Entropy of the Quantum Scalar Field in Static Black Holes

The quantum corrections to the entropy of staticblack holes are investigated by two methods: the brickwall method of 't Hooft and the Euclidean path integralapproach of Gibbons and Hawking. Two general formulas for the entropy are obtained and someexamples are considered. It is shown that if thecontribution from the vacuum surrounding the system isignored, then the two approaches give the same results. It is found that the entropy of the quantumscalar field in a general static black hole consists oftwo parts: a quadratically divergent term which takes ageometric character and a logarithmically divergent term which is not proportional to the horizonarea. The logarithmically divergent term, in general,depends on the black hole characteristics (inparticular, the whole entropy is determined only by this term for some extreme cases) and thereforecannot be neglected as a nonessential additive constant.The renormalization of the entropy is also discussedbriefly.