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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?

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  • $\begingroup$ @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. $\endgroup$ – aitfel Aug 29 '19 at 18:25
  • $\begingroup$ which part exactly are you struggling with? The concept of parallel transport or the notion of constant phase? Please elaborate a bit. $\endgroup$ – Rudyard Sep 5 '19 at 14:47

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