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What does $p^2_theta$ mean in the context of a photon described by its four momentum?

Physics Asked on August 11, 2021

The paper Null Geodesics of the Kerr Exterior shows the equation

$$Q = p_ theta^2 – cos^2theta(a^2 p_t^2 – p_phi^2 csc^2theta),$$

where $p$, the four-momentum of a photon, has superscript 2 and subscript $theta$. I understand the 2 symbolizes the third part of the momentum, but what does the $theta$ part mean? If it is the $theta$ of the Boyer-Lindquist coordinate, does $p_theta^2$ mean $p^2times(theta text{ of the position of the photon})$?

One Answer

The answer is a bit boring: it's just a square. It's the covariant component $p_theta$, squared. I guess they thought their readers would be familiar enough with the subject to understand it, but there's really no other option anyway. Since $theta$ can only be a subscript, 2 can only be a square.

Correct answer by Javier on August 11, 2021

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