| 摘要: | 我們計畫持續這幾年來關於動力時空幾何的研究路線，特別是引力的局部時空對稱的規範理論(獨立的度規和聯絡，特別是Poincare 規範理論(PGT)、度規仿射(MA)理論、 teleparallel 理論和Einstein 的廣義相對論)。我們的興趣著重在準局域引力能量。主要的方法是透過我們的協變哈米爾頓數學形式。我們首要的工具有微分幾何、變分原理、正則Hamiltonian分析、旋量技巧、符號和數值的電腦計算。我們預期有新的結果包括，可傳播模式的動力學、拘束條件、規範(特別是非線性的效應)、引力的能量-動量(也包括角動量和質心矩)準局域性、能量的正定性和長距離的效應。我們正在研究可替換的引力理論，該理論能夠說明銀河尺度的觀測而不用大量的暗物質和暗能量。藉由符號的電腦計算我們能夠尋找PGT、MA和teleparallel理論新的解；藉由數值計算，我們可能可以做出宇宙模型或完整愛因斯坦方程的動力學演化，用來確認能量和輻射。我們將繼續從事電動力學的輻射反作用力研究。我們希望更進一步發展一般協變正則分析和準局域能動的表示式、檢驗準則、和別人的結果比較，並且應用到黑洞熱力學上，除此之外，探索更多他們與膺張量和邊界值選擇的關係，我們計畫更進一步探討Bondi極限、協變性、準局域角動量、物質的質心和能量流，以及旋量形式的限制和應用範圍。我們將持續改進協變正則表示式，並更深入研究引力能量表達式的小範圍極限。應用上，我們會繼續追求對其他可替換理論的正能量測試，本質的要求(概略地說，儘管有暗能量，但重力本質上是完全相吸的)已經顯示大部分可替換的理論不能說明銀河尺度的動力學，也顯示PGT對大多數的參數值是不可行的。對於特殊PGT和MA為例的拘束分析透露了它們的規範結構，隱藏的規範對稱性可能導致嚴重的問題。我們發現PGT有著不尋常的拘束條件分支現象是緊密的和快子傳播的問題(可能是決定性的)相關聯。這兩者都來自於非線性拘束，它們對可替換的引力理論提供一個強的測試，這個測試給PGT和MA的參數嚴格的條件，我們發現動力學扭量的純量模式是可行的；他們也許可以解釋宇宙的加速膨脹效應。對於有希望的理論，我們企圖使用我們曾在廣義相對論的工作中使用的技巧給出一個正能量的證明：旋量場和我們特殊正交標架(SOF)的規範條件(現在，我們已證明出它的存在性及唯一性)。旋量和幾何(Clifford)代數的新發展，使一個用於動力幾何和一個新的引力規範理論的時空代數的數學架構成為可能。我們想更進一步研究該理論和這些技巧的其他應用，我們發現一類重要的旋量曲率恆等式。一方允許一個新的正能量的證明，另一方提供一個廣義相對論的二次旋量拉格朗日函數，也因此連接自身對偶引力、Ashtekar 變數、 teleparallel 理論、特殊正交標架、正定性證明和局域化的關係。但是，這些美麗的旋量形式對於角動量和質心卻有著嚴苛的限制，對此，我們正在研究一些新的想法。這些 Clifford 代數、幾何的計算、旋量方法和我們的SOF 規範條件的實用性和效能將被探索。我們主要的考量在於重力系統的適切能量表示。最近我們驚訝的發現，許多宇宙學模型具有負能量，這和我們對正能量的預期有很大的衝突。瞭解它是我們優先處理的工作之一。 ; Geometrodynamics, Gravitation, Gauge Theory, Observable Effects, Hamiltonian, vravitational Energy, Quasi-local, Positive Energy, Energy Flux, Symbolic Computation, Numerical Computation, Clifford Algebra, Spinors We propose to continue our lines of investigation of recent years into dynamic spacetime geometry, mainly for gauge theories of gravity based on local spacetime symmetry (independent metric & connection, especially the Poincar´e Gauge Theory (PGT), the metric-affine (MA) theory, and teleparallel theories as well as Einstein’s GR). Our major interest is quasi-local gravitational energy. The main approach is via our covariant Hamiltonian formalism. The principal tools are differential geometry, the variational principle, the canonical Hamiltonian analysis, spinor techniques, symbolic and numerical computer calculations. We expect some new results re propagating mode dynamics, constraints & gauges (especially re nonlinear effects), gravitational energy-momentum (also angular momentum & center-of-mass) and its quasi-localization, energy positivity, and long range effects. We are interested in alternate gravity theories which can account for the galactic scale observations without large amounts of “dark matter” a/o account for the accelerating universe and “dark energy”. Using symbolic computer calculations we can seek new solutions for the PGT, MA and teleparallel theories. With numerical calculations we could do the dynamical evolution of cosmological models or the full Einstein equations, identifying the energy and radiation. We are also interested in electrodynamic radiation reaction. We hope to further develop our general covariant canonical analysis and our quasi-local energymomentum expressions, examining criteria, comparing with other expressions, refining their application to black hole thermodynamics, probing more their relationship with pseudotensors and the choice of boundary and reference values. We intend to look deeper into the Bondi limit, covariance, quasi-local angular momentum, center-of mass, energy flux and the scope and limitations of spinor formulations. We will continue developing our manifestly covariant canonical formalism and will further refine our study of the under-appreciated small region limit. Applications of our positive energy test to alternate theories will be pursued. This essential requirement (roughly, notwithstanding dark energy, gravity is innately purely attractive) has already shown that a large class of alternate theories cannot account for the dynamics at the galactic scale and that most PGT parameter values are not viable. The constraint analysis of special PGT and MA cases reveals their gauge structure. Hidden gauge symmetries can pose serious problems. We found that most PGT models have the curious conditional constraint bifurcation phenomena, intimately connected to another (probably fatal) problem: tachyonic propagation. Both arise from non-linear constraints; they provide a strong alternate gravity test, imposing severe conditions on the PGT & MA parameters. We found that the scalar dynamic torsion modes are ok; and in fact they could account for the accelerating universe. For promising theories we will seek a positive energy proof using techniques which worked for GR: spinor fields and our special orthonormal frame (SOF) gauge conditions (now we can show existence and uniqueness). New developments in spinor and geometric (Clifford) algebra enabled a geometric spacetime algebra formulation for dynamic geometry and a new gravitational gauge theory. We want to further study this theory and other applications of these techniques. We found an important class of spinor-curvature identities. One permitted a new positive energy proof, another provided a quadratic spinor Lagrangian for GR, thereby links between self dual gravity, Ashtekar variables, teleparallel theory, special orthonormal frames, positivity proofs and localizations. But these beautiful spinor formulations have severe limitations re angular momentum and the center-of-mass. We are investigating some new ideas that might get around this. The utility and efficacy of these Clifford algebra/geometric calculus/spinor methods and our SOF gauge conditions will be explored. Our main concern is the proper formulation of energy for gravitating systems. Recently we found, surprisingly, that many cosmological models have negative energy, in sharp contrast with our positivity expectations. Understanding this is one of our priorities. ; 研究期間 9808 ~ 9907 |
