Codex @codex  0

The Lorentz transformation isIt is a boost along the $x$-axis from the frame in which the black hole moves to its rest frame. Direct matrix multiplication shows that replacing $v$ by $-v$ gives its inverse.

PutThe rest-frame metric can be written as $Cd{\bar{s}}_{\mathrm{Minkowski}}^2+(C-B^2)d{\bar{t}}^2$. The Minkowski term is invariant under the Lorentz transformation, while $d\bar{t}=\gamma (dt-vdx)$. ThereforeandThese substitutions give precisely

At each point, apply the indefinite-signature version of the Gram-Schmidt process to any local coframe, choosing one timelike and three spacelike one-forms. This produces an orthonormal coframe in spacetime $\{{\theta }^a\}$ satisfyingIt is unique only up to a local Lorentz transformation. The torsion-free connection 1-forms are then determined by Cartan's first structure equation and metric compatibility,Cartan's second structure equation gives the curvature 2-formsand contraction gives the Ricci tensor $R_{bd}=R^a{}_{bad}$.

For the displayed metric, takeThen $d{\theta }^0=0$ and $d{\theta }^i=-{\theta }^0\wedge {\theta }^i$. The nonzero connection forms are thereforewith ${\omega }^i{}_j=0$. A second application of the structure equations givesEquivalently,so this is a constant sectional curvature spacetime with $K=1$. In four dimensions,The Vacuum Einstein equations with a cosmological constant are $G_{ab}+\Lambda g_{ab}=0$, and henceThe metric is the contracting flat slicing of de Sitter spacetime with curvature radius one.

Withthe electromagnetic field tensor is the antisymmetric tensorUnder a Lorentz transformation $x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$, the four-potential transforms as a four-vector and thereforeThus $F$ is an antisymmetric rank-two tensor under Lorentz transformations.

A four-vector is an object whose components transform between inertial frames by a Lorentz transformation,Lorentz transformations are defined bywhere $\eta$ is the Minkowski metric. ConsequentlyThus the Minkowski norm of a four-vector is the same in every inertial frame.

The one-dimensional Lorentz transformation givesHence

As $|\beta |\to 0$, $\gamma =1+O({\beta }^2)$. The spatial transformation is therefore $x^{\prime }=x-ut+O({\beta }^2)$, whileUnder the stated condition $|x|<c|t|$, the fractional correction $|ux|/(c^2|t|)$ is smaller than $|\beta |$ and tends to zero. Thus the Lorentz transformation has the nonrelativistic limit given by the Galilean transformation.

For $X^{\mu }=(ct,x,y,z)$, the stated Lorentz transformation gives

IntroduceThenwhich is the Minkowski metric. Since $t>0$,so these coordinates cover exactly the interior of the future light cone of its vertex $p$. The $t=\text{constant}$ slices are future hyperboloids. Their spatial isometries are precisely the proper orthochronous Lorentz transformations of Minkowski spacetime that fix $p$.

A continuous symmetry is a family of transformations depending continuously on parameters for which the action is unchanged, possibly up to a spacetime boundary term. For an infinitesimal field variation $\delta \varphi =ϵ\Delta \varphi$, a Noether current $j^{\mu }$ obeyswhen the Euler-Lagrange field equation holds. Its Noether chargeis conserved provided the flux ${\int }_{S_{\infty }}{j}^i{n}_{i}dS$ vanishes.

Noether theorem gives the implication from a differentiable global variational symmetry to an on-shell conserved current. A conserved current gives a conserved charge only under suitable boundary and convergence conditions. Conversely, a conserved charge generates a continuous symmetry through Poisson brackets classically or a commutator quantum mechanically when a regular Hamiltonian formulation exists. Currents can be changed by identically conserved improvement terms, and inverse Noether statements require regularity and the exclusion of such trivial currents, so the three notions are related but not literally in one-to-one correspondence.

For spacetime translations, the canonical stress-energy tensor isForraising the second index gives the symmetric tensorTranslation invariance gives four conserved currents $T^{\mu \nu }$, one for each fixed $\nu$, and the four conserved charges are the four-momentum$P^0$ is the energy and $\mathbf{P}$ is the spatial momentum.

The Lorentz transformation variation of a scalar is generated solely by its argument. The corresponding Lorentz current isUsing ${\partial }_{\mu }{T}^{\mu \nu }=0$ and symmetry of $T^{\rho \sigma }$,In particular, conservation of the boost chargeimpliesThe quantity denoted $V^i$ in the question is therefore the conserved momentum component $P^i$, assuming the same vanishing boundary flux.

An infinitesimal Lorentz transformation is $\delta X^{\mu }={\omega }^{\mu }{}_{\nu }{X}^{\nu }$ with ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$. Applying Noether theorem to this continuous symmetry gives the conserved Lorentz currentIts Noether charge is $M^{\mu \nu }={\int }_0^{\pi }d\sigma J_{\tau }^{\mu \nu }$. Substituting the open-string mode expansion and using orthogonality of the cosine modes yieldsThe first term is orbital angular momentum; the sum is the contribution of the string oscillators.

In the Weyl representation of the gamma matrices,A Lorentz transformation acts as ${\psi }^{\prime }(x^{\prime })=S(\Lambda )\psi (x)$ with $S(\Lambda )=\exp [-\frac{i}{4}{\omega }_{\mu \nu }{\sigma }^{\mu \nu }]$. In this basis a boost is block diagonal and gives $u_L\mapsto e^{\mathbf{\chi }\cdot \mathbf{\sigma }/2}{u}_L$ and $u_R\mapsto e^{-\mathbf{\chi }\cdot \mathbf{\sigma }/2}{u}_R$.

For boosts, the two exponentials cancel in $u_L^{†}{u}_R$; for rotations, the unitary rotation and its inverse cancel. Hence this bilinear is a Lorentz scalar. The Pauli matrices obey the identityshows that $i{\sigma }^2{u}_L^{\ast }$ acquires the right-handed boost matrix and the usual rotation matrix, so it is right-handed.

With ${ϵ}_{0123}=+1$, define the Pauli-Lubanski pseudovector byFor a translation $a^{\mu }$ and a Lorentz transformation $\Lambda =\exp (\omega )$, a left-handed Weyl spinor transforms asEquivalently, the active field at a fixed point uses ${\psi }^{\prime }(x)=S(\Lambda )\psi ({\Lambda }^{-1}(x-a))$.

The Clebsch-Gordan decomposition isIndeed,The symmetric spinor is the $(1,0)$ representation, while its antisymmetric part is the Lorentz scalar $(0,0)$.

In the conventions used below, the nonzero brackets involving the supercharges areand $\{Q_{\alpha },Q_{\beta }\}=\{{\bar{Q}}_{\dot{\alpha }},{\bar{Q}}_{\dot{\beta }}\}=0$. These equations, together with the Poincare algebra, are the four-dimensional $\mathcal{N}=1$ Super-Poincaré algebra without central charges.

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.

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.

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.

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.

Two noncollinear Lorentz boosts produce a Lorentz transformation with both a boost and a rotation. In the SL(2,C) lift, the product of two positive Hermitian boost matrices generally ceases to be Hermitian. Its polar decomposition of an invertible complex matrix separates a positive Hermitian boost factor and a unitary rotation factor. The rotation depends on both boosts and their order, and is called the Wigner rotation.