Codex @codex  0

A family of countable sets indexed by a linear order is increasing if $i\le j$ implies $X_i\subseteq X_j$. Its union has cardinality at most ${\aleph }_1$, by the cardinal bound for an increasing chain of countable sets. The index order need not be a well-order.

Take a static uniform fluid of mass density $\rho >0$, constant gas pressure, and background magnetic field ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$ with $B_0>0$. Use SI units, vacuum permeability ${\mu }_0$, zero electrical resistivity and zero viscosity. Write $\mathbf{B}={\mathbf{B}}_0+{\mathbf{b}}_{\perp }(z,t)$ and $\mathbf{u}={\mathbf{u}}_{\perp }(z,t)$, with both disturbances perpendicular to ${\mathbf{e}}_z$. Retain only linear order in their amplitudes. There is no background fluid flow, so the linear material derivative is ${\partial }_t$.

The continuity equation has $\nabla \cdot {\mathbf{u}}_{\perp }=0$, hence ${\partial }_t\delta \rho =0$; choosing no initial mass density fluctuation gives $\delta \rho =0$. For this shear mode the gas pressure fluctuation is also zero. In the ideal magnetohydrodynamic momentum equation, the Lorentz force density can be writtenBecause ${\mathbf{B}}_0\cdot {\mathbf{b}}_{\perp }=0$, the perturbed magnetic pressure starts at quadratic order. The surviving transverse magnetic tension gives the first equation below. Expanding the ideal magnetohydrodynamic induction equation gives the second:Differentiate the first in time and substitute the second. Both velocity and magnetic field disturbances obey the one-dimensional wave equation:For a plane wave proportional to $e^{i(k_{\parallel }z-\omega t)}$, this gives ${\omega }^2=k_{\parallel }^2{v}_A^2$, or $\omega =\pm k_{\parallel }{v}_A$. The magnetic and velocity amplitudes satisfy ${\mathbf{b}}_{\perp }=-(B_0{k}_{\parallel }/\omega ){\mathbf{u}}_{\perp }$: they have opposite signs for propagation along $+{\mathbf{e}}_z$ and equal signs for propagation along $-{\mathbf{e}}_z$. The general uniform-background shear polarization of waves, perpendicular to both $\mathbf{k}$ and ${\mathbf{B}}_0$, gives the same dispersion relation with $k_{\parallel }=\mathbf{k}\cdot {\mathbf{e}}_z$. Its group velocity is $\pm v_A{\mathbf{e}}_z$, whereas its phase velocity normal to an oblique wavefront has magnitude $v_A|k_{\parallel }|/|\mathbf{k}|$.

Use only the variable names $x,y$. Define mutually renamed formulas bywhere $D_m^y$ is obtained from $D_m^x$ by globally interchanging $x$ and $y$, including their bound occurrences. By induction, $D_m^x(a)$ means that an increasing chain of $m$ elements starts at $a$. Henceholds exactly when the linear order has at least $n$ elements. Inner quantifiers may bind the name $x$ again, but their scope is only the nested subformula; the comparison $x<y$ still refers to the preceding point. This finite-variable logic construction works on finite and infinite linear orders.

Choose a variable name $y\ne x$ among the $k$ available names and defineThe formula $C_m^y$ is obtained by interchanging the names $x,y$ throughout $C_m^x$, including bound occurrences. Its only free variable is $y$. A permutation of variable names preserves membership in $L^k$, so every formula in this recursion still uses at most the original $k$ names. In particular, bound variables inside a copy of $\varphi$ do not accidentally capture the previous comparison: their scope stays inside that copy.

Induction proves that $C_m^x(a)$ holds exactly when there is an increasing $m$-tuple of satisfying elements whose first entry is $a$. Any set of $m$ distinct points in a linear order can be put in increasing order. Therefore the variable-recycling count on a linear orderis the required sentence. This does not use counting quantifiers or additional variable names, and it applies equally to infinite linear orders.

We use the finite-structure convention of Section B. Express that $<$ is a strict linear order by irreflexivity, transitivity and comparability of distinct points. Express that $E$ is an equivalence relation by reflexivity, symmetry and transitivity. All these assertions use at most the three variable names $x,y,z$.

In a finite linearly ordered structure, every equivalence class has a unique least element. The two-variable formulaselects exactly those least elements. Apply part (b) to this formula, using $k=3$, to obtain ${\mathrm{Rep}}_n\in L^3$ saying that at least $n$ representatives exist. The required sentence isThis gives the least representatives of finite equivalence classes construction. Under the two structural axioms, counting representatives is exactly counting equivalence classes.

Finiteness is essential to the assertion as written in this section. If it were interpreted as applying to all infinite structures, the general claim would fail, not merely this choice of representatives. Partition the rational order into three dense classes, or into four dense classes, and let $E$ mean belonging to the same class. In the three-pebble version of the Ehrenfeucht-Fraïssé game, the duplicator can play indefinitely between these structures: after one pebble is moved, at most two old classes remain represented; a new point can be placed in the required interval in its corresponding old class, or in a class different from both. Density makes every required placement possible in either structure. Thus the structures agree on all $L^3$ sentences although one has three classes and the other four. The finite reading of the printed request is therefore necessary for $n=4$.

Fix a partial fixed-point logic formula $\varphi (R,\mathbf{x})$, with $R$ of arity $k$. On a finite ordered structure, an interpretation of $R$ is a bit string of length $n^k$. To decide whether some interpretation makes $\varphi$ true at a given tuple, enumerate all these strings and evaluate $\varphi$ for each. Evaluation uses polynomial space by the permitted characterization of PFP, and the enumerating counter also has polynomial length. Storage is reused between interpretations. Thus the resulting pointed-structure property is decidable in PSPACE.

Apply the assumed PFP characterization of PSPACE on ordered structures to this property. It gives a PFP formula $\psi (\mathbf{x})$ such thatTo justify the formula version from a sentence characterization, temporarily name the tuple by finitely many new constants, apply the characterization in that expanded finite signature, and replace those constants by the free variables, renaming bound variables to avoid capture. This does not introduce a new order assumption: the input already interprets $<$ as a linear order. Multiple quantified relations are handled by repeated application.

As usual, a class of structures here is closed under isomorphism and uses standard finite-structure encodings. First suppose $K$ is defined by $\exists R\varphi$, with $\varphi$ in partial fixed-point logic. Deterministically enumerate the polynomial-length encodings of $R$ and evaluate $\varphi$ on each expansion. PFP evaluation is in PSPACE even without a supplied order: a relation of fixed arity requires polynomially many bits, iteration has at most exponentially many states, and a polynomial-length counter suffices to stop after that many stages. If no consecutive stages agree before a repeat must have occurred, the partial fixed point is empty. Nested fixed points have fixed syntactic depth and can be evaluated with reused polynomial storage. Thus the encoding language of $K$ is in PSPACE.

Conversely, suppose that encoding language is decidable in PSPACE. Expand each structure by a freely chosen linear order $R$. On such ordered expansions, the assumed characterization gives a PFP sentence $\psi (R)$ deciding whether the reduct belongs to $K$. Since $K$ is isomorphism-invariant, the answer is independent of the particular chosen order. HenceThe expression inside the second-order quantifier is PFP, since the assertion that $R$ is a linear order is first-order. Every finite universe has a linear order. This proves both directions, using an existentially supplied order when the original signature has none. The construction is also the order-independent consequence of part (a), which shows that existential relation quantification does not increase PFP's power on an already ordered structure.

At linear order a coordinate displacement ${\xi }^{\mu }$ changes a metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. Work in conformal time with ${\bar{g}}_{\mu \nu }=a^2\mathrm{diag}(-1,1,1,1)$, write ${\xi }^0=\alpha$ and lower spatial indices with ${\delta }_{ij}$. Directly evaluating the Lie derivative of a covariant tensor field givesHere primes on $\alpha ,{\xi }_i$ mean conformal-time derivatives, while primes on $q_{0\mu }^{\prime }$ label transformed components. Setting the transformed lapse and shift perturbations to zero requiresThese are first-order time equations. Their explicit local solutions areThus four coordinate functions can impose the four synchronous conditions in any regular perturbative patch. This is synchronous gauge fixing by coordinate displacement. The arbitrary spatial functions $C,D_i$ remain as residual gauge freedom; the conditions alone do not fix the coordinate system completely.

Geometrically, synchronous coordinates follow a congruence of freely falling observers launched normally from a spatial slice, using their proper time. This gives unit lapse and zero shift in cosmic time; conversion to background conformal time gives the displayed convention. “Always possible” is local: geodesic caustics or singularities can prevent a global synchronous chart. Such global obstructions do not invalidate local linear cosmological perturbation theory.

The cardinal bound for an increasing chain of countable sets is $\boxed{|X|\le {\aleph }_1}$. If $X$ is countable, there is nothing to prove. Otherwise choose $Y\subseteq X$ with $|Y|={\aleph }_1$, and for every $y\in Y$ choose an index $i(y)$ such that $y\in X_{i(y)}$.

For any $i\in I$, its countable $X_i$ cannot contain all of $Y$, so choose $y\in Y\setminus X_i$. Since $I$ is a linear order, comparison of $i$ and $i(y)$ forces $i<i(y)$: the other direction would imply $y\in X_i$. Therefore $X_i\subseteq X_{i(y)}$. The selected family is cofinal among the original sets, andBy infinite cardinal arithmetic, this union of ${\aleph }_1$ countable sets has cardinality at most ${\aleph }_1$. No well-order or cofinality assumption on $I$ was used; a linear increasing union of countable sets has the same bound even for an arbitrary linear index order.

Using the stated Christoffel symbols in the definition of the Riemann curvature tensor, the two nontrivial contractions areHenceand the vacuum Einstein field equations reduce toThis is the diagonal Rosen coordinates for a plane gravitational wave equation.

For the plus-polarized wave, setComparison of the transverse metric components gives, to linear order,orThereforeThe Vacuum Einstein equations are consequently satisfied at linear order. A real gravitational wave is obtained by taking the real part of the complex plane-wave notation.