What is the Relationship Between Energy and Time in Different Frames?

[chewwy]

something which seems so fundamental that i can't find anywhere that derives it is the following:

E' = gamma * (E - p.v)

where E is the energy in one frame, p the momentum, v the relative velocity of the other frame, and E' the energy in the other frame.

i.e. energy transforms like time. i can't quite see where this comes from though.. help?

 

[jtbell]

It comes from the fact that energy is the "zeroth" component of the momentum four-vector $(p_0, p_1, p_2, p_3) = (E/c, p_x, p_y, p_z)$, just like time is the "zeroth" component of the position four-vector $(x_0, x_1, x_2, x_3) = (ct, x, y, z)$.

 

[tiny-tim]

Welcome to PF!

Hi chewwy! Welcome to PF!

(have a gamma: γ )

energy transforms like time. i can't quite see where this comes from though.. help?

Yes, it's because (E,p) is defined as the derivative of (t,x),

so E' = γE - γ(p.v),

just as t' = γt - γ(x.v)

 

[chewwy]

It comes from the fact that energy is the "zeroth" component of the momentum four-vector $(p_0, p_1, p_2, p_3) = (E/c, p_x, p_y, p_z)$, just like time is the "zeroth" component of the position four-vector $(x_0, x_1, x_2, x_3) = (ct, x, y, z)$.

okie doke. thanks everyone!