论文标题
迈向一般相对论的地球流量束形式主义：Riemannian案例
Towards a Geodesic Flow Bundle Formalism of General Relativity: Riemannian case
论文作者
论文摘要
重力是由于时空几何形状引起的现象。描述重力的主要方程是爱因斯坦方程。为了了解这些场方程的后果，我们需要计算到几何形状的自由粒子世界，这些粒子可以解决这些场方程，例如Schwarzschild指标求解了爱因斯坦方程，我们需要将地球方程求解到该度量标准。如果我们用大地测量而不是度量来描述时空几何形状，我们可以跳过求解度量的步骤，直接求解大地测量学。在这项工作中，我们开发了一种形式主义：我们使用歧管中所有点的Arclength参数化测量学（测量流束束）的束来描述Riemannian的几何形状。我们的形式主义使用无限的球形三角形作为产生元素来解决几何问题。我们将测量流捆绑包与高斯曲率相关联，并开发一种从高斯曲率场开始计算几何统计学的方法。结果相当于对恒定曲率的余弦和正弦法对变化的曲率磁场的概括。在这项工作中，我们将自己限制在Riemannian案例和积极的曲率领域。我们在功率系列中扩展了三角剖分问题，并将主要结果计算为二阶。该方法本身可以扩展以治疗更多的通用情况，特别是与Einsteins方程直接相关的伪riemannian几何形状。 我们针对球体测试结果，并执行一致性检查，以获取一些不同曲率场的示例。作为副产品，我们将整合的概念推广到产品中，并得出与积分积分的主要定理的关系类似物。
Gravity is a phenomenon which arises due to the space-time geometry. The main equations that describe gravity are the Einstein equations. To understand the consequences of these field equations we need to calculate the free particle worldlines to the geometries, which solve these field equations e.g. the Schwarzschild metric solves the Einstein equations and we would need to solve the geodesic equations to this metric. If we were to describe the space-time geometry in terms of geodesics instead of the metric, we could skip the step of solving for the metric and solve for the geodesics directly. In this work we have developed a formalism doing that: we use the bundle of the arclength parametrized geodesics (geodesic flow bundle GFB) from all points in the manifold to describe a Riemannian geometry. Our formalism uses infinitesimal spherical triangles as generating elements, to solve geometric problems. We relate the geodesic flow bundle to Gaussian curvature and develop a method to calculate the geodesics geometrically starting from a Gaussian curvature field. The result amounts to a generalization of the cosine- and sine-laws for constant curvature to varying curvature fields. In this work we restrict ourselves to the Riemannian case and positive curvature fields. We expand the triangulation problem in power series and calculate the main result up to second order. The method itself could be extended to treat more generic cases in particular pseudo-Riemannian geometries which relate directly to Einsteins equations. We test our results against the sphere and perform consistency checks for some examples of varying curvature fields. As a by-product we generalize the notion of integration to products and derive a relation analogue to the main theorem of calculus, for product integrals.
