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It is well known that extremal black holes do not Hawking radiate, which isusually realized by taking an extremal limit from the nonextremal case.However, one cannot perceive the same phenomenon using the Bogoliubovtransformation method starting from an extremal black hole itself, i.e.,without the limiting case consideration. In that case, the Bogoliubovcoefficients do not satisfy the required normalization condition. In canonicalformulation, which closely mimics the Bogoliubov transformation method, one canconsistently reproduce the vanishing number density of Hawking quanta for anextremal Kerr black hole. In this method, the relation between the spatialnear-null coordinates, imperative in understanding the Hawking effect, wasapproximated into a sum of linear and inverse terms only. In the present work,we first show that one can reach the same conclusion in canonical formulationeven with the complete relationship between the near-null coordinates, whichcontains an additional logarithmic term. It is worth mentioning that in thenonextremal case, a similar logarithmic term alone leads to the thermal Hawkingradiation. Secondly, we study the case with only the inverse term in therelation (i.e., when the spatial near-null coordinates associated to the pastand future observers are inversely related to each other) to understand whetherit is the main contributing term in vanishing number density. Third, for aqualitative realization, we consider a simple thought experiment to understandthe corresponding Hawking temperature and conclude that the inverse term indeedplays a crucial role in the vanishing number density.

Hawking effect in an extremal Kerr black hole spacetime