, we observe that there are known physically satisfactory fluid sources
for the LC spacetime satisfying boundary conditions for both ranges of
$\sigma$ (see for example [29]). On the other hand the fact that the scalar
Kretschmann decreases with increasing $\sigma$ may not be associated with the
strength of the gravitational field. Instead it could be associated with the
acceleration of a test particle that would measure more suitably the strength
of the gravitational field, which is the case for $1/4<\sigma<1/2$.

A possible interpretation of the LC solution is a spacetime generated by a
cylinder whose radius increases with $\sigma$ and tends to infinity as
$\sigma$ approaches $1/2$. This interpretation suggest that when $\sigma=1/2$
the cylinder becomes a plane.

It might be instructive to consider a static cylinder filled with anisotropic
perfect fluid and calculate its mass per unit length by using the junction
conditions to its exterior LC spacetime. The Whittaker formula [30] for the
active gravitational mass per unit length $\nu$ of the static distribution of
perfect fluid with energy density $\mu$ and principal stresses $P_{r}$,
$P_{z}$ and $P_{\phi}$ inside a cylinder of surface $S$ is

$\nu=2\pi\int^{r_{S}}_{0}(\mu+P_{r}+P_{z}+P_{\phi})\sqrt{-g}dr.$ (40)

Considering a static cylindrically symmetric metric

$ds^{2}=Adt^{2}-dr^{2}-Cdz^{2}-Dr^{2}d\phi^{2},$ (41)

in which $A$, $C$ and $D$ are only functions of $r$, and from Einstein’s field
equations we obtain

$\frac{A_{,rr}}{A}-\frac{1}{2}\frac{A_{,r}}{A}\left(\frac{A_{,r}}{A}-\frac{2}{r}-\frac{C_{,r}}{C}-\frac{D_{,r}}{D}\right)=\kappa(\mu+P_{r}+P_{z}+P_{\phi}).$
(42)

Substituting (42) into (40) we have the simple expression

$\nu=\frac{1}{4}\left(\frac{A_{,r}}{A}\sqrt{-g}\right)_{S},$ (43)

where regularity on the axis of symmetry [29] have been assumed. Now taking
the LC metric (16) for the exterior spacetime of the cylindrical surface $S$,
and imposing that it satisfies Darmoisj́unction conditions, [31] and [29],
with its interior spacetime described by (41) it amounts to impose the
continuity of the metric functions and its derivatives on $S$. By doing so,
from (43) we obtain

$\nu=\frac{\sigma}{\sqrt{a}},$ (44)

where $a$ is the constant defined in (16). From (44) we have that when $a>1$,
there is a topological angle deficit, then $\nu<\sigma$, and if $a<1$ there is
an angle excess producing $\nu>\sigma$. However, when there is no topological
defect, $a=1$, then $\nu$ can in fact be interpreted as mass per unit length
of its source. Furthermore, since with cylindrical sources no black holes are
formed, one might conclude that the minimum mass per unit length to form a
black hole satisfies the constraint $\nu>1/2$. This result would fulfil the
present knowledge of black holes that a lower mass limit is required for its
formation. We are aware that $\nu$ is model depended and cannot be given a
general meaning, but nonetheless it fulfils some of the expected properties.

A last comment on sources for the LC spacetime is the fact that conformally
flat static sources do not admit Darmois’ matching conditions satisfied for an
exterior LC spacetime. This result is proved in [32]. Conformal flatness of a
metric means that its corresponding Weyl tensor vanishes. The interpretation
of the Weyl tensor is that it describes the purely gravitational
characteristics of the source. This interesting result means that a static
cylindrical source deprived of its purely gravitational character cannot be
smoothly matched to the exterior LC spacetime. It is conspicuous that for
spherical symmetry this result does not stand, since there are conformally
flat static spherical sources matched to the Schwarzschild spacetime [33].

##  4 Lewis vacuum spacetime

The extension of the LC static cylindrically symmetric vacuum spacetime to a
stationary cylindrically symmetric vacuum spacetime was obtained independently
by Lanczos in 1924 [11] and Lewis in 1932 [12]. We consider the spacetime
described by the cylindrically symmetric stationary metric

$ds^{2}=fdt^{2}-2kdtd\phi-e^{\mu}(dr^{2}+dz^{2})-ld\phi^{2},$ (45)

where $f$, $k$, $\mu$ and $l$ are functions only of $r$. The ranges of the
coordinates are $-\infty<t<\infty$ for the time coordinate, $0\leq r<\infty$
for the radial coordinate, $-\infty<z<\infty$ for the axial coordinate and
$0\leq\phi\leq 2\pi$ for the angular coordinate with the hypersurfaces
$\phi=0$ and $\phi=2\pi$ being identified. The coordinates are numbered
$x^{0}=t$, $x^{1}=r$, $x^{2}=z$ and $x^{3}=\phi$. The general vacuum solution
$R_{\alpha\beta}=0$ for metric (45), in the notation given by [20] and [22],
is

$\displaystyle f$ $\displaystyle=$ $\displaystyle
ar^{1-n}-a\delta^{2}r^{1+n},$ (46) $\displaystyle k$ $\displaystyle=$
$\displaystyle-(1-ab\delta)\delta r^{1+n}-abr^{1-n},$ (47) $\displaystyle l$
$\displaystyle=$
$\displaystyle\frac{(1-ab\delta)^{2}}{a}r^{1+n}-ab^{2}r^{1-n},$ (48)
$\displaystyle e^{\mu}$ $\displaystyle=$ $\displaystyle r^{(n^{2}-1)/2},$ (49)

with

$\delta=\frac{c}{an}.$ (50)

The constants $n$, $a$, $b$ and $c$ can be either real or complex, and the
corresponding solutions belong to the Weyl class or Lewis class, respectively.
For the Lewis class these constants are given by

$\displaystyle n$ $\displaystyle=$ $\displaystyle im,$ (51) $\displaystyle a$
$\displaystyle=$ $\displaystyle\frac{1}{2}(a_{1}+b_{1})^{2},$ (52)
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{a_{2}+ib_{2}}{a_{1}+ib_{1}},$ (53) $\displaystyle c$
$\displaystyle=$ $\displaystyle\frac{m}{2}(a_{1}^{2}+b_{1}^{2}),$ (54)

where $m$, $a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$ are real constants satisfying

$a_{1}b_{2}-a_{2}b_{1}=1.$ (55)

The metric coefficients (46-49) with (51-54) become

$\displaystyle f$ $\displaystyle=$ $\displaystyle
r(a_{1}^{2}-b_{1}^{2})\cos(m\ln r)+2ra_{1}b_{1}\sin(m\ln r),,$ (56)
$\displaystyle k$ $\displaystyle=$
$\displaystyle-r(a_{1}a_{2}-b_{1}b_{2})\cos(m\ln
r)-r(a_{1}b_{2}+a_{2}b_{1})\sin(m\ln r),$ (57) $\displaystyle l$
$\displaystyle=$ $\displaystyle-r(a_{2}^{2}-b_{2}^{2})\cos(m\ln
r)-2ra_{2}b_{2}\sin(m\ln r),$ (58) $\displaystyle e^{\mu}$ $\displaystyle=$
$\displaystyle r^{-(m^{2}+1)/2}.$ (59)

A simple deduction of the Lewis metric, where one does not need to consider
complex parameters to obtain the Lewis class is given in [34]. There a
physical interpretation of the field equations is also provided, which permits
to have some understanding of the four parameters appearing in the Lewis
solution (46-49). Another derivation of the Lewis metric is given in [35]. It
is found that three parameters are essential, of which one characterizes the
local gravitational field, while the remaining two provide information about
the topological identification made to produce cylindrical symmetry.

In Newtonian physics the potential due to cylindrical matter source, being
static or stationary, has the same dependence, that is, it depends only on one
parameter, the mass per unit length. For the static vacuum cylindrical field
in General Relativity the solution is the LC metric, the one we studied in the
previous section, revealing two essential parameters, while for the stationary
cylindrical rotating source it has, in its usual form four parameters
reducible to three essential parameters [35].

In the next section we study the meaning of the parameters appearing in the
Lewis metric.

###  4.1 The parameters of the Weyl class

The transformation

$\displaystyle d\tau$ $\displaystyle=$ $\displaystyle\sqrt{a}(dt+bd\phi),$
(60) $\displaystyle d\bar{\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{n}[cdt-(n-bc)d\phi],,$ (61)

casts the metric (45) with (46-49) in the form

$ds^{2}=r^{1-n}dt^{2}-r^{(n^{2}-1)/2}(dr^{2}+dz^{2})-\frac{r^{n+1}}{a}d\phi^{2}.$
(62)

This is locally the LC metric. Nevertheless, since $\phi=0$ and $\phi=2\pi$
are identified, $\tau$ defined in (60) attains a periodic nature unless $b=0$.
On the other hand, the new coordinate $\bar{\phi}$ ranges from $-\infty$ to
$\infty$. A more detailed account of this subject can be found in [36]. In
order to globally transform the Weyl class of the Lewis metric into the static
LC metric we have to make $b=0$ and $c=0$. Note that in this case, from the
transformations (60) and (61), $\tau$ and $\bar{\phi}$ become true time and
angular coordinates. Hence we can say that $b$ and $c$ are responsible for the
non-staticity of this family of solutions of the Lewis metric.

As mention previously, the Cartan scalars provide the local characteristics of
a metric. They are obtained through the components of the Riemann tensor and
its covariant derivatives calculated in a constant frame. Two metrics are
equivalent if and only if there is a coordinate and Lorentz transformations
which transform the Cartan scalars of one of the metrics into the Cartan
scalars of the other. By performing these calculations for both metrics, the
LC metric and the Weyl class of the Lewis metric we find that both are
equivalent locally and indistinguishable, which confirms the coordinate
analysis made in the beginning of this subsection. Furthermore, we showed that
only the parameter $n$ curves spacetime for both static and stationary Weyl
class metrics. However, we shall see that the two metrics possess very
different topological behaviour.

Details of the calculations of the Cartan scalars are given in [22].

Considering a cylindrical matter source for the Weyl class metric consisting
of a rigidly rotating anisotropic fluid, one of Einsteinś field equations can
be integrated. This integration produces

$fk_{,r}-kf_{,r}=\xi r,$ (63)

where $\xi$ is an integration constant. Calculating the rotation of the source
as given in [22] produces the rotation magnitude given by $\xi/(2fe^{\mu}/2)$.
Now using the matching conditions on the surface of the source cylinder as
given by Darmois [31] we find

$c=-\frac{\xi}{2}.$ (64)

Note that this constant $c$ is different from the speed of light used in other
places of this chapter. In fact, now it measures the rotation of the
cylindrical source, as can be seen from (64).

In the Newtonian limit, the velocity term is negligible, then from (64)
$c\approx 0$, and recalling (46), we find that

$f=e^{2U}.$ (65)

Then the Newtonian potential is

$U=2\sigma\ln r+\frac{\ln a}{2},$ (66)

where $\sigma$ is given by

$\sigma=\frac{1-n}{4}.$ (67)

In Newtonian theory, (66) is the gravitational potential of an infinite
uniform line mass with mass per unit length $\sigma$. The constant $(\ln a)/2$
represents the constant arbitrary potential that exists in the Newtonian
solution. The metric (46-49) has infinite curvature, according to its Cartan
scalars, only at $r=0$ for all $n$ except $n=\pm 1$, i.e., $\sigma=0$ and
$1/2$. Thus the Weyl class metric has a singularity along the axis $r=0$, then
we can say that this spacetime is generated by an infinite uniform line source
for $0<\sigma<1/2$.

Considering the static limit of the Weyl class metric when $n=1$ ($\sigma=0)$
and $b=c=0$, we have from (46-49)

$ds^{2}=d\tau^{2}-dr^{2}-dz^{2}-\frac{r^{2}}{a}d\phi^{2},$ (68)

which is the limit of the LC metric when $\sigma=0$. In the previous section
it has been pointed that it generates strings when $a>1$ with mass per unit
length $\mu=\delta/4$ and how $a$ is directly linked to the gravitational
analog of the Aharanov-Bohm effect [18].

Considering $c=0$ and $n=1$ ($\sigma=0$) in (46-49) we have

$ds^{2}=d\tau^{2}+2b\sqrt{a}d\tau d\phi-
dr^{2}-dz^{2}-\left(\frac{r^{2}}{a}-b^{2}a\right)d\phi^{2},$ (69)

producing a locally flat spacetime. In this cas (69) represents the exterior
spacetime of a spinning string along the axis of symmetry [37] with the same
mass per untit length $\mu=\delta/4$ but with an angular momentum $J$ given by

$J=-\frac{b\sqrt{a}}{4},$ (70)

for $a>1$.

It has been shown [37] that a quantum scalar particle moving around a spinning
cosmic string as given by (69), exhibits a phase factor proportional to $J$ in
its angular momentum. It is a reminiscence of the Aharonov-Bohm effect. It is
also worth mentioning that even if $b=0$, an Aharonov-Bohm like effect exists,
thought of a different kind, appears (as commented in the static case), since
the angular momentum spectrum differs from the usual one, if only $a>1$.

###  4.2 Geodesics of Weyl class spacetime

The Weyl class of solutions is given by (45) with (46-49) and all parameters
being real, as shown above. Then for circular geodesics [38]
$\dot{r}=\dot{z}=0$, we find that
$f_{,r}{\dot{t}}^{2}-2k_{,r}{\dot{t}}{\dot{\phi}}-l_{,r}{\dot{\phi}}^{2}=0$,
where the dot stands for differentiation with respect to an affine parameter
$s$. The geodesic angular velocity is defined by
$\omega={\dot{\phi}}/{\dot{t}}$, and the velocity of the test particle has
only two non zero components, $W^{t}=k\omega/(f^{3/2}-{\sqrt{f}}k\omega)$ and
$W^{\phi}={\sqrt{f}}\omega/(f-k\omega)$, with

$\omega=\frac{c\pm n\omega_{0}}{n-b(c\pm n\omega_{0})},\;\;W=\frac{\delta
r^{n}\pm W_{0}}{1\pm\delta r^{n}W_{0}},$ (71)

where $\omega_{0}$ is the LC angular velocity and $W_{0}$ is the LC tangential
velocity,

$\omega_{0}^{2}=\left(\frac{1-n}{1+n}\right)\frac{a^{2}}{r^{2n}},\;\;W_{0}^{2}=\frac{1-n}{1+n}.$
(72)

We note that $\omega$ and $W$ vanish for $\omega_{0}=\mp c/n$ and
$W_{0}=\mp\delta r^{n}$, respectively, which are equivalent to say that the
free particle in the present case is simply static. This could come about if
the “centrifugal repulsion” balances the gravitational attraction.

The geodesic motion of a particle along the axis of symmetry $z$ for metric
(45) produces

$\ddot{z}=\frac{(1-n^{2}){\dot{r}}{\dot{z}}}{2r}.$ (73)

It is interesting to note that for this geodesic the parameters $b$ and $c$,
due to the stationarity of spacetime, do not appear in (73) and in fact it is
indistinguishable from the static limit, the LC spacetime. There is a force
that tends to damp the motion along the axis, $\ddot{z}<0$, whenever the
particle approaches the axis, $\dot{r}<0$, and reverses this tendency in the
opposite case. In the flat case, $n=1$ or $\sigma=0$, such an effect vanishes,
exposing its non-Newtonian nature.

For quasi-spherical objects it has been shown [39] that positive radial
acceleration can be produced along its axis of symmetry.

It is also worth noticing that the non-Newtonian forces parallel to the $z$
axis, also appear in the field of axially symmetric rotating bodies [40].
However the force parallel to $z$ found in [40], unlike the current case, is
directly related to the spin of the source. For the Kerr black hole, it is
shown that particles produced by the Penrose process, can be ejected from the
ergosphere surface, covering the the black hole through repulsive
gravitational fields. In this case too, unlike in the Lewis spacetime, the
gravitational repulsion is created by the spin of the Kerr black hole. These
ejected particles are highly collimated and might be a mechanism for the
observed extragalactic jets [41, 42, 43].

###  4.3 The parameters of the Lewis class

Using the transformation

$d\phi=d{\bar{\phi}}+\omega dt,\;\;\omega=-\frac{k}{l},$ (74)

the metric (56-59) can be diagonalized. In order to have an intergal
coordinate transformation $\omega$ must be a constant, therefore, from
(56-59), $m=0$. This implies, from (51-54) that $n=0$ and $c=0$. Thus the line
element becomes

$ds^{2}=-\frac{rdt^{2}}{a_{2}^{2}-b_{2}^{2}}-\frac{(dr^{2}+dz^{2})}{\sqrt{r}}-(a_{2}^{2}-b_{2}^{2})rd\bar{\phi}^{2}.$
(75)

This is a particular case of the static LC metric with the energy density per
unit length $\sigma$ given by (67) being equal to $1/4$. Nevertheless, the
transformation (74) is not global, since the new coordinate $\bar{\phi}$
ranges from $-\infty$ to $\infty$, instead of ranging from $0$ to $2\pi$ [36].

Considering, as in the case of the Weyl class, a rigidly rotating anisotropic
fluid, one of the Einstein’s field equations can be integrated producing (63).
With (56-59) the matching conditions given by Darmois [31] yield

$\xi=-m(a_{1}^{2}+b_{1}^{2}),$ (76)

and so

$c=-\frac{\xi}{2}.$ (77)

Hence, in order to have the rotation equal to zero, i.e. $\xi=0$, we need
$m=0$ since $a_{1}^{2}+b_{1}^{2}\neq 0$.

Observe the difference, at this point, between the Weyl class and the Lewis
class. In the latter the vanishing of the rotation yields a locally LC
spacetime, whereas in the former the vanishing of the rotation does not
necessarily imply that the metric can be reduced either globally or locally to
a static spacetime.

For the Cartan scalars that produce the local characteristics of a metric, we
have the following results for the Lewis class. In the Lewis class as in the
Weyl class, only the constant $n$ appears in the Cartan scalars. Nevertheless,
now $n$ must be substituted by its complex value $im$ (51). However, contrary
to the Weyl class, the Cartan scalar for the Lewis class are distinguishable
from the LC metric, except for $m=0$. Furthermore, there is no value of $m$
for which the Cartan scalars are all zero, implying at once that the Lewis
class does not include the Minkowski spcetime as a special case. This fact
implies too that there must be a lower limit to the energy per unit length of
its source. The Cartan scalars impose a upper bound on the parameter $m$,
given by

$m\leq\sqrt{3},$ (78)

since for larger values of $m$ than this, the singularity is ar $r=\infty$ and
not at $r=0$.

###  4.4 Sources producing Lewis spacetime

In a fine paper, well ahead of his time, as observed by Bill Bonnor [44], in
1937 van Stockum [45] completely solved the problem of a rigidly rotating
infinitely long cylinder of dust, including the application of adequate
boundary conditions. The solution is a remarkable one. The metric for the
interior is simple and unique depending on one parameter $w$ in our notation.
But, for the vacuum exterior, $r>R$, where $R$ is the coordinate radius of the
cylinder, there are three cases depending on the mass per unit length of the
interior. For the metric (45) we have the following results.

Case $wR<1/2$:

$\displaystyle f$ $\displaystyle=$ $\displaystyle-r\left[2\beta\cosh(2N\ln
r)+\frac{\alpha^{2}+\beta^{2}}{\alpha}\sinh(2N\ln r)\right],$ $\displaystyle
k$ $\displaystyle=$ $\displaystyle-r\left[\cosh(2N\ln
r)+\frac{\beta}{\alpha}\sinh(2N\ln r)\right],$ $\displaystyle l$
$\displaystyle=$ $\displaystyle\frac{r}{\alpha}\sinh(2N\ln r),$ $\displaystyle
e^{\mu}$ $\displaystyle=$
$\displaystyle\lambda\left(\frac{r}{R}\right)^{(4N^{2}-1)/2},$ (79)

with

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-4w^{2}R^{2})^{1/2},\;\;\alpha=\frac{(1-4w^{2}R^{2})^{1/2}}{2w^{3}R^{4}},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{2w^{2}R^{2}-1}{2w^{3}R^{4}},\;\;\lambda=e^{-w^{2}R^{2}}.$
(80)

Case $wR>1/2$:

$\displaystyle f$ $\displaystyle=$ $\displaystyle r\left[2\beta\sin(2N\ln
r)+\frac{\alpha^{2}-\beta^{2}}{\alpha}\cos(2N\ln r)\right],$ $\displaystyle k$
$\displaystyle=$ $\displaystyle r\left[\sin(2N\ln
r)-\frac{\beta}{\alpha}\cos(2N\ln r)\right],,$ $\displaystyle l$
$\displaystyle=$ $\displaystyle\frac{r}{\alpha}\cos(2N\ln r),$ $\displaystyle
e^{\mu}$ $\displaystyle=$
$\displaystyle\lambda\left(\frac{r}{R}\right)^{(1-4N^{2})/2},$ (81)

with

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{1}{2}(4w^{2}R^{2}-1)^{1/2},\;\;\alpha=\frac{(4w^{2}R^{2}-1)^{1/2}}{2w^{3}R^{4}},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{2w^{2}R^{2}-1}{2w^{3}R^{4}},\;\;\lambda=e^{-w^{2}R^{2}}.$
(82)

For the case $wR=1/2$ one obtains the relations from the limits either from
$wR<1/2$ or $wR>1/2$ which are equal. The solution for $wR<1/2$ belongs to the
Weyl class and its real parameters $n$, $a$, $b$ and $c$ assume the values

$\displaystyle n$ $\displaystyle=$
$\displaystyle(1-w^{2}R^{2})^{1/2},\;\;a=\frac{(\alpha-\beta)^{2}}{2\alpha},$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\pm\frac{1}{\alpha-\beta},\;\;c=\frac{\alpha^{2}-\beta^{2}}{\alpha}N.$
(83)

The solution $wR>1/2$ belong to the Lewis class and its real parameters $m$,
$a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$ assume the values

$\displaystyle m$ $\displaystyle=$
$\displaystyle(4w^{2}R^{2}-1)^{1/2},\;\;a_{1}=\frac{\beta}{b_{1}},\;\;a_{2}=-\frac{1}{b_{1}},$
$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle(-\alpha)^{1/2},\;\;b_{2}=0.$ (84)

For the Weyl class we have the Newtonian mass per unit length given by
$\sigma=(1-n)/4$ which implies, for Case $wR<1/2$, from (83),

$\sigma=\frac{1}{4}\left[1-(1-4w^{2}R^{2})^{1/2}\right].$ (85)

Hence (85) establishes a lower limit for $\sigma$ in the Lewis class and being
$\sigma=1/4$. This value is the frontier between the Weyl class metric and the
Lewis class metric, at least for the rotating dust solution obtained by van
Stockum [45].

For the Lewis class metric the Cartan scalars, as it was remarked, do not
admit the Minkovski spacetime. This is in accordance with the existence of a
lower limit for $\sigma$ in the van Sockum solution [45] $wR>1/2$, since with
a lower limit the source cannot be made a vacuum, and therefore the exterior
solution cannot be Minkovski.

The Cartan scalars also impose an upper bound on the parameter $m$, given by

$m\leq\sqrt{3},$ (86)

since for values of $m$ larger than this, the singularity is at $r=\infty$ and
not in $r=0$. When we substitute this value in (84), considering the equality,
we have $wR=1$.

Van Stockum solution is studied at length in [44], its properties concerning
gravitoelectric and gravitomagnetic fields in [40] and [47], its confinement
properties in [46], its extension to non-rigid rotation in [48]. A range of
stationary cylindrically symmetric perfect fluid sources are presented in [49]
and an anisotropic cylindrical stationary source can be found in [50].

##  5 Translation of cylinders

It has been proved [14] that the vacuum field produced by a rotating mass
cylinder is mathematically closely related to the field produced by a
translating mass cylinder along its axis of symmetry. Nonetheless, its
physical and geometrical properties differ significantly since the
relativistic frame dragging for rotation and translation physically differ
considerably.

We assume the general cylindrically symmetric metric with its source
translating parallel to its axis of symmetry given by

$ds^{2}=Adt^{2}-2Kdtdz-Bd\rho^{2}-Cdz^{2}-D\rho^{2}d\phi^{2},$ (87)

with the usual properties of its coordinates and $A$, $K$, $B$, $C$ and $D$
functions only of $\rho$. Making $\rho=e^{r}$ and after rescaling, the metric
(87) can be written as

$ds^{2}=fdt^{2}-2kdtdz-e^{\mu}(dr^{2}+d\phi^{2})-dz^{2}.$ (88)

The general vacuum solution for (88) is the stationary Lewis metric. So, the
vacuum solution corresponding to (87) is simply the lewis solution with the
coordinates $z$ and $\phi$ interchaged. Hence the metric coefficients in (88)
are the same as in (46-49).

In spite of the mathematical similarity between the vacuum solutions for the
fields produced by rotating and translating cylinder filled with perfect fluid
they differ substantially. Unlike the rotating case, the translating cylinder
cannot be filled with pressure free dust, as there is nothing here equivalent
to a centrifugal force that would prevent the matter collapsing to the axis.
The pressure must therefore be nonzero. Furthermore, unlike the rotating case
where matter can be rigidly rotating, which means with shear free rotation,
the translating matter case if translating rigidly can always be transformed
to a frame where the system is static [14].

The field of a cylinder of matter that is in translational motion along its
length has not been studied in detail, and any differences with the static
case are unknown. It is therefore of interest to determine whether or not
frames are dragged by motion along the cylinder in a way similar to that in
which they are dragged around it.

##  6 Gravitational collapse of cylinders

Gravitational collapse and the emission of gravitational waves has been one of
the most important problems in Einstein’s theory. However, due to the
complexity of Einstein’s field equations, even in simple cases it is not well
understood. It is well known, due to the Birkhoff’s theorem it is not possible
to have gravitational radiation in spherically symmetric spacetimes. The next
simplest symmetry assumption is cylindrical symmetry. In this sense
gravitational collapse of a cylindrical distribution of perfect fluid mass
matched to an exterior time depended vacuum field might be important since it
can be well stated and might give us hope to obtain exact solutions satisfying
matching conditions with parameters that can help us to understand the
mechanism of the eventual production of gravitational waves.

For the exterior of a cylindrically anisotropic perfect fluid under
gravitational collapse one considers the Beck-Einstein-Rosen metric, [15] and
[16], for vacuum (2) given by

$ds^{2}=e^{2(\gamma-\psi)}(dT^{2}-dR^{2})-e^{2\psi}dZ^{2}-e^{-2\psi}R^{2}d\phi^{2},$
(89)

where $\gamma$ and $\psi$ are functions of $T$ and $R$, and vacuum field
equations become

$\psi_{,TT}-\psi_{,RR}-\frac{\psi_{,R}}{R}=0,$ (90)

and

$\gamma_{,T}=2R\psi_{,T}\psi_{,R},\;\;\gamma_{,R}=R(\psi_{,T}^{2}+\psi_{,R}^{2}).$
(91)

Equation (90) is the cylindrically symmetric wave equation in an Eucledean
spacetime, suggesting the presence of a gravitational wave field. All the set
of equations that (89) has to satisfy in order to be matched, following
Darmois’ conditions, to an anisotropic perfect fluid under gravitational
collapse are given in [17]. There it is demonstrated that for a shearfree
cylindrically symmetric anisotropic fluids if the exterior spacetime is
static, i.e. the LC spacetime, the cylindrical source must be static too.
However, if the fluid is isotropic and shearfree, only a Robertson-Walker dust
interior. It is concluded as well that there is no energy transport in its
exterior.

It is still an open question the more general behaviour of a cylindrically
collapsing anisotropic fluid producing shear.

Further studies with collapsing dust have been undertaken by considering
nonzero shear but with zero expansion [55] and other simplifying assumptions
see references in [2] and [55].

##  7 Conclusions

With the beginning of the era of the gravitational wave astronomy, the strong
gravitational field regime will be soon explored observationally in various
aspects. Theoretical studies can be carried out analytically and/or
numerically. In the former, due to the complexity of Einstein’s field
equations symmetries of spacetimes are often imposed such as spherical, plane
and cylindrical [56]. Although they are all ideal models, and in realistic
situations any of these may not exist, they do provide solvable problems from
which some fundamental issues of physics can be addressed. A good example is
the Schwarzschild solution which plays a central role in the studies of black
hole physics.

In this brief review concerning different cylindrically symmetrical perfect
fluid sources, static, stationary, translating and collapsing, we present what
is so far achieved in understanding the physics and geometry. We hope that
this brief adventure into cylindrical systems in General Relativity helps to
justify its importance and motivate further studies on unsolved issues like
the ones that we called the attention in this review.

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# A fake Klein bottle with bubble

W.H. Mannan

###### Abstract

We resolve the question of the existence of a finite 2-complex with the same
fundamental group and Euler characteristic as a Klein bottle with a bubble,
but homotopically distinct to it.

MSC classes Primary: 57M05, 57K20. Secondary: 20C07, 16S34, 20C10, 55P15,
55N25

Keywords: 2-complex, Klein bottle, group presentation, homotopy equivalence,
homotopy group, Wall’s D(2) problem

Acknowledgments Thanks to Johnny Nicholson for productive discussions while
attending“A Festival Remembering Victor Snaith”, supported by The Heilbronn
Institute, The London Mathematical Society and the Snaith family. Thanks also
to the referee of the published version, for pointing out some relevant and
exciting recent work in geometric group theory.

##  1 Introduction

Among the first homotopy invariants that students of topology are typically
introduced to are the Euler characteristic and fundamental group. Among the
first classes of spaces such students may study (after finite graphs) will
often be finite 2-complexes. As such it is not uncommon for the more adept
students to question if these two invariants can distinguish between all
homotopy classes of finite 2-complexes.

It has long been known that they cannot [6, §1.7]. Lustig [10] produced
infinitely many finite 2-complexes with the same Euler characteristic and
fundamental group, but pairwise distinct second homotopy modules. Harlander
and Jensen did this in the case where the fundamental group is the trefoil
group [7], building on the initial examples of Dunwoody and Berridge [3, 4].
See [13] for more recent examples. For finite fundamental groups,
homotopically distinct finite 2-complexes with the same Euler characteristic
were found by Metzler for fundamental group ${C_{5}}^{3}$ [12, 9, p.105].
Recently the present author and Popiel did this for fundamental group $Q_{28}$
[11].

None of these spaces are easy to visualise. Nor is it easy for a novice to see
exactly what property distinguishes the spaces in question. A space that is
easy to visualise is a Klein bottle with a bubble $K^{\circ}$: formally, take
a Klein bottle $K$, remove a small disk and replace it with a $2$-sphere,
identifying the equator of the $2$-sphere with the boundary of the removed the
disk. Note $K^{\circ}$ is homotopy equivalent to the wedge $K\vee S^{2}$. For
aesthetic reasons, we prefer to work with $K^{\circ}$.

The finite $2$-complex $K^{\circ}$ has an easily visualised homotopy invariant
property. Its universal cover is the plane, with standard square tiling by
translates of a fundamental cell, and a “bubble” in each square (e.g. small
disk removed and 2-sphere inserted). See Figure 1 below.

Figure 1: The universal cover of $K^{\circ}$

Any bubble can be translated to any other bubble by a unique element of the
covering group. Collectively, the bubbles generate the second homology group
$H_{2}(\widetilde{K^{\circ}})$ over $\mathbb{Z}$. Thus
$H_{2}(\widetilde{K^{\circ}})$ is freely generated by a single element, as a
module over $\mathbb{Z}[\pi_{1}(K^{\circ})]$.

Motivated by the work of Harlander and his student Misseldine [5], [6, §1.8],
an open question in low dimensional topology for the last 14 years has been:
Is there a finite 2-complex, $X$ with the same fundamental group and Euler
characteristic as $K^{\circ}$, but with $H_{2}(\widetilde{X})$ not freely
generated as a $\mathbb{Z}[\pi_{1}(X)]$ module? We resolve this:

Theorem A. There exists a finite 2-complex $X$, with
$\pi_{1}(X)\cong\pi_{1}(K^{\circ})$ and $\chi(X)=\chi(K^{\circ})$, but with
$H_{2}(\widetilde{X})$ not freely generated as a $\mathbb{Z}[\pi_{1}(X)]$
module.

Many questions regarding the existence of a finite 2-complex with specified
(homotopy invariant) properties arise from the discovery of a finite $D(2)$
complex with those properties (a $D(n)$ complex is one with no cohomology -
twisted or otherwise - above dimension $n$). In the present case Harlander and
Misseldine discovered a finite $D(2)$ complex satisfying the conditions of
Theorem A [5].

The motivation for these questions is Wall’s $D(2)$ problem. In 1965 Wall
showed that for $n>2$, every finite $D(n)$ complex is homotopy equivalent to a
finite $n$-complex [17]. Swan and Stallings subsequently verified this when
$n=1$ [15, 16]. The case $n=2$ remains one of the major open questions in low
dimensional topology, known as Wall’s $D(2)$ problem: Is there a finite $D(2)$
complex, which is not homotopy equivalent to a finite 2-complex? If a finite
$D(2)$-complex is found with homotopy invariant properties not possessed by
any finite 2-complex, Wall’s $D(2)$ problem would be solved.

It had been hoped that Theorem A would be false, (so $H_{2}(\widetilde{X})$
would always be free for a finite 2-complex, sharing fundamental group and
Euler characteristic with $K^{\circ}$), meaning that Harlander and
Misseldine’s $D(2)$ complex would resolve the $D(2)$ problem. The hope that
$H_{2}(\widetilde{X})$ is always free came from the fact that unlike other
one-relator groups (such as the trefoil group) the relation modules for the
Klein bottle group are indeed free (see [6, Theorem 1.15] and [8]). Theorem A
shows that this approach does not suffice.

More generally, given a surface $Y$ and integer $r$, one could ask if there is
a finite 2-complex with the same fundamental group and Euler charecteristic as
$Y\vee\vee_{i=1}^{r}S^{2}$, but with non-isomorphic second homotopy module. If
$Y$ is a torus then the Quillen-Suslin theorem tells us that this cannot
happen, as the second homotopy module would be stably free, hence free.

The finite $2$-complex $X$ of Theorem A is the presentation complex of a 2
generator, 2 relation presentation. From Schanuel’s lemma we may conclude that
$H_{2}(\widetilde{X})\oplus\mathbb{Z}[\pi_{1}(X)]\cong\mathbb{Z}[\pi_{1}(X)]^{2}$.
We thank the referee for drawing attention to the recent beautiful result,
that for $Y$ a closed surface of genus greater than $e^{1000000}$, there are
no non-free modules $N$, satisfying
$N\oplus\mathbb{Z}[\pi_{1}(Y)]\cong\mathbb{Z}[\pi_{1}(Y)]^{2}$ [1, Corollary
4]. Thus for hyperbolic surfaces of high genus, there is no immediate analogue
of our construction for the Klein bottle [1, Theorem 6]. The possibility of
stably free modules with larger minimal generating sets has not been ruled
out. See [2] for further developments in this direction.

##  2 Presentations of the Klein bottle group

A finite presentation of a group $G$ consists of a set of generators
$g_{1},\cdots,g_{n}$, and a set of relators $R_{1},\cdots,R_{m}$: words in the
generators that represent the trivial element of the group. The $g_{i}$ must
generate $G$, and the $R_{j}$ must normally generate the subgroup of relators
in the free group on the $g_{i}$. In this case we write:

$\displaystyle\mathcal{P}=\langle
g_{1},\cdots,g_{n}\,|\,R_{1},\cdots,R_{m}\rangle$

is a presentation for $G$.

The Cayley complex $X_{\mathcal{P}}$ of a finite presentation $\mathcal{P}$ is
the union of a wedge (over a point) of directed loops, indexed by the
generators of $\mathcal{P}$, with a set of disks indexed by the relators of
$\mathcal{P}$, where the boundary of the disk indexed by relator $R_{j}$ is
identified with the composition of loops which $R_{j}$ expresses (reading
right to left).

By Van Kampen’s theorem, if $\mathcal{P}$ presents a group $G$, then
$\pi_{1}(X_{\mathcal{P}})=G$. Now let:

$\displaystyle\mathcal{P}=\langle x,y\,|\,y^{-1}xyx\rangle$

We have $X_{\mathcal{P}}$ is the Klein bottle $K$ (see Figure 2), so
$\mathcal{P}$ presents the fundamental group of the Klein bottle $\pi_{1}(K)$.

Figure 2: The Klein bottle as a Cayley complex

In this section we will show that:

$\displaystyle\mathcal{Q}=\langle
x,y\,|\,\,y^{-2}xy^{2}x^{-1},\,\,x^{-3}(y^{-1}xy)x^{2}(y^{-1}x^{-2}y)\rangle$

also presents $\pi_{1}(K)$. By Van Kampen’s theorem, we know that attaching a
disk to a finite complex $X$, by identifying the boundary of the disk with a
contractible loop in $X$, does not alter the fundamental group of $X$. Thus we
will be able to conclude that:

$\pi_{1}(X_{\mathcal{Q}})\cong\pi_{1}(K)\cong\pi_{1}(K^{\circ}).$

Also note that $\chi(X_{\mathcal{Q}})$ is the number of 2-cells (disks) in
$\mathcal{Q}$ minus the number of 1-cells (loops) plus the number of 0-cells
(points) which is
$2-2+1=1$. Performing the same calculation for $K=X_{\mathcal{P}}$ gives
$\chi(K)=1-2+1=0$. Finally attaching a single 2-cell to get $K^{\circ}$ we
conclude $\chi(K^{\circ})=\chi(K)+1=1$, so:

$\chi(X_{\mathcal{Q}})=1=\chi(K^{\circ}).$

That is, once we have shown that $\mathcal{P}$ and $\mathcal{Q}$ present the
same group, we will be able to conclude that $X_{\mathcal{Q}}$ and $K^{\circ}$
have the same fundamental group and Euler characteristic. In the next section
we will show that $H_{2}(\widetilde{X_{\mathcal{Q}}})$ is not freely generated
by a single element over $\mathbb{Z}[\pi_{1}(K)]$, thus proving Theorem A.

For the rest of this section we focus on showing that $\mathcal{Q}$ presents
the same group as $\mathcal{P}$. At first sight, the relators in $\mathcal{Q}$
look long and complicated. However the first relator merely tells us that
$y^{2}$ commutes with $x$. This is clearly true in the group presented by
$\mathcal{P}$ (conjugating $x$ by $y$ twice takes $x\mapsto x^{-1}\mapsto x$).
Let $a=y^{-1}xy,b=x$. The second relation of $\mathcal{Q}$ then merely states
that $b^{-3}ab^{2}a^{-2}$ is trivial, or equivalently:

$\displaystyle ab^{2}=b^{3}a^{2}.$ (1)

Again this holds in the group presented by $\mathcal{P}$, as there we have
$ab=1$. To show that $\mathcal{P}$ and $\mathcal{Q}$ present the same group,
it remains only to show that $ab=1$ holds in the group presented by
$\mathcal{Q}$. We know by definition that $y^{-1}by=a$ holds in this group.
From the first relator we then know that

$y^{-1}ay=y^{-2}xy^{2}=x=b.$

Thus conjugating both sides of (1) by $y$, we merely interchange the $a$’s and
$b$’s, to get:

$\displaystyle ba^{2}=a^{3}b^{2}.$ (2)

To complete our proof that $ab=1$ we use (1) and (2) and invoke [11, Lemma
3.1]. For completeness, we repeat the proof here.

######  Lemma 2.1.

[11, Lemma 3.1] For a group $G$, let $a,b\in G$ satisfy (1), (2). Then $ba=1$.

###### Proof.

Multiplying (1) through by $a^{2}$ on the left we get:

$a^{2}b^{3}a^{2}=a^{3}b^{2}=ba^{2},$

from (2). Thus $a^{2}b^{2}=1$ so $b^{2}a^{2}=1$ and (2) reduces to $b^{-1}=a$.
∎

Thus $\mathcal{P}$ and $\mathcal{Q}$ present the same group and we may
conclude:

######  Lemma 2.2.

We have $\chi(X_{\mathcal{Q}})=\chi(K^{\circ})$ and
$\pi_{1}(X_{\mathcal{Q}}))\cong\pi_{1}(K^{\circ}).$

##  3 Non-freeness of $H_{2}(\widetilde{X_{\mathcal{Q}}})$

Consider the cellular decomposition induced on the universal cover
$\widetilde{X_{\mathcal{P}}}$. This is a standard square tiling of the plane
(see Figure 3).

Figure 3: Cellular decomposition lifted to universal cover of Klein bottle

As the plane is contractible, the resulting cellular chain complex gives us an
exact sequence of (right) modules over $\mathbb{Z}[\pi_{1}(K)]$:

$0\to C_{2}(\widetilde{X_{\mathcal{P}}})\stackrel{{\scriptstyle
d_{2}}}{{\to}}C_{1}(\widetilde{X_{\mathcal{P}}})\stackrel{{\scriptstyle
d_{1}}}{{\to}}C_{0}(\widetilde{X_{\mathcal{P}}})\stackrel{{\scriptstyle}}{{\to}}\mathbb{Z}\to
0$

Here $C_{2}(\widetilde{X_{\mathcal{P}}})$ is freely generated by a 2-cell $D$
indexed by the relator $y^{-1}xyx$. Then $d_{2}$ is injective and exactness at
$C_{1}(\widetilde{X_{\mathcal{P}}})$ tells us that $d_{2}(D)$ freely generates
the kernel of $d_{1}$.

Now consider the cellular decomposition induced on the universal cover
$\widetilde{X_{\mathcal{Q}}}$. As $\widetilde{X_{\mathcal{Q}}}$ is simply
connected, from its cellular chain complex we obtain the following exact
sequence of (right) modules over $\mathbb{Z}[\pi_{1}(K)]$:

$\displaystyle 0\to H_{2}(\widetilde{X_{\mathcal{Q}}})\to
C_{2}(\widetilde{X_{\mathcal{Q}}})\stackrel{{\scriptstyle
d_{2}^{\prime}}}{{\to}}C_{1}(\widetilde{X_{\mathcal{Q}}})\stackrel{{\scriptstyle
d_{1}^{\prime}}}{{\to}}C_{0}(\widetilde{X_{\mathcal{Q}}})\stackrel{{\scriptstyle}}{{\to}}\mathbb{Z}\to
0$ (3)

As $\mathcal{P}$ and $\mathcal{Q}$ have the same generating set, we can
identify (3)with:

$\displaystyle 0\to H_{2}(\widetilde{X_{\mathcal{Q}}})\to
C_{2}(\widetilde{X_{\mathcal{Q}}})\stackrel{{\scriptstyle
d_{2}^{\prime}}}{{\to}}C_{1}(\widetilde{X_{\mathcal{P}}})\stackrel{{\scriptstyle
d_{1}}}{{\to}}C_{0}(\widetilde{X_{\mathcal{P}}})\stackrel{{\scriptstyle}}{{\to}}\mathbb{Z}\to
0$ (4)

Here $C_{2}(\widetilde{X_{\mathcal{Q}}})$ is freely generated over
$\mathbb{Z}[\pi_{1}(K)]$ by 2-cells $D_{1},D_{2}$ indexed by the relators of
$\mathcal{Q}$.

In the previous section we noted that the relators of $\mathcal{Q}$ are
trivial in the group presented by $\mathcal{P}$, where $y^{-1}xyx$ is the only
relator. In fact we can express the relators of $\mathcal{Q}$ as products of
conjugates of $y^{-1}xyx$.

######  Lemma 3.1.

The following identities hold in the free group on $x,y$:

$\displaystyle y^{-2}xy^{2}x^{-1}$ $\displaystyle=$
$\displaystyle(y^{-1}(y^{-1}xyx)y)(x(y^{-1}xyx)^{-1}x^{-1}),$ $\displaystyle
x^{-3}y^{-1}xyx^{2}y^{-1}x^{-2}y$ $\displaystyle=$
$\displaystyle(x^{-3}(y^{-1}xyx)x^{3})(x^{-1}(y^{-1}xyx)^{-1}x)(y^{-1}xyx)^{-1}.$

###### Proof.

Multiply out the right hand sides, and they will reduce to the left hand
sides. ∎

Conjugating a relator by a word representing the element $g$ of the presented
group corresponds to translating the boundary of its associated 2-cell by $g$
(see Figure 4). Multiplying two relators results in a relator whose associated
2-cell has boundary the sum of the boundaries of the 2-cells associated to the
two relators.

Figure 4: Conjugation action on relator / boundary of associated 2-cell

Thus from lemma 3.1 we conclude:

$\displaystyle d_{2}^{\prime}(D_{1})$ $\displaystyle=$ $\displaystyle
d_{2}(D)(y-x^{-1})$ $\displaystyle d_{2}^{\prime}(D_{2})$ $\displaystyle=$
$\displaystyle d_{2}(D)(x^{3}-x-1).$

We noted earlier that $d_{2}(D)$ freely generates the kernel of $d_{1}$. Thus:

######  Lemma 3.2.

The module $H_{2}(\widetilde{X_{\mathcal{Q}}})$ is precisely the kernel of the
module homomorphism
$\psi\colon\mathbb{Z}[\pi_{1}(K)]^{2}\to\mathbb{Z}[\pi_{1}(K)]$, mapping

$\left(\begin{array}[]{c}u\\\
v\end{array}\right)\mapsto(y-x^{-1})u+(x^{3}-x-1)v.$

Also, from the exactness of (4) we know $\psi$ is surjective.

It remains to show that $\text{ker}\,\psi$ is not a free module over
$\mathbb{Z}[\pi_{1}(K)]$.

Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of (finite) Laurent polynomials in
$x$. Then $R$ is a Noetherian domain. Let $\sigma\colon R\mapsto R$ denote the
ring automorphism mapping $x\mapsto x^{-1}$. In general we write $\sigma\colon
r\mapsto r^{\sigma}$. Let $S=R[y,y^{-1},\sigma]$ denote the ring of (finite)
Laurent polynomials in $y$, with coefficients in $R$ (on the right), where
multiplication is determined by the identity

$y^{-1}ry=r^{\sigma},$

for all $r\in R$. We have $S=\mathbb{Z}[\pi_{1}(K)]$.

######  Lemma 3.3.

We have an isomorphism of $S$-modules:

$H_{2}(\widetilde{X_{\mathcal{Q}}})\stackrel{{\scriptstyle\sim}}{{\to}}\\{v\in
S|\,\,(x^{3}-x-1)v\in(y-x^{-1})S\\}=V,$

mapping $\left(\begin{array}[]{c}u\\\ v\end{array}\right)\mapsto v$.

###### Proof.

. Given $\left(\begin{array}[]{c}u\\\ v\end{array}\right)\in\text{ker}\,\phi$,
we know $(y-x^{-1})u+(x^{3}-x-1)v=0$, so $(x^{3}-x-1)v=-(y-x^{-1})u$ and $v\in
V$, so the map is well defined. Conversely given $v\in V$, by construction we
have $u\in S$ with $(x^{3}-x-1)v=(y-x^{-1})u$, so
$\left(\begin{array}[]{c}-u\\\ v\end{array}\right)\in\text{ker}\,\phi$, and
the map is surjective.

The map is injective, as $(y-x^{-1})$ is not a zero-divisor in $S$. ∎

We follow Harlander and Misseldine [5, §2] in employing the following result
of Stafford to show that modules like $V$ are not free:

######  Theorem 3.4.

(Stafford, [14, Theorem 1.2 (Case (ii), $\delta=0$)] Let $R$ be a Noetherian
domain, $\sigma\colon R\to R$ be a ring automorphism and
$S=R[y,y^{-1},\sigma]$ be the ring of (finite) Laurent polynomials in $y$,
with coefficients in $R$, where multiplication is determined by the identity
$y^{-1}ry=r^{\sigma},$ for all $r\in R$. Then the module

$V=\\{v\in S|\,\,rv\in(y+s)S\\}$

is a non-free stably free module as long as $r,s\in R$ satisfy:

  * i)

$rS+(y+s)S=S$,

  * ii)

$sr^{\sigma}\notin rR$.

The proof is an exercise in elementary ring theory. Condition (ii) guarantees
that $V$ does not contain any degree $1$ monic polynomials in $y$. However it
is easy to show that $V$ does contain degree $1$ polynomials in $y$, as well
as monic polynomials in $y$. Thus a generator of $V$ must necessarily divide
an element that was both monic and degree $1$, contradicting (ii).

######  Theorem 3.5.

The module $H_{2}(\widetilde{X_{\mathcal{Q}}})$ is not free over
$\mathbb{Z}[\pi_{1}(K)]$.

###### Proof.

We must show that $r=x^{3}-x-1$ and $s=-x^{-1}$ satisfy the conditions of
Theorem 3.4.

  * i)

We know $rS+(y+s)S=S$ as $\psi$ is surjective (see Lemma 3.2 for geometric
reasoning). Explicitly we may write:

$(x^{3}-x-1)(x^{-3}-x^{-4}-y^{-1})-(y-x^{-1})(x^{4}-x^{2}-x)y^{-1}=1.$

  * ii)

To show $sr^{\sigma}\notin rR$ we must show that $x^{3}-x-1$ does not divide
$x^{3}+x^{2}-1$ in the ring $\mathbb{Z}[x,x^{-1}]$. Note if $x^{3}-x-1$ did
divide $x^{3}+x^{2}-1$, then it would also divide the difference between the
cubics: $x^{2}+x$. This is impossible, as $x^{2}+x$ has shorter length
(difference between highest and lowest exponents on $x$) than $x^{3}-x-1$.

∎

Theorem A then follows from theorem 3.5 and lemma 2.2.

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