Home Some anisotropic and perfect fluid plane symmetric solutions of Einstein's field equations using killing symmetries
Article Open Access

Some anisotropic and perfect fluid plane symmetric solutions of Einstein's field equations using killing symmetries

  • Muhammad Farhan , Tahir Hussain , Kamran EMAIL logo , Ioan-Lucian Popa , Jose Francisco Gomez Aguilar , Zareen A. Khan EMAIL logo and Zeeshan Ali EMAIL logo
Published/Copyright: September 30, 2025
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

The Einstein’s field equations (EFEs), central to the theory of general relativity, often require spacetime symmetries such as those defined by Killing vector fields to simplify their solutions and derive physically meaningful results. Killing vector fields preserve the metric of spacetime and yield vital conservation laws. This article presents a comprehensive study of Killing vector fields in the background of nonstatic plane symmetric spacetime using a novel method, the Rif tree approach. Using a Maple algorithm, this approach provides conditions on the metric coefficients that lead to additional Killing vector fields than the minimum ones. A detailed analysis yields a variety of spacetime metrics that admit different dimensional Killing algebras. The physical implications of the obtained metrics are discussed by finding the associated energy-momentum tensors. Several metrics are found to describe physically realistic models, including anisotropic and perfect fluid solutions of EFEs.

Keywords: spacetime symmetries; killing vector fields; plane symmetric spacetime; Rif tree approach

1 Introduction

The Einstein’s field equations (EFEs) relate the spacetime curvature with energy, momentum, and stress within that spacetime. These equations are ten nonlinear partial differential equations, given by [1]:

(1.1) R a b − 1 2 R g a b = k T a b ,

where R a b , g a b , and T a b are the Ricci, metric, and energy-momentum tensors, respectively, R is the Ricci scalar and k is a coupling constant. The nonlinearity of EFEs poses problems in solving these equations, and it is not possible to obtain a solution of these equations without some assumption. However, in literature, different approaches are used to find many solutions of these equations [1–6].

Symmetries of spacetimes have been a central theme in the study of general relativity, providing profound insights into both the mathematical structure of EFEs and the physical properties of gravitational fields. Among the various symmetries, vector fields associated with isometries, conformal transformations, and homotheties are particularly significant. These symmetries not only simplify the EFEs but also serve as a tool for generating their exact solutions, revealing underlying conservation laws, and offering insights into the causal and geometric structure of spacetimes. A Killing vector field (KVF) η = ( η 0 , η 1 , η 2 , η 3 ) is defined by the relation [7]:

(1.2) ℒ η g a b = 0 ;

ℒ being the Lie derivative operator and g a b is the spacetime metric. This condition corresponds to the preservation of the spacetime metric along the flow of the vector field, indicating an isometry or symmetry of the spacetime. KVFs are fundamental in describing conserved quantities associated with spacetime symmetries, such as energy, and linear and angular momentum.

In some cases, the metric is preserved up to a local scaling factor, reflecting a conformal symmetry that is defined in terms of a vector field η satisfying the condition [7]:

(1.3) ℒ η g a b = 2 ψ ( x a ) g a b ,

where ψ ( x a ) is a conformal factor that depends on the spacetime coordinates x a . Conformal symmetries are crucial in a variety of physical contexts, including the study of asymptotic structures in spacetimes and scale-invariant field theories.

In between the aforementioned defined two symmetries is the notion of homothetic vector field (HVF), defined by the condition:

(1.4) ℒ η g a b = 2 λ g a b ,

where λ is a constant. A HVF generates a scaling symmetry of the spacetime, where the metric is preserved up to a constant factor. These symmetries, also referred to as self-similar symmetries of first kind, are particularly relevant in cosmological models, self-similar gravitational collapse, and other scenarios where scale invariance plays a fundamental role.

The symmetries of other tensors, like Ricci, stress-energy and curvature tensors are defined in a similar way by replacing the metric tensor in Eqs. (1.2)–(1.4) by R a b , T a b , and R b c d a , respectively.

The focus of the current study is only on KVFs. The significance of KVFs lies in their ability to reduce the complexity of the EFEs, making the search for their exact solutions more tractable. In terms of their physical significance, KVFs are closely connected to conservation laws in spacetime, which are fundamental not only in general relativity but also in other fields of science. In the literature, KVFs have been explored for various spacetime metrics [8–13], where it is shown that ten independent KVFs exist for a spacetime that is flat or it has constant curvature, but non-flat geometries generally admit fewer KVFs. For example, in cylindrically symmetric spacetimes, the dimension of Killing algebra ranges from 3 to 10 [8]. In addition to the KVFs, numerous studies have classified spacetimes based on other symmetries including HVFs, conformal vector fields (CVFs) and Ricci collineations [14–20]. In recent literature, these symmetries are also explored for different spacetimes in modified theories of gravity. The CVFs of locally rotationally symmetric Bianchi type I spacetime in f ( Q ) gravity were found by Ayyoub et al. [21]. Qazi et al. [22] investigated CVFs of Bianchi type II spacetime in f ( T ) theory of gravity. Mehmood et al. [23] worked on CVFs of Bianchi type I spacetime in f ( R , T ) gravity.

These previous studies on Killing and other symmetries have focused on specific cases such as static plane, static spherically and static cylindrically symmetric spacetimes, and certain other simple cosmological models. However, non-static spacetimes remain a fertile area of exploration, with relatively fewer comprehensive studies because of the difficulties one faces while solving the highly nonlinear equations defining spacetime symmetries.

A plane symmetric spacetime is a class of spacetime models characterized by symmetry along two spatial dimensions, resembling the geometry of a plane. This spacetime is useful for studying systems that exhibit uniform properties in two directions, such as gravitational waves, cosmological models, or certain exact solutions of EFEs. The metric of this spacetime remains invariant under translations and rotations in the plane, which simplifies the study of gravitational fields and their effects. Due to its symmetry, plane symmetric spacetime is an ideal framework for investigating the properties of gravitational fields, conservation laws, and the behavior of matter in general relativity.

The study of KVFs in non-static spacetimes, particularly those with plane symmetry, is of considerable interest due to the rich geometric structure these spacetimes exhibit. Nonstatic plane symmetric spacetimes describe scenarios where the spacetime evolves with time while maintaining planar symmetry, making them suitable models for investigating gravitational waves, cosmological phenomena, and anisotropic gravitational fields. Solutions to EFEs in these spacetimes are often complex, and finding explicit KVFs can aid in simplifying their structure and providing insight into their solutions.

To explore the symmetries of a spacetime, one always needs to solve a system of determining equations representing these symmetries. The conventional method used to solve these determining equations is known as direct integration technique. In this method, the determining equations are decoupled and integrated directly to find the explicit form of symmetry vector fields. The process usually gives rise to a number of cases depending upon the conditions on the metric functions under which the spacetime under consideration admits the desired symmetries. It is a quite lengthy and cumbersome technique which may result in lack of potential spacetime metrics admitting the required symmetries.

In recent literature, the Rif tree approach has emerged as a powerful computational tool for analyzing systems of partial differential equations that govern the existence of symmetries in spacetimes. This algorithmic method transforms the system of determining equations into an involutive form, systematically solving them and allowing for the classification of vector fields such as KVFs. This method relies on a Maple algorithm (Rif algorithm), which is implemented using the Exterior package in Maple. The process begins by loading the “Exterior” package. Next, the system of differential equations defining KVFs is inserted using the command “sysDEs.” The third step involves applying the “symmetry, eq ≔ findsymmetry” command, which analyzes the symmetry equations and identifies the conditions on the metric functions that allow for KVFs. The algorithm then displays these conditions. A graphical representation of these conditions can be viewed using the “caseplot(eq, pivots)” command, resulting in a tree shape, called Rif tree. The branches of the Rif tree illustrate the conditions under which the spacetime may admit KVFs. Finally, the symmetry equations are solved under these branch-specific conditions, yielding the explicit form of the KVFs. The novelty of this approach is that it gives a better classification of the spacetime through its symmetries. Recently, the Rif tree approach has been applied in the classification of KVFs and other symmetries, leading to the discovery of additional spacetime metrics that were previously unidentified using the conventional direct integration technique [24–29]. In this article, we focus on finding the KVFs of non-static plane symmetric spacetimes, using the Rif tree approach. We aim to identify all the nonstatic plane symmetric metrics for which Killing symmetries exist and to derive the corresponding solutions to the EFEs for anisotropic or perfect fluid sources.

2 Killing symmetries

The metric of nonstatic plane symmetric spacetime is given by [1]:

(2.1) d s 2 = − f 2 ( t , x ) d t 2 + g 2 ( t , x ) d x 2 + h 2 ( t , x ) [ d y 2 + d z 2 ] ,

and the set of its minimum KVFs is given by M 3 = { ∂ y , ∂ z , z ∂ y − y ∂ z } . Several well-known and significant classes of solutions to EFEs are included in the metric (2.1). For example, this metric defines a static plane symmetric spacetime with an extra KVF, ∂ t , when f , g , and h depend only on x . Such metrics are crucial for the derivation of the famous Taub solution and Kasner’s spatially homogenous solutions of EFEs, which makes them significant in a variety of physical contexts. Similarly, the metric (2.1) turns into the Bianchi type I metric possessing the extra KVF, ∂ x , for f = f ( t ) , g = g ( t ) , and h = h ( t ) . It is commonly known that Bianchi type metrics are homogeneous, while not necessarily isotropic, cosmological models that solve the EFEs; the kind of metrics varies depending on the scale factor selection. Moreover, if f = g = h = f ( t ) , then the metric (2.1) simplifies to the Friedmann metric, which is commonly used in cosmology.

We apply the definition of KVFs, given in Eq. (1.2), to the metric (2.1) to derive the set of symmetry equations. The explicit form of Eq. (1.2) is given by:

(2.2) g a b , c η c + g a c η , b c + g b c η , a c = 0 .

The commas in the subscript denote partial derivatives with respect to spacetime coordinates. From Eq. (2.1), we can see that the components of the metric tensor are g 00 = f 2 ( t , x ) , g 11 = g 2 ( t , x ) , and g 22 = g 33 = h 2 ( t , x ) . For a = b = 0 , Eq. (2.2) gives g 00 , c η c + 2 g 0 c η , 0 c = 0 . Here, c is the dummy index. As g 00 = f 2 ( t , x ) that depends on t and x only and the nondiagonal components of the metric tensor are zero, thus using the summation convention for c , the last equation becomes g 00 , t η 0 + g 00 , x η 1 + 2 g 00 η , t 0 = 0 . Consequently, the first symmetry equation is derived as follows:

(2.3) f , t η 0 + f , x η 1 + f η , t 0 = 0 .

Similarly, giving different values to a and b , and applying summation convention on the dummy index c , Eq. (2.2) gives rise to the following nine more symmetry equations.

(2.4) g , t η 0 + g , x η 1 + g η , x 1 = 0 ,

(2.5) h , t η 0 + h , x η 1 + h η , y 2 = 0 ,

(2.6) h , t η 0 + h , x η 1 + h η , z 3 = 0 ,

(2.7) f 2 η , x 0 − g 2 η , t 1 = 0 ,

(2.8) f 2 η , y 0 − h 2 η , t 2 = 0 ,

(2.9) f 2 η , z 0 − h 2 η , t 3 = 0 ,

(2.10) g 2 η , y 1 + h 2 η , x 2 = 0 ,

(2.11) g 2 η , z 1 + h 2 η , x 3 = 0 ,

(2.12) η , z 2 + η , y 3 = 0 .

The aforementioned symmetry equations must be solved in order to find the exact form of the KVF η . The metric functions f ( t , x ) g ( t , x ) , and h ( t , x ) must be subject to specific constraints for these equations to be solved generally. The Rif algorithm is used to examine these equations, which offers various metrics admitting different dimensional Killing algebras. Figure 1 displays the generated Rif tree. The Rif tree’s nodes, also known as pivots, are derived as given in (2.13).

Figure 1 Rif Tree.
Figure 1

Rif Tree.

The dependent and independent variables ordering is crucial while employing the Rif algorithm, since it has a visible impact on how complicated the resulting Rif tree is. There is no universal rule for selecting an optimal variable order to simplify the Rif tree. However, through trial and error, we found that ordering the dependent variables as η 0 > η 1 > η 2 > η 3 and the independent variables as t > x > y > z yields the most simplified Rif tree in our case.

(2.13) p 1 = h , t p 2 = h , t g , x − h , x g , t p 3 = h , t f , x − h , x f , t p 4 = h , t h , t x − h , t t h , x p 5 = ( h , t ) 2 h , x x − 2 h , t x h , x + h , t t ( h , x ) 2 p 6 = ( h , t ) 2 h , t t x − 2 h , t h , t t h , t x − h , t h , x h , t t t + 2 ( h , t t ) 2 h , x p 7 = ( h , t ) 2 h , x x − h , t t ( h , x ) 2 p 8 = h , x p 9 = g , t p 10 = f , t p 11 = g , t f , x − g , x f , t p 12 = g , t g , t x − g , x g , t t p 13 = − ( g , t ) 2 g , x x + 2 g , t g , x g , t x − ( g , x ) 2 g , t t p 14 = g , t t t g , t g , x − ( g , t ) 2 g , t t x + 2 g , t g , t t g , t x − 2 g , x ( g , t t ) 2 p 15 = ( g , t ) 2 g , x x − ( g , x ) 2 g , t t p 16 = g x p 17 = f , t f , t x − f , x f , t t p 18 = ( f , t ) 2 f , x x − 2 f , t f , x f , t x + ( f , x ) 2 f , t t p 19 = ( f , t ) 2 f , t t x − f , t f , x f , t t t − 2 f , t f , t x f , t t + 2 f , x ( f , t t ) 2 p 20 = ( f , t ) 2 f , x x − ( f , x ) 2 f , t t p 21 = f , x .

The Rif tree uses the symbols “=” and “<>” to indicate whether the associated p i is zero or nonzero. In branch 1, for example, we have p 1 = h , t ≠ 0 and p 2 = h , t g , x − h , x g , t ≠ 0 . In branch 2, we have p 1 = h , t ≠ 0 , p 3 = h , t f , x − h , x f , t ≠ 0 and p 2 = h , t g , x − h , x g , t = 0 . Similar restrictions are placed on the metric coefficients by each branch of the Rif tree. It can be seen that each branch of the Rif tree restricts the metric coefficients in a different way. These restrictions are then used to solve Eqs. (2.3)–(2.12), that explicitly determines the components of KVFs and the values of the metric coefficients. In such a way, one obtains a complete classification of the spacetime under consideration via KVFs. In the present case the solution of Eqs. (2.3)–(2.12), under the restrictions of all branches of the Rif tree, yields various plane symmetric metrics admitting 3-, 4-, 5-, 6-, 7-, and 10-dimensional Killing algebras. The branches labeled by 4 and 6 yield only three KVFs, as given in the set M 3 . The metrics of the remaining branches produce extra KVFs in addition to those given in the set M 3 . Tables 1, 2, 3, 4 provide a summary of these branches’ outcomes, where the first two columns of each table show the metric numbering, the metric coefficients and the corresponding branch of the Rif tree that produces the metric. The last column of each table contains extra symmetries admitted by the associated metric.

Table 1

Metrics with 4-dimensional Killing algebra

Metric no./branch no. Metric coefficients Additional KVFs
4a f = const., g = a 1 t + a 2 , h = 1 a 2 ( a 3 e a 1 x + a 4 e − a 1 x ) ( a 1 t + a 2 ) , V ( 4 ) = − a 2 a 1 a 4 h , x g ∂ t + a 2 a 4 h g 2 ∂ x
1 where a i ≠ 0 ; i = 1 , … , 4
4b f = const . , g = g ( x ) , h = a 1 t + a 2 , V ( 4 ) = 1 g ( x ) ∂ x
1 where g , x ( x ) ≠ 0 and a 1 ≠ 0
4c f = const., g = g ( x ) , h = ( a 1 t + a 2 ) 1 + a 3 a 1 a 1 + a 3 , V ( 4 ) = 1 g ( x ) ∂ x
1 where g , x ( x ) ≠ 0 , a 1 ≠ 0 and a 1 ≠ − a 3
4d f = f ( t ) , g = const., h = a 1 x + a 2 ∫ f ( t ) d t , V ( 4 ) = 1 f ( t ) ∂ t − a 2 a 1 ∂ x
2 where f , t ( t ) ≠ 0 , a 1 ≠ a 2 , a 1 ≠ 0 , and a 2 ≠ 0 .
4e f = 2 a 1 x + 2 a 2 , g = const. h = e a 3 t , V ( 4 ) = − 1 a 3 ∂ t + y ∂ y + z ∂ z ,
2 where a 1 ≠ 0 and a 3 ≠ 0
4f f = const., g = const., h = a 1 2 ( t 2 − x 2 ) + a 2 x + a 3 t + a 4 , V ( 4 ) = − − a 1 x + a 2 a 1 ∂ t + a 1 t + a 3 a 1 ∂ x ,
3 where a 1 , a 2 , and a 3 are nonzero
4g f = const., g = a 2 t , h = ( a 1 t ) 1 − 2 a 3 a 1 , V ( 4 ) = ∂ x
7 where a 1 ≠ 0 , a 2 ≠ 0 and a 1 ≠ 2 a 3
4h f = const . , g = ( a 1 t + 2 a 2 ) 1 − 2 a 3 a 1 , h = ( a 1 t + 2 a 2 ) 1 − 2 a 4 a 1 , V ( 4 ) = ∂ x
7 where a 1 ≠ 0 , a 1 ≠ 2 a 3 and a 1 ≠ 2 a 4
4i f = const., g = const., h = ( a 1 t + 2 a 2 ) 1 − 2 a 3 a 1 , V ( 4 ) = ∂ x
7 where a 1 ≠ 0 , a 1 ≠ 2 a 3
4j f = const., g = const., h = a 1 t + a 2 , V ( 4 ) = ∂ x
7 where a 1 ≠ 0
Table 2

Metrics with 4- and 6-dimensional Killing algebra

Metric no./branch no. Metric coefficients Additional KVFs
4k f = const., g = const., h = a 1 x + a 2 t , V ( 4 ) = − a 1 a 2 ∂ t + ∂ x
7 where a 1 ≠ a 2 ≠ 0
4l f = const., g = 2 a 1 t + 2 a 2 , h = e a 3 x , V ( 4 ) = − 1 a 3 ∂ x + y ∂ y + z ∂ z ,
8 where a 1 ≠ 0 , a 3 ≠ 0
4m f = f ( t ) , g = const . , h = ( a 1 x + 2 a 2 ) 1 − 2 a 3 a 1 , V ( 4 ) = 1 f ( t ) ∂ t
9 where a 1 ≠ 0 , a 1 ≠ 2 a 3 and f , t ( t ) ≠ 0 ,
4n f = f ( t ) , g = const . , h = a 1 x + a 2 , V ( 4 ) = 1 f ( t ) ∂ t
9 where f , t ( t ) ≠ 0 and a 1 ≠ 0 ,
4o f = ( a 1 x + 2 a 2 ) 1 − 2 a 3 a 1 , g = const . , h = ( a 1 x + 2 a 2 ) 1 − 2 a 4 a 1 , V ( 4 ) = ∂ t
10 where a 1 ≠ 0 , a 1 ≠ 2 a 3 and a 1 ≠ 2 a 4
4p f = const., g = const., h = ( a 1 x + 2 a 2 ) 1 − 2 a 3 a 1 , V ( 4 ) = ∂ t
10 where a 1 ≠ 0 , a 1 ≠ 2 a 3
4q f = const . , g = const . , h = a 1 x + a 2 , V ( 4 ) = ∂ t
10 where a 1 ≠ 0 ,
4r f = ( a 1 ∫ g ( x ) d x + 2 a 2 ) 1 − 2 a 3 a 1 , g = g ( x ) , h = const. V ( 4 ) = ∂ t
18 where g , x ( x ) ≠ 0 , a 1 ≠ 2 a 3 , and a 1 ≠ 0 .
4s f = ( a 1 x + 2 a 2 ) 1 − 2 a 3 a 1 , g = const., h = const., V ( 4 ) = ∂ t
24 where a 1 ≠ 2 a 3 and a 1 ≠ 0 .
6a f = const . , g = a 1 t + a 2 , h = 1 a 2 ( a 1 t + a 2 ) e a 3 x , V ( 4 ) = ∂ x − a 3 y ∂ y − a 3 z ∂ z
1 where a 1 , a 2 , a 3 are nonzero V ( 5 ) = y a 3 ∂ x + ( z 2 − y 2 2 + a 2 2 2 a 3 2 e 2 a 3 x ) ∂ y − y z ∂ z
V ( 6 ) = − z a 3 ∂ x + y z ∂ y + ( z 2 − y 2 2 − a 2 2 2 a 3 2 e 2 a 3 x ) ∂ z
6b f = const . , g = h = ( a 1 t + 2 a 2 ) 1 − 2 a 3 a 1 , V ( 4 ) = x ∂ y − y ∂ x
7 where a 1 ≠ 2 a 3 , a 1 ≠ 0 , V ( 5 ) = x ∂ z − z ∂ x ,
V ( 6 ) = ∂ x
6c f = const . , g = a 1 t , h = 1 a 2 t , V ( 4 ) = ∂ x
7 where a 1 ≠ 0 , and a 2 ≠ 0 . V ( 5 ) = x ∂ y − y a 1 2 a 2 2 ∂ x
V ( 6 ) = x ∂ z − z a 1 2 a 2 2 ∂ x
6d f = h = ( a 1 x + 2 a 2 ) 1 − 2 a 3 a 1 , g = const., V ( 4 ) = t ∂ y + y ∂ t
10 where a 1 ≠ 2 a 3 and a 1 is nonzero V ( 5 ) = t ∂ z + z ∂ t ,
V ( 6 ) = ∂ t
6e f = h = a 1 x + a 2 , g = const., V ( 4 ) = ∂ t
10 where a 1 ≠ 0 . V ( 5 ) = t ∂ y + y ∂ t
V ( 6 ) = t ∂ z + z ∂ t
6f f = a 1 e a 2 x + a 3 e − a 2 x , V ( 4 ) = f , x ( x ) m f ( x ) cos ( 2 a 2 a 1 a 3 t ) ∂ t + sin ( 2 a 2 a 1 a 3 t ) ∂ x ,
24 g and h are constant functions V ( 5 ) = − f , x ( x ) m f ( x ) sin ( 2 a 2 a 1 a 3 t ) ∂ t + cos ( 2 a 2 a 1 a 3 t ) ∂ x ,
and a i s are nonzero V ( 6 ) = ∂ t ,
where m = 2 a 2 a 1 a 3 ,
Table 3

Metrics with 7 and 10-dimensional Killing algebra

Metric no./branch no. Metric coefficients Additional symmetries
7a f = const., g = g ( x ) , h = e a 1 t , V ( 4 ) = 1 g ( x ) ∂ x
1 where g , x ( x ) ≠ 0 , and a 1 ≠ 0 . V ( 5 ) = y a 1 ∂ t + ( z 2 − y 2 2 − e − 2 a 1 t 2 a 1 2 ) ∂ y − y z ∂ z
V ( 6 ) = − z a 1 ∂ t + y z ∂ y + ( z 2 − y 2 2 + e − 2 a 1 t 2 a 1 2 ) ∂ z
V ( 7 ) = ∂ t − a 1 y ∂ y − a 1 z ∂ z
7b f = const., g = const., h = e a 1 t , V ( 4 ) = ∂ x
7 where a 1 ≠ 0 . V ( 5 ) = y a 1 ∂ t + ( z 2 − y 2 2 − e − 2 a 1 t 2 a ) ∂ y − y z ∂ z
V ( 6 ) = − z a 1 ∂ t + y z ∂ y + ( z 2 − y 2 2 + e − 2 a 1 t 2 a 1 ) ∂ z
V ( 7 ) = ∂ t − a 1 y ∂ y − a 1 z ∂ z
7c f = f ( t ) , g = const . , h = e a 1 x , V ( 4 ) = 1 f ( t ) ∂ t
9 where f , t ( t ) ≠ 0 and a 1 ≠ 0 V ( 5 ) = y a 1 ∂ x + ( z 2 − y 2 2 + e − 2 a 1 x 2 a 1 2 ) ∂ y − y z ∂ z
V ( 6 ) = − z a 1 ∂ x + y z ∂ y + ( z 2 − y 2 2 − e − 2 a 1 x 2 a 1 2 ) ∂ z
V ( 7 ) = − 1 a 1 ∂ x + y ∂ y + z ∂ z
7d f = const., g = const., h = e a 1 x , V ( 4 ) = ∂ t
10 where a 1 ≠ 0 , V ( 5 ) = y a 1 ∂ x + ( z 2 − y 2 2 + e − 2 a 1 x 2 a 1 2 ) ∂ y − y z ∂ z
V ( 6 ) = − z a 1 ∂ x + y z ∂ y + ( z 2 − y 2 2 − e − 2 a 1 x 2 a 1 2 ) ∂ z
V ( 7 ) = ∂ x − a 1 y ∂ y − a 1 z ∂ z
10a f = const., V ( 4 ) = ∂ x − a 1 y ∂ y − a 1 z ∂ z
1 g = a 1 t + a 2 , V ( 5 ) = e a 1 x ∂ t − e a 1 x a 1 t + a 2 ∂ x
h = a 1 t + a 2 a 2 e a 1 x , V ( 6 ) = a 1 2 a 2 2 ( y 2 + z 2 ) e a 1 x + e − a 1 x ∂ t + − a 1 2 a 2 2 ( a 1 t + a 2 ) ( y 2 + z 2 ) e a 1 x + e − a 1 x a 1 t + a 2 ∂ x
where a 1 and a 2 − 2 a 1 y e − a 1 x a 1 t + a 2 ∂ y − 2 a 1 z e − a 1 x a 1 t + a 2 ∂ z ,
are nonzero
V ( 7 ) = y e a 1 x ∂ t − y e a 1 x ( a 1 t + a 2 ) ∂ x − a 2 2 a 1 ( a 1 t + a 2 ) e − a 1 x ∂ y ,
V ( 8 ) = z e a 1 x ∂ t − z e a 1 x ( a 1 t + a 2 ) ∂ x − a 2 2 a 1 ( a 1 t + a 2 ) e − a 1 x ∂ z ,
V ( 9 ) = y a 1 ∂ x + ( z 2 − y 2 2 + a 2 2 2 a 1 2 e − 2 a 1 x ) ∂ y − y z ∂ z
V ( 10 ) = − z a 1 ∂ x + y z ∂ y + ( z 2 − y 2 2 − a 2 2 2 a 1 2 e − 2 a 1 x ) ∂ z
10b f = f ( t ) , g = const., V ( 4 ) = − x f ( t ) ∂ t − ∫ f ( t ) d t ∂ x + y ∂ y + z ∂ z
2 h = a 1 x + a 1 ∫ f ( t ) d t , V ( 5 ) = a 1 2 ( y 2 + z 2 ) 2 f ( t ) + 1 f ( t ) ∂ t − a 1 2 ( y 2 + z 2 2 ) ∂ x − a 1 y h ( t , x ) ∂ y − a 1 z h ( t , x ) ∂ z
where f , t ( t ) ≠ 0 V ( 6 ) = ( y 2 + z 2 2 ) a 1 2 f ( t ) ∂ t + − a 1 2 ( y 2 + z 2 ) 2 + 1 ∂ x − a 1 y h ( t , x ) ∂ y − a 1 z h ( t , x ) ∂ z
and a 1 ≠ 0 V ( 7 ) = y f ( t ) − ∫ f ( t ) d t + C a 1 ∂ t + y ∫ f ( t ) d t ∂ x + z 2 − y 2 2 + ∫ f ( t ) d t a 1 h ( t , x ) ∂ y − y z ∂ z
V ( 8 ) = z f ( t ) ∫ f ( t ) d t − C a 1 ∂ t + z ∫ f ( t ) d t ∂ x + y z ∂ y + z 2 − y 2 2 − ∫ f ( t ) d t a 1 h ( t , x ) ∂ z
V ( 9 ) = y f ( t ) ∂ t − y ∂ x − 1 a 1 h ( t , x ) ∂ y
V ( 10 ) = z f ( t ) ∂ t − z ∂ x − 1 a 1 h ( t , x ) ∂ z
10c f = const . , V ( 4 ) = y ∂ t − y x ∂ x + ( z 2 − y 2 2 − e − 2 t 2 + x 2 2 ) ∂ y − y z ∂ z
7 g = h = e t . V ( 5 ) = − z ∂ t + x z ∂ x + y z ∂ y + ( z 2 − y 2 2 − e − 2 t 2 − x 2 2 ) ∂ z ,
V ( 6 ) = − x ∂ t − ( y 2 + z 2 2 − e − 2 t 2 − x 2 2 ) ∂ x + x y ∂ y + x z ∂ z
V ( 7 ) = − ∂ t + x ∂ x + y ∂ y + z ∂ z
V ( 8 ) = − y ∂ x + x ∂ y ,
V ( 9 ) = − z ∂ x + x ∂ z ,
V ( 10 ) = ∂ x
Table 4

Metrics with 10-dimensional Killing algebra

Metric no./branch no. Metric coefficients Additional symmetries
10d f = h = e a 1 x , a 1 ≠ 0 . V ( 4 ) = t 2 + y 2 + z 2 2 + e − 2 a 1 x 2 a 1 2 ∂ t − t a 1 ∂ x + y t ∂ y + z t ∂ z ,
10 g is a constant function, V ( 5 ) = − y t ∂ t + y a 1 ∂ x + z 2 − y 2 − t 2 2 + e − 2 a 1 x 2 a 1 2 ∂ y − y z ∂ z ,
V ( 6 ) = z t ∂ t − z a 1 ∂ x + y z ∂ y + t 2 + z 2 − y 2 2 − e − 2 a 1 x 2 a 1 2 ∂ z ,
V ( 7 ) = − t ∂ t + 1 a 1 ∂ x − y ∂ y − z ∂ z
V ( 8 ) = y ∂ t + t ∂ y ,
V ( 9 ) = z ∂ t + t ∂ z ,
V ( 10 ) = ∂ t
10e f = const., V 4 = y e a 1 x ∂ t − y e a 1 x g ∂ x + g e a 1 x a 1 ∂ y
16 g = a 1 t + a 2 , V 5 = y e − a 1 x ∂ t + y e − a 1 x g ∂ x + g e − a 1 x a 1 ∂ y
h = const., V 6 = z e a 1 x ∂ t − z e a 1 x g ∂ x + g e a 1 x a 1 ∂ z
where a 1 ≠ 0 V 7 = z e − a 1 x ∂ t + z e − a 1 x g ∂ x + g e − a 1 x a 1 ∂ z
V 8 = e a 1 x ∂ t − e a 1 x g ∂ x
V 9 = e − a 1 x ∂ t + e − a 1 x g ∂ x
V 10 = ∂ x
10f f = f ( t ) , g = g ( x ) , V ( 4 ) = y f ( t ) ∂ t + ∫ f ( t ) d t ∂ y ,
17 h = const., V ( 5 ) = z f ( t ) ∂ t + ∫ f ( t ) d t ∂ z ,
where f , t ( t ) ≠ 0 V ( 6 ) = ∫ g ( x ) d x f ( t ) ∂ t + ∫ f ( t ) d t g ( x ) ∂ x ,
and g , x ( x ) ≠ 0 V ( 7 ) = − y g ( x ) ∂ x + ∫ g ( x ) d x ∂ y ,
V ( 8 ) = − z g ( x ) ∂ x + ∫ g ( x ) d x ∂ z ,
V ( 9 ) = 1 f ( t ) ∂ t
V ( 10 ) = 1 g ( x ) ∂ x
10g f = a 1 ∫ g ( x ) d x , g = g ( x ) , h = const., V ( 4 ) = a 1 y e a 1 t f ( x ) ∂ t − a 1 y e a 1 t g ( x ) ∂ x + f ( x ) e a 1 t ∂ y ,
18 where a 1 ≠ 0 and g , x ( x ) ≠ 0 V ( 5 ) = − a 1 y e − a 1 t f ( x ) ∂ t − a 1 y e − a 1 t g ( x ) ∂ x + f ( x ) e − a 1 t ∂ y ,
V ( 6 ) = a 1 z e a 1 t f ( x ) ∂ t − a 1 z e a 1 t g ( x ) ∂ x + f ( x ) e a 1 t ∂ z ,
V ( 7 ) = − a 1 z e − a 1 t f ( x ) ∂ t − a 1 z e − a 1 t g ( x ) ∂ x + f ( x ) e − a 1 t ∂ z ,
V ( 8 ) = − e a 1 t ∫ g ( x ) d x ∂ t + a 1 e a 1 t g ( x ) ∂ x ,
V ( 9 ) = − e − a 1 t ∫ g ( x ) d x ∂ t − a 1 e − a 1 t g ( x ) ∂ x ,
V ( 10 ) = ∂ t
10h f = const., g = g ( x ) , h = const., V ( 4 ) = t ∂ y + y ∂ t ,
18 where g , x ( x ) ≠ 0 V ( 5 ) = t ∂ z + z ∂ t ,
V ( 6 ) = ∫ g ( x ) d x ∂ t + t B ∂ x ,
V ( 7 ) = − y g ( x ) ∂ x + ∫ g ( x ) d x ∂ y ,
V ( 8 ) = − z g ( x ) ∂ x + ∫ g ( x ) d x ∂ z ,
V ( 9 ) = ∂ t
V ( 10 ) = 1 g ( x ) ∂ x
10i f = f ( t ) , g = const. h = const., V ( 4 ) = y f ( t ) ∂ t + ∫ f ( t ) d t ∂ y ,
23 where f , t ( t ) ≠ 0 V ( 5 ) = z f ( t ) ∂ t + ∫ f ( t ) d t ∂ z ,
V ( 6 ) = x f ( t ) ∂ t + ∫ f ( t ) d t ∂ x ,
V ( 7 ) = x ∂ y − y ∂ x ,
V ( 8 ) = x ∂ z − z ∂ x ,
V ( 9 ) = 1 f ( t ) ∂ t
V ( 10 ) = ∂ x
10j f = a 1 x + a 2 , g = const., V ( 4 ) = y e a 1 t f ∂ t − y e a 1 t ∂ x + f e a 1 t a 1 ∂ y
24 h = const., V ( 5 ) = − y e − a 1 t f ∂ t − y e − a 1 t ∂ x + f e − a 1 t a 1 ∂ y
where a 1 ≠ 0 V ( 6 ) = z e a 1 t f ∂ t − z e a 1 t ∂ x + f e a 1 t a 1 ∂ z
V ( 7 ) = − z e − a 1 t f ∂ t − z e − a 1 t ∂ x + f e − a 1 t a 1 ∂ z
V ( 8 ) = − e a 1 t f ∂ t + e a 1 t ∂ x
V ( 9 ) = e − a 1 t f ∂ t + e − a 1 t ∂ x
V ( 10 ) = ∂ t

Each of the metrics 4a-4s admits four-dimensional Killing algebra. The dimension of Killing algebra for the metrics 6a-6f is 6, while its dimension for the metrics 7a-7d is 7. Finally, the metrics 10a-10j admit 10 KVFs. Of the derived metrics, those labeled by 6b, 6c and 10c are Friedmann metrics, while the metrics 4g-4j and 7b represent locally rotationally symmetric Bianchi type I models. All these metrics are same as given in ref. [25], where Bianchi type I spacetime was classified via its KVFs. Similarly, the metrics 4o-4s, 6d-6f, 7d, 10d, 10g, 10h, and 10j are static plane symmetric metrics. All the remaining derived metrics are nonstatic plane symmetric metrics. Though the KVFs of plane symmetric spacetime were explored in an earlier study [12], these metrics were not listed there. This shows the significance of Rif tree approach for achieving a complete classification of the spacetimes.

It is remarkable that the number of KVFs admitted by a spacetime reflects the degree of its symmetry. A higher number of KVFs corresponds to a larger isometry group, which constrains the curvature tensors and is helpful in reducing the complexity of the EFEs. A four-dimensional Lorentzian manifold can admit a maximum of 10 independent KVFs. This maximum number of KVFs is admitted by maximally symmetric spacetimes such as Minkowski, de Sitter, and anti-de Sitter spacetimes. Physically, the presence of higher number of KVFs implies more conservation laws via Noether’s theorem, which are essential in understanding the motion of particles and the behavior of fields. Hence, the existence of an extended symmetry algebra, such as a 10-dimensional Killing algebra in a nonstatic spacetime indicates that the spacetime is of constant curvature and plays a role in fundamental models in general relativity.

3 Solutions of EFEs

For nonstatic plane symmetric spacetime, we have constructed various Lorentzian metrics with different dimensional Killing algebras by solving the Killing symmetry equations. Of these metrics, those satisfying EFEs and having the energy-momentum tensor associated with some known matter provide the exact solutions to the EFEs. The corresponding energy-momentum tensor T a b for each of these metrics can be found using the EFEs. Moreover, T a b can be used to assess the physical realism of the obtained metrics and to check various energy conditions satisfied by these metrics. In this section, we follow this procedure to check which of the obtained metrics are physically realistic solutions of EFEs. The metric (2.1) has four diagonal and one off-diagonal nonvanishing components of T a b , given by:

(3.1) T 00 = − 1 g 3 h 2 [ 2 g f 2 h h , x x − g 3 ( h , t ) 2 − 2 g 2 h g , t h , t + f 2 g ( h , x ) 2 − 2 f 2 h g , x h , x ] , T 01 = 1 f g h [ 2 f g h , t x − g f , x h , t − f g , t h , x ] , T 11 = − 1 f 3 h 2 [ 2 f g 2 h h , t t + f g 2 ( h , t ) 2 − 2 g 2 h f , t h , t − f 3 ( h , x ) 2 − 2 f 2 h f , x h , x ] , T 22 = T 33 = h f 3 g 3 [ f 3 g h , x x − f g 3 h , t t − f g 2 h g , t t + f 2 g h f , x x + g 3 f , t h , t − f g 2 g , t h , t + g 2 h f , t g , t + f 2 g f , x h , x − f 3 g , x h , x − f 2 h f , x g , x ] .

In the aforementioned expressions, f , t , g , t , h , t , f , x , g , x , and h , x signify the first order partial derivatives of the metric functions f ( t , x ) , g ( t , x ) and h ( t , x ) with respect to t and x , respectively, while the terms like h , x x , f , x x , g , t t , h , t t , and h , t x define the second-order partial derivatives. To find the above components of T a b , first, we have found the Christoffel symbols for the metric (2.1). These symbols are used to find the components of Riemann curvature tensor, which are then contracted to find the components of the Ricci tensor. The Ricci tensor components are contracted with the inverse of the metric tensor to obtain the Ricci scalar. Finally, all these expressions are used in the EFEs, Eqs. (1.1), with k = 1 to obtain the desired components of T a b , given in (3.1).

The structure of T a b varies for several known sources of matter. Assuming that the matter source for the metric (2.1) is an anisotropic fluid, the components of T a b are obtained as T 00 = ρ f 2 , T 11 = p ‖ g 2 , T 22 = T 33 = p ⊥ h 2 , and T 01 = 0 , where ρ is the density, while p ‖ and p ⊥ are pressures in two directions. A perfect fluid is obtained when p ‖ = p ⊥ = p . Thus, among the classified metrics, those for which T 01 = 0 indicate anisotropic or perfect fluids. It is simple to calculate ρ , p ‖ , and p ⊥ for such metrics as follows:

(3.2) ρ = T 00 f 2 , p ‖ = T 11 g 2 , p ⊥ = T 22 h 2 .

Of all the obtained metrics in our classification, for only three metrics (labeled by 4 a , 4 e , and 4 l ), we have T 01 ≠ 0 , indicating that these models do not describe perfect or anisotropic fluids. We exclude these three metrics from our discussion. For all other classified metrics, we have T 01 = 0 , giving anisotropic or perfect fluid solutions to EFEs. For all these models, the components of T a b can be found using Eq. (3.1), and these components can be used in Eq. (3.2) to find the physical terms ρ , p ‖ , and p ⊥ . Consequently, these terms can be used to check different energy conditions, including weak energy condition (WEC, ρ ≥ 0 , ρ + p ‖ ≥ 0 , ρ + p ⊥ ≥ 0 ), null energy condition (NEC, ρ + p ‖ ≥ 0 , ρ + p ⊥ ≥ 0 ), strong energy condition (SEC, ρ + p ‖ ≥ 0 , ρ + p ⊥ ≥ 0 , ρ + p ‖ + 2 p ⊥ ≥ 0 ), and dominant energy condition (DEC, ρ ≥ 0 , ρ ≥ ∣ p ‖ ∣ , ρ ≥ ∣ p ⊥ ∣ ) [30].

For some of the obtained metrics, labeled by 10a, 10b, and 10e-10j the components of T a b vanish, giving vacuum solutions. Consequently, the terms ρ , p ‖ , and p ⊥ vanish for all these metrics. All the energy conditions are identically satisfied for these metrics. On the other hand, the metrics labeled by 7a–7d do not satisfy any of these energy conditions. For the metrics 7a and 7b, we have ρ = a 1 2 , p ‖ = − 3 a 1 2 , and p ⊥ = − a 1 2 . Though none of the energy conditions is satisfied by these terms, the positive value of energy density indicates that it is a physically realistic model. Similarly, for metrics 7c and 7d, we have ρ = − 3 a 1 2 , p ‖ = a 1 2 , and p ⊥ = a 1 2 . These terms do not satisfy any energy condition and the negative value of ρ shows that it is not a physically realistic model. In Tables 5, 6, 7, we present the physical terms ρ , p ‖ and p ⊥ for all the remaining metrics and give the details of energy conditions satisfied by these metrics. For metrics 6a, 6b, 6c, 10c, and 10d, we have p ‖ = p ⊥ , giving perfect fluids, while all the remaining metrics denote anisotropic fluids.

Table 5

Energy conditions

Metric no. Physical terms Energy conditions
4b, 4j ρ = a 1 2 ( a 1 t + a 2 ) 2 All energy conditions are identically satisfied
p ‖ = − ρ
p ⊥ = 0
4c ρ = ( a 1 + a 3 ) 2 ( a 1 t + a 2 ) 2 WEC, NEC and SEC are satisfied if
a 3 ( a 1 + a 3 ) < 0 and a 1 ( a 1 + a 3 ) > 0
p ‖ = − ( a 1 + a 3 ) ( a 1 + 3 a 3 ) ( a 1 t + a 2 ) 2 DEC is satisfied if a 3 ( a 1 + a 3 ) < 0 ,
p ⊥ = − a 3 ( a 1 + a 3 ) ( a 1 t + a 2 ) 2 a 1 ( a 1 + a 3 ) > 0 and ( a 1 + a 3 ) ( a 1 + 2 a 3 ) > 0
4d ρ = a 2 2 − a 1 2 [ a 1 x + a 2 ∫ f ( t ) d t ] 2 All energy conditions are satisfied if a 2 2 − a 1 2 > 0
p ‖ = − ρ
p ⊥ = 0
4f ρ = 4 [ 2 a 1 2 ( t 2 − x 2 ) + 4 a 1 ( a 2 x + a 3 t ) + 2 a 1 a 4 − a 2 2 + a 3 2 ] [ a 1 ( t 2 − x 2 ) + 2 ( a 2 x + a 3 t + a 4 ) ] 2 WEC is satisfied if 2 a 1 2 ( t 2 − x 2 ) + 4 a 1 ( a 2 x + a 3 t )
p ‖ = − ρ + 2 a 1 a 4 − a 2 2 + a 3 2 ≥ 0 and
a 1 2 ( t 2 − x 2 ) + 2 a 1 ( a 2 x + a 3 t ) − a 2 2 + a 3 2 ≥ 0
p ⊥ = − 4 a 1 a 1 ( t 2 − x 2 ) + 2 ( a 2 x + a 3 t + a 4 )
NEC is satisfied if
a 1 2 ( t 2 − x 2 ) + 2 a 1 ( a 2 x + a 3 t ) − a 2 2 + a 3 2 ≥ 0
SEC is satisfied if
a 1 2 ( t 2 − x 2 ) + 2 a 1 ( a 2 x + a 3 t ) − a 2 2 + a 3 2 ≥ 0
and 8 a 1 a 1 ( t 2 − x 2 ) + 2 ( a 2 x + a 3 t + a 4 ) < 0
DEC is satisfied if 2 a 1 2 ( t 2 − x 2 ) + 4 a 1 ( a 2 x + a 3 t )
+ 2 a 1 a 4 − a 2 2 + a 3 2 ≥ 0 and
8 a 1 2 ( t 2 − x 2 ) + 24 a 1 ( a 2 x + a 3 t )
+ 16 a 1 a 4 − 4 a 2 2 + 4 a 3 2 ≥ 0
Table 6

Energy conditions

Metric no. Physical terms Energy conditions
4g ρ = ( a 1 − 2 a 3 ) ( 3 a 1 − 2 a 3 ) a 1 2 t 2 WEC is satisfied if ( a 1 − 2 a 3 ) ( 3 a 1 − 2 a 3 ) > 0
( a 1 − 2 a 3 ) ( a 1 + 2 a 3 ) > 0 and a 1 ( a 1 − 2 a 3 ) > 0
p ‖ = − ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) a 1 2 t 2 NEC is satisfied if ( a 1 − 2 a 3 ) ( a 1 + 2 a 3 ) > 0
and a 1 ( a 1 − 2 a 3 ) > 0
p ⊥ = − ( a 1 − 2 a 3 ) 2 a 1 2 t 2 SEC is satisfied if ( a 1 − 2 a 3 ) ( a 1 + 2 a 3 ) > 0
a 1 ( a 1 − 2 a 3 ) > 0 and a 3 ( a 1 − 2 a 3 ) > 0
DEC is satisfied if ( a 1 − 2 a 3 ) ( 3 a 1 − 2 a 3 ) > 0
and a 1 ( a 1 − 2 a 3 ) > 0
4h ρ = 3 a 1 2 + 4 a 4 2 − 4 a 1 a 3 − 8 a 1 a 4 + 8 a 3 a 4 ( a 1 t + 2 a 2 ) 2 WEC is satisfied if
3 a 1 2 + 4 a 4 2 − 4 a 1 a 3 − 8 a 1 a 4 + 8 a 3 a 4 ≥ 0 ,
p ‖ = − a 1 2 + 12 a 4 2 − 8 a 1 a 4 ( a 1 t + 2 a 2 ) 2 2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0
and 2 a 1 2 − 4 a 1 a 4 − 4 a 3 2 + 4 a 3 a 4 ≥ 0
p ⊥ = − a 1 2 + 4 a 3 2 + 4 a 4 2 − 4 a 1 a 3 − 4 a 1 a 4 + 4 a 3 a 4 ( a 1 t + 2 a 2 ) 2
NEC is satisfied if 2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0
and 2 a 1 2 − 4 a 1 a 4 − 4 a 3 2 + 4 a 3 a 4 ≥ 0
SEC is satisfied if 2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0
2 a 1 2 − 4 a 1 a 4 − 4 a 3 2 + 4 a 3 a 4 ≥ 0
and 4 a 1 a 3 + 8 a 1 a 4 − 8 a 3 2 − 16 a 4 2 ≥ 0
DEC is satisfied if
3 a 1 2 + 4 a 4 2 − 4 a 1 a 3 − 8 a 1 a 4 + 8 a 3 a 4 ≥ 0 ,
2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0 ,
4 a 1 2 + 16 a 4 2 − 4 a 1 a 3 − 16 a 1 a 4 + 8 a 3 a 4 ≥ 0
2 a 1 2 − 4 a 1 a 4 − 4 a 3 2 + 4 a 3 a 4 ≥ 0 and
4 a 1 2 − 8 a 1 a 3 − 12 a 1 a 4 + 12 a 3 a 4 + 8 a 4 2 + 4 a 3 2 ≥ 0
4i ρ = ( a 1 − 2 a 3 ) 2 ( a 1 t + 2 a 2 ) 2 WEC, NEC and SEC are satisfied if a 3 ( a 1 − 2 a 3 ) > 0
and a 1 ( a 1 − 2 a 3 ) > 0
p ‖ = − ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) ( a 1 t + 2 a 2 ) 2 DEC is satisfied if a 3 ( a 1 − 2 a 3 ) > 0 , a 1 ( a 1 − 2 a 3 ) > 0
( a 1 − 2 a 3 ) ( a 1 − 4 a 3 ) > 0
p ⊥ = 2 a 3 ( a 1 − 2 a 3 ) ( a 1 t + 2 a 2 ) 2
4j ρ = a 1 2 ( a 1 t + a 2 ) 2 All energy conditions are identically satisfied
p ‖ = − ρ
p ⊥ = 0
4k ρ = a 2 2 − a 1 2 ( a 1 x + a 2 t ) 2 All energy conditions are satisfied if a 2 2 − a 1 2 > 0
p ‖ = − ρ
p ⊥ = 0
4m, 4p ρ = − ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) ( a 1 x + 2 a 2 ) 2 WEC is satisfied if ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) > 0 , a 3 ( a 1 − 2 a 3 ) > 0
( a 1 − 2 a 3 ) ( 4 a 3 − a 1 ) > 0
p ‖ = ( a 1 − 2 a 3 ) 2 ( a 1 x + 2 a 2 ) 2 NEC and SEC are satisfied if a 3 ( a 1 − 2 a 3 ) > 0
and ( a 1 − 2 a 3 ) ( 4 a 3 − a 1 ) > 0
p ⊥ = − 2 a 3 ( a 1 − 2 a 3 ) ( a 1 x + 2 a 2 ) 2 DEC is satisfied if a 3 ( a 1 − 2 a 3 ) > 0
( a 1 − 2 a 3 ) ( 4 a 3 − a 1 ) > 0 and ( a 1 − 2 a 3 ) ( 8 a 3 − a 1 ) > 0
4n, 4q ρ = − a 1 2 ( a 1 x + a 2 ) 2 An un-physical model with ρ < 0 and
p ‖ = − ρ satisfying none of the energy conditions
p ⊥ = 0
Table 7

Energy conditions

Metric no. Physical terms Energy conditions
4o ρ = 8 a 1 a 4 − a 1 2 − 12 a 4 2 ( a 1 x + 2 a 2 ) 2 WEC iis satisfied if 8 a 1 a 4 − a 1 2 − 12 a 4 2 ≥ 0 ,
2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0
and 4 a 1 a 4 − 8 a 4 2 − 4 a 1 a 3 + 4 a 3 2 + 4 a 3 a 4 ≥ 0
p ‖ = 3 a 1 2 + 4 a 4 2 − 4 a 1 a 3 − 8 a 1 a 4 + 8 a 3 a 4 ( a 1 x + 2 a 2 ) 2 NEC is satisfied if 2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0
and 4 a 1 a 4 − 8 a 4 2 − 4 a 1 a 3 + 4 a 3 2 + 4 a 3 a 4 ≥ 0
p ⊥ = a 1 2 + 4 a 3 2 + 4 a 4 2 − 4 a 1 a 3 − 4 a 1 a 4 + 4 a 3 a 4 ( a 1 x + 2 a 2 ) 2 SEC is satisfied if 2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0 ,
4 a 1 a 4 − 8 a 4 2 − 4 a 1 a 3 + 4 a 3 2 + 4 a 3 a 4 ≥ 0
and 3 a 1 2 − 8 a 1 a 3 − 4 a 4 2 + 12 a 3 a 4 − 4 a 1 a 4 + 4 a 3 2 ≥ 0
DEC is satisfied if 8 a 1 a 4 − a 1 2 − 12 a 4 2 ≥ 0 ,
2 a 1 2 − 8 a 4 2 − 4 a 1 a 3 + 8 a 3 a 4 ≥ 0
4 a 1 a 4 − 8 a 4 2 − 4 a 1 a 3 + 4 a 3 2 + 4 a 3 a 4 ≥ 0
16 a 1 a 4 − 4 a 1 2 − 12 a 4 2 + 4 a 1 a 3 − 8 a 3 a 4 ≥ 0
and 12 a 1 a 4 − 2 a 1 2 − 16 a 4 2 + 4 a 1 a 3 − 4 a 3 2 − 4 a 3 a 4 ≥ 0
4r ρ = 0 All energy conditions are satisfied if a 3 ( 2 a 3 − a 1 ) > 0
p ‖ = 0
p ⊥ = 2 a 3 ( 2 a 3 − a 1 ) [ a 1 ∫ g ( x ) d x + 2 a 2 ] 2
4s ρ = 0 All energy conditions are satisfied if a 3 ( 2 a 3 − a 1 ) > 0
p ‖ = 0
p ⊥ = 2 a 3 ( 2 a 3 − a 1 ) [ a 1 x + 2 a 2 ] 2
6a ρ = 3 ( a 1 2 − a 3 2 ) ( a 1 t + a 2 ) 2 All energy conditions are satisfied if a 1 2 − a 3 2 > 0
p ‖ = p ⊥ = − ρ 3
6b ρ = 3 ( a 1 − 2 a 3 ) 2 ( a 1 t + 2 a 2 ) 2 WEC and NEC satisfied if a 1 ( a 1 − 2 a 3 ) > 0
p ‖ = p ⊥ = − ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) ( a 1 t + 2 a 2 ) 2 SEC is satisfied if a 1 ( a 1 − 2 a 3 ) > 0
and a 3 ( a 1 − 2 a 3 ) > 0
DEC is satisfied if a 1 ( a 1 − 2 a 3 ) > 0
and ( a 1 − 3 a 3 ) ( a 1 − 2 a 3 ) > 0
6c ρ = 3 t 2 All energy conditions are identically satisfied
p ‖ = p ⊥ = − 1 t 2
6d ρ = − ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) ( a 1 x + 2 a 2 ) 2 WEC is satisfied if ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) < 0
a 1 ( a 1 − 2 a 3 ) > 0
p ‖ = 3 ( a 1 − 2 a 3 ) 2 ( a 1 x + 2 a 2 ) 2 NEC is satisfied if a 1 ( a 1 − 2 a 3 ) > 0
p ⊥ = − ρ SEC is satisfied if a 1 ( a 1 − 2 a 3 ) > 0
and ( a 1 − 2 a 3 ) ( a 1 − 3 a 3 ) > 0
DEC is satisfied if ( a 1 − 2 a 3 ) ( a 1 − 6 a 3 ) < 0
and ( a 1 − 2 a 3 ) ( 3 a 3 − a 1 ) > 0
6e ρ = − a 1 2 ( a 1 x + a 2 ) 2 WEC and DEC are failed, while NEC and
p ‖ = 3 a 1 2 ( a 1 x + a 2 ) 2 SEC are satisfied
p ⊥ = − ρ
6f ρ = p ‖ = 0 DEC is failed, while
WEC, NEC and SEC are satisfied if
p ⊥ = a 2 2 [ a 1 4 e 8 a 2 x + 4 a 1 3 a 3 e 6 a 2 x + 6 a 1 2 a 3 2 e 4 a 2 x + 4 a 1 a 3 2 e 2 a 2 x + a 3 2 ] ( a 1 e 2 a 2 x + a 3 ) 4 a 1 4 e 8 a 2 x + 4 a 1 3 a 3 e 6 a 2 x + 6 a 1 2 a 3 2 e 4 a 2 x
+ 4 a 1 a 3 2 e 2 a 2 x + a 3 2 > 0
10c ρ = 3 WEC, NEC and DEC are satisfied, while SEC is failed.
p ‖ = p ⊥ = − 3
10d ρ = − 3 a 1 2 NEC and SEC are satisfied, while WEC and DEC are failed.
p ‖ = p ⊥ = 3 a 1 2

4 Physical interpretation and connection to known models

While the work presented in this article focuses the classification of nonstatic plane symmetric spacetime using KVFs, many of the derived metrics during the classification have clear relevance in cosmological and astrophysical contexts. For example, consider the metrics labeled by 4j, 4n, and 4q, where the metric coefficient h ( t ) or h ( x ) is linear. These metrics exhibit directional anisotropy and are closely related to the Kasner-type or Bianchi type I cosmological models, which are commonly used to describe anisotropic expansion in the early universe. On the other hand, the metrics 4d and 4k involve mixed dependencies in both space and time and can be interpreted as models with shear or inhomogeneous anisotropic expansion, making them applicable in anisotropic gravitational collapse or structure formation scenarios. Next, in the metric 10c, all scale factors evolve exponentially and it resembles the vacuum inflationary models, like de Sitter spacetime. The metric 10a admitting ten KVFs gives a vacuum solution with plane symmetry and resembles the pp-wave or plane gravitational wave model that is widely used in theoretical studies of exact wave propagation in general relativity.

The energy-momentum tensor components for the derived metrics further support their physical relevance. For example, the metric 6c describes a perfect fluid with time-dependent energy density ρ = 3 ∕ t 2 , typical of radiation-dominated cosmological eras. Hence, these solutions offer not only mathematical classifications but also idealized models for gravitational waves, anisotropic cosmological evolution, and collapsing systems.

5 Conclusion

We have presented a novel approach, the Rif tree method, for the classification of KVFs in nonstatic plane symmetric spacetime. By systematically applying this method, several distinct spacetime metrics with varying dimensional Killing algebras were derived. These metrics offer deeper insights into spacetime symmetries and provide valuable tools for simplifying EFEs in the context of general relativity.

Our analysis has shown that nonstatic plane symmetric spacetime can admit additional KVFs beyond the minimal set, depending on the imposed conditions on the metric coefficients. This expansion of symmetries opens up new avenues for exploring exact solutions to EFEs, with potential applications in cosmological models and the study of gravitational fields.

Moreover, the physical implications of the obtained metrics were explored by deriving the corresponding energy-momentum tensors and evaluating the energy conditions. Several metrics were found to describe physically realistic models, including anisotropic and perfect fluid solutions, while few were shown to be unphysical due to negative energy densities.

Overall, this work extends the understanding of EFEs in nonstatic spacetimes and highlights the usefulness of the Rif tree approach in identifying and classifying spacetime symmetries. Future research may focus on applying this method to other classes of spacetimes and investigating the physical significance of the derived metrics in more specific gravitational and cosmological scenarios.

Acknowledgments

The author Z.A. Khan expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R8), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  1. Funding information: Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R8), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

References

[1] Stephani H, Kramer D, Maccallum M, Hoenselaers C, Herlt E. Exact solutions of Einstein’s field equations. 2nd ed. Cambridge: Cambridge University Press; 2003. 10.1017/CBO9780511535185Search in Google Scholar

[2] Sharif M, Naseer T. Charged anisotropic models with complexity-free condition. Ann Phys. 2023;453:169311. 10.1016/j.aop.2023.169311Search in Google Scholar

[3] Sharif M, Naseer T. Charged stellar models possessing anisotropic interiors. Eur Phys J Plus. 2024;139:296. 10.1140/epjp/s13360-024-05068-4Search in Google Scholar

[4] Siza B, Andrade J, Santana D, Naseer T. Anisotropic extension of the Kohler-Chao-Tikekar cosmological solution with like Wyman IIa complexity factor. Eur Phys J C. 2024;84:1203. 10.1140/epjc/s10052-024-13591-ySearch in Google Scholar

[5] Naseer T, Said JL. Existence of non-singular stellar solutions within the context of electromagnetic field: a comparison between minimal and non-minimal gravity models. Eur Phys J C. 2024;84:808. 10.1140/epjc/s10052-024-13172-zSearch in Google Scholar

[6] Naseer T, Sharif M. Implications of vanishing complexity condition in f(R) theory. Eur Phys J C. 2024;84:554. 10.1140/epjc/s10052-024-12916-1Search in Google Scholar

[7] Hall GS. Symmetries and curvature structure in general relativity. United Kingdom: World Scientific; 2004. 10.1142/1729Search in Google Scholar

[8] Qadir A, Ziad M. Classification of static cylindrically symmetric space-times. Nuovo Cimento B. 1995;110:277–90. 10.1007/BF02741369Search in Google Scholar

[9] Bokhari AH, Qadir A. Symmetries of static spherically symmetric space-times. J Math Phys. 1987;28:1019–22. 10.1063/1.527594Search in Google Scholar

[10] Bokhari AH, Qadir A. Killing vectors of static spherically symmetric metrics. J Math Phys. 1990;31:1463. 10.1063/1.528737Search in Google Scholar

[11] Bokhari AH, Karim M, Al-Sheikh DN, Zaman FD. Circularly symmetric static metric in three dimensions and its Killing symmetry. Int J Theor Phys. 2008;47:2672–78. 10.1007/s10773-008-9704-3Search in Google Scholar

[12] Feroze T, Qadir A, Ziad M. The classification of plane symmetric spacetimes by isometries. J Math Phys. 2001;42:4947–55. 10.1063/1.1385175Search in Google Scholar

[13] Ziad M. The classification of static plane-symmetric spacetime. Nuovo Cimento B. 1999;114:683–92. Search in Google Scholar

[14] Ahmad D, Ziad M. Homothetic motions of spherically symmetric space-time. J Math Phys. 1997;38:2547–52. 10.1063/1.531994Search in Google Scholar

[15] Camci U, Turkyilmaz I. Ricci collineations in perfect fluid Bianchi V spacetime. Gen Relat Grav. 2004;36:2005–19. 10.1023/B:GERG.0000038467.16961.04Search in Google Scholar

[16] Moopanar S, Maharaj SD. Conformal symmetries of spherical spacetimes. Int J Theor Phys. 2010;49:1878–85. 10.1007/s10773-010-0366-6Search in Google Scholar

[17] Maartens R, Maharaj SD. Conformal Killing vectors in Robertson-Walker spacetimes. Class Quantum Grav. 1986;3:1005–11. 10.1088/0264-9381/3/5/027Search in Google Scholar

[18] Moopanar S, Maharaj SD. Relativistic shear-free fluids with symmetry. J Eng Math. 2013;82:125–31. 10.1007/s10665-012-9573-xSearch in Google Scholar

[19] Duggal KL, Sharma R. Conformal Killing vector fields on spacetime solutions of Einsteinas equations and initial data. Nonlin Anal. 2005;63:447–54. 10.1016/j.na.2004.09.034Search in Google Scholar

[20] Sharif M. Symmetries of the energy momentum tensor of cylindrically symmetric static spacetimes. J Math Phys. 2004;45:1532–60. 10.1063/1.1668335Search in Google Scholar

[21] Ayyoub MJ, Alhouiti N, Ramzan M, Ali A, Ali A. Conformal motions of locally rotationally symmetric Bianchi type I space-times in f(Q) gravity. Resul Phys. 2024;57:107413. 10.1016/j.rinp.2024.107413Search in Google Scholar

[22] Qazi S, Hussain F, Ramzan M, Haq S. Conformal motions of anisotropic exact Bianchi type II models admitting energy conditions in f(T) gravity. Int J Mod Phys D. 2023;32:2350057. 10.1142/S0218271823500578Search in Google Scholar

[23] Mehmood AB, Hussain F, Ramzan M, Faryad M, Khan MJ, Malik S. A note on proper conformal vector fields of Bianchi type-I perfect fluid space-times in f(R,T) gravity. Int J Geom Meth Mod Phys. 2023;20:2350012. 10.1142/S0219887823500123Search in Google Scholar

[24] Albuhayr MK, Bokhari AH, Hussain T. Killing vector fields of static cylindrically symmetric spacetime-A Rif tree approach. Symmetry. 2023;15:1111. 10.3390/sym15051111Search in Google Scholar

[25] Bokhari AH, Hussain T, Hussain W, Khan F. Killing vector fields of Bianchi type I spacetimes via Rif tree approach. Mod Phys Lett A. 2021;36:2150208. 10.1142/S0217732321502084Search in Google Scholar

[26] Ahmad S, Hussain T, Saqib AB, Farhan M, Farooq M. Killing vector fields of locally rotationally symmetric Bianchi type V spacetime. Sci Rep. 2024;14:10239. 10.1038/s41598-024-58560-3Search in Google Scholar PubMed PubMed Central

[27] Hussain T, Bokhari AH, Munawar A. Lie symmetries of static spherically symmetric spacetimes by Rif tree approach. Eur Phys J Plus. 2022;137:1322. 10.1140/epjp/s13360-022-03526-5Search in Google Scholar

[28] Khan J, Hussain T, Mlaiki N, Fatima N. Symmetries of locally rotationally symmetric Bianchi type V spacetime. Resul Phys. 2023;44:106143. 10.1016/j.rinp.2022.106143Search in Google Scholar

[29] Khan J, Hussain T, Bokhari AH, Farhan M. Lie symmetries of Lemaitre-Tolman-Bondi metric. Int J Geom Meth Mod Phys. 2024;21:2450132. 10.1142/S0219887824501329Search in Google Scholar

[30] Coley AA, Tupper BOJ. Spherically symmetric spacetimes admitting inheriting conformal Killing vector fields. Class Quantum Grav. 1990;7:2195–214. 10.1088/0264-9381/7/12/005Search in Google Scholar
Received: 2025-05-05
Revised: 2025-07-08
Accepted: 2025-07-21
Published Online: 2025-09-30

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Research Articles
  2. Single-step fabrication of Ag2S/poly-2-mercaptoaniline nanoribbon photocathodes for green hydrogen generation from artificial and natural red-sea water
  3. Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation
  4. A novel gold and SiO2 material based planar 5-element high HPBW end-fire antenna array for 300 GHz applications
  5. Explicit exact solutions and bifurcation analysis for the mZK equation with truncated M-fractional derivatives utilizing two reliable methods
  6. Optical and laser damage resistance: Role of periodic cylindrical surfaces
  7. Numerical study of flow and heat transfer in the air-side metal foam partially filled channels of panel-type radiator under forced convection
  8. Water-based hybrid nanofluid flow containing CNT nanoparticles over an extending surface with velocity slips, thermal convective, and zero-mass flux conditions
  9. Dynamical wave structures for some diffusion--reaction equations with quadratic and quartic nonlinearities
  10. Solving an isotropic grey matter tumour model via a heat transfer equation
  11. Study on the penetration protection of a fiber-reinforced composite structure with CNTs/GFP clip STF/3DKevlar
  12. Influence of Hall current and acoustic pressure on nanostructured DPL thermoelastic plates under ramp heating in a double-temperature model
  13. Applications of the Belousov–Zhabotinsky reaction–diffusion system: Analytical and numerical approaches
  14. AC electroosmotic flow of Maxwell fluid in a pH-regulated parallel-plate silica nanochannel
  15. Interpreting optical effects with relativistic transformations adopting one-way synchronization to conserve simultaneity and space–time continuity
  16. Modeling and analysis of quantum communication channel in airborne platforms with boundary layer effects
  17. Theoretical and numerical investigation of a memristor system with a piecewise memductance under fractal–fractional derivatives
  18. Tuning the structure and electro-optical properties of α-Cr2O3 films by heat treatment/La doping for optoelectronic applications
  19. High-speed multi-spectral explosion temperature measurement using golden-section accelerated Pearson correlation algorithm
  20. Dynamic behavior and modulation instability of the generalized coupled fractional nonlinear Helmholtz equation with cubic–quintic term
  21. Study on the duration of laser-induced air plasma flash near thin film surface
  22. Exploring the dynamics of fractional-order nonlinear dispersive wave system through homotopy technique
  23. The mechanism of carbon monoxide fluorescence inside a femtosecond laser-induced plasma
  24. Numerical solution of a nonconstant coefficient advection diffusion equation in an irregular domain and analyses of numerical dispersion and dissipation
  25. Numerical examination of the chemically reactive MHD flow of hybrid nanofluids over a two-dimensional stretching surface with the Cattaneo–Christov model and slip conditions
  26. Impacts of sinusoidal heat flux and embraced heated rectangular cavity on natural convection within a square enclosure partially filled with porous medium and Casson-hybrid nanofluid
  27. Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge
  28. Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method
  29. Bilinear form and soltion solutions for (3+1)-dimensional negative-order KdV-CBS equation
  30. Solitary chirp pulses and soliton control for variable coefficients cubic–quintic nonlinear Schrödinger equation in nonuniform management system
  31. Influence of decaying heat source and temperature-dependent thermal conductivity on photo-hydro-elasto semiconductor media
  32. Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition
  33. Bifurcation, chaotic behavior, and traveling wave solutions for the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model
  34. New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis
  35. Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
  36. Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term
  37. Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber
  38. Chaotic dynamics and some solutions for the (n + 1)-dimensional modified Zakharov–Kuznetsov equation in plasma physics
  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Influence of a magnetic field on double-porosity photo-thermoelastic materials under Lord–Shulman theory
  78. Soliton-like solutions for a nonlinear doubly dispersive equation in an elastic Murnaghan's rod via Hirota's bilinear method
  79. Analytical and numerical investigation of exact wave patterns and chaotic dynamics in the extended improved Boussinesq equation
  80. Nonclassical correlation dynamics of Heisenberg XYZ states with (x, y)-spin--orbit interaction, x-magnetic field, and intrinsic decoherence effects
  81. Exact traveling wave and soliton solutions for chemotaxis model and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation
  82. Unveiling the transformative role of samarium in ZnO: Exploring structural and optical modifications for advanced functional applications
  83. On the derivation of solitary wave solutions for the time-fractional Rosenau equation through two analytical techniques
  84. Analyzing the role of length and radius of MWCNTs in a nanofluid flow influenced by variable thermal conductivity and viscosity considering Marangoni convection
  85. Advanced mathematical analysis of heat and mass transfer in oscillatory micropolar bio-nanofluid flows via peristaltic waves and electroosmotic effects
  86. Exact bound state solutions of the radial Schrödinger equation for the Coulomb potential by conformable Nikiforov–Uvarov approach
  87. Some anisotropic and perfect fluid plane symmetric solutions of Einstein's field equations using killing symmetries
  88. Nonlinear dynamics of the dissipative ion-acoustic solitary waves in anisotropic rotating magnetoplasmas
  89. Curves in multiplicative equiaffine plane
  90. Exact solution of the three-dimensional (3D) Z2 lattice gauge theory
  91. Propagation properties of Airyprime pulses in relaxing nonlinear media
  92. Symbolic computation: Analytical solutions and dynamics of a shallow water wave equation in coastal engineering
  93. Wave propagation in nonlocal piezo-photo-hygrothermoelastic semiconductors subjected to heat and moisture flux
  94. Comparative reaction dynamics in rotating nanofluid systems: Quartic and cubic kinetics under MHD influence
  95. Laplace transform technique and probabilistic analysis-based hypothesis testing in medical and engineering applications
  96. Physical properties of ternary chloro-perovskites KTCl3 (T = Ge, Al) for optoelectronic applications
  97. Gravitational length stretching: Curvature-induced modulation of quantum probability densities
  98. The search for the cosmological cold dark matter axion – A new refined narrow mass window and detection scheme
  99. A comparative study of quantum resources in bipartite Lipkin–Meshkov–Glick model under DM interaction and Zeeman splitting
  100. PbO-doped K2O–BaO–Al2O3–B2O3–TeO2-glasses: Mechanical and shielding efficacy
  101. Nanospherical arsenic(iii) oxoiodide/iodide-intercalated poly(N-methylpyrrole) composite synthesis for broad-spectrum optical detection
  102. Sine power Burr X distribution with estimation and applications in physics and other fields
  103. Numerical modeling of enhanced reactive oxygen plasma in pulsed laser deposition of metal oxide thin films
  104. Dynamical analyses and dispersive soliton solutions to the nonlinear fractional model in stratified fluids
  105. Computation of exact analytical soliton solutions and their dynamics in advanced optical system
  106. An innovative approximation concerning the diffusion and electrical conductivity tensor at critical altitudes within the F-region of ionospheric plasma at low latitudes
  107. An analytical investigation to the (3+1)-dimensional Yu–Toda–Sassa–Fukuyama equation with dynamical analysis: Bifurcation
  108. Swirling-annular-flow-induced instability of a micro shell considering Knudsen number and viscosity effects
  109. Review Article
  110. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  111. Erratum
  112. Erratum to “On Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model (Open Physics 2021;19:679–682)”
  113. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  114. Possible explanation for the neutron lifetime puzzle
  115. Special Issue on Nanomaterial utilization and structural optimization - Part III
  116. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  117. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  118. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  119. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  120. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
  121. Novel exact solitons to the fractional modified mixed-Korteweg--de Vries model with a stability analysis
https://doi.org/10.1515/phys-2025-0206
Keywords for this article
spacetime symmetries; killing vector fields; plane symmetric spacetime; Rif tree approach
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Single-step fabrication of Ag2S/poly-2-mercaptoaniline nanoribbon photocathodes for green hydrogen generation from artificial and natural red-sea water
  3. Abundant new interaction solutions and nonlinear dynamics for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation
  4. A novel gold and SiO2 material based planar 5-element high HPBW end-fire antenna array for 300 GHz applications
  5. Explicit exact solutions and bifurcation analysis for the mZK equation with truncated M-fractional derivatives utilizing two reliable methods
  6. Optical and laser damage resistance: Role of periodic cylindrical surfaces
  7. Numerical study of flow and heat transfer in the air-side metal foam partially filled channels of panel-type radiator under forced convection
  8. Water-based hybrid nanofluid flow containing CNT nanoparticles over an extending surface with velocity slips, thermal convective, and zero-mass flux conditions
  9. Dynamical wave structures for some diffusion--reaction equations with quadratic and quartic nonlinearities
  10. Solving an isotropic grey matter tumour model via a heat transfer equation
  11. Study on the penetration protection of a fiber-reinforced composite structure with CNTs/GFP clip STF/3DKevlar
  12. Influence of Hall current and acoustic pressure on nanostructured DPL thermoelastic plates under ramp heating in a double-temperature model
  13. Applications of the Belousov–Zhabotinsky reaction–diffusion system: Analytical and numerical approaches
  14. AC electroosmotic flow of Maxwell fluid in a pH-regulated parallel-plate silica nanochannel
  15. Interpreting optical effects with relativistic transformations adopting one-way synchronization to conserve simultaneity and space–time continuity
  16. Modeling and analysis of quantum communication channel in airborne platforms with boundary layer effects
  17. Theoretical and numerical investigation of a memristor system with a piecewise memductance under fractal–fractional derivatives
  18. Tuning the structure and electro-optical properties of α-Cr2O3 films by heat treatment/La doping for optoelectronic applications
  19. High-speed multi-spectral explosion temperature measurement using golden-section accelerated Pearson correlation algorithm
  20. Dynamic behavior and modulation instability of the generalized coupled fractional nonlinear Helmholtz equation with cubic–quintic term
  21. Study on the duration of laser-induced air plasma flash near thin film surface
  22. Exploring the dynamics of fractional-order nonlinear dispersive wave system through homotopy technique
  23. The mechanism of carbon monoxide fluorescence inside a femtosecond laser-induced plasma
  24. Numerical solution of a nonconstant coefficient advection diffusion equation in an irregular domain and analyses of numerical dispersion and dissipation
  25. Numerical examination of the chemically reactive MHD flow of hybrid nanofluids over a two-dimensional stretching surface with the Cattaneo–Christov model and slip conditions
  26. Impacts of sinusoidal heat flux and embraced heated rectangular cavity on natural convection within a square enclosure partially filled with porous medium and Casson-hybrid nanofluid
  27. Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge
  28. Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method
  29. Bilinear form and soltion solutions for (3+1)-dimensional negative-order KdV-CBS equation
  30. Solitary chirp pulses and soliton control for variable coefficients cubic–quintic nonlinear Schrödinger equation in nonuniform management system
  31. Influence of decaying heat source and temperature-dependent thermal conductivity on photo-hydro-elasto semiconductor media
  32. Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition
  33. Bifurcation, chaotic behavior, and traveling wave solutions for the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model
  34. New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis
  35. Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
  36. Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term
  37. Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber
  38. Chaotic dynamics and some solutions for the (n + 1)-dimensional modified Zakharov–Kuznetsov equation in plasma physics
  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Influence of a magnetic field on double-porosity photo-thermoelastic materials under Lord–Shulman theory
  78. Soliton-like solutions for a nonlinear doubly dispersive equation in an elastic Murnaghan's rod via Hirota's bilinear method
  79. Analytical and numerical investigation of exact wave patterns and chaotic dynamics in the extended improved Boussinesq equation
  80. Nonclassical correlation dynamics of Heisenberg XYZ states with (x, y)-spin--orbit interaction, x-magnetic field, and intrinsic decoherence effects
  81. Exact traveling wave and soliton solutions for chemotaxis model and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation
  82. Unveiling the transformative role of samarium in ZnO: Exploring structural and optical modifications for advanced functional applications
  83. On the derivation of solitary wave solutions for the time-fractional Rosenau equation through two analytical techniques
  84. Analyzing the role of length and radius of MWCNTs in a nanofluid flow influenced by variable thermal conductivity and viscosity considering Marangoni convection
  85. Advanced mathematical analysis of heat and mass transfer in oscillatory micropolar bio-nanofluid flows via peristaltic waves and electroosmotic effects
  86. Exact bound state solutions of the radial Schrödinger equation for the Coulomb potential by conformable Nikiforov–Uvarov approach
  87. Some anisotropic and perfect fluid plane symmetric solutions of Einstein's field equations using killing symmetries
  88. Nonlinear dynamics of the dissipative ion-acoustic solitary waves in anisotropic rotating magnetoplasmas
  89. Curves in multiplicative equiaffine plane
  90. Exact solution of the three-dimensional (3D) Z2 lattice gauge theory
  91. Propagation properties of Airyprime pulses in relaxing nonlinear media
  92. Symbolic computation: Analytical solutions and dynamics of a shallow water wave equation in coastal engineering
  93. Wave propagation in nonlocal piezo-photo-hygrothermoelastic semiconductors subjected to heat and moisture flux
  94. Comparative reaction dynamics in rotating nanofluid systems: Quartic and cubic kinetics under MHD influence
  95. Laplace transform technique and probabilistic analysis-based hypothesis testing in medical and engineering applications
  96. Physical properties of ternary chloro-perovskites KTCl3 (T = Ge, Al) for optoelectronic applications
  97. Gravitational length stretching: Curvature-induced modulation of quantum probability densities
  98. The search for the cosmological cold dark matter axion – A new refined narrow mass window and detection scheme
  99. A comparative study of quantum resources in bipartite Lipkin–Meshkov–Glick model under DM interaction and Zeeman splitting
  100. PbO-doped K2O–BaO–Al2O3–B2O3–TeO2-glasses: Mechanical and shielding efficacy
  101. Nanospherical arsenic(iii) oxoiodide/iodide-intercalated poly(N-methylpyrrole) composite synthesis for broad-spectrum optical detection
  102. Sine power Burr X distribution with estimation and applications in physics and other fields
  103. Numerical modeling of enhanced reactive oxygen plasma in pulsed laser deposition of metal oxide thin films
  104. Dynamical analyses and dispersive soliton solutions to the nonlinear fractional model in stratified fluids
  105. Computation of exact analytical soliton solutions and their dynamics in advanced optical system
  106. An innovative approximation concerning the diffusion and electrical conductivity tensor at critical altitudes within the F-region of ionospheric plasma at low latitudes
  107. An analytical investigation to the (3+1)-dimensional Yu–Toda–Sassa–Fukuyama equation with dynamical analysis: Bifurcation
  108. Swirling-annular-flow-induced instability of a micro shell considering Knudsen number and viscosity effects
  109. Review Article
  110. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  111. Erratum
  112. Erratum to “On Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model (Open Physics 2021;19:679–682)”
  113. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  114. Possible explanation for the neutron lifetime puzzle
  115. Special Issue on Nanomaterial utilization and structural optimization - Part III
  116. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  117. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  118. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  119. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  120. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
  121. Novel exact solitons to the fractional modified mixed-Korteweg--de Vries model with a stability analysis
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 3.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2025-0206/html
Scroll to top button