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Michael Lee's Mathematics and Physics

Why the speed of light is constant and why can't bodies go faster than light.


Suppose you are aboard a train travelling at say 100 kilometres per hour.  Suppose further you walk towards the front of the train at 5 kilometres per hour.  Your speed relative to the ground would simply be the sum of those two figures or 105 kilometres per hour.  Likewise, suppose you are moving towards the back of the train, then your speed relative to the ground would be 95 kilometres per hour.  That's rather uncontroversial, but suppose you were just standing still on the train with a flashlight pointed towards the front of it.  Surely the speed of the light would be going faster with respect to the ground.  Likewise, if you were to stand with a flashlight pointed towards the rear of the train, wouldn't the speed of the light be slower, relative to the ground?   Well, the answer is no in both cases; instead, the speed of light does not change regardless of the motion of the source emitting it or the motion of its observers.  


Light isn't the only wave that has this property, so do water and sound waves under equal conditions.  Here's an experiment you can do at home.  Fill the kitchen sink with about three inches of water say.  Next, tap the surface of the water with your finger and observe the speed of the wave produced.  Next, tap the water three times say as you move your hand to the right.  How fast are the waves travelling outwards?  They are all moving outwards at the same speed aren't they?   Now the speed of water waves do vary under different conditions.  For example, the speed of the waves created in the experiment are much slower than one created by an undersea earthquake that produces a tsunami that in deep water travels at the speed of a jet plane.  


Likewise, sound at a constant air pressure, travels at about 340 metres per second.  Suppose we have a source of sound like that of a musical metronome that goes, tick, tock, tick, tock, etc.   For each tick or tock, a sound wave is emitted from the location of the metronome and it goes away in all directions (up, down, left, right, forwards, backwards, etc.) like a balloon being filled with air.  This sound wave has a limited amount of energy in it and as it expands out in all directions, the "surface area" of the wave increases and hence the further away you are from the source of sound, the weaker it is.  Now suppose you are in your home listening to the metronome, the sound is approaching you at about 340 metres per second, as we would expect, but what if you are moving towards or away from the metronome.  What would happen then?  Well, if you are moving towards the metronome at say 5 metres per second, the sound waves still travel at 340 metres per second and not 345 metres per second.   Likewise, suppose you are moving away from the metronome at 5 metres per second, again, the speed of the sound waves are still 340 metres per second and not 335 metres per second.  Suppose it is the metronome that is moving.  If it is coming towards you at 5 metres per second, the speed of the sound waves are still travelling at 340 metres per second and not 345 metres per second.  Likewise, if the metronome is moving away, the speed of the waves are still travelling at 340 metres per second and not 335 metres per second. This is because the speed of sound or light doesn't slow down or speed up just because you or the source of it are moving around.  


What does change is the frequency of the ticks and tocks.  If one moves towards the source at a constant velocity, the frequency of ticks and tocks will be higher because the time interval between each wave is smaller.  Whereas if one moves away from the source, the time interval between each wave is larger and hence its frequency will be lower.  The frequency of something is the number of occurrences, events, or happenings per unit of time.  For example, middle C on the piano is at 256 Hz (Hertz: instances per second), which means there are 256 waves per second, and blue light is roughly 631 THz or Terahertz ( if you understand exponential notation, one Terahertz is 1 x 10^14  Hz).


In music, all notes have a particular pitch;  higher notes have a higher frequency and low notes have a lower frequency.  Suppose there is a very loud piano mounted on top of a very quiet flatbed truck and a musician aboard the truck hits middle C say.  Suppose you are a keen musician on the ground and safely away from the oncoming truck.  As the truck approaches, you would report it being slightly "sharper" than middle C, whereas the musician playing the note on the truck (he or she is in the same inertial frame of reference as the piano) would just hear a good middle C.  After the truck passes by, you would hear the pitch suddenly decrease to a note "flater" than middle C.  I may as well mention this now, it's called the Doppler Effect and applies to all kinds of waves including light.


Unavoidably, I must use some algebra to make this point.   The speed of light is denoted as C.





Why bodies cannot travel at or faster than the speed of light


Why is it not possible for a body to travel at or faster than the speed of light?  According to Newton, if we take a body and we apply a uniform net force to it in a near vacuum, like in outer space, the body shall accelerate uniformly.  Newton thought this uniform acceleration will continue indefinitely (see Derivation of Kinetic Energy), but that is not the case according to Einstein.  Einstein reasoned that any additional momentum induced in the body by the force goes towards both making a body's velocity greater and making its mass greater.  Newton thought only its velocity would increase and its mass would remain the same.  Special relativity, on the other hand, says at slow speeds much of the momentum behind the force goes towards increasing the body's velocity and some goes towards making its mass greater but only imperceptibly so, however at higher speeds, near the speed of light, any additional momentum transfered by the force on the body just makes it more massive and not much faster; its velocity will still go up a little but in diminishing quantities.  Besides, if the velocity of the body is at the speed of light C, the gamma value of the Lorentz transformation (see Derivation of the Lorentz Transformation) becomes one divided by zero which is mathematical nonsense.