On the hyperbolicity and stability of $$3+1$$ 3 + 1 formulations of metric f(R) gravity

Mongwane, Bishop

doi:10.1007/s10714-016-2147-x

Cited by 12 publications

(45 citation statements)

References 107 publications

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“…where a b = n c ∇ c n b is the acceleration of the normal n a , and it is related to the lapse function α via a b = D b ln α. By these definitions, the ADM decomposition of the full system is expressed as follows [34,35]…”

Section: The Adm Decomposition Of Equations Of Motion In F (R) Gravitymentioning

confidence: 99%

“…He investigated the hyperbolicity of f (R) gravity but only for a given shift function in both the ADM formulation and the BSSN formulation. Therefore, by adding all first-order derivatives as variables, he proved the ADM version of f (R) gravity is just weakly hyperbolic while the BSSN version of f (R) gravity is strongly hyperbolic [35].…”

Section: Introductionmentioning

confidence: 99%

“…On the one hand, it is known that there are many other formulations with different gauge conditions instead of the BSSN formulation with Bona-Masso slicing condition [36] which is used in the Ref. [35] for the f (R) gravity. For example, ADM formulation with harmonic gauge conditions and the Z4 formulation are common formulations.…”

Section: Introductionmentioning

confidence: 99%

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A Note on the Strong Hyperbolicity of $f(R)$ Gravity with Dynamical Shifts

Cao,

2021

Preprint

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The well-posedness of the gravitational equations of f (R) gravity are studied in this paper. Three formulations of the f (R) gravity with dynamical shifts (which are all based on the Arnowitt-Deser-Misner (ADM) formalism of the equations) are investigated. These three formulations are all proved to be strongly hyperbolic by pseudodifferential reduction. The first one is the Baumagarte-Shapiro-Shibata-Nakamura (BSSN) formulation with the so-called "hyperbolic K-driver" condition and the "hyperbolic Gamma driver" condition. The second one is the ADM formulation with modified harmonic gauge conditions. We find that the equations are not strong hyperbolic in traditional Z4 formulation for f (R) gravity. So, in the third formulation, we improve the Z4 formulation, and show these equations are strong hyperbolic with modified harmonic gauge conditions.

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Section: The Adm Decomposition Of Equations Of Motion In F (R) Gravitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Note on the Strong Hyperbolicity of $f(R)$ Gravity with Dynamical Shifts

Cao,

2021

Preprint

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“…Equation (5) governs the dynamics of the scalar degree of freedom inherent in the theory. As in the 3 + 1 formulation [56], it is convenient to use the equivalent form for the field equations…”

Section: A Field Equationsmentioning

confidence: 99%

Characteristic formulation for metric f(R) gravity

Mongwane

2017

Phys. Rev. D

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In recent years, the Characteristic formulation of numerical relativity has found increasing use in the extraction of gravitational radiation from numerically generated spacetimes. In this paper, we formulate the Characteristic initial value problem for f (R) gravity. We consider, in particular, the vacuum field equations of Metric f (R) gravity in the Jordan frame, without utilising the dynamical equivalence with scalar-tensor theories. We present the full hierarchy of non-linear hypersurface and evolution equations necessary for numerical implementation in both tensorial and eth forms. Furthermore, we specialise the resulting equations to situations where the spacetime is almost Minkowski and almost Schwarszchild using standard linearization techniques. We obtain analytic solutions for the dominant ℓ = 2 mode and show that they satisfy the concomitant constraints. These results are ideally suited as testbed solutions for numerical codes. Finally, we point out that the Characteristic formulation can be used as a complementary analytic tool to the 1 + 1 + 2 semi-tetrad formulation. * Electronic address: bishop.mongwane@uct.ac.za Within the numerical relativity community, there are now a number of Characteristic codes being used, with differing levels of sophistication. For instance, some codes employ second order finite difference schemes [38], others use higher order schemes [68] while others have adopted Spectral methods [41]. Another point of distinction among different codes is the coordinate system used to cover the sphere labelling the null directions of the light cones. Common choices range from stereographic coordinate system [15] to multi-patch coordinate systems [38,67]. There has also been efforts to introduce Adaptive Mesh Refinement schemes to Characteristic evolution codes [64,78]. Overall, these codes have made it possible to demonstrate the versatility of Characteristic methods in numerical relativity and have found extensive applications in, for example, binary black hole mergers [4,15,42,65], stellar core collapse [61,66,73], Einstein-Klein-Gordon systems [6,39,62], Observational Cosmology [9, 79, 80] etc. These systems represent potential astrophysical laboratories for testing general relativity in the non-linear regime.Over the years, the theory of general relativity has been subjected to a wide range of experimental tests and has no doubt emerged as one of the most successful theories in Physics. However, there has been considerable interest in the literature to study gravity theories whose Lagrangians contain higher order curvature invariants such as 29,71,74]. The motivation for these alternative theories of gravity stems from a variety of grounds, most notably from within the dark sector in Cosmology [28]. Moreover, the inflationary paradigm arises naturally in alternative theories of gravity without postulating additional inflaton fields [29,74,75]. These higher order corrections also arise in the effective action of quantum gravity. For example, in the low energy limit of string theory o...

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“…The ADM decomposition has been generalised for some of the modified gravity theories as well, e.g. for the f (R) gravitational theories [12].…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian Dynamics of Doubly-Foliable Space-Times

2018

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Abstract:The 2 + 1 + 1 decomposition of space-time is useful in monitoring the temporal evolution of gravitational perturbations/waves in space-times with a spatial direction singled-out by symmetries. Such an approach based on a perpendicular double foliation has been employed in the framework of dark matter and dark energy-motivated scalar-tensor gravitational theories for the discussion of the odd sector perturbations of spherically-symmetric gravity. For the even sector, however, the perpendicularity has to be suppressed in order to allow for suitable gauge freedom, recovering the 10th metric variable. The 2 + 1 + 1 decomposition of the Einstein-Hilbert action leads to the identification of the canonical pairs, the Hamiltonian and momentum constraints. Hamiltonian dynamics is then derived via Poisson brackets.

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On the hyperbolicity and stability of $$3+1$$ 3 + 1 formulations of metric f(R) gravity

Cited by 12 publications

References 107 publications

A Note on the Strong Hyperbolicity of $f(R)$ Gravity with Dynamical Shifts

A Note on the Strong Hyperbolicity of $f(R)$ Gravity with Dynamical Shifts

Characteristic formulation for metric f(R) gravity

Hamiltonian Dynamics of Doubly-Foliable Space-Times

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