Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge

Let $(M,g,N)$ be the flat model of The flat model spacetime: the Minkowski metric on R^4 and its inertial chart with inertial chart $x$ and constant metric components ${\eta }_{ab}=\mathrm{diag}(-1,1,1,1)$, and let $\mathrm{\Omega }\subseteq M$ be a nonempty open set, identified with its image under $x$; all components below are taken in the inertial chart. Let $c>0$ be a real parameter.

(a) Flat linearized family. A flat linearized family over $\mathrm{\Omega }$ consists of a number ${\epsilon }_1>0$, a smooth symmetric covariant $2$-tensor field $h$ on $\mathrm{\Omega }$ with components $h_{ab}$ (Smooth covariant 2-tensor fields on a smooth manifold) — the first-order perturbation — and a smooth family $q$ of symmetric component fields $q_{ab}(\lambda ,p)$, defined for $\lambda \in (-{\epsilon }_1,{\epsilon }_1)$ and $p\in \mathrm{\Omega }$, such that for every $\lambda$ with $0<\mathrm{\mid }\lambda \mathrm{\mid }<{\epsilon }_1$ the component field

$g_{ab}(\lambda ,p):={\eta }_{ab}+\lambda h_{ab}(p)+{\lambda }^2{q}_{ab}(\lambda ,p)$

is the component field of a smooth Lorentzian metric $g(\lambda )$ on $\mathrm{\Omega }$ of signature $(-,+,+,+)$; $q$ is the normalized remainder. The family is thus normalized so that $g(0)=\eta$ is its value at $\lambda =0$ and $h={\mathrm{\partial }}_{\lambda }g(\lambda ){\mathrm{\mid }}_{\lambda =0}$. No assumption is made that $g(\lambda )$ solves any field equation; the metrics $g(\lambda )$ are the mathematical input of the linearization.

(b) First-order coefficients. Let $A(\lambda )$ stand for any of the component families ${\mathrm{\Gamma }}^a{}_{bc}(g(\lambda ))$, ${\mathrm{Ric}}_{ab}(g(\lambda ))$, $S(g(\lambda ))$, $G_{ab}(g(\lambda ))$ of the metric-selected connection of $g(\lambda )$; these are smooth in $(\lambda ,p)$ because the Christoffel symbols, the curvature components and the inverse metric are smooth functions of the metric components and their derivatives (Christoffel symbols of an affine connection, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Define the first-order coefficients

${\mathrm{\Gamma }}^{(1)a}{}_{bc}:={\mathrm{\partial }}_{\lambda }{\mathrm{\Gamma }}^a{}_{bc}(g(\lambda )){\mid }_{\lambda =0},{\mathrm{Ric}}_{ab}^{(1)}:={\mathrm{\partial }}_{\lambda }{\mathrm{Ric}}_{ab}(g(\lambda )){\mid }_{\lambda =0},$ $S^{(1)}:={\mathrm{\partial }}_{\lambda }S(g(\lambda )){\mid }_{\lambda =0},G_{ab}^{(1)}:={\mathrm{\partial }}_{\lambda }{G}_{ab}(g(\lambda )){\mid }_{\lambda =0}\mathrm{.}$

(c) Index conventions. Indices on first-order quantities are raised and lowered with the flat metric $\eta$ and its inverse ${\eta }^{ab}$ (The inverse metric, musical isomorphisms and index raising): ${\mathrm{\partial }}^a:={\eta }^{ab}{\mathrm{\partial }}_b$, $h^{ab}:={\eta }^{ac}{\eta }^{bd}{h}_{cd}$, and

$\mathrm{\square }:={\eta }^{ab}{\mathrm{\partial }}_a{\mathrm{\partial }}_b=-{\mathrm{\partial }}_0^2+{\mathrm{\partial }}_1^2+{\mathrm{\partial }}_2^2+{\mathrm{\partial }}_3^2$

is the d'Alembertian of the flat metric, applied componentwise to component families. The trace of $h$ and the trace-reversed perturbation are

$h:={\eta }^{ab}{h}_{ab},{\bar{h}}_{ab}:=h_{ab}-\frac{1}{2}h{\eta }_{ab}\mathrm{.}$

The inverse passage is the same formula, ${\bar{h}}_{ab}$'s trace being ${\eta }^{ab}{\bar{h}}_{ab}=h-2h=-h$: explicitly

$h_{ab}={\bar{h}}_{ab}-\frac{1}{2}{\eta }_{ab}{\eta }^{cd}{\bar{h}}_{cd}\mathrm{.}$

(d) Gauge transformation. For a smooth covector field ${\xi }_b$ on $\mathrm{\Omega }$ define the gauge-transformed first-order perturbation

$h_{ab}^{\mathrm{\prime }}:=h_{ab}-{\mathrm{\partial }}_a{\xi }_b-{\mathrm{\partial }}_b{\xi }_a\mathrm{.}$

This is the first-order perturbation of the pullback family ${\mathrm{\Phi }}_{-\lambda }^{\ast }g(\lambda )$ along the local flows ${\mathrm{\Phi }}_s$ of the vector field ${\xi }^a={\eta }^{ab}{\xi }_b$, that is, the perturbation produced by the infinitesimal chart change $x^{\mathrm{\prime }a}=x^a+\lambda {\xi }^a$ and re-read on the fixed inertial chart; the transformation of the metric components displayed here is the linearization of the Jacobian transformation law, and its consequences for $\bar{h}$ and for ${\mathrm{Ric}}^{(1)}$, $S^{(1)}$ and $G^{(1)}$ are proved separately on this page (the first-order curvature and gauge theorem that follows this definition). The perturbation $h$ is in harmonic gauge (also called de Donder gauge) when

${\mathrm{\partial }}^a{\bar{h}}_{ab}=0.$

(e) Scope. This item fixes the objects of the flat linearization: the normalized family, the first-order coefficients defined as exact $\lambda$-derivatives, the trace reversal, the d'Alembertian and the gauge transformation. It asserts no explicit formula for the first-order coefficients, no existence of a gauge representative satisfying (d), no solution of any linearized field equation and no physical interpretation; those are separate results. The notation is tied to the inertial chart: a change of chart is a different operation from the gauge transformation (d), which changes the metric family inside the fixed chart.

• Flow calculation for the gauge formula. Let $X:={\xi }^a{\mathrm{\partial }}_a$ be the smooth vector field associated with the covector ${\xi }_b$ by ${\xi }^a={\eta }^{ab}{\xi }_b$. Its local flow ${\mathrm{\Phi }}_s$ is smooth on an open domain and satisfies ${\mathrm{\Phi }}_0(p)=p$ and ${\mathrm{\partial }}_s{\mathrm{\Phi }}_s(p)=X({\mathrm{\Phi }}_s(p))$ (Local and global flows generated by a vector field, The fundamental theorem on flows). Therefore, at every point where these local expressions are defined, ${{\mathrm{\partial }}_{\lambda }{x}^c({\mathrm{\Phi }}_{-\lambda }(p))\mid }_{\lambda =0}=-{\xi }^c(p),{{\mathrm{\partial }}_{\lambda }{\mathrm{\partial }}_a{\mathrm{\Phi }}_{-\lambda }^c(p)\mid }_{\lambda =0}=-{\mathrm{\partial }}_a{\xi }^c(p),$ where smoothness of $\mathrm{\Phi }$ permits interchanging these coordinate derivatives (Clairaut--Schwarz theorem for continuous second partial derivatives). The pullback formula for a covariant $2$-tensor is $({\mathrm{\Phi }}_{-\lambda }^{\ast }g(\lambda ){)}_{ab}(p)=g_{cd}(\lambda ,{\mathrm{\Phi }}_{-\lambda }(p)){\mathrm{\partial }}_a{\mathrm{\Phi }}_{-\lambda }^c(p){\mathrm{\partial }}_b{\mathrm{\Phi }}_{-\lambda }^d(p)\mathrm{.}$ Since $g_{cd}(\lambda ,x)={\eta }_{cd}+\lambda h_{cd}(x)+{\lambda }^2{q}_{cd}(\lambda ,x)$ and $\eta$ is constant in the inertial chart, the chain and product rules give, on differentiating at $\lambda =0$, $h_{ab}-{\eta }_{cb}{\mathrm{\partial }}_a{\xi }^c-{\eta }_{ac}{\mathrm{\partial }}_b{\xi }^c=h_{ab}-{\mathrm{\partial }}_a{\xi }_b-{\mathrm{\partial }}_b{\xi }_a$, the formula in clause (d). This is a local-in-flow calculation; it makes no claim that $X$ is complete or that the flow exists for a common nonzero parameter on all of $\mathrm{\Omega }$ (Local and global flows generated by a vector field, A smooth vector field is a smooth section of the tangent bundle, The pullback of a covariant tensor field).