A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr. :
“Einstein and Yang–Mills theories in hyperbolic form
without gauge fixing Phys. Rev. Lett.
75 : 19
(1995 ),
pp. 3377–3381 .
MR
1356171 Zbl
1020.83503 article
Abstract
People
BibTeX
The evolution of physical and gauge degrees of freedom in the Einstein and Yang–Mills theories are separated in a gauge-invariant manner. We show that the equations of motion of these theories can be written in flux-conservative first-order symmetric hyperbolic form where the only nonzero characteristic speed is that of light. This dynamical form is ideal for global analysis, analytic approximation methods such as gauge-invariant perturbation theory, and numerical solution.
@article {key1356171m,
AUTHOR = {Abrahams, Andrew and Anderson, Arlen
and Choquet-Bruhat, Yvonne and York,
Jr., James W.},
TITLE = {Einstein and {Y}ang--{M}ills theories
in hyperbolic form without gauge fixing},
JOURNAL = {Phys. Rev. Lett.},
FJOURNAL = {Physical Review Letters},
VOLUME = {75},
NUMBER = {19},
YEAR = {1995},
PAGES = {3377--3381},
DOI = {10.1103/PhysRevLett.75.3377},
NOTE = {MR:1356171. Zbl:1020.83503.},
ISSN = {0031-9007},
}
A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York :
“Un système hyperbolique non strict pour les équations d’Einstein ,”
C. R. Acad. Sci., Paris, Sér. II, Fasc. b
323 : 12
(1996 ),
pp. 835–841 .
Zbl
0916.35126 article
Abstract
People
BibTeX
We obtain a system for the spatial metric and extrinsic curvature of a spacelike slice that is hyperbolic non-strict i the sense of Leray and Ohya and is equivalent to the Einstein equations. Its characteristics are the light cone and the normal to the slice for any choice of lapse and shift functions, and it admits a well-posed causal Cauchy problem in a Gevrey class of index \( \alpha = 2 \) . The system becomes quasidiagonal hyperbolic if we posit a certain wave equation for the lapse function, and we can then relate the results to our previously obtained first order symmetric hyperbolic system for general relativity.
@article {key0916.35126z,
AUTHOR = {Abrahams, Andrew and Anderson, Arlen
and Choquet-Bruhat, Yvonne and York,
James W.},
TITLE = {Un syst{\`e}me hyperbolique non strict
pour les {\'e}quations d'{Einstein}},
JOURNAL = {C. R. Acad. Sci., Paris, S{\'e}r. II,
Fasc. b},
FJOURNAL = {Comptes Rendus de l'Acad{\'e}mie des
Sciences. S{\'e}rie II. Fascicule b},
VOLUME = {323},
NUMBER = {12},
YEAR = {1996},
PAGES = {835--841},
NOTE = {Zbl:0916.35126.},
ISSN = {1251-8069},
}
A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr. :
“Einstein–Bianchi hyperbolic system for general relativity Topol. Methods Nonlinear Anal.
10 : 2
(1997 ),
pp. 353–373 .
MR
1634577 Zbl
0917.35145 article
People
BibTeX
@article {key1634577m,
AUTHOR = {Anderson, Arlen and Choquet-Bruhat,
Yvonne and York, Jr., James W.},
TITLE = {Einstein--{B}ianchi hyperbolic system
for general relativity},
JOURNAL = {Topol. Methods Nonlinear Anal.},
FJOURNAL = {Topological Methods in Nonlinear Analysis},
VOLUME = {10},
NUMBER = {2},
YEAR = {1997},
PAGES = {353--373},
DOI = {10.12775/TMNA.1997.037},
NOTE = {MR:1634577. Zbl:0917.35145.},
ISSN = {1230-3429},
}
A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr. :
“Geometrical hyperbolic systems for general relativity and
gauge theories Classical Quantum Gravity
14 : 1A
(1997 ),
pp. A9–A22 .
MR
1691883 Zbl
0866.58059 article
Abstract
People
BibTeX
The evolution equations of Einstein’s theory and of Maxwell’s theory — the latter used as a simple model to illustrate the former — are written in gauge-covariant first-order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom (the spatial shift vector \( \beta^i(t, x^j) \) and the spatial scalar potential \( \phi(t, x^j) \) , respectively) are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of \( (t, x^j) \) from subsidiary equations that are not part of the evolution equations. Propagation of certain (“radiative”) dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by
taking a further time derivative of the equation of motion of the canonical momentum, and
adding a covariant spatial derivative of
the momentum constraints of general relativity (Lagrange multiplier \( \beta^i \) ) or of the Gauss law constraint of electromagnetism (Lagrange multiplier \( \phi \) ).
General relativity also requires a harmonic time-slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first-order symmetric form. The dynamically propagating gravity fields straightforwardly determine the “electric” or “tidal” parts of the Riemann tensor
@article {key1691883m,
AUTHOR = {Abrahams, Andrew and Anderson, Arlen
and Choquet-Bruhat, Yvonne and York,
Jr., James W.},
TITLE = {Geometrical hyperbolic systems for general
relativity and gauge theories},
JOURNAL = {Classical Quantum Gravity},
FJOURNAL = {Classical and Quantum Gravity},
VOLUME = {14},
NUMBER = {1A},
YEAR = {1997},
PAGES = {A9--A22},
DOI = {10.1088/0264-9381/14/1A/002},
NOTE = {MR:1691883. Zbl:0866.58059.},
ISSN = {0264-9381},
}
Y. Choquet-Bruhat, J. W. York, Jr., and A. Anderson :
“Curvature-based hyperbolic systems for general relativity ,”
pp. 112–121
in
The Eighth Marcel Grossmann Meeting, Part A, B
(Jerusalem, 1997 ).
Edited by T. Piran .
World Sci. Publ. (River Edge, NJ ),
1999 .
MR
1891864 Zbl
0970.83005 incollection
Abstract
People
BibTeX
@incollection {key1891864m,
AUTHOR = {Choquet-Bruhat, Yvonne and York, Jr.,
James W. and Anderson, Arlen},
TITLE = {Curvature-based hyperbolic systems for
general relativity},
BOOKTITLE = {The {E}ighth {M}arcel {G}rossmann {M}eeting,
{P}art {A}, {B}},
EDITOR = {Tsvi Piran},
PUBLISHER = {World Sci. Publ.},
ADDRESS = {River Edge, NJ},
YEAR = {1999},
PAGES = {112--121},
NOTE = {({J}erusalem, 1997). MR:1891864. Zbl:0970.83005.},
}
A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr. :
“Einstein’s equations and equivalent hyperbolic dynamical systems 30–54
in
Mathematical and quantum aspects of relativity and cosmology
(Pythagoreon, Samos, 1998 ).
Edited by S. Cotsakis and G. W. Gibbons .
Lecture Notes in Phys. 537 .
Springer (Berlin ),
2000 .
MR
1843032 Zbl
0987.83006 incollection
Abstract
People
BibTeX
We discuss several explicitly causal hyperbolic formulations of Einstein’s dynamical \( 3 + 1 \) equations in a coherent way, emphasizing throughout the fundamental role of the “slicing function,” \( \alpha \) — the quantity that relates the lapse \( N \) to the determinant of the spatial metric \( \bar{g} \) through \( N = \bar{g}^{1/2}\alpha \) . The slicing function allows us to demonstrate explicitly that every foliation of spacetime by spatial time-slices can be used in conjunction with the causal hyperbolic forms of the dynamical Einstein equations. Specifically, the slicing function plays an essential role
in a clearer form of the canonical action principle and Hamiltonian dynamics for gravity, and leads to a recasting
of the Bianchi identities \( \nabla_{\beta} G^{\beta} \alpha \equiv 0 \) as a well-posed system for the evolution of the gravitational constraints in vacuum, and also
of \( \nabla_{\beta} T^{\beta}_{\alpha} \equiv 0 \) as a well-posed system for evolution of the energy and momentum components of the stress tensor in the presence of matter,
in an explicit rendering of four hyperbolic formulations of Einstein’s equations with only physical characteristics, and
in providing guidance to a new “conformal thin sandwich” form of the initial value constraints.
@incollection {key1843032m,
AUTHOR = {Anderson, Arlen and Choquet-Bruhat,
Yvonne and York, Jr., James W.},
TITLE = {Einstein's equations and equivalent
hyperbolic dynamical systems},
BOOKTITLE = {Mathematical and quantum aspects of
relativity and cosmology},
EDITOR = {Spiros Cotsakis and Gary W. Gibbons},
SERIES = {Lecture Notes in Phys.},
NUMBER = {537},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2000},
PAGES = {30--54},
DOI = {10.1007/3-540-46671-1_2},
NOTE = {({P}ythagoreon, Samos, 1998). MR:1843032.
Zbl:0987.83006.},
}
