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Introducing Einstein's Theories of Relativity

This introduction is not a comprehensive introduction to Einstein's theories of relativity; it is an attempt to give a brief overview. To read more about Einstein's theories, consult the references at the end of this document.

Introduction

Relativity in physics was recognized a long time ago. Around 1632, Galileo Galilei made the observation that physical laws are independent of the chosen frame of reference. For example, the physics on the main cabin of a ship remain the same as long as the ship moves with constant velocity; birds in a cage fly around unaffected by the ship moving, so the distribution of birds inside the cage is uniform. The birds are not clumped into a preferred corner or a side of the cage.

In 1865, James Clark Maxwell recognized that electromagnetic waves always propagate at the speed of light. Light was just a manifestation of electronic magnetic waves. For the theory to work, waves always had to propagate at the same speed, the speed of light. However, this fact was not controversial as waves were believed to be propagated within a medium or an aether. If an observer was at rest with the aether, then everyone would perceive the speed of light as being constant.

This world view changed in a famous experiment, the Michelson-Morley experiment in 1887. Albert Michelson and Edward Morley studied how emitted light was affected by the Earth's motion through space. Contrary to expectations, the speed of light was the same in all directions and showed no sign that Earth was moving.

Some physicists tried to reconcile Newton's laws of motion by proposing a partial drag of the aether around the Earth. Other physicists proposed that the length of the arms (used in the Michelson-Morley experiment) were contracted by the aether. All these effects conspired together so that the speed of light would remain constant in the experiment.

In 1905, Albert Einstein proposed a new revolutionary idea, based on modified principles of space and time that could explain the same effects without the superfluous aether. This idea was later known as Einstein's Special Theory of Relativity. It is called special because it does not include accelerated motion and gravity. In 1915, Einstein made another revolutionary leap with his General Theory of Relativity, which included gravity and accelerated frames of reference.

Einstein's Special Theory of Relativity

The special theory of relativity is based upon the constraint that nothing can travel faster than light, that is,

$$ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 \leq c^2$$

The second constraint is that the speed of light is the same for all reference frames. Given these preconditions, you can derive that the property I

$$c^2t^2 - x^2 - y^2 - z^2 = I$$

must remain invariant when changing the frame of reference. This means that applying any transformation such as rotation or "speed boosts" will leave this value unaffected. In particular if the velocity is going straight along the X-axis (assuming that the Y-axis and Z-axis are parallel) then we have that the following system of equations:

$$\left\{ \begin{array}{rcl}
           c^2t^2 - x^2 & = & I \\
           c^2(t')^2 - (x')^2 & = & I
         \end{array}\right. $$

where x' and t' are translated coordinates. Solving this for x',y',z' and t' produces the Lorentz transformation:

$$\left\{ \begin{array}{rcl}
         t' & = & \gamma \left( t - vx/c^2 \right)            \\
         x' & = & \gamma \left( x - vt \right)                \\
         y' & = & y                                           \\
         z' & = & z                                           \\
         \gamma & = & 1 / \sqrt{1-v^2/c^2}                    \\
         v      & = & \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2} \\
         \end{array} \right.
$$

An observer will notice that a clock ticks slower for a moving frame of reference.

Einstein's Famous Equation

Contrary to what many people believe, there is no "QED proof" for E=mc^2. Physics is about observation, Mathematics is a tool to model observation. In 1905, when Einstein first published the paper speculating about the property E=mc^2. It was a bold and inspired guess. Today there are numerous experiments that unambiguously confirm this relationship. Let's see how Einstein reasoned.

The clock of a moving reference is slower by a factor of

$$\left\{ \begin{array}{rcl}
              \frac{dt}{d\tau} & = & 1 / \sqrt{1-v^2/c^2} \\
              d\tau^2 & = & c^2 dt^2 - x^2 - y^2 - z^2
          \end{array} \right.
$$

In Newtonian mechanics, momentum is preserved in all physical processes. So we need to reinterpret Newtonian mechanics with this new notion of time.

$$\left\{ \begin{array}{l}
         m \frac{dv}{d\tau} = m \frac{dv}{dt} \frac{dt}{d\tau} = \frac{dv}{dt} \frac{m}{\sqrt{1-v^2/c^2}} \\
         dv \in \{ dt, dx, dy, dz \} \\
         \end{array} \right.
$$

For dv = dx, dy, dz the conserved entities remind us of a slightly modified version of Newtonian momentum given that v << c. The conserved entity for dt is quite interesting. After applying Taylor expansion, we get:

$$ \frac{m}{\sqrt{1-v^2/c^2}} = m + \frac{1}{2}\frac{mv^2}{c^2} + \frac{3}{8}\frac{mv^4}{c^4} + ... $$

Einstein recognized that the second term in this expression entails the kinetic energy apart from the scaling factor 1/c^2. Could it be that this invariant is a generalized version of the energy laws? If we multiply the preceding equation with c^2, to make sure that the second term actually becomes the kinetic energy in standard units, we get:

$$ \frac{mc^2}{\sqrt{1-v^2/c^2}} = m c^2 + \frac{1}{2}mv^2 + \frac{3}{8}\frac{mv^4}{c^2} + ... $$

If we apply this data to the rest frame (v = 0), we get the conserved entity "mc^2." What does this mean? Einstein reasoned that if this equation is a generalized form of conservation of energy, then mass itself is energy scaled by c^2. If this energy was just a mathematical artifact (like potentials in electrical fields; only the difference is observable and meaningful) or if this energy was actually real was not clear to Einstein in 1905. The first convincing experimental proof came when the atom was split which ultimately led to the invention of the atomic bomb. The loss of mass could be directly translated to the gain of kinetic energy using E = mc^2.

Einstein's Theory of General Relativity

The general relativity is also based on an axiomatic proposition. The axiomatic proposition is that accelerated motion can be transformed to rest by pure coordinate transformation. The idea comes from free fall in a gravitational field, the laws of physics are invariant (and unaware) of accelerated motion (until one hits the ground!). This proposition is called the principle of equivalence. An observer cannot distinguish accelerated motion from rest because an observer is allowed to observe only the immediate (infinitesimal) neighborhood (for both space and time). In general, a person can distinguish between accelerated or unaccelerated motion and gravitational fields.However, all such observations are non-local. If non-local effects are ignored, then gravity is just a pure geometrical effect.

The following equations describe the motion of an object:

$$\frac{dx^2_i}{dt}={\sum^3_{j=0} {\sum^3_{k=0} -\Gamma^{i}_{jk} \frac{dx^i}{dt} \frac{dx^j}{dt}}} $$

Or as a convention in tensor calculus, if indices are repeated they are implicitly summed:

$$\frac{dx^2_i}{dt}=-\Gamma^{i}_{jk} \frac{dx^i}{dt} \frac{dx^j}{dt} $$

Assume that dimension 0 is time and that all the other dimensions are orthogonal axes of space (usually expressed in polar coordinates.) The Gamma symbol on the right hand side of the equation of motion is a 4x4x4 matrix that describes the acceleration as a function from curvature of space-time and current velocity. The upper indices are called contra-variant indices, which is what most people would consider standard coordinates. The lower indices are coordinates in co-variant format. The co-variant coordinates are coordinates in an orthogonal coordinate system, that is, for each coordinate we move orthogonally along all the other axes except the designated one. The co-variant coordinates can be computed from the contra-variant coordinates (and vice versa) using a metric 'g':

$$ \begin{array}{rcl}&#xA;  x_j &amp; = &amp; x^i g_{ij} \\&#xA;  g_{ij} g^{kl} &amp; = &amp; \left( \begin{array}{cccc}&#xA;                   1 &amp; 0  &amp; 0  &amp; 0 \\&#xA;                   0 &amp; -1 &amp; 0  &amp; 0 \\&#xA;                   0 &amp; 0  &amp; -1 &amp; 0 \\&#xA;                   0 &amp; 0  &amp; 0  &amp; -1 \\&#xA;                  \end{array} \right)&#xA;  \end{array}&#xA;$$

The metric is a 4x4 matrix that describes the orientation of the coordinate system for a particular point in space (and time). If space-time is curved, the coordinate system varies from point to point. The time axis is also part of the coordinate system, which makes it 4x4. Without time, the more familiar 3x3 matrix determines stretch and rotation.

If there is no gravity present, then the metric is flat:

$$g_{ij} = \left( \begin{array}{cccc} 1 &amp; 0  &amp; 0  &amp; 0 \\&#xA;                                      0 &amp; -1 &amp; 0  &amp; 0 \\&#xA;                                      0 &amp; 0  &amp; -1 &amp; 0 \\&#xA;                                      0 &amp; 0  &amp; 0  &amp; -1 \\&#xA;                  \end{array} \right) $$

Then the space-time invariant for special relativity is easily obtained as:

$$dx^i g_{ij} = c^2dt^2-dx^2-dy^2-dz^2$$

because g_{ij} are all constant. Compute Gamma and you get:

$$\Gamma^i_{jk} = 0$$

The equation of motion reduces to

$$\frac{dx^2}{dt^2} = 0$$

which means that objects move in Euclidean straight lines.

When mass/energy is present, objects no longer move in Euclidean straight lines; instead they move in geodesics ("curved lines") that can be modeled by curved space-time. The differential equations that determine how space-time is curved by mass are:

$$\left\{ \begin{array}{rcl} R_{ab} - \frac{1}{2} R g_{ab} &amp;&#xA;                             = &amp; T_{ab} (8 \pi G)/c^4 \\&#xA;                             R_{ab} &amp;&#xA;                             = &amp; \frac{\partial}{\partial_x} \Gamma^x_{ab} -&#xA;                               \frac{\partial}{\partial_a} \Gamma^x_{xb} +&#xA;                               \Gamma^x_{xy} \Gamma^y_{ab} -&#xA;                               \Gamma^x_{ay} \Gamma^y_{xb} \\&#xA;                             \Gamma^{i}_{jk} &amp;&#xA;                             = &amp; \frac{1}{2}&#xA;                             g^{ja}(\frac{\partial}{\partial_{j}} g_{ka} +&#xA;                                    \frac{\partial}{\partial_{k}} g_{aj} -&#xA;                                    \frac{\partial}{\partial_{a}} g_{jk})&#xA;          \end{array}&#xA;  \right. $$

The 4x4 matrix T determines the mass/energy of the system using the proportional constant (8 pi G)/c^4, where G is Newton's constant of gravity. The proportional constant is obtained by making the equations of motion match observed behavior for weak gravitational fields. The R_{ab} is the Ricci curvature tensor which is obtained from the full Riemann curvature tensor R_{ijkl}. The Ricci tensor expresses volume reduction for a stream of neighboring geodesics on a curved surface. The equations are carefully chosen so that conservation of energy is preserved (Bianchi identities).

It took approximately 10 years for Einstein to obtain these equations. Einstein's quest was to express gravity as curvature, but getting the precise curvature to match with observation was difficult.

If electromagnetic fields are ignored, as well as internal pressures, then only the (0,0) component of the matrix is non-zero, i.e.

$$T = \left( \begin{array}{cccc} M &amp; 0 &amp; 0 &amp; 0 \\&#xA;                                 0 &amp; 0 &amp; 0 &amp; 0 \\&#xA;                                 0 &amp; 0 &amp; 0 &amp; 0 \\&#xA;                                 0 &amp; 0 &amp; 0 &amp; 0&#xA;             \end{array} \right) $$

Solving Einstein's non-linear differential equations of gravity is very difficult. Not even Einstein himself thought that this would ever be possible. However, Karl Schwarzschild, a German physicist, was able to solve the equations for a spherical symmetric static body. Static means that the mass is kept unchanged (or independent over time.) Today, the Schwarzschild solution is used for most simple scenarios. Einstein himself did not use the methods of Schwarzschild but instead used direct approximation techniques. Still he was able to conclude that planet Mercury had a precession of 43 arc seconds (43/3600 degrees) per century, which was in stunning agreement with observation.

The Schwarzschild Solution

The Schwarzschild solution is an exact solution of Einstein's differential equations for gravity. The famous Schwarzschild line element is:

$$ \left\{ \begin{array}{rcl}&#xA;    g_{ij} &amp; = &amp; \left(&#xA;            \begin{array}{cccc}&#xA;            1-2M/r &amp;                 0 &amp; 0    &amp; 0 \\&#xA;                 0 &amp; -1/(1-2M/r)       &amp; 0    &amp; 0 \\&#xA;                 0 &amp;                 0 &amp; -r^2 &amp; 0 \\&#xA;                 0 &amp;                 0 &amp; 0    &amp; -r^2 sin(\theta)^2 \\&#xA;            \end{array}&#xA;            \right) \\&#xA;    M &amp; = &amp; mG/c^2&#xA;    \end{array} \right.&#xA;$$

Polar coordinates are used, so the line element is:

$$&#xA;ds^2 = [dt, dr, d\theta, d\phi] g_{ij}&#xA;$$

For this equation to have a meaningful solution, r > 2M. r > 2M is called the exterior Schwarzschild solution. For a black hole, there is no interior solution and any object with coordinates r < 2M vanishes into the singularity of the black hole.

To compute Gamma using the Schwarzschild metric requires a computer with symbolic math. After embedding the Gamma curvature into the equations of motion and simplifying, the acceleration vector is:

$$ \left\{&#xA;\begin{array}{rcl}&#xA;  a_t &amp; = &amp; (-2 M v_t v_r)/(-2Mr+r^2) \\&#xA;  a_r &amp; = &amp; ((-4M^3+4M^2r-Mr^2)v_t^2+Mr^2v_r^2+(4M^2r^3-4Mr^4+r^5) \dots \\&#xA;      &amp;   &amp;  v_\theta^2+(4sin(\theta)^2M^2r^3-4sin(\theta)^2Mr^4+&#xA;             sin(\theta)^2r^5) v_\phi^2)/ (-2Mr^3+r^4) \\&#xA;  a_\theta &amp; = &amp; (-2v_rv_\theta+cos(\theta)sin(\theta)rv_\phi^2)/r \\&#xA;  a_\phi   &amp; = &amp; (-2sin(\theta)v_r-2cos(\theta)rv_\theta)v_\phi/(sin(\theta)r) \\&#xA;\end{array}&#xA;\right. $$

Different simplifications lead to slightly different systems of equations (that are all equivalent), but the preceding equations are the equations used by General Relativity with MATLAB Coder.

References

Textbooks

Einstein et al. "The Principle of Relativity", ISBN 0-486-60081-5

Wolfgang Rindler, "Introduction to Special Relativity", ISBN 0-19-853952-5

Ray D'Inverno, "Introducing Einstein's Relativity", ISBN 0-19-859686-3

Robert M. Wald, "General Relativity", ISBN 0-226-87033-2

Arthur I. Miller, "Albert Einstein's Special Theory of Relativity", ISBN 0-201-04679-2

Roger Penrose, "The Road to Reality", ISBN 0-679-45443-8

Roger Penrose, "The Emperor's New Mind", ISBN 0-19-286198-0

Online resources

Kevin Brown, "Reflections on Relativity", http://www.mathpages.com/rr/rrtoc.htm

Edward L. Wright, "Relativity Tutorial", http://www.astro.ucla.edu/~wright/relatvty.htm

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