# what is the physical meaning of the 4-vector inner product?

Given two 4-vectors, $$P^\mu, Q_\mu$$, what is the physical meaning of the inner product, $$P^\mu Q_\mu = a$$? Specifically I would like to know what it means when the vectors are orthogonal ($$a=0$$) with respect to this inner product.

I know that this product is Lorentz invariant. I also know when the inner product of a four-vector with itself is zero, then the momentum is on the light/null cone.

If you have a way of defining the norm (squared magnitude) of vectors, then you automatically get an inner product from the relation $$(P+Q)^2=P^2+Q^2+2P\cdot Q$$. It works that way for the relativistic inner product, and it also works that way for the quantum-mechanical inner product.
The norm of a timelike displacement vector is the square of the time measured by a clock that travels inertially along that vector. The norm of a spacelike displacement vector is minus the square of the proper length along that vector (i.e., the length of a ruler that's at rest in the frame of an observer who says the vector connects simultaneous points). (This is for the $$+---$$ metric. You can also flip the signs if you prefer $$-+++$$.)