# Lorentz Invariance proof

1. Sep 15, 2016

### nmsurobert

1. The problem statement, all variables and given/known data
I'm asked to prove that Et - p⋅r = E't' - p'⋅r'

2. Relevant equations
t = γ (t' + ux')
x = γ (x' + ut')
y = y'
z = z'

E = γ (E' + up'x)
px = γ (p'x + uE')
py = p'y
pz = p'z

3. The attempt at a solution
Im still trying to figure out 4 vectors. I get close to the solution but I have some values hanging around.
For the first two terms, E and t, i just multiple them out.
(γ (E' + up'x))(γ (t' + ux') )

Next I work with the p and r. The way i understand them is that that p is equal to the three different equations i have listed for px,py, and pz. And the same thing for r but with x,y, and z. Im guessing that because i don't a lorentz transformation formula for just p or r.

I then multiply px with x, py with y, and pz with z. adding the products of each along the way.

am i on the right track? I start canceling terms but ultimately I'm left with a γ2ut'uE' and γ2ux'up'. I'm also left with a bunch of γ2's.

2. Sep 15, 2016

### nmsurobert

i just figured it out. the squared gamma factor helps me get rid of the left over terms. i forgot that gamma was something more than just a variable.