In canonical tetrad gravity, it is possible to identify the gauge variables, describing relativistic inertial effects, in Einstein general relativity. One of these is the York time, the trace of the extrinsic curvature of the instantaneous non‐Euclidean 3‐spaces (global Euclidean 3‐spaces are forbidden by the equivalence principle). The extrinsic curvature depends both on gauge variables and on dynamical ones like the gravitational waves after linearization. The fixation of these gauge variables is done by relativistic metrology with its identification of time and space. Till now, the International Celestial Reference Frame ICRF uses Euclidean 3‐spaces outside the Solar System. It is shown that York time and non‐Euclidean 3‐spaces may explain the main signatures of dark matter in ordinary space‐time before using cosmology. Also dark energy may be connected to these inertial gauge effects, because both red‐shift and luminosity distance depend on them.
- dark matter
An extremely important, till now not explicitly clarified, point in Einstein general relativity (GR) (and in every generally covariant theory of gravity), whose gauge group is the group of diffeomorphisms of the Lorentzian 4‐dimensional space‐time,1 is that the fixation of the gauge freedom is nothing else than
According with the International Astronomic Union IAU inside the Solar System, the choice of the 4‐coordinates is
These frames are compatible with the usual interpretation as
In astronomy, data like luminosity, light spectrum and angles are used to determine the positions of stars and galaxies and their temporal evolution in a 4‐dimensional nearly Galilei space‐time with the International Celestial Reference System ICRS [2, 3], a frame considered as a “quasi‐inertial frame” and with all galactic dynamics described by PN gravity.
This is in accord with the smallness of the intrinsic 3‐curvature of the 3‐spaces as implied by the CMB data, a property included in the standard Friedmann‐Lemaitre‐Robertson‐Walker (FLRW) CDM cosmological model with its isotropy and homogeneity symmetries. However, to reconcile all the existing data with this 4‐dimensional description, one must postulate the existence of
After a description of ICRS and of the measurements in ordinary astrophysics (not cosmology) of quantities like luminosity distance, rotation curves of galaxies, gravitational lensing,…. implying “darkness,” we will study canonical ADM tetrad gravity and its gauge freedom after a suitable but arbitrary 3 + 1 splitting of the space‐time in a family of Einstein space‐times able to include the extension of the models of particle physics to GR. We will identify which are the
2. Astrophysical metrology
Reference data for positional astronomy, such as the data in astrometric star catalogs, are specified in the International Celestial Reference System ICRS [2, 3] with origin in the solar system barycenter and with kinematically non‐rotating spatial axes fixed with respect to space according to the IAU conventions [2, 3]. It is based on the position of extragalactic radio sources that are distant enough to be considered stationary, in the limit of today’s capabilities, and whose position is known with a precision of 0.001 arcsec, thanks to the Very Long Baseline Interferometry technique . These sources are assumed to have no observable intrinsic angular momentum. The International Celestial Reference Frame ICRF is a realization of ICRS obtained by supposing that the origin is a quasi‐inertial observer and that we have a quasi‐inertial (essentially non‐relativistic) reference frame with rectangular 3‐coordinates in a nearly Galilean space‐time whose 3‐spaces are Euclidean.
However, a number of different categories of astronomical observations are explained in the usual Euclidean 3‐space only in terms of so far undetermined dark matter and dark energy: rotational curves of galaxies [14–17], gravitational lensing [18–20], application of the virial theorem to galaxy clusters [21–23] and the acceleration of the expansion of the universe [24–29]. This already happens before the transition from the ordinary space‐time to the cosmological one, the FLWR space‐time which is not a Galilean space‐time but has nearly internally flat 3‐spaces and uses a theoretical cosmic time. What is still not explored is the possibility that in Einstein GR one can use non Euclidean 3‐spaces with small internal 3‐curvature, but with an extrinsic curvature (as 3‐submanifolds of the space‐time) depending on the gauge variables, namely on the metrology conventions.
In all the astronomical observations, the distance of the objects needs to be known. Measuring distances in astronomy is a difficult task, especially when dealing with extragalactic objects. Different methods must be applied at increasing distances, which need to be inter‐calibrated appropriately. To get relevant quantities like distances and absolute luminosity of stars from the directly measured quantities, that is, apparent luminosity, angles and red‐shift, it is important to know the geometry of the 3‐spaces crossed by the propagating rays of light on null 4‐geodesics of the space‐time.
The most important methods rely on the absolute intrinsic luminosity of a
Also used is the
In both luminosity distance and angular diameter distance, the terms which depart from Euclidean geometry enter only at higher orders, which depend on the rate of expansion of the universe and on the curvature parameter. For the galaxies with the most reliable rotation curves that are within a range of a few tens of Mega‐parsec, they can be neglected, and we can consider the 3‐space to be Euclidean. Higher order terms need instead to be considered when the objects have a distance of hundreds of Mega‐parsec or more.
For larger , one has to take into account a model of cosmology: In a FLWR metric, one has with and with depending also on .
Assuming that all supernovae (SN) Ia have the same intrinsic luminosity, it was found [26–29] that the SN1a’s at are about 10 per cent fainter than expected, and this has been interpreted as evidence of an accelerated expansion of the universe and dark energy has been invoked to take care of the accelerated expansion.
3. Einstein general relativity
We shall use the formulation of Einstein GR in a 4‐dimensional Lorentzian space‐time (the one used in classical astrophysics, not in cosmology, after the recombination surface for the propagation of light) with the Lagrangian description implied by the ADM action principle [30, 31], because it allows to make the transition to the canonical formalism and to use Dirac theory of constraints , in particular to use the Shanmugadhasan canonical transformation [33, 34] to find canonical bases adapted to the constraints (see Ref.  for reviews). Light and visible stars and galaxies constitute the matter.
We will restrict ourselves to
While in the family of spatially compact without spatial boundary space‐times4, considered in loop quantum gravity [44, 45], the Dirac Hamiltonian is a combination of constraints because the canonical Hamiltonian vanishes, in our space‐times there is not a frozen picture, because the canonical Hamiltonian is the weak ADM energy 5 plus a combination of constraints. In the absence of matter, Christodoulou‐Klainermann space‐times  are compatible with this description.
In the ADM Lagrangian, the basic variable is the 4‐metric of the space‐time ( are local 4‐coordinates with an arbitrary origin): it determines the dynamical chrono‐geometrical structure of space‐time by means of the line element , and it teaches to massless particles which are the allowed trajectories in each point.
However, to include the coupling of gravity to the spin of fermions, we must use ADM
A. Metrology as the Fixation of the Gauge Freedom of General Relativity
While the ADM action for metric gravity is invariant under space‐time diffeomorphisms, the decomposition of the 4‐metric on the cotetrads gives an ADM action  invariant not only under the space‐time diffeomorphisms but also on a local O(3,1) Lorentz group describing the freedom in the orientation and transport of the gyroscopes along the time‐like world lines of observers. Let us remark that the same gauge freedoms are present in all the generally covariant formulations of GR proposed as modifications of Einstein GR.
In electromagnetism and in Yang‐Mills theories, the Lagrangian description in terms of potentials implies the presence of a gauge group acting on an internal space and implying the gauge nature of certain scalar and longitudinal components of the potentials: the gauge fixings imply the description of physics in terms of electric and magnetic fields or of their non‐abelian analogues. Instead, in the metric formulation of GR, the gauge freedom is connected with the freedom in the choice of the metrology conventions, described in the previous section, for the definitions of clocks (i.e., time) and 3‐space in each point of the space‐time. As we shall see a metrology convention implies the fixation of 8 of the 10 components of the 4‐metric, so that the remaining two components describe the physical degrees of freedom of the gravitational field (the gravitational waves (GW) of its linearization in the case of weak fields). In tetrad gravity, we have 16 fields, but the extra 6 fields are fixed by metrology conventions on the orientation of three gyroscopes and on their transport along time‐like world lines in each point of the space‐time.
In special relativity, the metrology conventions amount to the choice of a standard atomic clock and of the instantaneous Euclidean 3‐spaces of a
In GR, due to the equivalence principle forbidding the existence of global inertial frames, one has to use the cited theory of global non‐inertial frames in the form of the so‐called
The inverse transformation defines the embeddings of the 3‐spaces into the space‐time and the induced 4‐metric is , where , while the cotetrads take the form . As shown explicitly in Ref. , the use of the 4‐scalar radar 4‐coordinates implies that
While the 4‐vectors are tangent to , so that in each point of the 3‐space, the unit normal is proportional to , we have , where and are the lapse and shift functions of canonical GR.
In the chosen family of space‐times, the foliation needed for the 3+1 splitting is nice and admissible if the lapse function satisfies in every point of ,9 if 10 and if the positive‐definite 3‐metric has three positive eigenvalues. These are the Møller conditions [52, 53].
Moreover, all the 3‐spaces must tend to the same space‐like hyperplane at spatial infinity. Due to the imposed absence of super‐translations [39, 40], the non‐Euclidean 3‐spaces are orthogonal to the conserved ADM 4‐momentum at spatial infinity; therefore, each 3‐space is a
In Refs. [39–43], there is a parametrization of tetrads, cotetrads and 4‐metric in the framework of the 3+1 splitting of space‐time. The basic configuration variables, that is, the cotetrads, are connected to cotetrads adapted to the 3+1 splitting of space‐time (so that the adapted time‐like tetrad is the unit normal to the 3‐space ) by standard Wigner boosts for time‐like vectors depending upon boost parameters : . The adapted tetrads and cotetrads have the expression11,12
From Eq. (5.5) of the third paper in Ref. , we assume the following (direction‐independent, so to kill super‐translations) boundary conditions at spatial infinity (; ; ): , , , , , , , .
As shown in Refs. [[39–43], due to the existence of the asymptotic ADM Poincare’ group, the isolated system
As a conclusion to fix the gauge in GR with a metrology convention, so to visualize the associated gauge‐dependent inertial effects, we need to separate the gauge variables from the dynamical ones, the so‐called Dirac observables (DO), and only the Hamiltonian formalism has the tools to face this problem. The usual criticism that this can be done only in a non‐covariant coordinate‐dependent way is avoided due to the use of the radar coordinates implying the existence of 4‐scalar tensors.
B. Canonical ADM Tetrad Gravity and Its Gauge Variables
The parametrization of cotetrads given in the previous subsection for ADM tetrad gravity implies  that the ADM action may be considered function of the 16 configurational variables , , , . At the Hamiltonian level, there is a phase space spanned by these 16 configuration variables and their conjugated 16 momenta, and there are
The basis of canonical variables for this formulation of tetrad gravity, naturally adapted to 7 of the 14 first‐class constraints, is (only the momenta conjugated to the cotriads are not vanishing)
In Ref. , a York canonical basis, adapted to 10 first‐class constraints (not to the super‐Hamiltonian and super‐momentum ones, whose solution is unknown), was identified by means of a Shanmugadhasan canonical transformation [33, 34]; this allows
In this York canonical basis, the
The momenta and the 3‐volume element have to be found as solutions of the super‐momentum () and super‐Hamiltonian (i.e., the Lichnerowicz equation  ) constraints, respectively.
Instead, the DO’s (gauge invariant under the Hamiltonian gauge transformations generated by all the first class constraints; see Ref. ) of the gravitational field are not known15; they would be the two pairs of 4‐scalar tidal variables in a Shanmugadhasan canonical basis adapted to all the 14 first class constraints.
The extra O(3,1) gauge freedom of the tetrads16 is described by the gauge variables , . In the
The gauge angles 17 describe the freedom in the choice of the axes for the 3‐coordinates on each 3‐space: their fixation implies the determination of the shift gauge variables , namely the appearances of gravitomagnetism in the chosen 3‐coordinate system . The
Only one momentum is a gauge variable (a reflection of the Lorentz signature): the
The Dirac Hamiltonian is , where the weak ADM energy is an explicit function of all the variables, and the ’s are arbitrary Dirac multipliers (to be determined as a consequence of the gauge fixings).
In the family of Schwinger time gauges, the fixation of the primary gauge variables , implies elliptic equations on the instantaneous 3‐space for the determination of the lapse and shift functions (the secondary gauge variables) and then of their Dirac multipliers ’s. Instead in the usually used harmonic gauges, one imposes the primary gauge fixing , whose stability in time, that is, , implies hyperbolic equations for the lapse and shift functions, namely the necessity of Cauchy conditions in the past for these metrology gauge variables.
This parametrization of canonical tetrad gravity clarifies the meaning of the metrology conventions.
The fixation of the York time determines the sequence of instantaneous non‐Euclidean 3‐spaces of the 3+1 splitting of space‐time centered on an observer either on the Earth or on the Space Station19: all the clocks on each 3‐space are synchronized with the atomic clock ( is its proper time) of the observer at the intersection of the 3‐space with the observer world line. This time metrology convention implies also the determination of the lapse function, which describes how the unit of time of the atomic clock changes when one goes from a 3‐space to an infinitesimally near successive one. The metrology conventions on the choice of the three space coordinates also imply the determination of the shift functions, which say in which point of the infinitesimally near next 3‐space there are the same 3‐coordinates of the chosen point on the original 3‐space.
C. Einstein Hamilton Equations of Tetrad Gravity and their Linearization
In the York canonical basis, the Hamilton equations generated by the Dirac Hamiltonian are divided into four groups after the fixation of the O(3,1) gauge variables with the Schwinger time gauges:
Four contracted Bianchi identities, namely the evolution equations for and (they say that given a solution of the constraints on a Cauchy surface, it remains a solution also at later times).
Four evolution equation for the four basic primary gauge variables and : these equations determine the lapse and the shift functions once four gauge fixings for the basic gauge variables are added.
four evolution equations for the tidal variables , ;
the Hamilton equations for matter, when present.
The Hamilton equations become completely deterministic after a fixation of the gauge freedom. In the York canonical basis, it is convenient to use a family of
In Ref. , this class of asymptotically Minkowskian space‐times without super‐translations is used to study the coupling of N charged scalar point particles (with the inertial and gravitational masses equal as required by the equivalence principle) plus the electromagnetic field to ADM tetrad gravity. The use of Grassmann‐valued electric charges and the signs of the energy of the particles allows to regularize the self‐energies. The theory can be reformulated in terms of transverse electromagnetic fields by using the non‐covariant radiation gauge; this allows to extract the generalization of the Coulomb interaction among the particles in the Riemannian instantaneous 3‐spaces of global non‐inertial frames.
From the Hamilton equations in the York canonical basis , followed by a Hamiltonian Post‐Minkowskian (HPM) linearization (disregarding terms of order in the Newton constant and using an ultra‐violet cutoff for matter) with the asymptotic flat Minkowski 4‐metric at spatial infinity as background, it has been possible to develop a theory of GW’s with asymptotic background propagating in the non‐Euclidean 3‐spaces of a family of
Since the celestial reference frame ICRS has diagonal 3‐metric, our 3‐orthogonal Schwinger time gauges are a good choice for celestial metrology.
The open problem is that the GCRS and BCRS conventions in the Solar System are using the special harmonic gauge of IAU [2, 3], in which the lapse function satisfies a hyperbolic equation like the tidal variables and needs initial data in the past, differently from what happens in the 3‐orthogonal Schwinger time gauges. See Subsection 3.3 of the third paper in Ref.  for the comparison of the IAU harmonic gauge for BCRS with the 3‐orthogonal gauges and Subsection 3.3 of the second paper in Ref.  for the equations identifying the 4‐coordinate transformation from the 3‐orthogonal gauges to the harmonic ones after the linearization, which have to be solved to get the reformulation of IAU conventions in our gauges.
4. Dark matter as a relativistic inertial effect
The linearized HPM Hamilton equations for point particles of mass , 20, whose world lines are identified by radar 3‐coordinates due to the 3+1 splitting, and for the electromagnetic field coupled to tetrad gravity have been written explicitly in Refs. : among the forces acting on matter, there are both the inertial potentials and the GW’s.
In the third paper of Ref. , electro‐magnetism is eliminated and there is a detailed studied of the HPM equations of motion of the particles. Then, the PN expansion of these regularized HPM equations of motion for the particles was studied, and it was shown that the particle 3‐coordinates (coinciding with the Newtonian coordinates of the world lines at this level of approximation) satisfy the equation of motion
where at the lowest order, there is the standard Newton gravitational force
Since Eqs. (4) imply
there is a 0.5 PN
If we put , the standard results about binaries are reproduced.
The term in the non‐local York time can be
in each instantaneous 3‐space. Since, in the Newton potential, there are the gravitational masses of the particles, the effect is due to a modification of the effective inertial mass in each non‐Euclidean 3‐space depending on its shape as a 3‐submanifold of space‐time. Therefore, we find
In the two‐body case, one gets that for Keplerian circular orbits of radius the modulus of the relative 3‐velocity can be written in the form with function only of .
The data on the
Therefore, this dark matter can be explained as a
A similar interpretation (see Subsections 6.2 and 6.3 of the third paper in Ref. ) can be given for the other two main signatures of the existence of dark matter in the observed masses of galaxies and clusters of galaxies, namely
Therefore, there is the possibility of describing part (or maybe all) dark matter as a
The quoted three main experimental signatures of dark matter are well‐defined functional of the time and space derivatives of the non‐local York time22 the inertial gauge variable describing the general relativistic remnant of the gauge freedom in clock synchronization.
Since the time evolution of the signatures of dark matter is not known, at best from the data, we can extract information only on a
Since there is no indication of dark matter in the voids existing among the clusters of galaxies, we can get an idea on the form of the local York time in the 3‐space (i.e., the whole 3‐universe) by summing its value for all the known galaxies and clusters of galaxies. This would produce an indication of which could be a metrology convention on the inertial gauge variable describing the general relativistic gauge freedom in clock synchronization in the Einstein space‐time outside the Solar System. One expects that, with this metrology convention, the resulting 3‐spaces (each one with all the clocks synchronized) are nearly Euclidean except where there is need of introducing dark matter.
In Ref. , there is a first attempt to fit some data of dark matter by using a Yukawa‐like ansatz on the non‐local York time of a galaxy. In each galaxy, the Yukawa‐like potential of theories [10–12] is put equal to a contribution to the extra potential depending on the non‐local York time present in the lapse function appearing in Eq. (8); in this way, the good fits of the rotation curves of galaxies obtainable with theories can be reproduced inside Einstein’s GR as an inertial gauge effect.
5. Metrology against darkness
In conclusion, a suitable metrology convention on the inertial gauge variable York time could reduce or maybe eliminate the necessity of introducing dark matter in the classical universe and in its extension to classical cosmology after the recombination surface.
A needed natural proposal is now to define a
In this way, one would get a solution to the gauge problem for the PM space‐times of GR: one chooses a reference system of 4‐coordinates in a 3‐orthogonal gauge selected by the observational conventions for matter. A PM definition of ICRS will be also useful for the ESA‐GAIA mission  (cartography of the Milky Way) and for the possible anomalies (different from the already explained Pioneer one) inside the Solar System .
Let us also remark that in Szekeres space‐times [61–63], that is, in inhomogeneous space‐times without Killing symmetries, the York time remains an inertial gauge variable.
As said in Section 2, the red‐shift and luminosity distance of SNIa is a signal of
Let us also remark that in the back‐reaction approach [64–69], in which to take into account the inhomogeneity of the observed universe when trying to get a cosmological description of it, one considers spatial mean values on large scales, dark energy in cosmology is a byproduct of the nonlinearities of GR. In this approach, one gets that the spatial average of the 4‐scalar gauge variable York time gives the effective Hubble constant of this approach.
Finally, as shown in Eq. (10) of the last paper in Ref. , it can be shown that the York time is responsible for the negative terms in the kinetic energy term in the ADM energy, whose existence was known but whose explicit form could be given only in the York canonical basis. It is therefore possible that the connected Landau‐Lifschitz energy‐momentum pseudo‐tensor  of GR could be reformulated as the energy‐momentum tensor of a viscous pseudo‐fluid, which could have a negative pressure for certain choices of the York time like the dark energy fluid in FLWR cosmology.
In conclusion, the York time has a central position in all the cases where darkness is required to fit the data!
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- See Ref.  for theoretical considerations concerning the nature of space and time in GR.
- A relativistic version of ITRS is not yet existing, so that one cannot yet connect the time of the atomic clocks in different laboratories to the clock on the Space Station with a suitable Lorentz transformations.
- See however the LATOR proposal  of measuring the deviation from 2π of the sum of the three angles of a triangle formed by the Space Station and two spacecrafts behind the Sun. When this non‐Euclidean nature will be measured, one will have to redefine the standard of length measurements .
- Therefore, it is not possible to define a Poincare’ group and to find a connection with particle physics.
- It is a volume integral over 3‐space of a coordinate‐dependent energy density. It is weakly equal to the strong ADM energy, which is a flux through a 2‐surface at spatial infinity.
- (α) are flat indices and η(α)(β) is the flat 4‐metric of Minkowski space‐time. The signature of the 4‐metrics is ϵ=± so that η(α)(β)=ϵ (1;−1,−1,−1). ϵ=1 is the convention of particle physics, whereas ϵ=−1 is the convention usually used in GR
- Instead the usually used 1+3 point of view using the world line of a time‐like observer leads only to local coordinate systems like the Riemann and Fermi ones valid only in a neighborhood of a time‐like world line, because locally the 3‐spaces are identified with the tangent spaces orthogonal to the observer 4‐velocity so that they intersect each other.
- They were introduced by Bondi in Ref. [49, 50].
- Therefore, the 3‐spaces never intersect, avoiding the coordinate singularity of Fermi coordinates.
- This property avoids the coordinate singularity of the rotating disk.
- N(τ,σr)=1+n(τ,σr) and n(a)(τ,σr)=(Nr 3e(a)r)(τ,σr)=∑b R(a)(b)(α(c)(τ,σr)) n¯(b)(τ,σr) are the lapse and shift functions respectively.
- 4E∘(β)A and 4E¯∘(o)A are tetrads adapted to the 3+1 splitting.
- G is Newton constant. The set of numerical parameters γa¯a satisfies ∑u γa¯u=0, ∑u γa¯u γb¯u=δa¯b¯, ∑a¯ γa¯u γa¯v=δuv−13. Each solution of these equations defines a different York canonical basis.
- α(a), φ(a), θi and 3K are the primary gauge variables, whereas n and n¯(a) are the secondary ones, which are determined as a consequence of the gauge fixing of the primary ones.
- Ra¯, Πa¯ are not gauge invariant under the Hamiltonian gauge transformations generated by the super‐Hamiltonian and super‐momentum constraints.
- The gauge freedom for each observer to choose three gyroscopes as spatial axes and to choose the law for their transport along the time‐like world line.
- They identify the direction cosines of the tangents to the three coordinate lines in each point of the 3‐space Στ.
- It is absent in the Galilean space‐time of Newtonian gravity with its absolute notions of time and Euclidean 3‐space.
- The detailed structure of these non‐Euclidean 3‐spaces depends on the extrinsic curvature 3‐tensor 3Krs, which depends not only from all the gauge variables but also on the tidal variables, so that it is determined by the chosen solution of Einstein equations.
- mi is both the inertial and the gravitational mass, since they coincide in Einstein GR due to the equivalence principle.
- In the case of gravitational lensing Einstein’s deflection angle, α=4 G M/c2 ξ (ξ is the impact parameter of the ray of light deflected at the position of the mass M) has M=Mbaryon+MDM with the dark matter term given by G MDM=−2 c2 |σ→| ∂τ 3K˜(1)(τ,σu).
- ∂τ 3K˜(1)(τ,σr) in the gravitational lensing case, ddt 3K˜(1)(c t,η˜→i(t))=(∂∂ t+η˜→˙i(t)⋅∂∂ η˜→i) 3K˜(1)(c t,η˜→(t)) in the rotation curve case and d2dt2 3K˜(1)(c t,η˜→i(t)) in the virial theorem case.
- To test this possibility, one has to study the transition from harmonic gauges to 3‐orthogonal ones in linearized Einstein GR.