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%% blackholes.tex
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%% Started on  Thu Aug 23 13:04:46 2007 Alex Dehnert
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\documentclass{article}
\usepackage{alexschool}
\title{Blackholes}
\author{Michael (ISSYP lecturer), filtered through Alex Dehnert}
\date{Thursday, August 23, 2007}
\begin{document}
\maketitle{}

Recommended book: \book{Gravity From the Ground Up} by Schutz (ISBN: 0
521 45506 5). How gravity works, starting with Newton and working
forward

Book from the lecture: \book{Gravitation} by Meisner (sp?) (available online)

\section{Laws of Gravity}

\subsection{Newton}
\label{sec:newton}

\begin{align}
  m_i \frac{d^2\vec{x}}{dt^2} &= \vec{F} = m_g \vec{g}\\
\vec{g} &= \sum_j
\frac{-Gm_i(\vec{x}-\vec{x_j})}{\left|\vec{x}-\vec{x_j}\right|} \\
&=  \nabla\phi(\vec{x_j}\\
\phi(\vec{x_j}) &= -G\int \frac{\rho(\vec{x_j})d^3x}{|\vec{x_j}-x|}\\
\nabla^2\phi &= 4\pi G\rho(\vec{x_j})
\end{align}

\subsection{Einstein}
\label{sec:einstein}

\begin{align}
r &= \frac{GM}{3c^2}\\
\mbox{Let } M&=\frac{4}{3}\pi r^2\rho \qquad\mbox{(where $\rho$ is uniform density
  (average over r))}\\
f_r &= f_e(1+\frac{gH}{c^2}\\
\end{align}

\section{Dark Stars}

Rev John Michell (1783) suggests that a giant star could keep even
light from escaping.

Escape velocity
\begin{align}
  \frac{1}{2}mv^2 &= \frac{GMm}{r}\\
v &= \sqrt{\frac{2GM}{r}}
\end{align}

\section{Einstein's Ideas}

Equivalence principle: Can't distinguish a smoothly accelerating elevator (or rocket) from
gravity pulling on you.

Gravity as curvature: Gravity curves light path and time (slows it
down): $G_{uv} + \Lambda g_{uv} = 8\pi T_{uv}$ (G = geometry
of space time, $\Lambda$ = cosmological constant, T = distribution of
matter, motion in spacetime)

\subsection{Spacetime}

Constant time = hyperbola

\begin{align}
  (c\,t_B)^2 &= (c\,t_A)^2 + X_A^2\\
\end{align}

Mathematical tool that handles space and time $P(t, x, y, z)$.

Ex: 2D space:

\begin{align}
(\Delta s)^2 &= (\Delta x)^2 + (\Delta y)^2 \qquad\mbox{Euclidean
  Cartesian space}\\
(\Delta s)^2 &= -(\Delta t)^2 + (\Delta x)^2 \qquad\mbox{Minkowski
  Cartesian space}\\
\end{align}

A body will move along the curve of shortest distance (geodesic) in \textit{spacetime}
unless a force acts on it (not generally shortest distance in normal
space).

In general, geodesics are difficult to calculate.

Easy if you embed in Euclidean space.

\emph{Metric}: converts flat map distances into actual distances. Denoted by
``g''.

Finding a useful metric is hard --- then you just slot it into a
standard equation.

Non-physics example: Taxi metric to convert time, location, etc. into a
cab ride cost.

\emph{Special relativity}: Matter moves at most as fast as
light. Stuck inside a cone limiting speed.

\subsection{Metric g in four dimensions}

In four dimensions, need 10 numbers to indicate curvature.

Need additional math tools to keep them straight.

Call it a \emph{tensor}. Organize groups of tensors.

\begin{itemize}
\item 0-tensor: single number, eg 5
\item 1 tensor: 4 numbers (vector), eg $A_j=(1,0,-\pi, 2)$
\item 2-tensor: 4x4=16 number (matrix), eg\dots{} eh, you know\dots
\end{itemize}

\begin{align}
G_{uv} + \Lambda g_{uv} &= 8\pi T_{uv}\\
G_{uv} &= R_{ij}-\frac{1}{2}g_{uv}R \qquad\mbox{ (Einstein's)}\\
T_{uv} &= \parbox{3in}{ stress-energy tensor --- describes the density, flux
  of energy and momentum in spacetime}\\
g_{uv} &= \mbox{metric tensor}
\end{align}

EFE is a tensor equation relating some 4x4 tensors. Einstein's
equations are actually 16 equations of form: $G_{11} = 8\pi GT_{11} +
\Lambda g_{11}$

\subsection{Verification of GR}
\label{sec:verifygr}

In 1919, the sun was seen to have distorted the light from a distant
star by about 1.64 arc-seconds. Needed to be done during a solar
eclipse because otherwise there would be too much light.

\end{document}