Special relativity for beginners

Although the Special theory of relativity was first proposed by Einstein in 1905 the modern approach to the theory depends upon the concept of a four dimensional universe that was first proposed by Hermann Minkowski in 1908 and further developed as a result of the contributions of the famous female mathematician Emmy Noether. This approach uses the concept of invariance to explore the types of coordinate system that are required to provide a full physical description of the location and extent of things.

The modern theory of Special Relativity begins with the concept of "length". In everyday experience it seems that the length of objects remains the same no matter how they are rotated or moved from place to place. We think that the simple length of a thing is "invariant". However, as is shown in the illustrations below, what we are actually suggesting is that length seems to be invariant in a three dimensional coordinate system.

The length of a thing in a two dimensional coordinate system is given by Pythagoras' theorem:

It seems that provided all the directions in which a thing can be tilted or arranged are represented within a coordinate system then the coordinate system can fully represent the length of a thing. However, it is clear that things may also be re-ordered over a period of time. This is shown in the following diagram:

The path taken by a thing in both space and time is known as the space-time interval.

Hermann Minkowski realised in 1908 that if things could be rearranged in time then the universe might be four dimensional. He boldly suggested that Einstein's recently discovered theory of Special Relativity was a consequence of this four dimensional universe. He proposed that the space-time interval might be related to space and time by Pythagoras' theorem in four dimensions:

If the universe is four dimensional then the space-time interval will be invariant rather than spatial length. Whoever measures a particular space-time interval will get the same value no matter how fast they are travelling. The invariance of the space time interval has some dramatic consequences.

The first consequence is the prediction that if a thing is travelling at a velocity of c metres per second then all observers, no matter how fast they are travelling will measure the same velocity for the thing. The velocity c will be a universal constant. This is explained below.

When an object is travelling at c, the space time interval is zero:

The space-time interval is

The distance travelled by an object moving at velocity v in the x direction for t seconds is:

So:

But when the velocity v'\' equals c'':

And hence the space time interval

The universal constant, c, is known for historical reasons as the "speed of light". In the first decade or two after the formulation of Minkowski's approach many physicists, although supporting Special Relativity, expected that light might not travel at exactly c but might travel at very nearly c. There are now few physicists who believe that light does not propagate at c.

The second consequence of the invariance of the space-time interval is that clocks will appear to go slower on objects that are moving relative to you. Suppose there are two people, Bill and John on separate planets that are moving away from each other. John draws a graph of Bill's motion through space and time. This is shown in the illustration below:

Being on planets, both Bill and John think they are stationary and just moving through time. John spots that Bill is moving through what John calls space as well as time when Bill thinks he is moving through time alone. Bill would also draw the same conclusion about John's motion. To John it is as if Bill's time axis is leaning over in the direction of travel and to Bill it is as if John's time axis leans over.

John calculates the length of Bill's space-time interval as:

whereas Bill doesn't think he has travelled in space so writes:

So:

and hence .

The combination of time dilation and the way that c, the speed of light, is constant for all observers means that when John observes measuring rods on Bill's planet they will seem to be smaller than his own measuring rods. A prediction known as "relativistic length contraction".
Bill calculates the distance travelled by a light ray as X = cT but John calculates T to be given by . John also calculates that the distance travelled by the light ray, so he can work out that distances measured by Bill (X) are related to distances that he measures by:

.

Distances between two points according to Bill are simple lengths in space (X) whereas John sees Bill's measurement of distance as a combination of a distance (x) and a time interval:

But from

So:

And

So

Great care is needed when interpreting space-time diagrams. Diagrams present data in two dimensions and cannot show faithfully how, for instance, a zero length space-time interval appears.

It is a common misconception that special relativity only applies to objects that are moving quickly. This is entirely untrue. In the main page it is shown that the kinetic energy of an object at all speeds is a relativistic quantity. Kinetic energy is relativistic because, although relativistic changes in mass are tiny, these result in large changes in energy due to where is about 90,000,000,000,000,000. Newtonian physics describes the interplay between kinetic and potential energy without explaining the origin of kinetic energy or inertia, it just assumes these things whereas Special Relativity explains them at a deeper level.

Caveats

External links