Theory of Nonlocal Point Transformations in General Relativity
A discussion of the functional setting customarily adopted in General Relativity (GR) is proposed. This is based on the introduction of the notion of nonlocal point transformations (NLPTs). While allowing the extension of the traditional concept of GR-reference frame, NLPTs are important because they permit the explicit determination of the map between intrinsically different and generally curved space-times expressed in arbitrary coordinate systems. For this purpose in the paper the mathematical foundations of NLPT-theory are laid down and basic physical implications are considered. In particular, explicit applications of the theory are proposed, which concern a solution to the so-called Einstein teleparallel problem in the framework of NLPT-theory; the determination of the tensor transformation laws holding for the acceleration 4-tensor with respect to the group of NLPTs and the identification of NLPT-acceleration effects, namely, the relationship established via general NLPT between particle 4-acceleration tensors existing in different curved space-times; the construction of the nonlocal transformation law connecting different diagonal metric tensors solution to the Einstein field equations; and the diagonalization of nondiagonal metric tensors.
The investigation carried out in this paper concerns basic theoretical issues and physical problems of critical importance in the classical field theory of gravity, that is, General Relativity (GR), as well as for both classical and quantum relativistic theories. Thus, while leaving the axiomatic framework of the Standard Formulation to General Relativity (SF-GR) unchanged which is based on the Einstein field equations, a new approach to SF-GR is proposed. This is obtained by introducing a family of nonlocal point transformations (NLPTs) which act between suitable sets of space-times and are referred to here as NLPT-theory. This concerns the extension of the customary functional setting which lies at the basis of SF-GR, which is realized by the notion of local point transformations (LPTs) and their inverse :which connect arbitrary GR-reference frames. In SF-GR the group (LPT-group) of these transformations is associated with in principle arbitrary possible parametrizations, that is, 4-dimensional curvilinear coordinate systems, of the physical space-time, the latter being identified with a 4-dimensional connected and time-oriented real metric space , with . This determines for each parametrization a unique representation of the space-time metric tensor [1–7].
Hence, by definition, the group leaves invariant , which must therefore be identified with a differential manifold. It is obvious that such a functional setting is intrinsic to SF-GR; that is, it is actually required for the validity of SF-GR itself. The same transformations defined by (1) are assumed also to warrant the global validity of the so-called Einstein General Covariance Principle (GCP) ; namely, they must be endowed with a suitable functional setting, referred to here as LPT-functional setting (see related discussion in Section 2), which permits in turn also the corresponding realization of GCP. Such a principle is therefore referred to as LPT-GCP. In particular, this means that LPT must be smoothly differentiable so as to uniquely and globally prescribe also the 4-tensor transformation laws of the displacement 4-vectors; namely,Here, and denote the direct and inverse Jacobian matrices which take the so-called gradient form; that is, which uniquely globally prescribe also the corresponding 4-tensor transformation laws of all tensor fields which characterize SF-GR.
However, in this work we intend to show that—based on compelling physical considerations—an alternative approach to GR based on NLPT-theory actually exists, which involves a departure from the standard route adopted in SF-GR. This is founded on the introduction of an extended functional setting, referred to here as NLPT-functional setting, which maps in each other intrinsically different space-times and , that is, space-times which cannot be otherwise connected by means of the group .
Background and Physical Motivations. An ongoing subject of theoretical investigations in GR concerns its possible nonlocal modifications. Recent literature investigations of this type are several. Examples can be found, for instance, in [9–14], where nonlocal generalizations of the Einstein theory of gravitation have been proposed. Such a kind of nonlocal GR model leads typically to suitably modified forms of the Einstein equation  in which nonlocal field interactions are accounted for, by analogy with corresponding nonlocal features of the electromagnetic field occurring in classical electrodynamics.
It is well-know that the LPT-functional setting characteristic of the original Einstein formulation of GR is uniquely founded on the classical theory of tensor calculus on manifolds. The historical foundations of the latter, in turn, date back to the so-called absolute differential calculus developed at the end of the 19th century by Gregorio Ricci-Curbastro and later popularized by his former student and collaborator Tullio Levi-Civita [2, 4]. However, a basic issue that arises in GR and more generally in classical and quantum mechanics as well as in the theory of classical and quantum fields is whether these theories themselves might exhibit possible contradictions with the validity of the LPT-GCP and consequently a more general functional setting should be actually adopted for the treatment of these disciplines.
To better elucidate the scope and potential physical relevance of the topics indicated above, it is worth highlighting some of the main related physical issues which are relevant for the present investigation and whose solution, as explained below, appears of critical importance in GR. These include the following:(1)Problem #1: Teleparallel Approach to GR. An example of violation of LPT-GCP occurs in the framework of the Einstein teleparallel approach to GR (see ) and possibly also in some of its recently proposed generalizations [16–18]. Indeed, such a theory is intended to map intrinsically different space-times. In the case of teleparallelism one of such space-times is identified, by construction, with the flat time-oriented Minkowski space-time. As discussed below (see Section 3), this is achieved by a suitable matrix transformation between the corresponding metric tensors, denoted as teleparallel transformation problem (TT-problem), which lies at the basis of such an approach (see (17) or equivalently (18)). A number of related issues arise which concern in particular the following:(i)Problem #P11. It is the realization and possible nonuniqueness feature of the mapping to be established between the two space-times occurring in the teleparallel transformation itself. This refers in particular to what might/should be the actual representation of the corresponding coordinate transformations, the prescription of possible nonlocal dependence, with particular reference to 4-velocity dependence, and the relationship between local and nonlocal coordinate transformations.(ii)Problem #P12. It is the fact that obviously such problems, and the TT-problem itself, cannot be solved in the framework of the validity of the LPT-GCP.(iii)Problem #P13. It is the physical implications of the theory, with particular reference to the explicit construction of special NLPT.(iv)Problem #P14. It is the possible existence/nonexistence of corresponding tensor transformation laws with respect to arbitrary NLPT and is referred to here as NLPT 4-tensor laws, for observable tensor fields and in particular for the metric tensors which are associated with a curved space-time and the corresponding Minkowski space-time .(2)Problem #2: Diagonalization of Metric Tensors and Complex Transformation Approaches to GR. A second notable example concerns the adoption in GR of complex-variable transformations, such as the so-called Newman-Janis algorithm [19–21]. This is used in the literature for the purpose of investigating a variety of standard or nonstandard GR black-hole solutions [22, 23], as well as alternative theories of gravitation, such as the one based on noncommutative geometry . Its basic feature is that of permitting one to transform, by means of a complex coordinate transformation, a diagonal metric tensor corresponding to a spherically symmetric and stationary configuration (like the Schwarzschild one) into a nondiagonal one corresponding to a rotating black-hole (like the Kerr solution). On the other hand, a number of issues arise concerning the Newman-Janis algorithm. These include the following:(i)Problem #P21. First, it is complex, so that the transformed coordinates are complex too. This inhibits their objective physical interpretation in terms of physical observables.(ii)Problem #P22. It is the fact that, as for the teleparallel transformation, the diagonalization problem at the basis of the same transformation cannot be solved in the framework of the validity of the LPT-GCP. Indeed, the Newman-Janis algorithm seems worth mentioning especially in view of the fact that it obviously represents a patent violation of the LPT-GCP.(iii)Problem #P23. The physical meaning of the transformation: one cannot ignore that fact that there is no clear understanding regarding its physical interpretation and ultimately as to why the algorithm should actually work at all.(iv)Problem #P24. Finally, despite the obvious fact that the teleparallel transformation provides in principle also a solution to the diagonalization problem, there is no clear connection emerging between the same transformation and the Newman-Janis algorithm.(3)Problem #3: Acceleration Effects in Relativistic Classical Electrodynamics. A third issue worth pointing out for its potential relevance in the present discussion concerns the role of acceleration on GR-reference frames as discussed, for example, in [25, 26]. These papers deal with the necessity of taking into account, in the context of both GR and Maxwell’s equations, possible acceleration-induced nonlocal effects. However, the precise mathematical formulation and physical mechanisms by which nonlocality should manifest itself must still be fully understood. In fact, a number of basic issues remain unanswered. These concern in particular the following ones:(i)Problem #P31. First, the precise prescription of the mathematical setting of the theory and in particular the implementation and possible functional realization of the nonlocal acceleration effects in the context of GR remain unclear.(ii)Problem #P32. Indeed, nonlocal acceleration effects are introduced by postulating directly “ad hoc” integral representations (or “transformation laws”) for appropriate tensor fields.(iii)Problem #P33. The validity of these transformation laws, namely, the reason why ultimately they should apply, and consequently their physical interpretation remain both unclear.(4)Problem #4: Nonlocal Effects in Classical Electrodynamics. A further intriguing example which is by itself sufficient to demonstrate the role of nonlocality in physics can be found in the framework of a special-relativistic treatment of classical electrodynamics. This concerns the so-called electromagnetic radiation-reaction (EM-RR) problem, that is, the dynamics of an extended charge in the presence of its self-generated EM field. As shown in [27, 28] such a problem can be rigorously treated in the framework of a first-principle approach based on the Hamilton variational principle. In such a context the source of nonlocality appears at once as being due to the finite size of charged particles. Indeed, its physical origin is related to the retarded EM interaction of the extended particle with itself [29–33]. However, further fundamental physical issues emerge which should be answered:(i)Problem #P41. First, the precise prescription of the transformation laws with respect to the group on NLPT should be achieved for the EM 4-potential and of the corresponding EM Faraday tensor .(ii)Problem #P42. Second, it remains to be ascertained whether the transformations indicated above are realized by means of 4-tensor NLPT-transformation laws, that is, in particular for , transformation laws formally identical to those determined by the 4-position infinitesimal displacement or the dyadic tensor
The key question which needs to be ascertained in the context of GR is whether these problems do actually require, as anticipated above, the introduction of a more general class of GR-reference frames. In fact, despite previous solution attempts [25, 26], a basic issue which still remains unsolved nowadays concerns the construction of the explicit general form and physically admissible realizations which the transformations occurring among arbitrary GR-frames should take. The problem matter refers therefore to possible nonlocal generalization of the customary local tensor calculus and coordinate transformations to be adopted in GR. This is actually the task which we intend to undertake in the present investigation.
Under such premises it must be noted that the present work departs, while being at the same time also in some sense complementary, from the nonlocal GR theories indicated above. In fact it belongs to the class of studies aimed at introducing in the context of GR a new type of nonlocal phenomenon based on the coordinate transformations established between GR-reference frames and at the same time extending the functional setting customarily adopted in such a context.
Goals and Structure of the Paper. The work-plan of the investigation is to address the problem of the nonlocal generalization of GR achieved by a suitable extension of its functional setting. This task concerns basic theoretical issues and unsolved physical problems whose solution presented in this investigation for the first time appears of critical importance in General Relativity (GR). In detail these include the following:(1) Goal #1. It is the identification of possible generalizations of the LPT-setting customarily adopted in GR, based on physical example-cases. A notable problem of this type is realized by Einstein’s approach to the so-called Einstein teleparallelism. The issue arises whether such a theory can be recovered from SF-GR by means of a suitable mathematical, that is, purely conceptual, viewpoint. This involves the introduction of appropriate nonlocal point transformations (or NLPTs). It must be stressed that the possible prescription of NLPT is by no means “a priori” obvious since they remain—it must be stressed—largely arbitrary and intrinsically nonunique. For this purpose Problems #P1–#P1 are addressed in Sections 2–5. Their solution is crucial for their identification. This goal can be reached based on the adoption of a suitable subset of NLPTs, referred to here as special NLPT-group acting on appropriate extended GR-frames which are defined with respect to prescribed space-times. For definiteness, in view of warranting the validity of suitable NLPT 4-tensor laws for the metric tensor which is associated with the teleparallel transformation (see (41) below), in the present treatment these transformations are assumed to preserve the line element (see Section 4 below); in other words they are required to map space-times and having the same line elements and .(2) Goal #2. In this context Problems #P2–#P2 are addressed. For such a purpose the determination is done of the group of general nonlocal point transformations (general NLPTs) connecting subsets of two generic curved space-times and This is referred to here as general NLPT-group (Section 6). The task posed here involves also their physical interpretation (Section 7). As an illustration of the theory, the explicit construction of possible physically relevant transformations of the group are addressed, with special reference to the problem of the NLPT between diagonal metric tensors (Section 8) and the diagonalization of metric tensors in GR (Section 9).(3) Goal #3. It is the investigation of physical implications of the general NLPT-functional setting in reference to the identification of possible acceleration effects both in GR and in classical electrodynamics. The goal of Sections 10 and 11 is to look for a possible solution to Problems #P3–#P3 and Problems #P4-#P4 indicated above as well as to point out relevant possible realizations of general NLPT. This involves in particular the investigation of the role of acceleration on GR-reference frames and the search of NLPT 4-tensor laws occurring, respectively, for the acceleration 4-tensor and the EM 4-vector potential, with respect to the group of NLPT established between suitable subsets of two arbitrary curved space-times and .
2. The LPT-Functional Setting
We first recall the functional setting which—as anticipated above—is usually adopted both in relativistic theories and in Einstein’s 1915 theory of gravitation , that is, SF-GR itself. In both cases the goal is, in principle, to predict all physically relevant realizations of the observables. In the case of GR these concern the physical space-time itself . As is well-known, in SF-GR this is identified with a 4-dimensional Lorentzian metric space on which is endowed with a prescribed metric tensor when the same set is represented in terms of a given set of curvilinear coordinates . Nevertheless, validity of GR and in particular of the Einstein equation itself requires couching them in a suitable mathematical framework.
As recently pointed out in  in the context of a variational treatment of SF-GR, this involves, besides the fulfillment of a suitable property of gauge invariance, also the adoption of Classical Tensor Analysis on Manifolds. In other words both GR and the same Einstein equation should embody by construction the validity of LPT-GCP, namely, formulated consistent with the so-called LPT-functional setting. More precisely, this means explicitly that the following mathematical requirements (A–C) should apply:(A)All physically observable tensor fields defined on space-time must be realized by means of 4-tensor fields with respect to a suitable ensemble of coordinate transformations connecting in principle arbitrary, but suitably related, 4-dimensional curvilinear coordinate systems, referred to as GR-reference frames, and .(B)The PDEs, together with their corresponding variational principles, which characterize all classical and quantum physical laws should satisfy the criterion of manifest covariance, whereby it should be possible to cast them in all their realizations in manifest 4-tensor form.(C)The set of coordinate transformations indicated above is identified with the group of transformations that in Eulerian form are prescribed by means of the invertible maps (1) which identify the group . For this purpose, suitable restrictions must be placed on the admissible GR-reference frames, that is, coordinate systems, prescribed by means of (1) which are realized by the following requirements:(i) LPT-Requirement #1. For the validity of GCP, the two space-times must coincide and be transformed into one another by means of LPT; that is, , so as to define a single -differentiable Lorentzian manifold with , that is, have either signature or analogous permutations.(ii) LPT-Requirement #2. These transformations must be assumed as purely local, so that in (1) and must depend only locally, respectively, on and . In other words, the local values and are required to be mutually mapped in each other by means of the same equations, with (resp., ) being a function of (and similarly ) only.(iii) LPT-Requirement #3. The coordinates and must realize physical observables and hence be prescribed in terms of real variables, while the functions relating them ( and must be suitably smooth in the sense that they are of class , with . This means that must realize a -differentiable Lorentzian manifold with .(iv) LPT-Requirement #4. Equations (1) generate the corresponding 4-vector transformation equations for the contravariant components of the displacement 4-vectors and (see (2)). Analogous transformation laws follow, of course, for the covariant components of the displacements; namely, . In view of (1), by construction and are considered, respectively, local functions of and only and must necessarily coincide with the gradient forms (3)-(4). Nevertheless, since and are mutually related being inverse matrices of each other and the point transformations are purely local, it follows that in view of (3) and (4) they can also both formally be regarded as functions, respectively, of the variables and .(v) LPT-Requirement #5. In terms of the Jacobian matrix and its inverse the fundamental LPT 4-tensor transformation laws for the group are set for arbitrary tensors. Consider, for example, the Riemann curvature tensor . In terms of an arbitrary LPT it obeys the 4-tensor transformation law The same transformation law also requires that 4-scalars must be left unchanged under the action of the group . Thus, by construction the 4-scalar proper-time element , that is, the Riemann-distance defined in terms of the equation , must satisfy the transformation law which can be equivalently expressed as Furthermore, the covariant and contravariant components of the metric tensor, that is, and and, respectively, and , must satisfy, respectively, the LPT 4-tensor transformation laws so that the validity of the scalar transformation laws (6) and (7) is warranted.(vi) LPT-Requirement #6. Introducing the corresponding Lagrangian form of the same equations, obtained by parametrizing both and in terms of suitably smooth time-like world-lines and , (1) take the equivalent form whereby the displacement 4-vectors and can be viewed as occurring during the proper-time . Then it follows that (10) imply also suitable transformation laws for the 4-velocities and , which by definition span the tangent space The latter are provided by the equations implying the simultaneous validity of the mass-shell constraints Notice that here also the Jacobian and its inverse must be considered as -dependent (but just only through and , resp.), that is, of the form(vii) LPT-Requirement #7. Finally, in terms of (10) and (11) one notices that LPT can be formally represented in terms of Lagrangian phase-space transformations of the type (LPT-phase-space map), with the vectors and to be viewed as representing the phase-space states, endowed by 4-positions and , respectively, and corresponding 4-velocities and . Hence, by construction transformation (15) warrants the scalar and tensor transformation laws (6) and (8) and preserves the structure of the space-time .
This concludes the prescription of the LPT-functional setting required for the validity of GCP. The set of assumptions represented by LPT-Requirements #1–#7 will be referred to here as LPT-theory.
It must be stressed that its adoption is of paramount importance in the context of GR and in particular for the subsequent considerations regarding the physical interpretations of Einstein teleparallelism. This happens at least for the following three main motivations. The first one is that, in validity of the LPT-requirements #1–#6, and in particular the gradient form requirement (3)-(4) for the Jacobian matrix, (11) are equivalent to the Eulerian equations (1) (and of course also to the corresponding Lagrangian equations (10)). Hence, both equations actually allow one to identify uniquely the group (Proposition #1).
The second one concerns the very notion of particular solution to be adopted in the context of GR for the Einstein equation. In fact, if denotes a parametrized-solution to the same equation obtained with respect to a GR-frame , the notion of particular solution for the same equation is actually peculiar. Indeed, it must necessarily coincide with the whole equivalence class of parametrized-solutions, represented symbolically as , which are mapped in each other by means of an arbitrary LPT of the group . Such a property, which is actually a consequence of GCP (and consequently of Classical Tensor Analysis on Manifolds), is usually being referred to in GR as the so-called principle of frame’s (or observer’s) independence (Proposition #2).
The third motivation concerns the very notion of curved space-time , compared to that of the Minkowski flat space-time , which when expressed in orthogonal Cartesian coordinates has the metric tensor . A generic space-time of this type is characterized, by definition, by a nonvanishing Riemann curvature 4-tensor . As a consequence of the 4-tensor transformation laws (8)-(9) it follows that two generic space-times and can be mapped in each other by means of LPTs and hence actually coincide, only provided the respective metric tensors, and hence also the corresponding Riemann curvature 4-tensors, are transformed into each other via the same equations (8)-(9). Hence, it is obvious that a generic curved space-time cannot be mapped into the said Minkowski space-time purely by means of LPT (Proposition #3).
3. Einstein’s Teleparallel Transformation Problem
Most of the historical developments achieved so far in GR since its original appearance in 1915 have been obtained in the framework of the GCP-setting of GR . Nonetheless for a long time the issue has been debated whether Relativistic Classical Mechanics and Relativistic Classical theory of fields might exhibit in each case (possibly different) nonlocal phenomena. In the literature there are several examples of studies aimed at extending in the context of GR the classical notions of local dynamics and local field interactions. A related question is, however, whether there actually exist additional nonlocal phenomena which might escape the validity of GCP and require the setup of a proper theoretical framework for their study.
As we intend to show, an instance of this type arises in the context of the so-called teleparallel approach to GR, also known as Einstein teleparallelism  (see also [16–18]). To state the issue in the appropriate physical context let us briefly highlight the basic ideas behind such an approach. This is based on the conjecture on Einstein part that at each point of the space-time manifold the corresponding tangent space can be “parallelized.” This means, in other words, that at all 4-positions it should be possible to cast each tangent 4-vector in the formwith being an invertible matrix with inverse . More precisely, according to Einstein’s approach the metric tensor of a generic curved space-time should satisfy an equation in the formor equivalentlywith being here the metric tensor associated with the flat Minkowski space-time having the Lorentzian signature . The goal is therefore to determine the mapknown as the teleparallel transformation (TT), while (17) (or equivalently (18)) will be referred to as the TT-problem. For definiteness, it must be stressed here what appears to be Einstein’s key assumption underlying these equations: it is understood in fact that in (17) and (18) manifestly identifies the metric tensor of the Minkowski space-time when expressed in terms of orthogonal Cartesian coordinates. On the other hand it is also understood that (17) and (18) should include the identity transformation among their possible solutions. This means that for consistency can always be identified with the metric tensor of the curved space-time when expressed as a local function of the same Cartesian coordinates. In the present paper such a viewpoint will be consistently adopted in the subsequent considerations to be developed below.
The following additional remarks must also be made regarding the TT-problem. The first one concerns the interpretation of (18) in the so-called tetrad formalism. It implies, in fact, that for the fields , and can simply be interpreted as a tetrad basis, that is, a set of four independent real 4-vector fields that are mutually orthogonal, that is, such that for Also, all basis 4-vectors are unitary, in the sense that, for all , , one of them () being time-like and the others being space-like; namely,together span the 4D tangent space at each point in the space-time .
The second remark is about the choice of the curved space-time in the TT-problem. It must be stressed, in fact, that the space-time should remain in principle arbitrary. Therefore, it should always be possible to identify with the curved space-time having signature different from that of the Minkowski space-time. Therefore, the solution to the TT-problem should be possible also in the case in which and have different signatures.
The third remark is about the ultimate goal of Einstein teleparallelism. This emerges perspicuously from (17) (or equivalently its inverse represented by (18)). The determination of the matrix solution to such an equation will be referred to here as TT-problem. In fact, (17) (i.e., if a solution exists to such an equation) should permit one to relate curved and flat space-time metric tensors, respectively, identified with and .
From these premises, therefore, the fundamental problem of establishing a map between the generic curved space-time indicated above and the Minkowski space-time emerges, which should have a global validity; namely, it should hold in the whole or at least in a finite subset of the same space-time. However, such a kind of transformation cannot be realized by means of LPT of type (1) in which is identified with the corresponding Jacobian (see (3) below). This happens because the teleparallel transformation cannot be realized by means of the group of LPTs (see also the related Proposition #3 indicated above). The issue arises whether in the context of GR the teleparallel transformation (17) (and equivalently its inverse, i.e., (18)) might actually still apply in the case of a more general type of nonlocal point transformations, with the matrix to be identified with a corresponding suitably prescribed Jacobian matrix.
The existence of such a class of generalized GR-reference frames and coordinate systems is actually suggested by the Einstein equivalence principle (EEP) itself. This is expressed by two separate propositions, which in the form presently known must both be ascribed to Albert Einstein’s 1907 original formulation  (see also ). The part of EEP which is mostly relevant for the current discussion is the one usually referred to as the so-called weak equivalence principle (WEP). This is related, in fact, to the fundamental notion of equivalence between gravitational and inertial mass as well as to Albert Einstein’s observation that the gravitational “force” as experienced locally while standing on a massive body is actually the same as the pseudoforce experienced by an observer in a noninertial (accelerated) frame of reference. Apparently there is no unique formulation of WEP to be found in the literature. However, the form of WEP which is of key importance in the following consists in the two distinct claims by Einstein stating (a) the equivalence between accelerating frames and the occurrence of gravitational fields (see also ) and (b) the fact that “local effects of motion in a curved space (gravitation)” should be considered as “indistinguishable from those of an accelerated observer in flat space” [34, 35]. Incidentally, it must be stressed that statement (b) is the basis of Einstein’s 1928 paper on teleparallelism.
From a historical perspective, the original introduction of WEP (and EEP) on the part of Albert Einstein was later instrumental for the development of GR. An interesting question concerns the conditions of validity of GCP and the choice of the class of LPTs to which WEP applies. In fact, based on the discussion above, the issue is whether it is possible to extend in such a framework the class of LPTs. In particular, here we intend to look for a more general group of point transformations, to be identified with NLPT. These are distinguished from the class introduced above and form a group of transformations denoted here as special NLPT-group . This new type of transformation connects two accelerating frames, namely, curvilinear coordinate systems mutually related by means of suitable acceleration-dependent and necessarily nonlocal coordinate transformations. The latter should permit one to connect globally two suitable subsets of Lorentzian spaces which realize accessible domains (in the sense indicated below) and are endowed with different metric tensors having intrinsically different Riemann tensors. Therefore, these transformations should have the property of being globally defined and, together with the corresponding inverse transformations, be, respectively, endowed with Jacobians and .
We intend to show that provided suitable “ad hoc” restrictions are set on the class of manifolds among which NLPTs are going to be established, a nontrivial generalization of GR by means of the general NLPT-group can be achieved. These will be shown to be realized in terms of a suitably prescribed diffeomorphism between 4-dimensional Lorentzian space-times and of the general formwith inverse transformationHere the squared brackets and denote possible suitable nonlocal dependence in terms of the 4-positions and and corresponding 4-velocities and , respectively. As a consequence, (22)-(23) identify a new kind of point transformation, which unlike LPTs (see (1)) is established between intrinsically different manifolds and , that is, which cannot be mapped in each other purely by means of LPTs.
4. Solution to the TT-Problem: The NLPT-Functional Setting
Let us now pose the problem of constructing explicitly the new type of point transformation, that is, the NLPT, which are involved in the representation problem of teleparallel gravity and identifying, in the process, the corresponding NLPT-functional setting.
For this purpose we introduce first the conjecture that, consistent with EEP, it should be possible to generate such a transformation introducing a suitable 4-velocity transformation which connects appropriate sets of GR-reference frames belonging to the two space-times indicated above. Indeed, the possibility of constructing “ad hoc” 4-velocity transformations which are not reducible to LPTs of type (1) is physically conceivable. To show how this task can be achieved in practice, we notice that the transformation laws for the 4-velocity which are realized, by assumption, by (16) necessarily imply the validity of corresponding transformation equations for the displacement 4-vectors and These read manifestlywhere for generality and are considered of the forms and . By analogy with (14), when evaluated along the corresponding world-lines, it follows that they take the general functional formwith and being now smooth functions of through the variables and . More precisely, by analogy to the LPT-requirements recalled above, the following prescriptions can be invoked to determine the NLPT-functional setting:(i) NLPT-Requirement #1. The coordinates and realize by assumption physical observables and hence are prescribed in terms of real variables, while and must both realize -differentiable Lorentzian manifolds, with (ii) NLPT-Requirement #2. The matrices and are assumed to be locally smoothly dependent only on 4-position, while admitting at the same time also possible nonlocal dependence. More precisely, in the case of the Jacobian the second variable which enters the same function can contain in general both local and nonlocal implicit dependence, the former one in terms of . Similar considerations apply to the inverse matrix , which, besides local explicit and implicit dependence in terms of , may generally include additional nonlocal dependence through the variable .(iii) NLPT-Requirement #3. The Jacobian matrix and its inverse are assumed to be generally nongradient. In other words, at least in a subset of the two space-times and while elsewhere they can still recover the gradient forms (3) and (4); namely, In both cases the partial derivatives are performed with respect to the local dependence only.(iv) NLPT-Requirement #4. Introducing the (proper-time) line elements and in the two space-times and defined, respectively, according to (7) so that the Riemann-distance conservation law is set. This implies that the equation must hold.(v) NLPT-Requirement #5. Finally, we will assume that the 4-positions and spanning the corresponding space-times and are represented in terms of the same Cartesian coordinates; that is,
Let us now briefly analyze the implications of these requirements. First, (24) (or equivalently (16)) can be integrated at once performing the integration along suitably smooth time- (or space-) like world-lines and :where the initial condition is set:Transformations (34) will be referred to as special NLPT in Lagrangian form, the family of such transformations identifying the special NLPT-group , that is, a suitable subset of the group of general NLPT-group . The subsets of two space-times and which are mapped in each other by a special NLPT, both assumed to have nonvanishing measure, will be referred to as accessible subdomains.
Notice that the Jacobians and remain still in principle arbitrary. In particular, in case they take the gradient forms (28) the Lagrangian LPT defined by (10) is manifestly recovered. Furthermore, (16) or equivalently (34) can be also represented in terms of the equations for the infinitesimal 4-displacements, given by (24). In particular, assuming the matrix to be continuously connected to the identity implies that the Jacobian matrix and its inverse can always be represented in the formwith and being suitable transformation matrices, which are mutually related by matrix inversion. Hence, in terms of (36)-(37), the special NLPT in Lagrangian form (34) yields then the corresponding Lagrangian and Eulerian forms:We stress that, in difference with the treatment of LPT, in the proper-time integral on the rhs of (34) and (38) the tangent-space curve (resp., ) must be considered as an independent variable. This is a peculiar feature of (34) which cannot be avoided. The reason lies in the fact that there is no way by which (and ) can be uniquely prescribed by means of the same equations. Indeed, (34) (or equivalently (38) and (39)) together with (16) truly establish a phase-space transformation of the following form:This will be referred to as NLPT-phase space map. The latter applies to a new type of reference frame, denoted as extended GR-frames, which are represented by the vectors and , respectively. These can be viewed as phase-space states (of the corresponding extended GR-frames) having, respectively, 4-positions and and 4-velocities and Finally, let us mention that transformation (40), in contrast with (15), obviously does not preserve the structure of the space-times and Nevertheless the scalar transformation law (6) is still by construction warranted, while at the same time the metric tensor satisfies by construction the TT-problem, that is, (17).
Let us now show how the matrices and can be explicitly determined in terms of the teleparallel transformation (17). The relevant results, which actually prescribe the general form of related NLPT, are summarized by the following proposition.
Theorem 1 (realization of the special NLPT-group for the TT-problem). Let one assumes that and identify, respectively, a generic curved space-time and the Minkowski space-time both parametrized in terms of orthogonal Cartesian coordinates (32) and (33).
Then, given validity of the NLPT-Requirements #1–#5, the following propositions hold:In the accessible subdomain of the teleparallel transformation (17) (or equivalently its inverse, i.e., (18)), relating with the Minkowski space-time , is realized by a nonlocal point transformation of type (34) or equivalently (38) and (39), with a Jacobian and its inverse being of forms (25) and (26), respectively. This is required to satisfy the NLPT 4-tensor laws prescribed by the matrix equation and similarly its inverse (see (18)) where identifies a prescribed symmetric metric tensor associated with the space-time , by assumption expressed in the Cartesian coordinates (32). Hence, necessarily coincides with the Jacobian matrix of the TT-problem (see (17)).The set of special NLPTs has the structure of a group.
Proof. Let us prove proposition (). For this purpose it is sufficient to construct explicitly a possible, that is, nonunique, realization of the NLPT and the corresponding set , satisfying (41). In fact, let us consider the equation for the infinitesimal 4-displacement (see (24)), which in validity of (37) becomesand similarlywhere the matrices and are suitably related. Substituting on the rhs of the last equation and invoking the independence of the components of the infinitesimal displacement , this means for consistency that the covariant components of the metric tensor, that is, and, respectively, , must satisfy the NLPT 4-tensor law (41). Such a tensor equation delivers, therefore, a set of 10 algebraic equations. Their solution can be determined in a straightforward way for the 16 components of the matrix . For example, one of these equations readsThe remaining equations following from (41) are not reported here for brevity.
One can nevertheless show that the solution to this set is nonunique. In fact, due to the freedom in the choice of the matrix elements of , the latter can in principle be chosen arbitrarily by suitably prescribing appropriate components of the same matrix. A particular solution is obtained, for example, by requiring validity of the constraint equationsThe surviving components of are then determined by the same algebraic equations of the set (41). From these considerations it follows that necessarily it must be . In particular, here we notice that all diagonal components for can be viewed as determined, up to an arbitrary sign, by the diagonal components of the metric tensor . Instead, the remaining nondiagonal matrix elements are then prescribed in terms of the nondiagonal components of the metric tensor, which follow analogously from the corresponding 6 equations of the set. Then, both the 4-displacement transformations (42) and their inverse ones (43) exist and can be nonuniquely prescribed. An example of possible realization is given bywith determinantto be assumed as nonvanishing, and with inverse transformationIn particular, from (48) one can easily evaluate in terms of the precise expression taken by the matrix . Hence one finds that necessarily , with being now considered as prescribed by means of the NLPT (38). Finally, the corresponding finite NLPT generated by (55) and (48) can always be equivalently represented in terms of (34).
Next, dropping the assumption of validity of (45), let us prove proposition (). For this purpose let us consider the two special NLPTswhich map the space-times (for ) onto and where by construction the Jacobians for admit the inverse matrices Requiring that both the corresponding admissible subsets of and their intersection have a nonvanishing measure the product of two special NLPTs is defined on such a set. Its Jacobian iswith . It follows that in such a circumstance the product of the two special NLPTs belongs necessarily to the same set , which is therefore a group.
Theorem 1 provides the formal solution to Einstein’s TT-problem in the framework of the theory of NLPT. This is achieved by means of the introduction of a nonlocal phase-space transformation of type (15), which is realized by means of a special NLPT (34) and the corresponding 4-velocity transformation law (16). In this reference the following comments must be mentioned:(i)First, the NLPT-functional setting has been prescribed in terms of the special NLPT-group , determined here by (34) together with the NLPT-Requirements #1–#5.(ii)Due to the nonuniqueness of the matrix solution to the TT-problem (see (41)) and of the related matrix , the realization of the NLPT-transformation (55) [and hence (48)] yielding the solution to the TT-problem is manifestly nonunique too. For a prescribed curved space-time which is parametrized in terms of the Cartesian coordinates, the ensemble of NLPT which provide particular solutions to the TT-problem will be denoted as .(iii)Both for (46) and for (48) the corresponding Jacobians determined by means of (36) and (37) take by construction and consistently with (27) a manifest nongradient form. This follows immediately from Proposition #1 thanks to the validity of (41) and the requirement that is a curved space-time.(iv)In terms of the Jacobian matrix (and its inverse ) (41) means that should actually satisfy the original Einstein equations (17) and (18). The latter can be interpreted as NLPT 4-tensor laws for the metric tensor .(v)Similarly and by analogy with (6) holding in the case of LPT, the validity of the scalar transformation law (7) is warranted also in the case of NLPT, thanks to the transformation law (41).(vi)The transformation law (41) for the metric tensor can be interpreted as tensor transformation law with respect to the special NLPT-group . This will be referred to as NLPL 4-tensor transformation law. In terms of the same Jacobian matrix and its inverse , analogous NLPT 4-tensor laws can be set in principle for tensors of arbitrary order. Nevertheless, it must be noted that—specifically because of the validity of the same transformation law (41)—such a type of tensor transformation law cannot be fulfilled by the Riemann curvature tensor , the reason being that it manifestly vanishes identically in the case of the Minkowski space-time.
A further issue concerns the identification of the physical domain of existence and the actual possible realization of NLPT which are implied by Theorem 1. In this regard it is obvious that NLPT, just like LPT, can actually be defined only in the accessible subdomains of , namely, the connected subsets which in the curved space-time can be covered by time- (or space-) like world-lines which are endowed with a finite 4-velocity. Nevertheless, the components of the same 4-velocity can still be in principle arbitrarily large, so that the corresponding world-line can be arbitrarily close to light trajectories (and therefore to the light cones).
Another aspect of the existence problem for NLPT is related to the solubility conditions of the algebraic equations arising in Theorem 1, which follow from the requirement that all components of the matrix should be real. For example, in the case of (44) the corresponding condition is determined by the inequalityIt must be stressed that the validity of inequalities of this type for the remaining equations in general cannot be warranted in the whole admissible subset of the space-times , that is, in particular in the subset in which . On the other hand, “a priori” the symmetric metric tensor must be regarded in principle as completely arbitrary. Hence it is obvious that such inequalities following from Theorem 1 cannot place any “unreasonable” physical constraint on the same tensor .
In fact, consider the case in which the metric tensor has the signature and is also diagonal; namely, . Then, necessarily the metric tensor must be such that everywhere in the same admissible subset , while . As a consequence the functional class contains transformations which may not exist everywhere in the same set. In fact, some of the inequalities of the group (51) which involve the spatial components, that is, (with ), must be considered as local, that is, are subject to the condition of local validity of the same inequalities. Although NLPTs of this kind are physically admissible, the question arises whether particular solutions actually exist which are not required to fulfill the same inequalities (51). These solutions, if they actually exist, have therefore necessarily a global character; that is, they are defined everywhere in the same admissible subset of . In view of these considerations, since the only acceptable physical restriction on concerns its signature, it can be shown that global validity is warranted everywhere in provided the following two sets of constraints are required to hold:and in validity of the signature indicated above The first equations actually require these 3 independent equations to apply separately. Particular solutions to the components of satisfying the 3 constraint equations (54) and either the 3 inequalities (53) or corresponding equations obtained replacing the inequality symbol with = will be denoted, respectively, as partially unconditional or unconditional solutions. In both cases it is immediately shown that these solutions are nonunique, even if in all cases the transformation matrix is again a local function of ; that is, . In particular, here we notice that all the diagonal components for can be viewed as determined, up to an arbitrary sign, by the diagonal components of the metric tensor . Instead, the remaining nondiagonal matrix elements are then prescribed in terms of the nondiagonal components of the metric tensor, which follow analogously from the set of equations mentioned in Theorem 1. In validity of the constraints given above, that is, both for partially unconditional or for unconditional particular solutions, the 4-displacement transformations (42) becomeSimilarly, one can show that also the corresponding inverse NLPTs exist.
5. Application of Special NLPT: Diagonal Metric Tensors
As pointed out above the theory of special NLPT must in principle hold also when the space-times and have different signatures. In particular, if coincides with a flat space-time, then it might still have in principle an arbitrary signature. To clarify this important point we present in this section a sample application. For definiteness, let us consider here a curved space-time which is diagonal when expressed in terms of Cartesian coordinate. The following two possible realizations are considered:(A),(B).
In both cases here the functions are assumed to be prescribed real functions which are strictly positive for all . Since by construction the Riemannian distance is left invariant by arbitrary NLPT, it follows that in the two cases either the differential identityorrespectively, must hold. Let us point out the solutions to the TT-problem, that is, (17) or equivalently (18), in the two cases.
5.1. Solution to Case A
In validity of (56), if one adopts a special NLPT of the formin terms of (18) this delivers for diagonal matrix elements for all the equationswith the formal solutionsNotice that here only the positive algebraic roots have been retained in order to recover from (60) the identity transformation when letting . From (38) one obtains therefore the special NLPTwhere in the integrand is to be considered as an implicit function of and, as indicated above, remains still arbitrary. Thus, explicit solution to (61) can be obtained by suitably prescribing
5.2. Solution to Case B
Let us now consider the solution to the TT-problem when (57) applies. For definiteness, let us look for a special NLPT of the typeIn terms of (18) this delivers for diagonal matrix elements the equationswith the formal solutionsHence, the corresponding NLPTs in integral form are found to be in this casewhere, again, in the integrands is to be considered as an implicit function of while has to be suitably prescribed.
Cases A and B correspond, respectively, to curved space-times having the same or different signatures with respect to the Minkowski flat space-time. Therefore, based on the discussion displayed above, it is immediately concluded that NLPT which maps mutually the two space-times indicated above must necessarily exist in all cases considered here.
Physical insight on the class of special NLPTs emerges from the following two statements, represented, respectively, by the following:(i)Proposition of Theorem 1.(ii)The explicit realization obtained by the 4-velocity transformation laws (16) which follows in turn from (24).
Let us briefly analyze the first one, that is, in particular the fact that the set is endowed with the structure of a group. For this purpose, consider two arbitrary connected and time-oriented curved space-times for and assume that the corresponding admissible subsets of , on which the same space-times are mapped by means of special NLPT, have a nonempty intersection with nonvanishing measure. The corresponding Jacobian matrices are by assumption of type (36) so that their product must necessarily belong to (Proposition ()). The conclusion is of outmost importance from the physical standpoint. Indeed, it implies that by means of two special NLPTs it is possible to mutually map in each other two, in principle arbitrary, curved space-times. Therefore, the same theory can be applied in principle to the treatment of arbitrary curved space-times in terms of products of suitable special NLPT.
The validity of the second statement indicated above is also perspicuous. In fact, the prescription of the “geometry” of the transformed space-time , namely, its metric tensor and the corresponding Riemann curvature tensor , is obtained by means of a suitable nonlocal point transformation mapping the two space-times and . This involves, in turn, the prescription of suitable nonuniform (i.e., position dependent) 4-velocity transformations between the same space-times. In particular, in the case of the solution indicated above for the transformation matrix , the transformed 4-velocity has the following qualitative properties. First, its time-component, besides depending on the corresponding time-component of the Minkowski space-time, in general may carry also finite contributions which are linearly dependent on all spatial components of the Minkowskian 4-velocity. Second, the spatial components of the same 4-velocity depend linearly only on the corresponding spatial components of the Minkowskian 4-velocity and hence remain unaffected by its time-component, that is, its energy content in the Minkowski space-time.
6. Theory of General NLPT
In this section the problem is posed of the search of possible generalizations of the nonlocal point transformations (22) and (23). In the following these will be referred to as general NLPT and general NLPT-theory, respectively. More precisely, besides NLPT-Requirements #1–#5, the new transformations should embody the following additional optional features:(i)NLPT-Requirement #6. They should realize a mapping between two in principle arbitrary connected and time-oriented 4-dimensional curved space-times and .(ii)NLPT-Requirement #7. The space-times and should be possibly referring to arbitrary curvilinear coordinate systems which may differ in the two space-times. In addition, as for special NLPT we will require again that also general NLPTs establish between and suitably prescribed real diffeomorphisms of forms (22) and (23), the square brackets denoting appropriate nonlocal dependence. In particular, here , , , and , while and identify as usual the covariant derivatives defined in the two space-times and , respectively.
For definiteness, we will also assume that (22) and (23) are also consistent with the NLPT-Requirements #1–#4. It is then immediately noticed that an obvious particular realization of these transformations can be obtained simply by considering explicitly -dependent smooth real direct and inverse transformations of the typedefined for all . Again, for and transformations of types (24) and (16) are implied. However, the Jacobians are of the types and and read, respectively,thus losing their gradient form (see (39) and (50) above). Nevertheless, it is obvious that transformations of the type indicated above generally imply the violation of the Riemann-distance constraint (30) (see NLPT-Requirement #4).
On the other hand, once the implications of the same equation are properly taken into account the representation problem posed here can be readily solved. Consider in fact again (30). Due to the arbitrariness of and of and it follows that the same equation requires simultaneously thatmust hold, with denoting a suitable and still undetermined real Jacobian matrix and being its inverse. Therefore, (69) imply thatthat is, the Jacobian matrices can only be functions of or, respectively, More precisely, on the rhs of the first (second) equation () must be considered as a function of (resp., of ) determined by means of an equation analogous to that holding for special NLPT. Hence (66) must recover the form with being a suitable Jacobian matrix and being its inverse. Such transformations will be referred to as general NLPT. The corresponding phase-space map analogous to (40), namely,will be denoted as general NLPT-phase space map. Then the following result holds.
Theorem 2 (realization of the general NLPT-group ). The group of general NLPTs of type (71) can always be realized by means of Jacobians and of the formwith and being suitable transformation matrices. As a consequence, an arbitrary general NLPT can be represented as
Proof. In fact, given validity of (74) it follows, for example, thatwhere manifestly . Now we notice that it is always possible to set the initial condition so that This implies the validity of the first of (75). The proof of the second one is analogous.
Notice that, in difference with the special NLPT defined by (34), transformations (75) (or equivalently (72)) now establish a diffeomorphism between two different, connected, and time-oriented space-times and For definiteness let us consider the possible optional choices:(A1) is an arbitrary curved space-time.(A2) is an arbitrary curved space-time.(B1)the space-times and are referred to as arbitrary GR-frames.(B2)the same space-times and are referred to as different GR-frames.
Let us consider possible particular realizations of the general NLPT given above.
The first one is obtained dropping assumption (B2), that is, requiring that the GR-frames of the two space-times and coincide. In fact, if the coordinate systems for and are the same ones while still remaining arbitrary, then one obtains that the constraint equationsmust hold identically. In such a case, denoting the transformations matrices astransformations (75) recover the same form given by (38) and (39) above. These can be conveniently written aswith and identifying the nonlocal displacementsTherefore (75) in validity of (77) identify again a special NLPT belonging to the group (see also Theorem 1). From this conclusion the relationship between general and special NLPT is immediately inferred. In fact, it is obvious that for an arbitrary general NLPT the relationship existing between the Jacobians and , as well as the corresponding transformation matrices and , is simply provided by the matrix equationwith being the Jacobian of a suitable LPT.
Another interesting realization occurs when the space-time is identified with the Minkowski space-time represented in terms of general curvilinear coordinates In such a case its metric tensor is of the formwith being the corresponding Minkowski metric tensor in orthogonal Cartesian coordinates. The corresponding NLPT 4-tensor laws (70) become nowwhich are analogous to (41) (see Theorem 1). However, remarkably, the corresponding coordinate transformations become now—in difference with the special NLPT introduced above—of the general NLPT type (75).
It is interesting to stress that the same conclusions, that is, in particular equations (72), can actually be recovered following an alternative route. This is obtained by introducing suitable prescriptions on transformations (22) and (23). Consider in fact the following possible realization of the said maps: