Evolution of Curvature in Riemannian Geometry
Abhishek Das
B. M. Birla Science Centre
Adarsh Nagar, Hyderabad
India, 500063
<EMAIL_ADDRESS>

###### Abstract

In this paper we shall follow two different routes that embody the existence
of a point of _exclusivity_ \- the opposite of a _singularity_. First, we show
that this follows from Raychaudhuri’s equation. Then, we substantiate that the
evolution of the Riemann-Christoffel tensor can be expressed entirely by an
arbitrary timelike vector field and that the curvature tensor returns to its
initial value with respect to change in a particular index. It has been shown
that geodesics can diverge just as they can converge, resulting in a point of
_exclusivity_ and point of _singularity_.

##  1 Introduction

General theory of relativity, the geometric theory of gravitation is based on
the plinth of Riemannian geometry that studies differentiable manifolds. There
is a plethora of literature on these two intricately related topics. As is
known, there are a multitude of implications resulting from Riemannian
geometry when applied to the general relativity - the Einstein field equations
[1], solutions to such equations in the form of Schwarzchild metric [2],
Friedmann-Lemaitre-Robertson-Walker metric [3, 4, 5, 6, 7, 8, 9, 10] and
others [11, 12], singularities [13] and black holes [14, 15].
However, in essence, the current paper is not innately related to the general
theory of relativity. It is essentially devoted to the geometric aspect of
space-time, somewhat in contradistinction to a recent paper that was based on
a Hamiltonian formulation [16]. The inception begins with the feasibility of
existence of a point of exclusivity (the opposite of a singularity), that is
corroborated considering the Raychaudhuri equation [13]. And then, we consider
parallel transported vectors and thereby manipulating of their connection with
the geometry of space-time we are led to an evolution equation for the
Riemann-Christoffel tensor. As a consequence, several interesting implications
are drawn out. Most importantly, it has been argued that with necessary
conditions fulfilled one can indeed discern the existence of a point of
exclusivity.

##  2 The positive and negative values of the expansion scalar

In his seminal paper of 1955 [13], Raychaudhuri had obtained an equation that
led to the focusing theorem which in turn substantiated the existence of
singularities and black holes, eventually. We would find in the present
section that the result derived by Raychaudhuri plays a significant role in
some novel aspects. Let us commence with the expansion scalar ($\theta$) of
the Raychaudhuri equation. We know that

$\theta=\frac{\partial}{\partial t}(\ln G)$

where, $G=\sqrt{-g}$. Also, we know that

$A_{;i}^{i}=\frac{1}{\sqrt{-g}}[A^{i}\frac{\partial}{\partial
x^{i}}\sqrt{-g}+\sqrt{-g}\frac{\partial}{\partial x^{i}}A^{i}]$

So, considering the coordinate $t$ ($i=0$) we may write

$A_{;0}^{0}=[A^{0}\theta+A^{0}_{,0}]$

Now, since the Kronecker delta function is independent of both the normal and
covariant derivatives the last equation can be equivalently expressed as

$A_{k;0}=A_{k}\theta+A_{k,0}$ (1)

Again, we also know that

$A_{k;i}=A_{k,i}-\Gamma_{ki}^{l}A_{l}$

Therefore, using equation (1) we obtain

$A_{k}\theta+\Gamma_{k0}^{l}A_{l}=0$ (2)

or,

$A_{k}\theta+\Gamma_{k0}^{l}\delta_{l}^{k}A_{k}=0$

Since, the vector fields $A_{k}$’s are arbitrary, we have the following
equation for the expansion scalar

$\theta+\Gamma_{k0}^{l}\delta_{l}^{k}=0$ (3)

One can make two immediate observations from equation (3). If $k\neq l$, we
have

$\theta=0$ (4)

And, if $k=l$, we have

$\theta+\Gamma_{l0}^{l}=0$ (5)

Again, on account of the relation

$\Gamma_{l0}^{l}=\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{0}}\sqrt{-g}=\frac{\partial}{\partial x^{0}}(\ln\sqrt{-g})$

we derive from equation (5)

$\theta+\theta=0$ (6)

which is, in essence, similar to equations (37). Either the expansion scalar
is zero, or

$|\theta|=\pm\theta$ (7)

which seems erroneous from a mathematical perspective. But, suppose $\theta$
can actually have the same value with opposite signs; then the last two
equations imply some physical meaning. This shall be further corroborated by
the following methodology. Equation (2) can also be written as (with,
$A_{0}^{k}=\Gamma_{k0}^{l}A_{l}$)

$A_{k}\theta+A_{0}^{k}=0$

Differentiating with respect to $x^{0}$ and using the above equation again we
have

$A_{k}\dot{\theta}-\frac{A_{k,0}A^{k}A^{k}_{0}}{A^{2}}+A^{k}_{0,0}=0$ (8)

where, $A^{2}=A^{k}A_{k}$. Again, Raychaudhuri’s equation is as follows:

$\dot{\theta}=\frac{\theta^{2}}{3}-2\sigma^{2}+2\omega^{2}-R_{s}$ (9)

where, the symbols have their usual meanings. Thus, from equations (8) and (9)
we have

$\frac{A^{2}\theta^{2}}{3}+A^{2}(2\omega^{2}-\sigma^{2}+R_{s})+A_{k,0}A^{k}\theta+A^{k}A^{k}_{0,0}=0$

Writing $\epsilon=2\omega^{2}-2\sigma^{2}-R_{s}$, and solving the quadratic
equation we finally derive

$\theta=\frac{3}{2}\left[-\frac{A_{k,0}A^{k}}{A^{2}}\pm\sqrt{\frac{(A_{k,0}A^{k})(A_{k,0}A^{k})}{A^{4}}-\frac{4}{3}\left\\{\frac{A_{k,0}A^{k}}{A^{2}}+\epsilon\right\\}}\right]$
(10)

Here, we can derive some interesting conclusions. When the discriminant of the
above solution is zero, we would have another quadratic equation of the form:

$p^{2}-\frac{4}{3}p-\frac{4}{3}\epsilon=0$

where, $p=\frac{A_{k,0}A^{k}}{A^{2}}$. The solution is of the form

$p=\frac{2}{3}\pm\frac{2}{3}\sqrt{1-3\epsilon}$

which implies that $\epsilon\geq\frac{1}{3}$. On the other hand, if we have

$\frac{(A_{k,0}A^{k})(A_{k,0}A^{k})}{A^{4}}=\frac{4}{3}\left\\{\frac{A_{k,0}A^{k}}{A^{2}}+\epsilon\right\\}$

then, in such a scenario

$\theta=-\frac{3}{2}\frac{A_{k,0}A^{k}}{A^{2}}$

This may be considered the initial value of $\theta$ during the birth of the
universe when all geodesics were focused at a particular point and then after
the big bang as the universe began to evolve the value of $\theta$ evolved -
accumulating the discriminant term. Again, if the Raychaudhri scalar is such
that

$R_{s}>\frac{A_{k,0}A^{k}}{A^{2}}+2\omega^{2}-2\sigma^{2}$

then

$\sqrt{\frac{(A_{k,0}A^{k})(A_{k,0}A^{k})}{A^{4}}-\frac{4}{3}\left\\{\frac{A_{k,0}A^{k}}{A^{2}}+\epsilon\right\\}}>\frac{A_{k,0}A^{k}}{A^{2}}$

So, $\theta$ can have positive values as well. Therefore, writing

$\varepsilon=-\frac{A_{k,0}A^{k}}{A^{2}}\pm\sqrt{\frac{(A_{k,0}A^{k})(A_{k,0}A^{k})}{A^{4}}-\frac{4}{3}\left\\{\frac{A_{k,0}A^{k}}{A^{2}}+\epsilon\right\\}}$

we would have

$\theta=\pm\frac{3}{2}\varepsilon$ (11)

This is what insinuates from equation (7) too. Hence, essentially,
$\dot{\theta}$ can diverge to both positive and negative infinity. The
physical significance of this conclusion is novel and significant - geodesics
can diverge just as they can converge. As a consequence, there will exist a
point of exclusivity akin to a point of singularity. This point of exclusivity
can be looked upon as the origin of all matter and energy and the formation of
spacetime as we know it. We understand immdiately how important the
Raychaudhuri scalar and expansion scalar are.
However, it should be borne in the mind that the notion and methodology
innovated here is in stark contrast with the theory of Big Rip [17],
particularly for the fact that no speculative, mysterious energy has been
talked about - here it is essentially the geometry trying to explain the
evolution of the universe.

##  3 The evolution equation

Now, we shall substantiate the novel result of the preceding section taking a
different route. At first, let us consider a covariant vector $B_{i}$ whose
transport between two different points of a Riemannian manifold is independent
of the path and that is not covariantly constant; then it is known that the
derivative of this vector field would be given as [18]

$\frac{\partial B_{i}}{\partial x^{k}}=\Gamma_{ik}^{l}B_{l}$ (12)

We assume that $B_{i}$ is thrice differentiable. Thus we have the following
from relation (12)

$\frac{\partial^{2}B_{i}}{\partial x^{n}\partial
x^{k}}=\frac{\partial\Gamma_{ik}^{l}}{\partial
x^{n}}B_{l}+\Gamma_{ik}^{l}\frac{\partial B_{l}}{\partial x^{n}}$ (13)

and

$\frac{\partial^{3}B_{i}}{\partial x^{m}\partial x^{n}\partial
x^{k}}=B_{i,mnk}=\frac{\partial^{2}\Gamma_{ik}^{l}}{\partial x^{m}\partial
x^{n}}B_{l}+\Gamma_{ik}^{l}\frac{\partial^{2}B_{l}}{\partial x^{m}\partial
x^{n}}+\frac{\partial\Gamma_{ik}^{l}}{\partial x^{n}}\frac{\partial
B_{l}}{\partial x^{m}}+\frac{\partial\Gamma_{ik}^{l}}{\partial
x^{m}}\frac{\partial B_{l}}{\partial x^{n}}$ (14)

Now, computing $B_{i,mkn}$, subtracting the resultant equation from (14) and
then changing the indices as $l\rightarrow j$, we have the equation

$B_{i,mnk}-B_{i,mkn}=\Gamma_{ik,mn}^{j}B_{j}-\Gamma_{in,mk}^{j}B_{j}+\Gamma_{ik}^{j}B_{j,mn}-\Gamma_{in}^{j}B_{j,mk}+\theta$
(15)

where,
$\theta=\Gamma_{ik,n}^{j}B_{j,m}+\Gamma_{ik,m}^{j}B_{j,n}-\Gamma_{in,k}^{j}B_{j,m}-\Gamma_{in,m}^{j}B_{j,k}$
and the ’comma’ implies normal derivative. Now, we know the following
relations regarding the Riemann-Christoffel (RC) tensor

$B_{i;kn}-B_{i,nk}=-R_{k\leavevmode\nobreak\ ni}^{m}B_{m}$

and

$B_{i;nk}-B_{i,kn}=-R_{n\leavevmode\nobreak\ ki}^{m}B_{m}$

where, the ’semi-colon’ implies covariant derivative. Therefore, we can write
the following from equation (15)

$\partial_{m}[B_{i,nk}-B_{i,kn}]=\partial_{m}[2R_{i\leavevmode\nobreak\
kn}^{j}B_{j}+\eta
B_{i}]=\Gamma_{ik,mn}^{j}B_{j}-\Gamma_{in,mk}^{j}B_{j}\Gamma_{ik}^{j}B_{j,mn}-\Gamma_{in}^{j}B_{j,mk}+\theta$

where, $\eta B_{i}=B_{i;kn}-B_{i;nk}$ ($\eta$ being a covariant derivative
operator). This yields the premature evolution equation for the RC tensor or
the curvature tensor as

$2\partial_{m}[R_{i\leavevmode\nobreak\
kn}^{j}]B_{j}+2R_{i\leavevmode\nobreak\
kn}^{j}B_{j,m}+[\Gamma_{in,mk}^{j}-\Gamma_{ik,mn}^{j}]B_{j}+\Gamma_{in}^{j}B_{j,mk}-\Gamma_{ik}^{j}B_{j,mn}+\partial_{m}(\eta
B_{i})-\theta=0$ (16)

It is worth noting that the vector fields $B_{i}$ can be related to all the
necessary features of the manifold and do not depend explicitly on the
Christoffel symbols, in this regard. Now, with the differential equation

$\frac{\partial B_{i}}{\partial x^{n}}=\Gamma_{in}^{m}B_{m}$

we shall have equation (13) as

$B_{i,kn}=\Gamma_{in,k}^{m}B_{m}+\Gamma_{in}^{m}\Gamma_{mk}^{p}B_{p}$

Differentiating this again and rearranging we get

$\Gamma_{in,jk}^{m}B_{m}=B_{i,jkn}-\Gamma_{in,k}^{m}\Gamma_{mj}^{r}B_{r}-\Gamma_{in}^{m}\Gamma_{mk,j}^{p}B_{p}-\Gamma_{in,j}^{m}\Gamma_{mk}^{p}B_{p}-\Gamma_{in}^{m}\Gamma_{mk}^{p}B_{p,j}$

Now, interchanging the indices $m$ and $j$ ($m\leftrightarrow j$) we have

$\Gamma_{in,mk}^{j}B_{j}=B_{i,mkn}-\Gamma_{in,k}^{j}\Gamma_{jm}^{r}B_{r}-\Gamma_{in}^{j}\Gamma_{jk,m}^{p}B_{p}-\Gamma_{in,m}^{j}\Gamma_{jk}^{p}B_{p}-\Gamma_{in}^{j}\Gamma_{jk}^{p}B_{p,m}$
(17)

Similarly, we would have

$\Gamma_{ik,mn}^{j}B_{j}=B_{i,mnk}-\Gamma_{ik,n}^{j}\Gamma_{jm}^{r}B_{r}-\Gamma_{in}^{j}\Gamma_{jn,m}^{p}B_{p}-\Gamma_{in,m}^{j}\Gamma_{jn}^{p}B_{p}-\Gamma_{ik}^{j}\Gamma_{jn}^{p}B_{p,m}$
(18)

Again, since
$\Gamma_{in,k}^{j}B_{j}=B_{i,kn}-\Gamma_{in}^{j}\Gamma_{jk}^{p}B_{p}$, the
equations (17) and (18) can be rewritten respectively as

$\displaystyle\Gamma_{in,mk}^{j}B_{j}$
$\displaystyle=B_{i,mkn}-\Gamma_{in}^{j}\Gamma_{jk}^{p}\Gamma_{pm}^{q}B_{q}-\delta_{r}^{j}\Gamma_{jm}^{r}\\{B_{i,kn}-\Gamma_{in}^{j}\Gamma_{jk}^{p}B_{p}\\}$
(19)
$\displaystyle\quad-\Gamma_{in}^{j}\\{B_{j,mk}-\Gamma_{jk}^{p}\Gamma_{pm}^{q}B_{q}\\}-\delta_{p}^{j}\Gamma_{jk}^{p}\\{B_{i,mn}-\Gamma_{in}^{j}\Gamma_{jm}^{p}B_{p}\\}$

and

$\displaystyle\Gamma_{ik,mn}^{j}B_{j}$
$\displaystyle=B_{i,mnk}-\Gamma_{ik}^{j}\Gamma_{jn}^{p}\Gamma_{pm}^{q}B_{q}-\delta_{r}^{j}\Gamma_{jm}^{r}\\{B_{i,nk}-\Gamma_{ik}^{j}\Gamma_{jn}^{p}B_{p}\\}$
(20)
$\displaystyle\quad-\Gamma_{ik}^{j}\\{B_{j,mn}-\Gamma_{jn}^{p}\Gamma_{pm}^{q}B_{q}\\}-\delta_{p}^{j}\Gamma_{jn}^{p}\\{B_{i,mk}-\Gamma_{ik}^{j}\Gamma_{jm}^{p}B_{p}\\}$

Now, subtracting equation (20) from equation (19) and rearranging, we have

$\displaystyle B_{j}[\Gamma_{in,mk}^{j}-\Gamma_{ik,mn}^{j}]$
$\displaystyle=(B_{i,mkn}-B_{i,mnk})+\Gamma_{jm}^{j}(B_{i,nk}-B_{i,kn})+\Gamma_{ik}^{j}B_{j,mn}-\Gamma_{in}^{j}B_{j,mk}$
(21)
$\displaystyle\quad+\delta_{p}^{j}(\Gamma_{jn}^{p}B_{i,mk}-\Gamma_{jk}^{p}B_{i,mn})+2\delta_{p}^{j}B_{j,m}(\Gamma_{in}^{j}\Gamma_{jk}^{p}-\Gamma_{ik}^{j}\Gamma_{jn}^{p})$

Let us consider the second term with parenthesis on the right hand side of the
last equation. Multiplying by $B^{2}B_{j}=B^{j}B_{j}B_{j}=B^{j}B^{2}_{j}$ we
shall have

$B^{2}B_{j}\Gamma_{jm}^{j}(B_{i,nk}-B_{i,kn})=B^{j}B_{j}B_{j}\Gamma_{jm}^{j}(B_{i,nk}-B_{i,kn})=B^{2}B_{j,m}(B_{i,nk}-B_{i,kn})$

Clearly, there is a breakdown of index notation, pertinent to the index ’j’.
We shall elaborate this scenario now and we shall find the above equation to
be useful subsequently. We make an ansatz that in this special scenario, the
first term in the parenthesis in equation (21), namely
$(B_{i,mkn}-B_{i,mnk})$, becomes explicitly independent of the index $j$
present in the left hand side. The rationale can be attributed to when some
structure preserving endomorphism that preserves the geometry of the manifold
breaks down and thereby the RC tensor accrues a new upper index and the index
notation breaks down.
The feasibility of the rationale introduced above will be evident later while
elucidating equation (30). So, essentially, the RC tensor arising from the
breakdown will be independent of the index $j$. This causes the breakdown of
the ’index notation’ mentioned earlier. Therefore, considering this ansatz we
may write

$B_{i,mkn}-B_{i,mnk}=-\partial_{m}[2R_{i\leavevmode\nobreak\ kn}^{p}B_{p}+\eta
B_{i}]=-2\partial_{m}[R_{i\leavevmode\nobreak\
kn}^{p}]B_{p}-2[R_{i\leavevmode\nobreak\ kn}^{p}]B_{p,m}-\partial_{m}(\eta
B_{i})$

where we have considered a new index - $p$, such that $j\neq p$. Thus, using
(21) and multiplying both sides of equation (16) by
$B^{2}=B^{j}B_{j}=\delta^{jj}B_{j}B_{j}$ we derive

$\displaystyle\quad 2B^{j}B^{2}_{j}[\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}B_{j}-\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{p}\\}B_{p}]+2B^{2}B_{j}[\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}B_{j,m}-\\{R_{i\leavevmode\nobreak\ kn}^{p}\\}B_{p,m}]$ (22)
$\displaystyle\quad+B^{2}B_{j,m}(B_{i,nk}-B_{i,kn})+\delta_{p}^{j}B^{j}B^{2}_{j}(\Gamma_{jn}^{p}B_{i,mk}-\Gamma_{jk}^{p}B_{i,mn})$
$\displaystyle\quad+2\delta_{p}^{j}B^{j}B^{2}_{j}B_{j,m}(\Gamma_{in}^{j}\Gamma_{jk}^{p}-\Gamma_{ik}^{j}\Gamma_{jn}^{p})-B^{j}B^{2}_{j}\theta=0$

Now, we shall use the relation
$\Gamma_{in,k}^{j}B_{j}=B_{i,kn}-\Gamma_{in}^{j}\Gamma_{jk}^{p}B_{p}$, and
compute $\theta$ as follows:

$\displaystyle B_{j}^{2}\theta$
$\displaystyle=B_{j}B_{j,m}(B_{i,nk}-\Gamma_{ik}^{j}\Gamma_{jn}^{p}B_{p})+B_{j}B_{j,n}(B_{i,km}-\Gamma_{im}^{j}\Gamma_{jk}^{p}B_{p})$
(23) $\displaystyle\quad-
B_{j}B_{j,m}(B_{i,kn}-\Gamma_{in}^{j}\Gamma_{jk}^{p}B_{p})-B_{j}B_{j,k}(B_{i,nm}-\Gamma_{im}^{j}\Gamma_{jn}^{p}B_{p})$

from which we obtain

$\displaystyle B_{j}^{2}\theta$
$\displaystyle=B_{j,m}(B_{j}B_{i,nk}-B_{i,k}B_{j,n})+B_{j,n}(B_{j}B_{i,km}-B_{i,m}B_{j,k})$
(24) $\displaystyle\quad-
B_{j,m}(B_{j}B_{i,kn}-B_{i,n}B_{j,k})-B_{j,k}(B_{j}B_{i,nm}-B_{i,m}B_{j,n})$

Rearranging the terms we have

$B_{j}^{2}\theta=B_{j}B_{j,m}(B_{i,nk}-B_{i,kn})+B_{j}(B_{j,n}B_{i,km}-B_{j,k}B_{i,nm})+B_{j,m}(B_{i,n}B_{j,k}-B_{i,k}B_{j,n})$

Also, we have

$B^{j}B_{j}\delta_{p}^{j}B_{j}(\Gamma_{jn}^{p}B_{i,mk}-\Gamma_{jk}^{p}B_{i,mn})=B^{2}(B_{j,n}B_{i,mk}-B_{j,k}B_{i,mn})$

and

$B^{j}\delta_{p}^{j}B^{2}_{j}B_{j,m}(\Gamma_{in}^{j}\Gamma_{jk}^{p}-\Gamma_{ik}^{j}\Gamma_{jn}^{p})=B^{j}B_{j,m}(B_{i,n}B_{j,k}-B_{i,k}B_{j,n})$

Therefore, the revised form of equation (22) is given as

$\displaystyle\quad 2B^{2}B_{j}[\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}B_{j}-\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{p}\\}B_{p}]+2B^{2}B_{j}[\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}B_{j,m}-\\{R_{i\leavevmode\nobreak\ kn}^{p}\\}B_{p,m}]$ (25)
$\displaystyle\quad+B^{2}B_{j}(B_{j,n}B_{i,mk}-B_{j,k}B_{i,mn})+2B^{j}B_{j,m}(B_{i,n}B_{j,k}-B_{i,k}B_{j,n})$
$\displaystyle\quad-B^{2}B_{j}(B_{j,n}B_{i,km}-B_{j,k}B_{i,nm})-B^{j}B_{j,m}(B_{i,n}B_{j,k}-B_{i,k}B_{j,n})=0$

Again, as we have seen before

$(B_{i,nk}-B_{i,kn})=(2R_{i\leavevmode\nobreak\ kn}^{j}+\eta B_{j})$

where, $\eta B_{j}=B_{j;kn}-B_{j;nk}$. And, we know the relation for the
contravariant vector as

$B_{\leavevmode\nobreak\ ;ik}^{n}-B_{\leavevmode\nobreak\
;ki}^{n}=R_{i\leavevmode\nobreak\ kl}^{n}B^{l}$

Now since, the metric tensor is invariant with respect to the covariant
derivative, lowering the index of the vector we obtain

$B_{l;ik}-B_{l;ki}=R_{i\leavevmode\nobreak\ kl}^{n}B_{n}$

Thus

$(B_{i,nk}-B_{i,kn})=2R_{i\leavevmode\nobreak\
kn}^{j}B_{j}+R_{k\leavevmode\nobreak\ ij}^{q}B_{q}$

Using this relation we finally derive the curvature evolution equation as
follows:

$\displaystyle\quad 2B^{2}B_{j}[\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}B_{j}-\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{p}\\}B_{p}]+2B^{2}B_{j}[\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}B_{j,m}-\\{R_{i\leavevmode\nobreak\ kn}^{p}\\}B_{p,m}]$ (26)
$\displaystyle\quad+B^{2}B_{j}B_{j,n}(2R_{i\leavevmode\nobreak\
km}^{j}B_{j}+R_{k\leavevmode\nobreak\
ij}^{q}B_{q})+B^{2}B_{j}B_{j,k}(2R_{i\leavevmode\nobreak\
mn}^{j}B_{j}+R_{m\leavevmode\nobreak\ ij}^{q}B_{q})$
$\displaystyle\quad+B^{j}B_{j,m}(B_{i,n}B_{j,k}-B_{i,k}B_{j,n})=0$

The mathematical significance is immediately apparent. The physical
significance will be manifest in the subsequent parts of the paper. Let us
consider the special case where $k=n$. In such a scenario, the preceding
equation reduces to

$2B^{2}B_{j}[\partial_{m}\\{R_{i}^{j}\\}B_{j}-\partial_{m}\\{R_{i}^{p}\\}B_{p}]+2B^{2}B_{j}[\\{R_{i}^{j}\\}B_{j,m}-\\{R_{i}^{p}\\}B_{p,m}]+B^{2}B_{j}B_{j,n}B_{q}(R_{n\leavevmode\nobreak\
ij}^{q}+R_{m\leavevmode\nobreak\ ij}^{q})=0$ (27)

Again, $R_{i}^{p}=\delta_{j}^{p}R_{i}^{j}$ and $B_{p}=\delta_{p}^{j}B_{j}$.
Also, taking into consideration another special case: $n=q=m$, we have

$2B^{2}B_{j}^{2}\partial_{m}R_{i}^{j}[1-\delta_{j}^{p}\delta_{p}^{j}]+2B^{2}B_{j}B_{j,m}R_{i}^{j}[1-\delta_{j}^{p}\delta_{p}^{j}]+B^{2}B_{j}B_{m}B_{j,m}R_{ij}=0$
(28)

which is another form of the evolution equation (LABEL:15), with $m=n=q=k$.
Now, if the ansatz breaks down and $j=p$ then we have from (28)

$\delta_{jm}\delta_{ij}B_{j,m}B^{2}R_{ij}B^{i}B^{j}=0$

where, $R_{s}=R_{ij}B^{i}B^{j}$ is the Raychaudhuri scalar. So

$B_{j,m}B^{2}R_{s}=\partial_{m}(B^{2}R_{s})-B^{2}\partial_{m}R_{s}=0$
$\Rightarrow B^{2}=const.$

Assuming that this constant term doesn’t change the structural form and
properties of $R_{s}$ we can write without loss of any generality

$B^{2}R_{s}\sim R_{s}$ (29)

which can be looked upon as an endomorphism of the set of Raychaudhuri scalar,
in this particular cosmology (with respect to the index $j$), given as

$B^{2}:R_{s}\longmapsto R_{s}$ (30)

which implies

$B^{2}\circ R_{s}=R_{s}$

So, when the ansatz breaks down it corresponds to the consideration of an
endomorphism. On the other hand, we assumed earlier that when an endomorphism
breaks down the ansatz will hold. This substantiates the plausibility of why
we had associated the breakdown of an endomorphism with the ansatz we
introduced. Essentially, these two can be considered to be correlated and
complementary.
Another interpretation with regard to the endomorphism is that $B_{j}$ is a
timelike unit vector field with the respect to the index $j$, which insinuates
that the Raychaudhuri scalar precludes all vector fields the self scalar
products of which are not constant.
Now, let us get back to equation (LABEL:15). Using the Kronecker delta it can
be rewritten as

$\displaystyle\quad 2B^{2}B_{j}^{2}\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}[1-\delta_{j}^{p}\delta_{p}^{j}]+2B^{2}B_{j}B_{j,m}\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}[1-\delta_{j}^{p}\delta_{p}^{j}]$ (31)
$\displaystyle\quad+B^{2}B_{j}B_{j,n}(2R_{i\leavevmode\nobreak\
km}^{j}B_{j}+R_{k\leavevmode\nobreak\
ij}^{q}B_{q})+B^{2}B_{j}B_{j,k}(2R_{i\leavevmode\nobreak\
mn}^{j}B_{j}+R_{m\leavevmode\nobreak\ ij}^{q}B_{q})$
$\displaystyle\quad+B^{j}B_{j,m}(B_{i,n}B_{j,k}-B_{i,k}B_{j,n})=0$

Again, we know that the Brouwer’s fixed point theorem [19] states: For any
continuous function $f$ mapping a compact convex set to itself there is a
fixed point.
Therefore, considering a continuous mapping $f$ in our Riemannian manifold and
a compact, geodesically convex vector field ($\mathfrak{B}$) comprised of
timelike vectors $B_{i}$, we shall have

$f:\mathfrak{B}\longmapsto\mathfrak{B}$

and a fixed point under this automorphism. Incidentally, choosing the index
$j$ we can infer that there is a fixed point with respect to this index in the
vector field (or the geodesic field), through which the family of vectors
$B_{j}$ are parallel transported. Consequently, the vectors $B_{j}$ will be
constant irrespective of the coordinates and the geometric structure of the
manifold. Thus, equation (LABEL:20) becomes

$2B^{2}B_{j}^{2}\partial_{m}\\{R_{i\leavevmode\nobreak\
kn}^{j}\\}[1-\delta_{j}^{p}\delta_{p}^{j}]=0$ $R_{i\leavevmode\nobreak\
kn}^{j}=const.$ (32)

Hence, the curvature tensor returns to its initial state with respect to the
index $j$. This corresponds to the statement of Poincare’s recurrence theorem
[20], if one considers the whole manifold to be a system.
Now, let us consider equation (28). Writing,
$1-\delta_{j}^{p}\delta_{p}^{j}=\delta$, we have

$2B^{2}B^{2}_{j}\delta\delta^{jk}\partial_{m}R_{ik}+2B^{2}B_{j}B_{j,m}\delta\delta^{jk}R_{ik}+B^{2}B_{j}B_{m}B_{j,m}\delta_{j}^{k}R_{ik}=0$

or,

$2B^{2}B^{2}_{j}\delta^{jk}\partial_{m}R_{ik}+2B^{j}B_{j,m}\delta_{ji}R_{ik}B^{i}B^{k}+B^{2}B_{j,m}\delta_{j}^{k}\delta_{ji}\delta_{mk}R_{ik}B^{i}B^{k}=0$

Since, we have considered the case where $j\neq p$, $\delta=1$. So, using the
expression for the Raychaudhuri scalar ($R_{s}$) and after rearranging we have
finally

$B_{j,m}R_{s}=-\rho B^{2}B_{j}^{2}R_{ik,m}$ (33)

where,
$\rho=2\delta^{jk}(2\delta_{ji}B_{j}^{-1}+\delta_{j}^{k}\delta_{ji}\delta_{mk})^{-1}=2\delta^{jk}(2\delta_{ji}B_{j}^{-1}+\delta_{mi})^{-1}$.
Thus, we have obtained the result that under certain conditions the
Raychaudhuri scalar depends on the derivative of the Ricci tensor ($R_{ik}$),
the associated timelike vector field ($B_{j}$) and its derivatives.
Since, it is known that $R_{s}$ is the trace of the tidal tensor epitomizing
the relative acceleration due to gravity of two objects separated by an
infinitesimal distance and that $R_{ik}$ measures the change in geometry as an
object moves along geodesics in the space, we can conclude: For some
particular timelike vector field in a Riemannian or pseudo-Riemannian
manifold, the relative acceleration due to gravity decreases with the increase
in curvature and vice versa.

##  4 Sectional curvature

In this section, we analyze and discuss the results of the preceding section.
Firstly, let us consider the parameter $\rho$ of equation (33). Now, for
$j=k\neq i$ and $m=i$ we have

$B_{j,i}R_{s}=-2B^{2}B_{j}^{2}R_{ij,i}$ (34)

where, $R_{s}=R_{ij}B^{i}B^{j}$. On the other hand, for $i=j=k$, $i\neq m$ we
have with $R_{s}=R_{jj}B^{j}B^{j}$. Hence, we shall obtain

$B_{j,m}R_{s}=-B^{2}B_{j}^{3}R_{jj,m}$ (35)

Now, equation (34) can also be written as

$B_{j,i}R_{s}=-B^{2}B_{j}^{2}[R_{ij,i}+R_{ij,i}]$ (36)

Now, if we take into consideration the automorphism introduced in the previous
section, then

$B_{j,i}=0$

Consequently, we have

$R_{ij,i}+R_{ij,i}=0$ (37)

The general implication of this equation is that the Ricci curvature,
$R_{ij}$, is constant with respect to the index $i$. However, there can be
another implication that the first term gives a positive value and the second
gives a negative value - a notion that might seem erroneous, but can be used
as an alternative explanation. To be precise

$|R_{ij,i}|=\pm R_{ij,i}$ (38)

which essentially insinuates that

$|R_{ij}|=\pm R_{ij}$ (39)

i.e. the Ricci curvature can have both positive and negative values. Again,
since $R_{ij}$ is obtained by contracting the RC tensor which in turn is
related to the sectional curvature of a Riemannian manifold (with respect to
the given manifold and two linearly independent tangent vectors at the same
point), we can conclude that the sectional curvature also will have both
positive and negative values. This bespeaks for both geodesic convergence and
divergence on account of Rauch’s comparison theorem [21, 22] that states: for
positive sectional curvature, geodesics tend to converge and for negative
sectional curvature, geodesics tend to diverge.
Essentially, under special circumstances, geodesics can diverge. Geodesic
convergence leads to a singularity; similarly geodesic divergence would lead
to a point of exclusivity as shown previously by resorting to Raychaudhuri’s
equation.
It is interesting to point out that during the inflationary era, curvature
played a significant role pertinent to the dynamics, as researchers have found
out [23].

##  5 Discussions

In the present article, we have established that an equation epitomizing the
evolution of curvature, resorting to an ansatz originating from the breakdown
of an endomorphism correlated to the Raychaudhuri scalar. It is also shown
that the ansatz and the endomorphism are interrelated in the sense that one
precludes the other.
From the aforementioned considerations we have also found that the Riemann-
Christoffel curvature tensor tends to follow Poincare’s recurrence theorem and
thereby the curvature returns to its initial value after certain period of
time.
Another interesting consequence is that the existence and feasibility of
negative curvature which entails geodesic divergence. This is also validated
by using the expansion scalar which is show to have both positive and negative
values. This negative value and that of the Riemann-Christoffel tensor
indicates that there might exist a point of _exclusivity_ which is the
opposite of the point of _singularity_ \- a result that has been derived from
the Raychaudhuri equation too.
Ostensibly, the notion of such a point is in a sense a speculative
extrapolation and demands ample amount of study and research. But, the
prospect is something worth investigating, at the very least.

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