Inertial Gauge Research Articles

Dark matter: a problem in relativistic metrology?

Besides the tidal degrees of freedom of Einstein general relativity (GR) (namely the two polarizations of gravitational waves after linearization of the theory) there are the inertial gauge ones connected with the freedom in the choice of the 4-coordinates of the space-time, i.e. in the choice of the notions of time and 3-space (the 3+1 splitting of space-time) and in their use to define a non-inertial frame (the inertial ones being forbidden by the equivalence principle) by means of a set of conventions for the relativistic metrology of the space-time (like the GPS ones near the Earth).The canonical York basis of canonical ADM gravity allows us to identify the Hamiltonian inertial gauge variables in globally hyperbolic asymptotically Minkowskian space-times without super-translations and to define the family of non-harmonic Schwinger time gauges. In these 3+1 splittings of space-time the freedom in the choice of time (the problem of clock synchronization) is described by the inertial gauge variable York time (the trace of the extrinsic curvature of the instantaneous 3-spaces). This inertial gauge freedom and the non-Euclidean nature of the instantaneous 3-spaces required by the equivalence principle need to be incorporated as metrical conventions in a relativistic suitable extension of the existing (essentially Galilean) ICRS celestial reference system.In this paper I make a short review of the existing possibilities to explain the presence of dark matter (or at least of part of it) as a relativistic inertial effect induced by the non- Euclidean nature of the 3-spaces. After a Hamiltonian Post-Minkowskian (HPM) linearization of canonical ADM tetrad gravity with particles, having equal inertial and gravitational masses, as matter, followed by a Post-Newtonian (PN) expansion, we find that the Newtonian equality of inertial and gravitational masses breaks down and that the inertial gauge York time produces an increment of the inertial masses explaining at least part of what is called dark matter in all its astrophysical signatures.

Dust in the York canonical basis of ADM tetrad gravity: The problem of vorticity

Brown's formulation of dynamical perfect fluids in Minkowski space-time is extended to ADM tetrad gravity in globally hyperbolic, asymptotically Minkowskian space-times. For the dust, we get the Hamiltonian description in closed form in the York canonical basis, where we can separate the inertial gauge variables of the gravitational field in the non-Euclidean 3-spaces of global non-inertial frames from the physical tidal ones. After writing the Hamilton equations of the dust, we identify the sector of irrotational motions and the gauge fixings forcing the dust 3-spaces to coincide with the 3-spaces of the non-inertial frame. The role of the inertial gauge variable York time (the remnant of the clock synchronization gauge freedom) is emphasized. Finally, the Hamiltonian Post-Minkowskian linearization is studied. This formalism is required when one wants to study the Hamiltonian version of cosmological models (for instance back-reaction as an alternative to dark energy) in the York canonical basis.

Canonical ADM tetrad gravity: From metrological inertial gauge variables to dynamical tidal Dirac observables

In this updated review of canonical ADM tetrad gravity in a family of globally hyperbolic asymptotically Minkowskian space-times without super-translations I show which is the status-of-the-art in the search of a canonical basis adapted to the first-class Dirac constraints and of the Dirac observables of general relativity (GR) describing the tidal degrees of freedom of the gravitational field. In these space-times the asymptotic ADM Poincaré group replaces the Poincaré group of particle physics, there is a York canonical basis diagonalizing the York–Lichnerowicz approach and a post-Minkowskian linearization is possible with the associated description of gravitational waves in the family of non-harmonic 3-orthogonal Schwinger time gauges. Moreover I show that every fixation of the inertial gauge variables (i.e. the choice of a non-inertial frame) of every generally covariant formulation of GR is equivalent to a set of conventions for the metrology of the space-time (like the GPS ones near the Earth): for instance the freedom in clock synchronization is described by the inertial gauge variable York time (the trace of the extrinsic curvature of the instantaneous 3-spaces). This inertial gauge freedom and the non-Euclidean nature of the instantaneous 3-spaces required by the equivalence principle are connected with the dark side of the universe and could explain the presence of dark matter or at least part of it by means of the adoption of suitable metrical conventions for the ICRS celestial reference system. Also some comments on a canonical quantization of GR coherent with this viewpoint are done.

The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 1. The equations of motion in arbitrary Schwinger time gauges 1This paper is one of three companion papers published in the same issue of Can. J. Phys.

We study the coupling of N charged scalar particles plus the electromagnetic field to Arnowitt–Deser–Misner (ADM) tetrad gravity and its canonical formulation in asymptotically Minkowskian space–times without super-translations. To regularize the self-energies, both the electric charge and the sign of the energy of the particles are Grassmann-valued. The introduction of the noncovariant radiation gauge allows reformulation of the theory in terms of transverse electromagnetic fields and to extract the generalization of the Coulomb interaction among the particles in the riemannian instantaneous 3-spaces of global noninertial frames, the only ones allowed by the equivalence principle. Then we make the canonical transformation to the York canonical basis, where there is a separation between the inertial (gauge) variables and the tidal ones inside the gravitational field and a special role of the eulerian observers associated with the 3+1 splitting of space–time. The Dirac hamiltonian is weakly equal to the weak ADM energy. The Hamilton equations in Schwinger time gauges are given explicitly. In the York basis they are naturally divided into four sets: (i) the contracted Bianchi identities; (ii) the equations for the inertial gauge variables; (iii) the equations for the tidal ones; and (iv) the equations for matter. Finally, we give the restriction of the Hamilton equations and of the constraints to the family of nonharmonic 3-orthogonal gauges, in which the instantaneous riemannian 3-spaces have a nonfixed trace 3K of the extrinsic curvature but a diagonal 3-metric. The inertial gauge variable 3K (the general-relativistic remnant of the freedom in the clock synchronization convention) gives rise to a negative kinetic term in the weak ADM energy vanishing only in the gauges with 3K = 0: is it relevant for dark energy and back-reaction? In the second paper will appear the linearization of the theory in these nonharmonic 3-orthogonal gauges to obtain hamiltonian post-minkowskian gravity (without post-newtonian approximations) with asymptotic Minkowski background, nonflat instantaneous 3-spaces and no post-newtonian expansion. This will allow the exploration of the inertial effects induced by the York time 3K in nonflat 3-spaces (they do not exist in newtonian gravity) and to check how well dark matter can be explained as an inertial aspect of Einstein’s general relativity: this will be done in a third paper on the post-minkowskian 2-body problem in the absence of the electromagnetic field and on its 0.5 post-newtonian limit.

The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 2. The weak field approximation in the 3-orthogonal gauges and Hamiltonian post-minkowskian gravity: the N-body problem and gravitational waves with asymptotic background 1This paper is one of three companion papers published in the same issue of Can. J. Phys.

In this second paper we define a post-minkowskian (PM) weak field approximation leading to a linearization of the Hamilton equations of Arnowitt–Deser–Misner (ADM) tetrad gravity in the York canonical basis in a family of nonharmonic 3-orthogonal Schwinger time gauges. The York time 3K (the relativistic inertial gauge variable, not existing in newtonian gravity, parametrizing the family, and connected to the freedom in clock synchronization, i.e., to the definition of the the shape of the instantaneous 3-spaces) is set equal to an arbitrary numerical function. The matter are considered point particles, with a Grassmann regularization of self-energies, and the electromagnetic field in the radiation gauge: an ultraviolet cutoff allows a consistent linearization, which is shown to be the lowest order of a hamiltonian PM expansion. We solve the constraints and the Hamilton equations for the tidal variables and we find PM gravitational waves with asymptotic background (and the correct quadrupole emission formula) propagating on dynamically determined non-euclidean 3-spaces. The conserved ADM energy and the Grassmann regularization of self-energies imply the correct energy balance. A generalized transverse–traceless gauge can be identified and the main tools for the detection of gravitational waves are reproduced in these nonharmonic gauges. In conclusion, we get a PM solution for the gravitational field and we identify a class of PM Einstein space–times, which will be studied in more detail in a third paper together with the PM equations of motion for the particles and their post-newtonian expansion (but in the absence of the electromagnetic field). Finally we make a discussion on the gauge problem in general relativity to understand which type of experimental observations may lead to a preferred choice for the inertial gauge variable 3K in PM space–times. In the third paper we will show that this choice is connected with the problem of dark matter.

The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 3. The post-minkowskian N-body problem, its post-newtonian limit in nonharmonic 3-orthogonal gauges and dark matter as an inertial effect 1This paper is one of three companion papers published in the same issue of Can. J. Phys.

We conclude the study of the post-minkowskian (PM) linearization of ADM tetrad gravity in the York canonical basis for asymptotically minkowskian space–times in the family of nonharmonic 3-orthogonal gauges parametrized by the York time 3K(τ, s) (the inertial gauge variable, not existing in Newton gravity, describing the general relativistic remnant of the freedom in clock synchronization in the definition of the shape of the instantaneous 3-spaces as 3-submanifolds of space–time). As matter we consider only N scalar point particles with a Grassmann regularization of the self-energies and with an ultraviolet cutoff making possible the PM linearization and the evaluation of the PM solution for the gravitational field. We study in detail all the properties of these PM space–times emphasizing their dependence on the gauge variable 3K(1) = (1/Δ)3K(1) (the nonlocal York time): Riemann and Weyl tensors, 3-spaces, time-like and null geodesics, red-shift, and luminosity distance. Then we study the post-newtonian (PN) expansion of the PM equations of motion of the particles. We find that in the two-body case at the 0.5PN order there is a damping (or antidamping) term depending only on 3K(1). This opens the possibility of explaining dark matter in Einstein theory as a relativistic inertial effect: the determination of 3K(1) from the masses and rotation curves of galaxies would give information on how to find a PM extension of the existing PN celestial frame used as an observational convention in the 4-dimensional description of stars and galaxies. Dark matter would describe the difference between the inertial and gravitational masses seen in the non-euclidean 3-spaces, without a violation of their equality in the 4-dimensional space–time as required by the equivalence principle.

Dark Matter: A Relativistic Inertial Effect in Einstein Canonical Gravity?

After consideration of non-inertial frames in special relativity with emphasis on the problem of clock synchronization (i.e. of how to define 3-space), an overview is given of Einstein canonical gravity in the York canonical basis and of its Hamiltonian Post-Minkowskian (PM) linearization in 3-orthogonal gauges. It is shown that the York time (the trace of the extrinsic curvature of 3-spaces) is the inertial gauge variable describing the general relativistic remnant of the clock synchronization gauge freedom. The dark matter implied by the rotation curves of galaxies can be explained with a choice of the York time implying a PM extension of the Newtonian International Celestial Reference System (ICRS).