本文探討了廣義相對論當中的精確解以及它們在無四維時空對稱理論中的推廣。第一類是Painleve-Gullstrand(PG)類的精確解。這些精確解在全域保持Lorentz signature(-,+,+,+)以及沒有視界座標奇異點。對於非旋轉黑洞具可調函數的廣義PG度規早先由林俊鈺與許祖斌建立。PG度規與其對應的標準型式度規之間的局域Lorentz變換在視界是一個無窮Lorentz boost。在局域插入可調函數容許我們建立免除座標奇異點以及全域實數值的物理廣義PG精確解。本文中建立的廣義PG精確解包含了完整的Newman-Penrose 帶電旋轉黑洞具非零宇宙常數的精確解。對於非旋轉黑洞，我們的PG精確解包含了定曲率解。我們比較了其他已知的精確解，其中一些在某些極端區域遭遇了複數值問題。這些問題在我們的PG精確解當中都不存在，且廣義PG理論也同時是最佳化的理論因其僅需一個額外的函數來克服問題。內稟時間幾何動力學(ITG) 由許祖斌及其合作者在一系列的工作中提倡。在量子的範疇，此理論由具非零Hamiltonian的薛丁格方程描述;愛因斯坦的廣義相對論是此由ITG描述較廣義的理論的一個特殊案例。在此無四維時空對稱的框架中，高階空間曲率包含Ricci項和Cotton-York項是被容許的，並且被導入用來改善紫外收斂性。愛因斯坦理論中定三維曲率精確解具有同時是ITG理論精確解的優勢。這些包含了Schwarzschild-de Sitter精確解的PG型式。除其他事項外，如同建構新精確解，我們確切演示了Schwarzschild-de Sitter PG型式通過所有廣義相對論的觀測測試，像是近日點移以及光偏折，而其他已知Horava理論的精確解極其偏離愛因斯坦理論的預測。

In this work, exact solutions of General Relativity and its
generalization to theories without space-time four-covariance are investigated. The first class of solutions studied are Painleve-Gullstrand(PG) type solutions. These maintain Lorentzian signature (-,+,+,+) globally and possess no coordinate singularities at the horizon(s). For non-rotating black holes, generalized PG metrics with adjustable functions were established previously by Lin and Soo. The local Lorentz transformation between the PG solution and the corresponding metric in standard form is an infinite Lorentz boost at the horizon(s). Inserting suitable adjustable functions in the local Lorentz
transformation allow us to construct physical generalized PG
solutions which are completely free of coordinate singularities and remain real everywhere. The generalized PG solutions constructed include the complete Newman-Penrose family of charged rotating black holes in the presence of non-trivial cosmological constant. For
non-rotating black holes, our PG solutions possess constant
curvature slicings. We compare with well known solutions, some of which become complex when certain extreme ranges of the parameters are encountered. These problems are absent in our PG solutions which are also optimal in that only one extra function for each solution is introduced to cure the problem.

Intrinsic time geometrodynamics (ITG) was advocated in a series of work by Soo and co-authors. In the quantum context, the theory is described by a Schrodinger equation with a non-trivial physical Hamiltonian; and Einstein's GR is a special case of a wider class of theories described by (ITG). In this scheme without paradigm of space-time four-covariance, higher order spatial curvatures, including Ricci and Cotton-York terms are permitted, and are
introduced to improve the ultra-violet convergence. Constant
three-curvature solutions of Einstein's theory have the advantage of being also exact solutions of ITG. These include the Schwarzschild-de Sitter solution in PG form. Among other things, as well as the construction of new solutions, we explicitly demonstrate that the Schwarzschild-de Sitter PG form passes all the observational tests of GR, such as perihelion shift and the bending of light, whereas other known solutions of Horava gravity theories depart starkly from the predictions of Einstein's theory.

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