

Michael Lee's Mathematics and Physics

Derivation of the Lorentz Transformation.


We all learned in primary school there is a time delay between the moment we see lightening flash and then hear its thunder.  This delay is due to the limited speed of sound.  It is approximately 340.29 metres per second or 372 yards per second; the speed of light is considerably faster at about 299,792 Kilometres per second or 881,991 times faster than the speed of sound.  What is important is it takes time for sound or light to travel from the moment an event happens to the moment it is perceived by the observers.  Events are things bumping into each other at some particular place and at a particular time; an atom hitting another atom is an event; a hammer hitting the floor is an event, you opening a door somewhere is an event, the birth of a child is an event, etc., etc.  So, different observers (people) at different distances from an event will report a different time that it happened in addition to reporting their different distances from it.


In the game of golf, for example, suppose a ball is about to be struck with a club (the event); other observers (spectators) at different distances from the ball strike will report different times of when it happened based upon their perception of its sound; remember the speed of sound is finite and about 372 yards per second.  Suppose I'm located 300 yards from where the ball shall be struck and another spectator is 150 yards from it.  Suppose further we synchronized our clocks before the event.  Both the other spectator and myself are too far away to actually see the ball being struck -- we can only hear it.   Furthermore, let's assume there is no human delay in recording the sound on the watch; for example, we could conceive of a clock that could be made to record it.  Suppose the ball strike shall happen at exactly twelve noon according to a synchronized clock located near the event (i.e. on the tee box).  The spectator at 150 yards would report the ball strike happening at 12:00 and 0.4032 seconds on his or her watch, whereas my watch, at 300 yards, will report it happening at 12:00 and 0.8064 seconds.  So both of our measurements of time were off a bit because the "actual" time of the ball strike was at 12:00 noon.  Suppose though, we can now see the ball being struck at twelve noon; even the speed of light is limited and so the spectator's watch at 150 yards will read the time of the ball strike as happening before my watch says so.  So how are we to account for these discrepancies?  Well, clearly there seems to be some relationship between space (distance) and time; in fact space and time are just two components of one thing we now call space-time. 


These slight disagreements of space and time owing to the limited speed of light doesn't sound like much of a difference, but if distances are greatly increased, it matters a lot.  Since the speed of light is limited, events in the heavens have a similar time delay like that of sound waves.  Suppose an event happens, a supernova say; it takes time for the light of the explosion to reach our telescopes, thus other observers (aliens?) at different distances from the explosion, no matter which direction they were going, would report different times of its occurrence, providing we are able to synchronize our atomic clocks long before the explosion happened.  The fascinating realization is the star light we see in our night skies is old light that left the star in the past;  so for example, upon seeing a supernova on earth from a distant star, the event actually happened long ago.


Before Galileo Galilei, most people, including Aristotle, thought the speed of light makes no sense and the time of the flash of lightening one sees is the same time for all observers no matter where they are.  As Aristotle put it, "light is due to the presence of something, but it is not a movement."  Galileo knew otherwise and reasoned the speed of light is indeed finite and measurable.  He tried to measure the speed of light by placing observers on two hills located at known distances.  Each observer was equipped with a lamp.  The idea of the experiment was for observer one to open up a shutter on his lamp, at such and such a time, and he instructed observer two to open his shutter upon seeing the light and once observer one sees the light from observer two, record the time elapsed on his clock.  Therefore the speed of light is equal to twice the distance between the observers on the hill tops, divided by the time elapsed on observer one's clock.  Upon completing the experiments, no matter what distance he placed the observers, his measurements of the speed of light were erratic and ridiculously low (owing to the erratic and slow reaction time of each observer), therefore Galileo argued the experiment is inconclusive.  Another difficulty was Galileo's limited ability to measure time accurately and with good precision.  According to what we know now, the time it would take for light to travel two miles is 0.000005 seconds, which is much too short to be measurable using Galileo's crude clocks and human observers.  Suppose the reaction time of the observers was somehow magically negligible, then Galileo would have needed an atomic clock to measure the time interval of the light's journey from one hilltop and back again.  Despite his difficulties, Galileo's design of the experiment was nonetheless sound.  All he needed was a much greater distance between the hills and a more accurate and precise instrument to measure time.  Despite these problems, rather than abandon his theory, he argued the speed of light must be very fast.  Shortly after Galileo's death in 1642, good measurements of the speed of light became known by the Danish astrophysicist Olaus Roemer whom in 1676, measured the speed of light as being 220,000 kilometres per second by timing the eclipses of the moon Io around Jupiter; recall it being 299,792 kilometres per second by today's measurements.


So far, the observers (the spectators) and the event (the golf strike) are stationary with respect to each other, but it is much more likely, if not downright certain, real observers and events move around each other; in other words, bodies in the universe are not stationary, and so things become a little more complicated when dealing with them.  Refer to the following diagram.  Suppose an event occurs at point A when two inertial frames of reference (S and S') are at the origin O; suppose it is an explosion of some kind.  Suppose further S is stationary while  S' is moving to the right with uniform velocity v towards the explosion.  So, we want to know where the explosion happened and when did it happen according to the inertial frame of reference one is in.




Galileo treated time as being universal in nature; that is, a clock in the frame S' appears to run at the same rate as one located in frame S.  Einstein disagreed and said measurements made of a clock in motion (i.e. a clock in frame S' when viewed from frame S), will read differently even though there is nothing wrong with the clocks. The clock moving to the right in frame S' will appear slower than one in frame S.  Suppose you are traveling away to the right in frame S', then your clock, as it appears to me, will read slower than mine located in frame S.  Today, we call that time dilation (see time dilation page).


Here is how Galileo saw time and space:




So this is what Einstein did, he added a prime symbol to represent the time of the event according to the S' clock as viewed from frame S.  It's added to the second equation to obtain the following.    Now all we need to know is where and when this explosion took place such that we can convert x and t to x' and t'    Well, the light coming from the explosion is travelling at a constant speed C and that is the same for all observers in all inertial frames of reference, no matter which direction or at what speed it is going.  So after Einstein, we obtain...













So, the Lorentz transformation is denoted above with lowercase gamma.  Below is how it affects our original equations.



                                                     For calculating time                           For calculating length