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Set the renormalized $g_3$ to zero. For a six-scalar vertex, b gives $\omega =-n_2$, so a new overall logarithmic divergence can only occur in graphs with $n_2=0$. With $b=6,f=0$, the line counts become $I_f=n_1$ and $I_b=n_1-3$, hence $L=n_1-2$.

At one loop this forces $n_1=3$ and $I_b=0$: the only candidate is a closed triangle Feynman diagram with two external scalar legs at each vertex. Its numerator at large loop momentum is proportional toThis vanishes for the specified two-component gamma matrices, each of which is traceless. Replacing one leading $/k$ by the fermion mass or an external momentum lowers the superficial degree of divergence from zero to at most $-1$. Thus the apparent logarithm cancels. Graphs with a $g_2$ vertex already have negative overall degree; the pure-scalar one-loop triangle, for instance, has $\omega =-3$. There is no one-loop ultraviolet subdivergence inside these proper one-loop vertices. ThereforeThe six-point correlation function need not vanish: finite terms are allowed. Nor is it correct to say that all odd gamma-matrix traces vanish in $2+1$ dimensions. In this representation,The cancellation above follows from repeating the same leading loop momentum, not from a false four-dimensional trace rule.

At two loops, $n_1=4$, $I_f=4$ and $I_b=1$ are possible. A closed fermion box with a scalar line joining two of its vertices leaves six external scalar legs. It has $L=2$, $\omega =0$, and four leading gamma matrices; the one-loop odd-numerator argument no longer applies.

A background-field check demonstrates that its logarithm is actually nonzero. Put $\varphi =v+\eta$ with constant $v$. Define the background fermion mass $\mu =m+g_1{v}^2/2$, the background scalar squared mass ${\mathcal{B}}^2=M^2+g_2{v}^2/2$, and the induced three-leg Yukawa coupling $y=g_{1}v$. The two-loop quantum effective potential contains the sunset diagram with two fermion lines and one scalar line. After a Wick rotation, its relevant integral, apart from a nonzero overall combinatorial factor, isHere the dots and gamma-matrix traces are Euclidean. With $A=k^2+{\mu }^2$, $B=l^2+{\mu }^2$, $C=(k-l{)}^2+{\mathcal{B}}^2$, its numerator is exactlyThe first three terms reduce to products of one-loop tadpole integrals, which have no logarithmic pole near three dimensions in dimensional regularization. The remaining term isFor positive regulator masses, the large-momentum region of $H$ has a positive, nonzero logarithmic divergence. One can see this directly by restricting the relative angles and the ratio $|k|/|l|$ to compact intervals away from zero: the remaining common radial integral is proportional to ${\int }^{\Lambda }dr/r$. Its proper one-loop subgraphs are ultraviolet convergent.

Finally, the coefficient multiplying $H$ containsThe two-loop sextic counterterm in quartic Yukawa theory follows because its $v^6$ coefficient is $g_1^4\ne 0$ for a nonzero coupling. Other two-loop vacuum topologies reduce to products of finite-in-dimensional-regularization tadpoles and do not cancel this logarithmic pole. ThusA sextic counterterm is required even if the renormalized sextic coupling is set to zero at a chosen renormalization scale. A theory constrained to $g_3=0$ at every scale lacks counterterm closure when $g_1\ne 0$. The decoupled exception $g_1=0$ is the three-dimensional scalar quartic interaction, which needs no sextic counterterm by power counting.

For $g=0$, the invariant scattering amplitude from the tree-level Feynman diagram in i has a pole at $s=M^2$. The intermediate heavy antiparticle can go on shell and live for a time of order its mean lifetime. Near this scattering resonance, inserting a single self-energy is no longer a uniformly small correction. Dyson resummation gives, to leading accuracy in a narrow decay width,Here $M$ is understood as the renormalized resonance mass; the real self-energy shifts the original pole position. The imaginary term removes the unphysical divergence of the fixed-order invariant scattering amplitude.

For equal incoming and outgoing light masses, the two-body momentum ratio cancels in the cross section. The angle-independent invariant scattering amplitude, integrated over the full solid angle with the identical final-state symmetry factor $1/2!$, gives $\sigma =|\mathcal{M}{|}^2/(32\pi s)$. Thus the Breit-Wigner resonance cross section isAt $s\gg M^2$ it reproduces $h^4/(32\pi s^3)$. At the resonance peak,This is also $8\pi /k_{\ast }^2$ for $k_{\ast }=M\beta /2$, the elastic identical-boson S wave partial-wave unitarity limit when this is the only open decay channel. The approximation assumes ${\Gamma }_{\mathrm{phys}}\ll M$ and a resonance sufficiently far from threshold for the width to vary slowly across it.

If one mechanically uses the unsymmetrized ${\Gamma }_{\mathrm{printed}}=2{\Gamma }_{\mathrm{phys}}$ from d, the same displayed Breit-Wigner resonance cross section would have that doubled width and predict a peak $8\pi /(M^2{\beta }^2)$. This is the literal-width substitution, rather than the consistent physical result: the given high-energy cross section already includes the identical-final-particle factor that the printed decay width integral omits.

The canonical anticommutation relations give $\{a_1^{†},a_2^{†}\}=0$, where each label includes both momentum and spin. ThereforeThus exchanging the complete labels of the two identical fermions changes the sign of their state. If the labels coincide, $(a_1^{†}{)}^2=0$; the state vanishes, which is the Pauli exclusion principle. A negative state vector describes the same isolated physical ray, but its relative sign matters when exchange alternatives interfere.

For a uniform continuum gas with pair interactions depending on relative positions, a Galilean boost changes every particle momentum by $p_0\hat{\mathbf{x}}$ and leaves the interaction energy unchanged. Write ${\hat{P}}_x={\sum }_j{\hat{p}}_{j,x}$. The transformed many-body Hamiltonian operator isThe resting ground state has $⟨{\hat{P}}_x⟩=0$, so the Galilean boost energy of an interacting gas isThe mass here is the bare particle mass, because this is an exact transformation of the microscopic kinetic energy. Substituting the Fermi-liquid effective mass $m^{\ast }$ would omit the change in interaction energy encoded by the Landau quasiparticle energy functional. This conclusion requires the Principle of Galilean relativity; a gas in a lattice or a fixed external potential energy landscape need not obey it.

Let $A$ be the real-space area and use ${\sum }_{\mathbf{p}}\to C\int d^{2}p$, with $C=A/(2\pi \hslash {)}^2$. There is one state per momentum cell, since the fermions are spinless. An annulus between $p$ and $p+dp$ contains $C2\pi pdp$ states. Near the Fermi surface, the quasiparticle dispersion relation has $dϵ/dp=v_F^{\ast }=p_F/m^{\ast }$, so the total density of states isThe density of states per unit area and the occupied-state count are, respectively,In particular, $D^{\ast }(\mu )=2N{m}^{\ast }/p_F^2$. No factor of two for spin degeneracy is present. This counts Landau quasiparticle levels, so it is not multiplied by the quasiparticle residue of the microscopic particle Green's function.

Identify the nonrelativistic mass with the rest mass, $m=m_0$. Expanding the relativistic kinetic energy in $p^2/(m^2{c}^2)$ givesFirst-order perturbation theory requires $\Delta E_0=⟨0|H^{\prime }|0⟩$. Put $\alpha =\sqrt{m\hslash \omega /2}$, so $p=i\alpha (a^{†}-a)$. The annihilation relations are $a|0⟩=0$ and $⟨0|a^{†}=0$. In particular, $a^{†}|0⟩$ does not vanish. A short calculation retaining the operator ordering givesEquivalently, of the ordered words in $(a^{†}-a{)}^4$, only $a{a}^{†}a{a}^{†}$ and $aa{a}^{†}{a}^{†}$ have nonzero ground-state expectation values, respectively $1$ and $2$. Words with a creation operator on the far left or an annihilation operator on the far right vanish; the remaining unbalanced words also vanish. ThusThe correction is negative because the relativistic kinetic energy is smaller than $p^2/(2m)$ at fixed nonzero momentum. This relativistic oscillator ground-state kinetic correction requires $\hslash \omega \ll m{c}^2$.

Use the shock frame, with upstream and downstream fluid velocities $u_1=u$ and $u_2=u/4$ pointing in the same direction. These are the Strong-shock Rankine-Hugoniot conditions for a nonrelativistic monatomic gas with adiabatic index $5/3$. The two scattering media approach one another at relative speed $U=u_1-u_2=3u/4$, to first order in $u/c$. An energetic electron repeatedly crosses the shock wave, scatters from magnetic turbulence carried by the local plasma, and returns. Each scattering approximately conserves its energy in the local fluid rest frame while changing its direction; comparing successive fluid frames produces a systematic energy and momentum gain. The energy comes from the converging bulk flow, through the moving magnetic field irregularities and their associated electric fields. This is diffusive shock acceleration, a first-order Fermi acceleration process.

Let $\eta$ be the positive cosine of the crossing angle with the shock wave's normal. The Lorentz transformation between fluid frames gives, for either crossing direction with the corresponding sign of the normal reversed,For an almost isotropic population, the number crossing a unit area is weighted by its normal velocity. Thus the normalized crossing distribution is $2\eta d\eta$, rather than a uniform distribution in $\eta$, andFor an ultrarelativistic electron, $E\simeq pc$ and its speed $v\simeq c$, soMore generally $dE/dp=v$ gives $⟨\Delta p⟩/p=2U/(3v)$, while a nonrelativistic kinetic energy has $⟨\Delta K⟩/K=4U/(3v)$. A complete upstream–downstream–upstream cycle contains two crossings and has ultrarelativistic fractional energy gain $u/c$.

The initially cold electrons need an injection stage: shock wave heating and small-scale plasma processes must first produce particles with speed much larger than $u$ that can return across the front. The $v\simeq c$ result is a description of that accelerated population; a cold electron cannot simply be assigned $E\simeq pc$ at its first encounter.

Work in the locally planar shock frame, taking $v_u,v_d>0$ towards the surface. For a stationary normal shock wave with negligible radiative loss inside the discontinuity, the Rankine-Hugoniot conditions for a perfect gas express conservation of mass, conservation of momentum, and conservation of energy per unit area asHere $h$ is specific enthalpy, including the pressure work; using only internal energy would omit that work. The gravitational potential is continuous across an infinitesimally thin front. For cold upstream gas, $P_u$ and $h_u$ are negligible, while a monatomic gas has $\gamma =5/3$ and $h=5P/(2\rho )$.

The hydrodynamic equations apply to flow parallel to the guiding magnetic field when its stress contributions cancel across the front. An oblique magnetohydrodynamic shock requires magnetic stresses and Poynting flux; a radiation-mediated front or appreciable escaping radiation requires the corresponding radiative momentum and energy flux terms. These are additional physics, not terms hidden in $h$ here.

accretion onto a massive black hole releases luminosity $L\simeq \eta \dot{M}{c}^2$. Even a small coupling of this energy to surrounding gas can matter because a galactic binding energy is only of order $M_{\mathrm{gas}}{\sigma }^2$, with stellar velocity dispersion $\sigma \ll c$. active-galactic-nucleus feedback means that the energy or momentum changes the gas supply, cooling or star formation, potentially regulating further accretion. Large energy availability alone does not establish efficient coupling.

In a radiatively efficient bright nucleus, radiation pressure and winds can push on dusty or ionized gas. Outflows can remove fuel or shock-heat it; the outcome depends on cooling, covering factor, clumping and whether the flow escapes. In a mechanically dominated state, relativistic jets inflate radio cavities in a hot halo or galaxy cluster. Their expansion, shocks, sound waves and subsequent motions can distribute energy and suppress a cooling flow. The two descriptions identify useful channels rather than a universal sharp division between all nuclei.

An approximately pressure-balanced cavity containing a relativistic fluid has enthalpyX-ray pressure estimates, cavity volume and buoyant or sound-crossing ages permit comparison with radiative cooling losses. Radio images check the jet association; X-ray fronts and ripples help locate transferred energy. Deep Chandra observations of Perseus reveal such structures and motivate mechanical heating of the cooling atmosphere.^[ref]

Projection, nonrelativistic cavity contents, pressure gradients, uncertain ages and intermittent duty cycles affect the power estimate. Injected cavity energy need not all dissipate where cooling occurs. A successful negative feedback loop requires that falling gas increases accretion, heating then reduces cooling or inflow, and the delay does not destroy regulation. This can explain reduced cooling and the shortage of excessively massive star-forming galaxies; correlations between black-hole mass and bulge properties are supporting context rather than standalone proof of causation. The key comparison is coupled heating versus local cooling and binding, including where and when energy is deposited.

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.

Without feedback, efficiently cooling gas would turn into stars too readily. Young stars return radiation, winds and supernova energy and momentum to their surroundings. This can heat gas, disrupt dense clouds and drive outflows, with the strongest fractional effect in shallow potential wells. The outflow mass-loading factor ${\eta }_w={\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\ast }$ is useful, but its dependence on escape speed differs between energy-limited and momentum-limited models. Some expelled gas later recycles; feedback also transports metals and changes subsequent cooling.

active-galactic-nucleus feedback is important in massive galaxies and hot haloes. Radiative or mechanical power from black-hole accretion can suppress the supply of cold gas. Collimated jets inflate radio/X-ray cavities, while broader winds can carry significant momentum and kinetic energy. For a relativistic cavity fluid with adiabatic index $4/3$, its enthalpy is approximately $4pV$, so $4pV/t_{\mathrm{age}}$ estimates mechanical power. Comparing it with the gas cooling luminosity tests whether that energy budget is sufficient, rather than merely showing that a radio source is present.

Deep Chandra observations of Perseus resolved bubbles, a density jump and weak surface-brightness ripples consistent with repeated mechanical disturbance of the intracluster gas. Their interpretation as heating requires a dissipation mechanism and a coupling history, not just a large instantaneous power.^[ref]

Observed outflow geometry, filling factors, cavity ages and cooling corrections create substantial uncertainties. Feedback can also compress gas and locally promote star formation. Simulations must track radiative losses and the scale on which energy or momentum couples: depositing unresolved thermal energy is not automatically equivalent to driving a resolved wind. A successful model should explain stellar masses, gas fractions, metallicity and circumgalactic or intracluster thermodynamics together.

An active galactic nucleus is a compact, exceptionally luminous central source powered chiefly by black-hole accretion rather than ordinary stellar fusion. Typical signatures include a broad spectral-energy distribution extending from radio to X-rays, rapid variability implying a small emitting region, strong optical or ultraviolet emission lines and, in some objects, radio jets and lobes. Broad permitted lines indicate rapidly moving dense gas; narrower forbidden lines can arise farther out. Obscuration and viewing orientation can hide the broad-line region, and many active nuclei are radio quiet, so no one signature is required in every object.

For the Schwarzschild circular-ring energy shift, set $r_g=GM/c^2$ and let $E_0$ be the photon energy in the orbiting gas frame. A local static observer in Schwarzschild spacetime measures $E_{\mathrm{stat}}$. For gas receding along the line-of-sight component of its velocity, special-relativistic frequency transformation givesPropagation from that static frame to a distant stationary observer contributes $E_{\infty }=E_{\mathrm{stat}}\sqrt{1-2r_g/r}$. HenceThe factors describe gravitational redshift, transverse time dilation and the longitudinal Doppler shift. Here the emitter is on the maximally receding azimuth of a circular disc ring. A general azimuth has a smaller projected velocity, and a fully ray-traced image must use the photon direction at emission rather than equating that local direction with the distant inclination. The supplied expression is the direct-projection transfer model, not a complete light-bending calculation.

The innermost stable circular orbit is the marginal stable timelike circular geodesic separating stable circular motion from radial instability. In Schwarzschild spacetime,For example, the circular-orbit specific angular momentum satisfies $j_c^2=GM{r}^2/(r-3r_g)$; its radial stability changes where $\mathrm{d}{j}_c^2/\mathrm{d}r=0$, at $6r_g$. This is outside both the horizon at $2r_g$ and the circular photon orbit at $3r_g$.

The printed local orbital-speed relation gives ${\beta }^2=r_g/(r-2r_g)$. Combining the gravitational and transverse factors,At the ISCO, $\beta =1/2$, soThe Edge-on Schwarzschild ISCO redshift envelope is reached for an edge-on disc on its receding side:The approaching edge has $1+z=\sqrt{2}(1-\frac{1}{2}\sin i)$. For a fixed inclination below $90$ degrees the maximum is smaller; the quoted number is an upper envelope, not the value for every observer. Within the stated ring model the receding factor increases inward across stable orbits, so the inner edge sets that envelope. Emission from a plunging region, occultation and returning rays are outside this ISCO-emitting model.

The Kerr ISCO efficiency limits follow from equatorial circular orbits in the Kerr metric. Prograde motion has a smaller ISCO than the non-spinning case, while retrograde motion has a larger one. A zero-torque thin disc radiating the orbital binding energy before the gas plunges has efficiency $\eta =1-E_{\mathrm{ISCO}}$, where $E_{\mathrm{ISCO}}$ is energy per unit rest-mass energy at infinity. The ideal limiting values areThus more orbital binding energy is available in the prograde inner disc. The extremal values are formal geodesic limits: photon capture, inner stresses, outflows and finite accretion thickness change the efficiency of real flows. An ISCO inside the ergoregion is not a claim that material or radiation escapes from inside the horizon.

For accretion output relative to galactic binding energy, distinguish accreted rest mass from the mass retained by the black hole. At constant radiative efficiency, building a hole from a negligible seed to $M_{\mathrm{BH}}=\alpha M$ emitsA non-negligible seed replaces $\alpha M$ by the actual mass increase. Let the first use of ${\sigma }_v$ mean the total three-dimensional rms random speed. For an approximately virialized galaxy with negligible boundary term, $K=\frac{1}{2}M{\sigma }_v^2$ and $W=-2K=-M{\sigma }_v^2$. The ratio to the magnitude of its gravitational potential energy is thereforeIf “binding energy” means the net energy needed to unbind the virialized system, $|K+W|=M{\sigma }_v^2/2$, the ratio is twice as large. The widely used order-of-magnitude estimate drops the retained-mass correction and writes $\eta \alpha (c/{\sigma }_v{)}^2$; it should not be confused with a precisely fixed structural coefficient.

For the numerical parameters given,No efficiency is specified, so this function is the numerical result. For an explicitly chosen illustration $\eta =0.1$, it is about $111$, or $222$ relative to the net virial binding energy. The non-spinning zero-torque efficiency gives about $60.6$ relative to $|W|$; the ideal retrograde and prograde limits give about $39.2$ and $731$, respectively. Spin affects the ratio through the adopted efficiency and orbital orientation, not through a universal rule that a larger unsigned spin always means a larger efficiency.

Such large energy ratios make even a modest coupling of the output to galactic gas dynamically important. Radiation and winds can heat or eject gas and reduce star formation or later black-hole feeding. For the $\eta =0.1$ illustration a percent-level coupling is already comparable to $|W|$. Energy available is not energy necessarily deposited: cooling, escape directions, gas geometry, photon opacity and the duration of activity determine the actual feedback. It does not follow that the entire stellar galaxy must be disrupted.

For the swept-shell calculation, the problem now explicitly redefines ${\sigma }_v$ as a one-dimensional line-of-sight dispersion. The specified isothermal profile hasNeglecting central-hole gravity and the shell's self-gravity, the inward force due to the remaining interior component isIn a single-scattering momentum-driven model with the shell intercepting the central luminosity, $F_{\mathrm{rad}}=L/c$. At the hydrogen Eddington luminosity this becomes $4\pi G{M}_{\mathrm{BH}}{m}_p/{\sigma }_T$. Sustained outward acceleration against that weight requiresand henceThe radius cancels because the swept isothermal mass is proportional to $R$. This is the isothermal-galaxy momentum-balance shell threshold, with its fourth-power dispersion scaling.

The coefficient identifies the approximations. Self-gravity in a thin spherical shell expansion adds the shell's inward pressure-equivalent force $G{M}_{\mathrm{sh}}^2/(2R^2)$ and replaces $f(1-f)$ by $f(1-f/2)$. A shell intercepting only a fraction of the photon momentum needs a larger luminosity; Thomson-thin material does not automatically receive $L/c$. Conversely, multiple scattering or an energy-driven hot wind changes the force model. A shell already carrying sufficient outward kinetic energy need not satisfy this instantaneous acceleration condition everywhere to escape. The printed inequality is the stated quasistatic, fully coupled momentum-balance estimate, not an assumption-free lower bound for every possible shell trajectory.

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.

Supernovae are explosive stellar events with different energy sources and progenitors; a remnant is the expanding ejecta and shocked environment left afterward. Their spectra classify the observed event, while explosion mechanisms give a related but distinct physical classification. Lack of hydrogen absorption defines TypeI spectroscopically; TypeII shows hydrogen. TypeIa typically shows silicon features, while stripped-envelope core-collapse events can be TypeIb or Ic.

In core collapse, an evolved massive star's central core loses support as nuclear processing ceases to supply stable pressure support, electron captures reduce electron pressure and photodisintegration consumes energy. Collapse can form a neutron star or black hole. A shock launched near core bounce loses energy, so a successful explosion requires additional processes, often modeled through neutrino heating aided by multidimensional flow; some collapses fail to eject a bright ordinary supernova. The compact remnant and neutrino emission distinguish this channel physically from a thermonuclear disruption.

TypeIa explosions involve thermonuclear burning of a carbon-oxygen white dwarf in a binary-related evolutionary channel. Both near-Chandrasekhar and sub-Chandrasekhar routes are studied; no single progenitor path is forced by the spectral classification alone. Radioactive nickel and cobalt decay can power much of the later optical emission. Empirical light-curve standardization makes these events useful distance indicators, but dust, population differences and calibration must be controlled; they are not intrinsically identical candles.

Initially ejecta may expand nearly freely, with radius $R_s\sim v_{\mathrm{ej}}t$, until swept-up external mass becomes dynamically important. The shock heats the surrounding gas and a reverse shock heats ejecta. For an approximately spherical, adiabatic explosion of fixed energy $E$ into a uniform density ${\rho }_0$, dimensional analysis gives the Sedov-Taylor stage:The dimensionless coefficient $\xi$ depends on the equation of state; no value is needed without specifying that model. For a strong nonrelativistic shock in a monatomic gas, an equilibrated post-shock temperature scale is $k_{B}T\simeq 3\mu m_p{v}_s^2/16$. Collisionless electron and ion temperatures may differ.

Later radiative losses can produce a dense cooled shell. Idealized pressure-driven and momentum-conserving snowplough models then have different expansion exponents; the environment and continuing interior pressure determine which description applies. For uniform ambient density and conserved shell momentum, $M_{\mathrm{swept}}{\dot{R}}_s\propto R_s^3{\dot{R}}_s$ gives $R_s\propto t^{1/4}$. Real remnants can be asymmetric and encounter nonuniform winds or clouds.

Thermal X-ray emission traces shocked plasma, optical lines trace cooling gas, and radio synchrotron emission traces relativistic electrons and magnetic fields. Shocks can accelerate cosmic rays and alter the energy budget. Ejecta abundances, a surviving compact object, pulsar-powered nebulae and the remnant's dynamics help reconstruct the explosion, but present morphology alone does not uniquely identify the progenitor.

Supernovae enrich the interstellar medium, stir and heat galactic gas and can drive outflows or trigger local compression. Their feedback couples small stellar scales to galaxy evolution. Quantifying that effect requires accounting for cooling and the collective environment rather than assigning all explosion energy to a permanently retained galactic reservoir.

In the rest frame, conservation of energy would require $E_{\gamma }=m{c}^2$, whereas conservation of momentum would require the photon momentum to vanish. These contradict $|{\mathbf{p}}_{\gamma }|=E_{\gamma }/c$. More generally, four-momentum conservation would equate the initial massive four-momentum to a null four-momentum, although their Minkowski norms differ:An isolated massive particle cannot decay into one photon in any inertial frame. Directly, a laboratory speed $u<c$ gives $E=\gamma m{c}^2$ and $|\mathbf{p}|=\gamma mu$; matching a photon would require $E=c|\mathbf{p}|$, hence $u=c$. Recoil against another object could change the kinematics, but would introduce an additional participant.

Write the four-velocity $U^{\mu }=\gamma (c,\mathbf{v})$ and four-acceleration $A^{\mu }=d{U}^{\mu }/d\tau$. Since $U_{\mu }{A}^{\mu }=0$, spatial orthogonality $\mathbf{U}\cdot \mathbf{A}=0$ implies $U^0{A}^0=0$. Thus $A^0=cd\gamma /d\tau =0$ and the speed is constant. If “acceleration” is read as the laboratory three-acceleration instead, $\gamma \mathbf{v}\cdot d\mathbf{v}/dt=0$ immediately gives the same conclusion.

For four-acceleration in uniform circular motion, the speed is $r|\omega |<c$, so $\gamma =(1-r^2{\omega }^2/c^2{)}^{-1/2}$ is constant. Differentiating the four-velocity using $d/d\tau =\gamma d/dt$ givesThe proper acceleration is $\sqrt{-A_{\mu }{A}^{\mu }}={\gamma }^{2}r{\omega }^2$. The vanishing time component reflects constant energy; the spatial components describe the changing momentum direction.

With $\gamma =(1-v^2/c^2{)}^{-1/2}$, four-momentum conservation gives total energy $E=(\gamma m_1+m_2)c^2$ and momentum $\mathbf{p}=\gamma m_1\mathbf{v}$. The compound system therefore hasHere $M^2{c}^4=E^2-c^2{p}^2$. The excess over $m_1+m_2$ is retained internal energy, assuming absorption without escaping radiation.

Let the inertial frame $S^{\prime }$ move at speed $v$ in the positive $x$ direction relative to $S$, with parallel axes and coincident origins at zero time. The inverse Lorentz transformation givesHere ${\gamma }_v=(1-v^2/c^2{)}^{-1/2}$ and ${\mathbf{u}}^{\prime }=d{\mathbf{x}}^{\prime }/d{t}^{\prime }$ is the particle's velocity in $S^{\prime }$. Dividing spatial differentials by $dt$ gives the relativistic velocity addition lawFor the collision, set $c=1$ during the algebra. Let the incident projectile have energy $E>m_1$ and momentum $p=\sqrt{E^2-m_1^2}$. The target is initially at rest. The squared invariant mass, boost speed and Lorentz factor of the center-of-momentum frame areIn that frame the projectile has $p_{\ast }=m_{2}p/\sqrt{s}$ and $E_{\ast }=(m_1^2+m_{2}E)/\sqrt{s}$. An elastic collision changes the direction of its momentum, through an angle $\phi$, while preserving these magnitudes. Transforming the outgoing four-momentum back to the laboratory givesThe maximum occurs at $\cos \phi =-1/B$. Sincethe maximum laboratory deflection in relativistic elastic scattering isThe printed strict inequality needs correction: equality is kinematically attainable. For example, in $c=1$ units take $m_1=2$, $m_2=1$, and incident $E=3$. A projectile scattered through $30^{\circ }$ with final energy $16/7$ has momentum $(3\sqrt{5}/7,\sqrt{15}/7)$. The target then has energy $12/7$ and momentum $(4\sqrt{5}/7,-\sqrt{15}/7)$; both outgoing particles satisfy their mass shells and conserve total four-momentum. This gives $\sin \theta =1/2=m_2/m_1$ exactly. The inequality is strict only away from the maximizing direction.

Use the $+---$ metric signature, with the time component of four-momentum equal to $E/c$. The incident particle has energy $\gamma m{c}^2$ and momentum of magnitude $mc\sqrt{{\gamma }^2-1}$. Adding the stationary target's four-momentum gives total laboratory quantitiesThe centre-of-momentum energy $W$ follows from the invariant Minkowski norm of the total four-momentum:At the relativistic collision mass threshold, the two final particles have zero relative velocity in their centre-of-momentum frame. Their total energy there is therefore $2\alpha m{c}^2$. Four-momentum conservation requires $W\ge 2\alpha m{c}^2$, soThe condition $\alpha >3/2$ ensures ${\gamma }_{\min }>1$. This is the minimum required Lorentz factor; larger values allow final kinetic energy in the centre-of-momentum frame. Equivalently, the incident laboratory kinetic energy at threshold is $({\alpha }^2-9/4)m{c}^2$.

The conserved initial four-momentum of a particle with positive rest mass satisfies $P_{\mu }{P}^{\mu }=m^2{c}^2>0$. A single photon instead has null four-momentum, $K_{\mu }{K}^{\mu }=0$. Four-momentum conservation for an isolated one-body decay would require $P^{\mu }=K^{\mu }$, contradicting these Lorentz invariants. Thus decay into a single photon is impossible in every inertial frame.

Equivalently, the incoming energy and momentum obey $E^2-c^2|\mathbf{p}{|}^2=m^2{c}^4$, whereas a single outgoing photon obeys $E=c|\mathbf{p}|$. Changing the laboratory speed cannot change this mismatch of the invariant mass shell.