Lorentz Transformations

First, I've to thank my ten-year friend, Sean Wu (used to be called Ted Wu) , his uncle and aunt with their two daughters, and my parents.Without their kindness, I couldn't trigger my advanture and blog. (My English isn't still fluent, and tell me if there is any correct or better way to make a sentence.)

This is only the sinfonia of Albert Einstein's Special Theory of Relativity and also of my blog.There are two main postulates on which "His" theory was established.

Postulate 1: the speed of light,"c" , is the same in all inertial frames of reference.

Postulate 2: the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.

"--------------O--------------> r" is an inertial frame of reference. We use light to measure space and time, like this: r=c*t, if r>=0. => r-c*t=0,
so if r<0, r+c*t=0.

"--------------O'--------------> r' " is another inertial frame of reference, and then
r'-c*t'=0, r'>=0,
r'+c*t'=0, r'<0.>=" means "is equal or more than", "=>" means ",so then...".

And we can write down: 0=r'-c*t'=(alpha)*(r-c*t), and the other is
r'+c*t'=(phaal)*(r+c*t). Then we have
r'=(((alpha)+(phaal))/2)*r-(((alpha)-(phaal))/2)*c*t, and
t'=-(((alpha)-(phaal))/2)*(r/c)+(((alpha)+(phaal))/2)*t.

Let a=(((alpha)+(phaal))/2) and b=(((alpha)-(phaal))/2).
r'=a*r-b*c*t
t'=(-b/c)*r+a*t
=>
r=(a*r'+b*c*t')/(a^2-b^2)
t=((b/c)*r'+a*t')/(a^2-b^2)

r'=a*r-b*c*t => dr'=a*dr-b*c*dt, where dr'=0, v=(dr/dt)=(b/a)*c => b=(v/c)*a
r'=a*(r-v*t)
t'=a*(t-(v/c^2)*r)

r'=a*(r-v*t) => dr'=a*(dr-v*dt) => l'=dr'=a*(dr-v*dt), as dt=0, l'=a*dr=a*l

dr'=a*(dr-v*dt) => v*dt=dr-dr'/a
t'=a*(t-(v/c^2)*r) => dt'=a*(dt-((v/c)^2)*dr)
=> v*dt'=a*(v*dt-((v/c)^2)*dr)=a*((dr-dr'/a)-((v/c)^2)*dr)
=a*(1-(v/c)^2)*dr-dr
dr'+v*dt'=a*(1-(v/c)^2)*dr=a*(1-(v/c)^2)*l,
as dt'=0, l=dr'/(a*(1-(v/c)^2))=l'/(a*(1-(v/c)^2))

As dt=0, l'=a*l.
As dt'=0, l=l'/(a*(1-(v/c)^2)).
According to Postulate 2, a=1/(a*(1-(v/c)^2)),
and l*l'>=0 => a>0, a=(1-(v/c)^2)^(-1/2)=(1-(beta)^2)^(-1/2)=(gamma)

As v=0, (gamma)=1;
As c>v>0, (gamma)>1;
As v=c, (gamma) doesn't exist.

r'=a*(r-v*t) t'=a*(t-(v/c^2)*r)
=>
r'=(gamma)*(r-v*t)
t'=(gamma)*(t-(v/c^2)*r)
=>
r=(gamma)*(r'+v*t') t=(gamma)*(t'+(v/c^2)*r')

(alpha)+(phaal)=2*a
(alpha)-(phaal)=2*b=2*(v/c)*a
=>
(alpha)=(1+v/c)*a=((1+(beta))/(1-(beta)))^(1/2)
(phaal)=(1-v/c)*a=((1-(beta))/(1+(beta)))^(1/2)
=>
(alpha)*(phaal)=1

r'^2-(c*t')^2=(r'-c*t')*(r'+c*t')=(alpha)*(r-c*t)*(phaal)*(r+c*t)=r^2-(c*t)^2=r^2+(i*c*t)^2

r'=(gamma)*(r-v*t) t'=(gamma)*(t-(v/c^2)*r)
=>
dr'=(gamma)*(dr-v*dt) dt'=(gamma)*(dt-(v/c^2)*dr)

Here, we reveal three facts in Special Theory of Relativity:

Fact 1: dt'=(gamma)*(dt-(v/c^2)*dr), as two things are simultaneous via an observer in the inertial frame of reference, O', meaning "dt'=0", but "dt=0" is not necessary depending on "dr".

Fact 2: l_o=l'=dr'=(gamma)*(dr-v*dt),
as dt=0, dr=l => l_o=(gamma)*l => l=(l_o)/(gamma) called "longitudinal contraction".

Fact 3: t=(gamma)*(t'+(v/c^2)*r') => dt=(gamma)*(dt'+(v/c^2)*dr'),
as dr'=0, dt=dt_o => dt=(gamma)*dt'=(gamma)*dt_o called "time dilation".

p.s. "l_o" means the orginal length and is the same as "dt_o" and "m_o".

The above-mentioned are only pieces of cake in Special Theory of Relativity. We don't still touch:

m=(gamma)*m_o called "mass dilation",

p=(gamma)*m_o*v called "new definition of momentum",

E=(m_o)*c^2 called"rest mass-energy equivalence".
And it'll be continued.Thank you for your reading my first article in my blog.