Research Article | Open Access
A Field Theory with Curvature and Anticurvature
The present work is an attempt to construct a unified field theory in a space with curvature and anticurvature, the PAP-space. The theory is derived from an action principle and a Lagrangian density using a symmetric linear parameterized connection. Three different methods are used to explore physical contents of the theory obtained. Poisson’s equations for both material and charge distributions are obtained, as special cases, from the field equations of the theory. The theory is a pure geometric one in the sense that material distribution, charge distribution, gravitational and electromagnetic potentials, and other physical quantities are defined in terms of pure geometric objects of the structure used. In the case of pure gravity in free space, the spherical symmetric solution of the field equations gives the Schwarzschild exterior field. The weak equivalence principle is respected only in the case of pure gravity in free space; otherwise it is violated.
Historically, it is well known that, during the second decade of the past century, Einstein has solved some of the problems of gravity theory by introducing a new philosophy: “the geometrization philosophy.” He has constructed a successful theory for gravity, the general relativity (GR). This philosophy and the quantization philosophy represent the two main philosophies, of the 20th century, in the domain of theoretical physics. Afterwards, it appeared that the geometrization philosophy is successful in the domain of macrophysics while microphysics is the domain of success of the quantization philosophy (the domain of short range forces).
Although GR is proved to be the best successful gravity theory in the domain of the solar and stellar systems, it has many problems in the domain of cosmological applications. The big bang model of the Universe suffers from such problems, for example, the initial singularity problem, particle horizons problem, and so on. Recently it is shown that GR cannot account for the results of observations, for example, SN type Ia, CMBR observations, the rotation curve of stars in spiral galaxies, and so forth, [1–3].
The need to modify GR, or to construct alternative theories, has increased in the past two or three decades. Some authors have suggested the use of the cosmological constant to solve such problems. But the cosmological constant itself has a big problem (cf. ). Another group has attempted to solve these problems in the context of Riemannian geometry by considering different functions of the curvature scalar , theories (cf. [5–7]). A third group has attempted to construct theories in the framework of geometries with nonvanishing torsion scalar , the theories (cf. [8–12]). Such scalars are usually used to construct the Lagrangian density of the theory suggested.
Authors almost use the absolute parallelism (AP) space as a space with nonvanishing torsion scalar . It can be easily shown that, apart from a divergence term (cf. ), . So, both and can reproduce the same theory. For this reason the theory resulting from is known in the literature as the “teleparallel equivalent of general relativity” (TEGR). It is tempting to expect that both GR and TEGR give rise to the same physics. A main difference between these two theories is that the latter is written in the AP-space which is much wider than the Riemannian one. Also, a difference between and theories is that the former gives fourth order differential equations while the latter gives second order differential equations for which it may be easier to find solutions.
In the present work, we draw the attention to the importance of the linear connection, of a certain space, in the physical contents of any geometric field theory, constructed in this space. We consider linear connection, of an appropriate geometry, as the most important geometric object when constructing geometric field theories. As an example, in the construction of GR, the role of Christoffel symbol is obvious in describing both field and motion. It is used in the definition of the curvature tensor (which represents the gravitational field) and in deriving the geodesic equation (describing motion in GR). So, one has to explore the role of any new linear connection, derived from the building blocks (BB) of the geometry used. In addition, torsion is defined as twice the skew part of the connection in geometries with nonsymmetric linear connections. So, in the context of geometrization philosophy, it is of importance to use any natural (here by natural we mean any geometric object defined in terms of the BB of the geometry used) modified linear nonsymmetric connection in constructing field theories. This may enable exploring the role of the above mentioned geometric objects in physics and predicting new physics, if any.
The use of a wider space with a general linear connection may give rise to new physics, which may be necessary to interpret recent observations. In the past two decades a wider type of geometry has been suggested and developed, the PAP-geometry [14–16]. Its linear connection is metric and nonsymmetric. This geometry has simultaneously nonvanishing torsion and curvature. Furthermore, a new geometric object called the “anticurvature” [17, 18] is defined which gives rise to “antigravity” . The path equations (curves) of this geometry give rise to a repulsive force, together with the ordinary attractive force of gravity, when it is written in its linearized form . Consequently, a theory written in the context of this geometry may help in solving some of the problems of GR especially the accelerating expansion of the Universe. Several successful applications have been done in the context of this geometry [18–21].
In the present paper we use the following terminology.
A geometry is a manifold () equipped with building blocks (BB).
A space is a geometry admitting a linear connection.
A geometric structure is a space admitting a symmetry group (SG) or more.
The aim of the present work is to construct and analyze a field theory in the context of PAP-space and also to explore the role of the parameter of the PAP-space. To achieve this aim, the paper is arranged as follows: in Section 2, we give a brief account on the PAP-geometry and show that it can be reduced to Riemannian and conventional AP-spaces in some special cases. In Section 3, we suggest a certain Lagrangian density to construct a set of field equations using an action principle. In Section 4, we show how physics is extracted from the suggested geometric field theory before solving its field equations. The spherically symmetric case is studied in Section 5. Discussion of the results obtained and some concluding remarks are given in Section 6.
2. The Underlying Geometry: PAP-Geometry
The present work is carried out in the context of the “parameterized absolute parallelism-” (PAP-) geometry. This geometry has many advantages over both Riemannian and conventional AP geometries. In the present section, we give a brief review of this type of geometry. For more details readers are referred to [14–16].
“The PAP-geometry is an -dimensional differentiable manifold equipped with a set of -linear independent vector fields ( represents the vector component and is used for the vector number. We use Greek indices for coordinate components written in a covariant or contravariant position. Latin indices are used to represent vector numbers, written in a lower position.).” These vector fields are the same BB of conventional absolute parallelism- (AP-) geometry (cf. [16, 22–24]). As a consequence of this independence, the determinant is nonvanishing; that is, . Now, we have the following (Einstein summation convention is taken over any repeated Greek index in the usual manner, while it is applied on repeated Latin indices wherever they appear.): where are the contravariant components of the vector fields , that is, the normalized cofactor of . One can define the following second order symmetric tensors: Now, since is nonvanishing, it is easy to show that and consequently we defined , the normalized cofactor of . Then we can write So, (3) and (4) can be used as the metric tensor of a Riemannian space in the framework of the PAP-geometry.
2.1. PAP-Linear Connection
Motivated by some arguments [14, 15], the PAP-linear connection has been defined as where is the Christoffel symbol of the second kind defined, as usual, by And is a third order tensor, called the parameterized contortion, defined by (Comma (,) is used for ordinary partial differentiation and (;) for covariant differentiation using Christoffel symbol.). Here is a dimensionless parameter, characterizing this type of geometry, to be discussed later. The space constructed using the parameterized linear connection (6) is called the PAP-space (, , and ).
2.2. Tensor Derivatives
Since is a nonsymmetric linear connection, one can define the following tensor derivatives: where is any arbitrary vector, and is the symmetric part of (6), defined as where
It can be easily shown that (9) is a symmetric linear connection under the effect of group of general coordinate transformation.
2.3. The Parameterized Torsion Tensor
Since parameterized linear connection is nonsymmetric, as stated above, it has a skew-symmetric part defined by which is called the parameterized torsion tensor and is the torsion tensor of the AP-space. The relation between the parameterized torsion and the parameterized contortion, is given by The contraction of the parameterized torsion (11) or contortion (12) of the PAP-space gives the parameterized basic vector , which is defined as
2.4. Second Order Tensors and Scalars
Table 1  gives the definition of a list of second order tensors: These tensors are defined using the above parameterized third order tensors and the parameterized basic vector. The table is similar to that constructed by Mikhail in 1962  in the context of AP-geometry.
Using tensors given in Table 1, we can define many scalars (e.g., , , , , , , and ).
Among these tensors, we have a set of second order traceless tensors (e.g., (), (), and ()) with possible linear combination. Such objects may be of physical interest.
2.5. Parameterized Curvature and Anticurvature Tensors
We can define many parameterized curvature tensors in the context of the PAP-space, none of which are vanishing. Here we are interested in the curvature corresponding to the linear connection (9). So, the definition of this parameterized curvature tensor may be written as
Substituting from (9) into (14), the parameterized curvature (14) can be split in two parts as follows: where is the conventional Riemann-Christoffel curvature tensor and is the parameterized anticurvature defined by which is a tensor of type (1,3). Note that the first term on the right-hand side of (15) is independent of the parameter . This is an important property that is discussed later. It is to be considered that the parameterized curvature (14) is, in general, a nonvanishing tensor. Equation (15) shows that it is split into two parts. The first (16) depends only on Christoffel symbol of the second kind, which coincides with Riemannian-Christoffel curvature tensor of the Riemannian space. The second (17) depends only on the contortion (or torsion via (11)) of the PAP-space. For certain properties, the tensor (17) is called the parameterized anticurvature tensor [17, 18, 26].
2.6. Path Equations
A general path equation for this type of space has been derived  and can be written in the following form: where is the parameter characterizing this path, and is the tangent of the path.
2.7. Special Cases
It is well known that for any AP-space there exists an associated Riemannian one, with metric tensor defined by (3). It is to be considered that Riemannian geometry is not a special case of the AP-space (cf. ). In PAP-space the situation is different. It can be easily shown that Riemannian space and the AP-space are just special cases of the PAP-space, corresponding to and , respectively. All features of the Riemannian geometry can be covered, by taking , in (6). Also, the PAP-space reduces to the AP-space if we take . So, Riemannian geometry is not associated with the PAP-geometry, but it is obtained as a special case. This important features facilitate comparison between the results obtained in the context of the PAP-geometry and those obtained in the context of Riemannian or AP geometries.
Figure 1 shows the relations between PAP-, AP-, and Riemannian spaces. Solid arrows give the direction for some special cases, while the dashed one gives the association case. It is to be considered that there is a wide range of spaces (a discrete spectra) between the above mentioned two special cases (i.e., between and ). This shows how wide the PAP-space is compared with Riemannian or conventional AP-spaces.
3. The Equations of the Theory
3.1. An Action Principle: Field Equations
In this section we derive a set of field equations corresponding to the scalar curvature (19) using an action principle. The Lagrangian density corresponding to the scalar (19) is defined by which can be written, using (19), as or equivalently, it can be written as Since, as it is well known, the term has no contribution to the variation; we write (24) in the following form;
The Euler-Lagrange equation for the function is For evaluating (26), we can define Multiplying both sides by we obtain, after necessary manipulations, where and any scalar of a second order symmetric tensor given in Table 1 is defined, as usual by . Using Table 1, the field equations (28) may be written in the following form: These are the field equations of the suggested theory corresponding to the Lagrangian density (25). In general, is nonsymmetric, and (30) is generally covariant and made of pure geometric objects defined in terms of the BB of the PAP-geometry. In four dimensions (30) represents 16 field equations in the 16 unknown functions (the BB of the PAP-geometry).
3.2. The Equations of Motion and Physical Meaning of
For the present theory, and in the framework of the geometrization of physics, we use the path equation (21) as an equation of motion. This equation can be used to describe the motion of a spinning test particle in the gravitational field given by the theory. In the case of motion of a charged spinning particle a modified version of (21) is to be used.
For new theories, the physical meaning for a geometric object, constant or parameter, is obtained with more ease using the equations of motion rather than using the field equations. Afterwards, this meaning is to be used for interpreting solutions of the field equations. For example, in the case of GR, the attribution of physical meaning to the metric tensor has been obtained by comparing the linearized geodesic equation with Newtonian motion. A similar procedure has been followed to get the physical meaning of in the present theory as follows.(i)In the context of geometrization philosophy, the path equations (21) have been used to represent trajectories of spinning test particles in a gravitational field .(ii)For several reasons  the parameter is suggested to take the form where is a natural number taking the values for particles with spin , respectively; is fine structure constant and is a free parameter depending on the system under consideration. In general is not a constant but a parameter depending on the spin of the moving particle and the system under consideration (the source of the gravitational field).(iii)The term on the right-hand side of (21) has been interpreted as an interaction between the spin of the moving particle and the torsion of the background field. It is called “spin-torsion interaction” term , with being the coupling of this interaction . Two conditions for this interaction to disappear are either using geometries with vanishing torsion (the case of Riemannian geometry) or the moving particle being spinless.(iv)Concerning the experimental results confirming the value and physical meaning of the parameter we caught here two experiments.(a)The COW-experiment: this experiment, carried out many times by Colella, Overhauser, and Wener [27–29] concerns quantum interference of thermal neutrons under the influence of the Earth’s gravitational field. The well known discrepancy of this experiment has been successfully interpreted  using the spin torsion interaction of (21) with for the Earth’s system.(b)The gravitomagnetism experiment: this experiment is also carried out in the Earth’s gravitational field using atomic clocks. The results of this experiment have been used to fix a value for the parameter . The results obtained confirmed that is of order unity for the Earth’s system.(v)It has been shown that reflects the existence of some quantum features for any nonsymmetric geometry [30, 31] (a geometry with nonvanishing torsion, that is, with nonsymmetric linear connection).
4. Extraction of Physics
The field equations of the theory (30) are pure geometric equations. In order to explore the physics covered by the suggested theory we are going to use three methods. In the first method, physics is gained by comparing the present theory with nonlinear field theories. More physics is obtained by using the second method, that is, comparing the theory with linear field theories. The third method is used to explore the capabilities of a geometric structure, used for applications, for representing physical fields. All three methods are used before solving the field equations of the theory.
4.1. Analysis of the Field Equations
For this analysis, we split the field equations into two parts. The symmetric part is to be compared with GR, while the skew symmetric part is to be compared with the electromagnetic sector of Einstein-Maxwell theory.
4.1.1. The Symmetric Part of the Field Equations
The symmetric part of the field equations (30) is Using (30), the above equations may be written in the form where the left-hand side is Einstein tensor and right-hand side is a second order pure geometric tensor defined by All second order tensors written on the right-hand side of (34) are defined in Table 1.
Now to gain some physical information from the field equations of the theory, we compare the symmetric part of the field equations (33) with Einstein field equations of GR, namely, We may attribute the following physical meaning, as a consequence of this comparison, to the symmetric tensors of (33).(1)The tensor can be considered as representing the gravitational potential in the suggested theory.(2)The left-hand side of (33) represents the status and evaluation of a gravitational field whose potential is given by as stated above.(3)The tensor may be used as a pure geometric representation of a material-energy distribution generating the field. Justifications of this suggestion are as follows.(a)It is a second order symmetric tensor.(b)It can be considered as a source of the gravitational field appearing on the left-hand side of (33).(c)The left-hand side of (33) satisfies Bianchi differential identity, and consequently the tensor will satisfy a generalized conservation law
Further justifications are given in the next subsections.
4.1.2. The Skew Part of the Field Equations
This part is given by Using (30), (37), the above equation may be written in the form where the right-hand side is a curl of the vector and the left-hand side is a second order, pure geometric, skew tensor defined by and all second order skew tensors, written on the right-hand side of (39), are defined in Table 1. Also, using equation (38) we can show that Using the skew tensor , we can define the following tensor density: from which we define the vector density Consequently, we get Also, we can introduce the quantities where . Considering the relations given in the present section, an apparent comparison with Einstein-Maxwell’s theory, we may attribute the following physical meaning to the geometric entities given above:(1)equations (38) (or consequently (40)) as a generalization of Maxwell’s dynamical equations,(2)the vector , as a generalization of electromagnetic potential of the electromagnetic field,(3)the tensor , as a geometric representation of the electromagnetic field strength,(4)the vector density given by (42) as a geometric representative of the current vector density,(5)the scalar , given by (44), as a geometric representative of the electromagnetic charge density,(6)the identity (43) as representing conservation of the charge density.
In general, we can take (38) as being responsible for the electromagnetic field sector of the suggested theory. Justification of the above attributions is given in the next subsections.
4.2. Weak Field Approximation
In this subsection, we give the second method used to attribute physical meaning to the geometric objects admitted by the theory. This is applied by comparing the present theory with linear field theories. In this method we assume a weak static field and a slowly moving test particle. All the abovementioned physical meanings (given in Section 4.1) can be more justified by considering weak field approximation. Although the scheme of weak fields is, in general, not covariant, it gives physical information about the geometric content of any nonlinear field theory. Let us assume that the field variables of the theory (the BB of the PAP-geometry) may be written as where is Kronecker delta, are functions of the coordinates representing deviation of the PAP-space from flat space, and is a small dimensionless parameter whose square can be neglected (weak field approximation). Then using (1) and (47) we can expand in the form
In what follows, we are concerned with the order of the parameter , not . Since the PAP-geometry has the same BB as AP-geometry, then all tensors of PAP will have the same order in as in the AP. The fundamental tensors in AP are expanded as given in . Table 2 gives the order of magnitude of connections and tensors, of the space concerned, expanded in terms of the parameter , where we write the general expansion form, of any geometric object , as where is the th order of the term.
Table 2 displays different orders of magnitude, of the parameter for each geometric object. The sign indicates that a certain order exists in the expansion formula, while the sign indicates the absence of this term.
4.2.1. The Linearized Symmetric Part of the Field Equations
This part may be written in the form or equivalently, where is the trace of and Consequently, The right-hand side of the linearized symmetric equation, (51), may be written in the following form, using (52) and (53), The left-hand side of the linearized symmetric field equations (51) can be written in the form since the field is static , and is defined as Using (54) and (55) we can write the linearized symmetric field equations (51) in the form As usually done, in order to compare the linear result (57) with Newton’s theory, we examine the case ; then since the field is static, . Also, we have , so, we may write (58) as Now define the function where is a parameter; then Substituting from (61) into (59), we get
The ultimate goal of linearizing any nonlinear field theory is to compare the theory with classical, partially successful linear field theory and also to attribute physical meaning to the geometric objects. In the present case the symmetric part of the field equations of the theory (62) is to be compared with Newton’s theory, that is, with Poisson’s equation. It seems from (62) that the tensor , in its linear form, is responsible for representing the material distribution. Let us make a one to one correspondence between the components of and those (the phenomenological matter tensor usually used in GR given in (35)). The spatial components of , for the isotropic perfect fluid, are attributed to the hydrostatic pressure , while the temporal component gives rise to the fluid density. Now, on one hand, pressure is associated with speed of the fluid particles and this is neglected since it would be of second order. On the other hand, classically speaking, pressure cannot produce gravity. So, from the one to one correspondence mentioned above and assuming that describe an isotropic perfect fluid, we can write Using this relation, then (62) can be written as Equation (64) can be considered as the geometric form of “Poisson’s equation” of Newton’s theory with representing the gravitational potential and the material source generating this potential. This justifies the physical attributions given in Section 4.1 which are (i) as the gravitational potential,(ii) as the geometric representation of the material distribution. Taking , then (64) can be written in the form This is “Laplace’s equation” giving the gravitational field in free space. This gives further justifications of the two attributions given above.
4.2.2. The Linearized Skew Part of the Field Equations
The linearized form of the skew part of the field equations (30) can be written as where
In the case of static electromagnetic field, the definition (42) gives and the field equations (66) give (note that since the field is static). Equation (69) can be written as or using (68), the above equation can be written as Now Poisson’s equation for an electric charge distribution with electric charge density and electric potential can be written as where is the dielectric constant. A comparison between (71) and (72) shows the following. (i)Equation (71) can be considered as the geometric form of Poisson’s equation.(ii) is the electric potential.(iii) is the geometric charge density.(iv)Conservation of charge density is guaranteed via (43).
It is clear that if , that is, , (71) reduces to the Laplace equation for the same problem Also, the current vector density (42) will give, using (67), The above physical attributions support those given at the end of Section 4.1.
4.3. The Type of a Certain Geometric Structure
Each field theory has its domain of applicability. A geometric field theory is constructed in the framework of a space while applications of the theory are carried out in the context of geometric structures. For example, the field theory suggested in the present work is constructed in the context of the PAP-space. The study given in the above two subsections shows that the theory suggested represents a unified field which is split into gravity and electromagnetic fields especially in the linearized form of the theory. A new feature of the theory is that it gives a geometric description of a material distribution. To support these representations several applications are needed. Such applications are to be carried out using geometric structures as stated above. A scheme is needed to explore the capabilities of a certain geometric structure to represent physical entities, in the context of the field theory suggested. As a trivial example, in the context of GR, a geometric structure with vanishing curvature cannot be used to represent gravity. The idea has been generalized in a scheme called “type analysis.” This scheme has been suggested by Mikhail and Wanas in 1981  in the context of generalized field theory (GFT) . This scheme is shown to be useful in applications (cf. [36–40]). In general it is a covariant scheme for analyzing any specific geometric PAP-structure (with certain symmetries) in the context of the theory suggested. In other words it gives the capabilities of a certain structure to represent physical entities. In Table 3, we give the physical type of the BB of any AP-structure in the context of the theory derived in the present work. The scheme depends mainly on the orders of magnitude of some tensors as given in Table 2.
It is to be considered that since the BB of the AP and the PAP-geometry are the same, the generalization of the scheme of type analysis is not difficult. The vanishing of a certain tensor in the AP-space implies the vanishing of its parameterized version in the PAP-space.
In constructing Table 3, we use the following definitions: which lead to
The type of a certain AP- (or PAP-) structure is written as a combination of the two groups of codes F, G, as will appear in the following section. Although the scheme of type analysis depends mainly on the expansion Table 2 (violating general covariance), this scheme is generally covariant. This is because it depends, as shown in Table 3, on the property of vanishing of some tensors. For example, the vanishing of Riemann-Christoffel curvature tensor indicates that the space is flat and consequently has neither matter-energy nor fields. A weak field depends on the existence (nonvanishing) of tensors with linear terms in its expansion formula. A strong field depends on the nonvanishing of tensors, describing the field, whose expansion formula starts with second order quantities and so on. In the following section we give an example to illustrate the application of this scheme.
5. The Case of Spherical Symmetry
In what follows we give an example for applying the theory suggested and for exploring some of the advantages of the scheme of type analysis in the case of spherical symmetry. Also, this example illustrates the role of the parameter () in this theory.
5.1. PAP Structure with Spherical Symmetry
Spherical symmetry represents the most simple case used to study many physical configurations. In 4-dimension, the most general AP-structure with spherical symmetry is that constructed by Robertson in 1932 , which, in spherical polar coordinates (, , , ), can be written in the following matrix: where , , and are unknown functions of only. Using relation (1), we can write the covariant components of (77) in the formThe Riemannian space defined from the PAP-structure (78) is given, using (3), by Let us first apply the scheme of type analysis, given in Section 4.3, to the tetrad (77). To get the type of this tetrad, we have to calculate the tensors listed in Table 3. In these calculations, we have used the algebraic manipulation language REDUCE 3.6. From definitions (75) and (76), we get the following nonvanishing components of the tensors necessary to get the type of the structure (77): Also, in addition to these values we get , . The values of the nonvanishing components of these tensors are not listed here, since these are too long.
Consequently, using Table 3 and the above results, the type of tetrad (77) in the context of present field theory is FIIGIV; that is, the BB (77) are capable of describing a unified strong electromagnetic and gravitational field within a material distribution.
In the present section, we aim to give an example, as simple as possible, to support the following points:(i)the physical meaning attributed to some geometric object given in Sections 4.1 and 4.2,(ii)the role of the scheme of type analysis in simplifying the problem,(iii)the present theory implying all the advantages of GR in similar cases. Further simplification will make the above points more clear. This is done in the following two subsections.
5.2. The Case of Pure Gravity
For simplicity, we consider pure gravity in the present subsection. If we take the type becomes F0GIV, because of the vanishing of all skew tensors of Table 1, identically, which leads to while This means that, with , the BB (77) can be used to study a strong gravitational field within material distribution. This shows the importance of type analysis. It represents a way for gaining some physical meaning of the geometric objects of the field equations before solving such equations. After taking (i.e. type F0GIV), the solution of the field equations will give rise to interior gravitational field, free from electromagnetism. The tetrad vector field (77) for the case may be written in the form and also The metric of the Riemannian space defined using the PAP-space (84) may be written, using (3), as According to tetrad vector field (84) and using tensors definition in Table 1, the field equations (30) will give rise to the following set of differential equations in the unknown functions , :