Content Special Relativity

## Special Relativity

Special relativity is developed by Albert Einstein in his article, "Zur Elektrodynamik bewegter Körper", June 1905, Annalen der Physik. An english translation of this article can be found here: special relativity. The special theory of relativity only deals with linear uniform motion. Later in 1916 Einstein developed the theory of general relativity in his article, "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, vierte Folge Band 49, p. 769-822, 1916. An english translation can be found here: general relativity. The general theory takes also into account all kind of motion such as accelerated and rotational motion.

Before starting it is important to realize that a physical theory can be quite contrary to normal day experience. A beginning student tries to understand a new theory by refering to his existing body of knowledge, common sense and experience. However a physical theory need to stand against only two types of tests: 1) the assumptions and predictions of a theory are confirmed by experimental facts and 2) the theory is logical consistent. A logical consistent theory implies that the theory could not lead to a state where something is true and false at the same time.

In the progress of developing scientific knowlegde a new theory normally extends existing knowledge instead of making this knowlegde obsolete. This is also true for the special relativity theory. This new theory in 1905 extended the theory of classical mechanics by taking into account the movement of electromagnetic objects (light). It made clear that the existing theory was not complete, but a special case, of a broader theory. Newtonian mechanics however is still a valid theory in our everyday world with low speeds and low masses.

## Reference Frames

Mechanics is based on the measurement of motion. Motion is described with position and time co-ordinates. To measure position a rigid body of reference is chosen with attached an imaginary co-ordinate system in three dimensional space. This whole configuration is called a reference frame. Newtonian physics is based on Euclidean geometry. This geometry defines the distance between any two points $A$ and $B$ as the length of the line segment $AB$ lying on the straight line defined by these two points. The length can be measured directly with a physical rod and its direction relative to the co-ordinate axis can be calculated with trigonometry. For measuring time a clock is used. A complete description of the motion of a body, its trajectory, is a set of position-time co-ordinates, $\{(x_0,t_0),\ldots,(x_n,t_n)\}$.

The trajectory depends on the body of reference of the observer. The trajectory of a stone dropped from the window of a railway carriage which is travelling uniformly, is seen as a straight line for an observer from the railway carriage and as a parabolic curve for an observer on the embarkment.

What should be the properties of the reference frame we choose. It turns out that classical mechanics and special relativity requires us to use so called inertial reference frames. By definition these are reference frames where the three laws of Newton hold. The laws of Newton hold if an object on which no force acts is at rest or in uniform motion in a straight line, the change of momentum is caused by a force and the total momentum of the system is conserved. This is only possible when no external force acts on the reference frame. By the same laws of Newton an inertial reference frame is at rest of in uniform translational motion. An inertial reference frame is a non-accelerating reference frame. Let $F$ be an inertial frame, then any other reference frame at rest of in uniform motion relative to $F$ is also an inertial frame. The visible fixed stars are bodies which are to a high degree of approximation in uniform translational motion.

This property that the mechanical laws have the same form in each inertial reference frame is called the Galilean principle of relativity.

## Einsteins Postulates

The Special Relativit theory of Einstein is based on two postulates. First Einstein extended the principle of Galilean relativity to the whole domain of physical laws including those of electromagnetic waves.

### Relativity

The laws of physics have the same form in all inertial reference frames.

Next Einstein postulated the constancy of the speed of light.

### Constancy speed of light

Light propagates through empty space with a definite speed $c$ independent of the source or inertial observer.

The speed of light is a constant in vacuo measured as $c= 299.792.458 m/s$ or rounded to $c \approx 3\times 10^8 m/s$.

Accepting both the principle of relativity and the constancy of the speed of light introduced an inconsistency in the existing theory of Newtonian mechanics. In Newtonian mechanics the time co-ordinate and distance between two locations is the same for observers of different reference frames. As a result the transformation of space time co-ordinates between a reference frame $x$ and $x'$ has a specific form, called the Galilean transformation.

## Galilean Transformations

An event is associated with a tuple of space and time co-ordinates $(x,t)$. A physical law is in essence a statement about the relation between these tuples of co-ordinates which always holds. It means that it holds at every location in space, at every direction in space and at all times. The physcial law must therefore be independent from the place and time of an observer. A physical law requires then also an assumption on the geometric behavior of this space-time structure. The geometric structure must transform co-ordinates from one reference frame to another in such a way that the physical law is preserved between the two reference frames. The theory of geometric structures is a mathematical theory. We will discuss it in more mathematical depth in the mathematics section. Before Einstein the geometric structure of the world was assumed as we perceive in our day to day living and physicists tried to fit all laws in this structure. Einstein realized that one must choose from the mathematical theory the geometric structure which fits the laws. Only by this abstraction we could discover the wonderfull relativistic nature of the space-time structure of our universe. We start now first by looking at the Galilean space time structure which underlies Newtonian or classical mechanics and we will see how it conflicts with the law of the constancy of the speed of light.

Suppose both $S$ and $S'$ are reference frames with the same oriented orthogonal co-ordinate axis, but are moving with regard to each other with a uniform velocity $u$ in the positive $x$ direction. Then the following relations hold in Newtonian mechanics between the co-ordinates of an event: \begin{align*} x' &= x - ut \\ y'&=y \\ z'&=z \\ t'&=t \end{align*} From this transformation follows: $dx'/dt = dx /dt - u$

This result is also stated as the law of addition of velocities.

Suppose that $S'$ is a riding car with speed $u$ and light from the rear is going past the car with speed $c$. According to the Galilean transformation the apparent speed of the passing light, as measured from the car, should not be $c$, but $c-u$. However this contradicts experiments that the speed of light is constant. So it is tempting to reject the principle of relativity. However Einstein found another way out by replacing the Galilean transformation, which is based on the assumption that time and space measurements are independent of the motion of the reference frame, with the Lorentz transformation of co-ordinates where distance and time are no longer independent of the motion of the reference frame.

## Lorentz transformation

To resolve the conflict from the previous paragraph, one must accept that in the transformation of co-ordiantes between reference frames, not only rotation and translation occurs, but also the lengths and time must be modified by a factor $\gamma$. This impact is called length contraction and time dilation.

### Transformation of space co-ordinate

We only consider the Lorentz transfomation for the simple case in which the relative motion of two reference frames is along their common x-axes. The most general case rather complicated, with all four quantities mixed up together.

Take again the two reference frames $S$ and $S'$ of the previous paragraph but introduce in the co-ordinate transformation this correction factor $\gamma$. \begin{align} x' &= \gamma ( x - ut) \label{eq1} \\ x &=\gamma(x'+ut') \end{align} Note that we used the same correction factor, because the two observers from these reference frames are equivalent, one is moving with speed $+u$ and the other with speed $-u$ relative to each other. So nature must use the same correction factor, otherwise it would have a preference for one or the other direction, which we assumed is not the case.

We multiply the LHS and RHS of both equations with each other and get

\begin{align} x'x&= \gamma^2 (x-ut)(x'+ut') \notag \\ &= \gamma^2(xx'+uxt'-ux't-u^2tt') \end{align} From the law of constancy of speed we have, $x=ct$ and $x'=ct'$, we substitute this in the above equation and get: \begin{align} c^2tt'&=\gamma^2(c^2tt'+uctt'-uct't-u^2tt') \notag \\ &=\gamma^2(c^2tt'-u^2tt') \\ \end{align} From this follows: $$\gamma^2=\frac{c^2}{(c^2-u^2)}=\frac{1}{1-\dfrac{u^2}{c^2}}$$

Finally we find $\gamma$: $$\label{eq2} \gamma=\frac{1}{\sqrt{1-\dfrac{u^2}{c^2}}}$$

Substitute this in $\ref{eq1}$ and we get the Lorentz transformation for the $x'$ co-ordinate:

#### Lorentz transformation $x'$

$$x'=\frac{x-ut}{\sqrt{1-\dfrac{u^2}{c^2}}}$$

Note that the reverse of this transformation is simply to take into account that the direction of $u$ is reversed:

#### Lorentz transformation $x$

$$x=\frac{x'+ut'}{\sqrt{1-\dfrac{u^2}{c^2}}} \label{eq8}$$

The Galilean transformation follows form the Lorentz transformation when we take $c=\infty$. We also see the velocity $c$ plays the part of an unattainable limiting velocity.

What do these transformations reveal about nature ? An observer from $S$ must correct the magnitude of the meters measured from the perspective of $S'$. So if these meters are $x'$ then for an observer in $S$ these meters are equivalent to $x'\sqrt{1-\frac{u^2}{c^2}}$ in his definition of meters. This factor is the length contraction. If the $S'$ system has travelled a distance $ut$ away from $S$ the observer in $S$ would say that the same point, measured in his co-ordinates, is at a distance $x= x'\sqrt{1-\frac{u^2}{c^2}} + ut$ or \begin{equation*} x'=\frac{x-ut}{\sqrt{1-\dfrac{u^2}{c^2}}} \end{equation*}

### Transformation of time co-ordinate

Now we analyze the transformation of the time co-ordinate. Let us start again with the following equations. \begin{align*} x' &= \gamma ( x - ut) \\ x &=\gamma(x'+ut') \end{align*} We eliminate $x'$ from these equations by multiplying the first equation with $\gamma$ and adding this result to the second equation and solve for $t'$. This gives the following result $$t'= \frac{1-\gamma^2}{\gamma u}x +\gamma t \label{eq5}$$

Let us first simply $1-\gamma^2$ by substituting the relation found for $\gamma$ in \ref{eq2}: \begin{align} 1-\gamma^2&=1-\frac{1}{1-\dfrac{u^2}{c^2}} \notag \\ &=\frac{-u^2}{c^2(1-\dfrac{u^2}{c^2})} \label{eq3} \end{align} We divide \ref{eq3} by $\gamma u$ and get: \begin{align} \frac{1-\gamma^2}{\gamma u} &= \frac{-u^2\sqrt{1-\dfrac{u^2}{c^2}}}{c^2(1-\dfrac{u^2}{c^2})u} \notag \\ &= \frac{-u}{c^2\sqrt{1-\dfrac{u^2}{c^2}}} \label{eq4} \end{align}

Now we substitute \ref{eq4} and \ref{eq2} into \ref{eq5} and get the transformation equation for $t'$:

#### Lorentz transformation $t'$

$$t'=\frac{t-\dfrac{u}{c^2}x}{\sqrt{1-\dfrac{u^2}{c^2}}}$$
The transformation for $t$ follows by reversing the sign of $u$.

#### Lorentz transformation $t$

$$t=\frac{t'+\dfrac{u}{c^2}x'}{\sqrt{1-\dfrac{u^2}{c^2}}} \label{eq9}$$

Let us assume a seconds-clock permanently situated at the origin of the $S'$ frame so $x'=0$. $t'=0$ and $t'=1$ are two successive ticks of the clock. As judged from $S$ the clock is moving with a velocity $u$ and based on the Lorentz transformation we have for the two successive ticks: \begin{align*} t&=0 , t=\frac{1}{\sqrt{1-\dfrac{u^2}{c^2}}} \end{align*} So the time elapsed between two successive ticks of the clock in $S'$ seen from $S$ is not one second, but $\frac{1}{\sqrt{1-\dfrac{u^2}{c^2}}}s$ , i.e. a somewhat larger time. As a consequence of its motion the clock goes more slowly than when at rest. This is called time dilation.

### Transformation veloctiy

Suppose again that $S'$ moves relative to $S$ with a speed $v$ in the $x$ direction, and that inside $S'$ an object moves with a uniform speed $v'_x$ relative to $S'$. We seek the transformation from $v'_x$ to $v_x$. We have: $$x'=v'_xt' \label{eq6}$$ We substitute \ref{eq6} into \ref{eq8} and get: $$x=\frac{v'_xt'+ut'}{\sqrt{1-\dfrac{u^2}{c^2}}}$$ Next we must finds the time $t$ by substituting \ref{eq6} into \ref{eq9} to get: $t=\frac{t'+\dfrac{u}{c^2}v'_xt'}{\sqrt{1-\dfrac{u^2}{c^2}}}$ Now we find $v_x$ by dividing $x/t$.

#### Transformation of velocity

$v_x=\frac{x}{t}=\frac{u+v'_x}{1+\dfrac{uv'_x}{c^2}}$
In the limiting case when $v'_x=c$, a light beam, we see that $v_x=c$ too. This is not surprising because the transformations have been designed to do so.