1. Introduction

An important and notable feature of solutions to Einstein theory of gravity, such as
Schwarzchild spacetime or the cosmological Friedmann-Lemaître-Robertson-Walker
solution, is that they all have apparent singularities. The question was, what would
be their physical implications? It was Raychaudhuri who tried to investigate these
solutions in both cosmological ^{[1]} and
gravitational contexts ^{[1]}. Later, in 1955, he
proposed his famous equations ^{[3]} which are
known as Raychaudhuri equations.

These equations appeared to be of great importance in describing gravitational
focusing and spacetime singularities, which are essentially explained by the
so-called Focusing Theorems. Assuming Einstein equations and using energy
conditions, it is established that time-like and null geodesics, if they are
initially converging, they will focus until reaching zero size in a finite time. One
of the most important notions about singularities, as pointed out by Landau and
Lifshitz ^{[7]}, is that a singularity would
always imply focusing of geodesics, while focusing itself does not imply a
singularity. Therefore in their work, the concept of geodesic focusing (although not
explicitly stated the same) was worked out. However, they did not introduce shear
and rotation, which are inseparable ingredients of Raychaudhuri equations.

Ever since the equations were published, they have been discussed and analyzed in
numerous frameworks of general relativity, quantum field theory, string theory and
relativistic cosmology. One should note that the concept of a singularity was first
defined in seminal works of Penrose and Hawking ^{[4}-^{6]}. Therefore, it was only then
that Raychaudhuri equations received their deserved acclaim.

In addition to general relativity, the Raychaudhuri equations have also been
discussed in the context of alternative theories of gravity, such as ^{[8]}. In this paper as
well, we consider characteristics of time-like flows in an alternative theory of
gravity which as well has cosmological implications, namely, the Weyl theory of
gravity. The paper is organized as follows: in Sec. 2 we briefly introduce the
Raychaudhuri equations and its kinematical parameters; in Sec. 3 we mention the Weyl
field equations and obtain the kinematical evolution of time-like radial and
rotational geodesic flows, and see how they behave by plotting the congruences of
the integral curves. Finally in Sec. 4, we summarize the results.

2. Raychaudhuri Equations

As it was mentioned, the Raychaudhuri equations are evolution equations for
expansion, shear and rotation of a time-like geodesic congruence of integral curves,
which is indeed pure geometrical and therefore, are independent of reference frames
of Einstein equations. Being parametrized in terms of the affine parameter of the
geodesics, ^{[9]}:

In Eqs. (1) to (3),

Generally speaking, the Raychaudhuri equations deal with the kinematics of flows
which are generated by vector fields. Such flows are indeed congruences of integral
curves which may or may not be geodesics. Actually, in the context of these
equations, we are interested in the evolution of the kinematical characteristics of
the so-called flows, not the origin of them. These characteristics which are
contained in the Raychaudhuri equations, may constitute one equation like ^{[10]}:

in which the trace-less symmetric part is defined as:

Also, the scalar trace is and the anti-symmetric part are:

Geometrically, these quantities are related to a cross-sectional area, which encloses
a definite number of integral curves and is orthogonal to them. Moving along the
flow lines, this area may isotropically changes its size or being sheared or
twisted, however, it still holds the same number of flow lines. There are some
analogies with elastic deformations which are discussed in Ref. ^{[11]}. Also, one can find explicit discussions
on these quantities in Refs.^{[12},^{13]}.

Here we should note that the Raychaudhuri’s equations may be essentially regarded as identities, which become equations when they are, for example, used in spacetimes defined by Einstein field equations.

Moreover, these equations are of first order and non-linear. Also, the expansion
equation in Eq. ([1]), is the same as Riccati equation in a mathematical point of
view ^{[14},^{15]}. The expansion is indeed the rate of change of the cross-sectional
area, which is orthogonal to the geodesic bundle.

In the next section, we will find the mentioned kinematical characteristics, for curve bundles on a definite spacetime background.

3. Time-like Geodesic Congruences in Weyl Gravity

The Weyl theory of gravity has had an interesting background and had received a great
deal of attention from those who believe that the dark matter/dark energy scenarios,
could be well-treated by altering general theory of relativity. This became more
elaborated after arguing that the extraordinary behavior of the galactic rotation
curves, could be extracted from Weyl gravity as a natural consequence of its vacuum
solutions ^{[16}-^{18]}. There were subsequently more attempts to make relations between the
theory’s anticipations and observational evidences ^{[19}-^{21]} and as the main course,
proposing gravitational alternatives to dark matter/dark energy ^{[23}-^{25]}.
Some good information about this 4th order theory of gravity can be found in Ref.
^{[22]} . Another important feature of the
Weyl theory of gravity, is its conformal invariance which, as it is stated in the
literature, could be considered as a tool of unification with the standard model by
creating the desired mass during the symmetry breaking ^{[26]}.

The Weyl theory of gravity, is a theory of 4th order with respect to the metric. Weyl gravity is characterized by the Bach action:

where

from which, using the total divergency of the Gauss-Bonnet term

Varying Eq. (11) with respect to ^{[16]}:

Accordingly, the Weyl field equations read as:

where ^{[16]}:

in which:

This solution has three important constants,

3.1 Radial Flows

Some features of pure radial time-like flows has been discussed in Ref. ^{[27]}. Firstly, for the spacetime
coordinates on a parametric integral curve:

we define the velocity 4-vector field:

which is supposed to be tangential to any integral curve inthe spacetime and for
time-like congruences, one must have

and the geodesic equations:

are respectively:

Integrating Eq. (21) we obtain:

using which in Eq. (20), gives

Taking the positive part, the tangential vector field in Eq. (17) for pure radial flows becomes:

The flows, which are formed by this vector field, are expanding by the following factor which is obtained by use of Eq. (7):

The vector field in Eq. (25) is
orthogonal to the hypersurface of the field crests. Therefore, it is of zero
rotation ^{[11]}. However, the shear tensor
is non-zero. From Eqs. (4), (6) and (26), we have:

To obtain a pictorial viewpoint of how radial flows will behave in a Weyl field,
we should find an expression for

Unfortunately, this is a non-linear second order equation. Even the first order
equation in Eq. (24) could not be
explicitly solved. Therefore, we consider a simpler case of

with:

where

*3.1.1 Focusing*

If spacetime singularities are concerned, Eq. (1) for the expansion receives the most central attention. As it was shown above (and also in Fig. 1), the expansion changes the cross-sectional area along the geodesic bundle. It is noted that, if the expansion approaches the negative infinity, the congruences will converge, whereas they diverge if the expansion goes to positive infinity. The convergence, is what we regard as the focusing of the geodesic bundles toward the singularities. However, to clarify whether any convergence occurs for a peculiar flow, one should examine the expansion equation. It has been put in the literature that, convergence occurs if:

or equivalently,

Therefore, one can observe that shear acts in favor of convergence, while
rotation opposes it. However, for zero rotations the condition in Eq. (30) reduces to ^{[10]}:

Now, for the zero-rotational flow defined by the vector field in Eq. (25), the condition in Eq. (32) gives:

This provides us a maximum for ^{[28},^{29]}Therefore, if galactic scales are of
interest, Eq. (30) implies
that for

3.2 Rotational Flows

Pure rotational flows in the equatorial plane, could be obtained by letting

Equation (35) implies

Therefore, the tangential vector field can be written as:

Together with Eq. (7), one can see
that

Moreover, using Eq. (8), one can obtain the rotation of the flow, which is characterized by the following non-zero components of the anti-symmetric part of the Raychaudhuri kinematical parameters:

The constant radial distance can be obtained from Eqs. (36) and (38). We have:

According to this, the pure rotational flow can be obtained which has been depicted in Fig. 2. One can note that the cross-sectional area will suffer a twist while holding the same number of integral curves. However, for each curve, the radial distance is a constant and the shear gradually will make the congruence bundle to become compressed.

3.3 Radial-Directional Flow

Now let us consider a geodesic flow, for which both radial and angular components of the tangent vector field of the congruence are supposed to be affine variables. We are still on the equatorial plane, so the time-like condition (18) and the geodesic Eqs. (19) read as:

The temporal part of the geodesic equations and consequently

Using Eqs. (23) and (45) in Eq.(43) to obtain

According to Eq. (46), one can obtain the kinematical characteristics of a Radial-Directional flow in a Weyl field. The expansion is:

where:

Moreover, the non-zero components of the shear tensor become:

Surprisingly, the rotation tensor vanishes;

Once again, dealing with a typical galaxy, the dimension-less term

Note also that, this is in agreement with the convergence condition for the purely radial flows, which was discussed above.

*3.1.1 Capture Zone*

One might even obtain the region in which there would be a possible capturing
of the flow, by use of an effective potential. To proceed with this method,
let us use the time-like condition in Eq. (43), exploiting the total energy definition

where:

This potential has been plotted in Fig.
3. However, in order to capture a time-like flow, it is pretty
useful to take care about the potential maximums. Such maximums may
represent unstable circular orbits. Hence, if the energy of a particle is
supposed to be the same as the potential maximum, it will be inevitably
captured and the corresponding time-like geodesics will ultimately
terminated where the potential originates from. The potential in Eq. (52) has a maximum around

4. Summary

In this paper, we dealt with the characteristics of time-like geodesic flows in a
Weyl field and we supposed that such field is formed in vacuum spacetime obtained
from vacuum Weyl field equations. We distinctly considered radial and rotational
flows, and obtained their corresponding expansion, shear and vorticity
(*i.e*. the rotation tensor). We noted that for radial flows,
only expansion and shear do contribute in the characterization and as it is obvious
in Fig. 1, the expansion is isotropic, however,
the shear makes the formation of the curve bundle an-isotropic. Moreover, for the
particular rotational flows considered here, we found that although it is not usual,
however, the expansion vanishes. Therefore, we are left with a simple shear tensor
and, of course, a non-vanishing vorticity. This may provide us a circular flow,
which is gradually getting compact. For further considerations, one may concern with
null-like geodesics in Weyl fields. Also, it is possible to take both rotation and
radial motions in same flows. In cosmological contexts, this may help us to discover
how real cosmic flows will behave in Weyl conformal gravity.