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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

(a) For a perfect flud in general relat?uty, $$ T_{\mu v}=\left(\rho c^{2}+P\right) U_{\mu} U_{v}+P g_{\beta v} \quad\left(U^{\mu} U_{y}=-1\right) $$ show that the conservation identities \(T^{\mu v}, v=0\) imply \(\rho_{v} U^{v}+\left(\rho c^{2}+P\right) U_{v^{2}}^{*}\) \(\left(\rho c^{2}+P\right) U_{,}^{\mu} U^{v}+P_{v}\left(g^{\mu \prime}+U^{\mu} U^{\nu}\right)\) (c) In the Newtonian approximatuon where $$ U_{\mu}=\left(\frac{1}{c},-1\right)+O\left(\beta^{2}\right), \quad P=O\left(\beta^{2}\right) \rho c^{2}, \quad\left(\beta=\frac{v}{c}\right) $$ where \(|\beta| \ll 1\) and \(g_{\mu \nu}=\eta_{h^{x}}+\epsilon h_{\mu \nu}\) with \(\epsilon \ll 1\), show that $$ h_{+\mu} \approx-\frac{2 \phi}{c^{2}}, \quad h_{i j} \approx-\frac{2 \phi}{c^{2}} \delta_{j} \quad \text { where } \quad \nabla^{2} \phi=4 \pi G \rho $$ and \(h_{t 4}=O(\beta) h_{44}\) Show in this approxumation that the equations \(T^{\mu t},=0\) approvimate to $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho v)=0, \quad \rho \frac{d v}{d t}=-\nabla P-\rho \nabla \phi $$

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}

In the Schwarzschild solution show the only possible closed photon path is a circular orbit at \(r=3 m\), and show that it is unstable.

(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$.

Let \((M, \varphi)\) be a surface of revolution defined as a submantfold of \(\mathbb{E}^{3}\) of the form $$ r=g(u) \cos \theta, \quad y=g(u) \sin \theta, \quad z=h(u) $$ Show that the induced metric (see Example 18.1) is $$ \mathrm{d} s^{2}=\left(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}\right) \mathrm{d} u^{2}+g^{2}(u) \mathrm{d} \theta^{2} $$ Picking the parameter \(u\) such that \(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}=1\) (interpret this choice!), and setting the basis 1-forms to be \(\varepsilon^{\prime}=\mathrm{d} u, \varepsilon^{2}=g d \theta\), calculate the connection I-forms \(\omega^{\prime}\), the curvature 1 -forms \(\rho_{\prime}^{\prime}\), and the curvature tensor component \(R_{1212}\).
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