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最新推荐文章于 2025-02-03 21:53:18 发布

原创最新推荐文章于 2025-02-03 21:53:18 发布 · 222 阅读

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这里记下一些疑问
1. 比如像 $\displaystyle \frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0$ 用特征线法可以裂成 $u_t \pm u_x=0$ ,如果从象征（or 傅立叶变换）来理解也对。 $\displaystyle g^{ij}g_{ab,ij}=0$ 可以裂开成类似于特征线那样的一阶方程组吗？问题的关键是这里的 $g^{ij}$ 是待解的，系数是变的，傅立叶变换以后没有好看的结果呢。
2. 考虑带有标量场的情况时,设 $\phi$ 是标量，它满足波方程 $g^{\alpha \beta}\partial_{\alpha}\partial_{\beta}\phi=0$ ,化成一阶方程的时候，我考虑最简单的 $k=\partial_t \phi,\xi_i=\partial_i\phi$ ,直接将标量场化简

\partial t k + 2 g 0 i g 00 \partial i k + g i j g 00 \partial j ξ i = 0 (1)

$\partial_t k+\frac{2g^{0i}}{g^{00}}\partial_ik+\frac{g^{ij}}{g^{00}}\partial_j \xi_i=0\tag{1}$

\partial t ξ i - \partial i k = 0 (i = 1, 2, 3) (2~4)

$\partial_t \xi_i -\partial_i k=0\ (i=1,2,3)\tag{2~4}$

\partial t ϕ \partial t k + 2 g 0 i g 00 \partial i k + g i j g 00 \partial j ξ i \partial t ξ i - \partial i k = k = \partial V \partial ϕ = 0

$\begin{aligned} \partial_t \phi&=k\\ \partial_t k+\frac{2g^{0i}}{g^{00}}\partial_ik+\frac{g^{ij}}{g^{00}}\partial_j \xi_i&=\frac{\partial V}{\partial \phi}\\ \partial_t \xi_i -\partial_i k&=0 \end{aligned}$
问题是：（1）这样做会不会产生非物理的速度？参考Kashif Alvi ：First-order symmetrizable hyperbolic formulations of Einstein’s equations including lapse and shift as dynamical fields （2）假如写成这样，加在Lee 的文章里面，会不会线性退化？检验一下。先看看一阶的情况

第二种格式：

检验特征速度：

假如令

k=∂tϕ+g0ig00∂iϕ,ξi=∂iϕ, $\displaystyle k=\partial_t \phi+\frac{g^{0i}}{g^{00}}\partial_i \phi,\xi_i=\partial_i \phi,$ 那么

∂tξi=∂i∂tϕ=∂i(k−g0ig00∂iϕ) $\displaystyle \partial_t \xi_i=\partial_i\partial_t \phi=\partial_i(k-\frac{g^{0i}}{g^{00}}\partial_i \phi)$
则

∂tk+g0ig00∂ik+γijg00∂iξj= $\displaystyle \partial_t k +\frac{g^{0i}}{g^{00}}\partial_i k+\frac{\gamma^{ij}}{g^{00}}\partial_i \xi_j=$ ?

∂tk=∂2tϕ+(−βi)t∂iϕ−βi∂i∂tϕ $\displaystyle \partial_t k=\partial^2_t \phi +(-\beta^i)_t \partial_i \phi-\beta^i\partial_i\partial_t \phi$

g0ig00∂ik=−βi∂t∂iϕ−βi(−βj),i∂jϕ+βiβj∂i∂jϕ $\displaystyle \frac{g^{0i}}{g^{00}}\partial_i k=-\beta^i\partial_t\partial_i\phi -\beta^i(-\beta^j)_{,i}\partial_j \phi+\beta^i\beta^j\partial_i\partial_j \phi$

γijg00∂iξj=γijg00∂i∂jϕ $\displaystyle \frac{\gamma^{ij}}{g^{00}}\partial_i \xi_j=\frac{\gamma^{ij}}{g^{00}}\partial_i \partial_j \phi$
最终可以写成

\partial t k + g 0 i g 00 \partial i k + γ i j g 00 \partial i ξ j = □ ϕ + (- β j), t ξ j + β i (β j), i \partial j ϕ

$\partial_t k +\frac{g^{0i}}{g^{00}}\partial_i k+\frac{\gamma^{ij}}{g^{00}}\partial_i \xi_j=\square \phi+(-\beta^j)_{,t}\xi_j+\beta^i(\beta^j)_{,i}\partial_j\phi$

= □ ϕ + N g i j [N (H j + g k l Γ i k l) - \partial j N]

$=\square \phi+Ng^{ij}[N(H_j+g^{kl}\Gamma_{ikl})-\partial_jN]$
宁一方面：

\partial t ξ i + g 0 j g 00 \partial j ξ i - \partial i k = \partial i (β j) ξ j

$\partial_t \xi_i+\frac{g^{0j}}{g^{00}}\partial_j \xi_i-\partial_i k=\partial_i(\beta^j)\xi_j$ 这里的

βj=−g0jg00 $\displaystyle \beta^j=-\frac{g^{0j}}{g^{00}}$
改写二阶为一阶：
令

k=∂tϕ+g0ig00∂iϕ,ξi=∂iϕ, $\displaystyle k=\partial_t \phi+\frac{g^{0i}}{g^{00}}\partial_i \phi,\xi_i=\partial_i \phi,$ 那么

∂tξi=∂i∂tϕ=∂i(k−g0ig00∂iϕ) $\displaystyle \partial_t \xi_i=\partial_i\partial_t \phi=\partial_i(k-\frac{g^{0i}}{g^{00}}\partial_i \phi)$
记

T=2Nφcd(gijΦicaΦjdb−ΠcaΠbd−φefΓaceΓbdf)−12NtctdΠcdΠab−NtcΠcigijΦjab−2N∇(aHb) $T=2N\varphi^{cd}(g^{ij}\Phi_{ica}\Phi_{jdb}-\Pi_{ca}\Pi_{bd}-\varphi^{ef}\Gamma_{ace}\Gamma_{bdf})-\frac{1}{2}Nt^ct^d\Pi_{cd}\Pi_{ab}-Nt^c\Pi_{ci}g^{ij}\Phi_{jab}-2N\nabla_{(a}H_{b)}$
令

Γb=Hb $\Gamma_b=H_b$ 就是广义调和令

Φiab=∂iφab, −NΠab=∂tφab−NiΦiab $\Phi_{iab}=\partial_i\varphi_{ab},\ \ -N\Pi_{ab}=\partial_t \varphi_{ab}-N^i\Phi_{iab}$ ，

t=(t0,t1,t2,t3)=(1N,−N1N,−N2N,−N3N) $t=(t^0,t^1,t^2,t^3)=(\frac{1}{N},\frac{-N^1}{N},\frac{-N^2}{N},\frac{-N^3}{N})$

\partial t φ a b - N k \partial k φ a b \partial t Π a b - N k \partial k Π a b + N g k i \partial k Φ i a b \partial t Φ i a b - N k \partial k Φ i a b \partial t ϕ - N i \partial i ϕ \partial t k - N i \partial i k - g i j N 2 \partial i ξ j \partial t ξ i - N j \partial j ξ i - \partial i k = - N Π a b = T + (\partial a ϕ) (\partial b ϕ) + φ a b V = 1 2 N t c t d Φ i c d Π a b + N g j k t c Φ i j k Φ k a b = k = N g i j [N (H j + g k l Γ i k l) - \partial j N] + \partial V \partial ϕ = \partial i (N j) ξ j

$\begin{aligned} \partial_t\varphi_{ab}-N^k\partial_k\varphi_{ab}&=-N\Pi_{ab}\\ \partial_t\Pi_{ab}-N^k\partial_k\Pi_{ab}+Ng^{ki}\partial_k\Phi_{iab}&=T+(\partial_a\phi )(\partial_b \phi)+\varphi_{ab}V\\ \partial_t\Phi_{iab}-N^k\partial_k\Phi_{iab}&=\frac{1}{2}Nt^ct^d\Phi_{icd}\Pi_{ab}+Ng^{jk}t^c\Phi_{ijk}\Phi_{kab}\\ \partial_t \phi-N^i\partial_i \phi&=k\\ \partial_t k -N^{i}\partial_i k-g^{ij}N^2\partial_i \xi_j&=Ng^{ij}[N(H_j+g^{kl}\Gamma_{ikl})-\partial_jN]+\frac{\partial V}{\partial \phi}\\ \partial_t \xi_i-N^j\partial_j \xi_i-\partial_i k&=\partial_i(N^j)\xi_j \end{aligned}$

k = \partial t ϕ + g 0 i g 00 \partial i ϕ

$\displaystyle k=\partial_t \phi+\frac{g^{0i}}{g^{00}}\partial_i \phi$

\partial t k + g 0 i g 00 \partial i k + γ i j g 00 \partial i ξ j = + N g i j [N (H j + g k l Γ i k l) - \partial j N] (1)

$\partial_t k +\frac{g^{0i}}{g^{00}}\partial_i k+\frac{\gamma^{ij}}{g^{00}}\partial_i \xi_j=+Ng^{ij}[N(H_j+g^{kl}\Gamma_{ikl})-\partial_jN]\tag{1}$

\partial t ξ i + g 0 j g 00 \partial j ξ i - \partial i k = \partial i (β j) ξ j (2~4)

$\partial_t \xi_i+\frac{g^{0j}}{g^{00}}\partial_j \xi_i-\partial_i k=\partial_i(\beta^j)\xi_j\tag{2~4}$
真空调和Einstein equation 令

Γb=0 $\Gamma_b=0$ 就是调和；令

Γb=Hb $\Gamma_b=H_b$ 就是广义调和令

Φiab=∂iφab, −NΠab=∂tφab−NiΦiab $\Phi_{iab}=\partial_i\varphi_{ab},\ \ -N\Pi_{ab}=\partial_t \varphi_{ab}-N^i\Phi_{iab}$ ，

t=(t0,t1,t2,t3)=(1N,−N1N,−N2N,−N3N) $t=(t^0,t^1,t^2,t^3)=(\frac{1}{N},\frac{-N^1}{N},\frac{-N^2}{N},\frac{-N^3}{N})$ 可得一阶方程组：

⎧ ⎩ ⎨ ⎪ ⎪ \partial t φ a b - N k \partial k φ a b \partial t Π a b - N k \partial k Π a b + N g k i \partial k Φ i a b \partial t Φ i a b - N k \partial k Φ i a b = - N Π a b = T = 1 2 N t c t d Φ i c d Π a b + N g j k t c Φ i j k Φ k a b

$\begin{cases} \partial_t\varphi_{ab}-N^k\partial_k\varphi_{ab}&=-N\Pi_{ab}\\ \partial_t\Pi_{ab}-N^k\partial_k\Pi_{ab}+Ng^{ki}\partial_k\Phi_{iab}&=T\\ \partial_t\Phi_{iab}-N^k\partial_k\Phi_{iab}&=\frac{1}{2}Nt^ct^d\Phi_{icd}\Pi_{ab}+Ng^{jk}t^c\Phi_{ijk}\Phi_{kab}\\ \end{cases}$

where:T=2Nφcd(gijΦicaΦjdb−ΠcaΠbd−φefΓaceΓbdf)−12NtctdΠcdΠab−NtcΠcigijΦjab $where:T=2N\varphi^{cd}(g^{ij}\Phi_{ica}\Phi_{jdb}-\Pi_{ca}\Pi_{bd}-\varphi^{ef}\Gamma_{ace}\Gamma_{bdf})-\frac{1}{2}Nt^ct^d\Pi_{cd}\Pi_{ab}-Nt^c\Pi_{ci}g^{ij}\Phi_{jab}$
这里用的是Lee的文章里的
Lee