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The Null Raychaudhuri equation is the light-ray version of the Raychaudhuri equation associated with Amal Kumar Raychaudhuri. His original work concerned relativistic cosmology.^[ref] For the normal null geodesic congruence before its first focal point, null twist vanishes. With an affine parameter $\lambda$, the four-dimensional equation reduces to the displayed ordinary differential equation. The null shear contraction is nonnegative on the transverse screen space, and the null convergence condition makes the other part of $Q$ nonnegative. Consequently ${\theta }^{\prime }\le -{\theta }^2/2$.

For $\theta (0)={\theta }_0<0$, put $w=-1/\theta >0$. ThenA regular negative null expansion therefore cannot persist beyond $\lambda =2/|{\theta }_0|$ if the null geodesic extends that far. The null focusing theorem turns this estimate into a focal-point bound. The number refers to an affine parameter, not the proper time of a light ray, which is zero.

For an original soluble model, prescribe constant $Q$ along one twist-free beam; this does not specify an entire solution of the Einstein field equations. Introduce a positive affine scale $L$ and the dimensionless variablesFor an infinitesimal area $A$ define $a=\sqrt{A/A_0}$ before the first zero. Since $y=2a^{\prime }/a$, where primes now mean $d/du$, the same model becomesTwo independently checkable solutions areThe nonnegative extra focusing term moves the first zero earlier. Direct substitution into both ordinary differential equations verifies the solutions; numerical integration of the nonlinear equation provides a separate comparison with the linear beam equation.

A focal point makes this beam description degenerate, but individual null geodesics may continue smoothly through it. Even converging light in Minkowski spacetime can form a caustic. Converting local focusing into null geodesic incompleteness requires the global step below.

A narrow fringe makes the center easier to resolve statistically; it does not certify an unbiased frequency. For example, with ideal $\pi /2$ pulses and a second-pulse oscillation phase offset ${\phi }_c$, the finite-pulse Ramsey transition probability approachesThis sign uses exactly the Rabi detuning convention and pulse Hamiltonian operator above, with the second pulse replacing ${\sigma }_x$ by $\cos {\phi }_c{\sigma }_x+\sin {\phi }_c{\sigma }_y$. The factor $2\pi$ converts angular frequency to ordinary frequency. For the original example, ${\phi }_c=0.010$ rad and $T=5\mathrm{ms}$ give a shift of about $-0.318\mathrm{Hz}$. For finite pulses or distributed flight times, use the full signal and its atomic clock frequency lock rather than inserting an unweighted mean $T$.

NIST-7 used optical pumping and fluorescence. Controls included field-sensitive transitions, independent beam-speed measurements, microwave power stabilization, and beam reversal for cavity quantum phase errors.^[ref]

The connection to special relativity is quantitative. A freely moving emitter's proper time advances more slowly than laboratory time by $\sqrt{1-v^2/c^2}$. With first-order propagation effects removed in the ideal comparison, its internal laboratory frequency is thereforeCorrecting an ensemble signal requires the effective velocity weights of preparation, interrogation and detection, not simply the oven's mean speed. Additional shifts from Zeeman splitting, unwanted light, collisions and cavity fields likewise require separate evaluation; feedback control locks to the observed line, including any uncorrected bias. Comparing clocks at different gravitational potentials additionally requires gravitational time dilation. The moving-clock experiments in twin paradox clock experiments compare calibrated clock readings along different world lines; the beam atoms' internal motion is part of the clock mechanism's own correction.

For a massive test particle with electric charge $e$, choose the electromagnetic four-potential $A_t=-Q/r$ and proper time $\tau$. The cyclic coordinates $t,\varphi$ in the equatorial Lagrangian give the conserved quantities $E=mF\dot{t}+eQ/r$ and $l=r^2\dot{\varphi }$. Here $l$ is angular momentum per unit mass. Substituting these quantities into the four-velocity normalization $g(U,U)=-1$ givesThe energy includes the gauge-dependent electrostatic term; the displayed convention fixes that gauge by taking the potential to vanish at infinity.

A device that supplies a reproducible measure of elapsed time. An ideal relativistic clock measures proper time along its world line; the clock hypothesis separates this geometric ideal from disturbances of a particular physical mechanism.

The clock hypothesis says that an ideal clock following a timelike curve measures its proper time: its instantaneous rate depends on velocity through $d\tau =dt\sqrt{1-v^2/c^2}$, with no separate acceleration term. Acceleration still changes the world line and hence the accumulated proper time. A physical clock may suffer disturbances under forces or environmental changes, so experiments must distinguish its mechanism's errors from the geometric prediction.

With positive curvature, zero cosmological constant and radiation density ${\rho }_0{a}^{-4}$ in units $8\pi G=1$, $A=\sqrt{{\rho }_0/3}$. The scale factor is positive for $0<\tau <\pi$, reaches its maximum at $\pi /2$, and has a Big Crunch at conformal time $\pi$ or proper time $2A$. This complements the closed matter-dominated Friedmann solution.

Let $X$ be a Conformal Killing vector field with conformal Killing factor $\lambda$, and let $U$ be the tangent vector of a geodesic with affine parameter $s$. The product rule, metric compatibility and geodesic equation giveEvery null geodesic therefore has the conserved quantity $g(X,U)$. On a timelike geodesic with proper time as parameter, the derivative is $-\lambda$ in metric signature $(-,+,+,+)$. With a nonaffine parameter satisfying ${\nabla }_{U}U=fU$, an extra term $fg(X,U)$ appears; the conserved expression for a null geodesic is tied to its affine parameter normalization.

Coordinate time labels events in a chosen spacetime chart. It is distinct from proper time, which is measured by a clock along its timelike path. For a stationary observer in a static metric with signature $(+---)$, the rate is $d\tau =\sqrt{g_{tt}}dt$ in units with light speed one. Different stationary observers can therefore accumulate different proper times during the same coordinate-time interval.

Cosmic time is the proper time measured by observers comoving with the homogeneous cosmological fluid. Constant-cosmic-time hypersurfaces are the homogeneous spatial slices of an FLRW spacetime.

In the 1923 note Waves and Quanta, Louis de Broglie proposed a periodic process associated with rest energy, remaining in phase with a wave along the moving particle's path. He distinguished the internal frequency slowed by time dilation from the wave frequency based on total energy.^[ref] The proposal developed into his 1924 thesis.^[ref] In the 1929 lecture The wave nature of the electron, he reconstructed the argument using a Lorentz transformation.^[ref]

Here is that quantum phase argument in modern notation. Set ${\omega }_0=m{c}^2/\hslash$ in the rest frame. A Lorentz transformation to a frame where the particle has velocity $v$ givesAlong $x=vt$, proper time is $\tau =t/\gamma$, soThis is the harmony: evaluating the spatial wave at the moving particle gives the slower clock phase. At fixed $x$, its angular frequency is instead $E/\hslash =\gamma {\omega }_0$. Since $k=p/\hslash$, the de Broglie wavelength follows as $\lambda =2\pi /k=h/p$. The proposed clock was a theoretical construction, not an observed mechanical vibration.

In a spherical collapse model a freely falling surface can cross its event horizon after finite proper time, while outgoing signals received far away have increasing delay and redshift. Schwarzschild-coordinate freezing does not mean collapse physically stops or leaves a permanently bright surface at the horizon.

In units $c=1$, a receiver on the hyperbolic motion worldline $t=a^{-1}\sinh (a\tau )$, $x=a^{-1}\cosh (a\tau )$ has retarded null coordinate $p=t-x=-a^{-1}{e}^{-a\tau }$. A right-moving light ray emitted at $(t_0,L)$ has $p=t_0-L$, so it meets the receiver only if $t_0<L$. The photon wave covector is proportional to $(1,-1)$, and contraction with the receiver's four-velocity gives its measured frequency as ${\nu }_0(\dot{t}-\dot{x})={\nu }_0{e}^{-a\tau }$. The intersection condition gives the second displayed equality. Emission at $t_0=L$ lies on the Rindler horizon and never reaches the receiver at finite proper time; $L$ is the supremum of receivable emission times, not an attained latest time.

For an unbound equatorial Schwarzschild spacetime geodesic, let $E_{\infty }$ and $L_{\infty }$ be its conserved energy and angular momentum measured relative to infinity. The ratio $J=c^2{L}_{\infty }/E_{\infty }$ has dimensions of area divided by time. For a massive test particle, $E_{\infty }=m{c}^2{\gamma }_{\infty }$ and $L_{\infty }=m\ell$, so $J=\ell /{\gamma }_{\infty }=b{v}_{\infty }$, where $\ell$ is specific angular momentum and $b$ the impact parameter. For a photon, $E_{\infty }=c{p}_{\infty }$ and $L_{\infty }=b{p}_{\infty }$, giving $J=bc$ without assigning a finite Lorentz factor or a proper time to the photon. This ratio removes the arbitrary normalization of a null affine parameter.

A timelike equatorial Schwarzschild geodesic with specific energy $E=1$ and angular momentum $J=4m$ obeys $(du/d\varphi {)}^2=2u(u-1/4{)}^2$ for $u=m/r$. Substituting $q=2\sqrt{u}$ reduces it to $(dq/d\varphi {)}^2=(q^2-1{)}^2/8$, integrating to the two displayed branches. One lies outside $r=4m$ and the other inside; each approaches the same unstable circular orbit exponentially in angle and only after infinite proper time. Linearization of $u^{\prime \prime }=1/16-u+3u^2$ about $u=1/4$ gives $\delta u^{\prime \prime }=\delta u/2$, confirming instability. These are separatrix approaches or departures, not stable oscillations around the circular orbit.

Replace instantaneous acceleration and braking by legs of constant proper-acceleration magnitude $a$, retaining the same cruise rapidity $\eta$ and fixed endpoint separation. Each leg takes proper time $c\eta /a$ and displaces $(c^2/a)(\cosh \eta -1)$ in the endpoint rest frame. Subtracting the proper cruise time displaced by both legs gives the displayed correction. A cruise segment exists only if the total endpoint separation exceeds twice this acceleration distance.

A freely falling source emits optical pulses at proper times $\delta +nP$, for integers $n\ge 0$. If its horizon-crossing proper time is finite, only pulses beginning before $s_H$ can escape. Their number is the displayed ceiling, for $0\le \delta <P<s_H$. Infinite Schwarzschild time at crossing does not give infinitely many emitted pulses. The pulse phase $\delta$ fixes both the count and the redshift of the last pulse. If a finite pulse straddles the horizon, its exterior portion has an arbitrarily late, increasingly redshifted received tail.

The four-acceleration is the covariant derivative of four-velocity along a massive observer's worldline: $A^{\mu }=d{U}^{\mu }/d\tau +{\Gamma }_{\nu \rho }^{\mu }{U}^{\nu }{U}^{\rho }$. In Cartesian Minkowski spacetime coordinates the Christoffel symbols vanish, leaving the ordinary proper time derivative. Differentiating the constant velocity norm gives $U\cdot A=0$; in the instantaneous rest frame, $A=(0,\mathbf{a})$, so $A\cdot A=-a^2$ with $+---$ metric signature. The magnitude $a$ is the proper acceleration.

The four-force is the proper time derivative of four-momentum. For constant rest mass and coordinate three-force $\mathbf{F}=d\mathbf{p}/dt$, its components are $K^{\mu }=\gamma (\mathbf{F}\cdot \mathbf{v}/c,\mathbf{F})$ in the convention $p^{\mu }=(E/c,\mathbf{p})$. Thus it packages the power transfer and three-force into one Lorentz-covariant vector.

A massive observer's four-velocity is the tangent $U^{\mu }=d{X}^{\mu }/d\tau$ to its worldline, parametrized by proper time. Its invariant squared Minkowski norm is $c^2$ in the $+---$ metric signature, and $-c^2$ in the opposite convention. Unlike the coordinate values $X^{\mu }$, the tangent transforms as a vector under a general coordinate change.

The inward radial geodesic congruence with unit Killing energy has $U=f^{-1}{\partial }_t-\sqrt{2M/r}{\partial }_r$, where $f=1-2M/r$. Its metric dual is $U^{♭}=-d\tau$, so $\tau$ is a synchronized proper time for the congruence. Locally $\tau =t+2\sqrt{2Mr}+2M\log |(\sqrt{r}-\sqrt{2M})/(\sqrt{r}+\sqrt{2M})|$. This yields Painlevé–Gullstrand coordinates with flat spatial slices and a regular future horizon.