Physical Quantities of Reissner-Nordström Spacetime with Arbitrary Function and Regularized Procedure

Table 1

Comparison between the results of the arbitrary function ℑ(𝑟), other constants, and the associated energy.

Spacetime	Arbitrary function 𝐵(𝑟)	The constants 𝑐1 and 𝑐2	Energy	Total conserved charge	Regularized conserved charge
	ℑ(𝑟)=0 or	𝑐1=0; ℑ(𝑟)=const.	𝐸=𝑀	𝒬(𝜉,𝜗)=𝑀𝜁^0	𝒬(𝜉,𝜗)=𝑀𝜁^0
Schwarzschild	ℑ(𝑟)=0 or; ℑ(𝑟)=const.	𝑐1=𝑀	𝑀𝐸=3	𝒬(𝜉,𝜗)=5𝑀3𝜁^0	𝒬(𝜉,𝜗)=𝑀𝜁^0
	ℑ(𝑟)=∫1−1−(2𝑐1/𝑟)𝑟𝑑𝑟	𝑐1≠0	𝐸=𝑀	𝒬(𝜉,𝜗)=𝑀𝜁^0	𝒬(𝜉,𝜗)=𝑀𝜁^0
	ℑ(𝑟)=0 or; ℑ(𝑟)=const.	𝑐1=0 and; 𝑐2=0	𝑞𝐸=𝑀−^22𝑅	𝒬(𝜉,𝜗); ≈𝑞𝑀−^2+𝑀^2𝜁2𝑟^0; 1+𝑂𝑟^2	𝒬(𝜉,𝜗); ≈𝑞𝑀−^2+𝑀^2𝜁2𝑟^0; 1+𝑂𝑟^2
Reissner-Nordström	ℑ(𝑟)=0 or; ℑ(𝑟)=const.	𝑐1=𝑀 and; 𝑐2=𝑞	1𝐸=3𝑞𝑀−^22𝑅	𝒬(𝜉,𝜗); ≈5𝑀3−5𝑞^2+𝑀^2𝜁6𝑟^0; 1+𝑂𝑟^2	𝒬(𝜉,𝜗); ≈𝑞𝑀−^2+𝑀^2𝜁2𝑟^0; 1+𝑂𝑟^2
	ℑ(𝑟)=∫1−1−(2𝑐1/𝑟)+(𝑐2^2/𝑟^2)𝑟𝑑𝑟	𝑐1≠0 and; 𝑐2≠0	𝑞𝐸=𝑀−^22𝑅	𝒬(𝜉,𝜗); ≈𝑞𝑀−^2+𝑀^2𝜁2𝑟^0; 1+𝑂𝑟^2	𝒬(𝜉,𝜗); ≈𝑞𝑀−^2+𝑀^2𝜁2𝑟^0; 1+𝑂𝑟^2