Difference between revisions of "Equations of General Relativity"
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= Equations of General Relativity = | = Equations of General Relativity = | ||
| − | <p align="right">''I have thought seriously about this question,<br/> | + | <p align="right">''I have thought seriously about this question,''<br/> |
| − | + | ''and have come to the conclusion that''<br/> | |
| − | + | ''what I have to say cannot reasonably be conveyed''<br/> | |
| − | + | ''without a certain amount of mathematical notation''<br/> | |
| − | + | ''and the exploration of genuine mathematical concepts.''</p> | |
| − | <p align="right">''Roger Penrose'' | + | <p align="right">''Roger Penrose''<br/> |
| − | The Road to Reality</p> | + | ''The Road to Reality''</p> |
<div id="equ_oto1"></div> | <div id="equ_oto1"></div> | ||
=== Problem 1. === | === Problem 1. === | ||
| − | + | Consider a spacetime with diagonal metric | |
| + | \[ds^2=g_{00}(dx^{0})^2 + g_{11}(dx^{1})^2 | ||
| + | +g_{22}(dx^{2})^{2}+g_{33}(dx^{3})^{2}.\] | ||
| + | Find the explicit expressions for the intervals of proper time and spatial length, and for the 4-volume. Show that the invariant $4$-volume is given by | ||
| + | \[\sqrt{-g}\;d^{4}x\equiv | ||
| + | \sqrt{-g}\;dx^{0}dx^{1}dx^{2}dx^{3},\] | ||
| + | where $g=\det(g_{\mu\nu})$. | ||
<div class="NavFrame collapsed"> | <div class="NavFrame collapsed"> | ||
<div class="NavHead">solution</div> | <div class="NavHead">solution</div> | ||
<div style="width:100%;" class="NavContent"> | <div style="width:100%;" class="NavContent"> | ||
| − | <p style="text-align: left;"> | + | <p style="text-align: left;">We assume that the coordinate $x^{0}$ is timelike, so $g_{00}>0$, and that $x^{1}$, $x^2$ and $x^3$ is spacelike, so $g_{11},g_{22},g_{33}<0$. |
| + | |||
| + | The interval along the worldline of a particle at rest ($dx^1 = dx^2 = dx^3 = 0$) takes the form $ds^{2}=g_{00}(dx^{0})^2$. On the other hand, in the reference frame of the particle $ds^{2}$ is the proper time interval: $ds^{2}=c^{2}d\tau^{2}$, so $d\tau=\sqrt{g_{00}}\;dx^{0}/c$. | ||
| + | |||
| + | The change of variables $\sqrt{-g_{ii}}dx^{i}=d\tilde{x}^{i}$ (there is no summation over indices here) transforms the metric to locally Lorentz form | ||
| + | \[ds^{2}=c^{2}d\tau^{2}-(d\tilde{x}^{1})^{2} | ||
| + | -(d\tilde{x}^{2})^{2}-(d\tilde{x}^{3})^{2},\] | ||
| + | and it is easy to see that the element of three-dimensional length (line element) is given by | ||
| + | \[dl^{2}=-\big[g_{11}(dx^{1})^2 | ||
| + | +g_{22}(dx^{2})^{2}+g_{33}(dx^{3})^{2}\big].\] | ||
| + | The element of $4$-volume is then | ||
| + | \[cd\tau d\tilde{x}^{1}d\tilde{x}^{2}d\tilde{x}^{3} | ||
| + | =\sqrt{-g_{00}g_{11}g_{22}g_{33}}\; | ||
| + | dx^{0}dx^{1}dx^{2}dx^{3} | ||
| + | =\sqrt{-g}\;dx^{0}dx^{1}dx^{2}dx^{3}.\]</p> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | |||
| + | <div id="equ_oto1a"></div> | ||
| + | === Problem 2. === | ||
| + | Let there be an observer with $4$-velocity $u^{\mu}$. Show that the energy of a photon with $4$-wave vector $k^\mu$ that he registers is $u^{\mu}k_{\mu}$, and the energy of a massive particle with $4$-momentum $p^{\mu}$ is $u^{\mu}p_{\mu}$. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"></p> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | |||
| + | <div id="equ_oto2"></div> | ||
| + | === Problem 3. === | ||
| + | The ''covariant derivative (or connection)'' $\nabla_{\mu}$ is a tensorial generalization of partial derivative of a vector field $A^{\mu}(x)$ in the curved space-time. It's action on vectors is defined as | ||
| + | \begin{equation}\label{nabla} | ||
| + | \nabla_{\mu}A^{\nu}=\partial_{\mu}A^{\nu}+ | ||
| + | {\Gamma^{\nu}}_{\lambda\mu}A^{\lambda};\qquad | ||
| + | \nabla_{\mu}A_{\nu}=\partial_{\mu}A_{\nu}- | ||
| + | {\Gamma^{\rho}}_{\nu\mu}A_{\rho}, | ||
| + | \end{equation} | ||
| + | where matrices ${\Gamma^{i}}_{jk}$ are called the connection coefficients, so that $\nabla_{\mu}A^{\nu}$ and $\nabla_{\mu}A_{\nu}$ are tensors. The connection used in GR is symmetric in lower indices (${\Gamma^{\lambda}}_{\mu\nu}= {\Gamma^{\lambda}}_{\nu\mu}$) and compatible with the metric $\nabla_{\lambda}g_{\mu\nu}=0$. It is called the Levi-Civita's connection, and the corresponding coefficients ${\Gamma^{\lambda}}_{\mu\nu}$ the Christoffel symbols. The action on tensors is defined through linearity and Leibniz rule. Express the Christoffel symbols through the metric tensor. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"></p> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | |||
| + | <div id="equ_oto3"></div> | ||
| + | === Problem 4. === | ||
| + | Derive the transformation rule for matrices ${\Gamma^{\lambda}}_{\mu\nu}$ under coordinate transformations. Show that for any given point of spacetime there is a coordinate frame, in which ${\Gamma^{\lambda}}_{\mu\nu}$ are equal to zero in this point. It is called a locally inertial, or locally geodesic frame. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"></p> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | |||
| + | <div id="equ_oto4"></div> | ||
| + | === Problem 5. === | ||
| + | Free falling particles' worldlines in General Relativity are \textit{geodesics} of the spacetime--- i.e the curves $x^{\mu}(\lambda)$ with tangent vector $u^{\mu}=dx^{\mu}/d\lambda$, such that covariant derivative of the latter along the curve equals to zero: | ||
| + | \[u^{\mu}\nabla_{\mu}u^{\nu}=0.\] | ||
| + | |||
| + | In a (pseudo-)Euclidean space the geodesics are straight lines. Obtain the general equation of geodesics in terms of the connection coefficients. Show that the quantity $u^{\mu}u_{\mu}$ is conserved along the geodesic. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"></p> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | |||
| + | <div id="equ_oto5"></div> | ||
| + | === Problem 6. === | ||
| + | Consider the action for a massive particle of the form | ||
| + | \[S_{AB}=-mc\int_{A}^{B} ds,\quad\text{where}\quad | ||
| + | ds=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}\] | ||
| + | and derive the geodesic equation from the principle of least action. Find the canonical $4$-momentum of a massive particle and the energy of a photon. | ||
| + | <div class="NavFrame collapsed"> | ||
| + | <div class="NavHead">solution</div> | ||
| + | <div style="width:100%;" class="NavContent"> | ||
| + | <p style="text-align: left;"></p> | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 20:12, 28 May 2012
Contents
Equations of General Relativity
I have thought seriously about this question,
and have come to the conclusion that
what I have to say cannot reasonably be conveyed
without a certain amount of mathematical notation
and the exploration of genuine mathematical concepts.
Roger Penrose
The Road to Reality
Problem 1.
Consider a spacetime with diagonal metric \[ds^2=g_{00}(dx^{0})^2 + g_{11}(dx^{1})^2 +g_{22}(dx^{2})^{2}+g_{33}(dx^{3})^{2}.\] Find the explicit expressions for the intervals of proper time and spatial length, and for the 4-volume. Show that the invariant $4$-volume is given by \[\sqrt{-g}\;d^{4}x\equiv \sqrt{-g}\;dx^{0}dx^{1}dx^{2}dx^{3},\] where $g=\det(g_{\mu\nu})$.
We assume that the coordinate $x^{0}$ is timelike, so $g_{00}>0$, and that $x^{1}$, $x^2$ and $x^3$ is spacelike, so $g_{11},g_{22},g_{33}<0$. The interval along the worldline of a particle at rest ($dx^1 = dx^2 = dx^3 = 0$) takes the form $ds^{2}=g_{00}(dx^{0})^2$. On the other hand, in the reference frame of the particle $ds^{2}$ is the proper time interval: $ds^{2}=c^{2}d\tau^{2}$, so $d\tau=\sqrt{g_{00}}\;dx^{0}/c$. The change of variables $\sqrt{-g_{ii}}dx^{i}=d\tilde{x}^{i}$ (there is no summation over indices here) transforms the metric to locally Lorentz form \[ds^{2}=c^{2}d\tau^{2}-(d\tilde{x}^{1})^{2} -(d\tilde{x}^{2})^{2}-(d\tilde{x}^{3})^{2},\] and it is easy to see that the element of three-dimensional length (line element) is given by \[dl^{2}=-\big[g_{11}(dx^{1})^2 +g_{22}(dx^{2})^{2}+g_{33}(dx^{3})^{2}\big].\] The element of $4$-volume is then \[cd\tau d\tilde{x}^{1}d\tilde{x}^{2}d\tilde{x}^{3} =\sqrt{-g_{00}g_{11}g_{22}g_{33}}\; dx^{0}dx^{1}dx^{2}dx^{3} =\sqrt{-g}\;dx^{0}dx^{1}dx^{2}dx^{3}.\]
Problem 2.
Let there be an observer with $4$-velocity $u^{\mu}$. Show that the energy of a photon with $4$-wave vector $k^\mu$ that he registers is $u^{\mu}k_{\mu}$, and the energy of a massive particle with $4$-momentum $p^{\mu}$ is $u^{\mu}p_{\mu}$.
Problem 3.
The covariant derivative (or connection) $\nabla_{\mu}$ is a tensorial generalization of partial derivative of a vector field $A^{\mu}(x)$ in the curved space-time. It's action on vectors is defined as \begin{equation}\label{nabla} \nabla_{\mu}A^{\nu}=\partial_{\mu}A^{\nu}+ {\Gamma^{\nu}}_{\lambda\mu}A^{\lambda};\qquad \nabla_{\mu}A_{\nu}=\partial_{\mu}A_{\nu}- {\Gamma^{\rho}}_{\nu\mu}A_{\rho}, \end{equation} where matrices ${\Gamma^{i}}_{jk}$ are called the connection coefficients, so that $\nabla_{\mu}A^{\nu}$ and $\nabla_{\mu}A_{\nu}$ are tensors. The connection used in GR is symmetric in lower indices (${\Gamma^{\lambda}}_{\mu\nu}= {\Gamma^{\lambda}}_{\nu\mu}$) and compatible with the metric $\nabla_{\lambda}g_{\mu\nu}=0$. It is called the Levi-Civita's connection, and the corresponding coefficients ${\Gamma^{\lambda}}_{\mu\nu}$ the Christoffel symbols. The action on tensors is defined through linearity and Leibniz rule. Express the Christoffel symbols through the metric tensor.
Problem 4.
Derive the transformation rule for matrices ${\Gamma^{\lambda}}_{\mu\nu}$ under coordinate transformations. Show that for any given point of spacetime there is a coordinate frame, in which ${\Gamma^{\lambda}}_{\mu\nu}$ are equal to zero in this point. It is called a locally inertial, or locally geodesic frame.
Problem 5.
Free falling particles' worldlines in General Relativity are \textit{geodesics} of the spacetime--- i.e the curves $x^{\mu}(\lambda)$ with tangent vector $u^{\mu}=dx^{\mu}/d\lambda$, such that covariant derivative of the latter along the curve equals to zero: \[u^{\mu}\nabla_{\mu}u^{\nu}=0.\]
In a (pseudo-)Euclidean space the geodesics are straight lines. Obtain the general equation of geodesics in terms of the connection coefficients. Show that the quantity $u^{\mu}u_{\mu}$ is conserved along the geodesic.
Problem 6.
Consider the action for a massive particle of the form \[S_{AB}=-mc\int_{A}^{B} ds,\quad\text{where}\quad ds=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}\] and derive the geodesic equation from the principle of least action. Find the canonical $4$-momentum of a massive particle and the energy of a photon.
