Theoretical Physics Institute, University of Alberta, Edmonton, Alberta,
Canada T6G 2E1

# Limiting curvature models of gravity

Valeri P. Frolovins:x ins:xins:x

###### Abstract

In this paper we discuss a limiting curvature condition and its application to
cosmological and black hole solutions.

##  1 Introduction

Existence of singularities is a generic property of general relativity. For
well known cosmological and black-hole solutions of Einstein equations these
singularities are related to infinite growth of curvature. One can either
“live” with a situation when the spacetime terminates its existence at the
singularity, or believe that this “decease” of general relativity can be cured
by proper modification of Einstein equations in the ultraviolet domain. There
exists a lot of publications where different modifications of general
relativity were proposed. At the same time one may try to answer a “simpler
question”: What might be properties of cosmological and black-hole models in a
theory which forbids the curvature singularity formation.

In 1982 Markov proposed a so called limiting curvature condition [1, 2].
According to this conjecture the spacetime curvature should be always
restricted by some universal value

$|{\cal R}|<\Lambda={B_{\cal R}\over\ell^{2}}\,.$ (1)

Here ${\cal R}$ is a scalar curvature invariant which has dimension of
$[length]^{-2}$ and $\Lambda$ is its limiting value. The parameter $\ell$
connected with the radius of the curvature plays a role of a fundamental
length of the theory. The dimensionless parameter $B_{\cal R}$ depends on a
choice of the curvature invariant ${\cal R}$ but it is universal in the
following sense. It does not depend on the parameters which enter solutions,
such as black hole mass or the total entropy of the universe.

There are a lot of publications where nonsingular cosmological and black hole
models are discussed. In principle, there exist two kind of approaches to this
problem

  * •

One can try to guess the form of the metric which satisfies the limiting
curvature condition and which in the spacetime domain where the curvature is
much smaller than the critical one properly reproduces corresponding solutions
of Einstein equations;

  * •

One can try to find such a modification of the Einstein equations which
automatically prevents infinite growth of the curvature.

In the present paper we discuss some aspects of these two approaches.

##  2 Metrics satisfying the limiting curvature condition

Let us first describe an interesting class of metrics which satisfy the
limiting curvature condition. Let us consider the following metric
&ds^2=-f(r,v) dv^2+2 dv dr +r^2 dω^2 ,
f(r,v)=1-m^2(v)ℓ2F(x), x= rm(v) . Here $m(v)$ and $F(x)$ are two arbitrary
functions and $d\omega^{2}$ is a metric on a unit round $2D$ sphere. The
parameter $\ell$ which enters this metric has the dimension of length. At the
moment it is arbitrary. Later $\ell$ will be identified with the radius
corresponding to the limiting curvature. The mass function $m$ controls the
mass of the black hole and when $dm/dv\neq 0$ it is related to the energy
density of the spherically symmetric null fluid flux, which changes the black
hole parameters. We shall demonstrate that for a proper choice of structure
function $F(x)$ metric (2) describes a nonsingular spherically symmetric black
hole and its spacetime obeys the limiting curvature condition.

A well known special case of (2) is Vaidya metric

$ds_{V}^{2}=-\left(1-{2M(v)\over r}\right)dv^{2}+2dvdr+r^{2}d\omega^{2}\,.$
(2)

It describes a black hole. Here $M(v)$ is the mass of the black hole as a
function of the advanced time $v$. For constant $M$ metric (2) coincides with
Schwarzschild metric. If $dM/dv\neq 0$ there exists a null fluid flux which
changes the black-hole mass. Vaidya metric (2) is often used to describe a
formation and consequent evaporation of a black hole. At the phase of
formation $dM/dv>0$, while during the evaporation phase $dM/dv<0$. It is easy
to check that metric (2) can be presented in the form (2) with the following
choice of the mass function

$m(v)=[2M(v)\ell^{2}]^{1/3}\,.$ (3)

Our next step is calculation of the curvature invariants for metric (2). We
use the following notations &C^2=C_αβγδC^αβγδ, S^2=S_αβS^αβ=R_αβR^αβ-14R^2 ,
K=R_αβγδR^αβγδ=C^2+2S^2+16R^2 . Here $S_{\alpha\beta}=R_{\alpha\beta}-{1\over
4}Rg_{\alpha\beta}$, and $R_{\alpha\beta}$ and $R$ are Ricci tensor and Ricci
scalar, respectively.

Calculation of the scalar curvature invariants gives C&=13ℓ2x2 (x^2
F”-2xF’+2F) ,
S=12ℓ2x2 (x^2 F”-2F) ,
R=1ℓ2x2 (x^2 F”+4xF’+2F) ,
K=1ℓ4[ (F”)^2+4(F’)^2x2+4F^2x4 ] . These relations demonstrate a remarkable
property of the metric (2): The curvature invariants written as functions of
$x$ do not depend on the mass function $m(v)$. This means that in order to
keep these invariants uniformly and universally bounded it is sufficient to
chose a properly bounded function $F(x)$.

Using expressions (2)–(2) one gets

$\ell^{2}\left(2\sqrt{3}C-6S+R\right)=12{F(x)\over x^{2}}\,.$ (4)

If the curvature invariants are finite at $x=0$ then the function $F(x)$ at
this point should have the following asymptotic form

$F(x)=F_{0}x^{2}+\ldots,\hskip 14.22636ptF_{0}=\mbox{const}\,,$ (5)

where dots denote terms of the higher order in $x$. Let us write $F(x)$ in the
form $F(x)=x^{2}U(x)$, then C&=13ℓ2(x^2 U”+2xU’) ,
S=12ℓ2(x^2U”+2xU’) ,
R=1ℓ2(x^2U”+8xU’+12U) . If the function $U(x)$ and its derivatives at $x=0$
are finite then the scalar curvature invariants are finite at this point. The
metric function $f$ at small $r$ has the following form

$f=1-U(0){r^{2}\over\ell^{2}}+\ldots\,.$ (6)

We use an ambiguity in the definition of the parameter $\ell$ to absorb the
factor $U(0)$ into it.

In order to have the asymptotically flat metric the function $U(x)$ at large
$x$ should have the following form $U(x)\sim c/x^{3}+\ldots$ . Then one has

$f\sim 1-{c\ m^{3}\over\ell^{2}r}+\ldots\,.$ (7)

The constant $c$ can be absorbed into a redefinition of the mass function $m$.
Using this ambiguity we choose $c=1$. For this choice the function $f$ at
large distance asymptotically takes the form (2) if

$m(v)=[2M(v)\ell^{2}]^{1/3}\,.$ (8)

Let us summarize. Using the above discussed ambiguities one can impose the
following conditions on the function $U(x)$

$U(0)=1,\hskip 14.22636ptU(x\to\infty)\sim 1/x^{3}\,.$ (9)

Using this asymptotic form of $U(x)$ at large $x$ it is easy to show that the
leading term of the Einstein tensor $G_{\mu\nu}$ calculated for the metric (2)
at large $r$ is

$G_{\mu\nu}\approx{2\dot{M}\over r^{2}}v_{,\mu}v_{,\nu}\,.$ (10)

If the following equation

$x^{2}U(x)={\ell^{2}\over m^{2}(v)}\,.$ (11)

has a solution $x=x(v)$ then the metric (2) has a trapped surface at $x=x(v)$.
Since the function $x^{2}U(x)$ grows from $0$ near $x=0$ and decrease as $1/x$
at the infinity it has one or more local maxima. We assume that there exists
only one maximum at some point $x=x_{*}$. If at this point

$x_{*}^{2}U(x_{*})>{\ell^{2}\over m(v)}\,,$ (12)

then equation (11) has two solutions, $x_{0}<x_{*}(v)$ and $x_{1}>x_{*}(v)$.
If the mass $M$ is constant and does not depend on advanced time $v$, the
outer surface $x=x_{1}$ is an event horizon and $x=x_{0}$ is the inner
horizon. In a time-dependent case solutions of the equation (11) describe
positions of the inner and outer branches of the apparent horizon. When the
mass $M$ decreases with time then at some time these branches can meet one the
other. This happens when $x_{0}=x_{1}=x_{*}(v)$. At this point the apparent
horizon terminates. In such a situation the information collected inside the
apparent horizon can become visible to an external observer. Technically, this
means that the event horizon does not exist.

Figure 1: Function $f$ for Hayward metric (13) (solid line) and its
generalization (17) with $n=2$ (dash line). In both cases the mass $M$ is
constant and its value is chosen to be equal $27/2$ in the units where
$\ell=1$. A coordinate $x$ along the horizontal axis is $x=r/m=r/3$.

Figure 2: Plots for the curvature invariants for Hayward metric (13) (left)
and its generalization (17) (right). The parameters of these metrics are
chosen the same as in Fig. 1. Solid, dash and point lines are used for $C$,
$S$ and $R$ invariants, respectively.

The simplest choice of function $U(x)$ satisfying conditions (9) is

$U={1\over 1+x^{3}}\,.$ (13)

It is easy to check that the function $f(r,v)$ is this case is

$f(r,v)=1-{2M(v)r^{2}\over r^{3}+2M(v)\ell^{2}}\,,$ (14)

and the corresponding metric (2) is nothing but a Vaidya-type generalization
of the Hayward metric [3]. Discussion of properties of this metric and its
generalizations as well as further references can be found in [4, 5, 6, 7].

A solid line in figure 1 shows a plot of function $f$ for metric (2) with $U$
given by (13). For illustration we choose the mass $M$ to be constant and put
$2M=27$ and $\ell=1$, so that $m=3$. This plot shows $f$ as a function of
$x=r/m=r/3$. The left plot in figure 2 shows invariants $C$, $S$ and $R$
calculated for the same parameters $M$ and $\ell$.

Metrics (2) with function $U(x)$ satisfying conditions (9) is a far-going
generalization of the Hayward-Vaidya metric (14). Their common property is
that their curvature invariants do not depend on mass function. If $U(x)$ and
its derivatives are bounded on the interval $x\in[0,\infty)$, and the
following relations are valid

$\max_{x\in[0,\infty)}|U(x)|=U_{0},\hskip
5.69046pt\max_{x\in[0,\infty)}|xU^{\prime}(x)|=U_{1},\hskip
5.69046pt\max_{x\in[0,\infty)}|x^{2}U^{\prime\prime}(x)|=U_{2}\,,$ (15)

where $U_{0}$, $U_{1}$ and $U_{2}$ are finite constants, then the curvature
invariants obey the limiting curvature conditions

$C^{2}\leq{B_{C}\over\ell^{4}},\hskip
5.69046ptS^{2}\leq{B_{S}\over\ell^{4}},\hskip
5.69046pt|R|\leq{B_{R}\over\ell^{2}}\,.$ (16)

Dimensionless constants $B_{C}$, $B_{S}$ and $B_{R}$ depend only on the choice
of the function $U(x)$.

There exists infinite number of functions $U(x)$ satisfying both conditions
(9) and (15). For example one can take

$U(x)={1+x^{n}\over 1+x^{n+3}}\,,$ (17)

where $n$ is a positive integer number. Figure 1 shows by a dash line the
metric function $f$ for $n=2$. The curvature invariants for this metric with
$n=2$ are shown in the right plots in figure 2.

##  3 Limiting curvature theory of gravity

Infinitely growing curvature in the black hole interior signals the formation
of singularity. Metrics presented in the previous section are nonsingular and
they obey the limiting curvature condition. However, they are not solutions of
some explicitly known modified gravity equations. How to modify gravity
equations in order to escape singularities? Recently a new approach has been
proposed which allows one to prevent infinite growth of the spacetime
curvature [8, 10, 9]. It was called a limiting curvature gravity (LCG) theory.
Its general idea is to incorporate inequality constraints restricting the
growth of the curvature into the gravity action.

Suppose we want not to allow some curvature invariant to be arbitrary large.
Let us denote this invariant by ${\mathcal{R}}$ and its limiting value by
$\Lambda$. We introduce a constraint function

$\Phi={\mathcal{R}}-\Lambda\,.$ (18)

Thus our goal is to include the following inequality constraint $\Phi\leq 0$
into our theory of gravity. This can be done as follows. Let the original
(unconstrained) action for the gravitational field be

$S_{g}=\int dx\sqrt{-g}L_{g}\,.$ (19)

$L_{g}$ is the function of the metric $\bf{g}$ and its derivatives. In the
standard general relativity $L_{g}={1\over 2\kappa}R$, where $R$ is a scalar
curvature, and $\kappa=8\pi G$. If there exist other fields beside the
gravitational one, then $S_{g}$ depends on them as well.

We modify this action by adding a term which is responsible for imposed
inequality constraint &S=S_g+S_c ,
S_c=∫dx -g χ (Φ+ζ^2) . The extended action $S$ besides the metric
${\boldsymbol{g}}$ contains two Lagrange multipliers, $\chi(x)$ and
$\zeta(x)$. Variation of $S$ with respect to these functions gives

$\Phi+\zeta^{2}=0,\hskip 14.22636pt\chi\zeta=0\,.$ (20)

If $\Phi<0$, then the first equation gives $\zeta^{2}=-\Phi$, while the second
equation implies that $\chi=0$. In this regime, which is called subcritical,
the gravitational equations are not modified

${\delta{S}\over\delta g_{\mu\nu}}\bigg{|}_{\chi=0}={\delta{S}_{g}\over\delta
g_{\mu\nu}}=0\,.$ (21)

When the curvature ${\mathcal{R}}$ reaches its critical value $\Lambda$, the
function $\zeta$ vanishes. In this so called supercritical regime the second
Lagrange multiplier, $\chi$ becomes non-zero. As a result this function enters
into the variation of the action $S$ and modifies the gravitational equations.
They contain now $\chi$ and its derivatives. The number of the gravitational
equations remain the same, but they contain one more variable. However, the
complete set of equations contains now an additional constraint equation

$\Phi=0\,.$ (22)

This extended system of equations is sufficient to determine both the metric
${\boldsymbol{g}}$ and the function $\chi$. Since a non-vanishing value of
$\chi$ indicates that a system is in its supercritical regime it is called a
control function.

Suppose a transition between initial subcritical solution to the supercritical
one happens at some junction surface $\Sigma$. One can use obtained system of
equations and known subcritical metric in order to find the initial conditions
on $\Sigma$ for the corresponding supercritical solution. It is sufficient to
use the continuity conditions which follow from the field equations.

The extended action $S$ can be easily generalize to the case of several
inequality constraints. For this purpose it is sufficient to substitute
instead of $\chi\ (\Phi+\zeta^{2})$ in the integrand in $S_{c}$ a sum

$\sum_{i}\chi_{i}\ (\Phi_{i}+\zeta_{i}^{2})\,.$ (23)

Here $\Phi_{i}$ is a constraint function for the constraint number $i$, and
$\chi_{i}$ and $\zeta_{i}$ are two Lagrange multipliers accompanying this
constraint.

##  4 Two-dimensional black holes in LCG theory

To illustrate how a described in the previous section approach works let us
consider a $2D$ black hole LCG model [8]. We start with the action of a 2D
dilaton gravity model which is known to have black-hole solutions and is
exactly solvable at the classical level

$\displaystyle S_{g}=\frac{1}{2}s\int
d^{2}x~{}|g|^{1/2}~{}\Big{(}\psi^{2}R+4(\nabla\phi)^{2}+4\lambda^{2}\Big{)}.$
(24)

Here $R$ is the curvature of the two-dimensional spacetime with metric
$g_{\alpha\beta}$ and $\psi$ is a dilaton field. Parameter $\lambda$ has the
dimension $[length]^{-2}$. This model naturally appears in the framework of
string theory and its properties have been extensively studied.

A limiting curvature model is obtained by adding to $S_{g}$ an action

$S_{c}=\frac{1}{2}\int
d^{2}x~{}|g|^{1/2}~{}\chi~{}\big{(}\Phi+\zeta^{2}\big{)},\hskip
14.22636pt\Phi=R-\Lambda\,.$ (25)

Here $\Lambda>0$ is a positive constant constraining the $2D$ curvature $R$.
Without the Lagrange multiplier $\zeta$ the action (25) looks exactly as that
of Jackiw-Teitelboim gravity model.

Variation of the action $S=S_{g}+S_{c}$ over $\zeta$ and $\chi$ gives the
constraint equations (20), while its variation over $\psi$ and
$g_{\alpha\beta}$ leads to the dilaton and gravitational field equations
&□ψ-(λ^2+14(Λ-ζ^2))ψ=0, ,
χ_;αβ+12g_αβRχ=Q_αβ ,
Q_αβ=4ψ_;αψ_;β +12g_αβ[- 4ψ_;ϵψ^;ϵ+(4λ^2+R)ψ^2].

In the subcritical domain these equations coincide with the standard equations
of $2D$ dilaton gravity and their solutions are well known. The solution of
the field equations (4) is &ds^2=-f dt^2+f^-1dr^2
f=1-Mλe^-2λr, ψ=e^λr . This metric describes a $2D$ black hole in an
asymptotically flat spacetime. Its horizon is located at
$r_{\scriptscriptstyle H}=\frac{1}{2\lambda}\ln\frac{M}{\lambda}$, and its
spacetime curvature is

$\displaystyle R=-\frac{\partial^{2}f}{\partial r^{2}}=4\lambda
M\,e^{-2\lambda r}.$ (26)

It takes value $R=4\lambda^{2}$ on the horizon. The Killing vector
${\boldsymbol{\xi}}=\partial_{t}$ is time-like in the black hole exterior, and
it is space-like inside the black hole.

To cover a complete spacetime of the $2D$ black hole we introduce coordinates
similar to the standard Kruskal coordinates. For this purpose we introduce
null coordinates

$U=-\sqrt{\lambda\over M}\exp(-\lambda u),\ V=\sqrt{\lambda\over
M}\exp(\lambda v)\,.$ (27)

where

$u={t-r_{*}},\hskip 5.69046ptv={t+r_{*}},\hskip 14.22636ptr_{*}=\int{dr\over
f}={1\over 2\lambda}\ln\left(\exp(2\lambda r)-{M\over\lambda}\right)\,.$ (28)

In these coordinates

$ds^{2}=-{1\over\lambda^{2}}{dU\,dV\over 1-UV},\hskip
14.22636pt-\infty<U<\infty\,,\quad-\infty<V<\infty\,,\quad UV<1\,.$ (29)

To construct the Carter-Penrose conformal diagram for the $2D$ metric we
introduce new null coordinates $(p,q)$

$U=\tan p\,,\quad V=\tan q\,,\quad\left(-{\pi\over 2}<p<{\pi\over
2}\,,-{\pi\over 2}<q<{\pi\over 2}\right).$ (30)

In these coordinates the metric takes the form

$ds^{2}=-{1\over\lambda^{2}}{dp\,dq\over\cos p\,\cos q\,\cos(p+q)}\,.$ (31)

Lines $p+q=\pm{\pi\over 2}$ correspond to spacelike curvature singularities.

At line $r=r_{\Lambda}=-(1/2\lambda)\ln(\Lambda/(4\lambda M))$ the curvature
$R$ reaches its critical value. We impose a condition
$\Lambda/(4\lambda^{2})>1$, so that $r_{\Lambda}<r_{H}$ and the transition
point is located inside the black hole. Beyond this point the supercritical
regime starts. In this regime the constraint equation takes the form
$R=\Lambda$. The spacetime metric in this supercritical domain is locally de
Sitter one. Analysis of the dilaton and gravitational equations shows that the
control function $\chi$ does not vanish there. This means that after the
solution enters its supercritical phase from the subcritical one it remains in
this phase forever. A detailed discussion of gluing sub- and supercritical
solutions and global structure of the $2D$ black holes in the LCG theory can
be found in [8]. Figure 3 from this paper shows a conformal diagram for an
eternal $2D$ black hole.

Figure 3: Conformal diagram for the spacetime of the solution of LCG theory
which consists of an ‘eternal’ $2D$ black hole domain glued to the de Sitter
spacetimes. Dashed lines represent junction surfaces where the curvature
reaches its maximal value $R=\Lambda$.

Similarly to the eternal $4D$ Schwarzschild black hole it has two domains
$R_{+}$ and $R_{-}$ with asymptotically flat infinities. Regions $T_{-}$ and
$T_{+}$ describe interiors of black and white holes, respectively. Instead of
singularities these domains contain expanding and contracting de Sitter-like
cores. As usual, such an eternal spacetime solution can be used to construct
solutions for different physically interesting cases. For example, when a
black hole is formed in gravitational collapse of a body, a boundary of this
body is represented by a timelike curve on the diagram 3. A spacetime domain
of the eternal black hole located to the left from this line should be cut and
substituted by the metric inside the collapsing body. Let us summarize: In the
LCG model of a $2D$ black hole its interior contains expanding de Sitter core
instead of the singulatity.

##  5 Four-dimensional spherically symmetric black holes in LCG theory

Study of the interior of four-dimensional black holes in LCG theory is much
more complicated [9]. One of the reasons is that the number of independent
curvature invariants for spherically symmetric metrics is 4 instead of one
scalar curvature invariant in $2D$ case.

When the curvature invariants are smaller than their critical value, a
solution is subcritical and coincides with the Schwarzschild metric

$ds^{2}=-\left(1-{2M\over r}\right)dt^{2}+{dr^{2}\over 1-{2M\over
r}}+r^{2}d\omega^{2}\,.$ (32)

We assume that a transition to the supercritical regime takes place inside the
black hole where the metric takes the form

$ds^{2}=-{dr^{2}\over f}+fdt^{2}+r^{2}d\omega^{2},\hskip 14.22636ptf={2M\over
r}-1\,.$ (33)

In the black hole interior $f>0$ and introducing a new coordinate $\tau$

$\tau=\sqrt{r(2M-r)}+M\arcsin({M-r\over r})+{1\over 2}\pi M\,,$ (34)

one can write (33) in the form

$\displaystyle ds^{2}=-d\tau^{2}+B^{2}dt^{2}+a^{2}d\omega^{2}\,,$ (35)

where $a$ and $B$ are functions of proper time $\tau$

$B^{2}=f(r(\tau)),\hskip 14.22636pta=r(\tau)\,.$ (36)

If the transition from sub- to supercritical regime takes place at
$\tau=\tau_{0}$ than after this time the corresponding supercritical solution
still has the same form. However, the metric functions $a(\tau)$ and $B(\tau)$
are different. They, as well as the control function $\chi$ should be
determined by solving the corresponding set of equations. The form of these
equations depends on how many inequality constraints are imposed and what is
their structure.

The Riemann curvature tensor for the metric (35) has four independent non-
vanishing components

$R_{\hat{\tau}\hat{t}\hat{\tau}\hat{t}}=-v,\hskip
14.22636ptR_{\hat{\tau}\hat{\theta}\hat{\tau}\hat{\theta}}=-q\,,,\hskip
14.22636ptR_{\hat{t}\hat{\theta}\hat{t}\hat{\theta}}=u,\hskip
14.22636ptR_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}}=p\,.$ (37)

The hat over the indices means that the components of this tensor are
calculated in the orthonormal tetrad formed by unit vectors along the
corresponding coordinate lines. The independent basic curvature invariants
$(p,q,u,v)$ are

$p=\frac{\dot{a}^{2}+1}{a^{2}},\hskip 14.22636ptq=\frac{\ddot{a}}{a},\hskip
14.22636ptu=\frac{\dot{a}\dot{B}}{aB},\hskip 14.22636ptv=\frac{\ddot{B}}{B}.$
(38)

A dot in these expressions means the derivative with respect to $\tau$. For
the subcritical Schwarzschild metric these curvature invariants are

$p=v=-2q=-2u={2M\over r^{3}}\,.$ (39)

Invariant $p$ is positive definite, while the other invariants do not have a
definite sign. This property makes the analysis of required inequality
constraints more complicated. In paper [9] we discussed a case of constraints
which are linear functions of curvature invariants (38). The main results of
this study are the following. The required property of the limiting curvature
can be achieved if the following two constraints are imposed &Φ_1=v-Λ=0 ,
Φ_2=p-μq-λΛ=0 . Here $\Lambda=1/\ell^{2}$ is the limiting curvature parameter,
while $\mu\in(0,1)$ and $\lambda$ are two dimensionless parameters. The
transition from sub- to supercritical regime happens at $\tau=\tau_{0}$ where
$r=r_{0}=(2M\ell^{2})^{1/3}$. For $\lambda>1+\mu/2$ the first constraint which
is saturated is $v=\Lambda$. For a supercritical solution in this regime the
invariants $p$ and $|q|$ still grow. When condition (5) is met the second
constraint becomes active. This happens at some time $\tau_{1}\geq\tau_{0}$.
Study of the behavior of the control function $\chi_{1}$ associated with
constraint (5) shows that it does not vanish at this point. This means that at
the second phase, that is beyond this point, both constraints (5) and (5) are
valid. In a general case such a supercritical solution does not leave this
second phase after it enters it.

Constraint (5) has a simple solution

$B(\tau)=B_{0}\cosh\left[\ell^{-1}(\tau-\tau_{0}+\phi)\right]\,.$ (40)

Parameters $B_{0}$ and $\phi$ are defined by continuity conditions at
$\tau=\tau_{0}$. Solution (40) has the same form for both supercritical
phases. Let us consider a $(\tau,t)$ sector of the metric (35). Its $2D$
metric is

$dS^{2}=-d\tau^{2}+B^{2}(\tau)dt^{2}\,.$ (41)

For $B(\tau)$ defined by (40) this is $2D$ de Sitter metric with curvature
${}^{(2})R=\ell^{-2}=\Lambda$. Thus, the supercritical solution for the
interior of $4D$ black hole in its $2D$ sector $(\tau,t)$ is the same as a
solution for the interior of $2D$ black hole discussed in the previous
section.

The second constraint (5) gives a nonlinear second order ordinary differential
equation for function $a(\tau)$. For the imposed condition $0<\mu<1$ a
solution of this equation has two turning points $a_{min}<a_{max}$, where
$\dot{a}=0$. If $M\gg\ell$, then $a_{min}$ is close but slightly bigger than
$\ell$, while $a_{max}\sim(2M\ell^{2}/\lambda)^{1/3}$. Let us denote

$d\Gamma^{2}=-d\tau^{2}+a^{2}(\tau)d\omega^{2}\,.$ (42)

Then the solution of the constraint equation (5) describes a $3D$ oscillating
closed universe.

To summarize, the interior of a $4D$ spherically symmetric black hole in the
LCG model with linear in curvature constraints is a $3D$ space which is
exponentially expanding in one direction, determined by its Killing vector
${\boldsymbol{\xi}}=\partial_{t}$, and it is oscillating in the other two
(spherical) directions. It is possible to check that all the invariants
$(p,q,u,v)$ for this solutions remain bounded and their values are uniformly
restricted by a quantity proportional to the limiting curvature $\Lambda$.
Derivation of these results and their detailed discussion can be found in [9].

##  6 Bouncing cosmologies in LCG theory

As the last example of application of the LCG theory let us discuss a case of
the cosmology. Namely, we assume that a homogeneous isotopic universe filled
with thermal radiation is in its contraction phase. Its metric is

$ds^{2}=-d\tau^{2}+a^{2}(\tau)d\gamma^{2}\,,$ (43)

where $d\gamma^{2}=\gamma_{ij}dx^{i}dx^{j}$ is a line element on a unit round
3D sphere $S^{3}$. The metric $ds^{2}$ is conformally flat so that its Weyl
tensor vanishes. We consider curvature invariants which are scalar functions
of the Ricci tensor $R_{\alpha\beta}$.

According to the General Relativity a collapse of such universe ultimately
causes the cosmic scale factor $a(\tau)$ to reach zero and the contraction of
the universe ends by the Big Crunch with the curvature singularity formation.
In paper [10] we demonstrated that in the LCG model such a singularity is
absent. Instead of it the universe has a Big Bounce after which it becomes
expanding. Let us briefly discuss these results.

The Ricci tensor calculated for the metric (43) is

$R^{\mu}_{\nu}=\mbox{diag}(3q,q+2p,q+2p,q+2p)\,,$ (44)

where

$q=\frac{\ddot{a}}{a},\hskip 14.22636ptp=\frac{\dot{a}^{2}+1}{a^{2}}.$ (45)

Scalar curvature invariants are functions of $p$ and $q$, and the
corresponding inequality constraint can be written in the form

$\Phi=\Phi(p,q,\Lambda)\leq 0\,.$ (46)

The simplest choice of the constraint function $\Phi$ is

$\Phi=p-\mu q-\Lambda=0,\hskip 14.22636pt0<\mu<1\,.$ (47)

Let us note that this constraint is similar to the constraint (5) which was
discussed in the previous section. We again define a critical length by
$\ell$, so that $\Lambda=\ell^{-2}$.

In the subcritical regime where $p-\mu q-\Lambda<0$ a solution coincides with
a solution of the Einstein equations. The stress-energy tensor of the thermal
radiation is

$\displaystyle T^{\mu}_{\nu}$
$\displaystyle=\mbox{diag}(-\varepsilon,P,P,P)\,,$

where $\varepsilon$ is the energy density and $P$ is the pressure. The
conservation law $T^{\mu}_{\ \nu;\mu}=0$ is satisfied if

$\displaystyle\varepsilon=Aa^{-4},$ (48)

where the factor $A$ is defined by the total (conserved) entropy of the
universe $S$

$A=\nu\hbar c\Big{(}\frac{S}{k_{\mbox{\tiny B}}}\Big{)}^{4/3},\hskip
14.22636pt\nu={3\over 16\pi^{3}}\Big{(}\frac{90}{n\pi}\Big{)}^{1/3}.$ (49)

For pure electromagnetic radiation $n=2$ and $\nu\approx 0.015$.

At the contraction phase of the radiation dominated universe the following
relation is valid

$q=-p\,.$ (50)

During the contraction of the universe both invariants $p$ and $|q|$ grow
until the constraint function $\Phi$ reaches its critical value. After this a
solution becomes supercritical and it follows the constraint equation (47). It
is possible to show that for this solution the invariant $p$ monotonically
grows until $a(\tau)$ reaches its minimal value $a_{min}$ which is of order of
$\ell$ but slightly bigger than it. After this the scale function $a(\tau)$
grows. Similarly to the case of the $4D$ black hole interior, there also
exists a second turning point where $\dot{a}=0$ and $a(\tau)$ has maximum.
However, before this the supercritical solution crosses the line (50) on the
$(p,q)$ plane. Because of the symmetry of the equation for the control
function $\chi$ it vanishes at this point, and instead of further motion along
the constraint (47) the solution returns to its subcritical regime. This
subcritical solution describes an expanding universe filled with the thermal
radiation. Thus in the LCG theory a contracting homogeneous isotropic universe
has a Big Bounce instead of a Big Bang singularity. Derivation of these
results and their detailed discussion can be found in a recent paper [10].

##  7 Discussion

In this paper we discussed two types of models which satisfy the limiting
curvature condition. In section 2 a wide class of spherically symmetric
metrics describing nonsingular black holes is presented. These metrics which
contain two arbitrary functions of one variable present a far-going
generalization of Hayward-Vaidya metrics. One of the functions, a mass
function $m(v)$, controls the black hole mass and the null fluid flux
responsible for its change. The other one, a structure function $F(x)$, is
responsible for properties of the nonsingular black hole interior. These
metrics have an interesting property: The curvature invariants calculated for
them do not depend on the mass function and after imposing simple restrictions
of the structure function they obey the limiting curvature condition.

In the second part of the paper we discuss a new recently proposed limiting
curvature theory of gravity. In this theory the Einstein-Hilbert gravitational
action is modified by adding to it terms which guarantee that inequality
constraints for chosen curvature invariants are satisfied. We discussed
application of LCG theory to three cases: $2D$ dilaton gravity black holes
[8], $4D$ spherically symmetric LCG modification of the Schwarzschild metric
[9] and to bouncing cosmologies [10]. As further applications of the LCG
theory it would be interesting to discuss such “hot” subjects as the large
scale structure formation in bouncing cosmological LCG models, possible
anisotropy suppression in anisotropic cosmologies and the interior structure
of charged and rotating $D$ black holes.

## Acknowledgments

The author is grateful to the Natural Sciences and Engineering Research
Council of Canada and the Killam Trust for their financial support. He also
thanks Andrei Zelnikov for useful remarks.

## References

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