1. **Problem Statement:** Verify the correctness of the Schwarzschild solution as a static, spherically symmetric vacuum solution to the Einstein Field Equations (EFE). 2. **EFE Overview:** The Einstein Field Equations relate spacetime curvature to energy and momentum: $$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$ For vacuum, $T_{\mu\nu} = 0$, so: $$G_{\mu\nu} = 0$$ 3. **Metric Ansatz:** Assume a static, spherically symmetric metric: $$ds^2 = -e^{2\Phi(r)} c^2 dt^2 + e^{2\Lambda(r)} dr^2 + r^2 d\Omega^2$$ where $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$. 4. **Solving EFE in Vacuum:** Setting $G_{\mu\nu} = 0$ leads to differential equations for $\Phi(r)$ and $\Lambda(r)$. One finds: $$\Phi(r) = -\Lambda(r)$$ and $$\Phi'(r) = \frac{GM}{c^2 r^2}$$ Integrating: $$\Phi(r) = -\frac{GM}{c^2 r} + C$$ Choosing asymptotic flatness sets $C=0$. 5. **Final Schwarzschild Metric:** $$ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2$$ 6. **Verification:** - Newtonian limit recovered for large $r$. - Event horizon at $r = \frac{2GM}{c^2}$ is a coordinate singularity. - Matches known exact vacuum solution. 7. **Conclusion:** Your understanding and derivation of the Schwarzschild solution as a static, spherically symmetric vacuum solution to the EFE is correct. This confirms your correctness in the problem scope and solution.