Embedded Files
TikZ iz a LaTeX graphic package I occasionally use to generate pictures in my talks and papers. Some examples + source code (expandable) are below. Needless to say, someone who really knows TikZ would surely do that in a vastly superior manner; for conformal diagrams, here's one, and here's a whole package. All of these assume tikz and float; any additional packages will be explicitly listed.
Forthcoming: the diagram from the paper on exotic smooth manifolds, few Penrose diagrams from the paper on black to white hole transitions.
Fully evaporating Schwarzschild black hole
Fully evaporating Schwarzschild black hole
\begin{figure}[H] \centering % to rescale everything proportionally by a factor n: replace n with a factor, uncomment the line below, uncomment } for closing the scaling command (this can be found in the line right after \end{tikzpicture}) \scalebox{0.75}{ \begin{tikzpicture}%fully evaporating Schwarzschild black hole\draw (1,5) -- (3,3);\draw (1,3) -- (1,5);\draw[decorate,decoration=zigzag] (0,3) -- (1,3);\draw (0,0) -- (3,3);\draw (0,0) -- (0,3);
\draw (1,3) -- (0,2);%horizon
\draw [dashed] (1,3) -- (2,4);%Cauchy horizon
\node [right] at (3,3) {$i^{o}$};\node [below] at (0,0) {$i^{-}$};\node [above] at (1,5) {$i^{+}$};\node [above] at (2.2,4) {$\cal{J}^+$};\node [below] at (2,1.5) {$\cal{J}^-$};
\draw[red] (0,0) arc (-30:30:3);\end{tikzpicture} % the bracket below is for closing the scaling command: }% \caption{Fully evaporating black hole.} \label{fig:fullyevaporating}\end{figure}
\draw (1,3) -- (0,2);%horizon
\draw [dashed] (1,3) -- (2,4);%Cauchy horizon
\node [right] at (3,3) {$i^{o}$};\node [below] at (0,0) {$i^{-}$};\node [above] at (1,5) {$i^{+}$};\node [above] at (2.2,4) {$\cal{J}^+$};\node [below] at (2,1.5) {$\cal{J}^-$};
\draw[red] (0,0) arc (-30:30:3);\end{tikzpicture} % the bracket below is for closing the scaling command: }% \caption{Fully evaporating black hole.} \label{fig:fullyevaporating}\end{figure}
Conformal diagram of (a portion of) the subextremal Kerr spacetime
Conformal diagram of (a portion of) the subextremal Kerr spacetime
\begin{figure}[H] \centering % to rescale everything proportionally by a factor n: replace n with a factor, uncomment the line below, uncomment } for closing the scaling command (this can be found in the line right after \end{tikzpicture}) \scalebox{0.48}{ \begin{tikzpicture}
\filldraw[fill=white, draw=white] (0,0) rectangle (8,14);
%true exterior copy of I\draw[black, thick] (4,4) -- (6,2) -- (8,4) -- (6,6) -- (4,4);%label I\node [thick, black] at (6,4) {I};
%true interior copy of II\draw[black, thick] (4,4) -- (6,6) -- (4,8) -- (2,6) -- (4,4);%label II\node [black] at (4,6) {II};
%true interior copy of III\draw[black, thick] (2,6) -- (4,8) -- (2,10) -- (0,8) -- (2,6);%label III\node [black] at (2,8) {III};
%p in III%\draw[fill=black] (3,8) circle [radius=0.05];%\node [below, black] at (3,8) {$p$};
%scri+\node [above, black] at (7.2,5) {$\mathcal{I}^{+}$};
%label EH\node [black] at (5,5) {EH};
%label CH\node [black] at (3,7) {CH};
%gamma in I%\draw[fill=black] (5.5,4) circle [radius=0.05];%\draw[black, thin] (5.5,4) to [out=105,in=270] (6,6);%\node[black] at (6,5) {$\gamma$};
%thin copy of II below\draw[gray, thin, dashed] (4,4) -- (2,2) -- (4,0) -- (6,2); %label II\node [thick, gray] at (4,2) {II};
%thin copy of I below\draw[gray, thin, dashed] (2,2) -- (0,4) -- (2,6);%label I\node [thick, gray] at (2,4) {I};
%thin copy of III above\draw[gray, thin, dashed] (4,8) -- (6,10) -- (8,8) -- (6,6);%label III\node [thick, gray] at (6,8) {III};
%thin copy of II above\draw[gray, thin, dashed] (2,10) -- (4,12) -- (6,10);%label II\node [thick, gray] at (4,10) {II};
%thin copy of I above\draw[gray, thin, dashed] (2,10) -- (0,12) -- (2,14) -- (4,12);%label I\node [thick, gray] at (2,12) {I};
%thin copy of I above\draw[gray, thin, dashed] (6,10) -- (8,12) -- (6,14) -- (4,12);%label I\node [thick, gray] at (6,12) {I};
\end{tikzpicture} % the bracket below is for closing the scaling command: } %\caption{Penrose diagram showing causal structure of a subset of subextremal Kerr spacetime. Region I represent the universe outside the black hole, region II the black hole interior, region III the deep interior. The maximal analytic extension continues infinitely in the manner shown on shaded copies of regions I, II, III. Acausal features of region III are not shown on this diagram. Event horizon is marked with EH, Cauchy horizon with CH, point $p$ and curve $\gamma$ show the Malament-Hogarth property.} \label{fig:kerrdiagramfirst}\end{figure}
\filldraw[fill=white, draw=white] (0,0) rectangle (8,14);
%true exterior copy of I\draw[black, thick] (4,4) -- (6,2) -- (8,4) -- (6,6) -- (4,4);%label I\node [thick, black] at (6,4) {I};
%true interior copy of II\draw[black, thick] (4,4) -- (6,6) -- (4,8) -- (2,6) -- (4,4);%label II\node [black] at (4,6) {II};
%true interior copy of III\draw[black, thick] (2,6) -- (4,8) -- (2,10) -- (0,8) -- (2,6);%label III\node [black] at (2,8) {III};
%p in III%\draw[fill=black] (3,8) circle [radius=0.05];%\node [below, black] at (3,8) {$p$};
%scri+\node [above, black] at (7.2,5) {$\mathcal{I}^{+}$};
%label EH\node [black] at (5,5) {EH};
%label CH\node [black] at (3,7) {CH};
%gamma in I%\draw[fill=black] (5.5,4) circle [radius=0.05];%\draw[black, thin] (5.5,4) to [out=105,in=270] (6,6);%\node[black] at (6,5) {$\gamma$};
%thin copy of II below\draw[gray, thin, dashed] (4,4) -- (2,2) -- (4,0) -- (6,2); %label II\node [thick, gray] at (4,2) {II};
%thin copy of I below\draw[gray, thin, dashed] (2,2) -- (0,4) -- (2,6);%label I\node [thick, gray] at (2,4) {I};
%thin copy of III above\draw[gray, thin, dashed] (4,8) -- (6,10) -- (8,8) -- (6,6);%label III\node [thick, gray] at (6,8) {III};
%thin copy of II above\draw[gray, thin, dashed] (2,10) -- (4,12) -- (6,10);%label II\node [thick, gray] at (4,10) {II};
%thin copy of I above\draw[gray, thin, dashed] (2,10) -- (0,12) -- (2,14) -- (4,12);%label I\node [thick, gray] at (2,12) {I};
%thin copy of I above\draw[gray, thin, dashed] (6,10) -- (8,12) -- (6,14) -- (4,12);%label I\node [thick, gray] at (6,12) {I};
\end{tikzpicture} % the bracket below is for closing the scaling command: } %\caption{Penrose diagram showing causal structure of a subset of subextremal Kerr spacetime. Region I represent the universe outside the black hole, region II the black hole interior, region III the deep interior. The maximal analytic extension continues infinitely in the manner shown on shaded copies of regions I, II, III. Acausal features of region III are not shown on this diagram. Event horizon is marked with EH, Cauchy horizon with CH, point $p$ and curve $\gamma$ show the Malament-Hogarth property.} \label{fig:kerrdiagramfirst}\end{figure}
A null truncated Minkowski spacetime has an epistemic hole
A null truncated Minkowski spacetime has an epistemic hole
\begin{figure}[H] \centering \begin{tikzpicture}
%draws a square\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%diagonal dashed line S\draw[black,loosely dashed, ultra thick] (2,6) -- (6,2);\node[black] at (2,5.5) {S};
%line from p to S\draw[black, thick] (3,1) -- (3,5);\draw[black, thick] (3,1) -- (4,4);
%labels to the line from p to S\node[black] at (3.6,2) {$\gamma$};\node[black] at (2.8,2) {$\gamma '$};
%label r\draw[fill=black] (3,5) circle [radius=0.05];\node [above, black] at (3,5) {r};%label q\draw[fill=black] (4,4) circle [radius=0.05];\node [above, black] at (4,4) {q};
%past lightcones of points on S\draw[black, thick, dashed] (3,5) -- (0,3);\draw[black, thick, dashed] (4,4) -- (0,1);\draw[black, thick, dashed] (3,5) -- (6,2);
%labels of I- of y and y'%\draw[fill=white] (2,3.5) circle [radius=0.05];\node [below, black] at (2,3.5) {$I^{-}(\gamma')$};%\draw[fill=white] (4.5,2) circle [radius=0.05];\node [below, black] at (4.5,2) {$I^{-}(\gamma)$};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%I+(S)%\draw[fill=white] (5,5) circle [radius=0.05];\node [below, black] at (5,5) {$I^{+}(S)$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{If $S$ has a null component, the truncated spacetime violates EHF(g): the case of a null-truncated Minkowski spacetime $\langle M^{'1}, g^{'1}_{ab} \rangle$.} \label{fig:hasehf}\end{figure}
%draws a square\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%diagonal dashed line S\draw[black,loosely dashed, ultra thick] (2,6) -- (6,2);\node[black] at (2,5.5) {S};
%line from p to S\draw[black, thick] (3,1) -- (3,5);\draw[black, thick] (3,1) -- (4,4);
%labels to the line from p to S\node[black] at (3.6,2) {$\gamma$};\node[black] at (2.8,2) {$\gamma '$};
%label r\draw[fill=black] (3,5) circle [radius=0.05];\node [above, black] at (3,5) {r};%label q\draw[fill=black] (4,4) circle [radius=0.05];\node [above, black] at (4,4) {q};
%past lightcones of points on S\draw[black, thick, dashed] (3,5) -- (0,3);\draw[black, thick, dashed] (4,4) -- (0,1);\draw[black, thick, dashed] (3,5) -- (6,2);
%labels of I- of y and y'%\draw[fill=white] (2,3.5) circle [radius=0.05];\node [below, black] at (2,3.5) {$I^{-}(\gamma')$};%\draw[fill=white] (4.5,2) circle [radius=0.05];\node [below, black] at (4.5,2) {$I^{-}(\gamma)$};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%I+(S)%\draw[fill=white] (5,5) circle [radius=0.05];\node [below, black] at (5,5) {$I^{+}(S)$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{If $S$ has a null component, the truncated spacetime violates EHF(g): the case of a null-truncated Minkowski spacetime $\langle M^{'1}, g^{'1}_{ab} \rangle$.} \label{fig:hasehf}\end{figure}
A zig-zag truncated Minkowski spacetime has an epistemic hole
A zig-zag truncated Minkowski spacetime has an epistemic hole
\begin{figure}[H] \centering \begin{tikzpicture}
%square\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%dashed diagonal line S\draw[black,loosely dashed, ultra thick] (2.5,5.5) -- (0,5.5);\draw[black,loosely dashed, ultra thick] (2.5,5.5) -- (5,3);\draw[black,loosely dashed, ultra thick] (5,3) -- (6,3);
%label r\draw[fill=black] (3,5) circle [radius=0.05];\node [above, black] at (3,5) {r};%label q\draw[fill=black] (4,4) circle [radius=0.05];\node [above, black] at (4,4) {q};
\node[black] at (5,3.5) {S};
%lines from p to S\draw[black, thick] (3,1) -- (3,5);\draw[black, thick] (3,1) -- (4,4);
%labels to lines from p to S\node[black] at (3.6,2) {$\gamma$};\node[black] at (2.8,2) {$\gamma '$};
%past lightcones of points on S\draw[black, thick, dashed] (3,5) -- (0,3);\draw[black, thick, dashed] (4,4) -- (0,1);\draw[black, thick, dashed] (3,5) -- (6,2);
%lables of I- of y and y'%\draw[fill=white] (2,3.5) circle [radius=0.05];\node [below, black] at (2,3.5) {$I^{-}(\gamma')$};%\draw[fill=white] (4.5,2) circle [radius=0.05];\node [below, black] at (4.5,2) {$I^{-}(\gamma)$};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%I+(S)%\draw[fill=white] (5,5) circle [radius=0.05];\node [below, black] at (5,5) {$I^{+}(S)$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{If $S$ has a null component, the truncated spacetime violates EHF(g): the case of a zig-zag-truncated Minkowski spacetime $\langle M^{'2}, g^{'2}_{ab} \rangle$.} \label{fig:hasehf2}\end{figure}
%square\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%dashed diagonal line S\draw[black,loosely dashed, ultra thick] (2.5,5.5) -- (0,5.5);\draw[black,loosely dashed, ultra thick] (2.5,5.5) -- (5,3);\draw[black,loosely dashed, ultra thick] (5,3) -- (6,3);
%label r\draw[fill=black] (3,5) circle [radius=0.05];\node [above, black] at (3,5) {r};%label q\draw[fill=black] (4,4) circle [radius=0.05];\node [above, black] at (4,4) {q};
\node[black] at (5,3.5) {S};
%lines from p to S\draw[black, thick] (3,1) -- (3,5);\draw[black, thick] (3,1) -- (4,4);
%labels to lines from p to S\node[black] at (3.6,2) {$\gamma$};\node[black] at (2.8,2) {$\gamma '$};
%past lightcones of points on S\draw[black, thick, dashed] (3,5) -- (0,3);\draw[black, thick, dashed] (4,4) -- (0,1);\draw[black, thick, dashed] (3,5) -- (6,2);
%lables of I- of y and y'%\draw[fill=white] (2,3.5) circle [radius=0.05];\node [below, black] at (2,3.5) {$I^{-}(\gamma')$};%\draw[fill=white] (4.5,2) circle [radius=0.05];\node [below, black] at (4.5,2) {$I^{-}(\gamma)$};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%I+(S)%\draw[fill=white] (5,5) circle [radius=0.05];\node [below, black] at (5,5) {$I^{+}(S)$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{If $S$ has a null component, the truncated spacetime violates EHF(g): the case of a zig-zag-truncated Minkowski spacetime $\langle M^{'2}, g^{'2}_{ab} \rangle$.} \label{fig:hasehf2}\end{figure}
Spacelike truncated Minkowski spacetime is EHF: case 1
Spacelike truncated Minkowski spacetime is EHF: case 1
\begin{figure}[H] \centering\begin{tikzpicture}
\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%straight dashed line S\draw[black,loosely dashed, ultra thick] (0,5) -- (6,5);\node[black] at (1,4.7) {S};
%I+(S)%\draw[fill=black] (3,5.75) circle [radius=0.05];\node [below, black] at (3,6) {$I^{+}(S)$};
%point q\draw[fill=black] (3,5) circle [radius=0.05];\node [above, black] at (3,5) {q};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%past ligthcones of a point on S\draw[black, thick, dashed] (3,5) -- (0,2);\draw[black, thick, dashed] (3,5) -- (6,2);
%two curves from p\draw[black, thick] (3,1) to [out=105,in=270] (3,5);\draw[black, thick] (3,1) to [out=60,in=290] (3,5);
%labels of these two curves\node[below, black] at (2.5,3) {$\gamma$};\node[below, black] at (4,3) {$\gamma'$};\node[black] at (4.5,0.5) {$I^{-}[\gamma] = I^{-}[\gamma']$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{Future-truncated Minkowski spacetime $\langle M^{'3}, g_{ab}^{'3} \rangle$, case 1: $I^{-}[\gamma] = I^{-}[\gamma']$.} \label{fig:noehf1}\end{figure}
\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%straight dashed line S\draw[black,loosely dashed, ultra thick] (0,5) -- (6,5);\node[black] at (1,4.7) {S};
%I+(S)%\draw[fill=black] (3,5.75) circle [radius=0.05];\node [below, black] at (3,6) {$I^{+}(S)$};
%point q\draw[fill=black] (3,5) circle [radius=0.05];\node [above, black] at (3,5) {q};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%past ligthcones of a point on S\draw[black, thick, dashed] (3,5) -- (0,2);\draw[black, thick, dashed] (3,5) -- (6,2);
%two curves from p\draw[black, thick] (3,1) to [out=105,in=270] (3,5);\draw[black, thick] (3,1) to [out=60,in=290] (3,5);
%labels of these two curves\node[below, black] at (2.5,3) {$\gamma$};\node[below, black] at (4,3) {$\gamma'$};\node[black] at (4.5,0.5) {$I^{-}[\gamma] = I^{-}[\gamma']$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{Future-truncated Minkowski spacetime $\langle M^{'3}, g_{ab}^{'3} \rangle$, case 1: $I^{-}[\gamma] = I^{-}[\gamma']$.} \label{fig:noehf1}\end{figure}
Spacelike truncated Minkowski spacetime is EHF: case 2
Spacelike truncated Minkowski spacetime is EHF: case 2
\begin{figure}[H] \centering\begin{tikzpicture}
%the usual square\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%straight dashed line S\draw[black,loosely dashed, ultra thick] (0,5) -- (6,5);\node[black] at (1,4.5) {S};
%I+(S)%\draw[fill=black] (3,5.75) circle [radius=0.05];\node [below, black] at (3,5.75) {$I^{+}(S)$};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%label r\draw[fill=black] (2,5) circle [radius=0.05];\node [above, black] at (2,5) {r};%label q\draw[fill=black] (5,5) circle [radius=0.05];\node [above, black] at (5,5) {q};
%lines from p to S\draw[black, thick] (3,1) -- (2,5);\draw[black, thick] (3,1) -- (5,5);
%labels of these lines p\node[black] at (5,4) {$\gamma'$};\node[black] at (2,4) {$\gamma$};
%intersection of I- of gamma and gamma'\node[black] at (4.5,0.5) {$I^{-}(\gamma) \cap I^{-}(\gamma')$};
%past lightcones of points on S\draw[black, thick, dashed] (2,5) -- (0,3);\draw[black, thick, dashed] (2,5) -- (6,1);
\draw[black, thick, dashed] (5,5) -- (6,4);\draw[black, thick, dashed] (5,5) -- (1,0);
%labels of I- of y and y'%\draw[fill=black] (2,3.5) circle [radius=0.05];\node [below, black] at (1.5,3.5) {$I^{-}(\gamma)$};
%\draw[fill=black] (4.5,2) circle [radius=0.05];\node [below, black] at (5.25,3.5) {$I^{-}(\gamma')$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{Future-truncated Minkowski spacetime $\langle M^{'3}, g_{ab}^{'3} \rangle$, case 2: $I^{-}[\gamma] \bigtriangleup I^{-}[\gamma'] \neq\varnothing$, and is not contained in $I^{-}[\gamma]$ or in $I^{-}[\gamma']$.} \label{fig:noehf2}\end{figure}
%the usual square\filldraw[fill=white, draw=black,loosely dashed,ultra thick,rounded corners] (0,0) rectangle (6,6);
%straight dashed line S\draw[black,loosely dashed, ultra thick] (0,5) -- (6,5);\node[black] at (1,4.5) {S};
%I+(S)%\draw[fill=black] (3,5.75) circle [radius=0.05];\node [below, black] at (3,5.75) {$I^{+}(S)$};
%point p\draw[fill=black] (3,1) circle [radius=0.05];\node [below, black] at (3,1) {p};
%label r\draw[fill=black] (2,5) circle [radius=0.05];\node [above, black] at (2,5) {r};%label q\draw[fill=black] (5,5) circle [radius=0.05];\node [above, black] at (5,5) {q};
%lines from p to S\draw[black, thick] (3,1) -- (2,5);\draw[black, thick] (3,1) -- (5,5);
%labels of these lines p\node[black] at (5,4) {$\gamma'$};\node[black] at (2,4) {$\gamma$};
%intersection of I- of gamma and gamma'\node[black] at (4.5,0.5) {$I^{-}(\gamma) \cap I^{-}(\gamma')$};
%past lightcones of points on S\draw[black, thick, dashed] (2,5) -- (0,3);\draw[black, thick, dashed] (2,5) -- (6,1);
\draw[black, thick, dashed] (5,5) -- (6,4);\draw[black, thick, dashed] (5,5) -- (1,0);
%labels of I- of y and y'%\draw[fill=black] (2,3.5) circle [radius=0.05];\node [below, black] at (1.5,3.5) {$I^{-}(\gamma)$};
%\draw[fill=black] (4.5,2) circle [radius=0.05];\node [below, black] at (5.25,3.5) {$I^{-}(\gamma')$};
%Minkowski-like lightcone\draw[black] (1,1) node{} %middle -- (1.5,1.5) node{}%right above -- (0.5,1.5) node{}%left above -- cycle;\draw[black] (1,1) node{} %middle -- (1.5,0.5) node{}%right below -- (0.5,0.5) node{}%left below -- cycle; \end{tikzpicture} \caption{Future-truncated Minkowski spacetime $\langle M^{'3}, g_{ab}^{'3} \rangle$, case 2: $I^{-}[\gamma] \bigtriangleup I^{-}[\gamma'] \neq\varnothing$, and is not contained in $I^{-}[\gamma]$ or in $I^{-}[\gamma']$.} \label{fig:noehf2}\end{figure}
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