= Solution
The physical <spherical coordinates> cover $t\in\mathbb R$, $r\in[0,\infty)$, $\theta\in[0,\pi]$, and $\phi\in[0,2\pi)$ with periodic identification. The center and the polar axes are the usual <coordinate singularities>, rather than physical <curvature singularities>; a regular polar chart restricts $r>0$ and $0<\theta<\pi$. For the <null coordinates>,
$$
u=t-r,\quad v=t+r,\qquad t=\frac{u+v}{2},\quad r=\frac{v-u}{2},\qquad u,v\in\mathbb R,\ v\geq u.
$$
Thus the physical <metric tensor> is
$$
d\hat s^2=du\,dv-\frac{(v-u)^2}{4}d\Sigma^2.
$$
The <conformal transformation> followed by $p=\arctan u$, $q=\arctan v$ gives $du=(1+u^2)dp$, $dv=(1+v^2)dq$, and
$$
\frac{v-u}{\sqrt{(1+u^2)(1+v^2)}}=\sin(q-p).
$$
Consequently $ds^2=4dp\,dq-\sin^2(q-p)d\Sigma^2$. Finally $T=p+q$, $R=q-p$ converts $4dp\,dq$ to $dT^2-dR^2$, yielding
$$
\boxed{ds^2=dT^2-dR^2-\sin^2R\,d\Sigma^2.}
$$
The ranges must retain their joint constraints:
$$
-\frac\pi2<p,q<\frac\pi2,\quad q\geq p;\qquad -\pi<T<\pi,\quad0\leq R<\pi,\quad |T|+R<\pi.
$$
Equivalently, $-\pi<T-R<\pi$ and $-\pi<T+R<\pi$, with $R\geq0$. The angles retain their original ranges and identifications. Extending $R$ to negative values would double-count the radial polar geometry. The <conformal factor> is
$$
\boxed{\Omega=2\cos p\cos q=\cos T+\cos R>0\ \text{in the physical region},\qquad g=\Omega^2\hat g.}
$$
The unphysical <metric tensor> extends onto the <Einstein static universe>, while physical <Minkowski spacetime> occupies the triangular region in its radial <Penrose diagram>. The edge $R=0$ is the regular center and belongs to the physical region away from its endpoints.
The <conformal boundary> consists of the following infinities. <Future null infinity> has $q=\pi/2$, hence $T+R=\pi$ with $0<R<\pi$, and corresponds to finite $u$ with $v\to+\infty$. <Past null infinity> has $p=-\pi/2$, hence $T-R=-\pi$, and corresponds to finite $v$ with $u\to-\infty$. <Future timelike infinity> is $(T,R)=(\pi,0)$, <past timelike infinity> is $(-\pi,0)$, and <spacelike infinity> is $(0,\pi)$. Although angles label points on null infinity, their two-spheres collapse at these three endpoint locations in the compactified picture.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-60-minkowski-compactification.png]
{title=The radial Minkowski conformal diagram and the endpoints of timelike, null and spacelike geodesics}
For the <geodesic endpoints in Minkowski conformal compactification>, one can check the limits directly. A timelike inertial trajectory is $\mathbf x=\mathbf b+\mathbf w t$ with $|\mathbf w|<1$. At $t\to+\infty$ both $t-r$ and $t+r$ tend to $+\infty$, so $p,q\to\pi/2$ and the trajectory ends at <future timelike infinity>. At $t\to-\infty$ both tend to $-\infty$, giving <past timelike infinity>. This includes a stationary inertial worldline with bounded $r$.
For a null inertial trajectory, $|\mathbf w|=1$. As $t\to+\infty$, $r=t+\mathbf w\cdot\mathbf b+o(1)$, so $u$ has a finite limit while $v\to+\infty$; its future endpoint lies on <future null infinity>. As $t\to-\infty$, $r=-t-\mathbf w\cdot\mathbf b+o(1)$, giving a finite limiting $v$ and $u\to-\infty$, hence a past endpoint on <past null infinity>. Radial <null geodesic> paths have $T\pm R$ constant. A ray passing through the center changes its angular direction; in the angle-suppressed radial diagram it reflects at $R=0$ rather than terminating there.
A <spacelike geodesic> is also a straight line, now parametrized so that $|d\mathbf x/d\lambda|>|dt/d\lambda|$. At either end, $r>|t|$ asymptotically, $u\to-\infty$, $v\to+\infty$, and the endpoint is <spacelike infinity>. A radial spacelike line at $t=0$ has two opposite-angle halves which project onto the same horizontal interval in the diagram.
The <conformal compactification> endpoints occur at finite $T$ but at infinite physical <proper time> or <affine parameter>; <Minkowski spacetime> is <geodesically complete>. The <conformal preservation of null geodesic paths> does not preserve an <affine parameter>: if $\lambda$ is physically affine, an unphysical <affine parameter> satisfies $d\widetilde\lambda/d\lambda=\Omega^2$ up to a constant. Along a complete null ray this can have a finite integral at infinity. Timelike and spacelike physical <geodesic> paths are not generally <geodesics> of the rescaled <metric tensor>, so their drawn curves must not be interpreted as freely falling unphysical observers.
Finally, every physical event can send an outgoing null ray to <future null infinity>, so $J^-(\mathcal I^+)$ is all of <Minkowski spacetime>. Its <black-hole region> and intrinsic <event horizon> are empty. The null edges in the diagram are infinity, not horizons concealing part of the physical manifold. Complete inertial observers likewise have no <observer event horizon>. Accelerated observers restricted to a <Rindler wedge> can have a <Rindler horizon>, but that is observer-dependent and does not contradict the <horizon-free Minkowski spacetime>. \b[The conformal diagram compactifies infinity; it does not introduce a physical horizon.]
Back to article page
