This page is intended as an introduction for beginners to the main page Special relativity

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:

h² = x² + y²

This two dimensional length is not invariant if the thing is tilted out of the two dimensional plane. In everyday life a three dimensional coordinate system seems to describe the length fully. The length is given by the three dimensional version of Pythagoras' theorem:

h² = x² + y² + z²

The derivation of this formula is shown in the illustration below.

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:

s² = x² + y² + z² + (ict)²

Where i is the square root of minus one, c is a constant and t is the time interval spanned by the space-time interval, s. In this equation the 'second' becomes just another unit of length. In the same way as centimetres and inches are both units of length related by centimetres = 'conversion constant' times inches, metres and seconds are related by metres = 'conversion constant' times seconds. The conversion constant, c has a value of about 300,000,000. Now i² (i.e.: ) is equal to minus one so the space-time interval is given by:

s² = x² + y² + z² - (ct)²

Minkowski's use of the square root of minus one has been superceded by the use of advanced geometry that uses a tool known as the "metric tensor" but his original equation survives, the space-time interval is still given by:

s² = x² + y² + z² - (ct)²

Space-time intervals are difficult to conceptualise; they extend between one place and time and another so the velocity of the thing that travels along the interval is already determined.

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 s² = x² + y² + z² - (ct)²

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

x = vt

So: s² = (vt)² - (ct)²

But when the velocity v equals c:

s² = (ct)² - (ct)²

And hence the space time interval s² = 0

A space-time interval of zero only occurs when the velocity is c. When observers observe something with a space-time interval of zero they all observe it to have a velocity of c no matter how fast they are moving themselves.

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:

s² = (vt)² - (ct)²

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

s² = (0)² - (cT)²

So: - (cT)² = (vt)² - (ct)²

and hence .

So if John measures a time interval of 10 seconds (t=10) he will see that Bill's clock measures an interval, T that is less than this. Clocks in motion are predicted to slow down relative to those on observers at rest. This is known as time dilation.

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, x = ct so he can work out that distances measured by Bill (X) are related to distances that he measures by:

.

The last consequence to be discussed here is that clocks will appear to be out of phase with each other along the length of a moving object. This means that if one observer sets up a line of clocks that are all synchronised so they all read the same time then another observer who is moving along the line at high speed will see the clocks all reading different times. This means that observers who are moving relative to each other see different events as simultaneous. This effect is known as "Relativistic Phase" or the "Relativity of Simultaneity". Relativistic phase is calculated below:

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:

X² = x² - (ct)²

But from

X² = x² - (v² / c²)x²

So: (ct)² = (v² / c²)x²

And ct = (v / c)x

So t = (v / c²)x

The net effect of the four dimensional universe is that observers who are in motion relative to you seem to have time coordinates that lean over in the direction of motion and consider things to be simultaneous that are not simultaneous for you. Spatial lengths in the direction of travel are shortened because they tip upwards and downwards relative to the time axis in the direction of travel, this is akin to a rotation out of three dimensional space.

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 E = mc² where c² 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

The discussion given above has been confined to what is known as "flat space-time". The general, differential form of the space-time interval is given in the article Special Relativity. The modern description of the universe uses the term (3+1)D rather than 4D to show how time is not like the spatial dimensions.

External links

• Sean Carroll's online Lecture Notes on General Relativity
• E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws by Nina Byers

See also

• Special Relativity
• Invariance