Spherically Symmetric Geometries in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.4924pt" id="M1" height="16.753pt" version="1.1" viewBox="-0.0657574 -12.2606 34.6663 16.753" width="34.6663pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-103" d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"/></g><g transform="matrix(.017,0,0,-0.017,11.059,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.017,0,0,-0.017,17.202,0)"><path id="g50-85" d="M655 676H628C608 656 600 650 571 650H156C118 650 114 651 102 676H77C63 615 45 548 24 485L60 486C81 531 97 562 114 580C134 603 148 611 252 611H308L217 127C201 41 196 38 105 32L98 0H384L393 32C300 38 288 41 304 127L393 611H466C553 611 571 604 582 580C592 559 597 530 595 485L630 489C636 548 645 624 655 676Z"/></g><g transform="matrix(.017,0,0,-0.017,28.276,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.4924pt" id="M2" height="16.753pt" version="1.1" viewBox="-0.0657574 -12.2606 34.5113 16.753" width="34.5113pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><use xlink:href="#g50-103"/></g><g transform="matrix(.017,0,0,-0.017,11.059,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.017,0,0,-0.017,17.202,0)"><path id="g50-83" d="M636 22C611 29 589 38 560 74C536 104 518 134 495 173L430 300C518 328 592 392 592 492C592 555 563 594 528 618C491 643 435 650 377 650H140L131 618C222 610 223 601 208 518L137 136C121 43 116 40 27 32L18 0H295L300 32C212 40 206 42 223 138L248 279H296C334 279 350 272 369 230C393 174 413 140 441 94C482 24 523 1 626 -8L636 22ZM497 480C497 422 468 374 417 343C387 324 353 315 304 315H255L298 556C303 583 309 598 319 603C331 610 348 613 367 613C448 613 497 573 497 480Z"/></g><g transform="matrix(.017,0,0,-0.017,28.116,0)"><use xlink:href="#g50-42"/></g></svg> Gravitational Theories

Nashed, Gamal G. L.

doi:https://doi.org/10.1155/2015/821519

Advances in High Energy Physics

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Research Article | Open Access

Volume 2015 | Article ID 821519 | https://doi.org/10.1155/2015/821519

Spherically Symmetric Geometries in and Gravitational Theories

Gamal G. L. Nashed^1,2

Academic Editor: Rong-Gen Cai

Received03 Apr 2015

Revised05 May 2015

Accepted05 May 2015

Published28 May 2015

Abstract

Using the well know relation between Ricci scalar, , and torsion scalar, , that is, , we show that, for any spherically symmetric spacetime whose (i) scalar torsion vanishing, that is, or (ii) total derivative term, that is, with is the contraction of the torsion, vanishing, or (iii) the combination of scalar torsion and total derivative term vanishing, could be solution for and gravitational theories.

1. Introduction

In the last few decades many theories of modified gravity have been constructed concerning different physical goals. In the 60s, Brans and Dicke coupled a scalar field to the metric to obatin a variable effective gravitational constant [1, 2]. In Brans-Dicke theory, the scalar field is a new degree of freedom of the gravitational field, which is not directly coupled to the matter, but it brings influence by getting in the dynamical equations that govern the spacetime geometry.

Also in the 60s, Einstein-Hilbert Lagrangian has been added with quadratic terms in the curvature to overcome the renormalization of the theory [3]. In 1971, Lovelock [4] examined terms of higher order in the curvature as well; but he was motivated by another motivation. While this kind of Lagrangians leads, generally, to fourth-order dynamical equations because they involve second-order derivatives, Lovelock derived a general Lagrangian polynomic in curvature that leads to conserved second-order equations for metric spacetime. Lovelock has shown that the bigger the spacetime dimension is, the bigger the number of terms these Lagrangians can contain is.

In the 70s, Buchdahl and Astron aimed at replacing the Einstein-Hilbert scalar Lagrangian with a function of the scalar curvature and studied its cosmological consequences [5]. This kind of amended gravity is for time being called theory. In 1983, Milgrom [6] thought that the galactic rotation curves were a guidance of the fail of Newtonian gravity to describe gravitation in the weak field regime. According to Milgrom, no dark matter was able to explain the data but a theory of amended gravity. In this regard, Milgrom built amended newtonian dynamics theory. In the last decade, Bekenstein [7] developed a relativistic theory of gravity named tensor-vector-scalar gravity (TeVeS) because it mixes the metric tensor, a vector field, and a scalar field. TeVeS involves Milgrom’s weakening of Newtonian gravity in the weak field regime and has also effects for lensing phenomena, cosmology, and so forth.

String theory has been a source of detection for theories of amended gravity. For example, Dvali, Gabadaze, and Poratti (DGP) gravity [8] describes the 4-dimensional universe as being immersed in a 5-dimensional manifold. Therefore, a “normal” 5D gravity can effect large scale effects in 4D as an accelerated expansion with no need of dark energy. These 4D consequences are driven by a scalar field named galileon due to its symmetries it obeys [9, 10].

For the present accelerating era of our universe, shown by the recent observations of type Ia supernovae [11–13], general relativity (GR) can only describe the dynamics of the universe through an exotic fluid, which is called dark energy (DE), that has a negative pressure. Therefore, this phenomena divide researchers into two groups. The first group has been trying to explain the late-time acceleration in the frame work of standard cosmology, supplying the existence of DE. However, till now, the nature of DE is still unknown and is not an easy task in modern physics [14–17]. The various cosmological observational data support the cold dark matter (CDM) model. At the present epoch, the CDM model seems to be a standard cosmological model. However, the theoretical root of the cosmological constant has not been fully catched. A number of models for DE to explain the late-time cosmic acceleration without the cosmological constant have been proposed (for more details of this topic readers are advised to return to the review [18] and references therein).

theories have been proposed in the last few years as an alternative to modified gravity [19, 20] and they constitute a very promising research area as is witnessed by the increasing interest in this field [21–30]. The dynamical object in theories is the field of frames (vierbeins or tetrads), which involve the basis of the tangent space and its respective cobasis . The Lagrangian density of this theory is built with the product of the determinant of the covariant form of the tetrad times a function of the scalar , which is quadratic in the set of 2-form , where measures the basis departing from a coordinate basis. actions are preserved only under global Lorentz transformations but not preserved under local Lorentz transformation. Therefore, theories are dynamical theories not only for the metric but also for the entire tetrad. As will be mentioned later, only in the special case of the so-called teleparallel equivalent of general relativity (TEGR) does the theory acquire invariance under local Lorentz transformations, thus becoming a dynamical theory just for the metric. In general, it could be said that theory provides the spacetime not only with a metric but also with an absolute parallelization [31–45]. It is shown that, based on a special frame, some static solutions can be derived with spherical symmetry in gravity theories [46]. It is proved that Schwarzschild spacetime remains as solution to ultraviolet deformations of GR in the framework of gravitational theories [47]. The existence of circularly symmetric solutions was examined in three dimensions [48]. For a special form of , a spherically symmetric solution which depends on the parameter is derived [49].

Within gravitational theory there are many solutions, spherically symmetric [50, 51], spherically symmetric charged [52], homogenous anisotropic [53], and stability of Einstein universe [54]. Some cosmological features of CDM model in the framework of the are investigated [35]. It is the aim of this study to show that any spherically symmetric geometries are solutions to the modified gravitational theories and .

This paper is organized as follows. In Section 2, basic equations of gravity theory are presented. In Section 3, a tetrad field with spherical symmetry is given and application to the field equation of is provided. New analytic, vacuum, and nonvacuum solutions are derived in Section 3. Final section is devoted to the main results.

2. Basic Equations of Gravity

The mathematical concept of the gravity theory is based on the Weitzenböck geometry. Our convention and nomenclature are the following: the Latin indices describe the components of the tangent space to the manifold (spacetime), while the Greek ones describe the components of the spacetime. For a general spacetime metric, we write the line element as , where is the Minkowski metric. is the covariant vector fields and its inverse is the contravariant vector fields satisfying the orthogonality and unitary conditions and . In a spacetime with absolute parallelism, the parallel vector fields define the nonsymmetric affine connection (cf., [55]) , where . The curvature tensor defined by the Weitzenböck connection, , is identically vanishing. The torsion and the contortion components are defined as and .

The skew symmetric tensor has the form and the torsion scalar is given by . The action of theory is given bywhere , is the Lagrangian of matter field, is the cosmological constant and are matter fields, and is the reduced Planck mass, which is related to the gravitational constant by . Similar to the theory, one can define the action of theory as a function of the tetrad fields and by putting the variation of the function with respect to the tetrad fields to be vanishing one can obtain the following equations of motion:whereand is the energy-momentum tensor.

Now we are going to rewrite the field equations (2) in another form. The field equations (2) are written in terms of the tetrad and its partial derivatives. These equations appear to be different from Einstein’s field equations. Following [19, 20, 56], one can obtain an equation relating the scalar torsion with the Ricci scalar . These will make the equivalence between teleparallel gravity and GR clear. On the other hand, the tetrad cannot be eliminated completely in favor of the metric in (2) because of the lack of local Lorentz symmetry but one can show that the latter can be brought in a form that closely resembles Einstein’s equation. This form is more suitable for constructing analytic solutions in the theory. To start writing the field equations in a covariant version, we must replace partial derivatives in the tensors by covariant derivatives compatible with the metric , that is, , where . Thus, the definition of the torsion tensor can be written asUsing (4) in the definition of contorsion and the skew symmetric tensor , one can obtain

On the other hand, from the relation between Weitzenböck connection and the Levi-Civita connection, one can write the Riemann tensor for the Levi-Civita connection, , in terms of the nonsymmetric connection, , in the formThe associated Ricci tensor can then be written asNow, by using the definition of the contorsion along with the relations and considering that , one has [24, 57–65]Equation (8) implies that the torsion scalar, , and Ricci scalar, , differ only by a covariant divergence of a spacetime vector. Therefore, the Einstein-Hilbert action and the teleparallel action (i.e., ) will both lead to the same field equations and are dynamically equivalent theories. However, the divergence term is the main reason that makes the field equations of not invariant under local Lorentz transformation (LLT). Let us explain this for some specific form of . ConsiderThe last term in the R.H.S. of (9) is not a total derivative term and therefore it is responsible to make when written in terms of and not invariant under LLT in contrast to the linear case, that is, the form of (8). Same discussion can be applied to the general form of and which shows in general a difference between and gravitational theories. It is known that the field equations of are of fourth-order and invariant under LLT, while the field equations of are of second-order and not invariant under LLT. Therefore, if the divergence term is vanishing then and hence one can show that any solution in TEGR can be a solution to GR and for a general case one can also show that any solution of can be a solution to . The aim of this study is to show that in the spherically symmetric case any solution of whose divergence term is vanishing will also be a solution to theory.

By using the equations listed above and after some algebraic manipulations, one can getwhere is the Einstein tensor. Finally, by using (10), the field equations of gravity, (2), can be rewritten in the form

Equation (11) can be taken as the starting point of the modified gravity model and it has a structure similar to the field equations of gravity. Note that in the more general case with the field equations are in covariant form. Nevertheless, the theory is not local Lorentz invariant. In case of and constant torsion, , GR is recovered and field equations are covariant and the theory is Lorentz invariant.

3. Spherically Symmetric Solution in Gravity Theory

The spherically symmetric tetrad has the formwhere and are two unknown functions of the radial coordinate, . Tetrad field (12) is a solution to the field equations (11) when , that is, TEGR, and for Schwarzschild black hole or for Reissner Nordström black hole or for Reissner Nordström-dS black hole [66]. However, our aim is to find a solution to gravitational theory. Tetrad field (12) gives a nonvanishing value of the scalar torsion; that is, , which means that the extra term appears in the field equation (2); that is, will have a nonvanishing value. Therefore, tetrad (12) does not fit to be a solution for gravitational theories. For this purpose we use the following tetrad:where and are two unknown functions of the radial coordinate, , and . Tetrad (13)¹ is a generalization of tetrad (12); that is, when we return to tetrad (12). Also we exclude the case in which to preserve the symmetry we assume, spherical symmetry. Using tetrad (13) we get the nonvanishing components of the torsion tensor and the skew symmetric tensor in the formwhere , , , , and .

Using (14) we get the scalar torsion in the formPutting the torsion scalar we get a solution for in the formwith being an arbitrary function of and . Solution (16) satisfied (11) under some constraints on the function and that leads to the vanishing of torsion scalar. We put the constraint to ensure that any solution of will be a solution to . Other constraints, in the spherically symmetric case, will not satisfy the equality of the two theories.

The other solution comes from the calculations of the total derivatives term, surface term. The nonvanishing components of the vector density of the torsion, that is, , have the form where , , and are defined after (14). Using (17) we get the divergence term in the formwhere , , and . Putting we get a solution for in the formwhere is an arbitrary function of and ². Using (19) in the relation , gives ; that is, any solution in GR will also be a solution to TEGR and vice versa. In general, we can say that any spherically symmetric solution of theories will also be a solution to theories.

Finally, we are going to calculate the term which has the formEquation (20) does not show any dependence on the unknown which is consistent with the fact that and Ricci scalar does not have any dependence on the unknown . Therefore, this constraint, , tells us that we work on GR theory which means that any GR solution will be a solution to TEGR or gravitational theories.

4. Main Results and Discussion

In this study, we have addressed the problem of finding spherically symmetric solution in the framework of and gravitational theories. To achieve this goal, we begin with the relation between Ricci scalar and torsion scalar tensors. From this relation we have studied three different cases.

Case 1. In this case we have applied a special³ spherically symmetric tetrad field having three unknown functions; one of them is a function of , , and , . The other two functions are responsible for reproducing all the known spherically symmetric solutions. We have calculated the scalar torsion of this tetrad and got a differential equation. By solving this differential equation with respect to we derived a solution that made the scalar torsion vanishing. Using this solution on the identity we get .

Case 2. In this case we have calculated the total derivative term, , and solved the resulting differential equation with respect to . Using this solution on the identity we get .

Case 3. In this case we have evaluated the equation and showed that the resulting differential equation did not show any dependence on . This is acceptable because we have which has no dependence on ; more precisely we work within the domain of GR.

Case 2 above shows that any spherically symmetric solution of GR will also be a solution to TEGR and vise versa. This means that, for example, for Schwarzschild spacetime whose Ricci scalar is vanishing, this solution will also be a solution to gravity. Let us explain this. The field equation of gravity has the form [67]where and are the energy momentum tensor of matter. For Schwarzschild solution we have and the field equation of for this solution takes the formso if we take the constraints that we get a solution within gravitational theory. At the same time for gravitational theories we have the problem of local Lorentz transformation which comes from the total derivative term. Therefore, Case 2 above means that if one succeeds to find the condition which makes the total derivatives term vanishing then and will be equivalent for the spherically symmetric case only, more precisely for the case where the Ricci scalar torsion has a zero or constant value.

All the above cases are related to the spherically symmetric geometry in which Ricci scalar either has a vanishing value or constant one like de Sitter. For other symmetries these studies can not be applied science the identity will not be in general satisfied. For example, in FRW models in which their symmetries are homogenous and isotropic, we can not apply the above procedure. This is due to the fact that Ricci scalar in those models has a nonvanishing value. Therefore, if one applies the procedure of Case 1 we get which is a contradiction. This contradiction comes from the fact that Ricci scalar is invariant under local Lorentz transformation while the total derivative term is not.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is partially supported by the Egyptian Ministry of Scientific Research under Project ID 24-2-12.

Endnotes

Tetrad (13) is a combination of the diagonal tetrad and so (3), that is,wherewhere is the diagonal form of tetrad field (13).
For the Schwarzschild solution in which , (18) takes the formand (19) takes the form
Tetrad (13) is not the general spherically symmetric one. Same procedure done for tetrad field (13) can be followed for the general spherically symmetric tetrad field [66].

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Copyright © 2015 Gamal G. L. Nashed. 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. The publication of this article was funded by SCOAP³.

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