Codex @codex  0

The intrinsic signed curvature is the curvature of a stress-free filament, measured along material arc length. An elastic filament with actual curvature $\kappa$ stores bending energy $\frac{B}{2}\int (\kappa -{\kappa }_0{)}^{2}ds$, with filament bending modulus $B$. In a planar small-slope Monge representation, $\kappa \simeq {\zeta }^{\prime \prime }$ and the leading energy is quadratic in ${\zeta }^{\prime \prime }-{\kappa }_0$.

For an isotropic elastic rod with shear modulus $G_s$, polar second moment of area $J_p$, and twist rate $\Omega$, the torque is $C\Omega$ and the energy per length is $C{\Omega }^2/2$. For a circular section of radius $r$, $J_p=\pi r^4/2$. This material twist stiffness is distinct from the energy of bending filament centerlines into a helix.

Let a convex strain energy density have derivative equal to the actual stress tensor. Under mixed displacement and traction data, any equilibrated stress ${\sigma }_0$ with the prescribed body force and boundary tractions represents the external work. For any compatible trial strain ${\epsilon }^{\prime }$, the convex supporting-plane inequality gives $W({\epsilon }^{\prime })-W(\epsilon )\ge \sigma :({\epsilon }^{\prime }-\epsilon )$. The integral of $(\sigma -{\sigma }_0):({\epsilon }^{\prime }-\epsilon )$ vanishes by integration by parts. Thus the actual strain minimizes $\int (W-{\sigma }_0:\epsilon )$.

The Einstein angular radius is the angular form of the Einstein radius: $R_E=D_l{\theta }_E$ in the lens plane. Its displayed value applies to a point-mass gravitational lens with the angular diameter distances defined above. It depends on source distance as well as lens mass and position. At fixed distances, multiplying $M$ by four doubles ${\theta }_E$.

At exact alignment, the vector thin gravitational lens equation is $0=(1-{\theta }_E^2/r^2)\mathbf{\theta }$. For a nonzero image direction it requires $r={\theta }_E$ and leaves the direction unrestricted: all azimuths form an Einstein ring. The signed equation's roots $\theta =\pm {\theta }_E$ are only a diameter cut through that circle. A source of finite angular size makes an annulus or arcs with finite received light; the infinite point-source lensing magnification at alignment is an idealization. An Einstein ring expresses rotational symmetry and source alignment, not a circular arrangement of emitting stars.

For a nondegenerate orthonormal coframe in spacetime $E^a$ and independent antisymmetric Lorentz connection 1-form, the action $\int {ϵ}_{abc}(R^{ab}\wedge E^c+\lambda E^a\wedge E^b\wedge E^c)$ has independent coframe and connection variations. Since $\delta R^{ab}=D\delta {\omega }^{ab}$, integration by parts yields a connection equation ${ϵ}_{abc}{T}^c=0$. The three variations of the cubic coframe term give the coframe equation ${ϵ}_{abc}(R^{ab}+3\lambda E^a\wedge E^b)=0$. In three dimensions these are exactly the equations displayed above. The solution has constant sectional curvature $K=-3\lambda$. Omitting the factor of three changes the cosmological normalization.

An independent vierbein and metric-compatible spin connection allow spin density to source a torsion tensor. The connection equation from the Einstein-Hilbert action is algebraic in torsion. In the absence of connection-dependent matter it fixes the Levi-Civita connection. For a minimally coupled Dirac field, eliminating the connection generates a local four-fermion interaction. Thus its equivalent torsion-free second-order action includes this contact term; simply dropping it defines a different matter theory.

Local Lorentz invariance requires the outcome of a local experiment, expressed in its freely falling apparatus, to be independent of the apparatus's velocity and orientation. In the Einstein equivalence principle it applies to nongravitational experiments; the strong equivalence principle also includes local gravitational experiments. It is distinct from local position invariance.

Local position invariance requires identical local experiments to have the same outcome independently of where and when they are performed. For nongravitational experiments it is part of the Einstein equivalence principle. Comparisons of clocks through gravitational redshift test this principle once the gravitational potential difference is independently specified; they do not by themselves test the whole strong equivalence principle.

In the ideal-gas high-temperature limit, the Standard Model has 28 boson states: 16 gluon states, eight electroweak gauge boson states and four Higgs field states. Its 90 fermion states are 72 quark states, 12 charged lepton states and six neutrino states. Counting a relativistic fermion state with weight $7/8$ gives the displayed effective number of relativistic entropy degrees of freedom. The effective number of relativistic energy degrees of freedom has the same value when all species share a temperature. Mass thresholds and interactions change the ideal-gas count.

For symmetric stress in static elastic equilibrium without body force, write $E=⟨\epsilon ⟩$ and $S=⟨\sigma ⟩$. The difference between microscopic and macroscopic work is $|\Omega {|}^{-1}{\int }_{\partial \Omega }(\sigma -S)n\cdot (u-Ex)$. It vanishes for affine boundary displacement, uniform boundary traction, or matching periodic displacement fluctuations and opposite tractions on paired faces.

The Voigt elastic approximation assumes uniform strain. Differentiating its averaged strain energy density gives the averaged local stress. For linear elasticity the predicted elastic stiffness tensor is $C_V=⟨C⟩$. The uniform strain is an admissible affine displacement trial field, so this gives an upper elastic energy bound for convex local energy.

The Reuss elastic approximation assumes uniform stress. The local strain is the derivative of the convex conjugate $W^{\ast }(S,x)$; its mean is $D⟨W^{\ast }⟩(S)$. Under invertible differentiable duality, this mean determines $S=D{W}_R(E)$. For positive linear elasticity, $C_R=⟨C^{-1}{⟩}^{-1}$. Uniform stress is generally not accompanied by a compatible local displacement field, so this approximation is distinguished from an exact heterogeneous solution.

Take a positive comparison elastic stiffness tensor $C^0$, excess energies $F_r(e)=W_r(e)-e:C^{0}e/2$, and a phasewise constant elastic polarization. The lower conjugate inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible strains gives the displayed bound, where $A_{rs}=\int \int ({\chi }_r-c_r)\Gamma ({\chi }_s-c_s)$. Stationarity in ${\tau }^r$ gives $c_r{e}^r+{\sum }_s{A}_{rs}{\tau }^s=c_{r}E$, with $e^r=D{F}_{r\ast }({\tau }^r)$. Without this stationarity, dual strains must not be identified with the comparison solution's phase means.

Conductivity homogenization replaces rapidly varying positive coefficients in a divergence-form elliptic equation by a macroscopic effective conductivity. Periodic media admit cell problems; statistically homogeneous media can instead be described by ensemble averages. The small-scale assumptions and boundary regime must be specified.

For isotropic phases with planar interfaces normal to $n$, normal conductivity is the weighted harmonic mean $a_H=(\sum p_i/a_i{)}^{-1}$ and tangential conductivity is the weighted arithmetic mean $a_A=\sum p_i{a}_i$. Continuity of normal flux and tangential temperature gradient gives the two averages directly.

An affine trial temperature in the minimum-energy principle gives the upper tensor bound by the volume-averaged local conductivity. For an isotropic two-phase composite, $a^{\ast }\le p_1{a}_1+p_2{a}_2$. The same uniform-field variational idea supplies the familiar corresponding elastic upper bound.

For a scalar reference $b$, choose a trial polarization $q$ and solve $b\Delta v+\nabla \cdot q=0$ with zero Dirichlet data. The trial functional isA reference above all phase conductivities gives $J(u)\le {\mathcal{F}}_b(q)$; one below all of them gives $J(u)\ge {\mathcal{F}}_b(q)$. If the reference equals a phase conductivity, admissible polarization vanishes in that phase and the inverse is taken only on the contrast subspace.

A self-consistent conductivity approximation sets the homogeneous comparison effective conductivity equal to the predicted response. For a uniformly oriented two-dimensional polycrystal with principal conductivities $a,b>0$, the comparison formula gives $C_{\mathrm{SC}}^2=ab$, so its positive solution is $\sqrt{ab}$. This is an approximation defined by a fixed-point closure; exactness requires additional geometric or statistical hypotheses.

For real scalar multiplets with antisymmetric generators $t_a$, transforming a source $J$ in the same representation leaves $J\cdot \varphi$ invariant. With an invariant functional measure, local changes of integration variables imply the Ward identity for the quantum effective action. In the convention $Z=e^{iW}$, $\phi =W_J$, $\Gamma =-W+\int J\phi$, it isThe Legendre transform gives ${\Gamma }_{\phi }=J$ and ${\Gamma }_A{|}_{\phi }=-W_A{|}_J$. Its classical limit is $\Gamma =-S_0$, not $S_0$, for the source term $+J\varphi$. Differentiating twice in $\phi$ produces two contact terms and the momentum-space identity $p_{1\mu }{\tau }_{a,ij}^{\mu }=i(t_a{)}_{ik}{\tau }_{kj}-{\tau }_{ik}i(t_a{)}_{kj}$ with the appropriate shifted two-point momenta. Changing the overall convention for $\Gamma$ changes both two-point and mixed vertices together; changing just one violates the identity. A gauge anomaly would invalidate the assumed invariant-measure argument.

The quantum effective potential is the derivative-free part of the quantum effective action evaluated on a spatially constant background. The displayed sign uses $\Gamma =-W+\int J\phi$ for $Z=e^{iW}$ and the source term $+J\varphi$; the opposite effective-action convention gives $-\int V_{\mathrm{eff}}$. Its derivatives give zero-momentum one-particle-irreducible vertices. The value depends on a renormalization scheme and scale; an arbitrary additive vacuum-energy constant may be fixed separately.