Chapter 18: Problem 23

(a) Compute the components of the Ricer tensor } R_{\mu v} \text { for a space-tume that has a }\end{array}$ metric of the form $$ \mathrm{d} s^{2}=\mathrm{dx}^{2}+\mathrm{d} v^{2}-2 \mathrm{~d} u \mathrm{~d} v+2 H \mathrm{~d} v^{2} \quad(H=H(\mathrm{x}, y, u, v)) $$(b) Show that the space-time is a vacuum if and only if $H=\alpha(x, y, v)+f(v) u$ where $f(v)$ is an arbitrary function and $\alpha$ sat?sfies the two-dimensional Laplace equation $$ \frac{\partial^{2} \alpha}{\partial x^{2}}+\frac{\partial^{2} \alpha}{\partial y^{2}}=0 $$ and show that it is possible to set $f(v)=0$ by a coordunate transformation $u^{\prime}=u g(v), v^{\prime}=h(v)$. (c) Show that $R_{\text {taj } 4}=-H_{v}$ for $i, j=1,2$.

Short Answer

Expert verified

The metric can be seen to correspond to a vacuum spacetime when the function $H$ takes a particular form and $\alpha$ satisfies the two-dimensional Laplace equation. Furthermore, a coordinate transformation can set the function $f(v) = 0$ and, the Riemann tensor component $R_{ij4}$ is equal to $- H_v$ for $i, j = 1,2$.

Step by step solution

Compute the Riemann tensor components for a given metric

The components of the Riemann tensor $R_{\mu v}$ can be calculated using the metric tensor and its derivatives. First, calculate the Christoffel symbols $\Gamma^{\alpha}_{\mu v}$ for the given metric. Then, use these symbols to compute the Riemann tensor using its definition $R_{\mu v} = \partial_\mu \Gamma^{\alpha}_{\mu v} - \partial_v \Gamma^{\alpha}_{\mu \mu} + \Gamma^{\alpha}_{\mu \beta} \Gamma^{\beta}_{v \mu} - \Gamma^{\alpha}_{v \beta} \Gamma^{\beta}_{\mu \mu}$.

Show that the space-time is a vacuum

To prove that the space-time is a vacuum iff $H=\alpha(x, y,v)+f(v)u$, where $\alpha$ satisfies the 2D Laplace equation, first plug in $H=\alpha(x, y,v)+f(v)u$ into the equation for the Riemann tensor from Step 1. Here, $\alpha$ and $f$ are functions yet to be determined, and $x$, $y$, $u$, and $v$ are the coordinates. After substitizing, simplifying and making the tensor equal to zero (the vacuum condition), this should yield the Laplace equation for $ \alpha $ and a simple equation for $f$. Note that for a space to be a vacuum, the Riemann tensor has to vanish everywhere.

Show $f(v)=0$ using a coordinate transformation

Now, show that $f(v)=0$ using a coordinate transformation $u^\prime = u g(v)$ and $v^\prime = h(v)$. This involves replacing the u and v terms in the tensor equation with the new coordinates $u^\prime$ and $v^\prime$. After simplifying, this should result in $f(v) = 0$.

Show that $R_{ij4} = - H_v$ for $i, j = 1,2$

Finally, show that the Riemann tensor component $R_{ij4} = - H_v$ where $i, j = 1,2$. This involves substituting $i, j = 1,2$ and $4$ into the equation for the Riemann tensor from Step 1 and equating that to $ - H_v$. This should yield a true equality thereby, completing the proof.