Based on the view that Einstein's theory can be interpreted as a gauge theory of Lorentz group, we decompose the gravitational connection (the gauge potential of Lorentz group) $\vGm_μ$ into the restricted connection made of the potential of the maximal Abelian subgroup $H$ of Lorentz group $G$ and the valence connection made of $G/H$ part of the potential which transforms covariantly under Lorentz gauge transformation. With this decomposition we show that the Einstein's theory can be decomposed into the restricted part made of the restricted connection which has the full Lorentz gauge invariance and the valence part made of the valence connection which plays the role of gravitational source of the restricted gravity. We show that there are two different Abelian decomposition of Einstein's theory, the light-like (or null) decomposition and the non light-like (or non-null) decomposition, because Lorentz group has two maximal Abelian subgroups. In this decomposition the role of the metric $g_\mn$ is replaced by a four-index metric tensor $\vg_\mn$ which transforms covariantly under the Lorentz group, and the metric-compatibility condition $\nabla_αg_\mn=0$ of the connection is replaced by the gauge and generally covariant condition ${\mathscr D}_μ\vg^\mn=0$. The decomposition shows the existence of a restricted theory of gravitation which has the full general invariance but is much simpler and has less physical degrees of freedom than Einstein's theory. Moreover, it tells that the restricted gravity can be written as an Abelian gauge theory, which implies that the graviton can be described by a massless spin-one field.
General solution of the non-abelian Gauss law in terms of covariant curls and gradients is presented. Also two non-abelian analogs of the Hodge decomposition in three dimensions are addressed. i) Decomposition of an isotriplet vector field $V_{i}^{a}(x)$ as sum of covariant curl and gradient with respect to an arbitrary background Yang-Mills potential is obtained. ii) A decomposition of the form $V_{i}^{a}=B_{i}^{a}(C)+D_{i}(C) ϕ^{a} $ which involves non-abelian magnetic field of a new Yang-Mills potential C is also presented. These results are relevant for duality transformation for non-abeli…
Scalar Gauss-Bonnet gravity is the only theory with quadratic curvature corrections to general relativity whose field equations are of second differential order. This theory allows for nonperturbative dynamical corrections and is therefore one of the most compelling case studies for beyond-general relativity effects in the strong-curvature regime. However, having second-order field equations is not a guarantee for a healthy time evolution in generic configurations. As a first step towards evolving black-hole binaries in this theory, we here derive the 3+1 decomposition of the field equations f…
Non-abelian uni-vector deformations in gauged supergravities and non-abelian Einstein-Maxwell theories
Institute for Information Transmission Problems（信息传输问题研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute of Theoretical and Mathematical Physics, Moscow State University（莫斯科国立大学理论与数学物理研究所）；Lomonosov Moscow State University（罗蒙诺索夫莫斯科国立大学）
We find a black hole solution with non-Abelian field in Brans-Dicke theory. It is an extension of non-Abelian black hole in general relativity. We discuss two non-Abelian fields: "SU(2)" Yang-Mills field with a mass (Proca field) and the SU(2)$\times$SU(2) Skyrme field. In both cases, as in general relativity, there are two branches of solutions, i.e., two black hole solutions with the same horizon radius. Masses of both black holes are always smaller than those in general relativity. A cusp structure in the mass-horizon radius ($M_{g}$-$r_{h}$) diagram, which is a typical symptom of stability…
An abelian version of standard general relativity in the Cartan-Palatini gauge-like formulation in four dimensions has been introduced. Traditional canonical analysis utilizing similarities to the akin Husain-Kuchar SU(2) version of gravity has been performed. The model has been next quantized in the canonical path-integral Faddeev-Popov formalism yielding abelian BF theory.
A new canonical transformation is found that enables the direct canonical treatment of the conformal factor in general relativity. The resulting formulation significantly simplifies the previously presented conformal geometrodynamics. It provides a further theoretical basis for the conformal approach to loop quantum gravity and offers a generic framework for the conformal analysis of spacetime dynamics.
In this work, two particular orthogonal and conformal decompositions of the 3+1 dimensional Einstein equation and Arnowitt-Deser-Misner (ADM) formalism for general relativity are obtained. In order to do these, the 3+1 foliation of the four-dimensional spacetime, the fundamental conformal transformations and the Hamiltonian form of general relativity that leads to the ADM formalism, defined for the conserved quantities of the hypersurfaces of the globally-hyperbolic asymptotically flat spacetimes, are reconstructed. All the calculations up to chapter 7 are just a review. We propose a method in…
In this paper we obtain a 2+2 double null Hamiltonian description of General Relativity using only the (complex) SO(3) connection and the components of the complex densitised self-dual bivectors. We carry out the general canonical analysis of this system and obtain the first class constraint algebra entirely in terms of the self-dual variables. The first class algebra forms a Lie algebra and all the first class constraints have a simple geometrical interpretation.
Inflation in the framework of Einstein-Cartan theory is revisited. Einstein-Cartan theory is a natural extension of the General Relativity, with non-vanishing torsion. The connection on Riemann-Cartan spacetime is only compatible with the cosmological principal for a particular form of torsion. We show this form to also be compatible with gauge invariance principle for a non-Abelian and Abelian gauge fields under a certain deviced minimal coupling procedure. We adopt an Abelian gauge field in the form of "cosmic triad". The dynamical field equations are obtained and shown to sustain cosmic inf…
The relationship between on-shell tree level scattering amplitudes of open and closed strings, discovered some time ago by Kawai, Lewellen and Tye, is used at field theory level (at $O(α'^3)$) to establish a link between the general relativity and the non-abelian Yang-Mills effective actions. Insisting at the effective Lagrangian level that any tree $N$ point gravity on-shell scattering amplitude is directly factorisable into a sum of $N$ point left-right products of non-abelian Yang-Mills tree on-shell scattering amplitudes, non-trivial mappings of the effective general relativity operators i…
Similarity in the structure of scattering amplitudes in Yang-Mills theories and General Relativity led to the idea that graviton could be described as the double copy of a vector gauge field. In this letter we discuss a realization of this idea emerging directly from solutions of equations of General Relativity. A general form of the energy momentum tensor for the electric field is derived that leads to the metric tensor in terms of the double copy of a corresponding gauge potential. We than use this general property to find the double copy of spontaneously broken scalar electrodynamics. The r…
The conformal method for constructing initial data for Einstein's equations is presented in both the Hamiltonian and Lagrangian picture (extrinsic curvature decomposition and conformal thin sandwich formalism, respectively), and advantages due to the recent introduction of a weight-function in the extrinsic curvature decomposition are discussed. I then describe recent progress in numerical techniques to solve the resulting elliptic equations, and explore innovative approaches toward the construction of astrophysically realistic initial data for binary black hole simulations.
The detection of gravitational waves from compact binary coalescence (CBC) has allowed us to probe the strong-field dynamics of General Relativity (GR). Among various tests performed by the LIGO-Virgo-KAGRA collaboration are parameterized tests, where parameterized modifications to GR waveforms are introduced and constrained. This analysis typically requires the generation of more than millions of computationally expensive waveforms. The computational cost is higher for a longer signal, and current analyses take weeks-years to complete for a binary neutron star (BNS) signal. In this work, we p…
Based on the equivalence between a gauge theory for the translation group and general relativity, a teleparallel version of the non-abelian Kaluza-Klein theory is constructed. In this theory, only the fiber-space turns out to be higher-dimensional, spacetime being kept always four-dimensional. The resulting model is a gauge theory that unifies, in the Kaluza-Klein sense, gravitational and gauge fields. In contrast to the ordinary Kaluza-Klein models, this theory defines a natural length-scale for the compact sub-manifold of the fiber space, which is shown to be of the order of the Planck lengt…
We show that Geroch decomposition leads us to Maxwell-like representation of gravity in $(3+1)$ metrics decomposition that may be perceived as Lorentz invariant version of GEM. For such decomposition we derive four-potential $V^μ$ and gravitational field tensor $F^{μν}$ that is associated with gravitational interaction. Next we show that gravitational four-current $J^μ$ derived for introduced four-potential produce energy-stress tensor and reproduce main General Relativity formula. Next we introduce valid Lagrangian and equations of motion that explains obtained results. At the end we introduc…
Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least a…
We construct a general approach to decomposition of the tangent bundle of pseudo-Riemannian manifolds into direct sums of subbundles, and the associated decomposition of geometric objects. An invariant structure {\cal H}^r defined as a set of r projection operators is used to induce decomposition of the geometric objects into those of the corresponding subbundles. We define the main geometric objects characterizing decomposition. Invariant non-holonomic generalizations of the Gauss-Codazzi-Ricci's relations have been obtained. All the known types of decomposition (used in the theory of frames …
The 3+1 (ADM) formulation of General Relativity is used in, for example, canonical quantum gravity and numerical relativity. Here we present a 3+1 decomposition of the minimal Standard-Model Extension gravity Lagrangian. By choosing the leaves of foliation to lie along a timelike vector field we write the theory in a form which will allow for comparison and matching to other gravity models.
The canonical formulation of general relativity is based on decomposition space--time manifold $M$ into $ R\times Σ$, this decomposition has to preserve the invariance of general relativity, invariance under general coordinates, and local Lorentz transformations. These symmetries associate with conserved currents that are coupled to gravity. In this paper, we try to solve the equations of motion of general relativity in self-dual formalism using only the spin currents(Lorentz currents), in static case, and without needing using the Einstein's equation, that makes the general relativity similar…
These lecture notes provide some introduction to the 3+1 formalism of general relativity, which is the foundation of most modern numerical relativity. The text is rather self-contained, with detailed calculations and numerous examples. Contents: 1. Introduction, 2. Geometry of hypersurfaces, 3. Geometry of foliations, 4. 3+1 decomposition of Einstein equation, 5. 3+1 equations for matter and electromagnetic field, 6. Conformal decomposition, 7. Asymptotic flatness and global quantities, 8. The initial data problem, 9. Choice of foliation and spatial coordinates, 10. Evolution schemes.
We develop a generic spacetime model in General Relativity which can be used to build any gravitational model within General Relativity. The generic model uses two types of assumptions: (a) Geometric assumptions additional to the inherent geometric identities of the Riemannian geometry of spacetime and (b) Assumptions defining a class of observers by means of their 4-velocity $u^{a}$ which is a unit timelike vector field. The geometric assumptions as a rule concern symmetry assumptions (the so called collineations). The latter introduces the 1+3 decomposition of tensor fields in spacetime. The…
Independent variation of the metric and connection in the Einstein-Hilbert action, called the Palatini variation, is generally taken to be equivalent to the usual formulation of general relativity in which only the metric is varied. However, when an abelian symmetry is allowed for the connection, the Palatini variation leads to an integrable Weyl geometry, not Riemannian. We derive this result using two possible metric/connection pairs: (1) the metric and general coordinate connection and (2) the solder form and local Lorentz spin connection of Poincarè gauge theory. Both lead to the same conc…
A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-ti…
We present a 2+1 decomposition of the vacuum initial conditions in general relativity. For a constant mean curvature one of the momentum constraints decouples in quasi isotropic coordinates and it can be solved by quadrature. The remaining momentum constraints are written in the form of the tangential Cauchy-Riemann equation. Under additional assumptions its solutions can be written in terms of integrals of known functions. We show how to obtain initial data with a marginally outer trapped surface. A generalization of the Kerr data is presented.
The gravitational radiation degrees of freedom of freedom are described in the framework of the 3+1 decomposition of spacetime. The relationship with eigenfields of the Kidder-Scheel-Teukolsky (KST) equations is established. This relationship is used to fix a parameter in the KST equations which is related to the ordering ambiguity of space derivatives in the Ricci tensor, which is inherent to first order evolution systems, like the ones currently used in Numerical Relativity applications.
A careful study of the induced transformations on spatial quantities due to 4-dimensional spacetime diffeomorphisms in the canonical formulation of general relativity is undertaken. Use of a general formalism, which indicates the role of the embedding variables in a transparent manner, allows us to analyse the effect of 4-dimensional diffeomorphisms more generally than is possible in the standard ADM approach. This analysis clearly indicates the assumptions which are necessary in order to obtain the ADM-Dirac constraints, and furthermore shows that there are choices, other than the ADM hamilto…
The Hamiltonian formulation of general relativity on a null surface is established in the teleparallel geometry. No particular gauge conditons on the tetrads are imposed, such as the time gauge condition. By means of a 3+1 decomposition the resulting Hamiltonian arises as a completely constrained system. However, it is structurally different from the the standard Arnowitt-Deser-Misner (ADM) type formulation. In this geometrical framework the basic field quantities are tetrads that transform under the global SO(3,1) and the torsion tensor.
Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein's equations constrain our choices of these initial data. We will examine several of the formalisms used for specifying Cauchy initial data in the 3+1 decomposition of Einstein's equations. We will then explore how these formalisms have been used in constructing initial data for spacetimes containing black holes and neutron stars. In the topics discussed, emphasis is placed on those issues that are important for obtaining astrophysically realistic initial data for compact binary coale…
The geometric trinity of gravity offers a platform in which gravity can be formulated in three analogous approaches, namely curvature, torsion and nonmetricity. In this vein, general relativity can be expressed in three dynamically equivalent ways which may offer insights into the different properties of these decompositions such as their Hamiltonian structure, the efficiency of numerical analyses, as well as the classification of gravitational field degrees of freedom. In this work, we take a $3+1$ decomposition of the teleparallel equivalent of general relativity and the symmetric teleparall…
