I didn’t pay attention in School and I regret that now. Please help me understand some aspects here. I read about Time being the fourth dimension but I am just having difficulty grasping this information.
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The idea is simple instead of thinking about locations in a 3D coordinate system you think in terms of events.
If we look at 1D of space and 1D of time we get an x-t coordinate system. This is very similar to an x(t) function. For a spring each point of its graph (the sin curve) is an event it tells you where the mass on the spring was at any given point in time.
So lets be more direct and we just say that we map events in a 4D spacetime. But the idea of using a 4D coordinate system comes from coordinate transformations.
We used to transform coordinates according to the Galilean transformation. Its what you use to calculate the difference in velocity between two movies objects. Lets say we are driving on the motorway and you want to overtake me. I travel with v1 and you with v2 so you’ll pass me with v=v2-v1. From my reference frame you move at v.
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So lets try to generalise a bit. If we both have a coordinate system K and K’ lets say at t=0 their origins are at one place and K’ is moving at v realtive to K along the x axis. So lets select a point in the coordinate system K, r0 and lets see how that point’s coordinates will change in K’. From K’ it’ll look as though r0’s x coordinate drifts in the negative direction. So the x coordinate of r0 will shift some distance in the negative direction for any t time vt is distance from v=s/t so x’=x-vt. So the origin of K in K’ will be at x=-vt y=0 and z=0. For v=3 units/s in 1 s r=(0,0,0) will appear in K’ as r’=(-3,0,0). And how does time transform well t=t’.
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So for some v velocity along the x axis K: (x,y,z,t) and K’: (x=x’-vt, y=y’, z=z’, t=t’). Ok great now transform EM waves accordingly and you get that there is some absolute rest medium through which EM waves propagate, we called it aether. Now this is the same for mechanical waves where you having some relative velocities to the medium effect how you see the waves. This is called Doppler shift.
So you do an experiment and you don’t see any of that and assume that then the speed of light must be the same for all frames.
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You create now two scenarios one where an event happens at the same place at different times. Like a clock, and lets use a beam of light bouncing off a mirror and coming bact to a detector. It travels l to and back from the mirror. So all together a distance of 2l. If we travel at some v velocity we see a triangle as for us the detector moves back vt’ lf a distance. And since c is constant we know that the the two sides are both ½ct’.
We got two event (t0,r0) = (0,0) and (t0′,r0′)=(0,0) since we can say for the two to start at (0,0) and r=(x,y,z) so r0=(0,0,0).
The second event is (t,r)=(2l/c, 0) and (t’,r’)=? 2l/c for simply v=s/t -> t=s/v.
Now we need pythagoras: l²=(½ct’)²-(½vt’)² and l is just ½ct.
From this we get t’=t/sqrt(1-v²/c²) and of course x’=vt’ and now we have it. The other exercise is to figure this out for two events that happen at the same time in different locations. And through a few additional steps we get the Lorentz transformation which is an upgrade for the Galilean one.
Now instead of doing geometry all the time we introduce the 4 vector formalism. We have vectors with the usual 3 spacial and a 0th time like component. The spacetime 4 vector is x_i = (ct,x,y,z) and we transform this into some coordinate system by applying the L Lorentz transformation which is a matrix it turns one vector into another.
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And from this the rest of the formalism unfolds and now we have special relativity. Its basically about using the Lorentz transformation instead of the classical one but the consequences are profound. (There is a lot more mathematical approach to this where we instead define rapidity and essentially arrive at the same conclusion.)
So we aren’t considering a 4D coordinate system with time since we are so bored but because time changes from coordinate system to coordinate system and we cant ignore it. And SR is how we account for that.
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