Example sheet 1 : Conserved charges in Einstein gravity

(* This example uses RG&TC-Code and SurfaceCharges-Code *)

Schwarzshild spacetime

metric = (({{-1+2GN M/r, 0, 0, 0}, {0, 1/(1-2 GN M/r), 0, 0}, {0, 0, (r^2), 0}, {0, 0, 0, (r^2) ((Sin[θ])^2)}}));

coord = {t,r,θ,ϕ};

xiU  = {1,0,0,0};

perturbationdd = D[metric,M] δM

assumptionsmetric = {r>0,M≥0,θ≥0,θ ≤π};

RGtensors[metric,coord,{0,0}];

Mass =ChargesEinstein[xiU,perturbationdd,-1,{1,2}]/GN

Integrate[Simplify[Mass,assumptionsmetric],{θ,0,π},{ϕ,0,2π}]

AngularMomentum = ChargesEinstein[{0,0,0,1},perturbationdd,-1,{1,2}]/GN

Myers-Perry 5d doubly rotating black hole

coord = {r,x,ψ,ϕ,t}

metric = {{(((b^2)+(r^2)+(a-b) (a+b) (x^2))/(-2 m+((((a^2)+(r^2)) ((b^2)+(r^2)))/(r^2)))),0,0,0,0},{0,(((b^2)+(r^2)+(a-b) (a+b) (x^2))/(1-(x^2))),0,0,0},{0,0,(x^2) ((r^2)+(b^2) (1+((2 m (x^2))/((b^2)+(r^2)+(a-b) (a+b) (x^2))))),-((2 a b m (x^2) (-1+(x^2)))/((b^2)+(r^2)+(a-b) (a+b) (x^2))),-((2 b m (x^2))/((b^2)+(r^2)+(a-b) (a+b) (x^2)))},{0,0,-((2 a b m (x^2) (-1+(x^2)))/((b^2)+(r^2)+(a-b) (a+b) (x^2))),(1-(x^2)) ((a^2)+(r^2)-((2 (a^2) m (-1+(x^2)))/((b^2)+(r^2)+(a-b) (a+b) (x^2)))),((2 a m (-1+(x^2)))/((b^2)+(r^2)+(a-b) (a+b) (x^2)))},{0,0,-((2 b m (x^2))/((b^2)+(r^2)+(a-b) (a+b) (x^2))),((2 a m (-1+(x^2)))/((b^2)+(r^2)+(a-b) (a+b) (x^2))),-1+((2 m)/((b^2)+(r^2)+(a-b) (a+b) (x^2)))}}

perturbationdd = D[metric,m] δm +D[metric,a] δa +D[metric,b] δb ;

assumptionsmetric = {r>0,m≥0,a≥0,b≥0,x≥-1,x ≤1};

RGtensors[metric,coord,{0,0}];

coord

δMass = Simplify[ChargesEinstein[{0,0,0,0,1},perturbationdd,-1,{5,1}]/GN,assumptionsmetric]

Integrate[δMass,{ϕ,0,2π},{ψ,0,2π}]

Series[%,{r,∞,0}]

Integrate[Normal[%],{x,-1,1}]

Mass = Integrate[%/.{δm-> 1},m]

δAngularMomentum = ChargesEinstein[{0,0,0,-1,0},perturbationdd,-1,{5,1}]/GN;

Series[δAngularMomentum,{r,∞,1}]

Integrate[Normal[%],{x,-1,1}]

AngularMomentum =Integrate[% /.{δa-> 1,δm-> 0},a]