- #1
Mihai_B
- 10
- 1
Can anyone help me find any mistake in this expansion ? (I've asked it also in other places but I got no answer))
Pα= e Fαβ Uβ
c = speed of light
m = "rest" mass
e = charge
a = sqr(1 - v2/c2)
v2 = vx2 + vy2 + vz2
dτ = dt a (proper time)
momentum 4 vector : Pα = [mc/a , mvx/a , mvy/a , mvz/a ]
velocity 4 vector : Uβ = [c/a , vx/a , vy/a , vx/a ]
electromagnetic tensor matrix Fαβ =
| 0 -Ex/c -Ey/c -Ez/c |
| Ex/c 0 -Bz By |
| Ey/c Bz 0 -Bx |
| Ez/c -By Bx 0 |
Expending Pα= e Fαβ Uβ we get
- for P0 :
d (m c/a) / dt = - e/(c a) (Ex vx+ Ey vy + Ez vz)
m c / a' = - e/(c a) (Ex vx+ Ey vy + Ez vz) + m c / a
- for P1 :
d (m vx)/(a dt) = e/a (Ex- Bz vy + By vz)
m v'x/a' = (e/a) (Ex- Bz vy + By vz) dt + m vx/a
- for P2 :
d (m vy)/(a dt) = (e/a) (Ey+ Bz vx - Bx vz)
m v'y/a' = (e/a) (Ey + Bz vx - Bx vz) dt + m vy/a
- for P3 :
d (m vz)/(a dt) = (e/a) (Ez- By vx + Bx vy)
m v'z/a' = (e/a) (Ez - By vx + Bx vy) dt + m vz/aAll up:
(0) m c/a' = - e/(c a) (Ex vx+ Ey vy + Ez vz) + m c / a = D
(1) m v'x/a' = (e/a) (Ex- Bz vy + By vz) dt + m vx/a = A
(2) m v'y/a' = (e/a) (Ey + Bz vx - Bx vz) dt + m vy/a = B
(3) m v'z/a' = (e/a) (Ez - By vx + Bx vy) dt + m vz/a = C
So:
A/v'x = B/v'y = C/v'z = D/c = m/a'
Are there any mistakes here ?
v'x = A c / D
v'y = B c / D
v'z = C c / D
a' = m c / D
where the new U'β = [c/a' , v'x/a' , v'y/a' , v'z/a'] and a' = sqr(1 - v'2/c2)
Thanks.
Pα= e Fαβ Uβ
c = speed of light
m = "rest" mass
e = charge
a = sqr(1 - v2/c2)
v2 = vx2 + vy2 + vz2
dτ = dt a (proper time)
momentum 4 vector : Pα = [mc/a , mvx/a , mvy/a , mvz/a ]
velocity 4 vector : Uβ = [c/a , vx/a , vy/a , vx/a ]
electromagnetic tensor matrix Fαβ =
| 0 -Ex/c -Ey/c -Ez/c |
| Ex/c 0 -Bz By |
| Ey/c Bz 0 -Bx |
| Ez/c -By Bx 0 |
Expending Pα= e Fαβ Uβ we get
- for P0 :
d (m c/a) / dt = - e/(c a) (Ex vx+ Ey vy + Ez vz)
m c / a' = - e/(c a) (Ex vx+ Ey vy + Ez vz) + m c / a
- for P1 :
d (m vx)/(a dt) = e/a (Ex- Bz vy + By vz)
m v'x/a' = (e/a) (Ex- Bz vy + By vz) dt + m vx/a
- for P2 :
d (m vy)/(a dt) = (e/a) (Ey+ Bz vx - Bx vz)
m v'y/a' = (e/a) (Ey + Bz vx - Bx vz) dt + m vy/a
- for P3 :
d (m vz)/(a dt) = (e/a) (Ez- By vx + Bx vy)
m v'z/a' = (e/a) (Ez - By vx + Bx vy) dt + m vz/aAll up:
(0) m c/a' = - e/(c a) (Ex vx+ Ey vy + Ez vz) + m c / a = D
(1) m v'x/a' = (e/a) (Ex- Bz vy + By vz) dt + m vx/a = A
(2) m v'y/a' = (e/a) (Ey + Bz vx - Bx vz) dt + m vy/a = B
(3) m v'z/a' = (e/a) (Ez - By vx + Bx vy) dt + m vz/a = C
So:
A/v'x = B/v'y = C/v'z = D/c = m/a'
Are there any mistakes here ?
v'x = A c / D
v'y = B c / D
v'z = C c / D
a' = m c / D
where the new U'β = [c/a' , v'x/a' , v'y/a' , v'z/a'] and a' = sqr(1 - v'2/c2)
Thanks.