Example sheet 3 : Energy of black holes in pp-wave spacetimes

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

Encoding of the pp - wave solution in 6 dimensions

(* Based on the paper hep-th/0306131  *)

(* In 6 dimensions *)

n = 6

f[r_] := 1 - M/r^(n-4)
k[r_] := 1 + β^2 M /r^(n-6)
αp = Sqrt[8/(n-2)];
nangles = n-3;

gtt = - f[r](1+β^2 r^2)/k[r];
gty = -β^2 r^2 f[r]/k[r];
gyy = 1-β^2 r^2/k[r];
grr = 1/f[r];
gss = -r^4 β^2 (1-f[r])/k[r];
phiString = -1/2 Log[k[r]];
ϕ = αp phiString;
Bts = β r^2 / k[r] f[r];
Bys = β r^2 / k[r] ;

(* dimension dependent subroutine : case n = 6 *)
If[n == 6,
Angles = {θ1,θ2,θ3};
VectX = {Cos[θ1],Sin[θ1] Cos[θ2], Sin[θ1]  Sin[θ2]Cos[θ3],Sin[θ1]  Sin[θ2]Sin[θ3]};
σ = Simplify[Table[VectX[[1]] D[VectX[[2]], Angles[[i]]] - VectX[[2]] D[VectX[[1]], Angles[[i]]]+VectX[[3]] D[VectX[[4]], Angles[[i]]] - VectX[[4]] D[VectX[[3]], Angles[[i]]],{i,1,nangles}]];
sA = r^2{1,Sin[θ1]^2,Sin[θ1]^2 Sin[θ2]^2};]

(* dimension dependent subroutine : case n = 5 *)
If[n == 8,
Angles = {θ1,θ2,θ3,θ4,θ5};
VectX = {Cos[θ1],Sin[θ1] Cos[θ2], Sin[θ1]  Sin[θ2]Cos[θ3],Sin[θ1]  Sin[θ2]Sin[θ3]Cos[θ4],Sin[θ1]  Sin[θ2]Sin[θ3]Sin[θ4]Cos[θ5],Sin[θ1]  Sin[θ2]Sin[θ3]Sin[θ4]Sin[θ5]};
σ = Simplify[Table[ VectX[[1]] D[VectX[[2]], Angles[[i]]] - VectX[[2]] D[VectX[[1]], Angles[[i]]]+VectX[[3]] D[VectX[[4]], Angles[[i]]] - VectX[[4]] D[VectX[[3]], Angles[[i]]]+VectX[[5]] D[VectX[[6]], Angles[[i]]] - VectX[[6]] D[VectX[[5]], Angles[[i]]],{i,1,nangles}]];
sA = r^2{1,Sin[θ1]^2,Sin[θ1]^2 Sin[θ2]^2,Sin[θ1]^2 Sin[θ2]^2 Sin[θ3]^2,Sin[θ1]^2 Sin[θ2]^2 Sin[θ3]^2 Sin[θ4]^2};]

Assumption = Join[Table[Angles[[i]] > 0 ,{i,1,nangles}],Table[Angles[[i]] < π ,{i,1,nangles-1}],{Angles[[nangles]] < 2 π},{r>0,M>0,β>0}]

AngleMatrix = gss Outer[Times,σ ,σ] +  DiagonalMatrix[sA]

coord = Flatten[{t,r,Angles,y}]

metricS = ConstantArray[0,{n,n}];
metricS[[1,1]] = gtt; 
metricS[[2,2]] = grr;
metricS[[1,n]] = gty;
metricS[[n,1]] = gty;
metricS[[n,n]] = gyy;
For[i=1,i⩽nangles,i++,For[j=1,j⩽nangles,j++,metricS[[2+i,2+j]] = AngleMatrix[[i,j]]]]
metric = metricS Exp[-4 phiString /(n-2)];

B = ConstantArray[0,{n,n}];
For[i=1,i⩽nangles,i++,B[[1,2+i]] = Bts σ[[i]]; B[[2+i,1]] = - Bts σ[[i]];B[[n,2+i]] = Bys σ[[i]];  B[[2+i,n]] =- Bys σ[[i]];  
]

metric // MatrixForm

Charges

(* Charge contribution from Einstein term and scalar field, based on hep-th/0703004 *)

(* Area[n_] := 2 π^((n-2)/2)/Gamma[(n-2)/2];*)

RGTensor

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

Charges

(* Infinitesimal perturbation of the pp-wave metric : *)

αp = Sqrt[8/(n-2)];

δmetric = Simplify[D[metric,M] δM + D[metric,β] δβ ];
δB = Simplify[D[B,M] δM + D[B,β] δβ ];
δϕ = Simplify[D[ϕ,M] δM + D[ϕ,β] δβ ];

(* Symmetry parameters : *)

xi =  ConstantArray[0,{n}]; 
xi[[1]] = 1;

λ = ConstantArray[0,{n}];

termEinstein = Simplify[ChargesEinstein[xi,δmetric,-1,{1,2}],Assumption]

Series[termEinstein,{r,∞,0}]

(* Charge contribution from 2-form *)

contrib2form = Simplify[ChargespForm[{xi,λ},{B,ϕ},{δmetric,δB,δϕ},-1,{1,2},{Exp[- αp ϕUneval],ϕUneval}],Assumption]

(* Contribution Scalar *)

ϕjd = Table[D[ϕ,coord[[i]]] ,{i,1, n}];ϕjU = Raise[ϕjd,1];
ContribScal[index1_,index2_]:=-1 /(16 Pi)Sqrt[-detg]δϕ(xi[[index1]] ϕjU[[index2]]-xi[[index2]] ϕjU[[index1]]) ;
contribScalar = ContribScal[1,2]

localδMass = Simplify[termEinstein +  contrib2form+contribScalar ]

δMass = Simplify[Integrate[localδMass /. {},{θ1,0,π},{θ2,0,π}] 2 π,Assumption]

(* which is the expected result *)