Conformally Flat Pseudoprojective Symmetric Spacetimes in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 54.0264 16.7875" width="54.0264pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g><g transform="matrix(.017,0,0,-0.017,11.025,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,16.962,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.017,0,0,-0.017,27.721,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,34.51,0)"><path id="g198-8" d="M417 653C377 677 335 687 290 687C171 687 83 622 83 541C83 462 150 415 250 398C231 363 224 320 224 292C224 212 282 165 361 165C409 165 455 180 499 209L500 208C439 89 369 16 246 16C183 16 127 48 122 79H124C131 68 141 66 153 66C181 66 197 90 197 117C197 142 173 164 146 164C111 164 87 134 87 95C87 39 157 -15 256 -15C421 -15 531 60 586 209C606 264 620 298 652 330L637 346C520 239 440 195 368 195C324 195 302 225 302 262C302 293 311 343 343 390C542 390 740 489 740 593C740 642 693 669 637 669C526 669 359 580 263 428C175 440 124 482 124 541C124 602 193 657 297 657C332 657 369 652 408 635L417 653ZM699 590C699 512 532 420 361 420C425 535 557 632 642 632C673 632 699 618 699 590Z"/></g><g transform="matrix(.017,0,0,-0.017,47.776,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> Gravity

De, Uday Chand; Shenawy, Sameh; Syied, Abdallah Abdelhameed; Turki, Nasser Bin

doi:https://doi.org/10.1155/2022/3096782

Advances in Mathematical Physics

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Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3096782 | https://doi.org/10.1155/2022/3096782

Conformally Flat Pseudoprojective Symmetric Spacetimes in Gravity

Uday Chand De,¹Sameh Shenawy,²Abdallah Abdelhameed Syied,³and Nasser Bin Turki⁴

Academic Editor: Sergey Shmarev

Received28 Dec 2021

Accepted07 Mar 2022

Published25 Mar 2022

Abstract

Sufficient conditions on a pseudoprojective symmetric spacetime whose Ricci tensor is of Codazzi type to be either a perfect fluid or Einstein spacetime are given. Also, it is shown that a is Einstein if its Ricci tensor is cyclic parallel. Next, we illustrate that a conformally flat spacetime is of constant curvature. Finally, we investigate conformally flat spacetimes and conformally flat perfect fluids in theory of gravity, and amongst many results, it is proved that the isotropic pressure and the energy density of conformally flat perfect fluid spacetimes are constants and such perfect fluid behaves like a cosmological constant. Further, in this setting, we consider some energy conditions.

1. Introduction

The notion of a pseudoprojective symmetric manifold, briefly denoted by , was first introduced and studied in 1989 by Chaki and Saha [1]. Such a manifold is a nonflat pseudo-Riemannian manifold whose projective curvature tensor [2] satisfies the condition where is the Riemann curvature tensor, is the Ricci tensor, is a nonzero -form, and denotes the covariant differentiation with respect to the metric . In [1], it was proved that a manifold is of constant scalar curvature, that is, and is an eigenvector of the Ricci tensor and the corresponding eigenvalue is that is,

Also, it was shown that if a manifold admits a unit parallel vector field, then it is reduced to a pseudosymmetric manifold [3].

An -dimensional Lorentzian manifold is said to be a pseudoprojective symmetric spacetime if its projective curvature tensor agrees with (2). A Lorentzian manifold is said to be perfect fluid if its Ricci tensor satisfies where and are scalar fields and that is, is a time-like velocity vector field [4, 5]. In differential geometry, a manifold satisfying the foregoing relation of the Ricci tensor is called a quasi-Einstein manifold without any restrictions on the velocity vector field [6, 7]. Throughout this paper, let be a unit timelike vector field.

The standard theory of gravity follows from Einstein’s field equations (EFE) [8, 9]. where , , and are the scalar curvature tensor and the Newtonian gravitational constant, and is the energy-momentum tensor describing the ordinary matter. These equations correlate the geometry of a spacetime with its matter content. That is, the geometry of a spacetime determines the matter content of the spacetime conversely. Many modifications of EFE have been introduced and studied on a large scale (see references [10–12] for examples of the modified gravity theories). Amongst these modified theories, there was one known under the name gravity theory [13], which is obtained by replacing the scalar curvature with a function of the scalar curvature and Gauss-Bonnet scalar . in the gravitational action term with being the action term of the standard matter fields. The field equations are given by where results from and , [14].

In this paper, we investigate spacetimes whose Ricci tensor is of Codazzi type or cyclic parallel. Next, a conformally flat spacetime is studied. After that, certain investigations of conformally flat spacetimes in modified gravity theory are carried out. Finally, we study conformally flat perfect fluid spacetimes in gravity.

2. On a Spacetime Whose Ricci Tensor Is of Codazzi Type or Cyclic Parallel

In this section, a spacetime whose Ricci tensor is of Codazzi type or cyclic parallel is considered. The Ricci tensor is called of Codazzi type if [15, 16] whereas is called cyclic parallel if

Transvecting equation (1) with , one gets where .

Contracting equation (2) with , we obtain

Using (1) and (12) in (13), we have

With the help of equations (3) and (4), one finds

First, suppose that the Ricci tensor of spacetime is of Codazzi type; thus, we have

The use of equation (15) in (16) implies that

It is to be noted that the Riemann curvature tensor has the following properties:

The use of the above properties of the Riemann curvature tensor in equation (17) implies

Contracting with and using (4), we have

We thus can state the following theorem:

Theorem 1. Let be a spacetime whose Ricci tensor is of Codazzi type; then, the Ricci tensor of is given by (20).

Suppose that , then (20) becomes

which means that a spacetime is perfect fluid.

Corollary 2. Let be a spacetime whose Ricci tensor is of Codazzi type. Then, is perfect fluid if .

The conformal curvature tensor is given by [17].

A contraction with implies

Equations (20) and (23) are combined to give where is the contracted Weyl tensor. Hence, we can state the following theorem:

Theorem 3. Let be a spacetime whose Ricci tensor is of Codazzi type; then, the Ricci tensor of is of the form (24).

In particular case, if , then equation (24) is reduced to be in the following form: which means a spacetime is Einstein.

Corollary 4. Let be a spacetime whose Ricci tensor is of Codazzi type. Then, is Einstein if the contracted Weyl tensor vanishes.

Assume that has cyclic parallel Ricci tensor, that is, the Ricci tensor agrees with (11). Then, using (15) in (11) infers

Contracting with and using equation (4), we obtain which means a spacetime whose Ricci tensor obeys (4) is Einstein. Hence, we motivate to state the following theorem:

Theorem 5. Let be a spacetime whose Ricci tensor is cyclic parallel; then is an Einstein spacetime.

3. Conformally Flat Spacetimes

The divergence of the conformal curvature is expressed as [18]

A spacetime is called conformally flat if the conformal curvature tensor vanishes, that is, . It is well-known that if , then . And consequently, the following equations hold

Since in spacetime the scalar curvature is constant, then equation (30) implies that which shows that the Ricci tensor is of Codazzi type [19]. We thus can conclude the following theorem:

Theorem 6. Let be a spacetime with a divergence-free conformal curvature tensor; then, the Ricci tensor of is of Codazzi type.

In view of Theorem 1, we can state the following corollary:

Corollary 7. Let be a spacetime with a divergence-free conformal curvature tensor; then, the Ricci tensor of is given by

From equation (29), we can get

Using (33) and (34) in (15), one obtains

It follows that

In a conformally flat spacetime, the Ricci tensor is of Codazzi type; therefore,

Contracting with and using equation (4), we get which illustrates that a conformally flat spacetime is Einstein.

Theorem 8. A conformally flat spacetime is Einstein.

The use of (38) in (1) implies that

Then, from (39) in (29), one infers

Hence, from (38), we get

From (38) and (41) in (1), we have which means that a conformally flat spacetime is of constant curvature.

In consequence of the above, we can state the following theorem:

Theorem 9. A conformally flat spacetime is projectively flat and of constant curvature.

4. Conformally Flat Spacetimes in Gravity

In this section, conformally flat spacetimes in theory of gravity are investigated. For , equation (38) becomes

It follows that

Multiplying equations (43) and (44), one gets

From equation (29), it follows that

Multiplying equations (29) and (46), we obtain

With the help of equation (47), the Gauss-Bonnet topological invariant is

The use of equation (45) implies that

Thus, we can state the following theorem:

Theorem 10. The Gauss-Bonnet scalar in a conformally flat spacetime is expressed as

In a conformally flat spacetime, equation (9) can be rewritten as where arises from the geometry of the spacetime. The tensor is given as [20]

Since in a conformally flat spacetime the scalar curvature is constant, the previous equation reduces

Utilizing equations (38) and (49) in equation (53), we get

The use of (43) and (54) in (51) implies that

The vector filed is called Killing if whereas is called conformal Killing if where is the Lie derivative with respect to the vector filed and is a scalar function [21, 22].

A spacetime is said to admit a matter collineation with respect to a vector field if the Lie derivative of the energy-momentum tensor with respect to satisfies while it is said that the energy-momentum tensor has the Lie inheritance property along the flow lines of the vector field if the Lie derivative of with respect to satisfies [21, 22]

Applying the Lie derivative on both sides of (55), one gets

If the vector field is Killing on a conformally flat spacetime , hence equation (60) implies that

In the contrast, if a conformally flat spacetime admits matter collineation with respect to , it follows from equation (60) that

Hence, we can state the following theorem:

Theorem 11. Let be a conformally flat spacetime obeying gravity theory; then, the vector field is Killing if and only if admits matter collineation with respect to .

Assume that the vector field is conformal Killing; then, after using (57) in (60) utilizing (55), we acquire that

Conversely, suppose that the energy-momentum tensor has the Lie inheritance property along the flow lines of , thus making use of (59) in (60) after that using (55)), we infer that

Thus, we can state the following theorem:

Theorem 12. Let be a conformally flat spacetime obeying gravity theory; then, has a conformal Killing vector filed if only if the energy-momentum tensor has the Lie inheritance property along .

5. Conformally Flat Perfect Fluid Spacetimes in Gravity

This section is mainly organized to study conformally flat perfect fluid spacetimes in modified gravity theory. For a perfect fluid spacetime, the energy-momentum tensor is given as where and are the isotropic pressure and the energy density of the ordinary matter, whereas and are the effective isotropic pressure and the effective energy density of the effective matter.

In view of (55) and (65), we have

Contracting twice with and , one finds

Utilizing (68) in (69), it arises

We thus motivate to state the following theorem:

Theorem 13. In a conformally flat perfect fluid spacetime obeying gravity, the isotropic pressure and the energy density are constants. Moreover, they are given by (68) and (70).

The combination of (68) and (70) gives which means that the spacetime represents inflation and fluid behaves as a cosmological constant [23].

Theorem 14. Let be a conformally flat perfect fluid spacetime obeying gravity; then, represents inflation and fluid behaves as a cosmological constant.

Using (54), (65), and (66) in (51), one infers

Making a comparison of both sides, we obtain

The use of (68) and (70) implies that

In the context of modified gravity, let us now deduce some energy conditions of a perfect fluid type effective matter. The energy conditions are obtained as follows [24, 25]: (1)Null Energy Condition (NEC). .(2)Weak Energy Condition (WEC). and .(3)Dominant Energy Condition (DEC). and .(4)Strong Energy Condition (SEC). and .

In view of (74) and (22), the energy conditions are always satisfied if .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was supported by the Researchers Supporting Project number RSP2022R413, King Saud University, Riyadh, Saudi Arabia.

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Copyright © 2022 Uday Chand De et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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