Can anyone help me find any mistake in this expansion ? (I've asked it also in other places but I got no answer))(adsbygoogle = window.adsbygoogle || []).push({});

P_{α}= e F_{αβ}U^{β}

c = speed of light

m = "rest" mass

e = charge

a = sqr(1 - v^{2}/c^{2})

v^{2}= v_{x}^{2}+ v_{y}^{2}+ v_{z}^{2}

dτ = dt a (proper time)

momentum 4 vector : P_{α}= [mc/a , mv_{x}/a , mv_{y}/a , mv_{z}/a ]

velocity 4 vector : U^{β}= [c/a , v_{x}/a , v_{y}/a , v_{x}/a ]

electromagnetic tensor matrix F_{αβ}=

| 0 -Ex/c -Ey/c -Ez/c |

| Ex/c 0 -Bz By |

| Ey/c Bz 0 -Bx |

| Ez/c -By Bx 0 |

Expending P_{α}= e F_{αβ}U^{β}we get

- for P_{0}:

d (m c/a) / dt = - e/(c a) (E_{x}v_{x}+ E_{y}v_{y}+ E_{z}v_{z})

m c / a' = - e/(c a) (E_{x}v_{x}+ E_{y}v_{y}+ E_{z}v_{z}) + m c / a

- for P_{1}:

d (m v_{x})/(a dt) = e/a (E_{x}- B_{z}v_{y}+ B_{y}v_{z})

m v'_{x}/a' = (e/a) (E_{x}- B_{z}v_{y}+ B_{y}v_{z}) dt + m v_{x}/a

- for P_{2}:

d (m v_{y})/(a dt) = (e/a) (E_{y}+ B_{z}v_{x}- B_{x}v_{z})

m v'_{y}/a' = (e/a) (E_{y}+ B_{z}v_{x}- B_{x}v_{z}) dt + m v_{y}/a

- for P_{3}:

d (m v_{z})/(a dt) = (e/a) (E_{z}- B_{y}v_{x}+ B_{x}v_{y})

m v'_{z}/a' = (e/a) (E_{z}- B_{y}v_{x}+ B_{x}v_{y}) dt + m v_{z}/a

All up:

(0) m c/a' = - e/(c a) (E_{x}v_{x}+ E_{y}v_{y}+ E_{z}v_{z}) + m c / a = D

(1) m v'_{x}/a' = (e/a) (E_{x}- B_{z}v_{y}+ B_{y}v_{z}) dt + m v_{x}/a = A

(2) m v'_{y}/a' = (e/a) (E_{y}+ B_{z}v_{x}- B_{x}v_{z}) dt + m v_{y}/a = B

(3) m v'_{z}/a' = (e/a) (E_{z}- B_{y}v_{x}+ B_{x}v_{y}) dt + m v_{z}/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}/c^{2})

Thanks.

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# I Electromagnetic field acting on a charged particle

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