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For an isotropically emitting rest-frame blob moving with Doppler factor $D$, take the observed target-band spectral flux density to satisfy $F_{\nu }\propto {\nu }^{-\alpha }$ over the relevant frequencies. The sign convention for $\alpha$ matters. At fixed cosmological redshift, absorb fixed distance and redshift factors into a reference optical depth ${\tau }_0$.

A light-crossing estimate gives $R^{\prime }\propto D$ for fixed observed variability time, while a corresponding logarithmic frequency bin has observed-to-rest-frame luminosity boost $D^4$. At a fixed observed high-energy photon energy, the internal pair-production target energy in a relativistic blob shifts the observed target energy by $D^2$. Combining these factors giveswhere $F_{\log }=\nu F_{\nu }$ and ${ϵ}_{20}$ is the reference target energy at $D=1$. For example, $\alpha =1$ gives $D^{-6}$, whereas a locally flat $F_{\nu }$ gives $D^{-4}$. The power-law assumption must cover the target energies swept by changing $D$. Spectral breaks, different emission geometries and external stationary illumination require a different calculation; $D^{-6}$ is not a universal pair-opacity law.

If two spectral lines come from the same uniformly moving source along the same line of sight, each wavelength is multiplied by the same longitudinal Doppler factor $D$. Subtracting ${\lambda }_{2,\mathrm{obs}}=D{\lambda }_{2,\mathrm{emit}}$ and ${\lambda }_{1,\mathrm{obs}}=D{\lambda }_{1,\mathrm{emit}}$ proves the displayed identity. Recession has $D>1$ and approach has $D<1$. Defining positive radial velocity for recession gives $\beta =(D^2-1)/(D^2+1)$. A measured increase in wavelength separation therefore cannot be reconciled with strictly approaching longitudinal motion merely by changing the speed's magnitude.

In this model, let $f_0$ denote the stationary source's emitted frequency and $f$ the cloud's resonance frequency in its own rest frame. Thus $q=f_0/f<1$ means that the source is redder than the cloud's resonance. Excitation requires the incoming Doppler factor to satisfy $D_{\mathrm{in}}{f}_0=f$, or $D_{\mathrm{in}}=1/q$. For the geometry below, the resonant position obeys $s=[1-1/(q\gamma )]/\beta$. At closest approach $s=0$, so $\gamma =1/q$ and $\beta =\sqrt{1-q^2}$. These definitions are essential: reversing the meaning of the frequency ratio would reverse the resonance condition.

For a compact cloud passing a stationary source at distance $L$, put $r(t)=\sqrt{L^2+v^2{t}^2}$ and $s=vt/r$. Incoming and returning rays have opposite Doppler factors. Source return time is $t_R=t+r/c$. If $q=f_0/f<1$, line-centre excitation is possible above $\beta =(1-q^2)/(1+q^2)$ and occurs at closest approach when $\beta =\sqrt{1-q^2}$. In a weak, optically thin, isotropic spontaneous emission model, the excited population obeys ${\dot{N}}_{\ast }=P_{\mathrm{abs}}-N_{\ast }/(\gamma \tau )$ and the received bolometric intensity scales as $D_{\mathrm{out}}^4{N}_{\ast }/(\tau r^2)$. Finite lifetime and line width smear the pulse and its frequency chirp; geometry and source strength are needed for an absolute prediction.

Let the relevant rest-frame photon-photon pair production energy product be $K={ϵ}_1^{\prime }{ϵ}_2^{\prime }$ for a specified collision-angle prescription. For fixed observed high-energy photon energy, ${ϵ}_1^{\prime }={ϵ}_{1,\mathrm{obs}}/D$, so the target has ${ϵ}_2^{\prime }=DK/{ϵ}_{1,\mathrm{obs}}$. Observing the same internally produced target population supplies a second Doppler factor, giving the displayed result. An external stationary radiation field requires its own angular transformation instead.

An approaching jet can be enhanced and its receding counterpart suppressed by different Doppler factors. A discrete moving emitter with rest-frame power-law spectrum $F_{\nu }\propto {\nu }^{-s}$ has the usual boost ${\delta }_D^{3+s}$, whereas a steady continuous-jet convention changes the exponent. Absorption and environmental asymmetry can also affect the contrast.

For a vacuum plane electromagnetic wave, a Lorentz boost multiplies both transverse Jones vector amplitudes by the same positive Doppler factor when the basis is carried to the aberrated ray. Their ratio and phase difference therefore remain unchanged. All Stokes parameters gain the same squared factor, leaving normalized parameters, ellipse shape, handedness and degree of polarization unchanged. This does not mean the propagation direction or the components in an arbitrarily fixed pair of spatial axes remain unchanged.

The photon phase-space occupation is a Lorentz scalar. For radiation propagating through vacuum, this makes specific intensity divided by cubed frequency invariant under a change of inertial observer. With Doppler factor $D$, $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)$. This describes corresponding ray directions; it does not assert that an angularly integrated radiation field is isotropic in both frames.

Coherent intrinsic luminosity variation on observed time $\Delta t_{\mathrm{obs}}$ bounds the characteristic emitting radius by a light-crossing argument. relativistic beaming shortens the observed interval by Doppler factor $D$, while cosmological redshift lengthens it by $1+z$. This bound assumes that the variation samples the emitting region causally; geometry, independent subregions and propagation-driven variation can change its interpretation.

The Doppler factors for the two discrete blobs are ${\mathcal{D}}_{\pm }=[\gamma (1\mp \beta \cos \theta ){]}^{-1}$. Their ratio is $(1+x)/(1-x)=77/33=7/3$. Lorentz-invariant specific intensity means $I_{\nu }/{\nu }^3$ is unchanged; with $\nu =\mathcal{D}{\nu }^{\prime }$ and the same rest-frame power-law spectrum,The brightness ratio at the common observed frequency therefore givesThe stated identical intrinsic specific intensities must refer to comparable intrinsic states. If the blobs fade with their proper ages, equal observer times sample different ages and need that correction. A steady continuous jet has a different boosting exponent, $2+\alpha$; the discrete ejecta and the specified specific intensity transformation justify $3+\alpha$ here.

For optically thin synchrotron radiation from an isotropic electron distribution $dN/dE\propto E^{-p}$, the characteristic emitted frequency scales as $B{E}^2$ and the total single-particle power as $B^2{E}^2$. Thus $j_{\nu }\propto E^{-p}{E}^2/(d\nu /dE)\propto {\nu }^{-(p-1)/2}$; integration over the full single-particle spectrum preserves this exponent. HenceThe test-particle strong nonrelativistic diffusive shock acceleration prediction is $p=(r+2)/(r-1)=2$ for compression ratio $r=4$, giving $\alpha =0.5$. The inferred spectrum is steeper. If interpreted literally within that uncooled formula, $p=2.50$ would correspond to $r\simeq 3$, but cooling, escape, and other acceleration conditions can also steepen it.

The observed Galactic cosmic ray differential energy spectrum below the knee is approximately $E^{-2.7}$, steeper still; its formal synchrotron radiation equivalent would be $\alpha \simeq 0.85$. Thus $p\simeq 2.50$ lies between the simplest source prediction and the propagated cosmic ray slope. The comparison is qualitative: most measured cosmic rays are nuclei, whereas the radio slope here probes electrons, and propagation and radiative losses alter the spectra differently.

Relativistic beaming concentrates the emission into a forward cone about the instantaneous velocity. The aberration of light maps a rest-frame ray at ${\theta }^{\prime }=\pi /2$ to $\cos \theta =\beta$, so the characteristic cone half-angle is $\theta \simeq {\gamma }^{-1}$. Thus its conventional full width is $2/\gamma =2\times 10^{-6}\mathrm{rad}$.

The isotropic-emission assumption has no beam nulls. For example, its received bolometric angular power is proportional to $D^4$, where the Doppler factor is $D=[\gamma (1-\beta \cos \theta ){]}^{-1}>0$. In particular it is nonzero at $\theta =1/\gamma$, disproving a literal null there. A true dipole pattern for an accelerated electron is not isotropic in its rest frame and needs a specified polarization and viewing plane. The width just quoted is a characteristic beaming angle, not a null-to-null width under the stated premise.

For a circular orbit seen in its orbital plane, let the velocity direction sweep past the observer through $-1/\gamma <\psi <1/\gamma$. The emission interval is $\Delta t_{\mathrm{em}}\simeq 2/(\gamma {\omega }_g)=1.14\times 10^{-2}\mathrm{s}$. The observed interval must also include light-travel compression:Its ratio to the gyromotion period is $\Delta t_{\mathrm{obs}}/T_g\simeq 2/(3\pi {\gamma }^3)=2.1\times 10^{-19}$. Since the boundary of a smooth beam is conventional, only the scale ${\omega }_g^{-1}{\gamma }^{-3}$ is invariant under reasonable pulse-width definitions. For helical motion the viewing geometry and pitch angle are additional inputs; the numerical estimate assumes transverse motion and a sightline intercepted by the cone.^[ref]

Polarization under a boost. Lorentz invariance of normalized plane-wave polarization preserves the polarization ellipse's shape, handedness and degree of polarization when its transverse basis is carried to the aberrated ray. The propagation direction changes; equality of a fixed laboratory Cartesian pair of field components is not the appropriate comparison.

Rotate the $yz$ axes so that a general incident vacuum plane wave has unit direction $\hat{\mathbf{n}}=(\cos \vartheta ,\sin \vartheta ,0)$. Choose the transverse basis ${\mathbf{e}}_p=(-\sin \vartheta ,\cos \vartheta ,0)$, ${\mathbf{e}}_s=(0,0,1)$, and arbitrary complex amplitudes $E_p,E_s$. ThenPutting $\beta =v/c$, the given transformation givesThe four-wavevector transforms with Doppler factor $\mathcal{D}=\gamma (1-\beta \cos \vartheta )>0$, givingIn the corresponding basis ${\mathbf{e}}_p^{\prime }=(-\sin {\vartheta }^{\prime },\cos {\vartheta }^{\prime },0)$, ${\mathbf{e}}_s^{\prime }={\mathbf{e}}_s$, the transformed field isBoth components of the Jones vector acquire the same positive real multiplier. Their amplitude ratio and relative wave phase are unchanged, proving preservation of arbitrary linear, circular or elliptical polarization. All Stokes parameters scale by ${\mathcal{D}}^2$, so their normalized values and the degree of polarization are unchanged. An unpolarized light ensemble remains unpolarized under this single-ray boost. A mixture of different ray directions requires separate transformations and is outside this statement.

Reflection from the moving slab. Take the upper interface as $z=0$, the material below it, and the incident ray along $-\hat{\mathbf{z}}$. This fixes the sign left implicit by “along the $z$-axis”; choosing the opposite surface convention changes no result below. Treat $n$ as the real refractive index in the slab's rest frame, with unit relative magnetic permeability. In the laboratory,Transform to the slab rest frame:The incidence is now oblique, in the $xz$ plane, at angle $i$ to the surface normal:The incoming ensemble remains unpolarized light. In this frame the ordinary stationary-interface Fresnel equations give the amplitude coefficients:For generic oblique incidence, $R_s=|r_s{|}^2$ and $R_p=|r_p{|}^2$ differ. Equal incident $s$ and $p$ intensities therefore produce unequal reflected intensities: the reflection is partially polarized. Its degree of linear polarization isThe inverse boost preserves this degree, so the same conclusion holds in the laboratory even though the laboratory incident ray is normal to the surface. The moving material cannot be treated as an isotropic stationary dielectric in that frame.

The reflected rest-frame wavevector is $(\omega /c)(\gamma ,-\gamma \beta ,0,+1)$. Boosting back gives $(\omega /c)(1,0,0,+1)$: the reflected laboratory ray is normal to the surface and has the incident frequency. There is no contradiction with the oblique rest-frame geometry. Translation tangential to a uniform interface does not itself introduce a moving normal mirror.

Complete linear polarization. For Brewster reflection from a tangentially moving dielectric, at Brewster's angle, $r_p=0$ and $\tan i_B=n$. Combining this with the aberration relation givesOnly the $s$ component is reflected. The plane of incidence in the slab frame is $xz$, so the reflected electric field is along $y$, perpendicular to the slab velocity. Boosting this $s$ field back keeps its electric direction along $y$. For example, the reflected rest-frame $s$ field has $B_z^{\prime }=-\beta E_y^{\prime }/c$; hence $E_y=\gamma (E_y^{\prime }+v{B}_z^{\prime })=E_y^{\prime }/\gamma$ and $E_x=E_z=0$.

This exact extinction assumes a transparent nonmagnetic dielectric with $n\ne 1$. At $v=0$ the reflected intensities are equal and the reflected beam is unpolarized. If $n=1$, both reflection coefficients vanish and there is no reflected beam whose polarization can be assigned. With dispersion, evaluate $n$ at the shifted frequency ${\omega }^{\prime }=\gamma \omega$; the Brewster-speed equation is then implicit. absorption generally prevents a real incidence angle from making the complex $r_p$ exactly zero.

Let $S^{\prime }$ move at $+v$ along $x$ relative to $S$. The Lorentz transformation of energy and momentum, from primed to unprimed components, isThe inverse equations have $v$ replaced by $-v$ and the primes interchanged. A photon making angle ${\theta }^{\prime }$ with $+x$ in $S^{\prime }$ has $p_x^{\prime }=(E^{\prime }/c)\cos {\theta }^{\prime }$. Thus its general Doppler factor isThe requested blueshifted expression assumes propagation parallel to the boost, ${\theta }^{\prime }=0$:The propagation-direction assumption matters: for an oppositely travelling photon, $D=\gamma (1-\beta )$, the reciprocal of the displayed factor. Relative frame speed alone does not select the longitudinal relativistic Doppler effect.

Take positive velocity towards $B$, and interpret $100\mathrm{MHz}$ as the emitted frequency in the probe's rest frame. The relativistic velocity-addition formula gives the probe's station-frame velocity:For reception at the station, photons propagate backwards while the probe recedes. The longitudinal relativistic Doppler effect givesBefore the probe overtakes $B$, their separation decreases. The probe's velocity in $B$'s inertial frame isThe approaching-source relativistic Doppler effect therefore givesAn independent frame calculation gives the same result: forward photons have station-frame frequency $300\mathrm{MHz}$, and a receiver moving at $0.4c$ measures $300\sqrt{0.6/1.4}\mathrm{MHz}$. If reception occurs after the probe passes $B$, backwards photons instead give $f_B=100\sqrt{7/27}\mathrm{MHz}\simeq 50.92\mathrm{MHz}$. The direction of propagation and the reception geometry are essential to choosing the Doppler factor.

For a blob in a relativistic jet, take two emission events separated by time $\Delta t_e$ in the galaxy frame. The blob moves transversely by $u\Delta t_e\sin \theta$, while its second photon has $u\Delta t_e\cos \theta$ less distance to travel to the distant observer. Therefore the arrival interval is $\Delta t_{\mathrm{obs}}=\Delta t_e(1-\beta \cos \theta )$ andDifferentiating with respect to viewing angle gives $d{\beta }_{\mathrm{app}}/d\theta =\beta (\cos \theta -\beta )/(1-\beta \cos \theta {)}^2$. The maximum occurs at $\cos \theta =\beta$ and equals $\beta \gamma$. Hence apparent superluminal motion is possible precisely when $\beta \gamma >1$, or $\beta >1/\sqrt{2}$. This is an arrival-time effect; the material and every local signal still travel slower than light.

For $\gamma \gg 1$, $1-\beta \simeq 1/(2{\gamma }^2)$ and $1-\beta \cos \theta \simeq ({\gamma }^{-2}+{\theta }^2)/2$. Thus the Doppler factor isThe Lorentz-invariant specific intensity gives $I_{\nu }/{\nu }^3=I_{{\nu }^{\prime }}^{\prime }/{\nu }^{\prime 3}$. Since $\nu =D{\nu }^{\prime }$, a power-law spectrum with $I_{{\nu }^{\prime }}^{\prime }\propto {\nu }^{\prime -\alpha }$ gives $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)\propto D^{3+\alpha }$. This is Doppler boosting of a moving blob: the spectral flux density of an unresolved, intrinsically isotropic blob has the same boost. Comparing identical approaching and receding blobs, with no differential absorption,A spatially steady continuous jet can have a different boost exponent because the emitting material contributing at one arrival time differs; it must not silently replace the specified blob model.

Using the stipulated peak of the photon-photon pair production cross section, not the exact head-on threshold, givesThe relevant photons are near infrared. Interpret the quoted $F_2$ as target-band energy flux, or energy flux per logarithmic frequency interval, consistent with its units $\mathrm{W}{\mathrm{m}}^{-2}$; it is not a spectral flux density per hertz. For isotropic emission $L_2=4\pi d^2{F}_2$. In a stationary source of radius $R$, the order-of-magnitude photon energy density is $L_2/(4\pi R^{2}c)$, soThe luminosity variability size bound gives $R\lesssim c\Delta t$. Taking the largest allowed radius gives the smallest stationary-source optical depth in this estimate:This assumes the gamma rays and target photons occupy the same region, with an approximately isotropic target field and an order-unity geometric escape factor. A smaller region makes the gamma-ray compactness problem worse.

Here $d=3\times 10^{25}\mathrm{m}$, $R\sim 3\times 10^{11}\mathrm{m}$, and ${ϵ}_{20}=1.67\times 10^{-19}\mathrm{J}$. ConsequentlySuch a stationary co-spatial source would absorb almost all of its high-energy photons.

Let primes denote the blob rest frame and continue to ignore cosmological redshift. For fixed observed gamma-ray energy, ${ϵ}_1^{\prime }={ϵ}_1/D$. The co-moving target energy at the assumed cross-section peak is therefore ${ϵ}_2^{\prime }=D{ϵ}_{20}$. The internal pair-production target energy in a relativistic blob acquires a second Doppler shift when this same target population is observed along our line of sight:It would be incorrect to transform only the gamma ray while treating the internal target radiation as a stationary external field. The target photons responsible for collisions span directions in the blob frame; the observed energy here labels the corresponding part of the internally emitted radiation spectrum.

relativistic beaming both increases the allowed rest-frame source radius, $R^{\prime }\lesssim Dc\Delta t$, and lowers the inferred rest-frame luminosity, $L_{\log }^{\prime }\sim L_{\mathrm{iso},\log }/D^4$. The beamed gamma-ray optical depth then follows from ${ϵ}_2^{\prime }=D{ϵ}_{20}$:For an unbroken power-law spectrum $F_{\nu }\propto {\nu }^{-\alpha }$, $F_{\log }=\nu F_{\nu }\propto {\nu }^{1-\alpha }$, so ${\tau }^{\prime }\propto D^{-(4+2\alpha )}$. In particular, $\alpha =1$ gives the stated $D^{-6}$. Spectral breaks, an external target field or non-coincident emission regions change this inference. Under the requested scaling, transparency requiresThis minimum uses the most favourable nearly axial view, $D\simeq 2\gamma$. The exact axial relation gives ${\gamma }_{\min }=(D_{\min }+D_{\min }^{-1})/2\simeq 8.94$; a less favourable viewing angle requires a larger Lorentz factor. The numerical bound is conditional on the stipulated opacity scaling and internal-source geometry.

An active galactic nucleus is powered by accretion onto a central supermassive black hole. Binding-energy release gives $L\sim \eta \dot{M}{c}^2$, so a compact source can outshine its host without converting its entire mass into radiation. A disc produces a thermal continuum, a hot corona can Comptonize disc photons into X-rays, and surrounding gas produces broad and narrow emission lines. Dust can obscure and reradiate the central emission. Orientation explains part of the diversity between unobscured and obscured nuclei, but accretion state and intrinsic variability also matter.

relativistic jets channel energy and momentum into collimated magnetized outflows. Disc rotation and, in some models, black-hole rotation supply the energy; magnetic stresses accelerate the plasma and connect the engine to the outflow. synchrotron radiation traces relativistic electrons and magnetic fields, while inverse Compton scattering can produce higher-energy emission. Bright knots, terminal hotspots and large radio lobes show that the deposited energy is not confined to the unresolved nucleus.

For bulk speed $\beta c$ at viewing angle $i$, the Doppler factor is $\mathcal{D}=[\Gamma (1-\beta \cos i){]}^{-1}$, and an approaching feature hasLight-travel delay can therefore give apparent superluminal motion without any material or signal travelling locally faster than light. Beaming also makes approaching jets appear brighter and more rapidly variable; a one-sided image is not proof of a physically one-sided outflow. Spectral, polarization and resolved-motion measurements constrain the geometry and emission mechanisms. Jets can inflate cavities and heat surrounding gas, linking black-hole accretion to galaxy and cluster feedback.

Astrophysical jets are directed plasma outflows whose bulk energy and momentum remain concentrated near an axis over distances much larger than the launch region. They occur around accreting black holes and neutron stars and in forming stellar systems, with very different characteristic speeds and scales. An accretion reservoir supplies binding energy; a rotating black hole can additionally provide spin energy through electromagnetic stresses. An observed jet therefore does not uniquely identify either a black hole or a particular launch mechanism.

Magnetic fields couple the rotating engine to the outflow. Gas tied to suitable open field lines can be accelerated centrifugally from a disc; a large-scale poloidal field threading a rotating hole can support outward electromagnetic energy flux. Farther out, magnetic stresses, hoop stresses and external pressure can aid collimation. Pressure-driven acceleration, magnetic-to-kinetic conversion and entrainment depend on loading and environment. A simple stationary axisymmetric picture does not guarantee stability against kinks, mixing or disruption.

The emission commonly contains synchrotron radiation from relativistic electrons spiraling in magnetic fields. Its broad continuum and linear polarization constrain the particles and ordered field component; inverse Compton scattering can extend radiation to higher photon energies. Compact knots can be shocks or moving patterns rather than individual material parcels. Interaction with the surroundings creates shocks, hotspots, cocoons and radio lobes, transporting energy far beyond the central emission region.

Relativistic viewing effects are crucial. If a feature moves at speed $\beta c$ at angle $\theta$ toward the observer, its Doppler factor isA comoving emission frequency is observed at ${\nu }_{\mathrm{obs}}={\delta }_D{\nu }^{\prime }/(1+z_{\cos })$. For a discrete isotropically emitting moving feature with spectral index $F_{\nu }^{\prime }\propto {\nu }^{\prime -s}$, the flux enhancement scales as ${\delta }_D^{3+s}$ in the usual unresolved-blob approximation. Continuous-jet conventions give a different exponent, so apparent brightness is not a direct orientation-free jet-power measurement.

Light-travel delays also produce apparent superluminal motion. In a source time interval $\Delta t$, the transverse displacement is $\beta c\Delta t\sin \theta$, but the observer receives the two signals separated by $\Delta t(1-\beta \cos \theta )$. ThereforeThe maximum is $\Gamma \beta$, at $\cos \theta =\beta$. An apparent speed above $c$ violates no local speed limit. The receding jet is deboosted, so an intrinsically two-sided source may look one-sided; absorption and environmental differences can also contribute.

Jets feed back on their host systems by depositing mechanical energy, inflating cavities, accelerating particles and modifying surrounding gas. Radio luminosity measures only part of that budget; unseen ions, magnetic fields, radiative losses and work on the environment all matter. Connecting the compact engine to a large lobe therefore requires both relativistic emission physics and the dynamics of its surroundings.

Assume an emitter at rest on the stellar surface and a static observer at infinity. Write $f(r)=1-2GM/(r{c}^2)$. The time-translation Killing energy of a photon is conserved, while the locally measured photon energy is proportional to $1/\sqrt{f(r)}$. ConsequentlyUsing $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, $M=4\times 10^{30}\mathrm{kg}$ and $R=1.1\times 10^4\mathrm{m}$,The Schwarzschild radius is approximately $5.94\mathrm{km}$, below the given surface radius. The positive shift is therefore a gravitational redshift of an escaping photon, with received wavelength about $1.475$ times the emitted wavelength. A moving emitter would introduce an additional Doppler factor; that is outside this static-surface model. The weak-field estimate $z\simeq GM/(R{c}^2)=0.270$ is insufficient at this compactness. These stated constants are numerical conventions, not a claim about an unavailable historical handbook's last digits.

The photon leaves the front rocket at $(t,x)=(0,a+L)$ and moves toward decreasing $x$, so its laboratory path is $x_{\gamma }(t)=a+L-ct$. Intersect it with the rear hyperbola, keeping the future root. Squaring and cancelling $c^2{t}^2$ givesThe rear rocket's speed there isAt emission the front rocket is instantaneously at rest in S, so the photon laboratory frequency is ${\omega }_H$. At reception the rear rocket moves in $+x$ while the photon travels in $-x$. The Doppler factor for this head-on reception isThe receiver sees a blueshift. It is essential to use the front velocity at emission and the rear velocity at reception; comparing the two velocities at one laboratory time would describe the wrong pair of measurements. The result also agrees with the Rindler light-transfer frequency ratio for static accelerated observers at radii $a$ and $a+L$. In the weak-acceleration limit $L/a\ll 1$, the shift is $1+L/a$, and the arrival time approaches $L/c$, providing useful checks.

Let $t$ label the reception event in the source reference frame. Its distance from the source and the relevant retarded time areThe received phase is $2\pi f_0{t}_e$. Because the observer's proper time obeys $d\tau =dt/\gamma$, its measured frequency isEquivalently, write $\beta =v/c$ and $s(t)=vt/r(t)$. The incoming Doppler factor is $D_{\mathrm{in}}=\gamma (1-\beta s)$, obtained also by contracting the photon four-momentum with the observer's four-velocity. The frequency decreases monotonically, withIn particular, the instant of closest approach is not the zero-shift point: a moving receiver looking perpendicular to its motion sees a factor $\gamma$, not $1/\gamma$. The latter factor applies to a different moving-emitter geometry. The graph uses source-frame reception time, with observer proper time $\tau =t/\gamma$.

An excited-state population lifetime $\tau$ gives a natural linewidth whose Lorentzian full width in ordinary frequency iswhen no extra dephasing broadens it. Additional thermal motion, collisions or inhomogeneity may increase the effective width. Therefore excitation is appreciable over $|D_{\mathrm{in}}{f}_0-f|\lesssim \Gamma /2$, rather than only at one event. A Lorentz line shape has nonzero wings, so ${\beta }_{\min }$ is a line-centre threshold, not a mathematically sharp cutoff of all signal. Below it, only a weak detuned wing may be excited; at it, finite-width illumination produces a broad weak precursor rather than a delta-function burst at infinity.

For a narrow line with a finite resonance time, a useful local excitation duration isNear the far-approach threshold this linear estimate becomes large and the full line shape and inverse-square illumination must be retained. After re-emission, the received full width is approximately $D_{\mathrm{out}}\Gamma$ at a fixed emission event. Lifetime memory and the changing return Doppler factor additionally produce a chirp and a redder tail.

The plots in part i use $\Gamma /f=0.02$ and $c\tau /L=0.4$, consistently corresponding to $fL/c=1/[2\pi (0.4)(0.02)]$ if the width is entirely natural. They are illustrative dimensionless parameters, not an asserted measurement for a specified atomic species.