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# Relativity 1

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%\hfill {ver. 1.01}
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\title{
Relativity
}
\author{Based on AIP The Physics Quick Reference Guide,
fixed by Kiyoshi Shiraishi
}
\date{ver. 1.0}
\begin{document}
\maketitle
\begin{abstract}
Yamaguchi University
\end{abstract}

\section{Special relativity} %%%%%%%%%%%%%%%%%%%%

\begin{itemize}

\item Lorentz transformations:

Transformation from an inertial reference frame $\Sigma$ to
an inertial frame $\Sigma'$ moving with respect to $\Sigma$ with
velocity $\beta c$ in the $+x$ direction:

\be
\gamma=(1-\beta^2)^{-1/2},~~\beta=\tanh\theta,~~\gamma=\cosh\theta,
\ee

\bea
x'&=&\gamma(x-\beta ct)=x\cosh\theta-ct\sinh\theta,~~y'=y,~~z'=z, \\
ct'&=&\gamma(ct-\beta x)=-x\sinh\theta+ct\cosh\theta.
\eea

The inverse transformation $(\Sigma'\rightarrow\Sigma)$ reverses the sign of $\beta$.

A body moving with velocity $\vv$ in $\Sigma$ moves with velocity $\vv'$ in $\Sigma'$:

\be
\tanh\phi_x=v_x/c,~~~\tanh{\phi'}_{x'}={v'}_{x'}/c,
\ee

\bea
{v'}_{x'}&=&\frac{v_x-c\beta}{1-\beta v_x/c}=c\tanh(\phi_x-\theta),~~~
{\phi'}_{x'}=\phi_x-\theta, \\
{v'}_{y'}&=&\frac{v_y}{\gamma(1-\beta v_x/c)}=v_y\frac{\cosh\phi_x}{\cosh{\phi'}_{x'}}, \\
{v'}_{z'}&=&\frac{v_z}{\gamma(1-\beta v_x/c)}=v_z\frac{\cosh\phi_x}{\cosh{\phi'}_{x'}}.
\eea

The transformation of forces on a moving particle:

\bea
{F'}_{x'}&=&F_x-\frac{\beta}{c-\beta v_x}\left(v_yF_y+v_zF_z\right), \\
{F'}_{y'}&=&\frac{c}{\gamma(c-\beta v_x)}F_y, \\
{F'}_{z'}&=&\frac{c}{\gamma(c-\beta v_x)}F_z.
\eea

The transformation of the electromagnetic field:

\bea
{E'}_{x'}&=&E_x,~~~{E'}_{y'}=\gamma(E_y-\beta B_z),~~~{E'}_{z'}=\gamma(E_z+\beta B_y), \\
{B'}_{x'}&=&B_x,~~~{B'}_{y'}=\gamma(B_y+\beta E_z),~~~{B'}_{z'}=\gamma(B_z-\beta E_y).
\eea

\item Energy:

\be
E=\gamma mc^2=mc^2+(kinetic~energy).
\ee

\item Energy-Momentum Transformation:

\bea
{p'}_{x}&=&\gamma(p_x-\beta E/c)=p_x\cosh\theta-E\sinh\theta /c, \\
{p'}_{y}&=&p_y,~~~{p'}_{z}=p_z, \\
{E'}&=&\gamma(E-\beta p_x c)=-p_x c\sinh\theta+E\cosh\theta, \\
\eea

\be
{E'}^2-(\vp'\cdot\vp') c^2=E^2-(\vp\cdot\vp) c^2=m^2 c^4,~~~~~~p=\beta E/c.
\ee

\end{itemize}

\section{General relativity}

\begin{itemize}

\item Einstein Postulates:

1.  Inertial mass is proportional to gravitational mass. (The
proportionality constant is set equal to 1.)

2.  The mass-energy density determines the metric tensor of space.
Empty space is flat; the Lorentzian metric can be expressed as

\be
ds^2=-c^2dt^2+dx^2+dy^2+dz^2.
\ee

3.  A particle moving in a gravitational field defined by the
distribution of mass-energy moves on a four-space geodesic satisfying the
differential equations

\be
\frac{d^2x^\alpha}{ds^2}+
\Gamma^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}=0.
\ee

4.  Light propagates along null geodesics, $ds=0$.

\item Mass-Energy Density Tensor:

\be
T_{\mu\nu}=(\rho c^2+p) g_{\mu\sigma}g_{\nu\tau}
\frac{dx^{\sigma}}{ds}\frac{dx^{\tau}}{ds}+p g_{\mu\nu},
\ee

where $\rho$ is the matter density and $p$ is the pressure (energy density per unit
volume):

\be
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}.
\ee

$\Lambda$ is the comsological constant introduced by Einstein to provide a static
solution for a non-empty universe:

\be
R_{\mu\nu}=\frac{8\pi G}{c^4}\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)+\Lambda g_{\mu\nu}.
\ee

\item Schwarzschild Line Element:

Spherically symmetric external solution with
a mass $m=Mc^2/G$ at the origin $(M=mG/c^2)$.
In spherically symmetric coordinates:

\be
ds^2=-c^2(1-2M/r)dt^2+(1-2M/r)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2).
\ee

In isotropic coordinates:

\be
ds^2=-c^2\left(\frac{1-M/(2R)}{1+M/(2R)}\right)^{2}dt^2+
(1+M/(2R))^{4}[dR^2+R^2(d\theta^2+\sin^2\theta d\phi^2)].
\ee

\item Clock Transformation:
TAI ({\it Temps Atomique International}) is a coordinate
time scale whose unit is the SI second as realized in a geocentric frame
by a cesium atomic clock at rest at sea-level on the rotating geoid.

a.  Transfer by portable clock: the coordinate time accumulated in moving
the clock from point $A$ to point $B$ is

\be
\Delta t=\frac{2\omega}{c^2}A_E+\int^B_A
\left[1-\frac{U(\vr)}{c^2}+\frac{v^2}{c^2}\right] d\tau,
\ee

where $\vr$ is the vector of the clock position in the earth frame with origin
at the center of the geoid; $v$ is the speed of the clock in that frame; $\omega$ is
the angular speed of rotation of the earth $(2\omega/c^2=1.623\times10^{-21} {\rm s/m^2})$;
$A_E$ is the equatorial projection of the area swept out by $\vr$ in the earth
frame, counted positive when the projected motion of $\vr$ is eastward; $d\tau$ is
the proper time of the moving clock; and $U(\vr)$ is the gravitational potential
relative to the geoid.

b.  Transfer by electromagnetic signal: the coordinate time interval be-
tween emission (at point $A$) to reception (at point $B$) is

\be
\Delta t=\frac{2\omega}{c^2}A_E+\frac{1}{c}\int^B_A ds,
\ee

where $ds$ is the increment of proper length along the path, and $A_E$ is the
equatorial projection of the triangle with vertices at $A$, $B$, and the center
of the geoid.

\end{itemize}

\section{Cosmology}

\begin{itemize}

\item Hubble Constant: $H\equiv 100h {\rm km s^{-1} Mpc^{-1}}$, $0.350$.

\item Critical Density: $\rho_c=3H^2/(8\pi G)$, the mass density required to close the
uniform isotropic universe, $\rho_c\approx 1.88\times10^{-26}h^2 {\rm kg/m^3}$:

\be
\Omega=\rho/\rho_c,~~~\Omega_0=\rho_0/\rho_c.
\ee

From large-scale velocity measurements $(>100 {\rm kpc}=3\times10^5 {\rm ly})$,
$0.1<\Omega<0.4$; approximately 95 percent of this density is apparently unobserved
(dark'') matter.

\end{itemize}

\section{General curvilinear spaces}

The Einstein convention is used: the explicit summation sign is suppressed
and an index appearing as both a subscript and a superscript is a dummy
value to be summed over its range.

\subsection{Metric tensor}

The metric of an $n-$space is given by

\be
ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu},
\ee

where $dx^{\mu}$ are generalized coordinates (a set of numbers locating a point
in the space) and $g_{\mu\nu}$ is the metric tensor:

\be
g^{\mu\sigma}g_{\sigma\nu}=g^{\mu}_{\nu}=\delta^{\mu}_{\nu}.
\ee
(The tensor $g^{\mu\nu}$ is the inverse of the tensor $g_{\mu\nu}$.)

\begin{itemize}

\item Raising and Lowering Indices: The dual vector is defined by
$A_{\mu}=g_{\mu\nu}A^{\nu}$; in terms of the components of the displacement
vector and its dual, the differential distance in the space is expressed as

\be
ds^2=dx_{\mu}dx^{\mu}.
\ee

For any tensor $T^{.\mu.\nu...}_{\alpha.\beta...}$,

\be
g_{\sigma\mu}T^{.\mu.\nu...}_{\alpha.\beta...}=T^{...\nu...}_{\alpha\sigma\beta...},~~~
g^{\gamma\alpha}T^{.\mu.\nu...}_{\alpha.\beta...}=T^{\gamma\mu.\nu...}_{..\beta...}.
\ee

\end{itemize}

\subsection{Characterization of intrinsic curvature}

\begin{itemize}

\item Christoffel Symbols: $[\mu\nu;\sigma]=[\nu\mu;\sigma]$,

\be
[\mu\nu;\sigma]=\frac{1}{2}\left(\frac{\partial g_{\sigma\nu}}{\partial x^{\mu}}+
\frac{\partial g_{\mu\sigma}}{\partial x^{\nu}}-
\frac{\partial g_{\mu\nu}}{\partial x^{\sigma}}\right),
\ee
\be
\Gamma^{\sigma}_{\mu\nu}=[\mu\nu;\lambda] g^{\lambda\sigma},
\ee
\be
\Gamma^{a}_{a\mu}=\frac{\partial\sqrt{|g|}}{\partial x^{\mu}},~~~
g=\det(g_{\mu\nu}).
\ee

\item Covariant Differentiation:

\be
\frac{dA^{a...}_{b...}}{dt}\equiv A^{a...}_{b...;c}\frac{dx^{c}}{dt},
\ee
\be
A^{a}_{;c}=\frac{\partial A^a}{\partial x^c}+\Gamma^{a}_{cd}A^{d},
\ee
\be
A_{b;c}=\frac{\partial A_b}{\partial x^c}-\Gamma^{d}_{bc}A_{d},
\ee
\bea
T^{ab...}_{cd...;s}&=&\frac{\partial T^{ab...}_{cd...}}{\partial x^s}+
\Gamma^{a}_{rs}T^{rb...}_{cd...}+\Gamma^{b}_{rs}T^{ar...}_{cd...}+\cdots \nn
&-&\Gamma^{r}_{cs}T^{ab...}_{rd...}-\Gamma^{r}_{ds}T^{ab...}_{cr...}-\cdots,
\eea
\be
g_{ab...;s}=\frac{\partial g_{ab}}{\partial x^s}-
\Gamma^{r}_{as}g_{rb}-\Gamma^{r}_{sb}g_{ar}\equiv 0.
\ee

This last condition is the defining relation for $\Gamma^{s}_{ab}$.

\item Riemann-Christoffel Tensor: Covariant differentiation is non-commutative
if the space is not flat.

\be
A_{c;ba}-A_{c;ab}=R^{d}{}_{cba}A{d},
\ee
\be
R^{d}{}_{cba}=\frac{\partial \Gamma^{d}_{ac}}{\partial x^b}-
\frac{\partial \Gamma^{d}_{bc}}{\partial x^a}+
\Gamma^{e}_{ac}\Gamma^{d}_{eb}-\Gamma^{e}_{bc}\Gamma^{d}_{ea},
\ee
\be
R^{d}{}_{abc}+R^{d}{}_{bca}+R^{d}{}_{cab}=0,
\ee
\be
R_{dcba}=g_{de}R^{e}{}_{cba}=-R_{cdba}=R_{abcd}.
\ee

As a result of these symmetries, only $n^2(n^2-1)/12$ of the $n^4$ components
in $n-$space are independent.

\item Ricci Tensor:

\be
R_{ca}=R^{d}{}_{cda}=\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial x^d}
\left(\sqrt{|g|}\Gamma^{d}_{ac}\right)-
\frac{\partial^2 \ln\sqrt{|g|}}{\partial x^a \partial x^c}-
\Gamma^{d}_{ae}\Gamma^{e}_{dc}=R_{ac}.
\ee

The choice of contraction is essentially unique since $R^{d}{}_{dbc}=0$ and
$R^{d}{}_{cbd}=-R^{d}{}_{cdb}$.

The contraction of the Ricci tensor: $R=R^{a}_{a}=g^{ac}R_{ac}$ is the mean
curvature (the {\it Gaussian} curvature of the surface for $n=2$).
\end{itemize}

\end{document}


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