## Thursday, April 02, 2009

### General Relativity Introduction Part 1

It's in Latex code, so just download Latex then copy and paste, and voila, you get to see my work. and above is the picture, just convert it to eps and save as LC.eps in the same folder as the tex file.

\documentclass[a4paper,12pt]{article}
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\title{Introduction to General Relativity}
\author{Ng Xin Zhao}
\newcommand{\eq}{}
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\newcommand{\tyh}{\begin{align}}
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\begin {document}
\maketitle

\section{Background}
The Special Theory of Relativity describes physical laws based on inertial reference frames, and successfully merged the consistency of the speed of light found by Maxwell with Newton's law of motion; however Newton's law of gravitation still conflicts with the consistency of the speed of light. This problem is solved by the General Theory of Relativity. The theory is based on the equivalence principle, that is the observation that the gravitational mass of an object is the same as the inertia mass of the object. Thus there is no distinction between locally gravitational frames with accelerated frames. The Theory of General Relativity states that gravity is nothing but curvature of four dimensional spacetime and the distribution of mass-energy bends spacetime, thus producing the familiar gravity. The sections below highlight the components of the General Relativity Theory and the physical meaning of them.

\subsection{Tensors}
Since the General Theory of Relativity is formulated to describe the laws of physics in the same form in all frames, it must be coordinate independent. Not only that, the theory should be able to describe how the change of the observer's frame affects how the observer looks at a physical phenomena. In the mathematical sense, the transformation of coordinates must be defined. And the mathematics for this kind of requirements exists in the form of Tensors. Therefore, the General Theory of Relativity must be expressed in the Tensor form. A Tensor then consists of the following properties:
\paragraph{}
It is a array of numbers in many dimensions relative to a choice of basis chosen. Here basis means a coordinate basis. However, tensors are geometrical in nature and thus are independent of any coordinate system that describes it.
\paragraph{}
A tensor has ranks; a tensor of rank 0 is a scalar, or a single number. A tensor of rank 1 is a vector or an array of numbers. A tensor of rank 2 is a matrix, or a 2 dimensional array of numbers. The definition extends this to any number of dimensions.
\paragraph{}
A tensor is usually denoted with subscripts and superscripts. The sum of subscripts and superscripts is the rank of a tensor. For example, $T^{\mu}$ is a rank one tensor and $T$ is a rank 0 tensor. A tensor with rank $n+s$ can be denoted as $T^{\mu_1 \mu_2 ...\mu_n}_{\nu_1 \nu_2 ...\nu_s}$.
\paragraph{}
A tensor is defined by the coordinate transformation
\eq
T^{\mu\prime_1...\mu\prime_n}_{\nu\prime_1...\nu\prime_s}=X^{\mu\prime_1}_{\mu_1}....X^{\mu\prime_n}_{\mu_n}X^{\nu_1}_{\nu\prime_1}...X^{\nu_s}_{\nu\prime_s}T^{\mu_1...\mu_n}_{\nu_1...\nu_s},
\qe
where
\eq
X^{\nu\prime}_{\nu}=\frac{\partial x^{\nu\prime}}{\partial x^{\nu}}.
\qe
\paragraph{Remarks on notation}
The notation used in General Relativity follows the Einstein notation where $\lambda^{\mu}\lambda_{\mu}$denotes a summation on $\mu$ for all the values of $\mu$. Or more simply,
\eq \lambda^{\mu}\lambda_{\mu}=\lambda^1_1+\lambda^2_2+\lambda^3_3+\lambda^4_4,
\qe
if $\lambda$ is the value 1 to 4.
Partial derivatives can be written as
\eq
\frac{\partial x^\alpha}{\partial x^\beta}=\partial_{\beta} x^{\alpha}=x^{\alpha}_{,\beta}.
\qe
The basis can be written in short notations of $x^\mu$ where $\mu$ range from 1 to 4 in 4 dimension spacetime. Thus,
\eq x^0=t, \hspace{3mm} x^1=r, \hspace{3mm} x^2=\phi, \hspace{3mm} x^3=z, \qe in cylindrical coordinates.
\section{Einstein's Equation}
\begin{align}
G_{\mu\nu} +\Lambda g_{\mu\nu} =R_{\mu\nu} -\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^2}T_{\mu\nu},
\end{align}
The Einstein Equation above is written in Tensorial form and it just equates the curvature of spacetime with the mass-energy that causes it. The terms are:
\begin{itemize}
\item $G_{\mu \nu}$, which represents the curvature of spacetime,
\item $T_{\mu\nu}$, the Stress Energy Tensor which describes completely the distribution of mass, energy and stress or pressure,
\item $\Lambda$, an extra term introduced to represent negative gravitation or more positive gravitation. Historically Einstein introduced it as a repellent force to balance against gravity to maintain a static Universe. Currently it is believed that this value for our Universe is small but non-zero, the force that is responsible for the acceleration of expansion of the Universe. It is called Dark Energy too due to ignorance of its nature.
\item $R_{\mu\nu}$ is called the Ricci Tensor, it tells how much the spacetime differs from Euclidean space, the traditional notion of space before Special Relativity.
\item $R$ is the Ricci Scalar, it denotes a single real value to every point of spacetime. If the value is positive, the volume of a object is smaller at that point in spacetime compared to the volume at Euclidean space and vice-versa.
\item $g_{\mu\nu}$, is called the metric tensor, it captures the geometric and casual structure of spacetime.
\end{itemize}

\subsection{The Metric Tensor, $g_{\mu\nu}$}
\paragraph{Euclidean Space} is just three dimensional space, when there is a rod with a length $l$, the length can be found in terms of a coordinate by Pythagorean theorem, that is $l^2=x^2+y^2+z^2$, and the infinitesimal form is $dl^2=dx^2+dy^2+dz^2$. In another coordinate basis, the length $dl^2$ remains the same that is, $dl^2=dx^2+dy^2+dz^2=dx\prime^2+dy\prime^2+dz\prime^2$.
\paragraph{Minkowski Spacetime} is the flat spacetime of Special Relativity where time is treated as another dimension and therefore the line element $ds^2$ is modified to include the time axis.
\eq ds^2=-dt^2+dx^2+dy^2+dz^2=-dt\prime^2+dx\prime^2+dy\prime^2+dz\prime^2. \qe
$ds^2$ is an invariant in spacetime, it does not change with the change of basis.
The time axis is negative to show that it is a special dimension.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.5]{LC.eps}
\label{Light Cones}
\vspace{-8cm}
\caption{Light Cone}
\end{center}
\end{figure}
The figure \ref{Light Cones}, is drawn in Minkowski spacetime. The horizonal plate is two of the three space dimension and the vertical axis is the time dimension. One more space dimension is compressed to draw the time axis. The cones are light paths tracing from the origin of the axis to both the posive time or future and the negative time or the past.

\paragraph{The Metric Tensor} is a rank 2 symmetric tensor that describes the geometry of spacetime. It can act on two vectors to produce a scalar number. The notation can be written as
\eq
g_{\mu \nu}[x^\mu ,x ^\nu]=g_{\mu \nu}x^\mu x ^\nu=c,
\qe
where $c$ is a scalar. In fact, when the metric tensor interacts with the infinitesimal change in the elements of the basis,$dx^\mu$, the scalar produced is called the line element, $ds^2$.
\eq
ds^2=g_{\mu \nu}dx^\mu dx ^\nu.
\qe
The metric tensor for Minkowski spacetime is then
\begin{eqnarray}
g_{\mu\nu}=
\left(
\begin{array}{cccc}
-1 & 0 & 0& 0 \\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{array}
\right).
\end{eqnarray}
Thus it can be seen that the metric tensor represents the form of the line element which in turns tells the properties of the spacetime in question.
The equation $ds^2=g_{\mu \nu}dx^\mu dx ^\nu$ arises because not all spacetime is flat, so curved spacetime can be described by just changing the metric tensor.
\paragraph{Metric signature} used here is $(-,+,+,+)$ and is just a convention for writing a metric. It comes from the signs used in Minkowski metric. The opposite signature of $(+,-,-,-)$ can be used but the whole system must be consistent in using just one signature.

\subsection{Casual Structure}
\paragraph{Manifold} is a mathematical space that resembles Euclidean space at a small enough scale. In General Relativity, a manifold,$M$ is then a curved spacetime that can be approximated to Minkowski spacetime locally. Here, locally is a term for a small enough scale so that the approximation holds and globally is used for big scales where the approximation fails.
\paragraph{Tangent Vector}

\end {document}