Yang–Mills Theory of Gravity

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

We show that color-kinematics duality is a manifest property of the equations of motion governing currents and field strengths. For the nonlinear sigma model (NLSM), this insight enables an implementation of the double copy at the level of fields, as well as an explicit construction of the kinematic algebra and associated kinematic current. As a byproduct, we also derive new formulations of the special Galileon (SG) and Born-Infeld (BI) theory.For Yang-Mills (YM) theory, this same approach reveals a novel structure — covariant color-kinematics duality — whose only difference from the conventional duality is that 1/□ is replaced with covariant 1/D2. Remarkably, this structure implies that YM theory is itself the covariant double copy of gauged biadjoint scalar (GBAS) theory and an F3 theory of field strengths encoding a corresponding kinematic algebra and current. Directly applying the double copy to equations of motion, we derive general relativity (GR) from the product of Einstein-YM and F3 theory. This exercise reveals a trivial variant of the classical double copy that recasts any solution of GR as a solution of YM theory in a curved background.Covariant color-kinematics duality also implies a new decomposition of tree-level amplitudes in YM theory into those of GBAS theory. Using this representation we derive a closed-form, analytic expression for all BCJ numerators in YM theory and the NLSM for any number of particles in any spacetime dimension. By virtue of the double copy, this constitutes an explicit formula for all tree-level scattering amplitudes in YM, GR, NLSM, SG, and BI.

A connection is established between Penrose's (1982) definition of quasi-local mass and the more conventional notions of mass, momentum etc arising from the canonical formalism of general relativity (which exist at least asymptotically). It is shown that each component of the 'angular momentum' twistor can be thought of as the value of a Hamiltonian which generates motions of regions of the spacetime which tend towards one of a collection of 'quasi-Killing vectors' on the bounding 2-surface on which the computations take place. The quasi-Killing vectors are obtained from solutions of the twistor equation, and essential use is made of the spinorial version of the gravitational Hamiltonian first employed in Witten's simplified proof of positive energy in general relativity. These ideas are then used to suggest a variation on Penrose's quasi-local mass definition using 'quasi-conformal Killing vectors' rather than quasi-Killing vectors. This has the advantage that there are only sixteen real quantities rather than the twenty real (ten complex) ones from Penrose's original definition.

We introduce a general approximation scheme in order to calculate gauge invariant observables in the canonical formulation of general relativity. Using this scheme we will show how the observables and the dynamics of field theories on a fixed background or equivalently the observables of the linearized theory can be understood as an approximation to the observables in full general relativity. Gauge invariant corrections can be calculated up to an arbitrary high order and we will explicitly calculate the first non-trivial correction. Furthermore, we will make a first investigation into the Poisson algebra between observables corresponding to fields at different spacetime points and consider the locality properties of the observables.

A CPT anomaly

We find a black hole solution with non-Abelian field in Brans-Dicke theory.\nIt is an extension of non-Abelian black hole in general relativity. We discuss\ntwo non-Abelian fields: "SU(2)" Yang-Mills field with a mass (Proca field) and\nthe SU(2)$\\times$SU(2) Skyrme field. In both cases, as in general relativity,\nthere are two branches of solutions, i.e., two black hole solutions with the\nsame horizon radius. Masses of both black holes are always smaller than those\nin general relativity. A cusp structure in the mass-horizon radius\n($M_{g}$-$r_{h}$) diagram, which is a typical symptom of stability change in\ncatastrophe theory, does not appear in the Brans-Dicke frame but is found in\nthe Einstein conformal frame. This suggests that catastrophe theory may be\nsimply applied for a stability analysis as it is if we use the variables in the\nEinstein frame. We also discuss the effects of the Brans-Dicke scalar field on\nblack hole structure.\n

The reduction of the Arnowitt-Deser-Misner canonical formulation of general relativity developed in the first paper of this series is applied to the full time-evolution problem for spherically symmetric charged dust shells. Detailed pictures of shell evolution are produced. Among other things, it is found that under certain well-defined circumstances the asymptotically flat spacelike hypersurfaces of constant time ''pinch off'' and become completely closed, the closure point being a locally naked singularity.

We give an example of a generally covariant geometric field theory which leads to the standard Gaussian and supermomentum constraints but which lacks the super-Hamiltonian constraint. This theory is closely linked to Ashtekar’s canonical formulation of general relativity. We discuss the Dirac constraint quantization of our model and comment on the issues which it raises regarding the quantization of generally covariant systems.

In general relativity, an inertial frame can only be established in a small region of spacetime, and local inertial frames are mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform Lorentz transformations in local inertial frames, and there exists freedom to choose the local inertial frame at each spacetime point. The local Lorentz transformations are known as non-Abelian gauge transformations for the tetrad field, and to fix the gauge freedom corresponding to the Lorentz gauge ∂μAμ=0 and the Coulomb gauge ∂iAi=0 in electrodynamics, the Lorentz gauge and Coulomb gauge for tetrad fields are proposed in the present work. Moreover, properties of the Lorentz gauge and the Coulomb gauge for tetrad fields are discussed to show their similarities to those in electromagnetic fields.

A nondiagonal tetrad field having six unknown functions plus an angle Φ, which is a function of the radial coordinate r, azimuthal angle θ and the polar angle ϕ, is applied to the charged field equations of modified teleparallel theory of gravity. A special nonvacuum solution is derived with three constants of integration. The tetrad field of this solution is axially symmetric and its scalar torsion is constant. The associated metric of the derived solution gives Kerr–Newman spacetime. We have shown that the derived solution can be described by a local Lorentz transformations plus a diagonal tetrad field that is the square root of the Kerr–Newman metric. We show that any solution of general relativity (GR) can be a solution in f(T) under certain conditions.

We give an exact solution of the gravitational and electromagnetic field equations with a charged rotating source in new general relativity. The solution has three parameters Q, h and a, and it gives a charged Kerr metric space-time. The parallel vector fields and the electromagnetic vector potential are axially symmetric. In this space-time, we cannot discriminate new general relativity from general relativity, so far as scalar, the Dirac and the Yang-Mills fields and macroscopic bodies are used as probes. The space-time does not have singularities at all, although it has an “effective singularity”. Two kinds of Reissner-Nordström metric solutions, one is our solution with h = 0 and the other is a solution given by Hayashi and Shirafuji, are physically equivalent with each other. Nevertheless, these are markedly different from each other with regard to the asymptotic behavior of the torsion tensor for r →∞ and the space-time singularities.

Klein-Weyl's program and the ontology of gauge and quantum systems

A brief analysis of the classical derivation of the continuity equation in general relativity is presented. The consequences of the equation obtained are reviewed and compared with similar results related with the conservation of particles (baryons and leptons). Emphasis is made on the fact that different transport equations may be derived from the possible choices of the mass balance equation in non-equilibrium formalisms. Nevertheless, the continuity equation is not at odds with the notion of particle conservation.

In general relativity, finding out the geodesics of a given spacetime manifold is an important task because it determines which classical processes are dynamically forbidden. Conserved quantities play an important role in solving the geodesic equations of a general spacetime manifold. Furthermore, knowing all possible conserved quantities of a system gives information about the hidden symmetries of that system since conserved quantities are deeply connected with the symmetries of the system. These are very important in their own right. Conserved quantities are also useful to capture certain features of spacetime manifold for an asymptotic observer. In this article, we show the existence of these conserved charges and their algebra in a generic curved spacetime for a class of dynamical systems with the Hamiltonians quadratic and linear in momentum and spin.

In this work we consider gravitational theories in which the effect of coupling characteristic classes, appropriately introduced as operators in the Einstein-Hilbert action, has been taken into account. As it is well known, this approach strays from the framework of general relativity since it results in theories in which torsion can be present. We consider here all the characteristic classes that are consistent with a four-dimensional space-time manifold. Then, we present explicit expressions for the contortion $1$-form and torsion $2$-form that can be obtained under general conditions in various cases of interest. Additionally, we use the same framework to study cosmological scenarios that are obtained mainly by selecting the flat FLRW metric and an ideal fluid.