Mass Shell in Light Cone Coordinates

Question

I'm reading Zweibach's introduction to string theory, and don't understand one of his claims.

He defined the mass shell to be the set of points in momentum space s.t. $p^2+m^2 = 0$. Then the physical mass shell is the subset of points where $E=p^0>0$. He claims that in light cone coordinates this may be parameterised by the transverse momenta $\vec{p}_T$ and the light cone momenta $p^+$ for which $p^+>0$.

Why the condition $p^+>0$? Surely $p^0>0$ doesn't immediately give us this? Shouldn't there be some $p^1$ dependence in the mix somewhere?

Many thanks in advance!

Answer 1

Zwiebach explains the inequality $p^{\pm}>0$ e.g. in Section 2.5 of his book A First course in String Theory:

$$\sqrt{2} p^{\pm}~=~p^0\pm p^1~=~\sqrt{\vec{p}\cdot\vec{p}+(mc)^2}\pm p^1~>~|\vec{p}|\pm p^1~\geq~0~ $$

if $m>0$.