Example sheet 4 : Charges of black branes in "asymptotically non-relativistic spacetimes"

(* Based on solutions provided in 0807.1099, 0807.1111, 0809.2020 *)

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

Encoding the 4d black brane solution

(* Solutions of 0809.2020 in 4d in coordinates (r, t, ξ, x) with r_{paper} = 1/r, u_{paper} = t, v_{paper} = -ξ *)

coord = {r,t,ξ,x}
z=2
dim  = Length[coord]
dd=dim-3;
coefa = 2(dd+2)/(dd+1)
f[r_]=1-r^(dd+2)/r0^(dd+2)
h[r_]=1 + β^2 r^dd/r0^(dd+2)

grr=Simplify[h[r]^(1/(dd+1))/ r^2/f[r]]
gtt=Simplify[r^(-2z) h[r]^(-dd/(dd+1)) ((f[r]-1)^2/(4(h[r]-1))-f[r])]
gξξ=Simplify[ h[r]^(-dd/(dd+1))(h[r]-1)]
gtξ=-Simplify[- 1/2 r^(-2) h[r]^(-dd/(dd+1))(f[r]+1)]
gxx = Simplify[h[r]^(1/(dd+1))/ r^2]

metric =L^2{{grr,0,0,0},{0,gtt,gtξ,0},{0,gtξ,gξξ,0},{0,0,0,gxx}}

vector =Simplify[ L {0,(1+f[r])/2/h[r] r^(-2),-(1-h[r])/h[r],0}]

phi =Simplify[ -1/2 Log[h[r]]]

Limit[metric, r0->∞]

SeriesWithNTerms[metric ,1,{r,0}]

RGtensors[SeriesWithNTerms[metric,{r,0},5],coord,{0,0}];

Computation of the charges

hdd = Simplify[Dt[metric ]/.{Dt[r]-> 0,Dt[L]-> 0,Dt[β] -> δβ,Dt[r0]-> δr0}]
ad =Simplify[ Dt[vector]/.{Dt[r]-> 0,Dt[L]-> 0,Dt[β] -> δβ,Dt[r0]-> δr0}]
varphi = Simplify[Dt[phi]/.{Dt[r]-> 0,Dt[L]-> 0,Dt[β] -> δβ,Dt[r0]-> δr0}]

coord

preNumberOp =Simplify[ChargesEinsteinMaxwellScalarAsympt[{{0,0,-1,0},0},{vector,phi},{hdd,ad,varphi},-1,{1,2},{coefa,Exp[-coefa ϕUneval],ϕUneval},{r,0},2] ,L>0]

δNumberOp = Simplify[Normal[preNumberOp]]

IntegrabilityCondition = Simplify[D[Coefficient[δNumberOp,δr0],β] - D[Coefficient[δNumberOp,δβ],r0]]

(* Obeyed. So, one can integrate the expression to get the charge : *)

NumberOp = Integrate[δNumberOp /.{δr0-> 0,δβ-> 1},β]

(* This is the expected result *)

CheckZero =Simplify[ D[NumberOp,β]δβ+D[NumberOp,r0]δr0-δNumberOp ]

δHamiltonian  = Simplify[ChargesEinsteinMaxwellScalarAsympt[{{0,1,0,0},0},{vector,phi},{hdd,ad,varphi},-1,{1,2},{coefa,Exp[-coefa ϕUneval],ϕUneval},{r,0},3],L>0 ]

Hamiltonian = Simplify[Integrate[Normal[δHamiltonian] /.{δr0-> 1},r0]]

(* This is the expected result *)