DescriptionSpecial relativity lecture.pdf	English: This file is the special relativity lecture of the Wikiversity:Special relativity and steps towards general relativity course. It is in pdf format for convenient viewing as a fullscreen, structured presentation in a classroom. It uses encapsulated postscript versions of many Wikimedia Commons diagrams, using their .fig or octave source code when available. Most diagrams are labelled with mathematical expressions, using LaTeX "specials". The pdf contains many links to Wikipedia resources for more in-depth learning. Mediawiki-text and svg might become directly usable in the long term, but do not yet seem ready to replace LaTeX.
Date	10 July 2011
Source	Own work plus Wikimedia Commons and Wikibooks files available under CC-BY-SA or public domain or GFDL licences, plus File:Commons-logo-en.svg (which is trademarked) to indicate to readers that images are from WM Commons. This logo usage seems likely to be acceptable, but the pdf file could easily be recreated without the logo if needed. A script for creating the pdf file and checking the licences of included (format converted) files will be added a few minutes after the main upload. The GFDL files seem to satisfy the CC-BY-SA compatibility operation.
Author	User:Boud
Permission (Reusing this file)	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original. https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue However, modifications need to either remove File:Commons-logo-en.svg from the file or be careful to continue following Wikimedia Foundation trademark policy regarding the logo.
Other versions	related:

script and LaTeX source

The pdf file can be produced from https://codeberg.org/boud/SRandGR_lectures (commit as of 2024-03-22: c25f50de), or it can (with a few minor hacks) be produced with the script and LaTeX source below.

shell script

This script is intended for users with the necessary GNU/Linux command line experience to understand what the script does and to modify it for personal usage.

• Do not use it without understanding what it does.
• Check this wiki page history to see who edited it, when they edited, and what changes they made.
• See the disclaimer linked to at the bottom of this page.

Having got through the warnings, you are most welcome to improve the script.

#!/bin/bash

## (c) CC-BY-SA-3.0 and/or GPLv3 at your choosing
## See the Wikimedia Foundation Commons website for authorship.
## http://commons.wikimedia.org/wiki/File:Special_relativity_lecture.pdf
## http://commons.wikimedia.org/wiki/File:Steps_towards_general_relativity_lecture.pdf

## Changelog
##
## 2012-05-15 download_non_svg, etc: generalise sed expression in each
##   URL=...  because some absolute URLs are written without "http:"
##   in (at least) the Commons html files; 
## download_pdf: make ampersand, greater than, less than to appear
##   correctly in the browser-rendered version of this script instead
##   of the source version.

DATE=`date +%y%m%d`
BASE_NAME=Special_relativity_lecture
BIG_TMP_DIR=`mktemp -d /tmp/tmp_sr_XXXXXX`
PDF_FINAL=sr_${DATE}.pdf


FILES_FIG_SVG="Rotation_Euclidean1.svg Rotation_Euclidean2.svg \
Rotation_Euclidean12.svg Lorentz_rotation1.svg Lorentz_rotation2.svg \
Lorentz_rotation_ds_spacelike.svg Lorentz_rotation_ds_timelike.svg \
Lorentz_rotation_high_beta.svg Time_dilation_derivation.svg \
Space_contraction_rocket_frame.svg \
Space_contraction_observer_frame.svg SR_redshift_derivation.svg \
Add_velocity_ark_POV.svg Add_velocity_Earth_POV.svg \
Relativistic_aberration1.svg Relativistic_aberration2.svg \
Relativistic_aberration3.svg Tachyonic_antitelephone.svg \
Tachyon_observer_frame_only.svg Tachyon_rocket_frame_also.svg \
Tachyon_world_line.svg Pole_barn_Minkowski.svg \
Multiply_connected_twins_paradox_0.svg \
Multiply_connected_twins_paradox_leftmoving_POV.svg \
Multiply_connected_twins_paradox_rightmoving_POV.svg \
Multiply_connected_twins_paradox_leftmoving_cylinder.svg \
Multiply_connected_twins_paradox_rightmoving_cylinder.svg \
Multiply_connected_twins_paradox_overview.svg \
Multiply_connected_twins_paradox_time_cutpaste.svg Four-velocity.svg \
Four-momentum.svg"

## override: debug or just download/convert a few files
#FILES_FIG_SVG="Tachyonic_antitelephone.svg Multiply_connected_twins_paradox_time_cutpaste.svg"
#FILES_FIG_SVG="Add_velocity_ark_POV.svg Add_velocity_Earth_POV.svg"


FILES_OCTAVE="Beta_versus_rapidity.svg"

## debug only
#FILES_SVG="Rotation_Euclidean1.svg" 

FILES_WIKIBOOKS="Inertialoverlay.GIF Rel2.gif Rel3.gif"


FILES_SVG_ONLY="148px-Commons-logo-en.svg \
500px-World_line2.svg \
500px-Ladder_Paradox_Overview.svg \
500px-Ladder_Paradox_GarageFrame.svg \
500px-Ladder_Paradox_LadderFrame.svg \
500px-Ladder_Paradox_GarageScenario.svg \
500px-Ladder_Paradox_LadderScenario.svg \
500px-LadderParadox2_Minkowski.svg"

FILES_ANIMATIONS="Lorentz_transform_of_world_line.gif"


FILE_PDF="${BASE_NAME}.pdf"

## non-svg non-animated Commons files to be downloaded and converted
FILES_NON_SVG="NONE"

## files uploaded in .svg output format plus  .fig source code.
## * create an .eps file with math labels using the full power of LaTeX
function download_fig_svg {
    set -x # help in debugging
    FILES=$1
    DSLASH="//" ## avoid mediawiki conversion to an anchored URL
    FILES_HTML=`echo ${FILES} | sed -e "s,\([^ ]*\),https:${DSLASH}commons.wikimedia.org/wiki/File:\1,g"`
    TMP_DIR=.
    mkdir -pv ${TMP_DIR}
    cd ${TMP_DIR} || exit 1
    pwd
    wget --random-wait --wait=2 --timestamping ${FILES_HTML}
    FILES_COLON=`echo ${FILES} | sed -e 's,\([^ ]*\),File\:\1,g'`

    TMP_FILE=tmp_xx
    for file in ${FILES_COLON}; do
        ## extract .fig source
        csplit -f ${TMP_FILE} ${file} '/<.*pre>/' '{1}'
        cat ${TMP_FILE}01 |sed -e 's/<pre>//' >  tmp_yy.fig

        # Some versions may not have the "<pre>" at the beginning.
        # If necessary, search to the 'Produced by xfig version' line':
        head -n1 tmp_yy.fig > tmp_firstline
        if (grep "Produced by xfig" tmp_firstline) ; then
            # If 'Produced by xfig version' is the first line, then use the whole file:
            sed -e 's/^ *//' \
                -e 's/ / /g' -e "s/'/'/g" tmp_yy.fig \
                > ${TMP_FILE}.fig
        else
            # Otherwise, split based on the first instance of 'Produced by xfig version'
            # and remove any leading spaces:
            csplit -f tmp_zz tmp_yy.fig '/Produced by xfig/'
            sed -e 's/^ *//' \
                -e 's/ / /g' -e "s/'/'/g" tmp_zz01 \
                > ${TMP_FILE}.fig
            ls -l ${TMP_FILE}.fig
        fi

        ## produce a reasonable size .eps figure
        fig2dev -L pstex -F ${TMP_FILE}.fig > ${TMP_FILE}.pstex
        fig2dev -L pstex_t -p ${TMP_FILE}.pstex ${TMP_FILE}.fig \
            > ${TMP_FILE}.pstex_t

        ## start a latex-able file
        cat >${TMP_FILE}.tex <<EOF 
\documentclass[12pt]{article}
\usepackage{graphicx,color}
\pagestyle{empty}
\begin{document} 
EOF

        echo "\\input{" ${TMP_FILE}.pstex_t "}" >> ${TMP_FILE}.tex

        cat >> ${TMP_FILE}.tex <<EOF 
\end{document}
EOF

        latex ${TMP_FILE}.tex 
        dvips ${TMP_FILE}.dvi -E -o ${TMP_FILE}.eps

        ## restore file names
        FILE_BASE=`echo ${file} |sed -e 's/\.svg//'|sed -e 's/File://'`
        
        mv ${TMP_FILE}.fig ${FILE_BASE}.fig
        mv ${TMP_FILE}.eps ${FILE_BASE}.eps
        echo "The following eps file should now exist:"
        ls -l ${FILE_BASE}.eps 
        #gv  ${FILE_BASE}.eps  # uncomment this if you want to check the .eps file interactively

        ## clean up
        rm ${TMP_FILE}0[012] 
        for i in aux dvi log pstex pstex_t tex; do
            rm ${TMP_FILE}.${i}
        done
        #ls -l ${TMP_FILE}*  ## uncomment to check for anything left
    done

    set +x
}

## files uploaded in .svg output format plus octave source code
function download_octave {
    FILES=$1
    DSLASH="//" ## avoid mediawiki conversion to an anchored URL
    FILES_HTML=`echo ${FILES} | sed -e "s,\([^ ]*\),https:${DSLASH}commons.wikimedia.org/wiki/File:\1,g"`

    TMP_DIR=.
    mkdir -pv ${TMP_DIR}
    cd ${TMP_DIR} || exit 1
    pwd
    wget --random-wait --wait=2 --timestamping ${FILES_HTML}
    FILES_COLON=`echo ${FILES} | sed -e 's,\([^ ]*\),File\:\1,g'`

    TMP_FILE=tmp_xx
    for file in ${FILES_COLON}; do
        ## extract .fig source
        csplit -f ${TMP_FILE} ${file} '/<.*pre>/' '{1}'
        tail -n +2 ${TMP_FILE}01 | sed -e "s,/tmp/,${TMP_DIR}/," >  ${TMP_FILE}.m

        ## produce a reasonable size .eps figure
        octave ${TMP_FILE}.m

        ## restore file names
        FILE_BASE=`echo ${file} |sed -e 's/\.svg//'|sed -e 's/File://'`
        
        mv ${TMP_FILE}.m ${FILE_BASE}.m
        echo "The following eps file should now exist:"
        ls -l ${FILE_BASE}.eps 
        #gv  ${FILE_BASE}.eps  # uncomment this if you want to check the .eps file interactively

        ## clean up
        # no intermediate files in this case
        #ls -l ${TMP_FILE}*  ## uncomment to check for anything left
    done

}

## files available from wikibooks in various non svg formats
function download_wikibooks {
    FILES=$1
    DSLASH="//" ## avoid mediawiki conversion to an anchored URL
    FILES_HTML=`echo ${FILES} | sed -e "s,\([^ ]*\),https:${DSLASH}en.wikibooks.org/wiki/File:\1,g"`

    TMP_DIR=.
    mkdir -pv ${TMP_DIR}
    cd ${TMP_DIR} || exit 1
    pwd
    wget --random-wait --wait=2 --timestamping ${FILES_HTML}
    FILES_COLON=`echo ${FILES} | sed -e 's,\([^ ]*\),File\:\1,g'`

    for file in ${FILES_COLON}; do
        URL=`grep fullMedia ${file} | sed -e 's/.*href=\"\(\(http\|\)[^ ]*\)\".*/\1/' | sed -e "s,^${DSLASH},http:${DSLASH},"`
        wget --random-wait --wait=2 --timestamping ${URL}
        ## recover file name
        FILE_ORIG=`echo ${file} |sed -e 's/^File://'`
        FILE_BASE=`echo ${FILE_ORIG}|sed -e 's/\.[^\.]*$//'`

        ## produce a brute-force .eps figure
        convert ${FILE_ORIG} ${FILE_BASE}.eps

        echo "The following eps file should now exist:"
        ls -l ${FILE_BASE}.eps 
        #gv  ${FILE_BASE}.eps  # uncomment this if you want to check the .eps file interactively

        ## clean up
        # no intermediate files in this case
        #ls -l ${TMP_FILE}*  ## uncomment to check for anything left
    done

}

## files available from Commons in various non svg formats
function download_non_svg {
    FILES=$1
    DSLASH="//" ## avoid mediawiki conversion to an anchored URL
    FILES_HTML=`echo ${FILES} | sed -e "s,\([^ ]*\),https:${DSLASH}commons.wikimedia.org/wiki/File:\1,g"`

    TMP_DIR=.
    mkdir -pv ${TMP_DIR}
    cd ${TMP_DIR} || exit 1
    pwd
    wget --random-wait --wait=2 --timestamping ${FILES_HTML}
    FILES_COLON=`echo ${FILES} | sed -e 's,\([^ ]*\),File\:\1,g'`

    for file in ${FILES_COLON}; do
        URL=`grep fullMedia ${file} | sed -e 's/.*href=\"\(\(http\|\)[^ ]*\)\".*/\1/' | sed -e "s,^${DSLASH},http:${DSLASH},"`
        wget --random-wait --wait=2 --timestamping ${URL}
        ## recover file name
        FILE_ORIG=`echo ${file} |sed -e 's/^File://'`
        FILE_BASE=`echo ${FILE_ORIG}|sed -e 's/\.[^\.]*$//'`

        ## produce a brute-force .eps figure
        convert ${FILE_ORIG} ${FILE_BASE}.eps

        echo "The following eps file should now exist:"
        ls -l ${FILE_BASE}.eps 
        #gv  ${FILE_BASE}.eps  # uncomment this if you want to check the .eps file interactively

        ## clean up
        # no intermediate files in this case
        #ls -l ${TMP_FILE}*  ## uncomment to check for anything left
    done

}


## files available from WMCommons in .svg format that do not have
## source code available for producing them; convert locally by brute force
## * input file names must include desired px size:  148px-Commons-logo-en.svg
function download_svg_only {
    FILES=$1
    ## URLs for downloaded full html pages, pixel sizes removed from string
    DSLASH="//" ## avoid mediawiki conversion to an anchored URL
    FILES_HTML=`echo ${FILES} | sed -e "s,[0-9]\+px-\([^ ]*\),https:${DSLASH}commons.wikimedia.org/wiki/File:\1,g"`

    TMP_DIR=.
    mkdir -pv ${TMP_DIR}
    cd ${TMP_DIR} || exit 1
    pwd
    wget --random-wait --wait=2 --timestamping ${FILES_HTML}
    
    ## store pixel sizes and html file names together in each string
    FILES_COLON=`echo ${FILES} | sed -e 's,\([0-9]\+\)px-\([^ ]*\),\1:::File\:\2,g'`

    for file_colon in ${FILES_COLON}; do
        ## html file name that should now be locally available
        FILE=`echo ${file_colon} |sed -e 's/^.*::://'`
        URL=`grep -i "fileInfo.*SVG file" ${FILE} |  sed -e 's/.*href=\"\(\(http\|\)[^ ]*\)\".*/\1/' | sed -e "s,^${DSLASH},http:${DSLASH},"`
        wget --random-wait --wait=2 --timestamping ${URL}
        ## recover file name
        FILE_ORIG=`echo ${FILE} |sed -e 's/^File://'`
        PIXEL_WIDTH=`echo ${file_colon}|sed -e 's/:::.*$//'`
        FILE_BASE=`echo ${FILE_ORIG}|sed -e 's/\.svg//'`

        ## produce a brute-force .eps figure
        convert ${FILE_ORIG} -resize ${PIXEL_WIDTH} ${PIXEL_WIDTH}px-${FILE_BASE}.eps

        echo "The following eps file should now exist:"
        ls -l ${PIXEL_WIDTH}px-${FILE_BASE}.eps 
        #gv  ${FILE_BASE}.eps  # uncomment this if you want to check the .eps file interactively

        ## clean up
        # no intermediate files in this case
        #ls -l ${TMP_FILE}*  ## uncomment to check for anything left
    done

}

## animation files available from WMCommons; downloaded, not converted locally
function download_animations {
    FILES=$1
    ## URLs for downloaded full html pages
    DSLASH="//" ## avoid mediawiki conversion to an anchored URL
    FILES_HTML=`echo ${FILES} | sed -e "s,\([^ ]*\),https:${DSLASH}commons.wikimedia.org/wiki/File:\1,g"`

    TMP_DIR=.
    mkdir -pv ${TMP_DIR}
    cd ${TMP_DIR} || exit 1
    pwd
    wget --random-wait --wait=2 --timestamping ${FILES_HTML}
    
    ## name of downloaded html files
    FILES_COLON=`echo ${FILES} | sed -e 's,\([^ ]*\),File\:\1,g'`

    for file_colon in ${FILES_COLON}; do
        ## html file name that should now be locally available
        URL=`grep -i "fullMedia" ${file_colon} | sed -e 's/.*href=\"\(\(http\|\)[^ ]*\)\".*/\1/' | sed -e "s,^${DSLASH},http:${DSLASH},"`
        wget --random-wait --wait=2 --timestamping ${URL}
        ## recover file name
        FILE_ORIG=`echo ${FILE} |sed -e 's/^File://'`

        echo "The following animation file should now exist:"
        ls -l ${FILE_ORIG}
    done

}


## download the LaTeX source of the pdf file and clean it from mediawiki
## interventions.
function download_pdf {
    FILES=$1
    ## URLs for downloaded full html pages
    DSLASH="//" ## avoid mediawiki conversion to an anchored URL
    FILES_HTML=$(echo ${FILES} | sed -e "s,\([^ ]*\),https:${DSLASH}commons.wikimedia.org/wiki/File:\1,g")

    TMP_DIR=.
    mkdir -pv ${TMP_DIR}
    cd ${TMP_DIR} || exit 1
    pwd
    wget --random-wait --wait=2 --timestamping ${FILES_HTML}
    
    ## name of downloaded html files
    FILES_COLON=$(echo ${FILES} | sed -e 's,\([^ ]*\),File\:\1,g')

    for file_colon in ${FILES_COLON}; do
        ## html file name that should now be locally available
        csplit ${file_colon} '/<.*pre>/' '{*}'
        tail -n +2 xx03 | \
            sed -e 's/>Template:\([^<]*\)</>{{\1}}</g' \
                -e 's,<a href[^>]*>,,gi' -e 's,</a>,,gi' \
                -e 's/&amp;/\&/gi' \
                -e 's/&lt;/</gi' -e 's/&gt;/>/gi' \
                -e 's/</</gi' -e 's/>/>/gi' \
                -e 's/ / /g' \
            > xx03.tex
        mv -v --backup xx03.tex ${BASE_NAME}.tex

        echo "The following pdf file should now exist:"
        ls -l ${BASE_NAME}.tex
    done

}

# produce the .pdf file
# WARNING: the intermediate postscript files may be big, so 
# make sure your have plenty of space in ${BIG_TMP_DIR}
function produce_pdf {
    set -x
    BASE_NAME=$1
    BIG_TMP_DIR=$2
    PDF_FINAL=$3
    printf "\n\n\n========== Will run LaTeX... ============\n\n\n"
    latex ${BASE_NAME}.tex
    printf "\n\n\n========== Will run LaTeX again... ============\n\n\n"
    latex ${BASE_NAME}.tex
    printf "\n\n\n========== Will run dvips... ============\n\n\n"
    dvips ${BASE_NAME}.dvi -o ${BIG_TMP_DIR}/${BASE_NAME}.ps
    printf "\n\n\n========== Will run ps2pdf... ============\n\n\n"
    # May need the following as a temporary hack around 2022/2023:
    # https://github.com/GenericMappingTools/gmt/issues/4453
    sed -e 's/\.setopacityalpha/.setfillconstantalpha/g' ${BIG_TMP_DIR}/${BASE_NAME}.ps > ${BIG_TMP_DIR}/${BASE_NAME}.eps
    ps2pdf -dALLOWPSTRANSPARENCY -dNORANGEPAGESIZE ${BIG_TMP_DIR}/${BASE_NAME}.eps ${BIG_TMP_DIR}/${BASE_NAME}.pdf
    cp -pi ${BIG_TMP_DIR}/${BASE_NAME}.pdf ${PDF_FINAL}
    echo " "
    echo "You should now have the pdf file:"
    ls -l ${PDF_FINAL}
    echo "You should keep this file and " ${FILES_ANIMATIONS} 
    echo " "
    echo "The other files can be safely removed, including those"
    echo "in ${BIG_TMP_DIR}/":
    ls -l ${BIG_TMP_DIR}
    set +x    
}

# interactive check of licences after downloading 
function check_licences {
    for i in `ls File\:*`; do 
        egrep -H --color "(is licensed.*Creative Commons.*/by-sa/|.*release.*public domain|Permission is granted.*GNU Free Documentation License)" $i
    done 

    echo "The following files might have problems. Check manually."  ## -L = files without
    for i in `ls File\:*`; do 
        egrep -L -H --color "(is licensed.*Creative Commons.*/by-sa/|.*release.*public domain|Permission is granted.*GNU Free Documentation License)" $i
    done 
}


download_octave "${FILES_OCTAVE}"

download_fig_svg "${FILES_FIG_SVG}"

download_wikibooks "${FILES_WIKIBOOKS}"

###download_non_svg "${FILES_NON_SVG}"

download_svg_only "${FILES_SVG_ONLY}"

download_animations "${FILES_ANIMATIONS}"

### 2024-03-17 Cleaning the source from the Wikimedia Commons download is not so
### easy to maintain for a full .tex file, so for the moment, use the .tex file
### in the git repository and comment out this download function:
# download_pdf ${FILE_PDF}

produce_pdf ${BASE_NAME} ${BIG_TMP_DIR} ${PDF_FINAL}

check_licences  ## prior to uploading a new .pdf

LaTeX source

This source is intended for users with the necessary LaTeX experience to understand it and to edit it after locally checking that the updated source is LaTeX-able and gives an improved pdf file.

%% (c) CC-BY-SA-3.0 
%% See the Wikimedia Foundation Commons website for authorship.
%% http://commons.wikimedia.org/wiki/File:Special_relativity_slides.pdf
%% http://commons.wikimedia.org/wiki/File:Steps_towards_general_relativity_slides.pdf

%%
%% howto prosper
%%
%% latex sr_gr.tex && latex sr_gr.tex && dvips sr_gr.dvi -o /tmp/s.ps && ps2pdf /tmp/s.ps /tmp/s.pdf
%% 
%% You might also try:
%% dvips -Ppdf -G0 file.dvi 
%% ps2pdf -dPDFsettings=/prepress file.ps
%%
%% see also: http://freshmeat.net/articles/view/667/

\RequirePackage{color}
\documentclass[pdf,ps2pdf,corners,slideColor,colorBG,accumulate,nototal]{prosper}
%\documentclass[pdf,contemporain,slideColor,colorBG,accumulate,nototal]{prosper}
%\documentclass{prosper}

\usepackage{color}
\definecolor{mygreen}{rgb}{0.0,0.4,0.0}

\usepackage{graphicx}
\usepackage{pstricks,pst-node,pst-text,pst-3d}  % for \rnode, \nccurve

%\usepackage{amsfonts}
%\usepackage{mathpi}

%\usepackage{ucs}  % for e.g. en-dash in unicode

%\usepackage[polish]{babel}
%\usepackage[T1]{fontenc}
%\usepackage[latin2]{inputenc}       % allow Latin1 characters
%\usepackage{polski}

%\usepackage{hyperref}
%\newcommand

%% comment out or include large sections, choose one of each definition
%\newcommand\SRslides[1]{}
\newcommand\SRslides[1]{#1}
\newcommand\GRslides[1]{}
%\newcommand\GRslides[1]{#1}

%% escape slash-slash so that mediawiki doesn't convert URLs to html anchored URLs
\newcommand\slashslash{//}


%% hyperlinks for astro-ph

\newcommand\aph[1]{ 
\href{http:\slashslash arxiv.org/abs/astro-ph/#1}{{\color{mygreen}
arXiv:astro-ph/#1}}}

\newcommand\arXiv[1]{ 
\href{http:\slashslash arxiv.org/abs/#1}{{\color{mygreen} arXiv:#1}}}

%% general command for making hyperlinks
\newcommand\myhref[1]{\href{#1}{{\color{mygreen} #1}}}
\newcommand\WikipediaEn[1]{\href{http:\slashslash en.wikipedia.org/wiki/#1}{{\color{mygreen} \underline{w:#1}}}}
\newcommand\WikipediaEnTwo[2]{\href{http:\slashslash en.wikipedia.org/wiki/#1}{{\color{mygreen} \underline{w:#2}}}}

\newcommand\WMCommons[1]{\href{http:\slashslash commons.wikimedia.org/wiki/File:#1}{\includegraphics[height=0.05\textheight]{148px-Commons-logo-en}}}

\newcommand\Wikibooks[1]{\href{http:\slashslash en.wikibooks.org/wiki/File:#1}{b:#1}}
\newcommand\WikiversityEn[1]{\href{http:\slashslash en.wikiversity.org/wiki/#1}{{\color{mygreen} \underline{wv:#1}}}}

% http://wikimediafoundation.org/wiki/Trademark_Policy#Things_You_Can_Do.2C_a_Summary
% http://www.webcitation.org/5wrHUrGON
% Things You Can Do, a Summary 

% link directly to Wikimedia's website(s) by using banners and buttons
% derived from Wikimedia trademarks and logos.

%


\newcommand\apj{ApJ.}                        % {Ap. J.}
\newcommand\apjs{ApJ. Supp.}                 % {Ap. J.}
\newcommand\aj{A.J.}                           % {A. J.}
\newcommand\aanda{A\&A}                      % {A. \& A.} 
%\newcommand\cqg{Class. Quant. Grav.}         % CQG
%\newcommand\mnras{Month. Not. R. Astron. Soc.}
\newcommand\cqg{Classical \& Quantum Gravity}         % CQG
\newcommand\mnras{Monthly Notices of the Royal Astronomical Society}
\newcommand\hMpc{\mbox{h$^{-1}$ Mpc}}
\newcommand\hGpc{\mbox{h$^{-1}$ Gpc}}
\newcommand\hkpc{\mbox{h$^{-1}$ kpc}}
\newcommand\ltapprox{\,\lower.6ex\hbox{$\buildrel <\over \sim$} \, }
\newcommand\ttimes{{\scriptstyle \times}}
\newcommand\exunit{{\bf e x}}  
\newcommand\eyunit{{\bf e y}}  
\newcommand\ezunit{{\bf e z}}  
\newcommand\Omm{\Omega_{\mbox{\rm \small m}}}
\newcommand\Omtot{\Omega_{\mbox{\rm \small tot}}}
\newcommand\wQ{\bf w_Q}
\newcommand\dpm{d_{\mbox{\rm \small pm}}}
\newcommand\dL{d_{\mbox{\rm \small L}}}
\newcommand\da{d_{\mbox{\rm \small a}}}
\newcommand\ddd{{\mbox{\rm d}}}
\newcommand\npairs{n_{\mbox{\rm \small pairs}}}%%% EDITOR modify as desired 
\newcommand\Lselec{L_{\mbox{\rm \small selec}}}%%% EDITOR modify as desired 
\newcommand\rinj{{r}_{\mbox{\rm \small inj}}}  %EDITOR: change font if want to.
\newcommand\mytanh{\mathrm{\,tanh\,}}
\newcommand\mytan{\mathrm{\,tan\,}}
\newcommand\epower[1]{\times 10^{#1}}
\newcommand\TTT{^{\mathrm{T}}}

\title{Special relativity and steps towards general relativity: SR}
\author{} 
\institution{\href{http:\slashslash en.wikiversity.org/wiki/Special_relativity_and_steps_towards_general_relativity}{(c) CC-BY-SA-3.0}}

%\newcommand\imagedir{images}
%\newcommand\gifmoviedir{gifmovie}

\begin{document}

\maketitle

%\renewcommand{\theequation}{}


%\leftheader{
%\fancyhead{}
\newcommand\SRGRcaption{
  {%\tiny 
    \SRslides{
    \hyperlink{Intro}{0}  $\cdot$
    \hyperlink{Mink}{M}  $\cdot$
    \hyperlink{Lorentz}{$\Lambda$} $\cdot$
    \hyperlink{beta}{$\beta$} $\cdot$
    \hyperlink{calib}{cal} $\cdot$
    \hyperlink{cconstant}{$c$} $\cdot$
    \hyperlink{addvel}{$\Sigma\phi$} $\cdot$
    \hyperlink{gamma}{$\gamma$} $\cdot$
    \hyperlink{tdilation}{$t+$} $\cdot$
    \hyperlink{xcontract}{$x-$} $\cdot$
    \hyperlink{dopp}{$z$} $\cdot$
    \hyperlink{aberr}{$xy$} $\cdot$
    \hyperlink{timecone}{tc} $\cdot$
    \hyperlink{RPPparadox}{Pen}  $\cdot$
    \hyperlink{tachy}{$\beta+$}  $\cdot$
    \hyperlink{polebarn}{PB}  $\cdot$
    \hyperlink{twins}{tw}  $\cdot$
    \hyperlink{fourvel}{$u^\mu$}  $\cdot$
    \hyperlink{fourmom}{$p^\mu$}  $\cdot$
    \hyperlink{fourinv}{$\|p\|$}  $\cdot$
    \hyperlink{emcsq}{E=m}  $\cdot$
    \hyperlink{photon}{E=p}  $\cdot$
    \hyperlink{srconclu}{SR} 
    }
    %% \, $\vert$
    \GRslides{
    \hyperlink{GRintro}{1} $\cdot$
    \hyperlink{coordtransf}{$\Lambda$} ~\hyperlink{summation}{$\Sigma$} $\cdot$
    \hyperlink{scalarfield}{$\tilde{\mathrm{d}}\phi$} $\cdot$
    \hyperlink{scalarprod}{$\left<\right>$} $\cdot$
    \hyperlink{metric}{{\bf g}} $\cdot$
    \hyperlink{coordinate}{$x^\mu$} $\cdot$
    \hyperlink{dstwo}{$\mathrm{d}s^2$} $\cdot$
    \hyperlink{gradvec}{$\nabla\vec{A}$} %% top of page
    \hyperlink{gradvecfinal}{;}  %% final formula
    $\cdot$
    %%%    \hyperlink{Gammanottensor}{$\Gamma$ warning}
    \hyperlink{manifold}{$M$} $\cdot$
    \hyperlink{dirderiv}{$\nabla_V$} 
    \hyperlink{paratransp}{$=0$} $\cdot$
    \hyperlink{riemannT}{{\bf R}} $\cdot$
    \hyperlink{stressEtensor}{$>$} 
    %%% \hyperlink{Gammafrommetric}{$\Gamma(\mathbf{g})$}
%    \hyperlink{Riem}{Riem} $\cdot$
%    \, $\vert$
%    \hyperlink{EFE}{EFE} $\cdot$ 
    %    \hyperlink{EquivP}{Eq} $\cdot$ 
%    \hyperlink{Schw}{Sch},  \hyperlink{FLRW}{FL}
    }
    %\hyperlink{maxima}{max}  

%    \, $\vert$
%    \hyperlink{ADM}{ADM} $\cdot$
%    \hyperlink{cactus}{cac}
%$\vert$
{{\bf {\quad} SR+$\epsilon$GR} }
}

%\ \ \hyperlink{canalu}{cU} 
%{}{}{}
}



\slideCaptionLeft{\SRGRcaption}

\slideCaption{}


%\rule{10cm}{0ex} 


\SRslides{
\parskip=12pt 
\overlays{9}{%
\begin{slide}{SR+GR }
\parskip=5pt  %\small 
\vspace{-10mm}
\onlySlide*{1}{
  $\bullet$ SR+GR: construct spacetime from set theory
    \hypertarget{Intro}{} 
    }

\fromSlide{2}{
  $\bullet$ SR+GR: construct spacetime from set theory

  $\bullet$ start with $X$ = {set} of points, no distances between
  points, angles, etc. defined
    }

\fromSlide{3}{
  $\bullet$ {no absolute simultaneity} = ``time is not absolute"
}

\fromSlide{4}{
  $\bullet$ add properties to $X$ that satisfy theorems 
}

\fromSlide{5}{
  $\bullet$ $\rightarrow$ differentiable 4-(pseudo-)manifold 
}

\fromSlide{6}{
  $\bullet$ point particle in space $\rightarrow$ \WikipediaEn{World line} 
  in spacetime
}

\fromSlide{7}{
  $\bullet$ point in spacetime $\rightarrow$ spacetime ``event"
}


\fromSlide{8}{
  $\bullet$ SR: spacetime = \WikipediaEn{Minkowski space}
}
 
\fromSlide{9}{
  $\bullet$ GR: spacetime = a solution of the 
  \WikipediaEn{Einstein field equations} 
}
 

\end{slide}
}


\parskip=12pt 
\overlays{4}{%
  \begin{slide}{SR: Minkowski spacetime}
    \parskip=5pt  %\small 
    \vspace{-15mm}
    \onlySlide*{1}{\includegraphics[height=0.80\textheight]{Rotation_Euclidean1} 
      \hypertarget{Mink}{} 

      $p$ at $(x,y)$, distance from observer at $O$ is $d$
    }

    \onlySlide*{2}{
      \begin{minipage}[b]{60mm}% width
        \includegraphics[height=0.80\textheight]{Rotation_Euclidean2} 
      \end{minipage}
      \begin{minipage}[b]{20mm}% width
        $\left( \begin{array}{c} p_{\mathrm{x'}} \\ p_{\mathrm{y'}} \end{array} \right)
        = \left( \begin{array}{cc}
            \cos\theta & \sin\theta \\
            -\sin\theta & \cos\theta 
          \end{array}
          \right) \, 
        \left( \begin{array}{c} p_{\mathrm{x}} \\ p_{\mathrm{y}} \end{array} \right)
        $ \\
        \vspace{20mm}
      \end{minipage}

      $p$ at $(x',y')$, distance from observer at $O$ is $d$ = {unchanged}
    }

    \onlySlide*{3}{
      \begin{minipage}[b]{51mm}% width
        \includegraphics[height=0.80\textheight]{Lorentz_rotation1} 
      \end{minipage}

      $p$ at $(x,t)$, \WikipediaEn{invariant interval}
      from observer at $O$ is $\Delta s $ where
      $(\Delta s)^2 = -(\Delta t)^2 + (\Delta x)^2$
    }

    \onlySlide*{4}{
      \begin{minipage}[b]{51mm}% width
        \includegraphics[height=0.80\textheight]{Lorentz_rotation2} 
      \end{minipage}
      \begin{minipage}[b]{20mm}% width
        $\left( \begin{array}{c} p_{\mathrm{x'}} \\ p_{\mathrm{t'}} \end{array} \right)
        = \left( \begin{array}{cc}
            \cosh\phi & -\sinh\phi \\
            -\sinh\phi & \cosh\phi 
          \end{array}
          \right) \, 
        \left( \begin{array}{c} p_{\mathrm{x}} \\ p_{\mathrm{t}} \end{array} \right)
        $ \\
        \vspace{20mm}
      \end{minipage}
      
      $p$ at $(x',t')$, invariant interval from observer at $O$ is $\Delta s$ 
      =  $(\Delta s)^2 = -(\Delta t')^2 + (\Delta x')^2$ = {unchanged}
      
    }

  \end{slide}
}


\parskip=12pt 
\overlays{8}{%
  \begin{slide}{SR: Lorentz transformation}
    \parskip=7pt  %\small 
    \vspace{-10mm}
    \onlySlide*{1}{
      \hypertarget{Lorentz}{}
      \begin{minipage}[b]{60mm}
      $\Lambda (\phi) := \left(
      \begin{array}{cc}
        \cosh\phi & -\sinh\phi \\
        -\sinh\phi & \cosh\phi 
      \end{array}
      \right) 
      $
      \end{minipage}
      \begin{minipage}[b]{40mm}
        $\cosh\phi := \frac{\mathrm{e}^\phi + \mathrm{e}^{-\phi}}{2}$

        $\sinh\phi := \frac{\mathrm{e}^\phi - \mathrm{e}^{-\phi}}{2}$

        \WikipediaEn{hyperbolic function}
      \end{minipage}
    }
    
    \fromSlide{2}{
      \begin{minipage}[b]{60mm}
        $\Lambda (\phi) := \left( 
        \begin{array}{cc}
          \cosh\phi & -\sinh\phi \\
          -\sinh\phi & \cosh\phi 
        \end{array}
        \right) 
        $
      \end{minipage}
      \begin{minipage}[b]{40mm}
        $\cosh\phi := \frac{\mathrm{e}^\phi + \mathrm{e}^{-\phi}}{2}$

        $\sinh\phi := \frac{\mathrm{e}^\phi - \mathrm{e}^{-\phi}}{2}$

        \WikipediaEn{hyperbolic function}
      \end{minipage}

      $
      \left( \begin{array}{c} p_{\mathrm{x'}} \\ p_{\mathrm{t'}} \end{array} \right) 
      = \Lambda \;
      \left( \begin{array}{c} p_{\mathrm{x}} \\ p_{\mathrm{t}} \end{array} \right)
      $
    }

    \fromSlide{3}{ units: $\Lambda$ only makes sense if same units for
      $x$, $t$, $x'$, $t'$
    }

    \fromSlide{4}{Definition: 1~m := (1/$2.99792458\epower{8}$)~s $\approx 10^{-8.5}$~s 
      $\approx$ 3~ns}


    \onlySlide*{5}{speed of light in a vacuum = $c$ 
      $= \frac{2.99792458\epower{8}~\mathrm{m}}{1~\mathrm{s}} $
    }

    \onlySlide*{6}{speed of light in a vacuum = $c$ 
      $= \frac{2.99792458\epower{8}~\mathrm{m}}{1~\mathrm{s}} $

      $= \frac{2.99792458\epower{8} \times 
        (1/2.99792458\epower{8})~\mathrm{s}}{1~\mathrm{s}}$
    }


    \onlySlide*{7}{speed of light in a vacuum = $c$ 
      $= \frac{2.99792458\epower{8}~\mathrm{m}}{1~\mathrm{s}}$

      %      $= \frac{2.99792458\epower{8} \times 
      %        (1/2.99792458\epower{8})~\mathrm{s})}{1~\mathrm{s}}$

      $= \frac{1~\mathrm{s}}{1~\mathrm{s}}$
    }
    

    \fromSlide{8}{speed of light in a vacuum = $c$ 
      $=\frac{2.99792458\epower{8}~\mathrm{m}}{1~\mathrm{s}}$ 

      %      $= \frac{2.99792458\epower{8} \times
      %        (1/2.99792458\epower{8})~\mathrm{s}}{1~\mathrm{s}}$
      
      $=\frac{1~\mathrm{s}}{1~\mathrm{s}} = 1$ (dimensionless) }

  \end{slide}
}


\parskip=12pt 
\overlays{10}{%
\begin{slide}{SR: rapidity $\phi$ vs velocity $\beta$ }
\parskip=10pt  %\small 
\vspace{-10mm}
    \onlySlide*{1}{ What is $\phi$ ? \hypertarget{beta}{} }

    \fromSlide{2}{ What is $\phi$ ? }

    \onlySlide*{2}{ observer {\cal A} has worldline $(x,t) = (0,t)$}

    \fromSlide{3}{ observer {\cal B} has worldline $(x',t') = (0,t')$}

    \onlySlide*{4}{ 
      in A's coordinate system, B's worldline is:

      $
      \left( \begin{array}{c} 0 \\ t' \end{array} \right)
      =
      \Lambda \; 
      \left( \begin{array}{c} x \\ t \end{array} \right) 
       $}

    \fromSlide{5}{ 
      in A's coordinate system, B's worldline is:

      $\Rightarrow \left( \begin{array}{c} x \\ t \end{array} \right) 
      =
      \Lambda^{-1} \; 
      \left( \begin{array}{c} 0 \\ t' \end{array} \right)
      = 
      \left( \begin{array}{c} t' \sinh\phi \\ t' \cosh\phi \end{array} \right)
      $
      }

    \onlySlide*{6}{ $\Rightarrow 
      x = t' \sinh\phi $}

    \onlySlide*{7}{ $\Rightarrow 
      x = t' \sinh\phi  = t' \cosh\phi \frac{\sinh\phi}{\cosh\phi}$}

    \onlySlide*{8}{ $\Rightarrow 
      x = t' \sinh\phi  = t' \cosh\phi \mytanh\phi$}

    \onlySlide*{9}{ $\Rightarrow 
      x = t' \sinh\phi  = t \mytanh\phi$}

    \fromSlide{10}{ $\Rightarrow 
      x = t' \sinh\phi  = t \mytanh\phi = \beta t$}

    \fromSlide{10}{
      where velocity $\beta := v/c \equiv v = \mytanh\phi$ }

%    \fromSlide{11}{ $\phi = \mathrm{artanh} \beta \equiv \frac{1}{2} \ln 
%      \frac{1+\beta}{1-\beta}$ if $|x| < 1$ }


  \end{slide}
}




\parskip=12pt 
\overlays{8}{%
  \begin{slide}{SR: calibration}
    \parskip=5pt  %\small 
    \vspace{-10mm}
    \fromSlide{1}{{\color{mygreen} Where does $(x',t') = (0,1)$ lie for observer A?}}
    \onlySlide*{1}{\hypertarget{calib}{}}

    \fromSlide{2}{{\color{red} Where does $(x',t') = (1,0)$ lie for observer A?}}


    \onlySlide*{3}{
      \begin{minipage}[b]{61mm}% width
        \includegraphics[height=0.80\textheight]{Rotation_Euclidean12} 
      \end{minipage}
      \begin{minipage}[b]{40mm}% width
        constant distance $d = 1$ from $(0,0)$ 

        \vspace{6ex}
      \end{minipage}
    }

    \onlySlide*{4}{
      \begin{minipage}[b]{55mm}% width
        \includegraphics[height=0.80\textheight]{Lorentz_rotation_ds_timelike} 
      \end{minipage}
      \begin{minipage}[b]{45mm}% width
        {\color{mygreen} constant interval \\
          $(\Delta s)^2 = -1$ from $(0,0)$}

        $\Lambda^{-1} \; 
        \left( \begin{array}{c} 0 \\ 1 \end{array} \right)  = 
        \left( \begin{array}{c}  \sinh\phi \\  \cosh\phi \end{array} \right)
        $

        \vspace{6ex}
      \end{minipage}

    }

    \onlySlide*{5}{
      \begin{minipage}[b]{55mm}% width
        \includegraphics[height=0.80\textheight]{Lorentz_rotation_ds_spacelike} 
      \end{minipage}
      \begin{minipage}[b]{50mm}% width
        {\color{red} constant interval \\ $(\Delta s)^2 = + 1$ from $(0,0)$}

        $\Lambda^{-1} \; 
        \left( \begin{array}{c} 1 \\ 0 \end{array} \right)  = 
        \left( \begin{array}{c}  \cosh\phi \\  \sinh\phi \end{array} \right)
        $

        \vspace{6ex}
      \end{minipage}
    }

    \fromSlide{6}{
      Can high $\phi$ push the $t'$ axis close to the $x$ axis?}

    \onlySlide*{7}{\includegraphics[height=0.70\textheight]{Lorentz_rotation_high_beta} }

    \fromSlide{8}{\includegraphics[height=0.70\textheight]{Beta_versus_rapidity} }
  \end{slide}
}



\parskip=12pt 
\overlays{8}{%
  \begin{slide}{SR: effect of $\Lambda$ on $x= t$ (photons)}
    \parskip=3pt  %\small 
    \vspace{-10mm}
    \onlySlide*{1}{What happens to a photon under Lorentz 
      transformation?\hypertarget{cconstant}{}}

    \fromSlide{2}{What happens to a photon under Lorentz transformation?}

    \fromSlide{3}{Observer A: photon spacetime path is the set of spacetime events
      $\{ (t,t) \,\vert\,  t_1 < t < t_2 \}$ for some $t_1, t_2$}
    
    \fromSlide{4}{
        $ 
        \left( \begin{array}{c} x' \\ t' \end{array} \right)
        = 
        \left( \begin{array}{cc}
            \cosh\phi & -\sinh\phi \\
            -\sinh\phi & \cosh\phi 
          \end{array}
          \right) \, 
        \left( \begin{array}{c} t \\ t \end{array} \right)
        $ \\
    }
      
    \fromSlide{5}{ 
      $ = 
        \left( \begin{array}{c} (\cosh\phi - \sinh\phi) t \\ 
          (-\sinh\phi + \cosh\phi) t \end{array} \right)
      $
    }

    \fromSlide{6}{
      $\Rightarrow$ path is $x' = (\cosh\phi - \sinh\phi) t = t'$
      i.e. the set of spacetime events $\{ (t',t') 
       \,\vert\, (\cosh\phi - \sinh\phi) t_1 < t' < (\cosh\phi - \sinh\phi) t_2 \}$
    }

    \fromSlide{7}{
      $\bullet$ photon speed same in both reference frames
    }

    \fromSlide{8}{
      $\bullet$ \WikipediaEn{Michelson-Morley experiment} (1887)
    }

  \end{slide}
}



\parskip=12pt 
\overlays{11}{%
  \begin{slide}{SR: adding velocities}
    \parskip=5pt  %\small 
    \vspace{-10mm} 
    \onlySlide*{1}{interstellar ark travels at $\beta_1 = 0.1$ from
      Sun, sends out rocket at $\beta_2 = 0.5$; rocket's speed
      $\beta_3$ in Sun frame =?\hypertarget{addvel}{}}

    \fromSlide{2}{interstellar ark travels at
      $\beta_1 = 0.1$ from Sun, sends out rocket at
      $\beta_2 = 0.5$; rocket's speed $\beta_3$ in Sun frame =?
      \vspace{-5mm}
    }

    \onlySlide*{2}{
      \begin{minipage}[b]{40mm}% width
        \includegraphics[height=0.60\textheight]{Add_velocity_ark_POV} 
      \end{minipage}
      \begin{minipage}[b]{60mm}% width
        {\tiny
        $
        \left( \begin{array}{c} x' \\ t' \end{array} \right) =
        \left( \begin{array}{cc}
            \cosh\phi_1 & -\sinh\phi_1 \\
            -\sinh\phi_1 & \cosh\phi_1 
          \end{array}
          \right) \, 
        \left( \begin{array}{c} x \\ t \end{array} \right) 
        $ \\
        where $\mytanh\phi_1 = \beta_1 = 0.1$
        }

        \vspace{2ex}
       \end{minipage}
    }
      
    \onlySlide*{3}{
      \begin{minipage}[b]{40mm}% width
        \includegraphics[height=0.60\textheight]{Add_velocity_ark_POV} 
      \end{minipage}
      \begin{minipage}[b]{60mm}% width
        {\tiny
        $
        \left( \begin{array}{c} x'{}' \\ t'{}' \end{array} \right) =
        \left( \begin{array}{cc}
            \cosh\phi_2 & -\sinh\phi_2 \\
            -\sinh\phi_2 & \cosh\phi_2 
          \end{array}
          \right) \, 
        \left( \begin{array}{c} x' \\ t' \end{array} \right) 
        $ \\
        where $\mytanh\phi_2 = \beta_2 = 0.5$
        }

        \vspace{2ex}
       \end{minipage}
    }

    \onlySlide*{4}{
      \begin{minipage}[b]{40mm}% width
        \includegraphics[height=0.60\textheight]{Add_velocity_Earth_POV} 
      \end{minipage}
      \begin{minipage}[b]{60mm}% width
        {\tiny
        $
        \left( \begin{array}{c} x'{}' \\ t'{}' \end{array} \right) =
        \left( \begin{array}{cc}
            \cosh\phi_2 & -\sinh\phi_2 \\
            -\sinh\phi_2 & \cosh\phi_2 
          \end{array}
          \right) \, 
        \left( \begin{array}{c} x' \\ t' \end{array} \right) 
        $ \\
        where $\mytanh\phi_2 = \beta_2  = 0.5$
        }

        \vspace{2ex}
       \end{minipage}
    }
      
    \fromSlide{5}{
      \begin{minipage}[b]{40mm}% width
        \includegraphics[height=0.60\textheight]{Add_velocity_Earth_POV} 
      \end{minipage}
      \begin{minipage}[b]{60mm}% width
        {%\tiny
        $
        \left( \begin{array}{c} x'{}' \\ t'{}' \end{array} \right)
        = \Lambda(\phi_2) \Lambda(\phi_1) \;
        \left( \begin{array}{c} x \\ t \end{array} \right) 
        $ 
        }

        \vspace{2ex}
       \end{minipage}
    }

    \fromSlide{6}{
      but $\Lambda(\phi_2) \Lambda(\phi_1)  = \Lambda(\phi_1 + \phi_2)$
    }
    \onlySlide*{7}{\hyperlink{Mink}{cf. rotation $\theta_1$ ``plus"
        rotation $\theta_2$ = rotation $\theta_1+\theta_2$}}

    \fromSlide{8}{
      so $\beta_3 = \mytanh (\phi_1 + \phi_2)$
      }
    \onlySlide*{9}{
      $= \frac{\mytanh\phi_1 + \mytanh \phi_2}{1 + \mytanh \phi_1 \mytanh \phi_2}$
    }
    \fromSlide{10}{
      $= \frac{\beta_1 + \beta_2}{1 + \beta_1 \beta_2}$
    }
    \fromSlide{11}{
      $= \frac{0.1 + 0.5}{1 + 0.1 \times 0.5} \approx 0.57$
    }
      
  \end{slide}
}


\parskip=12pt 
\overlays{3}{%
  \begin{slide}{SR: Lorentz factor}
    \parskip=12pt  %\small 
    \onlySlide*{1}{\hypertarget{gamma}{}$\Lambda$: alternative to hyperbolic trig
      functions}

    \fromSlide{2}{$\Lambda$: alternative to hyperbolic trig
      functions}

    \onlySlide*{2}{
      \begin{minipage}[b]{60mm}
        $\Lambda (\phi) := \left(
        \begin{array}{cc}
          \cosh\phi & -\sinh\phi \\
          -\sinh\phi & \cosh\phi 
        \end{array}
        \right) 
        $
      \end{minipage}
      \begin{minipage}[b]{40mm}
        $\cosh\phi := \frac{\mathrm{e}^\phi + \mathrm{e}^{-\phi}}{2}$
        
        $\sinh\phi := \frac{\mathrm{e}^\phi - \mathrm{e}^{-\phi}}{2}$
        
        \WikipediaEn{hyperbolic function}
      \end{minipage}
    }
    

    \fromSlide{3}{
      \begin{minipage}[b]{60mm}
        $\Lambda (\beta) := \left(
        \begin{array}{cc}
          \gamma & -\beta\gamma \\
          -\beta\gamma & \gamma
        \end{array}
        \right) 
        $
      \end{minipage}
      \begin{minipage}[b]{40mm}
        $\beta = \mytanh\phi$

        $\gamma := (1-\beta^2)^{-1/2}$ = Lorentz factor

        $\gamma = \cosh\phi$

        $\beta\gamma = \sinh\phi$
      \end{minipage}
    }
    
\end{slide}
}




\parskip=12pt 
\overlays{7}{%
  \begin{slide}{SR: worldline time dilation}
    \parskip=10pt %\small
    \vspace{-20mm}
    \begin{minipage}[b]{50mm}
    \onlySlide*{1}{\hypertarget{tdilation}{}\includegraphics[height=1.2\textheight]{Time_dilation_derivation}}


    \fromSlide{2}{\includegraphics[height=1.2\textheight]{Time_dilation_derivation}
      \vspace{-20mm}
    }
    
    \onlySlide*{2}{$\cosh\phi \equiv \gamma $ }

    \onlySlide*{3}{$\cosh\phi \equiv \gamma \equiv \frac{1}{\sqrt{1-\beta^2}}$ }

    \fromSlide{4}{$\cosh\phi \equiv \gamma \equiv \frac{1}{\sqrt{1-\beta^2}} > 1$ }

    \end{minipage}
    \begin{minipage}[b]{50mm}
    \fromSlide{5}{worldline ``time dilation" }
    \parskip=20pt %\small

    \fromSlide{6}{ {\color{blue} muons}: mean lifetime 2197~ns $\ll 15$~km} 

    \fromSlide{7}{ time dilation $\Rightarrow$ muons can hit the ground}
    \end{minipage}

  \end{slide}
}

\parskip=12pt 
\overlays{9}{%
  \begin{slide}{SR: worldsheet space contraction}
    \parskip=12pt %\small
    \vspace{-18mm}
    \onlySlide*{1}{\hypertarget{xcontract}{}\includegraphics[height=0.7\textheight]{Space_contraction_observer_frame}}

    \fromSlide{2}{\includegraphics[height=0.7\textheight]{Space_contraction_observer_frame}}

    \onlySlide*{2}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = $}

    \onlySlide*{3}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = \cosh\phi - \beta \sinh \phi $}

    \onlySlide*{4}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = \gamma - \beta \beta \gamma $}

    \onlySlide*{5}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = (1 - \beta^2) \gamma $}

    \onlySlide*{6}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = (1 - \beta^2)^{1 + (-1/2)}  $}

    \onlySlide*{7}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = (1 - \beta^2)^{1/2}  $}

    \fromSlide{8}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = \gamma^{-1} < 1 $}
    \fromSlide{9}{\quad worldsheet ``space contraction" }
  \end{slide}
}


\parskip=12pt 
\overlays{7}{%
  \begin{slide}{SR: worldsheet space contraction}
    \parskip=12pt %\small
    \vspace{-10mm}
    \fromSlide{1}{\includegraphics[height=0.7\textheight]{Space_contraction_rocket_frame}}

    \onlySlide*{2}{   $\Lambda^{-1} \; 
        \left( \begin{array}{c} 1 \\ -\beta \end{array} \right)  = 
        \left( \begin{array}{c}  \cosh\phi - \beta \sinh\phi \\  \sinh\phi - \beta \cosh\phi \end{array} \right)
        $}

    \onlySlide*{3}{   $\Lambda^{-1} \; 
        \left( \begin{array}{c} 1 \\ -\beta \end{array} \right)  = 
        \left( \begin{array}{c}  \cosh\phi (1 - \beta^2) \\  \cosh\phi(\mytanh\phi - \mytanh\phi) \end{array} \right)
        $}

    \onlySlide*{4}{   $\Lambda^{-1} \; 
        \left( \begin{array}{c} 1 \\ -\beta \end{array} \right)  = 
        \left( \begin{array}{c}  \gamma (1 - \beta^2) \\  0  \end{array} \right)
        $}

    \onlySlide*{5}{   $\Lambda^{-1} \; 
        \left( \begin{array}{c} 1 \\ -\beta \end{array} \right)  = 
        \left( \begin{array}{c}  \gamma^{-1} \\  0  \end{array} \right)
        $}

    \fromSlide{6}{$\sqrt{\Delta s^2(\mathrm{q},\mathrm{p})} = \gamma^{-1} < 1 $}
    \fromSlide{7}{\quad worldsheet ``space contraction" }
  \end{slide}
}


\parskip=12pt 
\overlays{12}{%
  \begin{slide}{SR: Doppler shift}
    \parskip=1pt  %\small 
    \vspace{-11mm}
    \onlySlide*{1}{\hypertarget{dopp}{}\includegraphics[height=0.8\textheight]{SR_redshift_derivation}}

    \fromSlide{2}{\includegraphics[height=0.8\textheight]{SR_redshift_derivation}}
    \vspace{-7mm}

    \fromSlide{2}{see \hyperlink{cconstant}{photon spacetime path} calculation}

    \onlySlide*{3}{$x' = (\cosh\phi + \sinh\phi) t $}

    \onlySlide*{4}{$x' = (\cosh\phi + \sinh\phi) x $}

    \onlySlide*{5}{$x'/x = \cosh\phi + \sinh\phi  $}

    \onlySlide*{6}{$x'/x = \gamma + \beta\gamma  $}

    \onlySlide*{7}{$x'/x = \gamma(1 + \beta) $}

    \onlySlide*{8}{$x'/x = \gamma(1 + \beta) = \frac{1+\beta}{\sqrt{1-\beta^2}}$}

    \onlySlide*{9}{$x'/x = \gamma(1 + \beta) = \frac{\sqrt{(1+\beta)^2}}{
          \sqrt{(1-\beta)(1+\beta)}} $} 

    \onlySlide*{10}{$x'/x = \gamma(1 + \beta) = \sqrt{\frac{1+\beta}{1-\beta}}$}
    
    \fromSlide{11}{$1+z := \lambda'/\lambda = \gamma(1 + \beta) \equiv
      \sqrt{\frac{1+\beta}{1-\beta}}$ \quad\quad  {\color{red} redshift} }
    
    \onlySlide*{12}{$\Rightarrow$ when $\beta \ll 1$, $z \approx \beta$}
    
  \end{slide}
}


\parskip=12pt 
\overlays{13}{%
  \begin{slide}{SR: relativistic aberration}
    \parskip=7pt  %\small 
    \vspace{-11mm}
    \onlySlide*{1}{\hypertarget{aberr}{}\includegraphics[height=0.8\textheight]{Relativistic_aberration1}}

    \onlySlide*{2}{\includegraphics[height=0.8\textheight]{Relativistic_aberration1}}

    \onlySlide*{2}{event B: $(x,y,t)=(\cos\theta,\sin\theta,1)$}

    \fromSlide{3}{event B: $(x,y,t)=(\cos\theta,\sin\theta,1)$}

    \onlySlide*{4}{ 
      $\Lambda^{-1} \; 
      \left( \begin{array}{c} \cos\theta \\ \sin\theta \\ 1 \end{array} \right) 
      $}

    \onlySlide*{5}{ 
      $\Lambda^{-1} \; 
      \left( \begin{array}{c} \cos\theta \\ \sin\theta \\ 1 \end{array} \right) 
      = 
      \left( \begin{array}{ccc} 
        \gamma & 0 & \beta\gamma \\
        0 & 1 & 0 \\
        \beta\gamma & 0 & \gamma \\
      \end{array} 
      \right) 
      \; 
      \left( \begin{array}{c} \cos\theta \\ \sin\theta \\ 1 \end{array} \right) 
      $}

    \onlySlide*{6}{ 
      $\Lambda^{-1} \; 
      \left( \begin{array}{c} \cos\theta \\ \sin\theta \\ 1 \end{array} \right) 
      = 
      \left( \begin{array}{c} 
        \gamma\cos\theta + \beta \gamma \\ 
        \sin\theta \\
        \beta \gamma \cos\theta + \gamma
      \end{array} \right) 
      $}

    \onlySlide*{7}{ 
      $\Lambda^{-1} \; 
      \left( \begin{array}{c} \cos\theta \\ \sin\theta \\ 1 \end{array} \right) 
      = 
      \left( \begin{array}{c} 
        \gamma(\cos\theta + \beta) \\ 
        \sin\theta \\
        \gamma(1 + \beta\cos\theta  )
      \end{array} \right) 
      $}


    \onlySlide*{8}{\includegraphics[height=0.8\textheight]{Relativistic_aberration2}}

    \onlySlide*{9}{\includegraphics[height=0.8\textheight]{Relativistic_aberration1}}

    \onlySlide*{10}{\includegraphics[height=1.0\textheight]{Relativistic_aberration3}}

    \fromSlide{11}{\includegraphics[height=0.7\textheight]{Relativistic_aberration1}}
    \vspace{-10mm}

    \onlySlide*{11}{$\mytan\theta' = 
      \frac{\sin\theta}{\gamma (\beta + \cos\theta)}$}

    \fromSlide{12}{$\mytan\theta' = 
      \frac{\sin\theta}{\gamma (\beta + \cos\theta)}$ {\color{blue} $< \mytan\theta$ if $0 < \beta < 1$}
      \quad \WikipediaEn{Relativistic aberration}}

    \fromSlide{13}{$\Rightarrow$ relativistic beaming, e.g. AGN jets}

  \end{slide}
}



\overlays{5}{%
\begin{slide}{SR: world line}
\parskip=3pt  %\small 
\vspace{-10mm}
    \onlySlide*{1}{\includegraphics[height=0.7\textheight]{500px-World_line2} \hypertarget{timecone}{}} 

    \fromSlide{2}{\includegraphics[height=0.7\textheight]{500px-World_line2}} 

    \fromSlide{2}{
      \vspace{-10mm}
      \WMCommons{World_line2.svg}

      lightlike interval = null interval: $(\Delta s)^2 = 0$}

    \onlySlide*{2}{spacetime =} 

    \fromSlide{3}{spacetime = on past 
      \WikipediaEn{light cone} + inside past light cone} 

    \fromSlide{4}{\quad\quad + on future light cone + inside future light cone} 

    \fromSlide{5}{\quad\quad + elsewhere}

\end{slide}
}


\overlays{8}{%
  \begin{slide}{SR: world line}
    \parskip=6pt  %\small 
    \vspace{-10mm}

% http://upload.wikimedia.org/wikipedia/commons/e/e4/Lorentz_transform_of_world_line.gif
    \fromSlide{1}{
      \href{run:mplayer -zoom -fs Lorentz_transform_of_world_line.gif}{Lorentz transform of world line}
    }
    
    \fromSlide{2}{
      \WMCommons{Lorentz_transform_of_world_line.gif}
    }
    
    \fromSlide{3}{ $\bullet$ coordinate time in spacetime model
      $\not=$ time in your brain (thinking) }
    
    \onlySlide*{4}{
      $\bullet$      $\frac{\mathrm{d}t}{\mathrm{d}t_{\mathrm{thinking}}}$ can be
      positive or negative
    }
    
    \fromSlide{5}{
      $\bullet$       $\frac{\mathrm{d}t}{\mathrm{d}\lambda}$ can be
      positive or negative, $\lambda$ arbitrary real parameter
    }
    

    \fromSlide{6}{
      $\bullet$       ``elsewhere" spacetime events can change from past
      to future even though   $\frac{\mathrm{d}t}{\mathrm{d}\lambda} > 0$ 
    }

    \fromSlide{7}{
      $\bullet$ \WikipediaEn{proper time} $\tau :=$ time along a worldline
      measured by clock following that worldline 
    }

    \fromSlide{8}{
      $\bullet$ often $\mathrm{d}\tau $ is useful for integrating
    }

  \end{slide}
}


\overlays{4}{%
  \begin{slide}{SR: Rietdijk--Putnam--Penrose p.}
    \parskip=3pt  %\small 
    \vspace{-10mm}
    \onlySlide*{1}{\includegraphics[height=0.8\textheight]{Inertialoverlay} \hypertarget{RPPparadox}{}} 

    \onlySlide*{2}{\includegraphics[height=0.8\textheight]{Inertialoverlay}} 


    \onlySlide*{2}{
      \Wikibooks{Inertialoverlay.GIF}

      $\bullet$ each observer can synchronise clocks + rods}

    \onlySlide*{3}{
      \includegraphics[height=0.5\textheight]{Rel2} 

%      \WikipediaEn{Rietdijk\unichar{\string"e28093}Putnam argument}
      \WikipediaEn{Rietdijk-Putnam argument}
      \Wikibooks{Rel2.gif}
    }

    \fromSlide{4}{
      \includegraphics[height=0.7\textheight]{Rel3} 

      \WikipediaEn{Rietdijk-Putnam argument}
      \Wikibooks{Rel3.gif}
    }

  \end{slide}
}


\overlays{12}{%
  \begin{slide}{SR: tachyons and causality}
    \parskip=3pt  %\small 
    \vspace{-15mm}
    \onlySlide*{1}{\includegraphics[height=0.7\textheight]{Tachyon_observer_frame_only}
      \hypertarget{tachy}{}

      observer ``at rest"
    } 

    \onlySlide*{2}{\includegraphics[height=0.7\textheight]{Tachyon_world_line} 

      add a tachyon with speed $\alpha > 1$
    }

    \onlySlide*{3}{\includegraphics[height=0.7\textheight]{Tachyon_world_line} 

      add a tachyon with speed $\alpha > 1$

      choose rocket at speed $\beta$ with $1/\alpha < \beta < 1$
    }

    \onlySlide*{4}{\includegraphics[height=0.7\textheight]{Tachyon_rocket_frame_also} 

      add a tachyon with speed $\alpha > 1$

      choose rocket at speed $\beta$ with $1/\alpha < \beta < 1$

      add axes $x', t'$ for the rocket
    }

    \onlySlide*{5}{\includegraphics[height=0.7\textheight]{Tachyon_rocket_frame_also} 

      add a tachyon with speed $\alpha > 1$

      choose rocket at speed $\beta$ with $1/\alpha < \beta < 1$

      add axes $x', t'$ for the rocket

      rocket frame: $(\alpha t,t)$ becomes $\Lambda \; (\alpha t,t)^{\mathrm{T}}$
    }

    \vspace{5mm}
    \onlySlide*{6}{
      $\left( \begin{array}{c} x' \\ t' \end{array} \right) =
      \Lambda \; 
      \left( \begin{array}{c} \alpha t \\ t \end{array} \right) 
      = $}

    \onlySlide*{7}{
      $\left( \begin{array}{c} x' \\ t' \end{array} \right) =
      \Lambda \; 
      \left( \begin{array}{c} \alpha t \\ t \end{array} \right) 
      = 
      \left( \begin{array}{c} 
        \gamma \alpha t - \beta\gamma t \\
        -\alpha \beta \gamma t + \gamma t
      \end{array} \right) 
      $}

    \fromSlide{8}{
      $\left( \begin{array}{c} x' \\ t' \end{array} \right) =
      \Lambda \; 
      \left( \begin{array}{c} \alpha t \\ t \end{array} \right) 
      = 
      \gamma t 
      \left( \begin{array}{c} 
        \alpha  - \beta \\
        -\alpha \beta  + 1
      \end{array} \right) 
      $}

    \fromSlide{9}{
      $ x' = \gamma t (\alpha - \beta) > 0$ since $\alpha > 1 > \beta$
      }

    \fromSlide{10}{
      $ t' = \gamma t (-\alpha \beta +1 ) <  0$ since we chose $\beta > 1/\alpha$
    }

    \fromSlide{11}{
      $\mathrm{d}t'/\mathrm{d}t < 0$

      same sequence of spacetime events = tachyon spacetime path:

      $t$ increases for observer ``at rest",

      $t'$ decreases for rocket observer (with $\beta > 1/\alpha$)
    }

    \fromSlide{12}{
      $\bullet$ observer at rest: tachyon emitted at origin

      $\bullet$ rocket: tachyon absorbed at origin
    }

  \end{slide}
}



\overlays{16}{%
  \begin{slide}{SR: tachyonic antitelephone}
    \parskip=3pt  %\small 
    \vspace{-15mm}
    \begin{minipage}[b]{45mm}
      \fromSlide{1}{\includegraphics[height=0.7\textheight]{Tachyonic_antitelephone}}

      \fromSlide{2}{B stationary: $(x,t)$ frame}

      \fromSlide{3}{A moving at speed $\beta$: $(x',t')$ frame}
    \end{minipage}
    \rule{5mm}{0ex}
    \begin{minipage}[b]{55mm}
      \onlySlide*{4}{A: tachyon at $\alpha >1$ to B}

      \onlySlide*{5}{B: tachyon at $\alpha >1$ to C}
      
      \onlySlide*{6}{\vspace{5mm}

        C:
        $\left( \begin{array}{c} L \\ t'_{\mathrm{C}} \end{array} \right) =
        \Lambda \; 
        \left( \begin{array}{c} \alpha t_{\mathrm{C}} \\ t_{\mathrm{C}} \end{array} \right) 
        $}

      \fromSlide{7}{
        $\left( \begin{array}{c} L \\ t'_{\mathrm{C}} \end{array} \right) =
        \gamma t_{\mathrm{C}} 
        \left( \begin{array}{c} 
        \alpha  - \beta \\
        -\alpha \beta  + 1
        \end{array} \right) 
        $}

      \onlySlide*{8}{$ t'_{\mathrm{C}} = \gamma t_{\mathrm{C}} (1-\alpha \beta)$}

      \onlySlide*{9}{$ t'_{\mathrm{C}} = \gamma \frac{L}{\gamma (\alpha-\beta)} (1-\alpha \beta)$}

      \fromSlide{10}{$ t'_{\mathrm{C}} = L \frac{ 1-\alpha \beta}{\alpha-\beta}$}

      \onlySlide*{11}{$ t'_{\mathrm{C}} - t'_{\mathrm{A}} = 
        L \left( \frac{ 1-\alpha \beta}{\alpha-\beta} + \frac{1}{\alpha} \right) $}
      
      \onlySlide*{12}{$ t'_{\mathrm{C}} - t'_{\mathrm{A}} = 
        L \frac{ \alpha -\alpha^2 \beta + \alpha - \beta }{\alpha(\alpha-\beta)}$}

      \fromSlide{13}{$ t'_{\mathrm{C}} - t'_{\mathrm{A}} = L \frac{ 2\alpha - (\alpha^2 +1) \beta}{\alpha(\alpha-\beta)}$}

      \fromSlide{14}{$ < 0 $ if $\beta > \frac{2\alpha}{\alpha^2+1}$ } 

      \fromSlide{15}{A receives tachyonic response at C before sending it} 

      \fromSlide{16}{\WikipediaEn{tachyonic antitelephone}} 
    \end{minipage}
  \end{slide}
}





\overlays{9}{%
  \begin{slide}{SR: pole-barn/ladder paradox}
    \parskip=3pt  %\small 
    \vspace{-10mm}
    \onlySlide*{1}{\includegraphics[height=0.7\textheight]{500px-Ladder_Paradox_Overview} \hypertarget{polebarn}{}} 

    \onlySlide*{2}{\includegraphics[height=0.7\textheight]{500px-Ladder_Paradox_Overview}
      \WMCommons{Ladder_Paradox_Overview.svg}
   
      $\bullet$ ladder of length $29.9\gamma$~ns, garage length 30~ns}


    \onlySlide*{3}{\includegraphics[height=0.7\textheight]{500px-Ladder_Paradox_GarageFrame}
      \WMCommons{Ladder_Paradox_GarageFrame.svg}
   
      $\bullet$ ladder of length $29.9\gamma$~ns, garage length 30~ns

      $\bullet$ instantaneously close front + back doors
    }

    \onlySlide*{4}{\includegraphics[height=0.7\textheight]{500px-Ladder_Paradox_GarageFrame}
      \WMCommons{Ladder_Paradox_GarageFrame.svg}
   
      $\bullet$ ladder of length $29.9\gamma$~ns, garage length 30~ns

      $\bullet$ instantaneously close front + back doors

      $\bullet$ $29.9\gamma$~ns $/\gamma$ $< 30$~ns $\Rightarrow$ OK
    }


    \onlySlide*{5}{\includegraphics[height=0.6\textheight]{500px-Ladder_Paradox_LadderFrame}
      \WMCommons{Ladder_Paradox_LadderFrame.svg}
   
      $\bullet$ ladder of length $29.9\gamma$~ns, garage length 30~ns

      $\bullet$ instantaneously close front + back doors

      $\bullet$ ladder frame: garage $30/\gamma$~ns long $\ll 29.9\gamma$~ns!!

      Is this possible or not? Make a spacetime diagram.

    }


    \onlySlide*{6}{\includegraphics[height=0.8\textheight]{Pole_barn_Minkowski}
    }


    \onlySlide*{7}{\includegraphics[height=0.9\textheight]{500px-Ladder_Paradox_GarageScenario}
      \WMCommons{Ladder_Paradox_GarageScenario.svg}
    }
    
    \onlySlide*{8}{\includegraphics[height=0.9\textheight]{500px-Ladder_Paradox_LadderScenario}
      \WMCommons{Ladder_Paradox_LadderScenario.svg}
    }
    

    \fromSlide{9}{
      \begin{minipage}[b]{30mm}
        \includegraphics[height=0.8\textheight]{500px-LadderParadox2_Minkowski}
        \WMCommons{LadderParadox2_Minkowski.svg}
      \end{minipage}
      \begin{minipage}[b]{70mm}
        \WikipediaEn{Ladder paradox}
      \end{minipage}
    }

  \end{slide}
}




\overlays{7}{%
  \begin{slide}{SR: twins paradox}
    \parskip=3pt  %\small 
    \vspace{-10mm}
    \onlySlide*{1}{\includegraphics[height=0.9\textheight]{Multiply_connected_twins_paradox_0} \hypertarget{twins}{}} 

    \onlySlide*{2}{\includegraphics[height=0.9\textheight]{Multiply_connected_twins_paradox_leftmoving_POV}} 

    \onlySlide*{3}{\includegraphics[height=0.9\textheight]{Multiply_connected_twins_paradox_rightmoving_POV}}

    \onlySlide*{4}{\includegraphics[height=0.8\textheight]{Multiply_connected_twins_paradox_leftmoving_cylinder}}

    \onlySlide*{5}{\includegraphics[height=0.9\textheight]{Multiply_connected_twins_paradox_rightmoving_cylinder}}

    \onlySlide*{6}{\includegraphics[height=0.9\textheight]{Multiply_connected_twins_paradox_overview}}

    \onlySlide*{7}{\includegraphics[height=0.7\textheight]{Multiply_connected_twins_paradox_time_cutpaste}
    
      Roukema \& Bajtlik 2008, MNRAS, 390, 655 \aph{0606559}

      $\bullet$ helps understand \WikipediaEn{Ehrenfest paradox}
    }


  \end{slide}
}



\parskip=12pt 
\overlays{8}{%
  \begin{slide}{SR: four-velocity, four-momentum}
    \parskip=2pt  %\small 
    \vspace{-10mm}
    \onlySlide*{1}{choose $x$ axis so that
      3-velocity $u_{\mathrm{Galilean}} = (\beta, 0, 0)\TTT$ for observer
      with $(t,x,y,z)\TTT$ coord system
      \hypertarget{fourvel}{}
    }

    \fromSlide{2}{choose $x$ axis so that
      3-velocity $u_{\mathrm{Galilean}} = (\beta, 0, 0)\TTT$ for observer
      with $(t,x,y,z)$ coord system
    }

    \begin{minipage}[b]{40mm}
      \vspace{-2cm}
      \fromSlide{2}{
        \includegraphics[height=0.60\textheight]{Four-velocity}
        
        $\bullet$ in $(t,x)$ spacetime 2-plane, extend from scalar
        speed $\beta$ to spacetime vector = tangent to worldline
      }
    \end{minipage}
    \rule{2ex}{0ex}
    \begin{minipage}[b]{60mm}
      \parskip=4pt
      \fromSlide{3}{
        $(u^t,u^x) := (\frac{\mathrm{d}}{\mathrm{d}\tau} t(\tau),
        \frac{\mathrm{d}}{\mathrm{d}\tau} x(\tau))$
        \WikipediaEn{four-velocity}
      }

      \onlySlide*{4}{similarly
        $(u^{t'},u^{x'}) = (\frac{\mathrm{d}}{\mathrm{d}\tau} t'(\tau),
        \frac{\mathrm{d}}{\mathrm{d}\tau} x'(\tau))$
      }

      \fromSlide{5}{similarly
        $(u^{t'},u^{x'}) = (\frac{\mathrm{d}}{\mathrm{d}\tau} t'(\tau),
        \frac{\mathrm{d}}{\mathrm{d}\tau} x'(\tau)) = (1,0)$
      }

      \onlySlide*{6}{
        want {\color{red} $\vec{u}$} Lorentz invariant $\Rightarrow$
        $(u^{t},u^{x})\TTT = \Lambda^{-1} (1,0)\TTT$
      }

      \fromSlide{7}{    
        want {\color{red} $\vec{u}$} Lorentz invariant $\Rightarrow$
        $(u^{t},u^{x})\TTT = \Lambda^{-1} (1,0)\TTT = \gamma (1,\beta)\TTT$ 
      }

      \fromSlide{8}{    
        4D: {\color{red} $\vec{u}$} $= \gamma (1,\beta^x, \beta^y, \beta^z)\TTT$ 

        notation in this pdf:  

        $\vec{u}= $  4-vector, 
        ${}^{(3)}\vec{u}= $  spatial component
      }
    \end{minipage}
    
  \end{slide}
}


\parskip=12pt 
\overlays{4}{%
  \begin{slide}{SR: four-velocity, four-momentum}
    \parskip=5pt  %\small 
    \fromSlide{1}{
      Is the ${}^{(3)}$-component (spatial component) of $\vec{u}$ the 
      same as the non-relativistic velocity?
    }

    \fromSlide{2}{
      $^{(3)}\vec{u} =
      \frac{\mathrm{d}}{\mathrm{d}\tau} (x,y,z)\TTT  
      $
    }

    \fromSlide{3}{
      $= \gamma \frac{\mathrm{d}}{\mathrm{d}t} (x,y,z)\TTT$ 
    }

    \fromSlide{4}{
      $\not= \frac{\mathrm{d}}{\mathrm{d}t} (x,y,z)\TTT$ except if $\beta=0 
      \Leftrightarrow \gamma=1$
    }
  \end{slide}
}



\parskip=12pt 
\overlays{6}{%
  \begin{slide}{SR: four-velocity, four-momentum}
    \parskip=5pt  %\small 

    \onlySlide*{1}{    
      momentum: $\vec{p} := m \vec{u} = m \gamma (1,\beta^x, \beta^y, \beta^z)\TTT$,
      where $m$ = constant \WikipediaEn{invariant mass}
      \hypertarget{fourmom}{}

      {\color{red} $^x$ \ldots = tensor-style component notation, not powers}
    }

    \fromSlide{2}{    
      momentum: $\vec{p} := m \vec{u} = m \gamma (1,\beta^x, \beta^y, \beta^z)\TTT$,
      where $m$ = constant \WikipediaEn{invariant mass}
    }


    \onlySlide*{3}{What does the time component of momentum = $p^0 =
      m\gamma$ mean physically?

      $\bullet$ first look at spatial component in a given ref. frame
    }


    \fromSlide{4}{
      $^{(3)}\vec{p} =
      m \frac{\mathrm{d}}{\mathrm{d}\tau} (x,y,z)\TTT  
      $
    }

    \fromSlide{5}{
      $= m \gamma \frac{\mathrm{d}}{\mathrm{d}t} (x,y,z)\TTT$ 
    }

    \fromSlide{6}{
      $\not= m \frac{\mathrm{d}}{\mathrm{d}t} (x,y,z)\TTT$ except if $\beta=0
      \Leftrightarrow \gamma=1$
    }
  \end{slide}
}


\overlays{8}{%
  \begin{slide}{SR: four-velocity, four-momentum}
    \parskip=5pt  %\small 

    \onlySlide*{1}{
      let us define 4-acceleration, 4-force
    }
    
    \onlySlide*{2}{
      $(u^t,u^x) := (\frac{\mathrm{d}}{\mathrm{d}\tau} t(\tau),
      \frac{\mathrm{d}}{\mathrm{d}\tau} x(\tau))$
    }

    \fromSlide{3}{
      $ \vec{a} := \frac{\mathrm{d}}{\mathrm{d}\tau} \vec{u}$
    }

    \onlySlide*{4}{
      $^{(3)}\vec{a} = \frac{\mathrm{d}^2}{\mathrm{d}\tau^2} {}^{(3)}\vec{x}$
    }

    \onlySlide*{5}{
      $^{(3)}\vec{a} = \frac{\mathrm{d}^2}{\mathrm{d}\tau^2} (x,y,z)\TTT$
    }

    \fromSlide{6}{
      $ ^{(4)}\vec{f} := m \; {}^{(4)}\vec{a}$ \quad defn \WikipediaEn{four-force}
    }

    \onlySlide*{7}{
      $ = m \frac{\mathrm{d}}{\mathrm{d}\tau} \vec{u}$
    }

    \fromSlide{8}{
      $ = \frac{\mathrm{d}}{\mathrm{d}\tau} \vec{p}$
    }

  \end{slide}
}

\overlays{8}{%
  \begin{slide}{SR: invariance of ${}^{(4)}u, {}^{(4)}a, {}^{(4)}f$}
    \parskip=5pt  %\small 
    \onlySlide*{1}{
      Euclidean norm: $\| \vec{x} \|^2 = \sum_\mu (x^\mu)^2$
      \hypertarget{fourinv}{}
    }

    \onlySlide*{2}{
      Minkowski pseudo-norm: $\| \vec{x} \|^2 = \sum_{\mu,\nu} \eta_{\mu\nu} x^\mu x^\nu$
    }

    \onlySlide*{3}{
      Minkowski pseudo-norm: $\| \vec{x} \|^2 = \eta_{\mu\nu} x^\mu x^\nu$
      
      \WikipediaEn{Einstein summation} sum is implicit 
    }

    \fromSlide{4}{
      Minkowski pseudo-norm: $\| \vec{x} \|^2 = -x^0 x^0 + \delta_{ij} x^i x^j$
      
      $\delta_{ij} = 1$ if $i=j$, otherwise $=0$; $i,j \in 1,2,3$

      invariance: $\| \vec{x} \|^2 $ = same in all reference frames

      sign convention: $(-,+,+,+)$ or $(+,-,-,-)$ are common
    }

    \onlySlide*{5}{
      non-rest frame: $\| \vec{u} \|^2 = -\gamma^2 + \gamma^2 \beta^2 $
    }

    \fromSlide{6}{
      non-rest frame: $\| \vec{u} \|^2 = -\gamma^2 + \gamma^2 \beta^2  = -1$
    }

    \fromSlide{7}{
      rest frame: $\| \vec{u} \|^2 = -1^2 + 0^2 = -1$  {\color{mygreen} invariant}
    }

    \fromSlide{8}{
      similarly: $\| \vec{a} \|^2$, $\| \vec{f}\|^2$  {\color{mygreen} invariant}
    }

  \end{slide}
}


\overlays{3}{%
  \begin{slide}{SR: energy: varies with ref frame}
    \parskip=5pt  %\small 
    \fromSlide{1}{

      Newtonian $K = (1/2) m \beta^2$ = 0 in rest frame
    }

    \fromSlide{2}{
      4-force $\vec{f}$ is invariant, but

      3-force usually {\em defined} to be frame-dependent:

      3-force $:= 
       \frac{\mathrm{d}}{\mathrm{d}t} {}^{(3)}\vec{p}$ \quad {\color{red} 
         $\not= \frac{\mathrm{d}}{\mathrm{d}\tau} {}^{(3)}\vec{p}$}
    }

    \fromSlide{3}{
      $\frac{\mathrm{d}}{\mathrm{d}t} 
      {}^{(3)}\vec{p} = \frac{^{(3)}\vec{f}}{\gamma}$ 
    }

    %TODO:    $\rightarrow$ $\vec{f} = \left( ^{(3)}\vec{f}\cdot \vec{u}, ^{(3)}\vec{f} \right)
    %http://en.wikipedia.org/wiki/Four-force
    %  \fromSlide{4}{
    %    $\rightarrow$
    %  }
  \end{slide}
}

\overlays{14}{%
  \begin{slide}{SR: energy: varies with ref frame}
    \parskip=2pt  %\small 
    \vspace{-4mm}
    \fromSlide{1}{

      in $(x,t)$ frame, 

      $K = $ work done 
    }

    \fromSlide{2}{
      \vspace{-2cm}
      $$ = \int_0^{\beta_2} \frac{^{(3)}\vec{f}}{\gamma} \cdot \mathrm{d}\vec{x}$$
    }

    \onlySlide*{3}{
      \vspace{-1cm}
      $$ = \int_0^{\beta_2} 
      \frac{\mathrm{d}}{\mathrm{d}t} \, {}^{(3)}\vec{p} \cdot
      \mathrm{d}\vec{x} $$
    }

    \onlySlide*{4}{
      \vspace{-1cm}
      $$ = \int_0^{\beta_2} 
      \frac{\mathrm{d}}{\mathrm{d}t} (m \beta \gamma)
      \mathrm{d} x $$ (assume ${}^{(3)}\vec{f}/\gamma$ $\|$ $\vec{x}\,$)
    }

    \onlySlide*{5}{
      \vspace{-1cm}
      $$ \;\;\;\; = \int_0^{\beta_2} 
      \frac{\mathrm{d}}{\mathrm{d}t} (m \beta \gamma) \mathrm{d} x  
      = m \int_0^{\beta_2} \mathrm{d}(\beta\gamma) \frac{\mathrm{d}x}{\mathrm{d}t}$$ 
    }

    \fromSlide{6}{
      \vspace{-1cm}
      $$ \; = \int_0^{\beta_2} 
      \frac{\mathrm{d}}{\mathrm{d}t} (m \beta \gamma) \mathrm{d} x  
      = m \int_0^{\beta_2} \mathrm{d}(\beta\gamma) \beta$$ 
    }

    \fromSlide{7}{
      \vspace{-1cm}
      $$ 
      = m \int_0^{\beta_2} \beta (\beta \mathrm{d}\gamma + \gamma \mathrm{d}\beta )$$ 
    }

    \fromSlide{8}{
      \vspace{-1cm}
      $$ 
      = m \int_{\gamma=1}^{\gamma=\gamma_2} (\beta^2 + \gamma^{-2}) \mathrm{d}\gamma \; 
      \Leftarrow \mathrm{d}\gamma = \beta \gamma^3 \mathrm{d}\beta
      $$ 
    }

    \onlySlide*{9}{
      \vspace{-1cm}
      $$ 
      = m \int_{\gamma=1}^{\gamma=\gamma_2} [\beta^2 + (1-\beta^2)] \mathrm{d}\gamma$$
    }

    \onlySlide*{10}{
      \vspace{-1cm}
      $$ 
      = m \int_{\gamma=1}^{\gamma=\gamma_2} \mathrm{d}\gamma$$
    }

    \fromSlide{11}{
      \vspace{-9mm}
      $$ 
      = m \int_{\gamma=1}^{\gamma=\gamma_2} \mathrm{d}\gamma = m\gamma_2 - m$$
    }

    \parskip=0pt  %\small 
    \onlySlide*{12}{
      $\Rightarrow$ $ K + m = m \gamma$ {\em drop ``$_2$"}
    }

    \fromSlide{13}{
      $\Rightarrow$ $ K + m = m \gamma = p^0$ 
    }

    \fromSlide{14}{
      so $p^0  =$ kinetic energy + rest mass 
    }
  \end{slide}
}


\overlays{4}{%
  \begin{slide}{SR: energy: varies with ref frame}
    \parskip=5pt  %\small 
    \fromSlide{1}{Does small $\beta$ limit agree with Newtonian $K$?}
    
    \onlySlide*{2}{
      momentum time component: 

      $p^0 = m \gamma = m (1 - \beta^2)^{-1/2}$

      $= m [ 1 -(1/2)(-\beta^2) + {\cal{O}}(\beta^4) ] $ if $\beta \ll 1$
    }

    \onlySlide*{3}{
      momentum time component: 

      $p^0 = m \gamma = m (1 - \beta^2)^{-1/2}$
      
      $\approx  m [ 1  + (1/2)\beta^2 ]$  if $\beta \ll 1$
    }

    \fromSlide{4}{
      momentum time component: 

      $p^0 = m \gamma = m (1 - \beta^2)^{-1/2}$
      
      $\approx  m  + (1/2) m \beta^2 $  if $\beta \ll 1$

      Yes.
    }

  \end{slide}
}



\parskip=12pt 
\overlays{6}{%
  \begin{slide}{SR: $\vec{p} \ldots$: invariant or not?}
    \parskip=5pt  %\small 
    \vspace{-1cm}
    \fromSlide{1}{
      momentum: $\vec{p} = m \gamma (1,\beta^x, \beta^y, \beta^z)\TTT$

      $p^0  = m  + K = m\gamma$  
    }

    \onlySlide*{2}{
      non-rest frame: $\| \vec{u} \|^2 = -\gamma^2 + \gamma^2 \beta^2  = -1$

      rest frame: $\| \vec{u} \|^2 = -1^2 + 0^2 = -1$  
      \quad {\color{mygreen} invariant}
    }

    \fromSlide{3}{
      non-rest frame: $\| \vec{p} \|^2 = -m^2 \gamma^2 + m^2 \gamma^2 \beta^2  = -m^2$

      rest frame: $\| \vec{p} \|^2 = -m^2 + 0^2 = -m^2$  
      \quad {\color{mygreen} invariant}
    }

    \fromSlide{4}{
      $m$ = \WikipediaEn{invariant mass} $\equiv$ rest mass: invariant 

      AND conserved (in interactions):
      $\| \vec{p} + \vec{q} \|^2 = \| \vec{r} \|^2$
    }


    \fromSlide{5}{
      where interaction is (4-momenta): 

      $\vec{p} + \vec{q} \rightarrow \vec{r}$
    }

    \fromSlide{6}{ {\color{red} WARNING: assume that 4-momentum
        vectors at different space-time positions can be
        parallel-transported; not the case in curved spacetime} }

  \end{slide}
}

\overlays{4}{%
  \begin{slide}{SR: $\vec{p} \ldots$: invariant or not?}
    \parskip=5pt  %\small 
    \vspace{-1cm}
    \fromSlide{1}{
      vector space $\Rightarrow$ $p^i + q^i = r^i$ ($i=1,2,3$)
    }

    \fromSlide{2}{
      = conservation of (relativistic) ${}^{(3)}$-momentum (Newtonian: conserved)
      
      but NOT invariant (Newtonian: not invariant)
    }

    \fromSlide{3}{
      vector space $\Rightarrow$ $p^0 + q^0 = r^0$
    }

    \fromSlide{4}{
      = conservation of (relativistic) ``total energy" = $m + K$ 
      (Newtonian: $m$ conserved, $K$ not conserved,
      $K+$ potential energy conserved)
      
      but NOT invariant (Newtonian: $m$ invariant, $K$ not invariant)
    }


  \end{slide}
}

\parskip=12pt 
\overlays{10}{%
  \begin{slide}{SR: $\vec{p} \ldots$: invariant or not?}
    \parskip=5pt  %\small 
    \begin{minipage}[b]{50mm}
      \onlySlide*{1}{
        \vspace{-5mm}
        \includegraphics[height=0.65\textheight]{Four-momentum}    
        \hypertarget{emcsq}{}
      }

      \fromSlide{2}{
        \vspace{-5mm}
        \includegraphics[height=0.65\textheight]{Four-momentum}    
      }

      \fromSlide{2}{
        $\vec{p} = m (\gamma, +\beta\gamma,0,0)\TTT$
      }

      \fromSlide{3}{
        $\vec{q} = m (\gamma, -\beta\gamma,0,0)\TTT$
      }

      \onlySlide*{4}{
        $\vec{r} = \vec{p} + \vec{q} $
      }

      \onlySlide*{5}{
        $\vec{r} = \vec{p} + \vec{q} $

        $= m [2 \gamma, (-\beta+\beta)\gamma,0,0]\TTT$
      }

      \fromSlide{6}{
        $\vec{r} = \vec{p} + \vec{q} 
        = 2 m \gamma (1,0,0,0)\TTT$
      }
    \end{minipage}
    \rule{1ex}{0ex}
    \begin{minipage}[b]{45mm}
      \fromSlide{7}{
        system rest mass before and after: $2m\gamma$
      }

      \fromSlide{8}{
        rest masses in many different frames: 

        $m + m \not= 2m\gamma $ if $\gamma \not=1$
      }

      \fromSlide{9}{
        $\sqrt{-\| \vec{p} \|^2} + \sqrt{-\|\vec{q} \|^2} 
        \not= \sqrt{-\| \vec{r} \|^2}$
      }

      \fromSlide{10}{
        system mass is invariant, but can be divided into
        $p^0$ and $p^i, i\in\{1,2,3\}$ components in many 
        different ways}

    \end{minipage}
  \end{slide}
}


\parskip=12pt 
\overlays{10}{%
  \begin{slide}{SR: $\vec{p} \ldots$: invariant or not?}
    \parskip=5pt  %\small 
    \vspace{-1cm}
    \fromSlide{1}{
      interaction: momenta: $\vec{p} + \vec{q} \rightarrow \vec{r}$
      
      moduli: $m$, $m$, $2m\gamma$
    }

    \fromSlide{2}{
      total energy $E := p^0$
    }
    
    \fromSlide{3}{
      rest frame:  $E := p^0$ = $t$ component of $m (1,0,0,0)\TTT$
      
      $E=m$ (the famous equation)
    }

    \fromSlide{4}{
      this means: in the rest frame, $K+m = m$ (trivial)
    }

    \fromSlide{5}{
      more interesting: non-rest frame:  
    }

    \fromSlide{6}{
      $-m^2 = \|\vec{p}\|^2$
    }


    \fromSlide{7}{
      $-m^2 = \|(E,p^x,p^y,p^z)\TTT \|^2  $
    }


    \onlySlide*{8}{
      $-m^2 = -E^2 +p^2  $
    }

    \onlySlide*{9}{
      $m^2 = E^2 -p^2  $
    }

    \fromSlide{10}{
      $m^2 = E^2 -\|^{(3)}\vec{p}\|^2  $
    }
  \end{slide}
}
    

\parskip=12pt 
\overlays{6}{%
  \begin{slide}{SR: null 4-momentum}
    \parskip=4pt  %\small 
    \onlySlide*{1}{
      photon: $m=0$
      \hypertarget{photon}{}
    }

    \fromSlide{2}{
      photon: $m=0$

      extend defn of 4-momentum to $\vec{p}$
      for a photon
    }
      
    \fromSlide{3}{
      $\not\Rightarrow$ $\vec{u} = \frac{\vec{p}}{m}$ 
      since $m=0$; no 4-velocity
    }
    
    \fromSlide{4}{
      so $0 = E^2 - p^2$
    }

    \fromSlide{5}{
      i.e. $\vec{p} = (E,E,0,0)\TTT = (p,p,0,0)\TTT $ (if $x$ direction)
    }

    \fromSlide{6}{
      so $E=p$ for any massless particle
    }

  \end{slide}
}


\parskip=12pt 
\overlays{3}{%
  \begin{slide}{SR: model summary}
    \parskip=12pt  %\small 

    \onlySlide*{1}{\hypertarget{srconclu}{}\hyperlink{Mink}{Minkowski
        spacetime}: draw a correct diagram}

    \fromSlide{2}{\hyperlink{Mink}{Minkowski spacetime}: draw a
      correct diagram}

    \fromSlide{2}{\hyperlink{Lorentz}{Lorentz transformation (boost)
        $\Lambda(\phi)$ or \hyperlink{beta}{$\Lambda(\beta)$}}}

    \fromSlide{3}{refuse the assumption of absolute simultaneity (time)
    }
\end{slide}
}

} %% \SRslides 

%------------------------------------------------------------------------------

%\overlays{3}{
%\begin{slide}{
%\small
%\end{slide}}

\end{document}