Flat And Newtonian Bridges

Let $M:={\mathbb{R}}^4$ carry its standard smooth structure (Smooth manifolds and their smooth charts), and let $x=(x^0,x^1,x^2,x^3):M\to {\mathbb{R}}^4$ be the identity chart, $x(p)=p$; it is a global chart of the maximal smooth atlas, and the coordinate derivations ${\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3$ form a basis of $T_{p}M$ at every $p\in M$ (Derivations at a point and the tangent space, Coordinate derivations form a basis of the tangent space). Write

$\eta :=\mathrm{diag}(-1,1,1,1)$

for the standard Lorentz form of inertia $(3,1,0)$ in this basis (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space).

(a) The metric. Define the covariant $2$-tensor field $g$ on $M$ by the requirement that its components in the identity chart are the constants

$g_{ab}:={\eta }_{ab}(a,b=0,1,2,3),$

that is, $g_p({\mathrm{\partial }}_a{\mathrm{\mid }}_p,{\mathrm{\partial }}_b{\mathrm{\mid }}_p)={\eta }_{ab}$ for every $p\in M$ (Smooth covariant 2-tensor fields on a smooth manifold clause (a)). The component functions are constant, hence $C^{\mathrm{\infty }}$, so $g$ is a smooth symmetric covariant $2$-tensor field; at every $p$ the form $g_p$ is nondegenerate with inertia $(3,1,0)$. By Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (a) the pair $(M,g)$ is therefore a smooth Lorentzian four-manifold of signature $(-,+,+,+)$.

(b) The inertial chart and frame. The chart $x$ is called the inertial chart of the flat model. In this chart the coordinate frame $({\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)$ is a $g$-orthonormal frame at every point, $g({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)={\eta }_{ab}$; it is simultaneously a coordinate frame and a tetrad field in the sense of Local frame fields, coordinate frames and tetrad fields, defined on all of $M$.

(c) Time orientation and causal classification. Define the smooth vector field $N:={\mathrm{\partial }}_0$ on $M$ (A smooth vector field is a smooth section of the tangent bundle). Since $g_p(N_p,N_p)={\eta }_{00}=-1<0$ for every $p$, the field $N$ is everywhere timelike, so $(M,g,N)$ is a smooth time-oriented Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c)). For a nonzero tangent vector $v=v^a{\mathrm{\partial }}_a{\mathrm{\mid }}_p$ the definitions of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (c) and Timelike, null, spacelike and causal vectors of a Lorentz vector space give

$g_p(v,v)=-(v^0{)}^2+{\sum }_{i=1}^3(v^i{)}^2,g_p(v,N_p)=-v^0\mathrm{.}$

so, for a nonzero causal $v$, $v$ is future-directed exactly when $v^0>0$, and

• $v$ is future timelike $⟺$ $v^0>0$ and $(v^0{)}^2>{\sum }_i(v^i{)}^2$;
• $v$ is future null $⟺$ $v^0>0$ and $(v^0{)}^2={\sum }_i(v^i{)}^2$;
• $v$ is spacelike $⟺$ $(v^0{)}^2<{\sum }_i(v^i{)}^2$;

consequently the future causal cone at every point is $\{v\mathrm{\ne }0:v^0\ge \sqrt{{\sum }_i(v^i{)}^2}\}$. These are the standard readings of the Minkowski form in the inertial chart; the chain of equivalences is the specialization of the causal classification to this pair $(g_p,N_p)$ and is recorded here for later use.

(d) Scope. This item defines geometric data only: a manifold, a metric field, a time orientation and a distinguished chart and frame. It makes no claim about curvature, about any field equation, about which geometry a physical model uses, or about nature; the physical reading of the data, the dimensions of the coordinates and the numerical values or signs of physical constants belong to the framework's unit conventions and postulates, which are separate items. In particular no relation between the coordinate functions $x^a$ and physical time or length measurements is asserted here.

Let $(M,g,N)$ be the flat model of The flat model spacetime: the Minkowski metric on R^4 and its inertial chart with its inertial chart $x$, and let $\mathrm{\nabla }$ be its metric-selected connection, whose existence and uniqueness are supplied by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Let $c>0$ be a real parameter.

(a) Vanishing connection. In the inertial chart all Christoffel symbols of $\mathrm{\nabla }$ vanish identically:

${\mathrm{\Gamma }}^a{}_{bc}\equiv 0(a,b,c=0,1,2,3)\mathrm{.}$

(b) Vanishing curvature. The curvature operator of $\mathrm{\nabla }$ vanishes identically, $R\equiv 0$; consequently the curvature components, the Ricci tensor, the scalar curvature and the Einstein tensor vanish identically,

$R^a{}_{bcd}\equiv 0,{\mathrm{Ric}}_{ab}\equiv 0,S\equiv 0,G_{ab}\equiv 0.$

(c) Translation Killing fields. Each coordinate field ${\mathrm{\partial }}_a$ is a Killing field of $(M,g)$ (Killing fields of a Lorentzian metric); in particular the time orientation field $N={\mathrm{\partial }}_0$ is a unit timelike Killing field.

(d) Straight geodesics. A smooth curve $\gamma :I\to M$ on an interval $I\subseteq \mathbb{R}$ with nonempty interior (Geodesic of an affine connection) is a geodesic of $\mathrm{\nabla }$ if and only if there are $x_0,v\in {\mathbb{R}}^4$ such that

$x(\gamma (s))=x_0+sv(s\in I)\mathrm{.}$

For such a geodesic the speed is constant, $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=\eta (v,v)=-(v^0{)}^2+{\sum }_{i=1}^3(v^i{)}^2$; the maximal geodesic with $x(\gamma (0))=x_0$ and $x(\gamma {)}^{\mathrm{\prime }}(0)=v$ has domain $\mathbb{R}$. If $\eta (v,v)<0$ and $v^0>0$, the geodesic is a future timelike worldline (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold) and, for any $s_0\in I$, the parameter $\tau (s):=\sqrt{-\eta (v,v)}(s-s_0)\mathrm{/}c$ is proper time up to an additive constant, with $d\gamma \mathrm{/}d\tau$ the physical four-velocity $U$, $g(U,U)=-c^2$ (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent clause 3).

(e) Inertial-chart velocity reading. Let $\gamma$ be a geodesic as in (d) with $x(\gamma (s))=x_0+sv$, $\eta (v,v)<0$ and $v^0>0$. Define $t(s):=x^0(\gamma (s))\mathrm{/}c=(x_0^0+s{v}^0)\mathrm{/}c$; this is strictly increasing, and on $t(I)$ define ${\gamma }_t(t):=\gamma ((ct-x_0^0)\mathrm{/}{v}^0)$. Then $x({\gamma }_t(t))=(ct,\mathbf{q}(t))$ for its spatial component $\mathbf{q}$, and the coordinate velocity is the constant vector $\mathbf{w}:=d\mathbf{q}\mathrm{/}dt=(c\mathrm{/}{v}^0)(v^1,v^2,v^3)$. With $\mathrm{\mid }\mathbf{w}\mathrm{\mid }:=\mathrm{\parallel }\mathbf{w}{\mathrm{\parallel }}_2$ from the Euclidean inner product (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$), timelikeness gives $\mathrm{\mid }\mathbf{w}\mathrm{\mid }<c$.

• Flatness is a computation, not a definition. The model is called flat in The flat model spacetime: the Minkowski metric on R^4 and its inertial chart; its curvature vanishing is proved here from the constant components, and no converse statement (that a flat Lorentzian manifold is isometric to this model) is asserted.
• Maximality. The geodesics of step 2.2 are complete as curves of the flat model; this completeness is special to the model and is not inherited by metrics only locally isometric to it.
• Use. Clause (b) supplies the curvature computation for the flat vacuum model on this page; clause (d) supplies the kinematic reading of that model and the kinetic input of the weak-field bridge.

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).

Let $\mathrm{\Omega }\subseteq M$ be an open set of the flat model The flat model spacetime: the Minkowski metric on R^4 and its inertial chart, let $({\epsilon }_1,h,q)$ be a flat linearized family over $\mathrm{\Omega }$ with first-order coefficients ${\mathrm{\Gamma }}^{(1)a}{}_{bc}$, ${\mathrm{Ric}}_{ab}^{(1)}$, $S^{(1)}$, $G_{ab}^{(1)}$ and with trace reversal $\bar{h}$, d'Alembertian $\mathrm{\square }$ and gauge transformation $h_{ab}^{\mathrm{\prime }}=h_{ab}-{\mathrm{\partial }}_a{\xi }_b-{\mathrm{\partial }}_b{\xi }_a$ as in Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge, and let $L>0$ be a length scale. All components are taken in the inertial chart and all contractions are with $\eta$ and ${\eta }^{-1}$.

(a) Christoffel symbols. For every compact $K\subseteq \mathrm{\Omega }$ there is a constant $C_K<\mathrm{\infty }$ such that for all $p\in K$ and $0<\mathrm{\mid }\lambda \mathrm{\mid }\le {\epsilon }_1\mathrm{/}2$,

$\mid {\mathrm{\Gamma }}^a{}_{bc}(\lambda ,p)-\lambda {\mathrm{\Gamma }}^{(1)a}{}_{bc}(p)\mid \le C_K{\lambda }^2,{\mathrm{\Gamma }}^{(1)a}{}_{bc}=\frac{1}{2}{\eta }^{ad}({\mathrm{\partial }}_b{h}_{cd}+{\mathrm{\partial }}_c{h}_{bd}-{\mathrm{\partial }}_d{h}_{bc})\mathrm{.}$

(b) Ricci tensor. With the same constants (enlarged if necessary),

$\mid {\mathrm{Ric}}_{ab}(\lambda ,p)-\lambda {\mathrm{Ric}}_{ab}^{(1)}(p)\mid \le C_K{\lambda }^2,{\mathrm{Ric}}_{ab}^{(1)}=\frac{1}{2}({\mathrm{\partial }}_c{\mathrm{\partial }}_a{h}^c{}_b+{\mathrm{\partial }}_c{\mathrm{\partial }}_b{h}^c{}_a-\mathrm{\square }{h}_{ab}-{\mathrm{\partial }}_a{\mathrm{\partial }}_{b}h)\mathrm{.}$

(c) Scalar curvature and Einstein tensor. With the same constants,

$\mid S(\lambda ,p)-\lambda S^{(1)}(p)\mid \le C_K{\lambda }^2,S^{(1)}={\mathrm{\partial }}_a{\mathrm{\partial }}_b{h}^{ab}-\mathrm{\square }h,$

and

$\mid G_{ab}(\lambda ,p)-\lambda G_{ab}^{(1)}(p)\mid \le C_K{\lambda }^2,G_{ab}^{(1)}=-\frac{1}{2}\mathrm{\square }{\bar{h}}_{ab}+{\mathrm{\partial }}^c{\mathrm{\partial }}_{(a}{\bar{h}}_{b)c}-\frac{1}{2}{\eta }_{ab}{\mathrm{\partial }}^c{\mathrm{\partial }}^d{\bar{h}}_{cd}\mathrm{.}$

(d) Gauge transformation. For every smooth covector field $\xi$ on $\mathrm{\Omega }$,

${\bar{h}}^{\mathrm{\prime }}{}_{ab}={\bar{h}}_{ab}-{\mathrm{\partial }}_a{\xi }_b-{\mathrm{\partial }}_b{\xi }_a+{\eta }_{ab}{\mathrm{\partial }}^c{\xi }_c,{\mathrm{\partial }}^a{\bar{h}}^{\mathrm{\prime }}{}_{ab}={\mathrm{\partial }}^a{\bar{h}}_{ab}-\mathrm{\square }{\xi }_b\mathrm{.}$

Consequently $h^{\mathrm{\prime }}$ is in harmonic gauge exactly when $\mathrm{\square }{\xi }_b={\mathrm{\partial }}^a{\bar{h}}_{ab}$, and in that case

$G_{ab}^{(1)}(h^{\mathrm{\prime }})=-\frac{1}{2}\mathrm{\square }{\bar{h}}^{\mathrm{\prime }}{}_{ab}\mathrm{.}$

(e) Gauge invariance of the first-order curvature. For every smooth covector field $\xi$ on $\mathrm{\Omega }$,

${\mathrm{Ric}}_{ab}^{(1)}(h^{\mathrm{\prime }})={\mathrm{Ric}}_{ab}^{(1)}(h),S^{(1)}(h^{\mathrm{\prime }})=S^{(1)}(h),G_{ab}^{(1)}(h^{\mathrm{\prime }})=G_{ab}^{(1)}(h)\mathrm{.}$

That is, subtracting the pure-gauge perturbation ${\mathrm{\partial }}_a{\xi }_b+{\mathrm{\partial }}_b{\xi }_a$ from the first-order perturbation leaves the first-order Ricci, scalar and Einstein tensors unchanged; their first-order coefficients depend on $h$ only through its gauge-equivalence class.

(f) Quadratic remainder. In this clause, suprema of the nonnegative component magnitudes over an empty set are defined to be $0$. For every compact $K\subseteq \mathrm{\Omega }$ there are ${\delta }_0>0$ and $C_K<\mathrm{\infty }$ such that for every smooth symmetric component field $A=(A_{ab})$ on a neighborhood of $K$ with

$B_K(A):={\sup }_K{\max }_{a,b}\mathrm{\mid }{A}_{ab}\mathrm{\mid }+L{\sup }_K{\max }_{a,b,c}\mathrm{\mid }{\mathrm{\partial }}_c{A}_{ab}\mathrm{\mid }+L^2{\sup }_K{\max }_{a,b,c,d}\mathrm{\mid }{\mathrm{\partial }}_c{\mathrm{\partial }}_d{A}_{ab}\mathrm{\mid }\le {\delta }_0,$

the metric $\eta +A$ is Lorentzian of signature $(-,+,+,+)$ on $K$ and

$\mid G_{ab}(\eta +A)(p)-G_{ab}^{(1)}(A)(p)\mid \le \frac{C_K{B}_K(A{)}^2}{L^2}(p\in K),$

where $G_{ab}^{(1)}(A)$ is the right-hand side of the displayed formula of (c) with $h$ replaced by $A$, and $G_{ab}(\eta +A)$ is the Einstein tensor of the metric with components ${\eta }_{ab}+A_{ab}$.