Computational aspects of weakly cyclic P-symmetric spacetime under semi-symmetric metric connection

Ever since the beginning of our civilized society, we have been on a scientific quest to understand the origin of our Universe. To this end, our ever-evolving thought process led to the birth of the popular subject of cosmology, along with the general theory of relativity, in short, GTR, which is considered the most elegant and sophisticated blessing of applied mathematics. The study of GTR, which disregards quantum effects, is pivotal to the understanding of cosmology. In GTR, the matter content of the universe is characterized by selecting the suitable energy-momentum tensor T, which is the most important part of the space-matter tensor P introduced by A. Z. Petrov [1]. Since the matter content of the universe is assumed to behave like a perfect fluid in the standard cosmological models, the physical motivation for studying Lorentzian manifolds is the assumption that a gravitational field may be effectively modeled by some Lorentzian metric defined on a suitable four-dimensional manifold. It is known that spacetime is considered as a four-dimensional torsion-free, time-oriented Lorentzian manifold of signature (+, +, +, −) as a particular sub-class of semi-Riemannian manifolds. Albert Einstein's field equations, in short, AFE, are fundamental in the construction of cosmological models, which means that the matter determines the geometry of the spacetime, and conversely, the motion of matter is determined by the metric tensor of the space, which is non-flat. Here, it should be noted that the space-matter tensor plays a perfect role in studying spacetime.

The semi-symmetric metric connection, in short, SSMC, also plays an important role in the study of semi-Riemannian manifolds. Different physical problems involving the SSMC are observed; for example, if a person is moving on the surface of the Earth always facing one definite point, say the South Pole, then this displacement is semi-symmetric and metric [2]. Again, during the mathematical congress in Moscow in 1934, one evening, mathematicians invented the “Moscow displacement”. The streets of Moscow are approximately straight lines through the Kremlin and concentric circles around it. If a person walks on the street always facing the Kremlin, then this displacement is semi-symmetric and metric [2].

The object of the present paper is to study a four-dimensional weakly cyclic P-symmetric spacetime, in short $(PWCS)4$⁠, admitting a semi-symmetric metric connection. In Section 2, we discuss the mathematical preliminaries about this connection. The next section is about the study of the space-matter tensor under the influence of SSMC. In Section 4, we define a four-dimensional weakly cyclic P-symmetric manifold admitting SSMC, in short $[(PWCS)4,▿¯]$⁠.

The natural question now arises whether such a spacetime exists or not with respect to a physical view point. We get its affirmative answer in Section 5 by constructing a non-trivial example. Also, this section is devoted to the solution of the mathematical model obtained here. The mathematical model involves a non-linear ordinary differential equation and its non-linear term belongs to the C^∞-class, so its solution exists uniquely in view of classical Cauchy's problem. Here we also study the behavior of the solution through numerical simulations in the context of spacetime anomaly, which are very important in the study of different cosmological models.

Let $(U4,g)$ be an four-dimensional spacetime of class C^∞ with the metric tensor g and ▿ be the connection of the space time $(U4,g)$⁠. A linear connection $▿¯$ on $(U4,g)$ is said to be semi-symmetric [3] if the torsion tensor τ of the connection $▿¯$ satisfies

for all vector fields X₁, X₂ on $U4$ and a is an associated 1-form of the semi-symmetric connection given by

where μ is called the associated vector field of the semi-symmetric connection.

A semi-symmetric connection $▿¯$ is called an SSMC [4] if, in addition, it satisfies

The relation between the SSMC $▿¯$ and the connection ▿ of (V⁴, g) is given by [5].

Also the covariant differentiation of a 1-form a with respect to $▿¯$ is given by [5].

for all vector fields X₁, X₂ in $U4$⁠. Again from Ref. [5], we have the relation between the curvature tensors R corresponding to ▿ and $R¯$ corresponding to $▿¯$ as follows

where B is a (0, 2) type tensor field defined by

for all vector fields X₁ and X₂. From (2.5) we find the relation between Ricci tensors $S¯$ corresponding to $▿¯$ and S corresponding to ▿ as

for all vector fields X₂ and X₃. Applying contraction over X₂ and X₃, we find the scalar curvature corresponding to $▿¯$ from (2.7) as

where α = Tr. B and r is the scalar curvature corresponding to ▿.

It is known that the AFE without a cosmological constant is

where λ is a gravitational constant, $TX1,X2$ is the energy-momentum tensor and r is the scalar curvature.

Contraction of (3.1) yields

where $T̃=Tr.gT$⁠. In 1949, Petrov [1] introduced a (0, 4) type tensor P and studied the gravitational fields on the basis of the algebraic structures of the tensor. This tensor is defined as follows

where

and σ is energy density, which is a scalar. P is known as the space-matter tensor of type (0, 4). The first part of this tensor represents the curvature of the space, and the second part represents the distribution and motion of the matter. By virtue of (3.3) and (3.4) we have

Contraction of the last relation over X₂ and X₄ yields

for all vector fields X₁ and X₃. If we assume that $P¯$ is the space-matter tensor with respect to $▿¯$⁠, then from (2.8) and (3.6), we get

for all vector fields X₁, X₃. So we have the following result:

In GTR and cosmology, weakly cyclic Ricci symmetric spacetime plays an important role and such a type of spacetime was studied by many authors [6]. In tune of these we define $(PWCS)4$ spacetime as follows:

where α, β and γ are 1-forms (non-vanishing simultaneously), and ▿ is the covariant differentiation operator with respect to the metric tensor g.

Also we define $(PWCS)4$ spacetime admitting SSMC as follows:

where $α¯$⁠, $β¯$⁠, $γ¯$ are 1-forms (non-vanishing simultaneously), and $▿¯$ denotes the operator of covariant differentiation operator with respect to the SSMC.

Here the 1-forms $α¯$⁠, $β¯$ and $γ¯$ are known as the associated 1-forms of the spacetime. A $(PWCS)4$ admitting an SSMC furnished with $▿¯$ will be denoted by $[(PWCS)4,▿¯]$⁠. Henceforth, geometric objects from $[(PWCS)4,▿¯]$ will also be denoted by “−”.

We now put the following:

Proof. In dust fluid spacetime, the energy-momentum tensor is of the form ([7, 8]):

where γ is the energy density of dust-like matter and E is a non-zero 1-form defined by E(X) = g(X, ψ) such that g(ψ, ψ) = −1, ψ being a unit vector field. Contracting (3.5) and using (4.3), we obtain

where $k=−m2−3σ$ and m + λϕ = 0. Thus, the spacetime is a PFS.

Conversely, in a PFS,

In light of (4.5), (3.5), we have

So if we assume that $T̃=2(k+3σ)λ$ then equation (4.6) yields the expression of the energy-momentum tensor of the dust fluid spacetime. □

This section deals with an example of $[(PWCS)4,▿¯]$ to show its existence. Also, graphical interpretations are provided for such a space-time.

We define $U4=R4$ equipped with the metric

where $g11=ex1=g22$⁠, g₃₃ = cosh x², g₄₄ = −1 and g_ij = 0 for i ≠ j. Then, the only non-vanishing components of the Ricci tensor and hence the scalar curvature as follows:

which shows that the scalar curvature is non-vanishing and non-constant. Using (3.6) we can compute the non-vanishing component of the space-matter tensor and its covariant derivatives as follows

where 2l + sech²x² + 1 = 0. We shall now show that $U4$ is $(PWCS)4$⁠. In terms of the local coordinate system, (4.1) can be written as

where α_i, β_i, γ_i are the local coordinates of 1-forms α, β, γ respectively. Let us define 1-forms as follows:

If i = j = 3 and k = 1, then by virtue of (5.3) and (5.5) we obtain the following relations for R.H.S. and L.H.S. of (5.4):

Similarly, it can be proved that (5.4) is true for other values of i, j and k. So our considered spacetime is a $(PWCS)4$⁠.

Now if in terms of the local coordinate system, the components of the connection ▿ and $▿¯$ are respectively $Γjki$ and $Γ¯jki$⁠, then in view of (2.4) we have

where δ is the Kronecker delta. Let us define the 1-form a as follows:

where $∂i=∂∂xi$ at any point $xi∈U$ and ξ is a C^∞-function of x¹ over an interval I ∈ R. Thus, we determine the function ξ such that the spacetime becomes $[(PWCS)4,▿¯]$⁠.

By virtue of (2.6), the only non-zero components of the Christoffel symbols $Γ¯jki$ and P_ij are

where $ξ̇=dξdx1$⁠. From (3.7) and (5.8), we can compute the non-vanishing component of the contracted space-matter tensor $P¯$ and their covariant derivatives as follows

Equation (4.2) can be written in terms of the local coordinate system as

where $α¯$⁠, $β¯$⁠, $γ¯$ are 1-forms and in terms of the local coordinates we define these as follows:

Using (5.8), (5.9) and (5.11) in (5.10), we have the following differential equation:

which is non-linear in nature. To obtain its solution, we have, after integrating once

where A is the constant of integration such that $A∈−∞,116$⁠. Applying the transformation $ξ(x1)=−u′(x1)u(x1)$ in the above differential equation, we get

It is worthy to mention here that the aforementioned reduced form becomes linear now. Employing $t=e−x1$ in the above equation, we have

It can be noticed that t = 0 is the regular singular point of the above differential equation (5.15). Now, without loss of generality, we assume that A = 0. Applying the Frobenius method [9, 10] the solution of the above differential equation (5.15) is derived as

Therefore, the solution of (5.14) is

where c₀ and d₀ are two arbitrary constants, and other parameters involved in the above equation are derived as follows:

and

Hence the desired solution is obtained as

Now once the arbitrary constants c₀ and d₀ are known, the solution of the differential equation (5.12) can be determined using the relations (5.17) - (5.20). So, we have completely determined the ξ(x¹) and the coefficients of SSMC. Thus, the most coveted torsion tensor τ can be calculated with these. Therefore, our considered manifold is a non-trivial example of a $[(PWCS)4,▿¯]$⁠.

It is worth mentioning here that the function ξ, described in (5.20), exists for certain values of x¹. This phenomenon occurs as the function ξ becomes undefined due to the occurrence of the form $∞∞$ for certain negative values of x¹ or of the form $00$ for certain positive values of x¹.

Noether's theorem [11] illustrates the relationship between symmetries and conserved quantities in dynamical systems. In curved spacetime models, the inclusion of gauge terms enhances the structure of allowable symmetries and results in altered conservation laws. In order to further understand the dynamical implications of the geometric structure introduced above, we briefly analyze Noether symmetries associated with a Lagrangian system compatible with the metric (5.1) and the 1-form field a_i = (ξ(x¹), 0, 0, 0).

Based on the metric provided in (5.1), the geodesic Lagrangian can be expressed in the standard quadratic form as

which explicitly becomes

Now, a vector field

is said to be a Noether symmetry with gauge function A, if it satisfies

where κ and ωⁱ describe infinitesimal symmetry transformations, s parametrizes the motion. X [1] is the first prolongation and D_s denotes the total derivative with respect to the parameter s.

Substituting the above Lagrangian (5.21) into the Noether condition yields a system of determining partial differential equations for the unknown κ(s, x), ωⁱ(s, x), and A(s, x). Compatibility of these equations leads precisely to the nonlinear differential equation (5.12). It is important to note that equation (5.12) governs the existence of gauge symmetry. This symmetry relates to the conservation of energy, conservation of momentum, and a modified invariant known as the Gauge-modified invariant involving ξ(x¹). Once ξ(x¹), the solution of the nonlinear differential equation (5.12), is known, it is possible to analyze the conditions for Noether's gauge symmetry. These conditions outline the behavior of a particle in curved spacetime influenced by a position-dependent gauge field. By applying Noether's theorem, we derive the first integrals as follows:

1. Energy-type invariant (invariance in x⁴):

1. Momentum-type invariant (invariance in x³):

1. Gauge-modified invariant:

which represents the canonical momentum conjugate to x¹, modified by the presence of the 1-form field.

The resulting system for determining the partial differential equations (5.12) is solved for symmetry coefficients and identifies the conditions under which the considered spacetime allows for nontrivial Noether's symmetries. A numerical solution of the constraint equation is shown in Figures 1 and 2. The first integral mentioned in equations (5.22)–(5.24) discussed above highlighting the physical interpretation concerning energy conservation and spacetime isotropy. Here, we have included a brief comparative study that demonstrates the relationship between Noether's symmetries and observed spacetime. It has been shown that Noether's gauge symmetries impose a nonlinear constraint on the gauge function. This analysis shows that the existence of Noether symmetries imposes nontrivial constraints on the admissible form of the function ξ(x¹), consistent with the geometric conditions derived earlier using the semi-symmetric metric connection. The additional invariant I₃ reflects the influence of the 1-form field and highlights how the underlying geometric structure modifies the dynamical conservation laws. In particular, the term ξ(x¹) acts as a position-dependent potential, altering the effective momentum in the x¹-direction. The conserved quantities that arise from this relationship offer insights into the interaction between geometry and gauge structure.

Though we have determined the coefficients of SSMC completely its practicability in the context of numerical simulation is yet to be seen. To inspect the differentiability of the solution function ξ, we demonstrate the graph of the solution ξ of the non-linear differential equation (5.12) in Figure 1. From this figure, we have observed that the derived solution is not smooth, which signifies that the solution function is not continuously differentiable. This is due to the presence of spacetime anomaly (or singularity) at various values of x¹. This observation signifies the existence of high-level energy (or a black hole) in the era of cosmological theory. The SSMC developed here can be taken into account, which is the futuristic scope of the present study, for various cosmological models are to be examined in the context of different energy conditions.

It is also noticed that the existence of the function ξ is limited to a certain region. For instance, when c₀ = 1, d₀ = −1, n = 60, the function ξ exists for x¹ ∈ [−118, 1,420] as can be seen from Figure 2. This phenomenon is exactly validated by the theoretical inspection discussed above. From this figure, one may think that there is only one anomaly. However this fact is untrue. To justify it we have shown a magnifying area for certain range of x¹ (in particular, −2 ≤ x¹ ≤ 2) in the inset. From this inset, it is clear that numerous spacetime anomalies are present.

The effects of c₀ and d₀ on the solution function ξ are demonstrated in Figure 3. From Figure 3(a), we have observed that the behavior of the function ξ (i.e. the presence of spacetime anomalies) remains the same as the values of c₀ change. However, the spacetime anomalies are shifted towards the increasing direction of x¹. A similar kind of phenomenon is also noticed for the case of d₀ (refer Figure 3(b)).

The authors express their deep sense of gratitude to the editor and the referees for their invaluable comments and suggestions which enabled them to carry out the desired revision of the manuscript.