The linearized-wave approximation regime and an ideal two-body detector with a prescribed finite arm path

Let $(M,g,N)=({\mathbb{R}}^4,\eta ,{\mathrm{\partial }}_0)$ be the flat model with its inertial chart and let $c>0$ be the parameter of SI unit and coordinate conventions of the classical gravitational framework. All objects below are component families in this chart.

(a) Linearized-wave datum. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^4$ be a nonempty open set and let $K\subseteq \mathrm{\Omega }$ be a nonempty compact set. Fix a length scale $L>0$, and use the numerical threshold ${\delta }_0=1\mathrm{/}16$ and chart control functional $B_K$ defined in Linearized field equations as the first-order consequence of the adopted field law, with the harmonic-gauge wave reduction and the quadratic residual control clause (d) for this $K,L$. A linearized-wave datum on $K$ consists of:

• a flat linearized family $({\epsilon }_1,h,q)$ over $\mathrm{\Omega }$ with perturbation $h$ (Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge) such that $B_K(h)<\mathrm{\infty }$ in the chart control functional of that clause;
• the real cosmological parameter $\mathrm{\Lambda }$ and a smooth family $\mathcal{T}(\lambda )$ of stress-energy tensor fields on $\mathrm{\Omega }$ for $\lambda \in (-{\epsilon }_1,{\epsilon }_1)$, with component fields smooth jointly in $(\lambda ,x)$ (Stress-energy tensor field of a model);
• the zeroth-order compatibility condition $\kappa {\mathcal{T}}_{ab}(0)=\mathrm{\Lambda }{\eta }_{ab}$, where $\kappa =8\pi G_N\mathrm{/}{c}^4$; define the linearized source field, with curvature units, by $S_{ab}:=\kappa {\mathrm{\partial }}_{\lambda }{\mathcal{T}}_{ab}(\lambda ){\mathrm{\mid }}_0$, and require $G_{ab}^{(1)}(h)+\mathrm{\Lambda }{h}_{ab}=S_{ab}$ in the sense of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (e);
• a nonzero real amplitude $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }<{\epsilon }_1$ for which $g(\lambda ):=\eta +\lambda h$ is a Lorentzian metric of signature $(-,+,+,+)$ on a neighborhood of $K$.

The parameter $\lambda$ is dimensionless, and in the adopted SI length coordinates $S_{ab}$, $G_{ab}^{(1)}(h)$ and $\mathrm{\Lambda }{h}_{ab}$ have units ${\mathrm{m}}^{-2}$, whereas ${\mathcal{T}}_{ab}$ has energy-density units and $\kappa {\mathcal{T}}_{ab}$ has units ${\mathrm{m}}^{-2}$ (SI unit and coordinate conventions of the classical gravitational framework). The $B_K$ and ${\delta }_0$ values in the chart Taylor estimate are numerical controls: evaluate component functions and their coordinate derivatives as SI numerical values in this fixed length-coordinate chart, then form the displayed real-valued functional. It is not a physical observable or an intrinsic norm. In this fixed numerical convention, $B_K$, ${\delta }_0$ and $C_{K,L}$ are real control numbers and $C_{K,L}{B}_K^2\mathrm{/}{L}^2$ has the residual component unit ${\mathrm{m}}^{-2}$ for $L$ in metres. The datum is a linearized vacuum wave datum when $S=0$ and $\mathrm{\Lambda }=0$. If in addition the perturbation is a TT planar perturbation with amplitudes $A_{+},A_{\times }$ (Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (b)), the datum is a TT linearized-wave datum. Its approximation defect is

$D(\lambda ,K):={\max }_{a,b}{\sup }_{p\in K}\mid G_{ab}(\eta +\lambda h)(p)+\mathrm{\Lambda }{g}_{ab}(\lambda )(p)-\kappa {\mathcal{T}}_{ab}(\lambda ,p)\mid ,$

where $G$ is the Einstein tensor of the exact metric $\eta +\lambda h$; this is the residual against the specified exact stress-energy family, not against its first-order coefficient alone. In length coordinates it has units ${\mathrm{m}}^{-2}$. The datum is within the quadratic approximation when $\mathrm{\mid }\lambda \mathrm{\mid }{B}_K(h)\le {\delta }_0$. This uses the chosen threshold rather than an unspecified existential ${\delta }_0\mathrm{/}{B}_K(h)$, and also covers $B_K(h)=0$. Whether a datum satisfies any of these conditions is a property of the given data; this definition names the data, the condition and the defect and asserts neither existence nor a bound.

(b) Ideal two-body detector. Let a TT linearized-wave datum be given. Choose a nonempty open interval $J\subseteq \mathbb{R}$, a spatial point $x_0\in {\mathbb{R}}^3$, and a nonzero spatial vector $\delta \in {\mathbb{R}}^3$ with Euclidean norm $\mathrm{\mid }\delta \mathrm{\mid }>0$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). Require the coordinate tube

${\mathcal{A}}_{J,\delta }:=\{(cs,x_0+q\delta ):s\in J,q\in [0,1]\}$

to lie in $\mathrm{\Omega }$. For $q\in [0,1]$ define the smooth curves ${\gamma }_q:J\to \mathrm{\Omega }$ by ${\gamma }_q(s):=(cs,x_0+q\delta )$, and require $g(\lambda )$ to be Lorentzian on a neighborhood of ${\mathcal{A}}_{J,\delta }$; the reference worldline is ${\gamma }_0$ and the companion worldline is ${\gamma }_1$. Whenever these endpoint curves are the ideal freely falling worldlines assigned by the postulate of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework, their pair with $(x_0,\delta )$ is the ideal two-body detector. Its arm direction is $e_{\delta }:=\delta \mathrm{/}\mathrm{\mid }\delta \mathrm{\mid }$, the spatial unit vector in the chosen inertial chart.

In SI length coordinates, the components of $x_0$ and $\delta$ and the arm length $\mathrm{\mid }\delta \mathrm{\mid }$ are in metres, while $e_{\delta }$ is dimensionless.

(c) Finite arm length and strain. For $s\in J$ the prescribed spatial arm path is ${\sigma }_s:[0,1]\to \mathrm{\Omega }$, ${\sigma }_s(q):={\gamma }_q(s)$, with tangent ${\dot{\sigma }}_s(q)={\delta }^i{\mathrm{\partial }}_i$. Its metric arm length is the metric length of this spacelike coordinate segment,

$L_{\delta }(s):={\int }_0^1\sqrt{g(\lambda {)}_{{\sigma }_s(q)}({\dot{\sigma }}_s(q),{\dot{\sigma }}_s(q))}dq,$

whenever the radicand is positive for every $q\in [0,1]$; the root is the nonnegative square root of Rational powers $a^r$ of a positive base. This is the length along the prescribed chart-straight path, not the infimum over spatial paths and not a tangent-vector norm between finite-separated endpoints. The strain function is $S_{\delta }(s):=L_{\delta }(s)\mathrm{/}\mathrm{\mid }\delta \mathrm{\mid }-1$. For a TT linearized-wave datum put

$a_{\delta }(s,q):=\frac{{\delta }^i{\delta }^j{h}_{ij}(cs,x_0+q\delta )}{\mathrm{\mid }\delta {\mathrm{\mid }}^2}\mathrm{.}$

Since $g_{0i}=0$ and $g_{ij}={\delta }_{ij}+\lambda h_{ij}$, wherever $1+\lambda a_{\delta }(s,q)>0$ for every $q\in [0,1]$ one has

$L_{\delta }(s)=\mathrm{\mid }\delta \mathrm{\mid }{\int }_0^1\sqrt{1+\lambda a_{\delta }(s,q)}dq,S_{\delta }(s)={\int }_0^1\sqrt{1+\lambda a_{\delta }(s,q)}dq-1.$

In the adopted SI length coordinates $L_{\delta }$ has units metres and $S_{\delta }$ is dimensionless.

The first-order strain response is

$S_{\delta }^{(1)}(s):={{\mathrm{\partial }}_{\lambda }{S}_{\delta }(s)\mid }_{\lambda =0}=\frac{1}{2}{\int }_0^1{a}_{\delta }(s,q)dq,$

which is the parameter derivative of this finite-path length ratio; the response theorem gives a uniform remainder estimate. If ${\delta }^3=0$, the phase $u=cs-x_0^3-q{\delta }^3$ is constant along the arm, so the integrand is constant in $q$ and the formula reduces to the usual single-point TT strain.

(d) Reading and scope. In the TT metric, $g({\gamma }_q^{\mathrm{\prime }},{\gamma }_q^{\mathrm{\prime }})=-c^2$, so $s=x^0\mathrm{/}c$ is proper time along each endpoint curve; their geodesic and free-fall properties are established in the response theorem of this page. The detector is the two ideal endpoint bodies plus the defined path length along a chosen coordinate segment at equal inertial-chart time; the length depends on the chosen path and chart and is not asserted to be a geodesic distance. Rigid bodies, elasticity, internal resonances, the propagation and readout of light, noise and calibration are outside this definition; no reading of a real apparatus is defined here, and no fact about nature is asserted.

• Why the regime conditions are separate from the detector. Clause (a) packages the approximation hypotheses — a linearized family, a length scale, a source and parameter, a compact set and an amplitude — so that later theorems can quantify the approximation defect and the response without repeating them. When $\mathrm{\Lambda }=0$ and the specified family satisfies $\mathcal{T}(\lambda )=0$ identically, the quadratic residual bound is the content of Linearized field equations as the first-order consequence of the adopted field law, with the harmonic-gauge wave reduction and the quadratic residual control clause (d) and TT plane waves with arbitrary bounded profiles solve the linearized vacuum equations, are in harmonic gauge, and give Lorentzian small-amplitude metrics with a quadratic Einstein-tensor residual clause (c). Merely having $S=0$ allows higher-order terms in $\mathcal{T}(\lambda )$, for which no residual bound is asserted here; the definition itself only names the defect.

• Simultaneity and the arm path are chart-defined. The length $L_{\delta }(s)$ is the metric length along the prescribed coordinate segment at equal inertial-chart time $x^0=cs$. It is not an intrinsic shortest distance between the endpoints or a statement about how any apparatus records a duration or length; its path and frame dependence are part of the model definition.