# Cosmologies with turning points

Bob Holdom<EMAIL_ADDRESS>Department of Physics, University of
Toronto, Toronto, Ontario, Canada M5S 1A7

###### Abstract

We explore singularity-free and geodesically-complete cosmologies based on
manifolds that are not quite Lorentzian. The metric can be either smooth
everywhere or non-degenerate everywhere, but not both, depending on the
coordinate system. The smooth metric gives an Einstein tensor that is first
order in derivatives while the non-degenerate metric has a piecewise FLRW
form. On such a manifold the universe can transition from expanding to
contracting, or vice versa, with the Einstein equations satisfied everywhere
and without violation of standard energy conditions. We also obtain a
corresponding extension of the Kasner vacuum solutions on such manifolds.

##  I The basic picture

A fundamental premise of general relativity is that spacetime can be modelled
as a 4-dimensional Lorentzian manifold. Among the defining characteristics of
a Lorentzian manifold is the requirement that its metric be simultaneously
smooth and non-degenerate everywhere. But the leading cosmological model based
on general relativity predicts the big bang singularity, where not only the
metric is degenerate, but the curvature invariants are singular. In the
interest of avoiding such a curvature singularity, we shall consider relaxing
the requirement that the metric be simultaneously smooth and non-degenerate.
We shall make use of manifolds that are not quite Lorentzian, defined by being
either smooth everywhere or non-degenerate everywhere, but not both
simultaneously. Which of these two possibilities is realized depends on the
coordinate system. Considering such manifolds from the start yields an
enlarged solution space, and we shall find spatially homogeneous and isotropic
solutions to the Einstein equations that avoid curvature singularities. All
curvature invariants, and the components of the Einstein tensor $G_{\mu\nu}$
as well, are nonsingular for all times in both coordinate systems.

At a turning point in the evolution of the scale factor, the derivative of the
scale factor vanishes. At these isolated times certain coordinate-dependent
artifacts show up. When the metric at the turning point is smooth but
degenerate, the degeneracy is due to the vanishing of $g_{\textit{tt}}$ and
the inverse metric is singular. Components of the various curvature tensors
with enough indices raised can then become singular even though the curvature
invariants remain finite. When the metric at a turning point is non-degenerate
but nonsmooth, due to a nonsmooth scale factor, all components of all
curvature tensors remain finite. Instead, both the tensor components and the
invariants can be be nonsmooth at the turning point. ‘Nonsmooth’ shall always
mean continuous with a noncontinuous first derivative. The Einstein equations
are identically solved at all times in both coordinate systems.

The full not-quite-Lorentzian manifold may be considered to be a collection of
Lorentzian manifolds, each bounded by, but not including, the time slices
where the metric is either degenerate or nonsmooth. But the proper time along
a timelike geodesic, between any point and a turning point, is finite, as are
the corresponding coordinate times for both coordinate systems. Thus any of
these Lorentzian manifolds is geodesically incomplete. Only the full not-
quite-Lorentzian manifold, where $-\infty<t<\infty$, is geodesically complete.
(We shall discuss the geodesics below.) This is another physically desirable
feature that emerges for these manifolds. The authors of [1] emphasized the
possible geodesic completeness of a bouncing cosmology versus its lack thereof
in inflationary cosmology [2]. They considered evolution through the
singularity itself, where the scale factor vanishes, and in this way connect a
contracting phase to an expanding phase. They argued that this could happen
with a Weyl-invariant matter sector. In our case the turning points occur at
finite values of the scale factor, the singularities are avoided altogether,
and matter can be standard.

We introduce the not-quite-Lorentzian spacetimes via the smooth metric

$\displaystyle\textit{ds}^{2}$
$\displaystyle=-Dd^{\prime}(\bar{t})^{2}d(\bar{t})^{u}\textit{dt}^{2}+d(\bar{t})(\textit{dx}^{2}+\textit{dy}^{2}+\textit{dz}^{2}).$
(1)

$\bar{t}=t/\ell$ is a dimensionless time and the nonconstant function
$d(\bar{t})$, its derivative $d^{\prime}(\bar{t})\equiv\partial
d(\bar{t})/\partial\bar{t}$, and the positive constant $D$ are also all
dimensionless. The exponent $u$ is another adjustable constant. This metric
turns out to generate nonsingular curvature invariants for $u>-2$ as long as
$d(\bar{t})>0$ for all $\bar{t}$. Turning points occur at
$d^{\prime}(\bar{t})=0$ where $g_{\textit{tt}}$ vanishes and the metric
becomes degenerate.

The violation of a basic premise of general relativity via a degenerate metric
has also been considered by other authors as we detail in Section VI. Novel to
our approach is the study of metric (1). This metric has some surprising
properties that show up when solving the Einstein equations. First, the
Einstein tensor turns out to involve no more than first derivatives. And
second, a solution amounts to finding the value of the exponent $u$ that is
appropriate for a given equation state of a perfect-fluid matter source. After
that, the function $d(\bar{t})$ is still free to choose, with or without
turning points.

After displaying these features of metric (1) in the next section, we go on to
show how coordinate transformations can transform the metric into Friedmann-
Lemaitre-Robertson-Walker (FLRW) form, that is where $g_{\textit{tt}}=-1$. In
Section III we give two examples of new solutions that describe nonsingular
bounce and oscillating universes with a normal equation of state for matter.
The transformation into FLRW form yields a representation with a piecewise set
of expanding and contracting FLRW cosmologies. Propagation on the spacetime is
explored via geodesics and a wave equation. In Section IV we consider the
addition of spatial curvature and a more general matter content. In Section V
we deviate from the main line of development to extend the Kasner vacuum
solutions to those with turning points. We conclude with comments in Section
VI.

##  II Solutions and transformations

We wish to solve the Einstein equations $G_{\mu\nu}=8\pi GT_{\mu\nu}$, where
$T_{\mu\nu}$ is that for a perfect fluid,

$\displaystyle T_{\mu\nu}$ $\displaystyle=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$
$\displaystyle=\textrm{diag}((-g_{\textit{tt}})\rho,g_{\textit{xx}}p,g_{\textit{yy}}p,g_{\textit{zz}}p),$
(2)

where in the last equality we have gone to the cosmic rest frame. The equation
of state parameter $w$ is defined by $p=w\rho$. The nonzero components of the
$G_{\mu\nu}$ for metric (1) are

$\displaystyle
G_{\textit{tt}}=\frac{3}{4\ell^{2}}\frac{d^{\prime}(\bar{t})^{2}}{d(\bar{t})^{2}}\quad
G_{\textit{xx}}=G_{\textit{yy}}=G_{\textit{zz}}=\frac{(1+2u)}{4D\ell^{2}}\frac{1}{d(\bar{t})^{1+u}}.$
(3)

The components are finite and smooth everywhere, including at any turning
point. There are no second derivatives and there are no differential equations
to be solved. The choice $u=(3w-1)/2$ is sufficient for a solution, for then
$G_{\mu\nu}=8\pi GT_{\mu\nu}$ with (2) immediately gives

$\displaystyle\rho(\bar{t})$ $\displaystyle=\frac{1}{8\pi
G\ell^{2}}\frac{3}{4D}\frac{1}{d(\bar{t})^{\frac{3}{2}(1+w)}},\quad
p(\bar{t})=w\rho(\bar{t}).$ (4)

Note that the particular power of $d(\bar{t})$ in (4) is the standard result
that also follows from energy conservation. We see that the Einstein equations
are solved for metric (1) without the occurrence of the Friedmann equations
and that $d(\bar{t})$ is still free to choose.

Curvature invariants also have a simple, non-derivative dependence on
$d(\bar{t})$, as illustrated by these two,

$\displaystyle R$
$\displaystyle=\frac{1}{\ell^{2}}\frac{3}{4D}(1-3w)\frac{1}{d(\bar{t})^{\frac{3}{2}(1+w)}},$
$\displaystyle R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
$\displaystyle=\frac{1}{\ell^{4}}\frac{3}{16D^{2}}(9w^{2}+6w+5)\frac{1}{d(\bar{t})^{3(1+w)}}.$
(5)

Thus the physical quantities in (4) and (5) and others are well behaved as
long as $d(\bar{t})>0$, whether or not it has turning points.

We now consider the behaviour of metric (1) under a coordinate transformation
of the form $\bar{t}=f(\bar{t}_{\rm new})$, where the function $f$ is one-to-
one and $\bar{t}_{\rm new}=t_{\rm new}/\ell$. Defining $d_{\rm
new}(\bar{t}_{\rm new})=d(f(\bar{t}_{\rm new}))$, the new metric is

$\displaystyle\textit{ds}^{2}$ $\displaystyle=-Dd_{\rm
new}^{\prime}(\bar{t}_{\rm new})^{2}d(\bar{t}_{\rm new})^{u}\textit{dt}_{\rm
new}^{2}+d_{\rm new}(\bar{t}_{\rm
new})(\textit{dx}^{2}+\textit{dy}^{2}+\textit{dz}^{2}).$ (6)

The derivative is now with respect to $\bar{t}_{\rm new}$, and so we can say
that the metric is form invariant under such transformations. Note that the
factors of $\partial\bar{t}/\partial\bar{t}_{\rm new}$ in the transformation
of the metric are absorbed by the transformation of the square of the
derivative in the metric. If the function $d$ is one-to-one then we can go
further and choose the transformation $\bar{t}=d^{-1}(d_{\rm new}(\bar{t}_{\rm
new}))$. We specify that $d_{\rm new}(\bar{t}_{\rm new})$ is one-to-one and
has the same range as $d$, but otherwise it is free to choose. If the function
$d$ has turning points and is thus many-to-one, we can make a transformation
as just described over each range of $\bar{t}$ where $d$ is one-to-one, that
is, in a piecewise fashion. Combining these transformation yields a full
transformation $\bar{t}=f(\bar{t}_{\rm new})$ that is one-to-one and
continuous as both $\bar{t}$ and $\bar{t}_{\rm new}$ range from $-\infty$ to
$\infty$.

We shall be interested in making such a coordinate transformation that takes a
solution in the form of metric (1) with $u=(3w-1)/2$ to FLRW form. We just
need to find a solution to $g_{\textit{tt}}=-1$, that is,

$\displaystyle Dd_{\rm new}^{\prime}(\bar{t}_{\rm new})^{2}d_{\rm
new}(\bar{t}_{\rm new})^{(3w-1)/2}=1,$ (7)

subject to the constraints on $d_{\rm new}$ as we have specified. We can label
such a solution by $a(\bar{t}_{\rm new})^{2}\equiv d_{\rm new}(\bar{t}_{\rm
new})$, thus giving the metric

$\displaystyle\textit{ds}^{2}$ $\displaystyle=-\textit{dt}_{\rm
new}^{2}+a(\bar{t}_{\rm
new})^{2}(\textit{dx}^{2}+\textit{dy}^{2}+\textit{dz}^{2}).$ (8)

$a(\bar{t}_{\rm new})$ is then the standard FLRW scale factor and
$a(\bar{t}_{\rm new})$ will satisfy the standard Friedman equations. We have
transformed from a degenerate metric to a non-degenerate metric, and this
requires a transformation $\bar{t}=f(\bar{t}_{\rm new})$ such that
$\partial\bar{t}/\partial\bar{t}_{\rm new}\to\infty$ at each turning point. As
we shall see, the result is that $a(\bar{t}_{\rm new})$ is nonsmooth at the
turning points.

##  III Cosmology and Propagation

We now consider an explicit example of a solution where $d(\bar{t})$ has a
turning point. We take

$\displaystyle d(\bar{t})=s+\bar{t}^{2},\quad s>0,$ (9)

for $-\infty<\bar{t}<\infty$. This gives a universe that is contracting for
negative times, expanding for positive times, and that bounces off its minimum
finite size at $\bar{t}=0$. The bounce is smooth and nonsingular. To go over
to FLRW form we solve (7) with the conditions $d_{\rm new}(\pm\infty)=\infty$
and $d_{\rm new}(0)=s$. This results in

$\displaystyle a(\bar{t}_{\rm new})^{2}$
$\displaystyle=\left(s^{v}+{\textstyle\frac{v}{\sqrt{D}}}|\bar{t}_{\rm
new}|\right)^{\textstyle\frac{1}{v}},\quad v=\frac{3w+3}{4}.$ (10)

We could have started with any $d(\bar{t})$ with a single turning point at
$\bar{t}=0$ with $d(0)=s$ and $d(\pm\infty)=\infty$, since they are all
solutions; and they all yield (10) in FLRW coordinates. The value of $\nu$
here is determined by the desired value of the equation of state parameter
$w$, for example $w=1/3$ or $w=0$ for radiation or matter domination and where
any $w>-1$ satisfies the null energy condition. We thus have a standard FLRW
expanding phase that starts at $t_{\rm new}=0$ when the universe has finite
size. This is preceded at negative times by a FLRW contracting phase.

The second example is oscillatory,

$\displaystyle d(\bar{t})=s+1-\cos(\bar{t}),\quad s>0.$ (11)

To transform to FLRW form we solve (7), again with $d_{\rm new}(0)=s$, but now
with the constraint that $d_{\rm new}(\bar{t}_{\rm new})$ only increases to
$2+s$ before turning back down. The result for one period of the oscillation
is

$\displaystyle a(\bar{t}_{\rm new})^{2}$
$\displaystyle=\begin{cases}\left(s^{v}+\frac{v}{\sqrt{D}}\bar{t}_{\rm
new}\right)^{\textstyle\frac{1}{v}}&0<\bar{t}_{\rm new}<{\bar{t}}_{\rm
half}\\\ \left((2+s)^{v}-\frac{v}{\sqrt{D}}(\bar{t}_{\rm new}-{\bar{t}}_{\rm
half})\right)^{\textstyle\frac{1}{v}}&{\bar{t}}_{\rm half}<\bar{t}_{\rm
new}<2{\bar{t}}_{\rm half}\end{cases}$ (12)

${\bar{t}}_{\rm half}$ is the value of ${\bar{t}}_{\rm new}$ at half the
period,

$\displaystyle{\bar{t}}_{\rm
half}={\frac{\sqrt{D}}{v}}\left((2+s)^{v}-s^{v}\right).$ (13)

Each period of the original oscillation is now represented by a standard FLRW
expanding phase followed by a FLRW contracting phase. The universe alternates
between growing up to size $a=\sqrt{2+s}$ and then shrinking down to size
$a=\sqrt{s}$. Both phases last the same amount of time, which is
${\bar{t}}_{\rm half}\ell$.

For both examples the expanding and contracting FLRW solutions meet at turning
points where $a^{\prime}(\bar{t}_{\rm new})$ is not continuous. Nevertheless,
FLRW time is the standard cosmic time, and so when the turning point arrives,
a cosmic expansion will seem to instantaneously change to a cosmic
contraction, or vice versa. The acceleration $a^{\prime\prime}(\bar{t}_{\rm
new})$ (which as usual is negative when $w>-\frac{1}{3}$) remains continuous,
as does $a^{\prime}(t_{\rm new})^{2}$. Thus the substitution of these
solutions into the standard Friedman equations introduces no discontinuity,
that is, the Einstein tensor $G_{\mu\nu}$ remains continuous. The matter
energy density and pressure are similarly nonsmooth at turning points (unlike
in (4) where they were smooth) in such a way that the equation of state
$p(\bar{t}_{\rm new})=w\rho(\bar{t}_{\rm new})$ continues to be satisfied at
all times.

We now consider the propagation of particles in these spacetimes. The particle
energy is defined by $E=-u_{\mu}(dx^{\mu}/d\lambda)$ where $\lambda$ is an
affine parameter in the case of massless particles, or the proper time in the
case of massive particles (in which case $E$ is replaced by $E/m$). We observe
the particle in the cosmic rest frame where only
$u_{0}=-\sqrt{-g_{\textit{tt}}}$ is nonvanishing. If the particle travels in
the $x$ direction we make use of the Killing vector $K=\partial_{x}$, or
$K_{\mu}=(0,g_{\textit{xx}},0,0)$, and the relations

$\displaystyle K_{\mu}\frac{dx^{\mu}}{d\lambda}=\kappa,\quad
g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=-\epsilon,$ (14)

to obtain the result

$\displaystyle\frac{dt}{d\lambda}=\sqrt{\frac{\kappa^{2}+\epsilon
g_{\textit{xx}}}{(-g_{\textit{tt}})g_{\textit{xx}}}},$ (15)

where $\epsilon=0$ or 1 and the constant $\kappa=p$ or $p/m$ for massless or
massive particles respectively. This relation can be used in either coordinate
system; in FLRW coordinates $t\to t_{\rm new}$.

The dependence on $g_{\textit{tt}}$ means that $dt/d\lambda$ diverges at a
turning point for metric (1), while it remains finite for FLRW coordinates.
But the dependence on $g_{\textit{tt}}$ cancels in the definition of energy,
and in particular for a massless particle we have $E\propto
d(\bar{t})^{-\frac{1}{2}}$ and $E\propto a(\bar{t}_{\rm new})^{-1}$ for the
two coordinate systems respectively. The difference is that $d(\bar{t})$ is
smooth, and so the transition between redshifting and blueshifting at a
turning point is smooth for metric (1), while it is nonsmooth in FLRW
coordinates. As we have seen, this mirrors the behaviour of other physical
quantities.

The apparent coordinate velocities $dx/dt$ and $dx_{\rm new}/dt_{\rm new}$ can
also be obtained from the above. For $dx/dt$, the velocity increases close to
the turning point but then drops to be instantaneously zero at the turning
point. $dx/dt$ is symmetric around the turning point and is nonsmooth at the
turning point. For $dx_{\rm new}/dt_{\rm new}$, the velocity picks up a small
nonsmooth component proportional to $|t_{\rm new}-t_{\rm new}^{\rm tp}|/\ell$
close to the turning point, where we have reinstated $\ell$. This nonsmooth
behaviour can be negligible since $\ell$ is related to the size of the
universe.

We may also consider the scalar wave equation $\Box\phi(\bar{t},\bar{X})=0$
where $X=(x,y,z)$. This equation with metric (1) implies that at the time of a
turning point $\bar{t}^{\rm tp}$ we must have
$\partial_{\bar{t}}\phi(\bar{t},\bar{X})|_{\bar{t}=\bar{t}^{\rm tp}}=0$, and
as a result of this, the scalar quantity
$g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ remains finite. In FLRW
coordinates this scalar is obviously finite, and instead, like other
invariants, it is nonsmooth. In these coordinates the wave itself will have a
nonsmooth component that is proportional to $|t_{\rm new}-t_{\rm new}^{\rm
tp}|/\ell$ close to the turning point. Again this is typically negligible
compared to the normal time variation of the wave.

##  IV Generalization

Thus far we have described cosmologies where $T_{\mu\nu}$ has a single
component described by $p=w\rho$, for some $w$. We shall generalize metric (1)
in this section, so as to accommodate the more realistic situation of having
several components contribute to $T_{\mu\nu}$. First let us show another need
for this generalization by modifying metric (1) to include a spatial curvature
parameterized by $k$. This is done in the usual way via spherical coordinates
as follows,

$\displaystyle\textit{ds}^{2}$
$\displaystyle=-Dd^{\prime}(\bar{t})^{2}d(\bar{t})^{u}\textit{dt}^{2}+d(\bar{t})\left(\frac{1}{1-k\bar{r}^{2}}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin(\theta)^{2}d\phi^{2}\right).$
(16)

Now when we take $u=(3w-1)/2$ and calculate $-G_{\textit{tt}}/g_{\textit{tt}}$
and $G_{\textit{xx}}/g_{\textit{xx}}$ we find two contributions to each. One
corresponds to the original equation of state $p=w\rho$ as before. The other
is proportional to $k$ and corresponds to an equation of state
$p=-\frac{1}{3}\rho$, and thus $\rho(\bar{t})\propto 1/d(\bar{t})$ for this
component. This corresponds to the spatial curvature term that appears in the
Einstein equations when the FLRW metric is used. In our case the Einstein
equations are algebraic and they are no longer solved (that is for arbitrary
$d(\bar{t})$ and assuming that the actual matter has $p=w\rho$ with
$w\neq-\frac{1}{3}$). In other words the additional effective contribution to
$T_{\mu\nu}$ spoils the solution.

Before turning to the required generalization of metric (1), let us briefly
consider another effective contribution to $T_{\mu\nu}$. This is due to
corrections to the field equations when curvature-squared terms are added to
the action. Metric (1) has a vanishing Weyl tensor and a vanishing Bach
tensor, and so a Weyl-squared term gives no corrections. An $R^{2}$ term
corrects the field equations with a term proportional to

$\displaystyle\left(R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R+g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)R.$
(17)

By evaluating this term with metric (1) with $u=(3w-1)/2$, we find that it
corresponds to a new effective contribution to $T_{\mu\nu}$ with

$\displaystyle\\!\rho(\bar{t})\propto\frac{1}{\ell^{4}}\frac{5+2\tilde{w}-3\tilde{w}^{2}}{d(\bar{t})^{\frac{3}{2}(1+\tilde{w})}},\quad\\!p(\bar{t})=\tilde{w}\\!\rho(\bar{t}),$
(18)

where $\tilde{w}=1+2w$. This contribution has $\\!\rho(\bar{t})\propto
1/d(\bar{t})^{3(1+w)}$ as compared to (4), and there is a suppression factor
of order $G/\ell^{2}$.

So let us turn to the generalization needed to deal with a $T_{\mu\nu}$ having
several components, each with its own equation of state. If we consider an
effective equation of state for the complete $T_{\mu\nu}$, it is sufficient to
allow this to depend on the size of the universe. We use

$\displaystyle p(\bar{t})=w(d(\bar{t}))\rho(\bar{t}),$ (19)

where $p$ and $\rho$ now include all contributions. We generalize metric (1)
to

$\displaystyle\textit{ds}^{2}$
$\displaystyle=-Dd^{\prime}(\bar{t})^{2}F(d(\bar{t}))\textit{dt}^{2}+d(\bar{t})(\textit{dx}^{2}+\textit{dy}^{2}+\textit{dz}^{2}),$
(20)

where we now have a function $F$ instead of a power of $d(\bar{t})$. It is
useful to define

$\displaystyle{\cal
D}(\bar{t})=\exp\left({\int^{\bar{t}}w(d(\tilde{t}))\frac{d^{\prime}(\tilde{t})}{d(\tilde{t})}\,d\\!\tilde{t}}\right)=\exp\left({\int^{d(\bar{t})}w(\tilde{d})\,d\\!\ln(\tilde{d})}\right).$
(21)

${\cal D}(\bar{t})\to d(\bar{t})^{w}$ when $w(d(\bar{t}))$ is simply a
constant $w$. We find that $G_{\mu\nu}=8\pi GT_{\mu\nu}$ is solved with
$T_{\mu\nu}$ incorporating the equation of state in (19) when

$\displaystyle F(d(\bar{t}))={\cal
D}(\bar{t})^{\frac{3}{2}}d(\bar{t})^{-\frac{1}{2}}.$ (22)

$G_{\mu\nu}$ is still first order in derivatives and these solutions can once
again be such that $d(\bar{t})$ has turning points.

The previous results in (4-5) can be updated for the generalized metric (20)
with (22),

$\displaystyle\rho(\bar{t})$ $\displaystyle=\frac{1}{8\pi
G\ell^{2}}\frac{3}{4D}\frac{1}{d(\bar{t})^{\frac{3}{2}}}\frac{1}{{\cal
D}(\bar{t})^{\frac{3}{2}}},$ $\displaystyle R$
$\displaystyle=\frac{1}{\ell^{2}}\frac{3}{4D}(1-3w(d(\bar{t})))\frac{1}{d(\bar{t})^{\frac{3}{2}}}\frac{1}{{\cal
D}(\bar{t})^{\frac{3}{2}}},$ $\displaystyle
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
$\displaystyle=\frac{1}{\ell^{4}}\frac{3}{16D^{2}}(9w(d(\bar{t}))^{2}+6w(d(\bar{t}))+5)\frac{1}{d(\bar{t})^{3}}\frac{1}{{\cal
D}(\bar{t})^{3}}.$ (23)

The metric (20) continues to be form invariant under the coordinate
transformations we have discussed. The transformation to FLRW form is once
again of the form $\bar{t}=d^{-1}(a(\bar{t}_{\rm new})^{2})$ for the finite
ranges of $t$ where $d(\bar{t})$ is one-to-one. Explicitly finding
$a(\bar{t}_{\rm new})^{2}$ amounts to finding a solution to
$g_{\textit{tt}}=-1$ using (20) and (22). Finding the scale factor this way is
entirely equivalent to starting with the FLRW metric and using the Einstein
equations to solve for the scale factor using the equation of state in (19).

##  V Extending Kasner

Similar to the way not-quite-Lorentzian manifolds extend the solution space
for homogeneous and isotropic metrics, they can also extend the solution space
for metrics that are homogeneous but not isotropic. The relevant known
solutions of this type are the Kasner solutions of the vacuum Einstein
equations. Let us give the new solutions first; the following smooth metric
solves $G_{\mu\nu}=0$ for arbitrary choice of the function $d(\bar{t})$ and
the constants $u_{1}$ and $u_{2}$,

$\displaystyle\textit{ds}^{2}$
$\displaystyle=-Dd^{\prime}(\bar{t})^{2}d(\bar{t})^{2v-2}\textit{dt}^{2}+d(\bar{t})^{-\frac{u_{2}u_{1}}{u_{1}+u_{2}}}\textit{dx}^{2}+d(\bar{t})^{u_{1}}\textit{dy}^{2}+d(\bar{t})^{u_{2}}\textit{dz}^{2},$
(24) $\displaystyle v=\frac{u_{1}^{2}+u_{2}u_{1}+u_{2}^{2}}{2(u_{1}+u_{2})}.$

At turning points of $d(\bar{t})$, the metric becomes degenerate.

In the special case of a power law $d(\bar{t})=\bar{t}^{\alpha}$ and
$D=1/\alpha^{2}$ we recover the original form of the Kasner metric (see eq.
(13.51) in [3]),

$\displaystyle\textit{ds}^{2}$
$\displaystyle=-\bar{t}^{2a_{4}}\textit{dt}^{2}+\bar{t}^{2a_{1}}\textit{dx}^{2}+\bar{t}^{2a_{2}}\textit{dy}^{2}+\bar{t}^{2a_{3}}\textit{dz}^{2},$
$\displaystyle a_{1}=-\frac{\alpha}{2}\frac{u_{2}u_{1}}{u_{1}+u_{2}},\quad
a_{2}=\frac{\alpha}{2}u_{1},\quad a_{3}=\frac{\alpha}{2}u_{2},\quad
a_{4}=\alpha v-1.$

These exponents satisfy the Kasner conditions,

$\displaystyle a_{1}+a_{2}+a_{3}=a_{4}+1,\quad
a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=(a_{4}+1)^{2}.$ (25)

The quadratic curvature invariants from metric (24) are

$\displaystyle R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
$\displaystyle=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=\frac{1}{\ell^{4}}\frac{2v}{D^{2}}\frac{u_{1}^{2}u_{2}^{2}}{u_{1}+u_{2}}d(\bar{t})^{-4v}.$
(26)

These curvatures remain bounded as long as $d(\bar{t})$ is positive and
bounded, for either sign of $v$. Since we are free to choose $d(\bar{t})$ in
metric (24) and still have a solution, we can thus extend the Kasner solutions
to solutions that are everywhere nonsingular. For example we can again choose
a bounce solution as in (9) (here we need $v>0$ since $d(\bar{t})$ is
unbounded from above in this case) or an oscillating solution as in (11).

Metric (24) is again form invariant under transformations of the form
$\bar{t}=f(\bar{t}_{\rm new})$, such that $d(\bar{t})$ is simply replaced by
$d_{\rm new}(\bar{t}_{\rm new})=d(f(\bar{t}_{\rm new}))$ in the transformed
metric. It is standard practice to transform the Kasner metric into a form
where $g_{\textit{tt}}=-1$, and we can do the same for metric (24). The new
function $d_{\rm new}(\bar{t}_{\rm new})$ in this case must be a solution of

$\displaystyle Dd_{\rm new}^{\prime}(\bar{t}_{\rm new})^{2}d_{\rm
new}(\bar{t}_{\rm new})^{2v-2}=1.$ (27)

The solution is

$\displaystyle d_{\rm new}(\bar{t}_{\rm
new})=\left(c\pm{\textstyle\frac{v}{\sqrt{D}}}\bar{t}_{\rm
new}\right)^{\textstyle\frac{1}{v}},$ (28)

for an arbitrary constant $c$, and so the transformed metric is

$\displaystyle\textit{ds}^{2}$ $\displaystyle=-\textit{dt}_{\rm
new}^{2}+(c\pm{\textstyle\frac{v}{\sqrt{D}}}\bar{t}_{\rm
new})^{2p_{1}}\textit{dx}^{2}+(c\pm{\textstyle\frac{v}{\sqrt{D}}}\bar{t}_{\rm
new})^{2p_{2}}\textit{dy}^{2}+(c\pm{\textstyle\frac{v}{\sqrt{D}}}\bar{t}_{\rm
new})^{2p_{3}}\textit{dz}^{2},$ (29) $\displaystyle
p_{1}=-\frac{u_{1}u_{2}}{u_{1}+u_{2}}\frac{1}{2v},\quad
p_{2}=\frac{u_{1}}{2v},\quad p_{3}=\frac{u_{2}}{2v}.$

These exponents satisfy the Kasner relations,

$\displaystyle p_{1}+p_{2}+p_{3}=1,\quad p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1,$
(30)

and thus we have ended up with the common form of the Kasner metric.

The result in (28) assumes that the original $d(\bar{t})$ is one-to-one. If it
is many-to-one, then as before, the transformation must be done in a piecewise
fashion over every subrange of $\bar{t}$ where $d(\bar{t})$ is one-to-one.
This will result in a set of expansions and contractions, each of the form in
(28), that meet at turning points where the evolution, as described by the
complete $d_{\rm new}(\bar{t}_{\rm new})$, is nonsmooth. In this section we
are dealing with the vacuum Einstein equations, and these equations are
identically solved by the nonsmooth metric at all times, just as they are for
(24). Thus we have found the analog of Kasner solutions on a not-quite-
Lorentzian manifold, where they can be singularity free and geodesically
complete.

The Kasner solutions are such that the dependence on $t$ can be traded for any
other coordinate, and the sign of each component of the metric is free to
choose. This is also true for our extension of the Kasner metric, and so
analogous alterations to the metric (24) will give solutions such as

$\displaystyle\textit{ds}^{2}$
$\displaystyle=-a(\bar{x})^{-\frac{u_{2}u_{1}}{u_{1}+u_{2}}}\textit{dt}^{2}+Aa^{\prime}(\bar{x})^{2}a(\bar{x})^{2v-2}\textit{dx}^{2}+a(\bar{x})^{u_{1}}\textit{dy}^{2}+a(\bar{x})^{u_{2}}\textit{dz}^{2}.$
(31)

##  VI Comments

We return to a universe that is close to being homogeneous and isotropic.
Depending on how small the universe was at the last turning point, the
universe could contain structure that is older than that time. Also, entropy
tends to increase from one phase to the next [4]. These complications were not
present in our discussion of Section III, where we had a single-component
perfect fluid and a time evolution that was symmetric around each turning
point. We generalized the equation of state parameter to be dependent on the
size of the universe in Section IV, and a further generalization would be to
assume that this function of size is different for each phase, and in
particular different between expanding and contracting phases. Work on the
bounce and oscillating universes has a long history; see [5] for a historical
review and [6] for some modern motivation.

We have not discussed the stability of our solutions. But when viewed in FLRW
coordinates, our solutions are locally just the standard FLRW solutions, up to
a set of times of measure zero. The matter at all times satisfies standard
energy conditions. It is thus difficult to see how instabilities can occur.
Stability has been more of a concern for attempts to describe a nonsingular
bounce cosmology on a Lorentzian manifold, where some violation of the null
energy condition during the bounce seems necessary. Constructions involving
nontrivial scalar field dynamics have been proposed as a way to solve that
stability problem [7].

Something similar to our bounce cosmology, also based on a metric degenerate
at $t=0$, has been studied in [8] (and in references therein). In that
approach, the FLRW metric is modified by replacing $g_{\textit{tt}}=-1$ by the
ansatz $g_{\textit{tt}}=-t^{2}/(t^{2}+b^{2})$ for some constant $b$, and then
obtaining the scale factor $a(t)$ via the Einstein equations that now depend
on $b$. The transformation of the metric to standard FLRW form was also
obtained; the new time coordinate is not continuous, jumping from a value of
$-b$ to a value of $b$ as the turning point is traversed. These two times are
then identified. Perturbations around the metric in the modified FLRW form
were studied in [9] with apparently acceptable results.

Rather than stability, our solutions raise the issue of determinism. We have a
metric that provides cosmological solutions to the Einstein equations for any
smooth and positive $d(\bar{t})$. Some such function could in principle have a
set of turning points that occur randomly as a function of time, and such a
cosmology would randomly switch between expansion and contraction. This could
be viewed as an indeterminism in the cosmic evolution as described by the
Einstein equations, when spacetime is a not-quite-Lorentzian manifold.

There may be implications for a quantum theory. It was argued in [10] that the
path integral for quantum gravity should at least include all metrics with a
finite action, whether or not they are degenerate at isolated points. This
reference also observed what we have noted, that degeneracy at isolated points
still allows the Einstein equations to be solved everywhere and it still
allows curvature invariants and other scalars to be finite everywhere. The
focus of that work was on degenerate metrics that could lead to topology
change, but metric we have investigated does not appear to be of that type.

## References

  * [1] I. Bars, P. Steinhardt and N. Turok, “Local Conformal Symmetry in Physics and Cosmology,” Phys. Rev. D 89, 043515 (2014) doi:10.1103/PhysRevD.89.043515 arXiv:1307.1848 [hep-th]
  * [2] A. Borde, A. H. Guth and A. Vilenkin, “Inflationary space-times are incomplete in past directions,” Phys. Rev. Lett. 90, 151301 (2003) doi:10.1103/PhysRevLett.90.151301, arXiv:gr-qc/0110012 [gr-qc].
  * [3] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E. (2003). Exact Solutions of Einstein’s Field Equations (2nd ed., Cambridge Monographs on Mathematical Physics). Cambridge: Cambridge University Press, doi:10.1017/CBO9780511535185
  * [4] R. C. Tolman, Relativity, Thermodynamics and Cosmology (Clarendon Press, Oxford, 1934).
  * [5] Kragh, H. (2009), Continual Fascination: The Oscillating Universe in Modern Cosmology. Science in Context, 22(4), 587-612, doi:10.1017/S0269889709990172
  * [6] A. Ijjas and P. J. Steinhardt, “Bouncing Cosmology made simple,” Class. Quant. Grav. 35, no.13, 135004 (2018) doi:10.1088/1361-6382/aac482, arXiv:1803.01961 [astro-ph.CO]
  * [7] A. Ijjas and P. J. Steinhardt, “Fully stable cosmological solutions with a non-singular classical bounce,” Phys. Lett. B 764, 289-294 (2017) doi:10.1016/j.physletb.2016.11.047, arXiv:1609.01253 [gr-qc]
  * [8] F. R. Klinkhamer and Z. L. Wang, “Nonsingular bouncing cosmology from general relativity,” Phys. Rev. D 100, 083534 (2019) doi:10.1103/PhysRevD.100.083534, arXiv:1904.09961 [gr-qc].
  * [9] F. R. Klinkhamer and Z. L. Wang, “Nonsingular bouncing cosmology from general relativity: Scalar metric perturbations,” Phys. Rev. D 101, 064061 (2020) doi:10.1103/PhysRevD.101.064061, arXiv:1911.06173 [gr-qc].
  * [10] G. T. Horowitz, “Topology change in classical and quantum gravity,” Class. Quant. Grav. 8, 587-602 (1991) doi:10.1088/0264-9381/8/4/007