论文标题
基于弯曲切线空间的正常坐标
Normal coordinates based on curved tangent space
论文作者
论文摘要
riemann普通坐标（RNC）在常规事件下$ p_0 $ p_0 $ p_0 $ p_0 $ \ nathcal {m} $是通过施加的：（i）$ g _ {$ g _ {\ textsf {ab}}} | _ {p_0}} $γ^\ textsf {a} _ {\ phantom {\ textsf a} \ textsf {bc}}} | _ {p_0} = 0 $。但是，在RNC的定义中，有第三个，$独立的$，它基本上修复了$密度$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ p_0 $从$ p_0 $发出的价值，即在平坦的时空中的价值。我们放松（iii）并获得普通坐标，以及$ \ nathcal {t} _ {p_0}（\ nathcal {m {m}）$的指标$ g _ {\ textsf {ab}} $ $ |λ|^{ - 1/2} $。通常，这些坐标定义的“休息”帧是非惯用性的，其附加加速度$ \ boldsymbol a = - （λ/3）\，\ boldsymbol x $，具体取决于切线空间的曲率。我们的几何设置提供了具有宇宙常数$λ$的局部物理的方便探测，现在嵌入到时空的本地结构中，作为与弯曲的切线空间相关的基本常数。我们讨论了同一的经典和量子含义。
Riemann normal coordinates (RNC) at a regular event $p_0$ of a spacetime manifold $\mathcal{M}$ are constructed by imposing: (i) $g_{\textsf{ab}}|_{p_0}=η_{ab}$, and (ii) $Γ^\textsf{a}_{\phantom{\textsf a}\textsf{bc}}|_{p_0}=0$. There is, however, a third, $independent$, assumption in the definition of RNC which essentially fixes the $density$ $of$ $geodesics$ emanating from $p_0$ to its value in flat spacetime, viz.: (iii) the tangent space $\mathcal{T}_{p_0}(\mathcal{M})$ is $flat$. We relax (iii) and obtain the normal coordinates, along with the metric $g_{\textsf{ab}}$, when $\mathcal{T}_{p_0}(\mathcal{M})$ is a maximally symmetric manifold $\widetilde{\mathcal M}_Λ$ with curvature length $|Λ|^{-1/2}$. In general, the "rest" frame defined by these coordinates is non-inertial with an additional acceleration $\boldsymbol a = - (Λ/3) \, \boldsymbol x$ depending on the curvature of tangent space. Our geometric set-up provides a convenient probe of local physics in a universe with a cosmological constant $Λ$, now embedded into the local structure of spacetime as a fundamental constant associated with a curved tangent space. We discuss classical and quantum implications of the same.
