Lorentz Transformation

Lorentz Transformation

( Given x, y, z ,x', y', z',|v|, c, find out V_x, V_{y ,}V_z)

velocity-i +j +	k	\|v\|=
velocity-i +j +	k	\|v\|=
velocity V --	km/s	velocity of signal	km/s
AngleΨ with x-axis (degree)		V_x
AngleΦ with y-axis (degree)		V_y
Angleθ with z -axis (degree)		V_z
β (V/c)		γ = 1/√(1-v²/c²)
x		x'
y		y'
z		z'
t		t'
origin in S frame	(0,0,0,0)
origin in S' frame	,	,
origin in S' frame	,	,,
x²+y²+z²		c²t²
R'²=x'²+y'²+z'²		T'²=c²t'²
R'² - T'²
[t²v²-(V_xx+V_yy+V_zz)²/c²]γγ*
(x'-s'(x))²+(y'-s'(y))²+(z'-s'(z))²		c²(t'-s'(t))²
A1x Bx	Dx	tot
A1y By	Dy	tot
A1z Bz	Dz	tot

Remark : From Lorentz1.htm, first determine x',y',z' for a given x,y,z, V_{x , ,}V_{y ,}V_zand V and c,. when x,y,z and x',y',z' ,V is put here,

the result is some what consistent for a set of V_{x ,}V_{y ,}V_z. But for arbitrary values of x,y,z,x',y',z',V,c, there is inconsistency.

The same has to be studied further.

On simplification of Lorentz transformation equn, we get

A_x1V²_x+B_xV_x + D_x =0 ; A_y1V²+B_yV_y + D_y =0 ; A_z1V²_z+B_zV_z + D_z =0

A_x=(c²t²+x'²) / x'² ; A_x1=A_x-1 ; B_x = -2c²xt / x'²; C_x = c²( x²- x'²) ; D_x = C_x + V²

A_{y =}(c²t²+y'²) / y'² ; A_y1=A_y-1 ; B_y =-2c²yt / y'²; C_y = c²( y²- y'²) ; D_y = C_y + V²

A_{z =}(c²t²+z'²) / z'² ; A_z1=A_z-1 ; B_z =-2c²zt / z'²; C_z = c²( z²- z'²) ; D_z = C_z + V²

and solve the quadratic equn.