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12024–2025 Semester 1

1.(30 Points) (1) We assume that ${\xi }^{\eta }$ is a Killing field, that is,$${\nabla }_{\alpha }{\xi }_{\beta }+{\nabla }_{\beta }{\xi }_{\alpha }=0.$$(1)We also assume that $c$ is a timelike geodesic with unit tangent vector $p^{\alpha }$. Show that along $c$, $p^{\alpha }{\xi }_{\alpha }$ is constant.

(2) Give $10$ examples of Killing fields of Minkowski space ${\mathbb{R}}^{3,1}$.

2. (30 Points) For Schwartzschield solution $$g=-(1-\frac{2M}{r})d{t}^2+(-\frac{2M}{r}{)}^{-1}d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2),$$(2)the Komar mass coincides with the ADM mass.

3. (40 Points) (1) If all the $g_{\alpha \beta }$ is independent of the fixed $x^{\mu }$, show that the equation of the energy momentum tensor $T_{\mu ;\nu }^{\nu }=0$ is equivalent to$$\frac{1}{\sqrt{-g}}(\sqrt{-g}{T}_{\mu }^{\nu }{)}_{,\nu }=0.$$(3)

(2) Assume further that for each fixed $x_0$, the set of $(x_1,x_2,x_3)$ such that $T_{\mu }^{\nu }\ne 0$ for some $\mu ,\nu$ is bounded. Show with equation (3) that$${\int }_{x^0=const}{T}_{\mu }^{\nu }\sqrt{-g}{n}_{\nu }{d}^{3}x$$(4)is independent of $x^0$, where $n_{\nu }$ is the unit cotangent vector of the constant time surface

(3) Consider the Minkowski spacetime with polar coordinates $(t,r,\theta ,\varphi )$, show with equation (4) that$$J={\int }_{t=const}{T}_{\varphi }^0{r}^2\sin \theta drd\theta d\varphi$$(5)is independent of $t$.

(4) Change equation (5) into the rectangular coordinates $(t,x,y,z)$ to conclude that$$J=\int (x{T}^{y0}-y{T}^{x0})dxdydz$$(6)is independent of $t$.