Why Do the Equations E.v = 0 and E + v.B = 0 Hold in the Rest Frame?

[deadringer]

Assuming the Lorentz force law and also that in the rest frame of the particle the 3 acceleration is zero, we need to explain why the following equations hold:

E.v = 0 and E + v.B = 0

where v is the velocity.

I think this is because g(A,A) = -a squared is invariant. Therefore if a=0, I think this means that A must equal zero in every frame. Is this true, or can A be non zero and we get g(A,A) = 0 (i.e A is null).

 

[Meir Achuz]

I am not sure what you mean by A and a, but acceleration is not a 4-vector, and has complicated LT properties.
In the rest frame for a=0, E=0, and B is unknown.
Since E=0, E' in a system moving with velocity v is given by
$${\vec E}=\gamma{\vec v}\times{\vec B}$$, so
$${\vec v}\cdot{\vec E'}=0.$$

 

[Meir Achuz]

$$E=|{\vec E'}|=\gamma v B$$,
but $${\vec v}\cdot{\vec B'}=v B$$.

Therefore $$E'+{\vec v}\cdot{\vec B'}$$
does not equal zero.