Difference between revisions of "Technical warm-up"
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==Metric in curved spacetime== | ==Metric in curved spacetime== | ||
+ | We see here, how, given an arbitrary metric tensor, to determine physical distance between points, local time and physical velocity of a particle in an arbitrary frame of reference. | ||
+ | This problem, though fundamentally important, is necessary in full form only for consideration of particle dynamics in the Kerr metric. In order to analyze the dynamics in the Schwarzschild metric, it suffices to answer all the questions with a substantially simplifying condition $g_{0\alpha}=0$, where $\alpha=1,2,3$ (see the last of the problems). | ||
+ | |||
+ | Let the spacetime metric have the general form | ||
+ | \[ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}.\] | ||
+ | Coordinates are arbitrary and do not carry direct metrical meaning. An observer, stationary in a given coordinate frame, has 4-velocity $u^{\mu}=(u^{0},0,0,0)$, and the interval determines his proper ``local'' time | ||
+ | \[c^2 d\tau^{2}=ds^{2}=g_{00}(dx^{0})^2.\] | ||
+ | An observer in point A, with coordinates $x^\mu$, determines the physical ``radar'' distance to an infinitely close point $B$, with coordinates $x^{\mu}+dx^{\mu}$, in the following way. She sends a light beam to $B$ and measures the time it takes for the reflected beam to come back. Then distance to $B$ is half the proper time she waited from emission to detection times $c$. It is also natural for her to consider the event of the beam reflection in $B$ to be simultaneous with the middle of the infinitely small 4-distance between the events of emission and detection of light beam in $A$. | ||
<div id="BlackHole04"></div> | <div id="BlackHole04"></div> |
Revision as of 07:49, 17 June 2012
Contents
Uniformly accelerated observer, Rindler metric
Einstein's equivalence principle states that locally a gravitational field cannot be dis\-tin\-guished from a non-inertial frame of reference. Therefore a number of effects of General Relativity, such as time dilation in a gravitational field and formation of horizons, can be studied in the frame of Special Theory of Relativity when considering uniformly accelerated observers.
Problem 1.
Derive the equation of motion $x(t)$ of a charged particle in Minkowski space in a uniform electric field without initial velocity. Show that its acceleration is constant.
The particle's equation of motion is $\dot{p}=eE=const$, so on choosing conveniently the zero point of time, we have $p(t)=eEt$. Then energy is \[\varepsilon(t)=\sqrt{p^{2}c^{2}+m^{2}c^{4}}= \sqrt{\mu^{2}+\tilde{t}^{2}},\quad\mbox{где}\; \mu=mc^2,\;\tilde{t}=eEct.\] As $p=\varepsilon v/c^{2}$, for velocity we get \[\beta(t)\equiv \frac{v(t)}{c}=\frac{cp}{\varepsilon}= \frac{\tilde{t}}{\sqrt{\mu^2+\tilde{t}^2}}.\] Integrating and restoring initial conditions $(t_{0},x_{0})$, we obtain \[(x+\rho-x_{0})^{2}-c^{2}(t-t_{0})^{2}=\rho^{2}, \qquad\mbox{где}\quad \rho=\frac{mc^{2}}{eE}.\] This is a hyperbola branch $x>0$, with asymptotes on the light cone, symmetry axis $OX$ and center $t=t_{0}$, $x=x_{0}-\rho$. Now calculate the $4$-acceleration of the particle: from $\dot{p}^{1}=\dot{p}=eE$, and $\dot{p}^{0}=\dot{\varepsilon}/c=eEv/c$ we obtain \[a_{\mu}a^{\mu}= \left(\frac{du^0}{ds}\right)^2- \left(\frac{du^1}{ds}\right)^2= \gamma^{2} \left[(\dot{u}^{0})^{2}-(\dot{u}^{1})^{2}\right]= \frac{\gamma^{2}}{(mc)^{2}} \left[\frac{\dot{\varepsilon}^{2}}{c^{2}} -\dot{p}^{2}\right]= -\frac{a^{2}}{c^{2}}.\] It is instructive to show that this acceleration is also the acceleration in the momentarily comoving frame. The latter is derived using the velocity composition law $v'=\frac{v+u}{1+vu}$, the expression for the proper time interval $d\tau=\gamma^{-1}dt$, and relation $\dot{p}=\gamma^{3}m\dot{v}$: \[a=\left.\frac{dv'}{d\tau}\right|_{v'=0}= \gamma\left.\frac{d}{dt}\frac{u+v}{1+uv} \right|_{u+v=0}=\gamma \frac{dv/dt}{1-\beta^{2}}= \gamma^{3}\dot{v}=\frac{\dot{p}}{m}= \frac{eE}{m}=\frac{c^{2}}{\rho}.\]
Problem 2.
Problem 3.
Metric in curved spacetime
We see here, how, given an arbitrary metric tensor, to determine physical distance between points, local time and physical velocity of a particle in an arbitrary frame of reference.
This problem, though fundamentally important, is necessary in full form only for consideration of particle dynamics in the Kerr metric. In order to analyze the dynamics in the Schwarzschild metric, it suffices to answer all the questions with a substantially simplifying condition $g_{0\alpha}=0$, where $\alpha=1,2,3$ (see the last of the problems).
Let the spacetime metric have the general form \[ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}.\] Coordinates are arbitrary and do not carry direct metrical meaning. An observer, stationary in a given coordinate frame, has 4-velocity $u^{\mu}=(u^{0},0,0,0)$, and the interval determines his proper ``local time \[c^2 d\tau^{2}=ds^{2}=g_{00}(dx^{0})^2.\] An observer in point A, with coordinates $x^\mu$, determines the physical ``radar distance to an infinitely close point $B$, with coordinates $x^{\mu}+dx^{\mu}$, in the following way. She sends a light beam to $B$ and measures the time it takes for the reflected beam to come back. Then distance to $B$ is half the proper time she waited from emission to detection times $c$. It is also natural for her to consider the event of the beam reflection in $B$ to be simultaneous with the middle of the infinitely small 4-distance between the events of emission and detection of light beam in $A$.
Problem 4.
Problem 5.
Problem 6.
Problem 7.
Problem 8.