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The Lagrangian density is Lorentz invariant, so a scalar-valued static solution can be composed with a Lorentz transformation. Let $X=\gamma (x-vt-x_0)$, with $\gamma =(1-v^2{)}^{-1/2}$ and $|v|<1$. If $\varphi (x,t)=f(X)$, the chain rule givesThe boosted equation of motion is therefore exactly the static one already verified in solution, andis a solution. The center moves as $x=x_0+vt$: $v$ is the physical velocity in units where the limiting speed is one. The kink width contracts from $k^{-1}$ to $(\gamma k{)}^{-1}$, while its topological charge remains unchanged.

The Proca action is invariant under spacetime translations and proper continuous Lorentz transformations, together forming the connected Poincaré group. The fixed nonzero mass excludes scale invariance. In four dimensions, a dimensionless action and the canonical kinetic term implyThese are mass dimensions. The field-strength term is invariant under $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\chi$, but the mass term is not: its change contains $m_A^2{A}^{\mu }{\partial }_{\mu }\chi +\frac{1}{2}{m}_A^2(\partial \chi {)}^2$. Thus the displayed massive-vector theory has no Maxwell-type local gauge invariance. The massless limit has a different constraint structure.

The action is dimensionless and $[d^{4}x]=-4$, so $[\mathcal{L}]=4$. A canonical scalar kinetic term then requires $2+2[\varphi ]=4$, and the electromagnetic kinetic term requires $2[F_{\mu \nu }]=4$. ThusThe brackets here denote mass dimension. The Lagrangian has no explicit coordinate dependence, and all indices are contracted as Lorentz scalars. Its continuous spacetime symmetry is the Poincaré group. Noether's theorem gives conserved four-momentum from translations and conserved angular momentum, including the boost generators, from Lorentz transformations. The nonzero mass $M$ and dimensionful couplings prevent a dilation symmetry.

Sign qualification: the printed electromagnetic kinetic term has the opposite healthy relative sign to the scalar kinetic terms. With the stated mostly-minus Minkowski metric, a physical Maxwell Lagrangian is $-F_{\mu \nu }{F}^{\mu \nu }/4$. The printed plus sign instead gives photon Hamiltonian density $-({\mathbf{E}}^2+{\mathbf{B}}^2)/2$; a nonzero purely electric configuration already has negative energy, unbounded below as its amplitude grows. The physical decay widths below use the healthy Maxwell sign with the given interaction normalization. The dimensional and spacetime-symmetry conclusions themselves do not depend on this sign correction.

A passive Lorentz transformation changes the components used to describe the same events and fields while preserving the standard Minkowski metric matrix: ${\Lambda }^T\eta \Lambda =\eta$. This expresses the relation between standard clock-normalized inertial frames. An arbitrary coordinate relabeling remains possible, but generally changes that metric matrix. Active boost invariance of a physical system additionally requires the relevant dynamical symmetry; a preferred physical background can obstruct it even when every equation is written in tensor notation.

Invariance under spacetime translations and Lorentz transformations. A scalar action with constant coefficients and contracted Lorentz indices has this symmetry when its domain and boundary conditions also permit the transformations.

The first postulate is Einstein's principle of relativity: nongravitational laws have the same form in every inertial frame. The second states that light in vacuum propagates at the same finite speed of light $c$ in every inertial frame, independently of the source's motion. Together with the usual spacetime symmetries and calibrated coordinates, these postulates yield the Lorentz transformation. The relativity principle alone also admits the Galilean transformation with its infinite invariant-speed limit.

A Lorentz transformation with determinant $+1$ that preserves future-directed timelike four-vectors. Equivalently, it belongs to the identity component of the Lorentz group. For the Minkowski metric $(-,+,+,+)$, its matrix satisfies ${\Lambda }^T\eta \Lambda =\eta$ and ${\Lambda }^0{}_0\ge 1$. Such transformations preserve the future light cone, so the retarded Green function of the d'Alembert operator transforms as a Lorentz scalar.

The Lorentz transformation changes the photon four-momentum and therefore its frequency, producing the displayed formula for a boost along the first axis. Here the primed ray angle is measured in the emitting frame and $\beta =v/c$. The angle and frequency transforms must be used together: aberration of light changes the direction, and a photon-number angular fraction need not equal the corresponding radiant-energy fraction.

Here $c$ is the vacuum speed of light, $\gamma$ is the Lorentz factor, and the two frequencies are measured in different inertial reference frames. Unlike the classical Doppler effect for sound, there is no material rest reference frame for light in vacuum. For small radial recession speed $v_r\ll c$, the wavelength shift satisfies $\Delta \lambda /\lambda \simeq v_r/c$. This small-speed limit underlies Doppler spectroscopy; it is not an exact sound formula.

Spatially separated simultaneous events in one inertial frame are generally not simultaneous in a relatively moving frame. The Lorentz transformation explicitly mixes position into time. This is essential when comparing distances between moving light pulses: one must compare their positions at the same time in the measuring frame.

At $v=0.6c$, the rod's simultaneous laboratory endpoint measurements in length contraction have $(ct,x)=(0,0)$ and $(0,8\mathrm{m})$. They become $(c{t}^{\prime },x^{\prime })=(0,0)$ and $(-6\mathrm{m},10\mathrm{m})$. In contrast, simultaneous rod-frame measurements at $t^{\prime }=0$ use $(ct,x)=(0,0)$ and $(7.5\mathrm{m},12.5\mathrm{m})$. The latter front-end event occurs later in the laboratory, after the front has moved. The sloping $t^{\prime }=0$ line in the figure is a set of simultaneous spacetime events, not a signal's trajectory. ^[ref].

The rest frame of a massive particle is an inertial frame in which its spatial momentum is zero. Its four-momentum is then $(mc,0)$ and its energy is $m{c}^2$. Such a frame exists for a future-directed timelike four-momentum: the Lorentz transformation with boost velocity $\mathbf{v}=c^2\mathbf{p}/E$ makes the spatial momentum zero, and the mass shell relation implies $|\mathbf{v}|<c$. A massless particle has no rest frame, because $E=c|\mathbf{p}|$ would require a boost with speed $c$.

A moving rod can have both ends inside a shorter shed on a single constant-time slice of the shed frame. Lorentz transformation sends the two simultaneous endpoint events to different times in the rod frame; it does not imply simultaneous containment there. Entrance and first collision can be spacelike separated and occur in reversed time order. After impact, length changes must propagate through the rod causally; a perfectly rigid instantaneous stopping model is not relativistically admissible.

Two spacetime events are simultaneous in a specified inertial frame when their time coordinates are equal. The Lorentz transformation changes this condition for spatially separated events. A spacelike separation admits an inertial frame in which the two events are simultaneous; a nonzero null separation does not.

With metric signature $(-,+,+,+)$, a nonzero vector $v$ is spacelike when its Lorentzian inner product with itself satisfies $g(v,v)>0$. In Minkowski spacetime, this means its spatial components have larger squared Euclidean length than its time component. The classification is preserved by Lorentz transformations. A timelike vector has negative squared Lorentzian inner product, and a null vector has zero squared Lorentzian inner product.

Special relativity describes physics in Minkowski spacetime, where inertial frames assign different space and time coordinates to the same spacetime events. Its two postulates are Einstein's principle of relativity and the same vacuum speed of light $c$ in every inertial frame, independent of the source's motion. Together they replace the Galilean transformation with the Lorentz transformation, coupling space and time rather than providing a universal clock. Einstein's 1905 paper developed this account through measuring rods, clocks, and light signals. ^[ref].

A spacetime event is an occurrence at a definite place and time: a pulse reaching a detector, for example. An inertial frame supplies a Cartesian coordinate system and clocks at rest throughout it. Einstein synchronisation sets a distant clock's reflection reading to the midpoint between a light signal's departure and return readings at the sending clock. The resulting coordinate time records the local synchronized clock at each spacetime event; it differs from the time when a distant observer receives its image. This distinction makes time dilation, length contraction, and the relativity of simultaneity precise measurements. ^[ref].

The beauty of special relativity is that these different measurements fit one geometry: all inertial frames agree on the Minkowski interval. It lets us translate an experiment between a laboratory and a moving apparatus while preserving the light-signal relations between spacetime events.

For a static rest-frame electromagnetic stress-energy tensor with no Poynting flux, a Lorentz transformation to a laboratory where it moves along $+x$ gives $u={\gamma }^2(u_0+{\beta }^2{T}_{\ast }^{xx})$ and $S_x={\gamma }^{2}v(u_0+T_{\ast }^{xx})$. Thus energy flux includes stress as well as rest energy density. In an ideal parallel-plate capacitor, $T_{\ast }^{xx}=+u_0$ for motion perpendicular to the electric field, but $T_{\ast }^{xx}=-u_0$ for motion parallel to it. The former has $S_x=2{\gamma }^{2}v{u}_0$; the latter has zero interior flux, with work at the moving plates transferring energy at the boundaries. The field subsystem need not have an integrated four-momentum independent of simultaneity surface; include material support stresses for a closed system.

The Super-Poincaré group combines Lorentz transformations and supertranslations. Its even part acts by spin-group transformations and ordinary spacetime translations; odd parameters generate the supercharges. Their graded commutator closes on translations through the Super-Poincaré algebra. A superspace tensor built from invariant coframes and Lorentz-invariant contractions is invariant under the full group.

With left Grassmann derivatives, define ${\mathcal{P}}_{\mu }=-i{\partial }_{\mu }$, ${\mathcal{Q}}_{\alpha }=-i{\partial }_{{\theta }^{\alpha }}-{\sigma }_{\alpha \dot{\gamma }}^{\mu }{\bar{\theta }}^{\dot{\gamma }}{\partial }_{\mu }$ and ${\bar{\mathcal{Q}}}_{\dot{\beta }}=i{\partial }_{{\bar{\theta }}^{\dot{\beta }}}+{\theta }^{\gamma }{\sigma }_{\gamma \dot{\beta }}^{\mu }{\partial }_{\mu }$. The anticommutators of the derivative parts and of the coordinate parts vanish. Each mixed derivative/coordinate pair contributes $-i{\sigma }^{\mu }{\partial }_{\mu }$ by the graded product rule, giving the displayed mixed anticommutator. Equal-chirality anticommutators vanish, and all these operators commute with translations. Under Lorentz transformations their spinor indices transform in conjugate Weyl spinor representations, completing the minimal Super-Poincaré algebra. A convention for left derivatives is essential to the signs.

A supertranslation is generated by translations and supercharges, excluding the Lorentz transformations. For flat Majorana superspace with ${\Pi }^m=d{X}^m+i\bar{\theta }{\gamma }^{m}d\theta$, the rigid odd transformation $\delta \theta =ϵ$, $\delta X^m=-i\overline{ϵ}{\gamma }^m\theta$ leaves ${\Pi }^m$ invariant. Its commutator gives an ordinary translation.

Translations and Lorentz transformations give Noether charges ${\mathcal{P}}_m=\int P_{m}d\sigma$ and ${\mathcal{J}}_{mn}=\int (X_m{P}_n-X_n{P}_m)d\sigma$. Their time derivatives are boundary fluxes, which vanish for a closed string or a free-ended open string.

Special relativity connects measurable clock rates, length contraction, light propagation, and four-momentum through the Lorentz transformation. Tests probe particular predictions rather than measuring a coordinate transformation in isolation. Examples include cosmic-ray muon time-dilation experiments, muon storage-ring time-dilation experiments, the Michelson-Morley experiment, the Fizeau experiment, and particle-collision energy and momentum balances.