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                                      Michael Lee's Mathematics and Physics

Physics and Special Relativity



The most incomprehensible thing about the world is that it is all comprehensible.  (Albert Einstein)


While learning about special relativity in college, I encountered such weird things as 'length contraction' and 'time dilation.'  As for the former, I was told that metre sticks in motion shall measure shorter than those at rest; that is, metre sticks are longest in their own inertial frame of reference.  As for the latter, clocks go "tick, tock, tick, tock, etc.." slower  (their period is longer) when in motion relative to our clocks at rest; that is, clocks go fastest in their own inertial frame of reference.  So, with reference to special relativity, the length of metre sticks in motion and the period of clocks in motion are different than if they are at rest.   






Our understanding of the universe, despite our fine tools such as telescopes, microscopes, magnetic imaging, x-rays, etc., ultimately consists of using our sense impressions to make conclusions about reality.  But Einstein was interested in knowing the nature of things beyond their casual appearances.  For example, according to Aristotle light bodies, feathers say, fall slower than heavy bodies like a lump of lead, and that certainly appears to be the case.  Who could be so bold to say otherwise?  Feathers falling at the same rate as lead?  Galileo argued that is precisely the case; feathers fall at the same rate as a lump of lead and any apparent "lag" of falling feathers behind that of falling lead can be explained as being due to air resistance; without this factor (e.g. a near vacuum), a feather will fall as fast as lead.  Nobody at the time believed him.  One bright sunny day, according to legend, he climbed to the top of the Tower of Pisa and dropped a musket ball and a canon ball at the same time and had observers down below watch what happens when they hit the ground.  The light musket ball reached the earth in almost the same time as the heavy cannon ball.  Still, people didn't believe Galileo because that would entail the great Aristotle was wrong about this matter.  Indeed Aristotle was wrong and the modern scientific community thinks Aristotle to be a huge blunder to science.  That is wrong as it was human politics that caused the problem because back then you could build on Aristotle's work but you couldn't disagree with him, except when he  contradicted the Bible by saying the universe is eternal, with no beginning nor an end; it has and forever shall be in existence.  He also developed a system of deductive and inductive logic unparalled even to this day; he knew every trick in the book about how people make arguments more persuasive than they should be.  His understanding of ancient presocratic philosophers are the best second hand accounts we have of who they really were and what they believed. 


Similarly, upon casual observation, metre sticks certainly appear to be the same length in motion and at rest and clocks in motion appear to run at the same speed as clocks at rest; it's only at very high speeds do special relativistic factors become significant.   What Einstein then did was formalize mathematically these relationships such that when it comes to length, using what is known as the Lorentz Transformation, we can use our measured lengths of metre sticks in motion and then calculate their lengths as if they  were at rest.   Likewise for time, using the measurements of time from clocks in motion and using the Lorentz Transformation, we can calculate its period or durations between 'ticks' and 'tocks', as if the clock were at rest.  (See the Lorentz Transformation page or the lecture below.)


I absolutely love viewing lectures on YouTube as it absolutely prevents me from asking questions.  Below is a great lecture of Professor R. Shankar of Yale University on the subject matter.