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Chapter 18: Problem 21

Show that every two-dimensional space-time metric (signature 0 ) can be expressed locally in confor mal coontinates $$ \mathrm{ds}^{2}=\mathrm{e}^{2 \varphi}\left(\mathrm{d} x^{2}-\mathrm{d} t^{2}\right) \text { where } \phi=\phi(x, t) $$ Calculate the Rucmann curvature tensor component \(R_{1212}\), and writc out the two-dimensional Enstein vacuum equations \(R_{u j}=0\). What is their general solunon?

Short Answer

Expert verified
The Riemann curvature tensor component \(R_{1212}\) for this metric is \(e^{-2\varphi}\varphi_{,xx}\) and the two-dimensional Einstein vacuum equations simplify to \(\varphi_{,tt} - \varphi_{,xx} = 0\) and \(2\varphi_{,xt} = 0\). Solving these, the general solution is \(\varphi = ax + bt\), where \(a\) and \(b\) are constants.

Step by step solution

01

Express the metric in conformal coordinates

Our given metric is \(ds^{2} = e^{2\varphi}(dx^{2} - dt^{2})\) where \(\varphi = \varphi(x, t)\). This is a conformal transformation of the Minkowski metric \(ds^{2} = dx^{2} - dt^{2}\). Thus, the metric is already expressed in conformal coordinates.
02

Calculate the Riemann curvature tensor component \(R_{1212}\)

The Riemann tensor in 2D is defined as \(R^{\mu}_{\nu\rho\sigma}=\partial_{\rho}\Gamma^{\mu}_{\nu\sigma}-\partial_{\sigma}\Gamma^{\mu}_{\nu\rho}\). In two dimensions, the \(R_{1212}\) component simplifies to \(R_{1212} = e^{-2\varphi}\varphi_{,xx}\). So, we need to calculate the second derivative of \(\varphi\) with respect to \(x\) to find this component.
03

Write out the two-dimensional Einstein vacuum equations

The two-dimensional Einstein vacuum equations are given as \(R_{ij} = 0\), where \(R_{ij}) is the Ricci tensor. In two dimensions, this simplifies to \(R_{00} = R_{11} = 0\). Thus, the two-dimensional Einstein vacuum equations are \(\varphi_{,tt} - \varphi_{,xx} = 0\) and \(2\varphi_{,xt} = 0\).
04

Solve the Einstein vacuum equations

Solving the two Einstein vacuum equations, we obtain \(\varphi = ax + bt\) as the general solution.

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Most popular questions from this chapter

Problem \(18.12\) (a) Show that in a pscudo-Riemannian space the action principle $$ \delta \int_{t_{1}}^{b_{2}} L, \mathrm{~d} t=0 $$ where \(L=g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}\) gives rise to geodesic equations with affine parameter \(t\). (b) For the sphere of radius \(a\) in polar coordinates, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols \(\Gamma_{v \rho^{*}}^{\mu}\) where \(L=g_{k v} \dot{x}^{\mu} \dot{x}^{v}\) gives rise to geodesic equations with affine parameter \(t\). (b) For the sphere of radius \(a\) in polar coordinates, $$ \mathrm{ds}^{2}=a^{2}\left(\mathrm{~d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}\right) $$ use this variation principle to write out the equations of geodesics, and read off from them the Christoffel symbols \(\Gamma_{i \rho^{*}}^{\mu}\) (c) Verify by direct substitution in the geodcsic equations, that \(L=\theta^{2}+\sin ^{2} \theta \dot{\phi}^{2}\) is a constant along the geodesics and use this to show that the general solution of the geodesic equations is grven by $$ b \cot \theta=-\cos \left(\phi-\phi_{0}\right) \text { where } b, \phi_{0}=\text { const. } $$ (d) Show that these curves are great circles on the sphere

(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$.

Consider two radual light s?gnals (null geodesics) received at the spatial origin of coordinates at times \(t_{0}\) and \(t_{0}+\Delta t_{0}\), emitted from \(x=x_{1}\) (or \(r=r_{1}\) in the case of the flat models) at time \(t=t_{1}

Show that a space is locally flat if and only if there exists a local basis of vector ficlds \(\left\\{e_{t} \mid\right.\) that are absolutely parallel, \(D e_{1}=0\).

Consider an oscillator at \(r=r_{0}\) emitting a pulse of light (null geodesic) at \(t=t_{0}\). If this is received by an observer at \(r=r_{1}\) at \(t=t_{1}\), show that $$ t_{1}=t_{0}+\int_{r_{0}}^{r_{1}} \frac{d r}{c(I-2 m / r)} $$ By considering a signal emitted at \(t_{0}+\Delta t_{0}\), received at \(t_{1}+\Delta t_{1}\) (assuming the radial positions \(r_{0}\) and \(r_{1}\) to be constant), shou that \(t_{0}=t_{1}\) and the gravitational redsbift found by comparing proper times at cmission and reception is given by $$ 1+z=\frac{\Delta t_{1}}{\Delta \tau_{0}}=\sqrt{\frac{1-2 m / r_{1}}{1-2 m / r_{0}}} $$ Show that for two clocks at different heights \(h\) on the Earth's surface, this reduces to $$ z \approx \frac{2 G M}{c^{2}} \frac{h}{R} $$ where \(M\) and \(R\) are the mass and radius of the Earth.
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