Initial assumptions

$Assumptions={r>0 ,rp>0,t>0,θ1>θ>0,z>0, u>0,θp>0,α>θ>0};

Wedge Tbar pieces for general alpha, in polar coordinates

wedgeTbar1=((Csch[u] Sinh[((π u)/(2 α))])/(2 π r rp (4 α Cos[((π (θ+θp))/(2 α))]+4 α Cosh[((π u)/(2 α))])));

wedgeTbar2=((Csch[u] Sinh[((π u)/(2 α))])/(8 π r rp α (-Cos[((π (θ+θp))/(2 α))]+Cosh[((π u)/(2 α))])));

wedgeTbar3 = ((-Csch[u] Sinh[((π u)/(2 α))])/(2 π r rp (4 α Cos[((π (θ-θp))/(2 α))]+4 α Cosh[((π u)/(2 α))])));

wedgeTbar4=((-Csch[u] Sinh[((π u)/(2 α))])/(8 π r rp α (-Cos[((π (θ-θp))/(2 α))]+Cosh[((π u)/(2 α))])));

Definition of u

u=ArcCosh[(2r*rp)^(-1)(r^2+rp^2+t^2+(z-zp)^2)];

For the time being, we specialize to alpha=pi/2.

zp:=0;

Wedge Tbar pieces for alpha=pi/2, in polar coordinates

wtbar1pi2=Simplify[wedgeTbar1/.α->Pi/2]

wtbar2pi2=Simplify[wedgeTbar2/.α->Pi/2]

wtbar3pi2=Simplify[wedgeTbar3/.α->Pi/2]

wtbar4pi2=Simplify[wedgeTbar4/.α->Pi/2]

Tbar pieces for planes at x=0 and y=0, in Cartesian coordinates

cartTbar1=((1/(2Pi^2))/((t^2)+(((x+xp))^2)+(((y-yp))^2)+(((z-zp))^2)));

cartTbar2=((1/(2Pi^2))/((t^2)+(((x-xp))^2)+(((y+yp))^2)+(((z-zp))^2)));

cartTbar3=-((1/(2Pi^2))/((t^2)+(((x+xp))^2)+(((y+yp))^2)+(((z-zp))^2)));

cartTbar4=-((1/(2Pi^2))/((t^2)+(((x-xp))^2)+(((y-yp))^2)+(((z-zp))^2)));

Tbar pieces for planes at x=0 and y=0, in polar coordinates

pctbar1=Simplify[cartTbar1/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pctbar2=Simplify[cartTbar2/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pctbar3=Simplify[cartTbar3/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pctbar4=Simplify[cartTbar4/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

Do these agree with the wedge pieces at alpha=pi/2?

FullSimplify[wtbar1pi2-pctbar1]

FullSimplify[wtbar2pi2-pctbar2]

FullSimplify[wtbar3pi2-pctbar3]

FullSimplify[wtbar4pi2-pctbar4]

Yes! Good.

Now, we will look at the energy density and perpendicular/radial pressure for each piece, in both cartesian and polar coordinates.

Cartesian T00 and pressures, in Cartesian coordinates

cartrho1=-1/2 (D[cartTbar1,{t,2}])//Simplify

cartrho2=-1/2 (D[cartTbar2,{t,2}])

cartrho3=-1/2 (D[cartTbar3,{t,2}])

cartrho4=-1/2 (D[cartTbar4,{t,2}])

px1=1/8(D[cartTbar1,{x,2}]+D[cartTbar1,{xp,2}]-2D[D[cartTbar1,x],xp])//Simplify

py1=1/8(D[cartTbar1,{y,2}]+D[cartTbar1,{yp,2}]-2D[D[cartTbar1,y],yp]) //Simplify

px2=1/8(D[cartTbar2,{x,2}]+D[cartTbar2,{xp,2}]-2D[D[cartTbar2,x],xp])//Simplify

py2=1/8(D[cartTbar2,{y,2}]+D[cartTbar2,{yp,2}]-2D[D[cartTbar2,y],yp])//Simplify

px3=1/8(D[cartTbar3,{x,2}]+D[cartTbar3,{xp,2}]-2D[D[cartTbar3,x],xp]) //Simplify

py3=1/8(D[cartTbar3,{y,2}]+D[cartTbar3,{yp,2}]-2D[D[cartTbar3,y],yp]) // Simplify

px4=1/8(D[cartTbar4,{x,2}]+D[cartTbar4,{xp,2}]-2D[D[cartTbar4,x],xp]) //Simplify

py4=1/8(D[cartTbar4,{y,2}]+D[cartTbar4,{yp,2}]-2D[D[cartTbar4,y],yp])//Simplify

Txy1=1/8(D[D[cartTbar1,x],y]+D[D[cartTbar1,xp],yp]-D[D[cartTbar1,x],yp]-D[D[cartTbar1,xp],y])//FullSimplify

Txy2=1/8(D[D[cartTbar2,x],y]+D[D[cartTbar2,xp],yp]-D[D[cartTbar2,x],yp]-D[D[cartTbar2,xp],y])FullSimplify

Txy3=1/8(D[D[cartTbar3,x],y]+D[D[cartTbar3,xp],yp]-D[D[cartTbar3,x],yp]-D[D[cartTbar3,xp],y])//FullSimplify

Txy4=1/8(D[D[cartTbar4,x],y]+D[D[cartTbar4,xp],yp]-D[D[cartTbar4,x],yp]-D[D[cartTbar4,xp],y])//FullSimplify

Txy4/.{xp->x,yp->y}

py1/.{xp->x,yp->y}

px2/.{xp->x,yp->y}

py4/.{xp->x,yp->y}

px4/.{xp->x,yp->y}

Cartesian T00 and pressures, in polar coordinates

pcrho1=Simplify[cartrho1/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcrho1d=pcrho1/.{rp->r, θp->θ}

pcrho2=Simplify[cartrho2/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}];

pcrho2d=pcrho2/.{rp->r, θp->θ}

pcrho3=Simplify[cartrho3/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}];

pcrho3d=pcrho3/.{rp->r, θp->θ}

pcrho4=Simplify[cartrho4/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}];

pcrho4d=pcrho4/.{rp->r, θp->θ}

pcpx1=Simplify[px1/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}];

pcpx1d=pcpx1/.{rp->r, θp->θ}

pcpy1=Simplify[py1/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcpy1d=pcpy1/.{rp->r, θp->θ}

pcpx2=Simplify[px2/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}];

pcpx2d=pcpx2/.{rp->r, θp->θ}//Simplify

pcpy2=Simplify[py2/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}];

pcpy2d=pcpy2/.{rp->r, θp->θ}

pcpx3=Simplify[px3/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcpy3=Simplify[py3/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcpx4=Simplify[px4/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcpx4d=pcpx4/.{rp->r, θp->θ}

pcpy4=Simplify[py4/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}];

pcpy4d=pcpy4/.{rp->r, θp->θ}

pcTxy1=Simplify[Txy1/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcTxy2=Simplify[Txy2/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcTxy3=Simplify[Txy3/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcTxy4=Simplify[Txy4/.{x->r Cos[θ],xp->rp Cos[θp],y->r Sin[θ],yp->rp Sin[θp]}]

pcTxy4d=pcTxy4/.{rp->r, θp->θ}

Wedge T00 and pressures, in polar coordinates, for alpha=pi/2

T001pi2=-(1/2)D[wtbar1pi2,{t,2}]

T001pi2d=T001pi2/.{rp->r, θp->θ}//Simplify

T002pi2=-(1/2)D[wtbar2pi2,{t,2}];

T002pi2d=T002pi2/.{rp->r, θp->θ}//Simplify

T003pi2=-(1/2)D[wtbar3pi2,{t,2}];

T003pi2d=T003pi2/.{rp->r, θp->θ}//Simplify

T004pi2=-(1/2)D[wtbar4pi2,{t,2}];

T004pi2d=T004pi2/.{rp->r, θp->θ}//Simplify

Tpp1pi2=(1/(4r))D[wtbar1pi2,r]+(1/(4r^2))(D[wtbar1pi2,{θ,2}]-D[D[wtbar1pi2,θ],θp])//Simplify

Tpp1pi2d=Tpp1pi2/.{rp->r, θp->θ}//Simplify

Tpp2pi2=(1/(4r))D[wtbar2pi2,r]+(1/(4r^2))(D[wtbar2pi2,{θ,2}]-D[D[wtbar2pi2,θ],θp]);

Tpp2pi2d=Tpp2pi2/.{rp->r, θp->θ}//Simplify

Tpp3pi2=(1/(4r))D[wtbar3pi2,r]+(1/(4r^2))(D[wtbar3pi2,{θ,2}]-D[D[wtbar3pi2,θ],θp])//Simplify

Tpp3pi2d=Tpp3pi2/.{rp->r, θp->θ}//Simplify

Tpp4pi2=(1/(4r))D[wtbar4pi2,r]+(1/(4r^2))(D[wtbar4pi2,{θ,2}]-D[D[wtbar4pi2,θ],θp]) //Simplify

Tpp4pi2d=Tpp4pi2/.{rp->r, θp->θ}//Simplify

Trr1pi2=-(1/4)(D[D[wtbar1pi2,rp],r]-D[wtbar1pi2,{r,2}]) //Simplify

Trr1pi2d=Trr1pi2/.{rp->r, θp->θ}//Simplify

Trr2pi2=-(1/4)(D[D[wtbar2pi2,rp],r]-D[wtbar2pi2,{r,2}]);

Trr2pi2d=Trr2pi2/.{rp->r, θp->θ}//Simplify

Trr3pi2=-(1/4)(D[D[wtbar3pi2,rp],r]-D[wtbar3pi2,{r,2}]) //Simplify

Trr3pi2d=Trr3pi2/.{rp->r, θp->θ}//Simplify

Trr4pi2=-(1/4)(D[D[wtbar4pi2,rp],r]-D[wtbar4pi2,{r,2}]) //Simplify

Trr4pi2d=Trr4pi2/.{rp->r, θp->θ}//Simplify

Tpr1pi2=-(1/4)((1/r)D[D[wtbar1pi2,r],θp]-(1/r)D[D[wtbar1pi2,θ],r])-(4r^2)^(-1)D[wtbar1pi2,θ]

Tpr1pi2d=Tpr1pi2/.{rp->r, θp->θ}//Simplify

Tpr2pi2=-(1/4)((1/r)D[D[wtbar2pi2,r],θp]-(1/r)D[D[wtbar2pi2,θ],r])-(4r^2)^(-1)D[wtbar2pi2,θ];

Tpr2pi2d=Tpr2pi2/.{rp->r, θp->θ}//Simplify

Tpr3pi2=-(1/4)((1/r)D[D[wtbar3pi2,r],θp]-(1/r)D[D[wtbar3pi2,θ],r]) -(4r^2)^(-1)D[wtbar3pi2,θ]//Simplify

Tpr3pi2d=Tpr3pi2/.{rp->r, θp->θ}//Simplify

Tpr4pi2=-(1/4)((1/r)D[D[wtbar4pi2,r],θp]-(1/r)D[D[wtbar4pi2,θ],r])-(4r^2)^(-1)D[wtbar4pi2,θ];

Tpr4pi2d=Tpr4pi2/.{rp->r, θp->θ}//Simplify

Do these energy densities agree with the Cartesian pieces?

FullSimplify[pcrho1-T001pi2]

FullSimplify[pcrho2-T002pi2]

FullSimplify[pcrho3-T003pi2]

FullSimplify[pcrho4-T004pi2]

Yes! Good.

Do the pressures agree? We will find the components of the stress-energy tensor for the wedge in terms of the pressure components px, py, and Txy from the Cartesian case, and then compare the two.

Now, the on-diagonal terms, separated into pieces. The format is ( Cartesian Tμν converted from Cartesian to polar components ) - ( wedge Tμν in polar components ), so all of these should be zero if the two things are equal.

pcpx1d Cos[θ]^2+pcpy1d Sin[θ]^2+2pcTxy1 Sin[θ] Cos[θ]-Trr1pi2d//FullSimplify

pcpx2d Cos[θ]^2+pcpy2d Sin[θ]^2+2pcTxy2 Sin[θ] Cos[θ]-Trr2pi2d//FullSimplify

pcpx3 Cos[θ]^2+pcpy3 Sin[θ]^2+2pcTxy3 Sin[θ] Cos[θ]-Trr3pi2d//FullSimplify

pcpx4d Cos[θ]^2+pcpy4d Sin[θ]^2+2pcTxy4d Sin[θ] Cos[θ]-Trr4pi2d//FullSimplify

pcpx1d Sin[θ]^2+pcpy1d Cos[θ]^2-2pcTxy1 Sin[θ] Cos[θ]-Tpp1pi2d//FullSimplify

pcpx2d Sin[θ]^2+pcpy2d Cos[θ]^2-2pcTxy2 Sin[θ] Cos[θ]-Tpp2pi2d//FullSimplify

pcpx3 Sin[θ]^2+pcpy3 Cos[θ]^2-2pcTxy3 Sin[θ] Cos[θ]-Tpp3pi2d//FullSimplify

pcpx4d Sin[θ]^2+pcpy4d Cos[θ]^2-2pcTxy4d Sin[θ] Cos[θ]-Tpp4pi2d//FullSimplify

Off-diagonal terms

pcTxy1(Cos[θ]^2-Sin[θ]^2)+Sin[θ] Cos[θ](-pcpx1d+pcpy1d)-Tpr1pi2d//FullSimplify

pcTxy2(Cos[θ]^2-Sin[θ]^2)+Sin[θ] Cos[θ](-pcpx2d+pcpy2d)-Tpr2pi2d//FullSimplify

Tpr1 and 2 aren't canceling for some reason? Now fixed!

pcTxy3(Cos[θ]^2-Sin[θ]^2)+Sin[θ] Cos[θ](-pcpx3+pcpy3)-Tpr3pi2d//FullSimplify

pcTxy4d(Cos[θ]^2-Sin[θ]^2)+Sin[θ] Cos[θ](-pcpx4d+pcpy4d)-Tpr4pi2d//FullSimplify

This final section has to do with the torque problem in the wedge. Point-splitting is along z.

Wedge T00 and pressures, in polar coordinates, for general alpha

T001=-(1/2)D[wedgeTbar1,{t,2}];

T002=-(1/2)D[wedgeTbar2,{t,2}];

T003=-(1/2)D[wedgeTbar3,{t,2}];

T004=-(1/2)D[wedgeTbar4,{t,2}];

Tpp1=(1/(4r))D[wedgeTbar1,r]+(1/(4r^2))(D[wedgeTbar1,{θ,2}]-D[D[wedgeTbar1,θ],θp]);

Tpp2=(1/(4r))D[wedgeTbar2,r]+(1/(4r^2))(D[wedgeTbar2,{θ,2}]-D[D[wedgeTbar2,θ],θp]);

Tpp3=(1/(4r))D[wedgeTbar3,r]+(1/(4r^2))(D[wedgeTbar3,{θ,2}]-D[D[wedgeTbar3,θ],θp]);

Tpp4=(1/(4r))D[wedgeTbar4,r]+(1/(4r^2))(D[wedgeTbar4,{θ,2}]-D[D[wedgeTbar4,θ],θp]);

Off-diagonal components

Trp1=-(1/4)((1/r)D[D[wedgeTbar1,r],θp]-(1/r)D[D[wedgeTbar1,θ],r])-(4r^2)^(-1)D[wedgeTbar1,θ];

Trp2=-(1/4)((1/r)D[D[wedgeTbar2,r],θp]-(1/r)D[D[wedgeTbar2,θ],r])-(4r^2)^(-1)D[wedgeTbar2,θ];

Trp3=-(1/4)((1/r)D[D[wedgeTbar3,r],θp]-(1/r)D[D[wedgeTbar3,θ],r])-(4r^2)^(-1)D[wedgeTbar3,θ];

Trp4=-(1/4)((1/r)D[D[wedgeTbar4,r],θp]-(1/r)D[D[wedgeTbar4,θ],r])-(4r^2)^(-1)D[wedgeTbar4,θ];

Calculations for T001 and Tpp1

wedgerho1=T001/.{rp->r,θp->θ,t->0,zp->0};

wedgepp1=Tpp1/.{rp->r,θp->θ,t->0,zp->0};

Frp1=Trp1/.{rp->r,θp->θ,t->0,zp->0,α->Pi/2}//Simplify

Derivative of rho with respect to alpha

drho1=D[wedgerho1,α]/.α->Pi/2;

Integral of that derivative over all theta

int1=Integrate[drho1,{θ,0,Pi/2}]

Integral of Frp over all theta

Frp1int=Integrate[Frp1*r,{θ,0,Pi/2}]

Energy density and pressure at θ=π/2

wr1=FullSimplify[wedgerho1/.{θ->Pi/2,α->Pi/2}]

wp1=FullSimplify[wedgepp1/.{θ->Pi/2,α->Pi/2}]

Integrate[wp1,{r,0,Infinity}]

Sum

FullSimplify[int1+wr1+wp1]

Energy density and pressure at θ=0

FullSimplify[wedgerho1/.{θ->0,α->Pi/2}]

FullSimplify[wedgepp1/.{θ->0,α->Pi/2}]

Calculations for T002 and Tpp2

wedgerho2=T002/.{rp->r,θp->θ,t->0,zp->0};

wedgepp2=Tpp2/.{rp->r,θp->θ,t->0,zp->0};

Frp2=Trp2/.{rp->r,θp->θ,t->0,zp->0,α->Pi/2}//Simplify

Frp1-Frp2//Simplify

So Frp is not the same for parts 1 and 2.  But replace θ by π/2 - θ; it’s OK!

Derivative of rho with respect to alpha

drho2=D[wedgerho2,α]/.α->Pi/2;

Integral of that derivative over all theta

int2=Integrate[drho2,{θ,0,Pi/2}]

Integral of Frp over all theta

Frp2int=Integrate[Frp2*r,{θ,0,Pi/2}]

Note that this is the negative of Frp1int. (Yes, still is.)

Energy density and pressure at θ=π/2

wr2=FullSimplify[wedgerho2/.{θ->Pi/2,α->Pi/2}]

wp2=FullSimplify[wedgepp2/.{θ->Pi/2,α->Pi/2}]

Integrate[wp2,{r,0,Infinity}]

Sum

FullSimplify[int2+wr2+wp2]

Energy density and pressure at θ=0

FullSimplify[wedgerho2/.{θ->0,α->Pi/2}]

FullSimplify[wedgepp2/.{θ->0,α->Pi/2}]

Calculations for T003 and Tpp3

wedgerho3=T003/.{rp->r,θp->θ,t->0,zp->0};

wedgepp3=Tpp3/.{rp->r,θp->θ,t->0,zp->0};

Frp3=Trp3/.{rp->r,θp->θ,t->0,zp->0,α->Pi/2}//Simplify

Derivative of rho with respect to alpha

drho3=D[wedgerho3,α]/.α->Pi/2;

Integral of that derivative over all theta

int3=Integrate[drho3,{θ,0,Pi/2}]//FullSimplify

Integral of Frp over all theta

Frp3int=Integrate[Frp3*r,{θ,0,Pi/2}]

Energy density and pressure at θ=π/2

wr3=FullSimplify[wedgerho3/.{θ->Pi/2,α->Pi/2}]

wp3=FullSimplify[wedgepp3/.{θ->Pi/2,α->Pi/2}]

Integrate[wp3,{r,0,Infinity}]

Sum

FullSimplify[int3+wr3+wp3]

Energy density and pressure at θ=0

FullSimplify[wedgerho3/.{θ->0,α->Pi/2}]

FullSimplify[wedgepp3/.{θ->0,α->Pi/2}]

Calculations for T004 and Tpp4

wedgerho4=T004/.{rp->r,θp->θ,t->0,zp->0};

wedgepp4=Tpp4/.{rp->r,θp->θ,t->0,zp->0};

Frp4=Trp4/.{rp->r,θp->θ,t->0,zp->0,α->Pi/2}//Simplify

drho4=D[wedgerho4,α]/.α->Pi/2;

int4=Integrate[drho4,{θ,0,Pi/2}]//FullSimplify

Integral of Frp over all theta

Frp4int=Integrate[Frp4*r,{θ,0,Pi/2}]

Energy density and pressure at θ=π/2

wr4=FullSimplify[wedgerho4/.{θ->Pi/2,α->Pi/2}]

wp4=FullSimplify[wedgepp4/.{θ->Pi/2,α->Pi/2}]

Integrate[wp4,{r,0,Infinity}]

Sum

FullSimplify[int4+wr4+wp4]

Energy density and pressure at θ=0

FullSimplify[wedgerho4/.{θ->0,α->Pi/2}]

FullSimplify[wedgepp4/.{θ->0,α->Pi/2}]

Sum of all parts from 1 + 2 + 3 + 4

FullSimplify[int1+int2+int3+int4+wr1+wr2+wr3+wr4+wp1+wp2+wp3+wp4]

Sum of all parts from 1 + 2

FullSimplify[int1+int2+wr1+wr2+wp1+wp2]

Sum of all parts from 3 + 4

FullSimplify[int3+int4+wr3+wr4+wp3+wp4]