Contracting a Levi-Civita tensor with vectors

Question

I have the following vectors:

p3 = {Subscript[m, τ]/√(1 - β^2), (Subscript[m, τ] β Sin[θ])/√(1 - β^2), 
   0, (Subscript[m, τ] β Cos[θ])/√(1 - β^2)};

p4 = {Subscript[m, τ]/√(1 - β^2), -((Subscript[m, τ] β Sin[θ])/√(1 - β^2)), 
   0, -((Subscript[m, τ] β Cos[θ])/√(1 - β^2))};

i3n = {0, 0, 1, 0};

i3r = {0, -Cos[θ], 0, Sin[θ]};

I want to contract them with a Levi Civita tensor of order 4 to get a scalar value.This is what i tried

Sum[LeviCivitaTensor[4][[μ, ν, α, β]] p3[[α]] p4[[β]] i3n[[μ]] i3r[[ν]], 
 {μ, 1, 4}, {ν, 1, 4}, {α, 1, 4}, {β, 1, 4}]

But I get errors that $\frac{1}{0}$ is encountered. How do I fix it? I am at a loss please help.

Answer 1

You are using beta as both the normalized relativistic velocity and the index symbol.

Sum[
 LeviCivitaTensor[4][[a, b, c, d]] p3[[a]] p4[[b]] i3n[[c]] i3r[[d]]
 , {a, 1, 4}
 , {b, 1, 4}
 , {c, 1, 4}
 , {d, 1, 4}]

Gives the same result as @Daniel Huber solution

LeviCivitaTensor[4] . p3 . p4 . i3n . i3r