# How does the connection of constant-phase surface in Schwarzschild spacetime work?

I am stuck while following the calculation of constant-phase surface of Hawking radiation in Hawking's original paper (Particle Creation by Black Holes, S.W. Hawking, Commun. Math. Phys. 43, 199 (1975)).

My problem is following while calculating $$p_\omega ^{(2)}$$ below figure 4. Hawking tells that if we parallely transport $$n^a$$ along the generator of the event horizon then $$-\epsilon n^a$$ will connect a surface of constant phase from the original point x on the event horizon. I cannot get around this idea using any equation ─ so, how should I proceed?

• @EmilioPisanty Thanks for the edit. It was my first time citing a problem from a paper so I will take note of this format while citing a paper. – aitfel Aug 29 '19 at 18:25
• which part exactly are you struggling with? The concept of parallel transport or the notion of constant phase? Please elaborate a bit. – Rudyard Sep 5 '19 at 14:47