%% %% gr.tex %% %% Made by Alex Dehnert %% Login %% %% Started on Fri Aug 24 09:20:14 2007 Alex Dehnert %% Last update Fri Aug 24 14:20:14 2007 Alex Dehnert %% \documentclass{article} \usepackage{alexschool} \usepackage{graphicx} \title{General Relativity} \author{Sean Gryb (ISSYP lecturer), as filtered through Alex Dehnert} \date{Friday, August 24, 2007} \standardmargins \begin{document} \maketitle{} \section{General Relativity} \label{sec:gr} \subsection{Why Force is Like Curvature} \label{sec:forcecurve} \begin{itemize} \item Alice and Bob on the earth's equator, at a large yelling distance away from each other \item Earth is frictionless \item Pushed straight north \item Eventually will impact each other as the great circles they move along collide with each other \item Possibilities \begin{description} \item[Force] Bob thinks his path is being curved by some outside force\\ \begin{align} F_0 &= -m \left(\frac{v}{B}\right) l(t)\\ B&=6400KM \end{align} \item[Curvature] Alice thinks they are travelling along a curved surface\\ \begin{align} \mbox{radius of curvature} = R_c &= B\\ \theta = \frac{l_0}{r_0} = \frac{l_o}{R_E} &= \frac{l(t)}{r(t)}\\ F=ma&=m\frac{d^2}{dt^2}l(t)\\ r(t) &= R_E\cos(\phi(t))\\ l(t)&=\frac{l_0}{R_E}r(t) \\ &= \frac{l_0}{R_E}R_E \cos(\phi(t)) \qquad\mbox{substitute in $r(t)$}\\ &= l_0 \cos \left(\frac{vt}{R_E}\right) \qquad\mbox{Cancel and substitute in $\phi(t)$}\\ \phi(t) = S/R_E &= vt/R_E\\ F &= -m \left(\frac{v}{R_E}\right)^2l(t) \end{align} \end{description} \end{itemize} \subsection{Equivalence Principle} \label{sec:equivprinc} \begin{itemize} \item It is always possible, for some object and force, to replace a force by some equivalent curvature \item For gravity, this works particularly well \begin{align} m_i &= \mbox{inertial mass}\\ m_g &= \mbox{gravitational mass}\\ F&=m_i a\\ F_N &= - \left(\frac{GM}{r^2}\right) m_g\\ m_{i}a &= \frac{GM}{r^2}m_g\\ a &= \left(\frac{GM}{r^2}\right) \frac{m_g}{m_i}\\ m_g &= m_i \\ \end{align} All bodies fall with equal acceleration \item GR = SR + Newton's Law + Straight Lines \end{itemize} \section{Topology and General Relativity} \subsection{Topology} \label{sec:topology} \begin{itemize} \item Intuition: Vectors --- more than just numbers, has other invariants even if you try to change the coordinates by rotating (eg, length) \item Imagine a surface like a sphere \begin{align} R^2 &= x^2 + y^2 + z^2 \\ x' &= zx \\ y' &= y/2 \\ z' &= 3z \\ R^2 &= \pfr{x'}{2}^2 + (2y')^2 + \pfr{z'}{3}^2 \end{align} Distorting the coordinates give you an ellipsoid \item Invariants \begin{description} \item [Handles] donut, coffee cup, torus all have one (basically a big hole that you could grab on to with \item [Holes] If you remove one point from a sphere, you get something equivalent to a plane \item[Twists] M\"obius strip has one \end{description} \end{itemize} \subsection{Requirements for GR} \label{sec:reqgr} \begin{itemize} \item All observers are equal (as long as gravity is the only force in effect) \item Alternatively, all coordinate systems are equivalent \item Hence, need things that are invariant with respect to coordinate system \begin{itemize} \item Topology \item Events \end{itemize} \item What is the difference between a donut and a coffee cup? Geometry \begin{itemize} \item Straight lines (or just distances) (metric = $g$) \item Derivatives (connection = $\Gamma$) \end{itemize} \item In GR, $g\leftrightarrow\Gamma$, but not in quantum gravity \item What you need for GR: \begin{itemize} \item Topology \item Matter \item Coordinate system \end{itemize} Plug into Einstein equations, get a a metric $g$ \end{itemize} \section{Black Holes} \label{sec:bh} \subsection{Schwarzschild Black Hole} \label{sec:schwarzbh} \begin{itemize} \item Stuff for GR \begin{description} \item[Topology] $\Rset^4 - \mbox{line}$ (three space minus the point for the the singularity, crossed with a line for time: ($\Rset^3-\{\mbox{point}\})\oplus\Rset^1$) \item[Matter] Spherically symmetric \item[Coordinate system] Far away from matter (fiducial observer = ``Fido'') \end{description} \item Get metric \begin{align} ds^2 &= -\left(1-\frac{2M}{r}\right)dt^2 + \pfr{1}{1-\frac{2M}{r}}dr^2 + \cdots \end{align} \item Meaning \begin{itemize} \item $ds$ = space-time distance \item $dr$ = physical distance \item $dt$ = time distance \end{itemize} \item Problems \begin{itemize} \item $r=0$: always an issue \item $r=2M$: only because of coordinates \end{itemize} \end{itemize} \subsection{Sonic Black Hole} \label{sec:sonicbh} \begin{itemize} \item Take a huge pond \item Drill a hole in the bottom, with a spike in the bottom \item Water flows out of the hole faster than the speed of sound in the water \item Take tadpoles in the pool \begin{description} \item[Freefo] Free fall observer \item[Fido] Fiduciary observer \end{description} \begin{itemize} \item Able to ``see'' solely using sonar \item Equipped with a regular noisemaker (eg, loud heartbeat) \end{itemize} \item As Freefo falls in, Fido hears Freefo's heart slowing down \item Past the horizon where the water starts moving faster than the speed of light, Freefo's heart beat (which travels as waves in the water) will never get out \item From Freefo's point of view, the horizon isn't anything too special \end{itemize} \section{Semi-classical GR} \label{sec:scgr} \begin{itemize} \item Hawking radiation \begin{itemize} \item $\Delta x \Delta p = \hbar$ \item Near the black hole, uncertainty about where exactly things are \item Because particles spontaneously appear and disappear, some energy will escape from the black hole \item Eventually the black hole will completely evaporate (maybe, if insufficient matter gets sucked in) \item Black holes have a ``temperature'': $T_{BH} = \frac{\hbar c^3}{\delta\pi GMk_B} \mbox{ is proportional to } \frac{1}{M}$ \item Information entropy \begin{itemize} \item Disorder \item Classical: Black hole is perfectly ordered, entropy = 0 \item Quantum: Black hole has screwy boundary: \begin{align} \mbox{entropy}=S=\mbox{Information (bits)}&=\frac{1}{4}\pfr{A}{l_{pl}^2}\\ l_{pl} &= \parbox{2.5in}{scale where spacetime loses its meaning}\\ \Delta x \Delta p &= \hbar \\ \Delta p &= p\\ \Delta x &= l\\ \therefore pl &=\hbar\\ \therefore p &= \hbar/l\\ p = \frac{E}{c} &= \frac{m_gc^2}{c} = m_gc\\ m_g = p/c &= \frac{\hbar}{ec}\\ l = \frac{R_{swh}}{2} = \frac{Gm_g}{c^2} &= \frac{G\hbar}{lc^3}\\ \parbox{2in}{Multiplying by $l$ and taking the root, }\:\: l = \sqrt{\frac{G\hbar}{c^3}} &= l_{pl}\\ \end{align} So, GR and quantum act strangely together, because we can randomly get black holes \end{itemize} \item Now we get things quantized at the size of $l_{pl}$, so we have just one bit per small region, instead of one bit per point \end{itemize} \end{itemize} \section{Quantum Gravity and Holography} \label{sec:qgandholo} \subsection{Holography} \label{sec:holography} \begin{itemize} \item All these black holes reduce our freedom \item Final entropy: \begin{align} S_f &= \frac{1}{4}\frac{A}{l_{pl}^2} \end{align} \item By the second law of thermodynamics, \begin{align} S_i &\le S_f = \frac{1}{4}\frac{A}{l_{pl}^2} \end{align} \item So, the information is proportional to the \emph{surface area}, so is the world 2D? \end{itemize} \subsection{Quantum Gravity} \label{sec:qg} \begin{itemize} \item Look at (2D space + 1D time = 3D spacetime) \item This has been solved \item Observables = integrals over path in spacetime\\ $H[\gamma] = \mbox{knots}$\\ \includegraphics[width=2in]{gr/gamma-circle} \item Quantum gravity in 2 space dimensions is holographic \end{itemize} \subsection{String Theory} \label{sec:string} \begin{itemize} \item Lives on a surface in 10 dimensions (4 space dimensions, 1 time dimension, plus a 5 spere at every point in space) \item Drop a dimension, because we look at the boundary (3+1 dim) \item Correspondence between String theory and Conformal Field Theory \end{itemize} \end{document}