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With the curvature conventionthe vacuum Schwarzschild metric satisfiesThe opposite Riemann-sign convention reverses this sign.

The cosmological redshift is the stretching of a freely propagating photon's wavelength between emission at cosmic time $t_e$ and reception at $t_0$ in an FLRW metric. For a comoving observer at each endpoint,Here ${\lambda }_e,{\lambda }_0$ are emitted and received wavelengths, and ${\nu }_e,{\nu }_0$ are the corresponding frequencies. A positive cosmological redshift means that the scale factor grew while the photon travelled. The name describes a shift toward longer wavelengths; it applies to light beyond the visible range as well.

For a computed example, let the scale factor double from $a(t_e)=0.5$ to $a(t_0)=1$. An emitted wavelength of $500\mathrm{nm}$ becomes $1000\mathrm{nm}$, giving $z=1000/500-1=1$. The received frequency is half the emitted frequency. This determines a ratio of spatial rulers, not a recession speed: using $cz$ as an actual velocity at $z=1$ is unjustified. The wavelength panel in the illustration under Hubble's law shows this doubling.

A short derivation follows successive wave crests along a radial null geodesic in FLRW spacetime. Their source and receiver have fixed comoving coordinates, so each crest traverses the same comoving distance:Shift both endpoint times to the next crest while keeping $\chi$ fixed. To first order in the very short periods $\delta t_e,\delta t_0$, this gives $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Locally, wavelength equals the speed of light times the crest period, yielding the displayed cosmological redshift relation. These periods are much shorter than the timescale on which the scale factor changes.

The relation isolates the smooth expansion contribution. An observed redshift can also contain a Doppler effect from source or observer peculiar velocity and a gravitational redshift. See combined kinematic and cosmological redshift for the multiplicative combination of endpoint motion and expansion. Treating every observed redshift as pure cosmological redshift can therefore misassign a nearby source's distance.

The useful insight is that a received wavelength ratio records how much the scale factor changed during the photon's journey. Turning that ratio into a distance requires an expansion history, through the Hubble parameter, and a specified convention such as luminosity distance or proper distance in cosmology; the ratio alone supplies neither.

A comoving coordinate remains fixed for an observer following the homogeneous cosmological expansion.

A comoving coordinate provides labels for the comoving and proper distances comparison; proper separation additionally contains the cosmological scale factor.

The Hubble parameter is the instantaneous fractional rate of cosmic expansion:where the dot is a derivative with respect to cosmic time. Thus a short interval $\Delta t$ changes any fixed-comoving proper distance in cosmology by the fraction $\Delta d/d\simeq H\Delta t$. Its dimensions are inverse time; $H>0$ describes expansion and $H<0$ contraction. The Hubble parameter identity gives $\dot{H}+H^2=\ddot{a}/a$.

Knowledge of a valid public and private exponent gives $ed-1$, a nonzero multiple of $\lambda (N)$. Writing it as $2^{s}t$ with $t$ odd and testing random bases produces a nontrivial square root of one modulo $N$ with high probability; a greatest common divisor then factors $N$. Thus one customer's private key under a shared semiprime modulus compromises every exponent using that modulus.

If the same invertible message $m$ is sent under one RSA modulus with coprime public exponents $e_1,e_2$, choose integers $a,b$ with $a{e}_1+b{e}_2=1$. From $c_i=m^{e_i}(\mathrm{mod}N)$ one recoversusing modular inverses for negative exponents.

At ${\omega }^2=g/R$, the stable downward equilibrium of a rotating-hoop bead loses stability and two stable equilibria withemerge symmetrically.

For $\omega =e^{2\pi i/N}$,This is the finite geometric series identity and expresses orthogonality of the characters of ${\mathbb{Z}}_N$.

A primitive $n$th root of unity has multiplicative order exactly $n$.

Equality in the Heisenberg uncertainty relation requires the centred vectors $x^{\prime }\psi$ and $p^{\prime }\psi$ to be imaginary scalar multiples, where $x^{\prime }=x-x_{0}I$, $p^{\prime }=p-p_{0}I$ and $x_0,p_0$ are the expectation values. Thus, for some $s>0$,In the position representation, using the momentum operator turns this into a first-order differential equation, whose normalizable solutions areHence the quantum states saturating the Heisenberg uncertainty relation are these Gaussian wave packets, with arbitrary position and momentum means and global phase. Their width is ${\sigma }^2=\hslash /(2s)$. A Gaussian wave packet with an additional quadratic quantum phase need not saturate this product bound.

For an admissible quantum state with zero position and momentum means, positivity of the squared norm $\Vert (p-isx)\psi {\Vert }^2$ for every real $s$ givesThe discriminant is nonpositive, hence $\Delta x\Delta p\ge \hslash /2$. Subtracting the expectation values from the position operator and momentum operator extends this outline to nonzero means.

Rolling without slipping requires the two contacting material points to have the same instantaneous velocity. For a rigid wheel of radius $R$ rolling straight on a stationary plane, this gives the familiar centre-speed relation $v=R\omega$, with compatible direction and spin conventions.

The contact point is instantaneously at rest in the plane's frame even while the wheel's centre moves. Combining the centre's translational velocity with the rotational contribution makes this cancellation explicit. A marked point on the rim therefore speeds up and slows down as it rotates, rather than moving with the centre's constant velocity.

This is a kinematic constraint, not a rule that friction must equal its maximum value. Static friction may supply the torque needed to enforce rolling, and its required magnitude must be checked against the contact limit. If the supporting surface moves, its own contact-point velocity must appear in the condition.

With the convention in which $\theta$ is inclination, $\varphi$ is precession, and $\psi$ is body-axis spin, an axisymmetric body's kinetic energy is

The moment of inertia about a specified axis measures how a mass distribution resists changes of angular velocity about that axis. For point masses $m_i$ at perpendicular distances $r_{\perp i}$, or a continuous distribution with mass density $\rho$,Its SI unit is $\mathrm{kg}{\mathrm{m}}^2$. The distance is to the entire axis, not to one point on it: moving a point mass along the axis leaves its contribution unchanged. The moment of inertia depends on both the distribution and the chosen axis; it is not determined by total mass alone. About an axis fixed in an inertial frame, a rigid body with larger moment of inertia has smaller angular acceleration under the same net axial torque, and greater rotational kinetic energy at the same angular speed.^[ref]

For a simple model, put four equal point masses of total mass $M$ on light rigid spokes at radius $a$. Their moment of inertia is $M{a}^2$. Doubling $a$ quadruples the moment of inertia without changing $M$. The illustrated placements $a=R/3$ and $a=R$ give $M{R}^2/9$ and $M{R}^2$. Actual spokes and finite-sized weights contribute their own moments of inertia; the point mass model neglects them. Repositioning weights during rotation also breaks the fixed-distribution assumption, so it requires a separate work and angular momentum calculation.

The moment of inertia of a uniform disc follows by dividing it into concentric annuli: $dm=(2M/R^2)rdr$ and $I={\int }_0^R{r}^2(2M/R^2)rdr=M{R}^2/2$. For an ideal thin ring every mass element is at $R$, so $I=M{R}^2$. A ring of finite radial thickness instead uses the moment of inertia of a uniform annulus, $I=M(R^2+b^2)/2$, with inner radius $b$. A moment of inertia is thus a compact summary of radial placement rather than a complete description of shape. For several joined pieces, their moments of inertia add only when each is calculated about the same axis.^[ref]

A physical pendulum makes the moment of inertia measurable through timing. Martín Monteiro, Cecilia Stari, Cecilia Cabeza and Arturo C. Martí used a metal bar with suspension holes $1.0\mathrm{cm}$ apart and an attached smartphone. They report bar length $1.199\mathrm{m}$, bar mass $0.2518\mathrm{kg}$ and phone mass $0.1311\mathrm{kg}$. Changing the suspension hole changes the pivot axis. Their readouts were phone gyroscope angular velocity records, analyzed by autocorrelation or a fit to a sinusoid, and an independent photogate; the logged records for the sinusoid fits lasted $10\mathrm{s}$. They observed agreement between the three timing methods and the geometry-based model, including a minimum of oscillation period as the suspension point varied. These are reported apparatus details and observations, not measurements made for this article.^[ref]

To infer a moment of inertia from such a measurement, determine the total mass $M_{\mathrm{tot}}$ including every attached part, locate the assembly's centre of mass $G$, and measure its distance $h>0$ below the horizontal pivot $O$. Record successive crossings in the same direction, or the elapsed time $t_N$ for $N$ complete oscillations, giving oscillation period $P=t_N/N$. The physical pendulum model with a fixed pivot, a uniform gravitational field, negligible friction and the small-angle approximation givesThe crossing times, masses and geometric distances are the measured inputs; $I_{O,\mathrm{inferred}}$ is an inference conditional on this model, while an integral over the assumed shape supplies an independent model prediction. The parallel axis theorem converts the inferred pivot moment of inertia to one about a parallel axis through $G$. The controls and interpretation of that comparison are described there. An axial measurement determines one scalar moment of inertia, not the complete inertia tensor needed for general rotation.

In principal-axis body coordinates, torque-free angular velocity satisfies

A principal body frame is an orthonormal basis fixed in the body and aligned with the principal axes of its inertia tensor. In this frame the inertia tensor is diagonal with principal moments $I_1,I_2,I_3$.

A fixed space frame is an inertial orthonormal basis whose axes do not rotate with the body.

Angular velocity is the vector whose direction gives the instantaneous rotation axis and whose magnitude gives the angular speed.

A symmetric top is a rigid body with two equal principal moments of inertia, conventionally $I_1=I_2$. Its third principal axis is its symmetry axis.

The Ricci tensor is the contraction $R_{\alpha \beta }=R^{\mu }{}_{\alpha \mu \beta }$ and is symmetric for the Levi-Civita connection.