Null Geodesics and Light Cones

A Lorentzian metric does one thing before it does anything else: at every event it sorts the tangent vectors into three classes by the sign of $g_{\mu\nu}V^\mu V^\nu$ — timelike (negative), spacelike (positive), and null (zero). Massive bodies thread the timelike interior; light and every other massless field ride the null boundary between what can be reached and what cannot. The curves that trace that boundary as economically as the geometry allows — everywhere null, and parallel-transporting their own tangent — are the null geodesics, and the surface they sweep out from an event is its light cone. Everything causal in general relativity is written in this one object. Two conditions define a light ray at once. It is null, $k_\mu k^\mu=0$ for the tangent $k^\mu=\mathrm{d}x^\mu/\mathrm{d}\lambda$ , so no proper time elapses along it and it carries zero interval between any two of its points. And it is a geodesic, $k^\nu\nabla_\nu k^\mu=0$ , its tangent carried forward unbent. Because there is no arc length to lean on, the only natural clock a photon has is an affine parameter, fixed by the geometry up to $\lambda\to a\lambda+b$ ; the physical observables are never $\lambda$ itself but ratios built from it — frequencies and redshifts. Three deep facts organize the rest. Null geodesics generate the boundaries $\partial J^{\pm}$ of the causal future and past, so they encode which events can signal which. They are conformally invariant, unchanged as point sets under $g_{\mu\nu}\to\Omega^2 g_{\mu\nu}$ — the reason light cones survive being drawn on a finite Penrose diagram. And their neighbouring rays focus under the Raychaudhuri equation, the engine behind trapped surfaces and the singularity theorems.

Key takeaways

A null geodesic is a light ray: a curve whose tangent $k^\mu$ is everywhere null, $g_{\mu\nu}k^\mu k^\nu=0$ , and parallel-transported along itself, $k^\nu\nabla_\nu k^\mu=0$ .
The null geodesics through an event sweep out its light cone; extended to the future they generate $\partial J^+(p)$ , the boundary of the causal future, so they carry the entire causal structure of a spacetime.
Proper time is useless as a label ( $\mathrm{d}\tau=0$ along the ray); an affine parameter $\lambda$ is supplied intrinsically by parallel transport, fixed only up to $\lambda\to a\lambda+b$ .
Null geodesics are conformally invariant — under $g_{\mu\nu}\to\Omega^2 g_{\mu\nu}$ they remain null geodesics (affine parameter rescaled) — which is exactly why light cones stay at 45° on a conformal diagram.
Observed frequency is the projection $\omega=-k_\mu u^\mu$ onto an observer's 4-velocity; only ratios of these — redshifts — are affine-independent physics.
Congruences of null geodesics obey the Raychaudhuri equation, and gravity always focuses light for matter obeying the null energy condition, driving beams to caustics and seeding the singularity theorems.
Key idea: A light ray is the unique curve that is both null and self-parallel; affineness fixes its parametrization, conformal invariance fixes that it depends only on the causal structure, and the Raychaudhuri equation governs how neighbouring rays focus.

The core insight

A light ray is the unique curve that is both null and self-parallel; affineness fixes its parametrization, conformal invariance fixes that the curve depends only on the causal structure, and the Raychaudhuri equation governs how neighbouring rays focus into caustics.

Where this lives today

Null geodesics are the working instrument of strong-gravity astronomy. Every image of a black hole — the Event Horizon Telescope's reconstructions of M87\ and Sagittarius A\, and every synthetic frame of Interstellar's Gargantua — is made by integrating null geodesics backward from a virtual camera, with the photon-sphere result of Worked Example 1 setting the shadow size at $b_{\rm crit}=3\sqrt3\,M$ . Gravitational lensing, from cluster arcs to weak-lensing shear maps, is null-geodesic propagation through inhomogeneous spacetime. In gravitational-wave astronomy, the multi-messenger event GW170817 sent a neutron-star merger's gravitational and electromagnetic signals along the same null cone; their arrival within seconds after 130 million years bounded the speed of gravity to that of light at the $10^{-15}$ level — a direct test that gravitational waves ride null geodesics. At the frontier, the focusing of null congruences underwrites the black-hole area theorem, the conformal boundary $\mathscr{I}$ hosts the BMS symmetries and the memory effect, and cosmic censorship — whether trapped-surface focusing always hides its singularity — remains one of the deepest open problems in classical relativity.

The physical picture

Set off a flash of light at an event $p$ . The spherical wavefront expanding at $c$ traces, in spacetime, a cone — the future light cone. Each individual photon worldline is one straight ruling of that cone, and each ruling is a null geodesic. The cone is not an abstraction laid over the physics; it is literally the union of the light rays, and its interior is the set of events a massive body launched from $p$ could still reach. The first figure makes this anatomy concrete: null generators on the surface, the timelike causal future inside, the spacelike elsewhere outside, and the mirror-image past cone below.

Figure

A flash at event p and its light cone. The future cone is the union of null-geodesic generators, each a photon worldline with ds² = 0. Its timelike interior is the causal future J⁺(p) reachable by massive bodies at event q; the spacelike exterior is the elsewhere that no signal from p can reach, such as event s. In Minkowski coordinates every generator runs at the speed of light, a 45° cone

Because light has no rest frame and no proper time elapses along its path, "how far along the ray" cannot be a length — it is an affine parameter, fixed only up to scale and origin. The observable is never the parameter but a ratio: project the tangent $k^\mu$ onto an observer's 4-velocity and $\omega=-k_\mu u^\mu$ is that observer's frequency; compare two observers and the ratio is a redshift, blind to how $\lambda$ was scaled. Conformal invariance is the deeper secret: stretching the metric by any positive factor cannot turn a null vector timelike or spacelike, so cones — and hence causality — survive rescaling, letting an infinite spacetime be bent onto a finite page. The last idea is focusing. Neighbouring rays in a beam converge or diverge at a rate called the expansion $\theta$ , and gravity, entering the Raychaudhuri equation through the Ricci tensor, always bends a bundle of light toward itself when matter obeys the null energy condition. Push a converging beam and it must reach a caustic where rays cross — the seed of trapped surfaces. The quantitative face of this lives in a black hole, where light orbits at a fixed radius; the second figure plots the effective potential that pins that orbit.

Figure

The Schwarzschild photon effective potential V(r) = (1/r²)(1 − 2M/r) for a light ray in geometric units M = 1, L = 1. It has a single maximum at the photon sphere r = 3M where V = 1/27 ≈ 0.0370, an unstable circular light orbit. A ray with E²/L² above the peak is captured; the critical value E²/L² = 1/27 gives the capture impact parameter b = 3√3 M ≈ 5.196 M, which sets the edge of a black-hole shadow

The core equations

A light ray obeys two statements simultaneously, and neither implies the other. The first is that the tangent has zero length everywhere — the ray lies on the local light cone at each point it passes.

g_{\mu\nu}\,k^\mu k^\nu = 0,\qquad k^\mu=\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda} \qquad (1)

where: $g_{\mu\nu}$ — the spacetime metric (signature $-+++$ ); $k^\mu$ — the tangent to the ray, the wave 4-vector in optics; $\lambda$ — an affine parameter along the ray. Read it as: the ray never leaves the null cone, so no observer ever measures it moving slower or faster than light locally. The second statement is that the tangent is carried forward by parallel transport with nothing left over — the geodesic condition, in the affine form that has zero right-hand side. Expanding the covariant derivative gives a second-order ODE for the coordinates.

k^\nu\nabla_\nu k^\mu = \frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\lambda^2}+\Gamma^\mu{}_{\alpha\beta}\frac{\mathrm{d}x^\alpha}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\beta}{\mathrm{d}\lambda}=0 \qquad (2)

where: $\nabla_\nu$ — the Levi-Civita covariant derivative; $\Gamma^\mu{}_{\alpha\beta}$ — the Christoffel symbols of $g$ . The clean zero on the right holds only for an affine $\lambda$ ; any other label puts a term $\propto k^\mu$ there, which bends nothing but spoils the bookkeeping. Metric compatibility makes the null value $k_\mu k^\mu=0$ propagate: it is set once and conserved for all $\lambda$ . Frequency is not intrinsic to the ray — it belongs to whoever measures it. Contracting the tangent with an observer's 4-velocity gives that observer's angular frequency; the ratio at two ends of the ray is the physical, affine-independent redshift.

\omega = -\,k_\mu u^\mu,\qquad \frac{\omega_{\rm rec}}{\omega_{\rm emit}}=\frac{(k_\mu u^\mu)_{\rm rec}}{(k_\mu u^\mu)_{\rm emit}} \qquad (3)

where: $u^\mu$ — the observer's 4-velocity, $u_\mu u^\mu=-1$ ( $c=1$ ); $\omega$ — the frequency that observer sees (its absolute value depends on the arbitrary scaling of $\lambda$ , but the ratio does not). Read it as: the affine ambiguity cancels in every observable, so only shifts, never bare frequencies, are physics. The single most powerful structural property is that the whole construction depends only on the metric up to a positive scale. Under a conformal rescaling the null cones are untouched, and null geodesics map to null geodesics.

\tilde g_{\mu\nu}=\Omega^2 g_{\mu\nu}\ \Rightarrow\ \tilde k^\nu\tilde\nabla_\nu k^\mu = 2\big(k^\nu\partial_\nu\ln\Omega\big)\,k^\mu\ \propto\ k^\mu \qquad (4)

where: $\Omega>0$ — the conformal factor; the right-hand side is purely tangential, so the curve is still a geodesic, merely re-labelled. Read it as: causality is a conformal invariant — which events can signal which is decided before any scale is chosen, and this is what makes Penrose–Carter diagrams faithful. For a bundle of neighbouring rays, the expansion $\theta=\nabla_\mu k^\mu$ measures the fractional growth rate of the beam's cross-section, and its evolution is governed by shear, twist, and curvature — the null Raychaudhuri equation.

\frac{\mathrm{d}\theta}{\mathrm{d}\lambda}=-\tfrac{1}{2}\theta^2-\sigma_{\mu\nu}\sigma^{\mu\nu}+\omega_{\mu\nu}\omega^{\mu\nu}-R_{\mu\nu}k^\mu k^\nu \qquad (5)

where: $\sigma_{\mu\nu}$ — shear (trace-free distortion); $\omega_{\mu\nu}$ — twist (rotation) of the congruence; $R_{\mu\nu}$ — the Ricci tensor, with $R_{\mu\nu}k^\mu k^\nu\ge0$ the null energy condition via Einstein's equations. Read it as: every term except the twist drives $\theta$ downward, so gravity focuses light — the mathematical root of the singularity theorems. In the high-frequency limit a wave $\psi\sim A\,e^{iS}$ has its phase surfaces governed by the eikonal equation, and the null geodesics are exactly the rays orthogonal to those surfaces — the bridge from wave optics to light rays.

g^{\mu\nu}\partial_\mu S\,\partial_\nu S=0,\qquad k_\mu=\partial_\mu S \qquad (6)

where: $S$ — the rapidly-varying phase; $k_\mu=\partial_\mu S$ — the wave covector, automatically null and (being a gradient) twist-free. Read it as: geometric optics is the leading term of wave optics, and its "rays" are precisely null geodesics of the metric.

Derivation ANull geodesic and null condition from the energy action

The goal is to derive the affine geodesic equation and the null condition together from a single variational principle, and to see why the usual arc-length action is helpless for light while a quadratic "energy" action succeeds. The square-root Lagrangian $\int\sqrt{-g_{\mu\nu}\dot x^\mu\dot x^\nu}\,\mathrm{d}\lambda$ integrates to zero along a null curve and its variation is ill-defined there; the quadratic action sidesteps this, gives the same geodesics for every signature, and hands us an affine parameter for free — the one in which the tangent norm is conserved, so the null value can be imposed once and holds everywhere.

Step 1. Adopt the energy action $S=\tfrac12\int g_{\mu\nu}\dot x^\mu\dot x^\nu\,\mathrm{d}\lambda$ with $\dot x^\mu\equiv\mathrm{d}x^\mu/\mathrm{d}\lambda$ , and vary $x^\mu$ to get the Euler–Lagrange equation $\frac{\mathrm{d}}{\mathrm{d}\lambda}(g_{\mu\nu}\dot x^\nu)-\tfrac12\partial_\mu g_{\alpha\beta}\dot x^\alpha\dot x^\beta=0$ . Why: unlike the arc length, this Lagrangian is smooth and nonzero on null curves, so its variation is legitimate for light — the price is that it is no longer reparametrization-invariant, which is exactly what pins down an affine parameter.

Step 2. Expand the total derivative and symmetrize: $g_{\mu\nu}\ddot x^\nu+\tfrac12(\partial_\alpha g_{\mu\beta}+\partial_\beta g_{\mu\alpha}-\partial_\mu g_{\alpha\beta})\dot x^\alpha\dot x^\beta=0$ , where the bracket is $g_{\mu\lambda}\Gamma^\lambda{}_{\alpha\beta}$ . Why: the particular antisymmetric-looking combination of metric derivatives is engineered to reassemble the Christoffel symbol, so the field equation is secretly the covariant one.

Step 3. Contract with the inverse metric to reach $\ddot x^\mu+\Gamma^\mu{}_{\alpha\beta}\dot x^\alpha\dot x^\beta=0$ , which is Eq. (2). Why: this is the affine geodesic equation with no right-hand side — it holds identically for timelike, spacelike, and null geodesics, so light is not a special case but the null member of one family.

Step 4. Note that the Lagrangian has no explicit $\lambda$ -dependence, so its value $g_{\mu\nu}\dot x^\mu\dot x^\nu$ is conserved along the solution; choose that constant to be $0$ . Why: the null condition $g_{\mu\nu}k^\mu k^\nu=0$ is therefore a first integral, imposed as an initial condition and preserved automatically — this is the precise sense in which "null" and "geodesic" are compatible but logically independent statements. *In flat space $\Gamma=0$ gives $\ddot x^\mu=0$ , so the solutions are straight lines $x^\mu=x_0^\mu+\lambda k^\mu$ with $\eta_{\mu\nu}k^\mu k^\nu=0$ ; the conserved norm is the affine hallmark.*

Go deeper · Derivation B · 1 minNull geodesics are conformally invariant

Derivation BNull geodesics are conformally invariant

The goal is to prove the claim that makes conformal diagrams honest: that multiplying the metric by any positive function $\Omega^2$ leaves the set of null geodesics unchanged, merely rescaling their affine parameter — and to isolate the single term whose vanishing (thanks to nullity) makes the result hold for light but fail for massive particles. We track how the Christoffel symbols transform under $\tilde g_{\mu\nu}=\Omega^2 g_{\mu\nu}$ , feed the change into the geodesic operator, and watch the metric-dependent piece disappear precisely because $k_\mu k^\mu=0$ .

Step 1. Under $\tilde g_{\mu\nu}=\Omega^2 g_{\mu\nu}$ the connection shifts by $\tilde\Gamma^\mu{}_{\alpha\beta}=\Gamma^\mu{}_{\alpha\beta}+\delta^\mu_\alpha\partial_\beta\ln\Omega+\delta^\mu_\beta\partial_\alpha\ln\Omega-g_{\alpha\beta}\partial^\mu\ln\Omega$ . Why: a conformal factor is not a mere coordinate change, so the connection genuinely changes; the three extra terms are the entire content of that change, and only the last one carries the metric $g_{\alpha\beta}$ .

Step 2. Insert this into $\tilde k^\nu\tilde\nabla_\nu k^\mu$ with the same tangent $k^\mu$ , giving $k^\nu\nabla_\nu k^\mu+2(k^\nu\partial_\nu\ln\Omega)k^\mu-(g_{\alpha\beta}k^\alpha k^\beta)\partial^\mu\ln\Omega$ . Why: the two $\delta$ -terms collapse into a single tangential piece $\propto k^\mu$ , while the $g_{\alpha\beta}$ -term is the only one pointing off the ray — it is the potential troublemaker.

Step 3. Impose nullity $g_{\alpha\beta}k^\alpha k^\beta=0$ so the off-ray term vanishes, leaving $\tilde k^\nu\tilde\nabla_\nu k^\mu=2(k^\nu\partial_\nu\ln\Omega)\,k^\mu\propto k^\mu$ . Why: a right-hand side proportional to $k^\mu$ is still a geodesic — just non-affinely parametrized — so the curve survives intact; for a timelike geodesic $g_{\alpha\beta}k^\alpha k^\beta=-1\ne0$ and the extra term bends the path, which is why only light cones are conformally invariant.

Step 4. Restore affineness by rescaling $\lambda\to\tilde\lambda$ with $\mathrm{d}\tilde\lambda/\mathrm{d}\lambda=\Omega^2$ (up to constants). Why: the residual $\propto k^\mu$ source is removable by a one-line reparametrization, so in the rescaled metric the very same points form an affinely-parametrized null geodesic — the light ray is a conformal invariant as a curve. *As a check, for $\Omega=$ const the result reduces to the identity; the surviving proportionality-to- $k^\mu$ is exactly the removable non-affineness.*

Go deeper · Derivation C · 1 minRaychaudhuri focusing and the inevitability of caustics

Derivation CRaychaudhuri focusing and the inevitability of caustics

The goal is to show that a bundle of light rays which is momentarily converging must, under gravity plus the null energy condition, focus to a caustic within a bounded affine distance — the local mechanism behind trapped surfaces and the Penrose singularity theorem. We start from the null Raychaudhuri equation for the expansion, specialize to the hypersurface-orthogonal (twist-free) congruences that generate null surfaces, drop everything that only helps focusing, and integrate the resulting inequality to bound the distance to $\theta\to-\infty$ .

Step 1. Take Eq. (5) and set the twist to zero, $\omega_{\mu\nu}=0$ , which holds for the generators of any null hypersurface such as $\partial J^+(p)$ or an event horizon. Why: a gradient wave covector $k_\mu=\partial_\mu S$ is automatically twist-free, so the surface-forming congruences that carry all the causal information are exactly the ones this simplification applies to.

Step 2. Discard the shear and curvature terms as manifestly non-positive: with the null energy condition $R_{\mu\nu}k^\mu k^\nu\ge0$ and $\sigma_{\mu\nu}\sigma^{\mu\nu}\ge0$ , the equation becomes the inequality $\mathrm{d}\theta/\mathrm{d}\lambda\le-\tfrac12\theta^2$ . Why: both dropped terms subtract from $\mathrm{d}\theta/\mathrm{d}\lambda$ , so ignoring them only weakens the bound — whatever focusing the inequality forces, the full equation forces at least as strongly.

Step 3. Rewrite as $\mathrm{d}(\theta^{-1})/\mathrm{d}\lambda\ge\tfrac12$ and integrate to $\theta^{-1}(\lambda)\ge\theta_0^{-1}+\tfrac12\lambda$ . Why: dividing by $-\theta^2$ turns the nonlinear inequality into a linear one for the reciprocal, which integrates immediately and makes the blow-up transparent.

Step 4. Suppose the beam starts converging, $\theta_0<0$ . Then $\theta^{-1}$ climbs from a negative value through zero, so $\theta\to-\infty$ at an affine distance no greater than $\Delta\lambda\le 2/|\theta_0|$ . Why: $\theta\to-\infty$ is a caustic — neighbouring rays cross and the cross-sectional area collapses — and this is precisely the ingredient Penrose combined with a non-compact Cauchy surface to prove geodesic incompleteness inside a trapped surface. *As a concrete illustration, a Schwarzschild ingoing null shell with initial expansion $\theta_0<0$ caustics within affine distance $\Delta\lambda\le2/|\theta_0|$ , in agreement with the reciprocal bound derived above.*

Limiting cases

Special regimes make the behaviour of null geodesics and their cones concrete:

Case	Result	Meaning
Flat spacetime (Minkowski)	$x^\mu(\lambda)=x_0^\mu+\lambda\,k^\mu,\ \eta_{\mu\nu}k^\mu k^\nu=0$	Null geodesics are straight lines along null vectors; light cones are exact 45° cones in every inertial frame.
Conformal rescaling	$g_{\mu\nu}\to\Omega^2 g_{\mu\nu}$ : same null geodesics, $\tilde\lambda$ rescaled	The rays are unchanged as curves; the new affine parameter obeys $\mathrm{d}\tilde\lambda/\mathrm{d}\lambda=\Omega^2$ . Causality is conformally invariant.
Geometric-optics (eikonal) limit	$k_\mu=\partial_\mu S,\ g^{\mu\nu}\partial_\mu S\,\partial_\nu S=0$	Surfaces of constant phase of a high-frequency wave obey the eikonal equation; their normals are null geodesics.
Schwarzschild photon sphere	$r=3M,\ b_{\rm crit}=3\sqrt3\,M\approx5.196\,M$	Unstable circular null geodesics orbit at $r=3M$ and set the black-hole shadow radius $3\sqrt3\,M$ .
FLRW cosmology	$1+z=\dfrac{a(t_{\rm rec})}{a(t_{\rm emit})}$	Light on a radial null geodesic in an expanding universe is redshifted exactly by the ratio of scale factors.

Common trapThe coordinate speed of light is always c, so light cones are always 45°

Only in Minkowski coordinates, or on a conformal diagram built to keep them there. What is truly invariant is that light travels on the local null cone, and that any physical observer measures its speed to be $c$ as it passes. The coordinate speed is a different animal and is routinely not $c$ : for a radial ray in Schwarzschild coordinates the null condition gives $\mathrm{d}r/\mathrm{d}t=\pm(1-2M/r)$ , which shrinks to zero as $r\to2M$ , so the light cones "close up" and appear to tip over near the horizon — the origin of the Shapiro time delay and of the coordinate divergence at $r=2M$ . Nothing pathological is happening to the light; the coordinates are stretching. Two companion traps: never parametrize a null geodesic by proper time or arc length — both vanish identically ( $\mathrm{d}\tau=\mathrm{d}s=0$ ), so you must use an affine $\lambda$ ; and "null" and "geodesic" are independent conditions — a curve can be null without being geodesic, and the measured frequency $-k_\mu u^\mu$ depends on the affine scale, so only frequency ratios are physical.

Worked examples

1 · The photon sphere and the shadow of a black hole (fully worked)

A light ray moves in the equatorial plane of a Schwarzschild black hole of mass $M$ (geometric units $G=c=1$ ). Find the radius of the unstable circular photon orbit, the critical impact parameter that separates capture from escape, and the angular radius of the resulting shadow for Sagittarius A\ and M87\.

Step 1. The Killing vectors $\partial_t$ and $\partial_\phi$ give two conserved charges along the affinely-parametrized ray, $E=(1-2M/r)\dot t$ and $L=r^2\dot\phi$ (dots are $\mathrm{d}/\mathrm{d}\lambda$ ). These are constant only because $\lambda$ is affine — a non-affine label would let each drift. The null condition $g_{\mu\nu}\dot x^\mu\dot x^\nu=0$ then collapses to a one-dimensional radial problem, $\dot r^2=E^2-V(r)$ with effective potential $V(r)=\dfrac{L^2}{r^2}\Big(1-\dfrac{2M}{r}\Big)$ .

Step 2. A circular orbit needs $\dot r=0$ and $\ddot r=0$ , i.e. $V'(r)=0$ . Differentiating, $V'(r)=L^2\big(-2r^{-3}+6Mr^{-4}\big)=0\Rightarrow -2r+6M=0$ , so $r_{\rm ph}=3M$ — the photon sphere. It is a maximum of $V$ , hence unstable: a ray perturbed inward plunges, one perturbed outward escapes.

Step 3. At $r=3M$ the potential is $V(3M)=\dfrac{L^2}{9M^2}\big(1-\tfrac23\big)=\dfrac{L^2}{27M^2}$ . A circular orbit has $E^2=V$ , so the critical impact parameter is $b_{\rm crit}=\dfrac{L}{E}=\sqrt{27}\,M=\mathbf{3\sqrt3\,M\approx5.196\,M}$ . Rays with $b<b_{\rm crit}$ are captured; $b>b_{\rm crit}$ escape; $b=b_{\rm crit}$ spirals asymptotically onto the photon sphere. This $b_{\rm crit}$ , not the horizon at $2M$ , is the apparent edge of the dark shadow.

Step 4. Project onto the sky. For Sagittarius A\, $M\approx4.15\times10^6\,M_\odot$ gives $GM/c^2\approx6.13\times10^6\ \mathrm{km}$ , so $b_{\rm crit}\approx3.18\times10^7\ \mathrm{km}$ ; at $D\approx8.15\ \mathrm{kpc}$ the angular radius is $b_{\rm crit}/D\approx\mathbf{26\ \mu as}$ (diameter $\approx52\ \mu\mathrm{as}$ ). For M87\, $M\approx6.5\times10^9\,M_\odot$ at $D\approx16.8\ \mathrm{Mpc}$ gives an angular radius $\approx\mathbf{20\ \mu as}$ (diameter $\approx40\ \mu\mathrm{as}$ ). Both match the bright rings imaged by the Event Horizon Telescope — the shadow is drawn by null geodesics, not by luminous matter.

2 · Gravitational redshift from a compact star (numbers)

Light of frequency $\omega_{\rm emit}$ leaves the surface of a static star at areal radius $R$ and is received by a distant static observer. Using the conserved quantity along the null geodesic, find the redshift for the Sun and for a $1.4\,M_\odot$ neutron star of radius $10\ \mathrm{km}$ .

Step 1. In a static spacetime the timelike Killing vector $\xi^\mu=\partial_t$ makes $E=-k_\mu\xi^\mu$ constant along the ray. A static observer's 4-velocity is the normalized Killing vector, $u^\mu=\xi^\mu/\sqrt{-g_{tt}}$ , so the measured frequency is $\omega=-k_\mu u^\mu=E/\sqrt{-g_{tt}}$ with $-g_{tt}=1-2M/r$ in Schwarzschild.

Step 2. Taking the ratio between emission at $R$ and reception at $r\to\infty$ (where $-g_{tt}\to1$ ), the affine scale and $E$ cancel: $\dfrac{\omega_{\rm rec}}{\omega_{\rm emit}}=\sqrt{\dfrac{1-2M/R}{1}}$ , hence $1+z=\dfrac{\omega_{\rm emit}}{\omega_{\rm rec}}=\dfrac{1}{\sqrt{1-2GM/Rc^2}}$ . The light climbs out of the potential well and loses frequency.

Step 3. For the Sun, $2GM/Rc^2=R_s/R=2.95/6.96\times10^{5}=4.24\times10^{-6}$ , so $z=(1-4.24\times10^{-6})^{-1/2}-1\approx\mathbf{2.1\times10^{-6}}$ — the classic solar redshift, equivalent to a $636\ \mathrm{m/s}$ Doppler shift and confirmed spectroscopically.

Step 4. For the neutron star, $2GM/Rc^2=(2\times1.4\times1.477\ \mathrm{km})/10\ \mathrm{km}=0.413$ , giving $z=(1-0.413)^{-1/2}-1\approx\mathbf{0.31}$ . A spectral line at 500 nm arrives at 654 nm — a 31% shift, and the reason neutron-star surface spectra are a probe of the mass-to-radius ratio. The same formula diverges as $R\to2M$ : light from the horizon is infinitely redshifted.

Physics of the portrait

Null geodesics became the central objects of modern relativity when a generation realized that the global organization of light cones — not the local metric — is where black holes, singularities, and infinity actually live. No one did more to make that shift than Roger Penrose, who turned the light cone from a picture into a calculus.

Origins · 1 minRoger Penrose and how this was discovered

Roger Penrose

b. 1931 · mathematical physicist who rebuilt causal structure from null geodesics and conformal methods

Penrose recognized that the causal skeleton of a spacetime is carried entirely by its null geodesics, and that this skeleton is conformally invariant — facts he forged into two of the most consequential tools in general relativity. In the early 1960s he introduced the conformal treatment of infinity, compactifying an infinite spacetime by a rescaling $\tilde g=\Omega^2 g$ that keeps all light cones at 45°, so that outgoing radiation could be read off at a sharp boundary he named null infinity, or "scri" ( $\mathscr{I}$ ); the same construction gives the Penrose–Carter diagrams that put a black hole's global structure on a single page. Then in 1965, in a three-page paper of astonishing economy, he combined the focusing of null-geodesic congruences with the topology of a trapped surface to prove that gravitational collapse must produce a singularity, with no symmetry assumed. This was the first of the singularity theorems; extended with Stephen Hawking, it showed the Big Bang and black-hole singularities to be generic features of Einstein's theory. Penrose also introduced spinor and twistor methods, the Penrose process, and the still-open cosmic-censorship conjecture, and shared the 2020 Nobel Prize in Physics for showing that black-hole formation is a robust prediction of general relativity. Nearly every idea in this article — that light cones generate causal structure, survive conformal rescaling, and focus into singularities — is one he set in its modern form.

History

1908

— Minkowski introduces spacetime and the light cone in his Cologne lecture "Raum und Zeit"

1955

— Raychaudhuri derives the focusing equation for geodesic congruences

1963

— Penrose brings infinity to a finite conformal boundary, defining null infinity

1965

— Penrose's singularity theorem uses null-geodesic focusing and trapped surfaces

2019

— the Event Horizon Telescope images the photon ring around M87\*

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §22 (geodesics) and §34 (causal structure) · Wald, General Relativity, §3.3, §8, §9 (causal structure, Raychaudhuri, singularity theorems) · Carroll, Spacetime and Geometry, §3.3 (geodesics), §5.4 (Schwarzschild null orbits) · Hartle, Gravity, §8–9 (geodesics, light bending) · Schutz, A First Course in General Relativity, §6 · Hawking & Ellis, The Large Scale Structure of Space-Time, §4 (null congruences). Key results collected here for reference: the affine geodesic equation and null first integral follow from the energy action; the conformal transformation of the connection leaves a surviving $\propto k^\mu$ source that nullity eliminates; the null Raychaudhuri equation yields the reciprocal focusing bound $\theta^{-1}\ge\theta_0^{-1}+\tfrac12\lambda$ ; the Schwarzschild photon sphere sits at $r=3M$ with $b_{\rm crit}=3\sqrt3\,M\approx5.196\,M$ , giving shadow angular radii $\approx26\ \mu\mathrm{as}$ (Sgr A\, $M=4.15\times10^6\,M_\odot$ , $D=8.15\ \mathrm{kpc}$ ) and $\approx20\ \mu\mathrm{as}$ (M87\, $M=6.5\times10^9\,M_\odot$ , $D=16.8\ \mathrm{Mpc}$ ); and the gravitational redshift $1+z=(1-2GM/Rc^2)^{-1/2}$ gives $z\approx2.1\times10^{-6}$ (Sun) and $z\approx0.31$ (1.4 $M_\odot$ , 10 km neutron star).

Prerequisites. Lorentzian Metric and the Geodesic Equation · Christoffel Symbols and the Covariant Derivative · Causal Structure of Spacetime. Leads to. Chronological and Causal Futures and Pasts · Achronal Sets and Edges · Conformal Transformations and Causal Invariance · Penrose–Carter Conformal Diagrams · Null Infinity (Scri) · Global Hyperbolicity.

Ask the tutor: Want to see how a converging beam of light is forced to a caustic, or why a black hole's shadow is set by rays that never touch it — ask, and we will trace a null geodesic through a real spacetime together.

Why can't proper time parametrize a light ray?

Along a null curve the line element vanishes identically: $\mathrm{d}s^2=g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu=0$ between any two of its points, so proper time $\mathrm{d}\tau$ never advances — a photon carries no clock of its own. You must use an affine parameter $\lambda$ , which is not borrowed from any ruler but supplied by the geometry through the demand that the tangent parallel-transport, $k^\nu\nabla_\nu k^\mu=0$ . It is fixed only up to $\lambda\to a\lambda+b$ , since there is no normalization $g_{\mu\nu}k^\mu k^\nu=\pm1$ to pin the scale as there is for massive or spacelike geodesics. Physically the affine parameter is the carrier of frequency: comparing the tangent against observers at emission and reception yields the redshift.

What does it mean that null geodesics are conformally invariant, and why does it matter?

It means that if you multiply the metric by any positive function, $g_{\mu\nu}\to\Omega^2 g_{\mu\nu}$ , the very same curves remain null geodesics — only their affine parameter rescales. The reason is that the one term in the transformed geodesic equation that could bend the ray is multiplied by $g_{\mu\nu}k^\mu k^\nu$ , which is zero for light. It matters because it means causality — which events can send signals to which — depends only on the metric up to scale. That is what lets a Penrose–Carter diagram compress an infinite spacetime onto a finite page while keeping every light cone at exactly 45°: the rescaling changes distances and times drastically but leaves the causal skeleton, carried by the null geodesics, perfectly intact.

Is the light cone the same thing as the event horizon?

No, though they are intimately related. A light cone is a local object: at every event, the null vectors there define which nearby events can be causally influenced. An event horizon is a global object: the boundary of the region from which null geodesics can ever escape to infinity — you cannot know where it is without following light rays all the way out. The horizon is in fact a special null surface, generated by null geodesics that neither escape nor fall in, so it is built from light rays. But whereas a light cone exists at every event in every spacetime, a horizon exists only when the global arrangement of cones traps a region. Confusing the two is a common source of error in reading black-hole diagrams.

Do gravitational waves travel on null geodesics too?

In the high-frequency (geometric-optics) limit, yes. A gravitational wave packet of wavelength short compared to the background curvature propagates along null geodesics of that background, exactly like light, with its own eikonal phase obeying $g^{\mu\nu}\partial_\mu S\,\partial_\nu S=0$ . This is why the multi-messenger event GW170817 could be used to compare the arrival of gravitational and electromagnetic signals: both traversed essentially the same null cone across 130 million light-years, arriving within seconds and bounding the fractional difference in their speeds to about $10^{-15}$ . Outside the short-wavelength limit gravitational waves also backscatter off curvature (tails), so they are not purely null, but the leading propagation is along the same light cones that carry electromagnetic radiation.

Why does gravity always focus light rather than defocus it?

Because of the sign structure of the Raychaudhuri equation. Every term on the right except the twist is non-positive: $-\tfrac12\theta^2$ , the shear $-\sigma_{\mu\nu}\sigma^{\mu\nu}$ , and the curvature $-R_{\mu\nu}k^\mu k^\nu$ . The last is non-negative precisely when matter obeys the null energy condition, $R_{\mu\nu}k^\mu k^\nu\ge0$ , which via Einstein's equations means energy densities as seen along a light ray are not negative — true for all ordinary matter and radiation. So an initially parallel or converging surface-forming beam can only be driven toward convergence; a passing mass acts like a converging lens, never a diverging one. Defocusing would require negative energy (a violation of the condition), which is why exotic energy is the ingredient every traversable-wormhole and warp-drive proposal must smuggle in.

How is a photon's frequency defined if it has no clock of its own?

Frequency is not a property of the ray alone — it belongs to whoever measures it. Take the ray's tangent $k^\mu$ and an observer's 4-velocity $u^\mu$ ; the projection $\omega=-k_\mu u^\mu$ is the angular frequency that observer records. Because the affine parameter is arbitrary up to a scale, the absolute value of $\omega$ is not physical, but the ratio between two observers is: $\omega_{\rm rec}/\omega_{\rm emit}=(k_\mu u^\mu)_{\rm rec}/(k_\mu u^\mu)_{\rm emit}$ , the redshift, is completely well-defined. In a static spacetime a conserved $E=-k_\mu\xi^\mu$ (from the timelike Killing vector) makes this ratio computable as $\sqrt{(-g_{tt})_{\rm emit}/(-g_{tt})_{\rm rec}}$ — the gravitational redshift. So a photon needs no clock; the observers supply the ticking.