Mathematical general relativity: A sampler

Chruściel, Piotr T.; Galloway, Gregory J.; Pollack, Daniel

doi:10.1090/s0273-0979-2010-01304-5

Cited by 72 publications

(75 citation statements)

References 348 publications

(410 reference statements)

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“…It is a consequence of Conclusion 3.1 and Theorems 3.1 and 3.2 for superpositions of ansatzes (39) and (40) resulting into (8). If we repeat the computations from Appendix Appendix A for geometric data (9) considering that contributions of type Γ · Γ are small, we can see that the equations (42)-(44) are derived to be, respectively, equivalent to sums of (29)-(31) and (34)-(36). The torsionless conditions (45) consist a sum of similar conditions (32) and (37).…”

Section: Decoupling With Effective Linearization Of the Ricci Tensorsmentioning

confidence: 96%

“…We can split such systems for more general parameterizations (than ansatzes (39) and (40)) in a form (8) with nontrivial ω and N-coefficients in (9). This is possible in any open region U ⊂ V where for computing the N-adapted coefficients of the Riemann and Ricci d-tensors, see formulas (15) and (16), we can neglect contributions from quadratic terms of type Γ · Γ but preserve values of type ∂ µ Γ.…”

Section: Decoupling With Effective Linearization Of the Ricci Tensorsmentioning

confidence: 99%

“…The bulk of known exact solutions with diagonalizable metrics and coefficients depending on two coordinates (in certain special frames of references) can be included as special cases for more general nonholonomic configurations. (8) with data ω = 1, h 3 = 1, w i = 0 and n i = 0 for (9), when h * a = 0, and the condition that the source…”

Section: Generating Solutions With Weak One Killing Symmetrymentioning

confidence: 99%

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Decoupling of Field Equations in Einstein and Modified Gravity

Vacaru¹

2013

J. Phys.: Conf. Ser.

129

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Abstract. This paper is concerned with giving the proof that there is a general decoupling property of vacuum and nonvacuum gravitational field equations in Einstein gravity and f (R, T )-modifications. The constructions are possible in terms of geometric and physical objects adapted to certain nonholonomic 2+2 splitting with local fibred structure. This allows us to generate exact and/or parametric depending solutions with generic off-diagonal metrics and generalized, or Levi-Civita, connections. Different classes of modified spacetimes are determined by corresponding generating and integration functions depending, in general, on all space and time coordinates and may possess, or not, Killing symmetries. The initial data sets for the Cauchy problem and their global properties are analyzed. There are formulated the criteria of evolution with spacetime splitting and decoupling of fundamental field equations. Examples of exact solutions defining ellipsoid deformations of black hole metrics and solitonic configurations are provided. IntroductionThe issue of constructing exact and approximate solutions of (modified) gravitational and matter field equations is of interest in mathematical physics and for various applications in modern cosmology and astrophysics. Mathematically, the fundamental field equations in gravitational physics are defined by sophisticate systems of nonlinear partial differential equations (PDE) which are very difficult to be integrated and studied in general forms.In this work, we address again and develop a geometric approach (the so-called anholonomic frame deformation method, AFDM, see [20,21,22]) to constructing exact solutions in gravity described by generic off-diagonal and nonlinear gravitational, gauge and scalar field interactions. Such a geometric method allows us to generate exact solutions when the coefficients of generic off-diagonal metrics and various classes of connections depend on all spacetime coordinates. Our goal is to provide new results on the decoupling property 1 of modified gravity equations and formal integration of such nonlinear systems of PDEs. Certain issues on the Cauchy problem and decoupling of gravitational equations and the initial data sets and nonholonomic evolution (with non-integrable constraints and/or with respect to anholonomic frames) will be analyzed. Examples of off-diagonal solutions for modified black holes/ellpsoids and solitonic configurations will be given.

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Section: Decoupling With Effective Linearization Of the Ricci Tensorsmentioning

confidence: 96%

Section: Decoupling With Effective Linearization Of the Ricci Tensorsmentioning

confidence: 99%

Section: Generating Solutions With Weak One Killing Symmetrymentioning

confidence: 99%

See 1 more Smart Citation

Decoupling of Field Equations in Einstein and Modified Gravity

Vacaru¹

2013

J. Phys.: Conf. Ser.

129

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“…These are, in 4-dimensions, (22) assuming that the cosmological constant Λ = 0 and µ, ν = 0, 1, 2, 3. Consider the null energy condition [35], which is (arguably) the most fundamental of the energy conditions [36],…”

Section: Why Convexity and Smoothness?mentioning

confidence: 99%

Convexity and the Euclidean Metric of Space-Time

Kalogeropoulos

2017

Universe

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Abstract:We address the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.

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“…Other important examples include the Schwarzschild and the Kerr spacetimes. We refer to the survey article [22] for basic materials in general relativity.…”

Section: Introductionmentioning

confidence: 99%

Energy, momentum, and center of mass in general relativity

Wang

2016

Surveys in Differential Geometry

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Abstract. These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein's time. I shall explain these difficulties and progresses that have been made. In particular, I shall introduce new definitions of center of mass and angular momentum at both the quasi-local and total levels, which are derived from first principles in general relativity and by the method of geometric analysis. With these new definitions, the classical formula p = mv is shown to be consistent with Einstein's field equation for the first time. This paper is based on joint work [14,15] with Po-Ning Chen and Shing-Tung Yau.

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Mathematical general relativity: A sampler

Abstract: Abstract. We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.

Cited by 72 publications

References 348 publications

Decoupling of Field Equations in Einstein and Modified Gravity

Decoupling of Field Equations in Einstein and Modified Gravity

Convexity and the Euclidean Metric of Space-Time

Energy, momentum, and center of mass in general relativity

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