Cosmological models based on relativity with a privileged frame
Special relativity (SR) with a privileged frame is a framework, which, like the standard relativity theory, is based on the relativity principle and the universality of the (two-way) speed of light but includes a privileged frame as an essential element. It is developed using the following first principles: (1) Anisotropy of the one-way speed of light in an inertial frame is due to its motion with respect to the privileged frame; (2) Space-time transformations between inertial frames leave the equation of anisotropic light propagation invariant; (3) A set of the transformations possesses a group structure. The Lie group theory apparatus is applied to define groups of transformations. The correspondingly modified general relativity (GR), like the standard GR, is based on the equivalence principle but with the properly modified space-time local symmetry in which an invariant combination differs from the Minkowski interval of the standard SR. That combination can be converted into the Minkowski interval by a change of space-time variables and then the complete apparatus of general relativity can be applied in the new variables. However, to calculate physical effects, an inverse transformation to the ’physical’ time and space intervals is to be used. Applying the modified GR to cosmology yields the luminosity distance – redshift relation corrected such that the observed deceleration parameter can be negative as it was derived from the data for type Ia supernovae. Thus, the observed negative values of the deceleration parameter can be explained within the matter-dominated Friedman-Robertson-Walker (FRW) cosmological model of the universe without introducing the dark energy. A number of other observations, such as Baryon Acoustic Oscillations (BAO) and Cosmic Microwave Background (CMB), that are commonly considered as supporting the late-time cosmic acceleration and the existence of dark energy, also can be well fit to the cosmological model arising from the GR based on the SR with a privileged frame.
Keywords: Special relativity, Light speed anisotropy, Lie groups of transformations, General relativity, FRW models, Late-time cosmic acceleration, Dark energy
Special relativity (SR) underpins nearly all of present day physics. The space-time symmetry of Lorentz invariance is one of the cornerstones of general relativity (GR) and other theories of fundamental physics. Nevertheless, the modern view is that, at least cosmologically, a privileged reference frame does exist. Modern cosmological models are based on the assumption of existence of a privileged frame in which the universe appears isotropic to a ”typical” freely falling observer. That typical (privileged) Lorentz frame is usually assumed to coincide with the frame in which the cosmic microwave background (CMB) temperature distribution is isotropic (’CMB frame’).
The view, that there exists a privileged frame of reference, seems to unambiguously lead to the abolishment of the basic principles of the special relativity theory: the principle of relativity and the principle of universality of the speed of light. Correspondingly, the modern versions of experimental tests of special relativity and the ”test theories” of special relativity ,  presume that a privileged inertial reference frame, identified with the CMB frame, is the only frame in which the two-way speed of light (the average speed from source to observer and back) is isotropic while it is anisotropic in relatively moving frames. Furthermore, it seems that accepting the existence of a privileged frame forces one to abandon the group structure for the set of space-time transformations between inertial frames – in the test theories, transformations between ”moving” frames are not considered, only a form of transformations between a privileged ”rest” frame and moving frames is postulated.
The initial motivation for this study was investigate fundamentals of relativity by developing a theory which incorporates the privileged frame into the framework of SR while retaining the basic principles of the theory, the relativity principle and universality of the speed of light, and also preserving the group structure of the set of transformations between inertial frames. However, after developing the theory that satisfies all those requirements, it was found that the special relativity with a privileged frame allows a straightforward extension to general relativity. Further, applying the modified general relativity to cosmology yields the luminosity distance versus redshift relation which provides an interpretation of the type Ia supernovae data differing from the common one. That relation allows negative values of the deceleration parameter in the matter-dominated Friedman-Robertson-Walker cosmological model of the universe and so it does not obligatory require introducing a dark energy. Considering those cosmological applications of the relativity with a privileged frame is the primary goal of this paper.
The main body of the paper consists of three parts. The first part is devoted to developing the special relativity, which, like the standard relativity theory, is based on the relativity principle and the universality of the (two-way) speed of light, but includes a privileged frame as an essential element. It is shown that the reconciliation and synthesis of those seemingly incompatible concepts is possible in the framework of the relativity theory. Since any one-way speeds of light, consistent with the two-way speed equal to , are acceptable, a privileged frame can be defined as the only frame in which the one-way speed of light is isotropic while it is anisotropic in any other frame moving with respect to a privileged frame. The analysis is based on invariance of the equation of anisotropic light propagation (preserving the property that the two-way speed of light is equal to ) with respect to the space-time transformations between inertial frames with the requirement that a set of the transformations possesses a group structure. The anisotropy parameter in the equation of light propagation is treated as a variable which takes part in the group transformations varying from frame to frame. In such a framework, the principle of constancy and universality of the two-way speed of light and the group property are preserved. The principle of relativity is also preserved since the privileged frame, in which the anisotropy parameter is zero, enters the analysis on equal footing with other frames – the transformations from/to that frame are not distinguished from other members of the group of transformations. However, the existence of a privileged frame is an essential element of the framework since an argument, that a size of the anisotropy in a specific frame is determined by its velocity with respect to the privileged frame, is used to specify the transformations.
At the first sight, that argument seems to be in conflict with a common view that, because of the inescapable entanglement between remote clock synchronization and one-way speed of light, the one-way speed of light is irreducibly conventional (see, e.g., – ). Nevertheless, the present paper analysis demonstrates that, in an anisotropic system, a specific value of the one-way speed of light (and the corresponding synchronization) is selected in some objective way as a measure of anisotropy – in the present context, it is the anisotropy caused by motion of a system relative to the privileged frame.
The space-time transformations between inertial frames derived as a result of the analysis differ from the Lorentz transformations. Correspondingly, the interval between two events, as distinct from the standard SR, is not invariant under the transformations but conformally transformed. In other terms, a combination, which is invariant under the transformations (a counterpart of the interval of the standard SR), differs from the Minkowski interval. In view of the fact that the theory is based on the special relativity principles, it means that the Lorentz invariance is violated without violation of relativistic invariance.
Since the local Lorentz invariance is one of the foundations of general relativity, the corresponding alterations need to be introduced into the framework of GR. The second part of the analysis is devoted to formulating a general relativity that is based on the equivalence principle but with modified space-time local symmetry in which an invariant combination differs from the Minkowski interval of the standard SR. That combination can be converted into the Minkowski interval by a change of space-time variables and then the complete apparatus of general relativity can be applied in the new variables. However, to calculate physical effects, an inverse transformation to the ’physical’ time and space intervals is to be used.
The third part of the paper is devoted to applying the modified GR to cosmological models which is a primary goal of this study. The cosmological models based on the modified GR allow an interpretation of the luminosity distance versus redshift relation for type Ia supernovae that is different from the common one. In the modern cosmology, that relation is interpreted as an indication that the present expansion of the universe is accelerated. This implies that the time evolution of the expansion rate cannot be described by a matter-dominated Friedman-Robertson-Walker cosmological model of the universe. In order to explain the discrepancy within the context of general relativity, dark energy, a new component of the energy density with strongly negative pressure that makes the universe accelerate, is introduced. In the relativity with a privileged frame, the deceleration parameter in the luminosity distance – redshift relation is corrected such that the deceleration parameter can be negative. Thus, in that framework, the observed negative values of the deceleration parameter can be explained within the Friedman model of the matter-dominated universe with no dark energy.
The late-time cosmic acceleration (and the existence of dark energy) is commonly considered to be supported by a number of other observations such as Baryon acoustic oscillations (BAO) and CMB. Nevertheless, the only observational data, that may be considered as providing a ”direct” evidence for a dark energy, is the Hubble diagram of distant supernovae. As a matter of fact, what is usually shown is that the BAO and CMB measurements can be made consistent with the supernovae observations by specifying the cosmological and dark energy parameters. The present’s paper analysis shows that both the SNIa data and the BAO results can be well fit to the model arising from the modified GR that is based on the relativity with a privileged frame. The analysis cannot be straightforwardly extended to calculating the CMB effects but it can be shown that the model is not contradictory with the data. In general, the cosmological model based on the relativity with a privileged frame can provide an alternative to the cosmology with a dark energy.
The paper is organized, as follows. In Section 2, following the Introduction, the special relativity with a privileged frame is constructed. In Section 3, an extension to the general relativity is considered. In Section 4, a cosmological model based on that extension is developed and fitting the observational data to the model is discussed. Concluding comments are furnished in Section 5. In Appendix A, the modified general relativity is applied to the astrophysical problem of collapse of a dust-like sphere. In Appendix B, some auxiliary calculations are placed.
2 Special relativity
2.1 Conceptual framework
The issue of anisotropy of the one-way speed of light is traditionally placed into the context of conventionality of distant simultaneity and clock synchronization – . Simultaneity at distant space points of an inertial system is defined by a clock synchronization that makes use of light signals. Let a pulse of light is emitted from the master clock and reflected off the remote clock. If and are respectively the times of emission and reception of the light pulse at the master clock and is the time of reflection of the pulse at the remote clock then the conventionality of simultaneity is a statement that one is free to choose the time to be anywhere between and . This freedom may be parameterized by a parameter , as follows
Any choice of corresponds to assigning different one-way speeds of light signals in each direction which must satisfy the condition that the average is equal to . Speed of light in each direction is therefore
The ”standard” (Einstein) synchronization entailing equal speeds in opposite directions corresponds to . If the described procedure is used for setting up throughout the frame of a set of clocks using signals from some master clock placed at the spatial origin, a difference in the standard and nonstandard clock synchronization may be reduced to a change of coordinates – 
where is the time setting according to Einstein (standard) synchronization procedure.
The analysis can be extended to the three dimensional case. If a beam of light propagates (along straight lines) from a starting point and through the reflection over suitable mirrors covers a closed part the experimental fact is that the speed of light as measured over closed part is always (Round-Trip Light Principle). In accordance with that experimental fact, if the speed of light is allowed to be anisotropic it must depend on the direction of propagation as , 
where is a constant vector and is the angle between the direction of propagation and . Similar to the one-dimensional case, the law (2.4) may be considered as a result of the transformation from ”standard” coordinatization of the four-dimensional space-time manifold, with , to the ”nonstandard” one with :
The freedom in the choice of synchronization has been repeatedly used in the literature to derive the transformations which are treated as replacing standard Lorentz transformations of special relativity if anisotropic one-way light speeds with are assumed – see, e.g.,  – . The derivations of those transformations (in what follows, they will be called the ”-Lorentz transformations”, the name is due to , ) are based on kinematic arguments and the requirement that, in the case of , the relations of the special relativity theory in its standard formulation were valid. The -Lorentz transformations can be equally obtained from the standard Lorentz transformations by a change of coordinates (2.3). The fact, that there can exist a variety of ”anisotropic” kinematics with different , is usually considered as supporting the view that the one-way speed of light is irreducibly conventional.
The purpose of the following discussion is to demonstrate that, in the case of an anisotropic system, that view is incorrect so that a specific value of the one-way speed of light (and the corresponding synchronization) is selected in some objective way as a measure of anisotropy. In particular, it is shown that (1) the variety of kinematics corresponding to the -Lorentz transformations, which are commonly considered as incorporating anisotropy, are in fact not applicable to an anisotropic system and (2) in the case of an isotropic system, the particular case of the transformations corresponding to the isotropic one-way speed of light and Einstein synchronization (standard Lorentz transformations) is privileged in some objective way.
The statement (1) is related to the issue of invariance of the interval. Invariance of the interval is traditionally considered as an integral part of the physics of special relativity which is used as a starting point for derivation of the space-time transformations between inertial frames. Nevertheless, invariance of the interval is not a straightforward consequence of the basic principles of the theory. The two principles constituting the conceptual basis of the special relativity, the principle of relativity, which states the equivalence of all inertial frames as regards the formulation of the laws of physics, and universality of the speed of light in inertial frames, taken together lead to the condition of invariance of the equation of light propagation with respect to the coordinate transformations between inertial frames. Thus, in general, not the invariance of the interval but invariance of the equation of light propagation should be a starting point for derivation of the transformations. Therefore, in the textbooks (see, e.g., , ), the use of the interval invariance is usually preceded by a proof of its validity based on invariance of the equation of light propagation. However, those proofs are not valid if an anisotropy is present and the same arguments lead to the conclusion that, in the presence of anisotropy, the interval is not invariant but modified by a conformal factor . The ”-Lorentz transformations”, like the standard Lorentz transformations, leave the interval invariant and therefore they are applicable only to an isotropic system.
The statement (2) relies on the correspondence principle. The correspondence principle was taken by Niels Bohr as the guiding principle to discoveries in the old quantum theory. Since then it was considered as a guideline for the selection of new theories in physical science. In the context of special relativity, the correspondence principle implies that Einstein’s theory of special relativity reduces to classical mechanics in the limit of small velocities in comparison to the speed of light. Being applied to the special relativity kinematics, the correspondence principle requires that the transformations between inertial frames should turn into the Galilean transformations in the limit of small velocities. The ”-Lorentz transformations” do not satisfy the correspondence principle unless  which means that the isotropic one-way speed of light and Einstein synchrony are selected in some objective way if no anisotropy is present in a physical system.
On the basis of the above discussion one can conclude that, in the case of an anisotropic system, there exists a privileged value of the one-way speed of light selected by the size of the anisotropy. Thus, a value of the one-way speed of light acquires meaning of a measure of a really existing anisotropy – in the present context, it is the anisotropy caused by motion of a system relative to the privileged frame.
In what follows, the special relativity kinematics applicable to an anisotropic system is developed based on the first principles of special relativity but without refereeing to the relations of the standard relativity theory. The principles constituting the conceptual basis of special relativity, the relativity principle, according to which physical laws should have the same forms in all inertial frames, and the universality of the speed of light in inertial frames, lead to the requirement of invariance of the equation of light propagation with respect to the coordinate transformations between inertial frames. In the present context, it should be invariance of the equation of propagation of light which incorporates the anisotropy of the one-way speed of light, with the law of variation of the speed with direction (2.4). The anisotropic equation of light propagation incorporating the law (2.4) has the form 
where are coordinates, is time and is a (constant) vector characteristic of the anisotropy. The change of notation, as compared with (2.4), from to is intended to indicate that is a parameter value corresponding to the size of the really existing anisotropy while defines the anisotropy in the one-way speeds of light due to the nonstandard synchrony equivalent to the coordinate change (2.5). Note that although the form (2.6) is usually attributed to the one-dimensional formulation, in the three-dimensional case, the equation has the same form if the anisotropy vector is directed along the -axis .
Further, in the development of the anisotropic relativistic kinematics, a number of other physical requirements, associativity, reciprocity and so on are to be satisfied which all are covered by the condition that the transformations between the frames form a group. Thus, the group property should be taken as another first principle. The formulation based on the invariance and group property suggests using the Lie group theory apparatus for defining groups of space-time transformations between inertial frames.
At this point, it should be clarified that there can exist two different cases: (1) The size of anisotropy does not depend on the observer motion and so is the same in all inertial frames; (2) The anisotropy is due to the observer motion with respect to a privileged frame and so the size of anisotropy varies from frame to frame. Groups of space-time transformations for the first case are studied in . The second case is relevant to the subject of the present study. In that case, the anisotropy parameter becomes a variable which takes part in the transformations so that groups of transformations in five variables are to be studied. In such a framework, the privileged frame, commonly defined by that the propagation of light in that frame is isotropic, is naturally present as the frame in which . However, it does not violate the relativity principle since the transformations from/to that frame are not distinguished from other members of the group. Nevertheless, the fact, that the anisotropy of the one-way speed of light in an arbitrary inertial frame is due to motion of that frame relative to the privileged frame, is a part of the paradigm which allows to specify the transformations.
The procedure of obtaining the transformations consists of the following steps: (1) The infinitesimal invariance condition is applied to the equation of light propagation which yields determining equations for the infinitesimal group generators; (2) The determining equations are solved to define the group generators and the correspondence principle is applied to specify the solutions; (3) Having the group generators defined the finite transformations are determined as solutions of the Lie equations; (4) The group parameter is related to physical parameters using some obvious conditions; (5) Finally, the conceptual argument, that the size of anisotropy of the one-way speed of light in an arbitrary inertial frame depends on its velocity relative to the privileged frame, is used to specify the results and place them into the context of special relativity with a privileged frame
The transformations between inertial frames derived in such a way contain a scale factor and thus do not leave the interval between two events invariant but modify it by a conformal factor (square of the scale factor). Applying the conformal invariance in physical theories originates from the papers by Bateman  and Cunningham  who discovered the form-invariance of Maxwells equations for electromagnetism with respect to conformal space-time transformations. Since then conformal symmetries have been successfully exploited for many physical systems (see, e.g., reviews , ). Transformations which conformally modify Minkowski metric have been introduced in the context of the special relativity kinematics in the presence of space anisotropy in  and  (see also ). As a matter of fact, those works are not directly related to the subject of the present study as they consider the case of a constant anisotropy degree, not dependent on the frame motion. Nevertheless, it is worthwhile to note that in the works ,  the assumption that the form of the metric changes by a conformal factor is imposed while, in the framework of the present analysis, conformal invariance of the metric arises as an intrinsic feature of special relativity based on invariance of the anisotropic equation of light propagation and the group property (see  for a more detailed discussion of the papers , ).
2.2 Space-time transformations with a varying anisotropy parameter
Consider two arbitrary inertial reference frames and in the standard configuration with the - and -axes of the two frames being parallel while the relative motion is along the common -axis. The space and time coordinates in and are denoted respectively as and . The velocity of the frame along the positive direction in , is denoted by . It is assumed that the frame moves relative to along the direction determined by the vector . This assumption is justified by that one of the frames in a set of frames with different values of is a privileged frame, in which , so that the transformations must include, as a particular case, the transformation to that privileged frame. Since the anisotropy is attributed to the fact of motion with respect to the privileged frame it is expected that the axis of anisotropy is along the direction of motion (however, the direction of the anisotropy vector can be both coinciding and opposite to that of velocity).
Transformations between the frames are derived based on the following first principles: invariance of the equation of light propagation (underlined by the relativity principle), group property and the correspondence principle. Note that the group property is used not as in the traditional analysis which commonly proceeds along the lines initiated by  and  which are based on the linearity assumption and relativity arguments. The difference can be seen from the derivation of the standard Lorentz transformations .
Invariance of the equation of light propagation. The equations for light propagation in the frames and are
where the anisotropy parameters and in the frames and are different. The relativity principle implies that the transformations of variables from to leave the form of the equation of light propagation invariant so that (2.7) is converted into (2.8) under the transformations.
Group property. The transformations between inertial frames form a one-parameter group with the group parameter (such that corresponds to ):
Remark that is a transformed variable taking part in the group transformations. Based on the symmetry arguments it is assumed that the transformations of the variables and do not involve the variables and and vice versa:
Correspondence principle. The correspondence principle requires that, in the limit of small velocities (small values of the group parameter ), the formula for transformation of the coordinate turns into that of the Galilean transformation111It should be noted that the relations , and , which are commonly included into the system of equations called the Galilean transformations, are not required to be valid in the limit of small velocities. Only the relation (2.11), which contains the first order term, provides a reliable basis for specifying the group transformations based on the correspondence principle (see more details in ).
Remark that the small limit is not influenced by the presence of anisotropy of the light propagation. It is evident that there should be no traces of light anisotropy in that limit, the issues of the light speed and its anisotropy are alien to the framework of Galilean kinematics.
The group property and the requirement of invariance of the equation of light propagation suggest applying the infinitesimal Lie technique (see, e.g., , ). The infinitesimal transformations corresponding to (2.10) are introduced, as follows
The correspondence principle is applied to specify partially the infinitesimal group generators. Equation (2.11) is used to calculate the group generator , as follows
It can be set without loss of generality since this constant can be eliminated by redefining the group parameter. Thus, the generator is defined by
Then equations (2.7) and (2.8) are used to derive determining equations for the group generators , , , and . Substituting the infinitesimal transformations (2.12), with defined by (2.14), into equation (2.8) with subsequent linearizing with respect to and using equation (2.7) to eliminate yields
where subscripts denote differentiation with respect to the corresponding variable. In view of arbitrariness of the differentials , , and, , the equality (2.2) can be valid only if the coefficients of all the monomials in (2.2) vanish which results in an overdetermined system of determining equations for the group generators.
The generators , and found from the determining equations yielded by (2.2) are
where , and are arbitrary constants. The common kinematic restrictions that one event is the spacetime origin of both frames and that the and axes slide along another can be imposed to make the constants , and vanishing (space and time shifts are eliminated). In addition, it is required that the and planes coincide at all times which results in and so excludes rotations in the plane .
The finite transformations are determined by solving the Lie equations which, after rescaling the group parameter as together with and omitting hats afterwards, take the forms
Because of the arbitrariness of , the solution of the system of equations (2.17), (2.18) and (2.19) contains an arbitrary function . Using (2.17) to replace in the second equation of (2.18) we obtain solutions of equations (2.18) subject to the initial conditions (2.20) in the form
where is defined by
To complete the derivation of the transformations the group parameter is to be related to the velocity using the condition
where is the value of calculated for given by (2.26).
Calculating the interval
Thus, the interval invariance of the standard relativity is replaced by conformal invariance with the conformal factor dependent on the relative velocity of the frames and the size of anisotropy in the frame .
Nevertheless, there exists a combination which is invariant under the transformations and can be considered as a counterpart of the interval of the standard special relativity. It is evident that the expression (2.24) for the scale factor can be represented in the form
is invariant under the transformations.
Furthermore, introducing the new variables
converts the invariant combination (2.33) into the Minkowski interval
The transformations defined by equations (2.2), (2.28) and (2.26) contain an indefinite function . The scale factor also depends on that function. In the next section, it is shown that incorporating the existence of a privileged frame into the analysis yields a formulation in which, instead of , a function , expressing dependence of the anisotropy size on the velocity of a frame with respect to the privileged frame, figures. A form of the latter function can be defined using some physical arguments which allows further specify the transformations.
2.3 Special relativity with a privileged frame
In derivation of the transformations in the previous section, nothing distinguishes a privileged frame, in which , from others and the transformations from/to that frame are members of a group of transformations that are equivalent to others. In this section, the transformations are specified using an argument, that anisotropy of the one-way speed of light in an inertial frame is due to its motion with respect to a privileged frame. The argument leads to the conclusion that the anisotropy parameter in a frame moving with respect to a privileged frame with velocity should be given by some (universal) function of that velocity. It follows from equations (2.17) and (2.26) which imply that so that for the transformation from the privileged frame to a frame we have or .
Next, consider three inertial reference frames , and . As in the preceding analysis, the standard configuration, with the - and -axes of the three frames being parallel and the relative motion being along the common -axis (and along the direction of the anisotropy vector), is assumed. The space and time coordinates and the anisotropy parameters in the frames , and are denoted respectively as , and . The frame moves relative to with velocity and velocities of the frames and relative to the frame are respectively and . The relation between , and can be obtained from the equation expressing a group property of the transformations, as follows
where , and are the values of the group parameter corresponding to the transformations from to , from to and from to respectively. Those values are expressed through the velocities and the anisotropy parameter values by a properly specified equation (2.26) which, upon substituting into equation (2.37), yields
Exponentiation of equation (2.38) yields
Let us now choose the frame to be a privileged frame. Then, and for the frames and we have
For a known function (and so for known ), the relation (2.42) defines the anisotropy parameter in the frame as a function of the anisotropy parameter in the frame and the relative velocity of the frames. and thus defines a form of the transformation of the anisotropy parameter. With expressed from (2.26), as follows
substituting (2.43) for and differentiating the result with respect to , with separated, yields
Equation (2.46) is the Lie equation defining (with the initial condition ) the group transformation which implies that the expression on the right-hand side is the group generator
A form of the function as an expansion in series of can be defined based on the argument that the expansion should not contain even powers of since it is expected that a direction of the anisotropy vector changes to the opposite if a direction of motion with respect to a privileged frame is reversed: . In particular, with accuracy up to the third order in , the dependence of the anisotropy parameter on the velocity with respect to a privileged frame can be approximated by
Thus, after the specification, the transformations between inertial frames incorporating anisotropy of light propagation are defined by equations (2.2) and (2.28) with given by (2.49) and the scale factor given by (2.51). It is readily checked that the specified transformations satisfy the correspondence principle. All the equations contain only one undefined parameter, a universal constant .
It should be clarified that, although the specification relies on the approximate relation (2.48), the transformations themselves, even with and defined by (2.49) and (2.51), are not approximate and they do possess the group property. The transformations (2.2) and (2.28) form a group, even with (or ) undefined, provided that the transformation of obeys the group property. Since the relation (2.42), defining that transformation, is a particular case of the relation (2.40) obtained from equation (2.37) expressing the group property, the transformation of satisfies the group property with any form of the function , and, in particular, with that defined by (2.48). It can be demonstrated by a straightforward check or, alternatively, we can calculate the group generator from (2.47) using the expression (2.48) for which yields
Then solving the initial value problem
An expression for the factor for arbitrary is derived in Appendix A.
3 General relativity
The basic principle of general relativity (The Equivalence Principle) asserts that at each point of spacetime it is possible to choose a ’locally inertial’ coordinate system in which the effects of gravitation are absent and the special theory of relativity is valid. It is evident that the principle, that it is always possible to choose a locally inertial frame in which objects obey Newton’s first law, is valid independently of the law of propagation of light assumed. It implies that the equivalence principle can be applied when the processes in the locally inertial frames are governed by the modified special relativity based on invariance of anisotropic equation of light propagation. Developing the general relativity using the equivalence principle in the latter case seems problematic since the interval is not invariant but conformally modified under the transformations. Nevertheless, the complete apparatus of general relativity can be applied based on that there exists the invariant combination (2.33) which takes the form of the Minkowski interval upon the change of variables (2.34). Thus, the general relativity equations in arbitrary coordinates are valid if the locally inertial coordinates are defined as
where , , and are defined in (2.34), and the invariant spacetime distance squared is equal to (repeated indices are summed and the common notation is used for the Minkowski metric). However, in the calculation of physical effects, the ’true’ time and space intervals in the ’physical’ variables are to be used.
It is worthwhile to remark that it does not influence a validity of the arguments based on the small velocity limits that are commonly used in developing a framework of the general relativity. In those limiting arguments, only the first order in terms are considered while the difference between the ’locally inertial’ coordinates and ’physical’ coordinates is, according to (2.34) and (2.56), of the second order in . Note, in addition, that the equation of a freely moving particle in a locally inertial frame, which plays an important role in developing a paradigm of general relativity, takes, in the ’locally inertial’ coordinates defined by (3.1) and (2.34), the same form as in the physical coordinates , as follows
where is the proper time.
In what follows, the notation for ’physical’ coordinates is changed to to leave freedom for using instead of in the contexts where it is traditionally done in the literature. For the sake of convenience, equations (2.34) relating the physical coordinates to the ’locally inertial’ coordinates are rewritten below with taking into account the relations (3.1) and (2.34), as follows
where is the velocity of a locally inertial (freely falling) observer relative to a privileged frame and, with an accuracy up to terms of order , the factor can be approximated by (2.56).
Let us now determine the relations of the ’true’ time and space intervals to the coordinates . First, recall the relation of the proper time interval to the interval . That relation is obtained by considering two infinitesimally separated events occurring at one and the same point in space (see, e.g., ) which yields
To obtain an expression for the element of ’true’ spatial distance consider, following , a light signal sent from some point B in space with coordinates to a point A with coordinates (here and below Greek indices run from 1 to 3, while Latin indices run from 0 to 3) and then back over the same part. The time required for this (as measured at the point B), when multiplied by , is twice the distance between the two points. Determining the interval between the departure of the signal and its return to B (see ) yields
The corresponding interval of the ’true’ proper time is obtained using (3.5) and the distance between the two points is obtained by multiplying it by which yields
In view of the fact that the time and the distance intervals are modified by the same factor , the expression for the proper velocity of a particle does not include that factor and so the proper velocity is calculated in a usual way.
Below we apply the modified GR to cosmology leaving aside other possible applications. The problem of a gravitational collapse of a dustlike sphere, which, in some aspects, is related to cosmological issues, is considered in Appendix B.
4 Cosmological models
4.1 General framework
Modern cosmological models are based on the assumption that the universe appear isotropic to ”typical” freely falling observers, those that move with the average velocity of typical galaxies in their respective neighborhoods. It is also assumed that such the typical (privileged) Lorentzian frame in which the universe appears isotropic coincides more or less with our own galaxy.
The metric derived on the basis of isotropy and homogeneity (the Robertson4–Walker metric) has the form
where a co-moving reference system, moving at each point of space along with the matter located at that point, is used. This implies that the coordinates are unchanged for each typical observer (presumably located at a galaxy). In (4.1), and in what follows, the system of units in which the speed of light is equal to unity, is used. The time coordinate is the synchronous proper time at each point of space. The constant (this notation is used, instead of common or , to avoid confusion with the symbols for the anisotropy parameter) by a suitable choice of units for can be chosen to have the value , , or . Introducing, instead of , the radial coordinate by the relation with
converts (4.1) into the form
Next, let us introduce, in place of the time , the conformal time defined by
Then the function may be treated as a function of (for what follows, it is convenient to leave the same notation for that function) and can be written as
4.2 The red shift
The information about the scale factor in the RobertsonWalker metric can be obtained from observations of shifts in frequency of light emitted by distant sources. To calculate such frequency shifts let us consider the propagation of a light ray in an isotropic space with the metric (4.5) adopting a coordinate system in which we are at the center of coordinates and the source is at the point with a coordinate . A light ray propagating along the radial direction obeys the equation . For a light ray coming toward the origin from the source, that equation gives
where corresponds to the moment of emission and corresponds to the moment of observation . Let is the time interval between departure of subsequent light signals from the point and is the time interval between arrivals of these light signals to the observer at the point . It follows from equation (4.6) that the corresponding increments of the variable are equal to each other (the co-moving coordinate is time-independent) which, upon using equation (4.4), gives
Next, the time intervals and are to be related to the intervals of physical time and using the relation
which requires calculation of the velocity with respect to a privileged frame. It is evident that for a ’typical’ freely falling observer all other typical observers (galaxies) are moving in radial direction and the privileged frame for such an observer is one in which the distribution of the velocities of galaxies appears isotropic. Thus, the observer at the origin of coordinates is at rest and the source is moving with the velocity with respect to the privileged frame so that, with the use of (3.5), equation (4.7) takes the form
If the signals are subsequent wave crests, the observed frequency is related to the frequency of the emitted light by
where the frequency of a spectral line coincides with that observed in terrestrial laboratories. The red-shift parameter is defined by
4.3 The red-shift versus luminosity distance relation
The relation expressing the Luminosity Distance of a cosmological source in terms of its redshift is one of the fundamental relations in cosmology. It has been exploited to get information about the time evolution of the expansion rate. Let is the energy emitted by the source during the interval of (’physical’) time . Then the absolute luminosity of the source is defined by
where the relation (4.8) has been used. Let us assume that the energy is emitted isotropically and imagine the luminous object to be surrounded with a sphere whose radius is equal to the distance between the source and the observer. With the metric (4.1), the area of the surface of the sphere at the moment of observation is equal to . Then the apparent luminosity (the energy passed per unit time per unit area of the surface of the sphere) is calculated as
where is the total energy passed through the surface during the time interval . The time interval is related to the time interval , within which the energy was emitted, by equation (4.7) and the energy received by the surface of the sphere is related to the emitted energy by
where and are the energies of the individual photons received by the observer and emitted by the source and is the number of photons emitted during the time interval . The luminosity distance is defined based on the relation from euclidian geometry ) by ,,
By eliminating using (4.12), as it is usually done, another form of the relation for is obtained, namely
This relation coincides with a common form of the relation for ,,. Nevertheless, even though it does not contain the factor , the dependence of on obtained by eliminating from equations (4.18) and (4.12) will differ from the common one since the relation (4.12) for does contain the factor .
To calculate the factor the value of is to be determined. The proper radial velocity of a remote object cannot be determined using the expression (3.7) for the distance passed by the object since the comoving coordinates do not change during the particle motion while (3.7) deals with the coordinate increments . The particle velocity with respect to the center can be calculated as where is the proper distance of the particle with the radial coordinate to the center. Commonly the quantity is called the proper distance , but, in fact, it is not the proper distance to the center as that relation is obtained by integrating the ’radial’ line element for constant which corresponds to simultaneous observation of all the points along the path of integration and so is physically not feasible ,. Nevertheless, the relation provides a small approximation for the proper distance . The corresponding relation for the velocity of an object with respect to the center (with respect to a privileged frame) is
Here, and in what follows, the derivatives with respect to are converted into derivatives with respect to using equation (4.4). Although the relation (4.19) contains only a term of the first order in , the approximation is sufficiently accurate. Since the expression (4.9) for the factor depends on , introducing the approximation (4.19) into (4.9) makes it valid up to the terms of the order . Within the same accuracy, the factor can be expressed as . Then incorporating this relation and the relation (4.19) in (4.12) yields
To derive the relation between luminosity distance and red-shift as a power series, and defined by equations (4.20) and (4.18) are expanded in series of (in the literature, the series in the ’look-back time’ are commonly used in that derivation), which, upon retaining terms up to the order , yields
where and are defined by