At every spacetime point in an arbitrary gravitational field, it is possible to choose a local inertial coordinate system, such that, within a sufficient small region of the point in question, the lows of the nature takes the same form as in un-accelerated cartesian coordinate in the absence if gravitation.

—— Weinberg 书中的等效原理表述

有了等效原理之后，定义一个在 XX 时空点的局域参考系 ξXα\xi^\alpha_X，没有引力时，在其中的固有时应该就是

dτ2=−ηαβdξXαdξXβ\text{d}\tau^2 = -\eta_{\alpha\beta}\text{d}\xi^\alpha_X\text{d}\xi^\beta_X

在另一个全局的参考系中，XX 这一点的坐标是 xμ(ξXα)x^\mu(\xi^\alpha_X). 把上面的固有时用 chain rule 重写，得到

dτ2=−ηαβ∂ξXα∂xμ∂ξXβ∂xνgμνdxμdxν=−gμν(X)dxμdxν\text{d}\tau^2 = -\underset{g_{\mu\nu}}{\boxed{\eta_{\alpha\beta}\frac{\partial\xi^\alpha_X}{\partial x^\mu}\frac{\partial\xi^\beta_X}{\partial x^\nu}}}\text{d}x^\mu\text{d}x^\nu = -g_{\mu\nu}(X)\text{d}x^\mu\text{d}x^\nu

前面的部分定义为 gμνg_{\mu\nu}，即为度规. 我们还可以计算在这个新的系统中，运动物体的方程的变化. 新的参考系里我们依然有

d2ξXαdτ2=0=ddτ(dξXαdτ)=ddτ(∂ξXα∂xμdxμdτ)=∂ξXα∂xμd2xμdτ2+∂2ξXα∂xμ∂xν∂xμ∂τ∂xν∂τ⟹d2xλdτ2+∂xλ∂ξα∂2ξα∂xμ∂xν∂xμ∂τ∂xν∂τ=0\begin{aligned} \frac{\text{d}^2\xi^\alpha_X}{\text{d}\tau^2} &= 0 = \frac{\text{d}}{\text{d}\tau}\left(\frac{\text{d}\xi^\alpha_X}{\text{d}\tau} \right) = \frac{\text{d}}{\text{d}\tau}\left(\frac{\partial\xi^\alpha_X}{\partial x^\mu}\frac{\text{d}x^\mu}{\text{d}\tau} \right)\\\\ &= \frac{\partial\xi_X^\alpha}{\partial x^\mu}\frac{\text{d}^2x^\mu}{\text{d}\tau^2}+\frac{\partial^2\xi_X^\alpha}{\partial x^\mu\partial x^\nu}\frac{\partial x^\mu}{\partial\tau}\frac{\partial x^\nu}{\partial\tau}\\\\ \Longrightarrow&\frac{\text{d}^2x^\lambda}{\text{d}\tau^2} + \frac{\partial x^\lambda}{\partial\xi^\alpha}\frac{\partial^2\xi^\alpha}{\partial x^\mu\partial x^\nu}\frac{\partial x^\mu}{\partial\tau}\frac{\partial x^\nu}{\partial\tau} = 0 \end{aligned}

下面再来算上节课算的东西：

∂gμν∂xρ=∂∂xρ(ηαβ∂ξXα(x)∂xμ∂ξXβ(x)∂xν)\begin{aligned} \frac{\partial g_{\mu\nu}}{\partial x^\rho} &= \frac{\partial}{\partial x^\rho}\left(\eta_{\alpha\beta}\frac{\partial\xi^\alpha_X(x)}{\partial x^\mu}\frac{\partial\xi^\beta_X(x)}{\partial x^\nu} \right) \end{aligned}

因为我们的假设 ∂ξXα(x)∂X∣x=X=0\displaystyle{\left.\frac{\partial\xi^\alpha_X(x)}{\partial X}\right|_{x=X}=0}，所以这个式子和上节课推出来的结果并无差异，

Γμρσ=12gμλ(gρλ,σ+gσλ,ρ−gρσ,λ)\Gamma^\mu{}_{\rho\sigma} = \frac{1}{2}g^{\mu\lambda}\left(g_{\rho\lambda,\sigma}+g_{\sigma\lambda,\rho}-g_{\rho\sigma,\lambda} \right)

同时我们可以知道在局域参考系中，gμν=ημνg_{\mu\nu}=\eta_{\mu\nu} (平直时空)，Γμλν=0\Gamma^\mu{}_{\lambda\nu} = 0，gμν,λ=0g_{\mu\nu,\lambda}=0.

为了更明确地理解度规的意义，我们来算一些熟悉的概念. 首先明确，Γ\Gamma 实际上是「力」的概念，引力包含在 Γ\Gamma 里面，而 Γ\Gamma 是 gg 的导数，所以合理的想法是 gg 对应着 potential. 考虑低速近似 v≪c=1v\ll c=1，弱场近似 gμν=ημν+hμνg_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} 且 ∣hμν∣≪1|h_{\mu\nu}|\ll1，并且场不随时间改变.

Newton 定律

d2xλdτ2+Γλρσdxρdτdxσdτ=0\frac{\text{d}^2x^\lambda}{\text{d}\tau^2}+\Gamma^\lambda{}_{\rho\sigma}\frac{\text{d}x^\rho}{\text{d}\tau}\frac{\text{d}x^\sigma}{\text{d}\tau}=0

因为 dxρ/dτ→1\text{d}x^\rho/\text{d}\tau\to1，dτ→dt\text{d}\tau\to\text{d}t，变为

d2xλdt2+Γλ00=0⟹d2xλdt2+12gλμ(g0μ,0+gμ0,0−g00,μ)=d2xλdt2−12gλμg00,μ=0\begin{aligned} &\frac{\text{d}^2x^\lambda}{\text{d}t^2}+\Gamma^\lambda{}_{00}=0\\\\ \Longrightarrow&\frac{\text{d}^2x^\lambda}{\text{d}t^2}+\frac{1}{2}g^{\lambda\mu}(g_{0\mu,0}+g_{\mu0,0}-g_{00,\mu})=\frac{\text{d}^2x^\lambda}{\text{d}t^2}-\frac{1}{2}g^{\lambda\mu}g_{00,\mu}=0 \end{aligned}

最终得到

d2xλdt2−12ηλμh00,μ=0\frac{\text{d}^2x^\lambda}{\text{d}t^2}-\frac{1}{2}\eta^{\lambda\mu}h_{00,\mu}=0

一共四个方程，其中，λ=0\lambda=0 的方程是 trivial，只用考虑 λ=i\lambda=i 的情况，为

d2xidt2−12h00,i=0⟺d2xidt2+∇ϕ=0\frac{\text{d}^2x^i}{\text{d}t^2}-\frac{1}{2}h_{00,i}=0\Longleftrightarrow \frac{\text{d}^2x^i}{\text{d}t^2}+\nabla\phi = 0

这就得到 h00=−2ϕ+const.h_{00}=-2\phi+\text{const.}，求出 (令常数为零)

g00=−(1−2GMr)g_{00} = -\left(1-\frac{2GM}{r}\right)

广义协变性原理：

1. The equation holds in the absence of gravitation. That is, is agrees with laws of special relativity when the metric tensor gαβg_{\alpha\beta} equals the Minkowski tensor ηαβ\eta_{\alpha\beta} and when the affine connection Γαβγ\Gamma^\alpha{}_{\beta\gamma} vanishes.
2. The equation is generally covariant; that is, it preserves its form under a general coordinate transformation.

我们需要把各个几何量和物理量变成协变的形式. 简单的例子：

dx′μ=(∂x′μ∂xα)dxα\text{d}x'^\mu = \left(\frac{\partial x'^\mu}{\partial x^\alpha} \right)\text{d}x^\alpha

对于度规，

gρσ′=gμν∂xμ∂x′ρ∂xν∂x′σg'_{\rho\sigma} = g_{\mu\nu}\frac{\partial x^\mu}{\partial x'^\rho}\frac{\partial x^\nu}{\partial x'^\sigma}

对于加速度：

dU′μdτ=ddτ(dx′μdτ)=ddτ(∂x′μ∂xνdxνdτ)=∂x′μ∂xνd2xνdτ2+dxνdτ∂2x′μ∂xν∂xρdxρdτ=∂x′μ∂xνdUνdτ+∂2x′μ∂xν∂xρUρUν\begin{aligned} \frac{\text{d}U'^\mu}{\text{d}\tau}&=\frac{\text{d}}{\text{d}\tau}\left(\frac{\text{d}x'^\mu}{\text{d}\tau} \right) = \frac{\text{d}}{\text{d}\tau}\left(\frac{\partial x'^\mu}{\partial x^\nu}\frac{\text{d}x^\nu}{\text{d}\tau}\right)\\\\ &=\frac{\partial x'^\mu}{\partial x^\nu}\frac{\text{d}^2x^\nu}{\text{d}\tau^2}+\frac{\text{d}x^\nu}{\text{d}\tau}\frac{\partial^2x'^\mu}{\partial x^\nu\partial x^\rho}\frac{\text{d}x^\rho}{\text{d}\tau}\\\\ &= \frac{\partial x'^\mu}{\partial x^\nu}\frac{\text{d} U^\nu}{\text{d}\tau}+\frac{\partial^2x'^\mu}{\partial x^\nu\partial x^\rho}U^\rho U^\nu \end{aligned}

这不是一个张量. 下面看联络的坐标变换.

Γμρσ=∂xμ∂ξα∂2ξα∂xρ∂xσ⟹Γ′νκλ=∂x′ν∂ξα∂2ξα∂x′κ∂x′λ=∂x′ν∂xμ∂xμ∂ξα∂∂x′κ(∂ξα∂xσ∂xσ∂x′λ)=∂x′ν∂xμ∂xμ∂ξα(∂2xσ∂x′κ∂x′λ∂ξα∂xσ+∂xσ∂x′λ∂2ξα∂xσ∂xρ∂xρ∂x′κ)=∂x′ν∂xμ∂xσ∂x′λ∂xρ∂x′κΓμσρ+∂x′ν∂xμ∂2xσ∂x′κ∂x′λ∂xμ∂xσ\begin{aligned} \Gamma^\mu{}_{\rho\sigma} &= \frac{\partial x^\mu}{\partial\xi^\alpha}\frac{\partial^2\xi^\alpha}{\partial x^\rho\partial x^\sigma}\\\\ \Longrightarrow \Gamma'^\nu{}_{\kappa\lambda}&= \frac{\partial x'^\nu}{\partial\xi^\alpha}\frac{\partial^2\xi^\alpha}{\partial x'^\kappa\partial x'^\lambda} = \frac{\partial x'^\nu}{\partial x^\mu}\frac{\partial x^\mu}{\partial\xi^\alpha}\frac{\partial}{\partial x'^\kappa}\left(\frac{\partial\xi^\alpha}{\partial x^\sigma}\frac{\partial x^\sigma}{\partial x'^\lambda} \right)\\\\ &= \frac{\partial x'^\nu}{\partial x^\mu}\frac{\partial x^\mu}{\partial\xi^\alpha}\left(\frac{\partial^2x^\sigma}{\partial x'^\kappa\partial x'^\lambda}\frac{\partial\xi^\alpha}{\partial x^\sigma}+\frac{\partial x^\sigma}{\partial x'^\lambda}\frac{\partial^2\xi^\alpha}{\partial x^\sigma \partial x^\rho}\frac{\partial x^\rho}{\partial x'^\kappa} \right)\\\\ &= \frac{\partial x'^\nu}{\partial x^\mu}\frac{\partial x^\sigma}{\partial x'^\lambda}\frac{\partial x^\rho}{\partial x'^\kappa}\Gamma^\mu{}_{\sigma\rho} + \frac{\partial x'^\nu}{\partial x^\mu}\frac{\partial^2x^\sigma}{\partial x'^\kappa\partial x'^\lambda}\frac{\partial x^\mu}{\partial x^\sigma} \end{aligned}