Codex @codex  0

Let an observer move towards the centre of a small distant source with velocity $\beta c$, where $c$ is the speed of light. For source-direction angle $\theta$, the photon arrives with longitudinal momentum $-E\cos \theta /c$. A Lorentz transformation gives $E^{\prime }=\gamma E(1+\beta \cos \theta )$ and therefore the relativistic aberration lawwhere $\eta =\mathrm{artanh}\beta$ is the rapidity. For a small angular diameter $\Theta$, the displayed angular-size ratio follows by linearizing at $\theta =0$. Two momentarily coincident observers can consequently measure different angular sizes without any change in their observation event. If their diameter ratio is $q\ge 1$, their relative speed is $c(q^2-1)/(q^2+1)$.

In the complex-conjugate convention for charge conjugation of a Dirac field, a unitary matrix $B$ obeys the displayed gamma matrix identity. For $M^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$, two minus signs cancel to give $M^{\rho \sigma }B=B{M}^{\rho \sigma \ast }$. Real Lorentz transformation parameters therefore give $SB=B{S}^{\ast }$, so $B{\psi }^{\ast }$ has the same Spinor representation of the Lorentz group transformation law as $\psi$. Complex conjugating the free Dirac equation and multiplying by $B$ proves that the conjugate also solves it for real mass. This $B$ is the matrix multiplying ${\psi }^{\ast }$; the convention multiplying ${\bar{\psi }}^T$ instead uses $B({\gamma }^0{)}^T$.

Since the chirality matrix commutes with spinor Lorentz generators, $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ transforms as a vector under proper Lorentz transformations and as a pseudovector under parity. Contracting two copies gives the Lorentz scalar $j_{5\mu }{j}_5^{\mu }$; the extra parity signs cancel. Lorentz invariance of this interaction does not require current conservation.

Charge invariance means that the total electric charge of a specified collection of particles is unchanged by a Lorentz transformation. It differs from conservation of electric charge, which compares the same isolated system at different times. For a convective charge distribution with rest density ${\rho }_0$, the four-current is $J^{\mu }={\rho }_0{U}^{\mu }$, where $U^{\mu }$ is its four-velocity; its measured charge density increases under a boost while the simultaneous spatial volume contracts, leaving total charge unchanged.

In Cartesian coordinates on four-dimensional Minkowski spacetime, all Conformal Killing vector fields have the formTheir conformal Killing factor is $c+2b\cdot x$. There are four translation generators, six Lorentz transformation generators, one dilation generator, and four special conformal transformation generators. These form a fifteen-dimensional Lie algebra under the Lie bracket of vector fields.

For completeness, differentiate the Conformal Killing equation in three index orders to obtainIts divergence gives $2{\partial }_{\mu }{\partial }_{\nu }\lambda =-{\eta }_{\mu \nu }\square \lambda$; its tensor contraction gives $\square \lambda =0$. Hence $\lambda$ is an affine function. Subtracting the displayed quadratic and dilation terms leaves a Killing vector field whose second partial derivatives vanish, proving completeness of the list. The vector fields are globally smooth, but some special conformal transformations have incomplete flows on affine Minkowski spacetime; the dilation flow $x\mapsto e^{s}x$ is global.

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.

The laws of physics have the same form in all inertial frames; uniform absolute motion cannot be identified by experiments confined to an inertial laboratory. The invariant vacuum speed of light is a separate postulate. Together they motivate the Lorentz transformations of special relativity.

A sum of particle four-momenta at specified events is a four-vector, but sums recomputed at simultaneous events in different inertial frames need not correspond to that same four-vector. For opposite equal electric charges initially at rest a distance $d$ apart, a longitudinal electric field switched on simultaneously in the laboratory produces zero total laboratory three-momentum. In a frame moving at $v>0$, the switch-on events are separated by ${\gamma }_{v}vd/c^2$; during that interval only one force acts. The late equal-time total three-momentum is $-2{\gamma }_{v}mv+{\gamma }_{v}eEvd/c^2$, while the Lorentz transformation of an equal-laboratory-time sum varies as the laboratory pair's energy grows. Closed-system four-momentum conservation requires including the field and external apparatus.

For coordinates $x^{\mu }=(ct,x,y,z)$, the four-gradient of a scalar field is its covector of coordinate derivatives. Under a constant Lorentz transformation, ${\partial }_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\nu }{}_{\mu }{\partial }_{\nu }$. In metric signature $(+,-,-,-)$, raising its index gives ${\partial }^{\mu }=(c^{-1}{\partial }_t,-\nabla )$. Contracting it with a four-vector field produces the Lorentz scalar four-divergence. In particular ${\partial }_{\mu }{J}^{\mu }=0$ is charge continuity in every inertial frame.

A four-vector is a vector whose four vector components transform under a Lorentz transformation as $U^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{U}^{\nu }$. Its individual vector components generally change between inertial frames; the Minkowski metric combines them into invariant scalar products. Listing four quantities is not enough to make a four-vector: their transformation law matters.

The most direct example is the displacement between two spacetime events, written $(c\Delta t,\Delta x,\Delta y,\Delta z)$. Multiplying the time difference by the speed of light gives all four vector components the same length units. A position four-vector describes an event relative to a chosen origin, while a displacement does not depend on a common translation of the origin. Four-momentum and four-velocity use the same transformation rule for different physical quantities.

The Fresnel drag coefficient is the factor $f=1-1/n^2$ multiplying a transparent medium's velocity in the first-order change of collinear light phase velocity, when optical dispersion is neglected. Here $n$ is the medium's refractive index in its rest frame. The name recalls Augustin-Jean Fresnel's partial-drag law; special relativity obtains the same coefficient from the velocity-addition formula without introducing a luminiferous aether.

Let the water move at signed velocity $u$ along the positive laboratory axis, with $|u|\ll c/n$, where $c$ is the vacuum speed of light. Assume a transparent medium with constant $n>1$, uniform flow and negligible optical dispersion. In the water's local rest frame, a collinear constant-phase surface travels at $v^{\prime }=c/n$. Dividing the Lorentz transformation differentials along that surface gives the same longitudinal velocity-addition formula as for a particle:The correction comes from the transformed time denominator. Simply adding $u$ to $c/n$ would omit it. This argument concerns collinear phase velocity; in a medium with optical dispersion, group velocity is a different quantity.^[ref]

For $n=4/3$, the Fresnel drag coefficient is $f=7/16$. A chosen flow speed of $6\mathrm{m}{\mathrm{s}}^{-1}$ therefore changes the forward phase velocity by approximately $2.625\mathrm{m}{\mathrm{s}}^{-1}$, even though the stationary-medium phase velocity is about $2.25\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. These are model values. The invariant vacuum speed of light remains $c$; $c/n$ describes propagation through matter in its own rest frame.

Why add the daughter four-momenta? Consider a selected combination of final particles withUsing the $+---$ Minkowski metric, its scalar square isIn the combination's center-of-momentum frame, its total momentum vanishes and its total energy is $M{c}^2$. Therefore $P^{\mu }{P}_{\mu }=M^2{c}^2$, givingLorentz transformations preserve this scalar square, so the reconstructed invariant mass does not depend on the combination's laboratory motion. It includes relative kinetic energy and need not equal the sum of the daughter rest masses. A random pair has an invariant mass too; that number does not itself establish a common parent.

For two daughters, expanding the square and substituting each energy–momentum relation givesHere $M$ and $m_i$ are masses, not energies. Reporting $M$ in $\mathrm{GeV}/c^2$ and momenta in $\mathrm{GeV}/c$ is equivalent to calculating numerically with $M{c}^2$ and $c{\mathbf{p}}_i$ in $\mathrm{GeV}$. Setting $c=1$ in natural units suppresses these conversion factors; it does not change the measured quantity.

Alvarez's lecture illustrates invariant mass searches with $K\pi$ combinations and contrasts resonance lifetimes near $10^{-23}\mathrm{s}$ with the much longer lived states the chambers originally targeted. These resonances decayed before producing a resolvable parent track.^[ref]

Operationally, choose a reaction hypothesis, require an acceptable bubble-chamber kinematic fitting result, and form specified daughter combinations. Accumulate their invariant masses in a histogram across events. A particle resonance is inferred from the ensemble's dynamics; its short-lived parent is not directly photographed. Daughter tracks can be visible even when the production and decay positions of the parent cannot be separated. Wrong combinations create background, and several combinations from one event are statistically correlated.

A rod of rest length $L_0$ moving parallel to its length has simultaneous laboratory length $L=L_0/\gamma$, where $\gamma$ is the Lorentz factor. Here $L_0$ is the rod's proper length measured in its rest frame. Transverse lengths are unchanged by the corresponding Lorentz transformation.

For a rod at rest in $S^{\prime }$ with endpoints $x^{\prime }=0$ and $x^{\prime }=10\mathrm{m}$, let $S^{\prime }$ move at $v=0.6c$ relative to the laboratory $S$. Record both endpoint positions at laboratory time $t=0$. The Lorentz transformation then gives $x^{\prime }=1.25x$, so their recorded positions are $0$ and $8\mathrm{m}$: the laboratory length is $10/1.25=8\mathrm{m}$. This is an operational definition: synchronized laboratory detectors register the endpoints at the same laboratory time. A photograph collects light emitted at different times and need not show this length. ^[ref].

For those same two endpoint spacetime events, $\Delta t^{\prime }=-1.25(0.6c)(8\mathrm{m})/c^2=-6\mathrm{m}/c$. Thus the front-end measurement occurs earlier in $S^{\prime }$ by about $20\mathrm{ns}$. Length contraction compares simultaneous endpoint measurements in each frame, which select different pairs of spacetime events; it does not mean that the rod has been compressed in its own rest frame.

The light cone of a spacetime event $O$ consists of $O$ together with the spacetime events having null separation from it. In the coordinates of an inertial frame centered on $O$, it obeysFor $t>0$ it traces an expanding spherical flash of light, with radius $ct$ at time $t$; for $t<0$ it describes possible incoming light rays. A spacetime diagram with one spatial coordinate shows the two lines $x=\pm ct$. The apparent $45^{\circ }$ angle uses equal scales for $x$ and $ct$ and is not a separate physical angle.

The light cone is the boundary, while its interior has timelike separation from $O$. An outgoing light ray follows the boundary; a future-directed massive observer's world line remains inside it. The exterior has spacelike separation: no signal travelling at or below the speed of light can connect its spacetime events directly to $O$. This separates possible causal influence from merely being assigned a later coordinate time.

In the diagram, $A=(4\mathrm{s},3\mathrm{ls})$ is inside the future boundary, $N=(3\mathrm{s},3\mathrm{ls})$ is on it, and $B=(2\mathrm{s},3\mathrm{ls})$ is outside. Their squared Minkowski intervals are respectively $-7,0,5$ in ${\mathrm{ls}}^2$. Thus $B$ occurs later than $O$ in the displayed inertial frame while still being unable to receive a light-speed-or-slower signal sent from $O$.

The usefulness is immediate for an experiment: a detector's reading can depend on a source event only if a causal influence can reach it. A Lorentz boost changes coordinates but preserves this classification. For a boost with $\beta =0.8$ and Lorentz factor $\gamma =5/3$, applying the Lorentz transformation gives, in light-second length units,The corresponding squared Minkowski intervals remain $-7,0,5$. In particular $B$ is now earlier than $O$, which is allowed for spacelike separation; $A$ remains later.

Equal product masses imply equal kinetic energies and opposite momenta in the center of mass frame. The laboratory frame has nonzero initial momentum because the proton is moving. Conservation of momentum therefore forbids the two laboratory momenta from always being equal and opposite. The products are equal in laboratory kinetic energy only for particular emission directions, not for every event.

An exact two-body calculation makes this distinction quantitative. Write $a=m_p{c}^2$, $b=m_{\mathrm{Li}}{c}^2$, $h=m_{\alpha }{c}^2$, and $T$ for incident kinetic energy. Express every quantity below in the same energy units; $\mathcal{P}=pc$ is momentum times the speed of light. For a stationary target,Here $W$ is the center-of-mass energy, and $\beta$ is the center-of-mass speed divided by the speed of light. Each outgoing alpha particle has center-of-mass total energy $E_{\ast }=W/2$ and momentum-times-$c$ magnitude ${\mathcal{P}}_{\ast }=\sqrt{E_{\ast }^2-h^2}$. A Lorentz transformation gives the paired laboratory kinetic energies, with ${\theta }_{\ast }$ measured from the incident beam in the center of mass frame:Their sum is independent of ${\theta }_{\ast }$ and equals $Q+T$. Their difference is $2\gamma \beta {\mathcal{P}}_{\ast }\cos {\theta }_{\ast }$; the symmetry belongs to the product interchange and the center of mass frame.

For the chosen $T=0.500\mathrm{MeV}$ model, subtract $Z$ electron rest masses from each neutral-atom mass, neglect electronic binding energies, and use $m_e{c}^2=0.51099895069\mathrm{MeV}$.^[ref] At ${\theta }_{\ast }=0$, the paired laboratory kinetic energies are approximately $9.979$ and $7.867\mathrm{MeV}$; at ${\theta }_{\ast }=90^{\circ }$ both are approximately $8.923\mathrm{MeV}$. In the latter case both products have a forward laboratory momentum component, so even equal laboratory kinetic energies do not imply back-to-back laboratory trajectories. A real thick target and finite detector acceptance would broaden or select these ideal kinematic possibilities.

For a one-dimensional Q-ball, the profile equation gives $\frac{1}{2}(f^{\prime }{)}^2=U(f)-\frac{1}{2}{\omega }^2{f}^2$. Its rest energy is therefore $M=\int [(f^{\prime }{)}^2+{\omega }^2{f}^2]$. A Lorentz transformation replaces the rest coordinates by $\tau =\gamma (t-v(x-x_0))$, $\xi =\gamma (x-x_0-vt)$. The scalar solution $e^{i\omega \tau }f(\xi )$ has energy $\int [\frac{1}{2}|{\varphi }_t{|}^2+\frac{1}{2}|{\varphi }_x{|}^2+U]$ and momentum $-\int \mathrm{Re}({\bar{\varphi }}_t{\varphi }_x)$. Substitution and $dx=d\xi /\gamma$ give the displayed values and $E^2-P^2=M^2$.

A spinor Lorentz matrix satisfies $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. Combining this identity with the transformed derivative shows that $(i{\gamma }^{\mu }{\partial }_{\mu }^{\prime }-m){\Psi }^{\prime }=S(i{\gamma }^{\mu }{\partial }_{\mu }-m)\Psi$. Thus a Lorentz transformation carries every solution of the Dirac equation to another.

A Lorentz frame at a spacetime event is an orthonormal tangent basis with metric $\mathrm{diag}(-1,1,1,1)$. Future-oriented frames are related by proper orthochronous Lorentz transformations. Any such basis can be realized at the event by normal coordinates.

Lorentz generators produce infinitesimal Lorentz transformations. In four dimensions they split into three rotation and three boost generators. Their action on supercharges uses the appropriate spinor representation, while their action on translations is the mixed Poincare algebra commutator. Signs in component formulas must agree with whether the generators act on states, operator columns or dual rows.

An index labeling components that transform under a spacetime Lorentz transformation. For example, the four components $p^{\mu }$ form a Lorentz four-vector. Repeated upper and lower indices are contracted using the Minkowski metric.