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A radial null geodesic in a spherically symmetric spacetime has zero angular momentum and fixed angular coordinates. Its projection is a null curve of the two-dimensional radial Lorentzian metric.

A connected nonvacuum perfect fluid with equation of state $p=\rho /3$ obeys the stationary radiation density law $\rho =C/N^4$ with $C>0$. On a compact region of a regular spacetime, a smooth Killing vector field has bounded norm. Consequently $\rho$ cannot tend to zero at a finite regular free surface. A vanishing timelike norm instead makes the density diverge. A radiation-filled cavity can have a material wall with nonzero boundary pressure; this is a different boundary condition.

Consider an ideal circular vacuum path of laboratory radius $R$, a limit of closely spaced mirrors constraining light around the circle at laboratory speed $c$. The source and detector rotate uniformly together with signed angular velocity $\Omega$, where $|\Omega |R<c$, in flat spacetime. Let $L=2\pi R$ be the laboratory circumference and $A=\pi R^2$ the positive area. The $+$ pulse travels counterclockwise and the $-$ pulse clockwise, independently of the sign of $\Omega$. Both depart from the detector at $t=0$.

Unwrap the circle into a laboratory arc coordinate. The two first-return conditions areConsequently the exact laboratory arrival times and their difference areThe denominators $c\mp \Omega R$ are closing rates in these laboratory equations; they are not local speed of light measurements by a rotating observer. The two arrival spacetime events lie on the same detector world line. Its constant speed is $|\Omega |R$, so proper time along that world line givesThus laboratory coordinate-time and rotating-detector proper time delays agree only to leading order at small $|\Omega |R/c$. The radius and circumference here are laboratory quantities, not lengths obtained by laying comoving rods around the rim. Finite mirror polygons, transparent media, and optical dispersion require their own path and propagation assumptions before using an exact formula.

An exact dimensionless check takes $\Omega R/c=1/5$. Relative to $L/c$, the laboratory arrival times are $5/4$ and $5/6$, giving $\Delta t/(L/c)=5/12$. The detector instead records $\Delta \tau /(L/c)=\sqrt{6}/6\simeq 0.408248$. The leading Sagnac area law would give $\Delta t/(L/c)\simeq 2/5$. These are model values chosen to make the relativistic distinction visible, not attainable parameters of the historical apparatus.

The rotating loop does not constitute a universal measurement of the one-way speed of light. It compares completed circuits at a single readout; assigning a velocity to each remote leg additionally requires a clock synchronization convention. Local Einstein synchronisation around a rotating rim does not close into a single globally synchronized clock network: the accumulated mismatch is another expression of the rotating-path asymmetry. Local comoving vacuum measurements still give $c$ in both directions. Special relativity predicts a nonzero closed-loop delay because the receiver moves during propagation, while no single comoving inertial frame covers the whole circuit.

The Sagnac effect is a difference in return times, or an associated interference fringe displacement, for light sent in opposite directions around a closed optical path in a rotating apparatus. The light leaves one beam splitter, travels around the same loop in opposite senses, and returns to the rotating beam splitter and readout. For a slowly rotating apparatus in flat spacetime, the leading laboratory arrival-time difference isHere $c$ is the vacuum speed of light, $\mathbf{\Omega }$ is the apparatus's angular velocity relative to an inertial frame, and $\mathbf{A}$ is the vector area oriented by the chosen positive direction around the closed curve $C$. With positive rotation about the upward normal vector, the counterclockwise beam is labelled $+$ and arrives later. Reversing rotation, or reversing the orientation convention alone, changes the sign of $\Delta t$.

The surprise is that a small loop can reveal its rotation by comparing two beams locally, even though each local vacuum speed of light measurement gives $c$. A rotating reference frame is not a single inertial frame: a beam must catch the moving receiver after a full circuit. The Sagnac area law makes this difference a practical way to measure angular velocity, while the Relativistic Sagnac delay on a circular ring distinguishes laboratory coordinate time from the receiver's proper time.

A scalar field assigns one scalar to each point of a specified space, such as Euclidean space or spacetime. For example, $F(x,y)=x^2+2y^2$ is a real-valued function on a plane, and its change along a moving observation is given by the derivative along a curve. Under a change of coordinates, a scalar field's value at the same physical point is unchanged.

A spacetime metric is the Lorentzian metric on a spacetime. It determines proper time along timelike paths and the null directions that describe local light propagation.

A time orientation is a continuous choice of future cone throughout a time-orientable spacetime. A continuous nonzero timelike vector field supplies such a choice. For metric signature $(-,+,+,+)$, a nonzero causal vector $Z$ is future-directed relative to a future timelike $W$ exactly when $g(W,Z)<0$.

A concrete example separates a trapped surface from a failure of coordinates. Use geometrized units $G=c=1$, metric signature $(-,+,+,+)$, and the regular ingoing Eddington-Finkelstein coordinates of a Schwarzschild black hole with mass $M>0$:For a round sphere, choose future null vector normalsThe spherical null expansion in ingoing coordinates follows directly from $A=4\pi r^2$: ${\theta }_k=2k(r)/r$. At $r=3M$, $({\theta }_{+},{\theta }_{-})=(1/(9M),-2/(3M))$; at $r=3M/2$, it is $(-2/(9M),-4/(3M))$. Both future light fronts shrink in the second case. At $r=2M$, ${\theta }_{+}=0$, so this sphere is marginally outer trapped surface, not a strict trapped surface. In the spherical model this boundary is an event horizon; that identification does not extend to arbitrary geometries.

Although $k_{-}$ is affine as written, $k_{+}$ generally obeys ${\nabla }_{k_{+}}{k}_{+}=(M/r^2)k_{+}$. These are valid initial normals for the sign test; use affine rescaling of a null normal before applying the affine Null Raychaudhuri equation. Regularity of the displayed metric tensor at $r=2M$ shows why a divergence in another chart would not establish null geodesic incompleteness.

On a compact trapped surface, both null expansions have a uniform negative margin. Sufficiently small smooth perturbations controlling the relevant metric tensor and surface derivatives preserve trapping. This conditional robustness does not prove that every nonspherical gravitational collapse creates a trapped surface.^[ref]

A trapped surface is a closed spacelike codimension-two surface whose two future-directed orthogonal null congruences both have negative expansion.

In four-dimensional spacetime, “closed” here means compact without boundary; “spacelike” means that the induced metric tensor is positive definite. Let $k_{+},k_{-}$ be the two future null vector normals and $q_{ab}$ the induced metric tensor. Their null expansions areHere $dA$ is an infinitesimal transverse area element and ${\lambda }_{\pm }$ an affine parameter along the respective normal null geodesics. A future trapped surface has ${\theta }_{+}<0$ and ${\theta }_{-}<0$ everywhere: even the outward-directed light front initially loses area. Multiplying either future normal by a positive function multiplies its initial null expansion by that function, preserving the sign. A small inward-converging patch is insufficient: the entire closed surface must satisfy both strict inequalities.

An instantaneous reversal is an idealization. A real traveller changes velocity over a finite time, follows a smooth world line, and has finite proper acceleration. Special relativity still calculates the elapsed proper time using the home inertial frame and the same integral; general relativity is not needed merely because the traveller accelerates in otherwise flat spacetime. The clock hypothesis identifies an ideal clock's reading with this geometric proper time.

One explicit smooth trip between the same $O$ and $R$ starts and ends at rest in the home inertial frame. ChooseIts velocity reverses continuously at $T/2$, and its proper acceleration is bounded because $v_{\max }<c$ and $dv/dt$ is finite. With $v_{\max }=0.6c$ and $T=10\mathrm{yr}$, numerical integration givesThis comparison fixes the reunion spacetime events and maximum speed, not the turnaround distance: the smooth trip reaches $v_{\max }T/\pi \simeq 1.910$ light-years, whereas the ideal trip reaches $3$ light-years. The smooth traveller spends less time at high speed and therefore accumulates more proper time than the ideal traveller, while still accumulating less than the home clock.

For a sufficiently short smooth reversal replacing the corner, the proper time approaches the piecewise inertial result. A reversal interval of home duration $\delta t$ contributes between zero and $\delta t$ to the clock reading; shrinking that interval cannot create a finite extra loss concentrated at the corner. The finite difference survives in the long coasting legs. Keeping the same reversal manoeuvre while lengthening those legs changes the age difference. Conversely, two travellers following mirrored paths with the same speed history accumulate equal proper times, even though both accelerate. Thus acceleration distinguishes paths and enables reunion here, but an amount of acceleration alone does not determine the age difference.

The changing simultaneity assignment also becomes continuous. At a traveller spacetime event $(t,x(t))$ with instantaneous velocity $v(t)$, the momentarily comoving inertial frame assigns home time $t_H=t-v(t)x(t)/c^2$. Differentiating givesDuring the reversal at positive $x$, the last term advances the assigned home time rapidly. It describes the traveller's changing relativity of simultaneity, not a locally measured alteration of the home clock's ticking.

The worldvolume is the history of a brane. An embedding $X:\Sigma \to \mathcal{M}$ maps its local coordinates into target spacetime. A string worldvolume is a worldsheet.