How do I prove Lorentz Invariance using 4-vectors?

[nmsurobert]

Homework Statement

I'm asked to prove that Et - p⋅r = E't' - p'⋅r'

Homework Equations

t = γ (t' + ux')
x = γ (x' + ut')
y = y'
z = z'

E = γ (E' + up'_x)
p_x = γ (p'_x + uE')
p_y = p'_y
p_z = p'_z

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 p_x,p_y, and p_z. And the same thing for r but with x,y, and z. I am guessing that because i don't a lorentz transformation formula for just p or r.

I then multiply p_x with x, p_y with y, and p_z 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.

 

[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.