Surface gravity is denoted by \(\kappa_1\) and is defined by where \(r_+\) is the horizon radius, \(r_+=R_S\) for the Schwarzschild spacetime, \(u^t\) is the time component of the 4-velocity. The 4-velocity of a free particle instantaneously at rest in the Schwarzschild spacetime: The only component of the 4-acceleration different from zero is \(a_r\)
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Grøn, Ø. (2009). Black Holes. In: Lecture Notes on the General Theory of Relativity. Lecture Notes in Physics, vol 772. Springer, New York, NY. https://doi.org/10.1007/978-0-387-88134-8_9
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