= Solution
{c}
First use the original <null coordinates> $u=t+m/\lambda$, $v=t-m/\lambda$. They obey $u-v=2m/\lambda$, and the radial <metric tensor> becomes
$$
g_{(2)}=-\frac{4m^2}{(u-v)^2}\,du\,dv.
$$
Apply the given substitutions $u=\tan(U/2)$ and $v=-\cot(V/2)$. The identities
$$
u-v=\frac{\cos((U-V)/2)}{\cos(U/2)\sin(V/2)},\qquad
du=\frac{dU}{2\cos^2(U/2)},\qquad
dv=\frac{dV}{2\sin^2(V/2)}
$$
give the <analytic extension of a spacetime>
$$
g=-\frac{m^2}{\cos^2((U-V)/2)}\,dU\,dV+m^2d\Omega_2^2.
$$
The poles of the individual tangent and cotangent substitutions are now ordinary coordinate lines wherever $\cos((U-V)/2)\ne0$. Set $T=(U+V)/2$ and $X=(U-V)/2$. Allowing these <coordinate functions> to continue across every such line gives
$$
\boxed{\widetilde M=\mathbb R_T\times(-\pi/2,\pi/2)_X\times S^2,\qquad
\widetilde g=m^2\sec^2X(-dT^2+dX^2)+m^2d\Omega_2^2.}
$$
This is the <universal cover> of <two-dimensional anti-de Sitter spacetime> of <radius> $m$, times a <two-sphere> of <radius> $m$: the simply connected <Bertotti-Robinson spacetime>. In defining this <maximal analytic extension>, \b[$T$ ranges over the whole real line without periodic identification]. A periodic identification would instead create <closed timelike curves> and would be an additional global choice.
For clarity, the original patch is embedded by the branches $-\pi<U<\pi$ and $0<V<2\pi$. The inequality $u>v$ is equivalent on these branches to $|U-V|<\pi$, so the original patch occupies
$$
-\frac\pi2<X<\frac\pi2,\qquad X<T<\pi-X.
$$
The first <spacetime extension> in part (a) occupies the larger band $0<T-X<2\pi$, with $|X|<\pi/2$. Its finite-$v$ <Killing horizon> is the interior line $T+X=\pi$. The edges $T-X=0,2\pi$ of that band are further regular <Poincaré horizons>; continuing through them produces the full <conformal strip of two-dimensional anti-de Sitter spacetime>.
The original coordinate $\lambda$ has a smooth continuation throughout the strip:
$$
\boxed{\lambda=\frac{2m\cos(U/2)\sin(V/2)}{\cos X}
=m\frac{\sin T-\sin X}{\cos X}.}
$$
Consequently the complete set of zero-level <Killing horizons> is
$$
\boxed{T=X+2\pi j\quad\text{or}\quad T=\pi-X+2\pi j,\qquad j\in\mathbb Z.}
$$
These are two alternating families of parallel <null lines>; different families meet only at the excluded <conformal boundaries>, never inside the <spacetime>. The continuation of the original <Killing vector field> is
$$
K=(1-\sin T\sin X)\partial_T+\cos T\cos X\,\partial_X,
\qquad K^2=-\lambda^2.
$$
The field never vanishes in the interior, including on the <Killing horizons>.
To obtain the <Penrose diagram>, suppress the <two-sphere> and multiply by the <conformal factor> $\cos^2X/m^2$. The resulting radial <metric tensor> is $-dT^2+dX^2$, so the diagram is an infinitely tall strip of width $\pi$, with <null lines> at 45 degrees and timelike edges $X=\pm\pi/2$. Generically,
$$
\boxed{X\to-\pi/2:\ \lambda\to+\infty,\qquad
X\to+\pi/2:\ \lambda\to-\infty.}
$$
At the special edge points where the two families of <Killing horizons> end, the displayed formula is a $0/0$ limit and $\lambda$ has no unique limiting value. Thus the infinity labels on the diagram refer to the open edge segments, not those vertices. Each interior point of the diagram represents a <two-sphere> of fixed <radius> $m$; neither edge is a centre where that <two-sphere> shrinks.
\Image[/past-exam-of-the-general-relativity-ii-course-of-the-university-of-oxford/2021-q3-penrose.png]
{title=<Penrose diagram> of the maximal <Bertotti-Robinson spacetime> extension, with the original patch and all zero-level <Killing horizons>}
Finally, these timelike edges are <conformal boundaries>, not omitted regular points of a further <spacetime extension>. One way to establish maximality is <geodesic completeness>. The <two-dimensional anti-de Sitter spacetime> factor embeds locally as $Z\cdot Z=-m^2$ in a flat space with <metric signature> $(-,-,+)$. Along an affinely parametrized <geodesic> with constant $\epsilon=\dot Z\cdot\dot Z$, the ambient equation is
$$
\ddot Z=\frac{\epsilon}{m^2}Z.
$$
Its oscillatory, linear or hyperbolic solutions for negative, zero or positive $\epsilon$ exist for all real <affine parameters> and lift to the <universal cover>. The <two-sphere> is also <geodesically complete>, and <geodesics> of the product split into <geodesics> of its factors. Hence $\widetilde g$ is <geodesically complete>. A proper regular <spacetime extension> would furnish a <geodesic> reaching its boundary in finite <affine parameter>, contradicting that completeness. This proves that the displayed simply connected analytic <spacetime> is maximal.
Back to article page
