3-orthogonal Gauges Research Articles

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 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 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).

Post-Minkowskian Gravity: Dark matter as a relativistic inertial effect?

Talk at the 1st Mediterranean Conference in Classical and Quantum Gravity, held in the Orthodox Academy of Crete in Kolymbari (Greece) from Monday, September 14th to Friday, September 18th, 2009.A review is given of the theory of non-inertial frames (with the associated inertial effects and the study of the non-relativistic limit) in Minkowski space-time, of parametrized Minkowski theories and of the rest-frame instant form of dynamics for isolated systems admitting a Lagrangian description. The relevance and gauge equivalence of the clock synchronization conventions for the identification of the instantaneous 3-spaces (Euclidean only in inertial frames) are described.Then this formalism is applied to tetrad gravity in globally hyperbolic, asymptotically Minkowskian space-times without super-translations, where the equivalence principle implies the absence of global inertial frames. The recently discovered York canonical basis, diagonalizing the York-Lichnerowicz approach, allows to identify the gauge variables (inertial effects in general relativity) and the tidal ones (the gravitational waves of the linearized theory) and to clarify the meaning of the Hamilton equations. The role of the gauge variable 3K, the trace of the extrinsic curvature of the non-Euclidean 3-space (the York time not existing in Newton theory), as a source of inertial effects is emphasized. After the presentation of preliminary results on the linearization of tetrad gravity in the family of non-harmonic 3-orthogonal gauges with a free value of 3K, we define post-Minkowskian gravitational waves (without post-Newtonian approximations on the matter sources) propagating in a non-Euclidean 3-space, emphasizing the non-graviton-like aspects of gravity. It is conjectured that dark matter may be explained as a relativistic inertial effect induced by 3K: it would simulate the need to choose a privileged gauge connected with the observational conventions for the description of matter.

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy ${\hat E}_{ADM}$, we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of {\it non-harmonic} 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) $r_{\bar a}(\tau ,\vec \sigma)$, $\pi_{\bar a}(\tau ,\vec \sigma)$, $\bar a = 1,2$. We define a Hamiltonian linearization of the theory, i.e. gravitational waves, {\it without introducing any background 4-metric}, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in ${\hat E}_{ADM}$. {\it We solve all the constraints} of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's $r_{\bar a}(\tau ,\vec \sigma)$, which replace the two polarizations of the TT harmonic gauge, and that {\it linearized Einstein's equations are satisfied} . Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the {\it rest-frame instant form} of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.