Fundamental Physics by Ramamurti Shankar at Yale_6_Four Vector

From Einstein’s time-space equation, people commonly asked if time is reversible. So if that is true, the following equation needs to be true too.

Because light of speed is the fastest in the universe, so the right side c*delta t is what light travels, over delta x, must be greater than 1, that infers u/c be greater than 1, which is impossible as again, u cannot exceed c.

As a result, the physicists sketched a light cone to make further interpretation of our life phenomena:

We live at present which is the surface in middle layer, tangible line to the cone has a theta of 45 degree, or pie/4 radian. The top and bottom cone space contains the future and past events that happen. But out of the cones, since no signal travels faster than light, stuff occur but won’t affect the observers in the center at all.

Now if we want to apply four-vector coordiante to denote the same concept – lorentz transformation, suppose X0 is the c*t, x1, is the distance or previous x, x2, and x3, we get

Another way to denote the four-vector system is as following, X is in capital.

In common coordinating system composed of x, y, if we turn or switch an angle of theta, the new system x’, y’, denoted by sin and cos, and we can deduce a pair of invariant x^2 + y^2 = x’^2 + y’^2. Is it possible to deduce such a pair in lorentz transformation. Yes, and it is not a sum but subtraction. The whole is given a name time-space interval, denoted as S^2.

Even it’s names as S^2, S^2 could be negative if the space summation is greater than the time X0^2. Now we can reinterpret the light cone:

Similarly, we get

Subsequently, deducing further vectors such as velocity, momentum:

Four vector momentum is as following.

Einstein was able to further fine cP0 somewhat like kinetic energy in Newtonian world. Which already looks similar to that famous energy mass equation E = mc^2.

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