Total Conserved Charges of Kerr-Newman Spacetimes in Gravity Theory Using a Poincaré Gauge Version of the Teleparallel Equivalent of General Relativity

Table 1

Comparison between the results of the total conserved charges and the other definitions.

Spacetime	Conserved charge in the tetrad gravity	Regularized conserved charge	Invariant conserved charge
Definition of conserved charge	𝒬(𝜉,𝜗)=∫𝜕𝑆𝜉^𝛼𝐻𝛼=1∫2𝜅^𝜕𝑆𝜉^𝛼Γ^𝛽𝛾∧𝜂𝛼𝛽𝛾	1𝒬(𝜉,𝜗)=∫2𝜅^𝜕𝑆𝜉^𝛼(Γ^𝛽𝛾−Γ^𝛽𝛾)∧𝜂𝛼𝛽𝛾	𝒬11(𝜉,𝜗)=∫2𝜅^𝜕𝑆[𝜉^𝛼(Γ^𝛽𝛾−Γ^𝛽𝛾)∧𝜂𝛼𝛽𝛾+Ξ^𝛼𝛽𝜂𝛼𝛽]
First coframe	𝑞(𝑀−^22𝑟)𝜁^0𝑟−(3+𝑀)𝑎𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−𝑀𝑎𝜁^3
Second coframe	𝑞−(𝑟−2𝑀−^2𝑟+𝑎^24𝑟)𝜁^0−𝑎𝑀𝜁^3	𝑞(𝑀−^2+𝑎2𝑟6𝑟^2)𝜁^0−(2𝑀+𝑟)𝑎3𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−𝑀𝑎𝜁^3
Third coframe	𝑞(𝑀−^22𝑟)𝜁^0−4𝑀𝑎3𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−(4𝑀−𝑟3)𝑎𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−𝑀𝑎𝜁^3
Fourth coframe	𝑞(𝑀−^22𝑟)𝜁^0−(2𝑀−𝑟)𝑎3𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−2𝑀𝑎3𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−𝑀𝑎𝜁^3
Fifth coframe	𝑞(𝑀−^22𝑟)𝜁^0−(𝑀−𝑟3)𝑎𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−𝑀𝑎3𝜁^3	𝑞(𝑀−^22𝑟)𝜁^0−𝑀𝑎𝜁^3