Plots of <phi^2> and <Tuu> for the 2D Sch paper

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

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.  However to avoid absolute value signs which are sometimes problematic, I’ll use “int” or “ext” at the end

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

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

The minimum value of t is obtained by finding its value on the null shell trajectory

tminint = v0-rstarint

tminext = v0-rstarext

This works both inside and outside the null shell

phi2inint= (1/(4*Pi))*(2*Log[v-uFint]+ Log[(v0-uFint)/(uFint-vH)]+ Log[(2-r)/(4*r)]+2*EulerGamma)

phi2inext= (1/(4*Pi))*(2*Log[v-uFext]+ Log[(v0-uFext)/(vH-uFext)]+ Log[(r-2)/(4*r)]+2*EulerGamma)

phi2ininttrs = phi2inint/.{v->t+rstarint,uint->rstarint-t}

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

TuuU = (1/(768*Pi))*(1-2/r)^2*(1+4/r+12/r^2)

Tuuinint = (1/(24*Pi))*(-1/r^3 + (3/2)*(1/r^4) -8/(uFint-v0)^3 - 24/(uFint-v0)^4)

Tuuinext = (1/(24*Pi))*(-1/r^3 + (3/2)*(1/r^4) -8/(uFext-v0)^3 - 24/(uFext-v0)^4)

Tuuininttrs = Tuuinint/.{uint->rstarint-t}

Tuuinexttrs = Tuuinext/.{uext->t-rstarext}

Plot[TuuU,{r,0.1,10},PlotRange->{{0,10},{0,0.0005}},PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

p1n=Plot[{Tuuininttrs/.{r->1}},{t,tminint/.{r->1},10},FrameStyle->Thickness[0.003],PlotRange->All,PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(Subscript[T, uu])⟩_in"},LegendMarkerSize->40],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},AxesStyle->Directive[Black, AbsoluteThickness[1.3]],TicksStyle->Directive[Bold,12],Frame->True]

p2n=Plot[{TuuU/.{r->1}},{t,tminint/.{r->1},10},FrameStyle->Thickness[0.003],PlotRange->All,PlotStyle->Directive[Darker[Red],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(Subscript[T, uu])⟩_U"},LegendMarkerSize->40],{Right,Top}]
,FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},AxesStyle->Directive[Black, AbsoluteThickness[1.3]],Frame->True]

Show[p1n,p2n]

p3n=Plot[{Tuuinexttrs/.{r->10}},{t,tminext/.{r->10},50},FrameStyle->Thickness[0.003],PlotRange->All,PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(Subscript[T, uu])⟩_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]]]

p4n=Plot[{TuuU/.{r->10}},{t,tminext/.{r->10},50},FrameStyle->Thickness[0.003],PlotRange->All,PlotStyle->Directive[Darker[Red],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{"⟨(Subscript[T, uu])⟩_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]]]

Show[p3n,p4n]

Plot[TuuU,{r,0.1,10},PlotRange->{{0,10},{0,0.0005}}]

{Directive[AbsoluteThickness[2],Blue]

Plot[TuuU,{r,0.1,10},PlotRange->{{0,10},{0,0.0005}},PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

pa1n= LogPlot[{Abs[((Tuuininttrs/.{r->1/2})-(TuuU/.{r->1/2}))/(TuuU/.{r->1/2})]},{t,tminint/.{r->1/2},20},PlotRange->All]

pa2 = LogPlot[{Abs[((Tuuininttrs/.{r->1})-(TuuU/.{r->1}))/(TuuU/.{r->1})]},{t,tminint/.{r->1},20},PlotRange->All]

pa3 = LogPlot[{Abs[((Tuuininttrs/.{r->3/2})-(TuuU/.{r->3/2}))/(TuuU/.{r->3/2})]},{t,tminint/.{r->3/2},20},PlotRange->All]

pa5 = LogPlot[{Abs[((Tuuininttrs/.{r->15/8})-(TuuU/.{r->15/8}))/(TuuU/.{r->15/8})]},{t,tminint/.{r->15/8},20},PlotRange->All]

pa1n=LogPlot[{Abs[((Tuuininttrs/.{r->1/2})-(TuuU/.{r->1/2}))/(TuuU/.{r->1/2})]},{t,tminint/.{r->1/2},20},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]]]

pa2n=LogPlot[{Abs[((Tuuininttrs/.{r->1})-(TuuU/.{r->1}))/(TuuU/.{r->1})]},{t,tminint/.{r->1},20},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]]]

pa3n=LogPlot[{Abs[((Tuuininttrs/.{r->3/2})-(TuuU/.{r->3/2}))/(TuuU/.{r->3/2})]},{t,tminint/.{r->3/2},20},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->20,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]]]

pa5n=LogPlot[{Abs[((Tuuininttrs/.{r->15/8})-(TuuU/.{r->15/8}))/(TuuU/.{r->15/8})]},{t,tminint/.{r->15/8},20},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]]]

N[15/8]

Show[pa1n,pa2n,pa3n,pa5n]

pb1n=LogPlot[{Abs[((Tuuinexttrs/.{r->9/2})-(TuuU/.{r->9/2}))/(TuuU/.{r->9/2})]},{t,tminext/.{r->9/2},20},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]],Axes->False]

pb2n=LogPlot[{Abs[((Tuuinexttrs/.{r->3})-(TuuU/.{r->3}))/(TuuU/.{r->3})]},{t,tminext/.{r->3},20},PlotRange->All,
PlotStyle->Directive[RGBColor[0.121,0.466,0.705],Thickness[0.0039],PointSize/@{0.005}],PlotLegends->Placed[LineLegend[{},LegendMarkerSize->40,LabelStyle->{FontFamily->"Times New Roman",FontSize->20,Bold}],{Right,Top}],FrameLabel->{"",""},LabelStyle->{FontSize->25,FontFamily->"Courier New",Black},Frame->True,FrameStyle->Directive[Thickness[0.003]],Axes->False]

pb3n=LogPlot[{Abs[((Tuuinexttrs/.{r->5})-(TuuU/.{r->5}))/(TuuU/.{r->5})]},{t,tminext/.{r->5},20},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]],Axes->False]

pb4n=LogPlot[{Abs[((Tuuinexttrs/.{r->10})-(TuuU/.{r->10}))/(TuuU/.{r->10})]},{t,tminext/.{r->10},20},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]],Axes->False]

Show[pb1n,pb2n,pb3n,pb4n]