Plots of <phi^2> for 2D Schwarzschild in the Unruh, HH, and Boulware states

phi2B = (1/(4*Pi))* Log[Abs[r-2]/r]

phi2U = (v-2*r)/(16*Pi) + (1/(4*Pi))*Log[2/r]

rstar = r + 2*Log[Abs[(r-2)]/2]

phi2Ualt = Expand[phi2U/.{v->u+2*rstar}]

phi2Ut = Expand[phi2U/.{v->t+rstar}]

phi2H =-r/(8*Pi) + (1/(4*Pi))*Log[2/r]

The plots labeled p1, p2, ... are the ones I propose to include in the paper

p1 = Plot[{phi2B,phi2U/.{v->10},phi2H},{r,10^(-2),2-10^(-3)}]

p2 = Plot[{phi2B,phi2U/.{v->10},phi2H},{r,2,10}]

Show[p1,p2,PlotRange->All]

p3 = Plot[{phi2B,phi2Ualt/.{u->10},phi2H},{r,2,10}]

This next plot is for t = 0

Next I’ll work with the null shell spacetime

uFext = vH - 4*ProductLog[Exp[(vH-uext)/4]]

uFint = vH - 4*ProductLog[-Exp[(uint+vH)/4]]

Let the null shell trajectory be v0 = 4

vH = 0

v0 = 4

Plot[uFext,{uext,-100,100}]

The null shell is at v = v0 = t + rstar  so if we fix the value of r and hence rstar, t > v0 - rstar.   This works both inside and outside
the future horizon

tmin = v0-rstar

Plot[tmin,{r,2.1,10}]

Plot[tmin,{r,0,1.9}]

This works both inside and outside the null shell

phi2in= (1/(4*Pi))*(2*Log[Abs[uF-v]]+ Log[Abs[(v0-uF)/(uF-vH)]]+ Log[Abs[(2-r)]/(4*r)]+2*EulerGamma)

phi2inextalt= (1/(4*Pi))*(2*Log[uFext-v]+ Log[(v0-uFext)/(vH-uFext)]+ Log[(r-2)/(4*r)]+2*EulerGamma)

phi2inext = phi2in/.uF->uFext

phi2inint = phi2in/.uF->uFint

phi2inextu = phi2inext/.{v->uext+2*rstar}

phi2U

I won’t use these for the paper but I’ll plot them to see what happens starting with uext = const.

p4 = Plot[{phi2inextu/.{uext->10},phi2U/.{v->10+2*rstar}},{r,2.1,10}]

This is for v = 10  since u = v - 2 rstar

phi2inext/.{v->10}

phi2U

p5 = Plot[{phi2inint/.{v->10,uint->2*rstar-10},phi2U/.{v->10}},{r,0.1,1.99}]

p6 =  Plot[{phi2inext/.{v->10,uext->10-2*rstar},phi2U/.{v->10}},{r,2.01,20}]

Show[p5,p6,PlotRange->All]

rstar1 = rstar/.{r->1}

p7 = Plot[{phi2inint/.{v->t+rstar1,uint->2*rstar1-t,r->1},phi2U/.{v->t+rstar1,r->1}},{t,tmin/.{r->1},10}]

OK so the extra log term in phi2in makes it so that phi2in grows faster in t than phi2U.  The same is true in the exterior region.

rstar3 = rstar/.{r->3}

p8 = Plot[{phi2inext/.{v->t+rstar3,uext->t-2*rstar3,r->3},phi2U/.{v->t+rstar3,r->3}},{t,tmin/.{r->3},100}]

p1 = Plot[{phi2B,phi2U/.{v->10},phi2H},{r,10^(-2),2-10^(-3)}]

p2 = Plot[{phi2B,phi2U/.{v->10},phi2H},{r,2,10}]

p1sin=Plot[phi2B,{r,10^(-2),2-10^(-3)},PlotRange->All,
PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(ϕ^2)⟩_B"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

p2sin=Plot[phi2U/.{v->10},{r,10^(-2),2-10^(-3)},PlotRange->All,
PlotStyle->Directive[Darker[Red],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(ϕ^2)⟩_U"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},AxesStyle->Directive[Black, AbsoluteThickness[1.3]],Frame->True,FrameStyle->Directive[Thickness[0.003]]]

p3sin=Plot[phi2H,{r,10^(-2),2-10^(-3)},PlotRange->All,
PlotStyle->Directive[Darker[Green],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(ϕ^2)⟩_HH"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

ptotin=Show[p1sin,p2sin,p3sin]

p1s=Plot[phi2B,{r,2,10},PlotRange->All,
PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

p2s=Plot[phi2U/.{v->10},{r,2,10},PlotRange->All,
PlotStyle->Directive[Darker[Red],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

p3s=Plot[phi2H,{r,2,10},PlotRange->All,
PlotStyle->Directive[Darker[Green],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

ptotout=Show[p1s,p2s,p3s]

Show[ptotin,ptotout]

Plot[{phi2inext/.{v->t+rstar3,uext->t-2*rstar3,r->3}},{t,tmin/.{r->3},100}]

pinr3=Plot[{phi2inext/.{v->t+rstar3,uext->t-2*rstar3,r->3}},{t,tmin/.{r->3},100},
PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(ϕ^2)⟩_in"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},TicksStyle->Directive[Bold,12],Frame->True,Axes->False,FrameStyle->Directive[Thickness[0.003]]]

pUr3=Plot[{phi2U/.{v->t+rstar3,r->3}},{t,tmin/.{r->3},100},
PlotStyle->Directive[Darker[Red],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(ϕ^2)⟩_U"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},AxesStyle->Directive[Black, AbsoluteThickness[1.3]],Frame->True,FrameStyle->Directive[Thickness[0.003]]]

Show[pinr3,pUr3]

Plot[{phi2inint/.{v->t+rstar1,uint->2*rstar1-t,r->1},phi2U/.{v->t+rstar1,r->1}},{t,tmin/.{r->1},10}]

pinr1=Plot[{phi2inint/.{v->t+rstar1,uint->2*rstar1-t,r->1}},{t,tmin/.{r->1},10},
PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(ϕ^2)⟩_in"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"((Subscript[t, s])/M)",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},AxesStyle->Directive[Black, AbsoluteThickness[1.3]],Frame->True,FrameStyle->Directive[Thickness[0.003]]]

pUr1=Plot[{phi2U/.{v->t+rstar1,r->1}},{t,tmin/.{r->1},10},
PlotStyle->Directive[Darker[Red],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(ϕ^2)⟩_U"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

Show[pUr1,pinr1,PlotRange->All]