Einstein's Theory of Special Relativity

Mark Schofield

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Introduction

The behavior of objects moving at everyday speeds is described in a set of laws called Newton's Laws of Motion. These laws, which remain in use today, were found to be an approximation of the real laws of motion that were discovered later by Einstein. Einstein's Special Theory of Relativity describes the relative motions of objects moving at speeds close to the speed of light. For objects moving at normal speeds, the differences between Newton's laws of motion and those derived by Einstein's theory are very small indeed.

So why study the Theory of Special Relativity ? The theory is interesting because of the consequences which flow from it. An understanding of this theory radically alters our understanding of both space and time. However, before we can understand this theory, we must first understand the science from which it came.

Background

Galileo and Newton (17th century)

The idea of relativity goes back to the 17th century and the work of Galileo. Galileo proposed that in a closed windowless room below decks in a smoothly moving ship, it was impossible to do an experiment that would tell you if the ship really was moving. In a ship moving at constant speed and direction it is impossible to feel the motion of the ship. Any mechanical experiment, such as measuring the speed of a moving object in the room would give precisely the same results as a similar experiment performed at rest on the shore. This is known as "Galilean relativity".

It follows that an observer in a house by the shore and an observer in the ship will not be able to determine that the ship is moving by comparing the results of experiments done inside the house and ship. In order to determine motion, these observers must look at each other. It is important to note that this is true only if the ship is sailing at constant speed and direction; should it speed up, slow down or turn the observer inside the ship can tell the ship is moving.

Generalizing these observations, Galileo postulated his relativity hypothesis: "any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments".

Lets go back to the example of the smoothly moving ship. To measure the speed of a moving object in the ship, we could mark two lines on the floor and measure the time taken for the object to travel the distance between the two lines (speed is the distance travelled per unit of time). The speed of a moving object relative to these lines is the same whether the ship is at rest or in uniform motion.

In this example, we are using the ship as our "frame of reference" (a set of coordinates to specify the precise position of an object at a given time). Of course, the ship is not our only possible frame of reference. We could measure the speed of a moving object in the ship relative to a fixed position on the shore. If the ship is in motion, the speed of the moving object measured from the shore will be different. Lets imagine we take the ship as our frame of reference and measure an object moving at 2 miles per hour. Now we take a point on the shore as our frame of reference. The ship is moving in the same direction as the object at 10 miles per hour. The speed of the object measured in this second reference frame is equal to the speed in the first reference frame plus the speed in the second reference frame, namely 12 miles per hour (10 + 2). This presents a problem. Which reference frame should be used to measure the absolute speed of the moving object ?

The problem does not end there. We could also use a frame fixed relative to the sun, our galaxy, even a point in space outside of our galaxy. In each case, the speed of an object in the ship would be different because the earth, the sun, the galaxy and all the other galaxies are also in motion. There is no way to calculate the absolute speed of a moving object, because there is no such thing as an absolute frame of reference. By saying absolute, what I actually mean is that there is no place in the universe that is completely stationary. Because there is no place or object in the universe that is stationary, there is no single place or object on which to base all other motion. All speed is relative.

The work of Newton, which introduced the concepts of forces acting on objects, ties in perfectly with the work of Galileo. As with Galilean relativity, Newton's laws require a frame of reference in which to measure distance and time. The relativity hypothesis of Galileo, still very much a part of scientific theory, was reworded to mean "the laws of physics are the same in a uniformly moving room as they are in a room at rest".

Michelson and Maxwell (19th century)

In 1879 a scientist named Michelson devised an experiment to measure the speed of light by timing a flash of light between mirrors. The speed of light was found to be 186,350 miles per second with a likely error of around 30 miles per second. This measurement, the most accurate yet made by direct measurement, agreed well with other measurements made from astronomical observations.

In the middle of the 19th century there was a substantial advance in the understanding of electric and magnetic fields. This new understanding was summarized into a single electromagnetic theory by a scientist called Maxwell (the theory is in fact a mathematical model consisting of 4 equations reffered to as Maxwell's equations). The theory demonstrated that the speed of electromagnetic waves is dependent only on the electric and magnetic constants of the medium through which they travel. As such, the speed of electromagnetic waves is independent of any motion of the source or the observer. Measurements from experiments using magnets on electrical fields showed the speed of these waves to be 186,300 miles per second.

Galilean Relativity vs Maxwell's Equations

Experiments had already demonstrated light can behave as a wave. Following the work of Maxwell, and the amazing similarity bewteen the speed of light and the speed of electromagnetic waves, it was immediately suggested that light is an electromagnetic wave (this is now held to be the case).

If Maxwell's theory applies to light, we are implying that the speed of light is independent of any motion of the source or the observer. This would appear to contradict Galilean relativity. A simple example proves the point. Imagine we perform Michelson's experiment to measure the speed of a light ray in a rocket travelling through space. Taking the rocket as our frame of reference we measure a light ray moving at 186,300 miles per second. Now we take the sun as our frame of reference. The rocket is moving in the same direction as the light ray at 300 miles per hour. What is the speed of the light ray in this second frame of reference ? Using Galilean relativity, we expect the speed of the light ray to be 186,600 miles per second (186,300 + 300). Maxwell's theory predicts the speed of the light ray to be 186,300 miles per hour. Which is correct ?

Special Relativity

Einstein made the simple but important insight that Galilean relativity does not apply to a moving source of light. It is possible he knew of another experiment performed by Michelson and Morley which had attempted to apply Galilean relativity to the speed of light and failed. Whatever the case, the insight was to be crucial to the theory of Special Relativity.

Theory of Special Relativity

The Theory of Special Relativity is deceptively simple, consisting of just 2 postulates (assumptions). Einstein took the principle of Galilean relativity and restated it as his first postulate :

1. The Laws of Physics are the same in all inertial (non-accelerating) reference frames.

Einstein then simply pointed out that the Laws of Physics must include Maxwell's equations describing electric and magnetic fields as well as Newton's laws describing motion of objects under forces. If Maxwell's equations apply in all reference frames, it follows that the speed of light is the same in all reference frames (the speed of light can be represented using the constant c). This is Einstein's second postulate :

2. The speed of light has the same value c in all inertial reference frames.

This then is the entire content of the Theory of Special Relativity: the Laws of Physics are the same in any inertial frame, and, in particular, any measurement of the speed of light in any inertial frame will always be 186,300 miles per second.

Thought experiments

It is difficult to conduct physical experiments which prove the validity of Einstein's theory. We need very precise instruments to measure motion and time when the speed of an object approaches the speed of light. Einstein often used "thought experiments" (theoretical experiments based on mathemetical calculations) to demonstrate his theory. Physical experiments conducted later would provide hard evidence to support his theory.

To conduct thought experiments using the Theory of Special Relativity, we need to create instruments to measure both motion and time. Speed is the distance covered per unit of time, so we need instruments to measure both distance (a ruler) and time (a clock).

The Clock

Let's take a simple clock, made of a light ray going back and forth between two mirrors. A light clock seems to be the best measure of time since the speed of light remains constant regardless of motion.

Let us consider what happens to a light ray within our light clock when the clock is stationary and the clock is in motion.

clock

In fig. A, Mr A and his light clock are stationary. In this frame of reference, Mr A sees a light ray travelling up and down between the two mirrors. The total distance travelled by the ray (marked in red) is twice the separation of the mirrors.

In fig. B, Mr B and his light clock are in uniform motion on a railway wagon. Mr B is in the frame of reference of the wagon. To Mr B, the total distance travelled by the ray in his clock is twice the separation of the mirrors. However, to Mr A things appear very different. In Mr A's frame of reference, the light ray in Mr B's clock travels a diagonal path from one mirror to the other (marked in red). This distance is greater than the distance travelled by the ray in his own clock. Since the speed of light is constant, and speed is the distance per unit time, Mr A must see Mr B's clock as ticking slower than his own !!!

This simple thought experiment demonstrates the suprising effect that time, like speed, is not absolute. Your measurement of time depends entirely on your frame of reference, and different inertial reference frames measure time differently (there is no single clock that tells the same time for everybody everywhere). This is one of the peculiar effects of relativity and is known as time dilation. The amount by which time slows down can be calculated from the above example using simple algebra.

Physical experiments have since shown the existence of time dilation. Atomic clocks, flown on high speed aircraft around the earth have been shown to slow down by a tiny but measurable amount when compared with similar clocks that had remained stationary on earth. The difference only becomes appreciable at speeds which are significant compared to the speed of light.

The Ruler

Lets go back to our previous example. Imagine the speed of the wagon relative to the track is v feet per second. Imagine it has a cutting device linked to the light clock to cut a notch in the track every second. How far apart are the notches ?

Mr B sees the track passing under his wagon at v feet per second. In the frame of reference of Mr B, the device cuts a notch every second and the notches are each v feet apart.

Mr A sees the wagon passing along the track at v feet per second (the relative motion of Mr B to Mr A will be the same as the relative motion of Mr A to Mr B). However, in the frame of reference of Mr A, Mr B's clock is running slow. One second for Mr B is greater than one second for Mr A. To Mr A, the notches are not cut at intervals of one second, but intervals of more than one second (the cutting device is linked to the light clock of Mr B, not Mr A). Since speed is the distance per unit time, and the speed remains the same, Mr A must see the wagon travel further than v feet in one second. The distance between notches will be greater for Mr A than for Mr B.

So who is right ? In fact, it is Mr A who is right. This is because the measurement of distance was made in his frame of reference (Mr A has the track which acts as the ruler). If Mr B stopped his wagon and joined Mr A by the track to measure the distance, they would confirm that the notches were more than v feet apart.

This implies that as a result of his motion, Mr B observes the notches to be closer together than they would be at rest. This effect is called length contraction (or Fitzgerald contraction) and can be calculated from the above example using simple algebra. The length contraction effect applies not just to the track but to Mr B, the wagon and everything else measured in this frame of reference. All lengths parallel with the direction of motion will contract so that objects have a 'squashed' appearance. This effect is only true for an observer outside of the reference frame. For an observer inside the reference frame, the wagon is the same whether it is in uniform motion or at rest. Length, like time and speed, is not an absolute quantity either !

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Mark Schofield
mark.schofield.free.fr
Last updated : 28 April 2007