我們考慮拉格朗日量為$\mathcal{L}=a_{0}+a_{1}R+a_{2}R^{2}+a_{3}R_{\mu\nu}R^{\mu\nu}$的修正引力理論, 其中 $a_{0}, a_{1}, a_{2}, a_{3}$ 為常數 ($a_{2}$, $a_{3}$ 不全為零), $R$ 為瑞奇純量, $R_{\mu\nu}$ 為瑞奇張量. 為求得基本的史瓦西外部解, 我們假設符合靜態條件並設定度規解 ${\rm d}s^{2}=g_{tt}(r){\rm d}t^{2}+g_{rr}(r){\rm d}r^{2}+r^{2}({\rm d}\theta^{2}+\sin^{2}\theta{\rm d}\phi)$ 滿足此條件 $-g_{tt}(r)=1/g_{rr}(r)$. 我們得到只有三種可能的史瓦析解形式: (i) 當 $a_{2}\ne0$ 並且 $a_{0}=a_{1}=a_{3}=0$ (相當於單純 $R^{2}$ 引力)時, $-g_{tt}(r)=1+\frac{b_{1}}{r}+\frac{b_{2}}{r^{2}}$ 或 $-g_{tt}(r)=1+\frac{b_{1}}{r}+c_{2}r^{2}$, (ii) 當 $3a_{2}+a_{3}=0$ 並且 $a_{0}=a_{1}=0$ (等價於共形引力), $-g_{tt}(r)=1+c_{0}+\frac{b_{1}}{r}+c_{1}r+c_{2}r^{2}$, (iii) 當其他 $(a_{0}\ne0, a_{1}\ne0, a_{2}, a_{3})$ 或 $(a_{0}=0, a_{1}=0, a_{2}, a_{3})$ 組合時, $-g_{tt}(r)=1+\frac{b_{1}}{r}+c_{2}r^{2}$. 其中這些 $b$ 與 $c$ 為由邊界條件決定的常數. 對於內部解, 我們考慮了理想流體的均勻星體. 利用射擊法, 我們可以求出此修正重力理論的數值解.

We consider modified gravity theory with Lagrangian $\mathcal{L}=a_{0}+a_{1}R+a_{2}R^{2}+a_{3}R_{\mu\nu}R^{\mu\nu}$, where $a_{0}, a_{1}, a_{2}, a_{3}$ are constants ($a_{2}$, $a_{3}$ not all vanishing), $R$ is the Ricci scalar, and $R_{\mu\nu}$ is the Ricci tensor. To obtain the basic exterior Schwarzschild solution, we assume static condition and parametrize the metric solution ${\rm d}s^{2}=g_{tt}(r){\rm d}t^{2}+g_{rr}(r){\rm d}r^{2}+r^{2}({\rm d}\theta^{2}+\sin^{2}\theta{\rm d}\phi)$ with the condition $-g_{tt}(r)=1/g_{rr}(r)$. We conclude that there are only three types of possible Schwarzschild metrics with: (i) $-g_{tt}(r)=1+\frac{b_{1}}{r}+\frac{b_{2}}{r^{2}}$ or $-g_{tt}(r)=1+\frac{b_{1}}{r}+c_{2}r^{2}$ for $a_{2}\ne0$ and $a_{0}=a_{1}=a_{3}=0$ (equivalent to pure $R^{2}$ gravity), (ii) $-g_{tt}(r)=1+c_{0}+\frac{b_{1}}{r}+c_{1}r+c_{2}r^{2}$ for $3a_{2}+a_{3}=0$ and $a_{0}=a_{1}=0$ (equivalent to conformal gravity), (iii) $-g_{tt}(r)=1+\frac{b_{1}}{r}+c_{2}r^{2}$ for other combinations of $(a_{0}\ne0, a_{1}\ne0, a_{2}, a_{3})$ or $(a_{0}=0, a_{1}=0, a_{2}, a_{3})$. The $b$'s and $c$'s are constants which can be determined by boundary conditions. For the interior solution, a uniform star of perfect fluid is considered. Via the shooting method, we could find the numerical solution for the modified gravity theory.

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