Isaacson Effective Stress–Energy

Gravitational waves do not possess a unique local stress–energy tensor in the same sense as matter: at one event, a coordinate choice can remove first derivatives of the metric. A rapidly varying weak wave on a slowly varying background nevertheless has a controlled, averaged energy–momentum. Isaacson’s construction makes that statement precise through scale separation, a second-order expansion of the Einstein tensor, and an average over several wavelengths.

Required background. Weak-Field Gravity is used to expand the Einstein tensor and track gauge dependence. Waves, Green Functions, and Geometric Optics supplies the eikonal hierarchy, wavelength, and transport along null rays.

Helpful background. Einstein Equations is used to interpret the averaged second-order term as a source for the slowly varying background. Equivalence Principles and Gravitational Measurement explains why an unaveraged local gravitational energy density is not a tensorial observable.

Two scales and one controlled average

Write the exact metric as

$$g_{\mu\nu}=\gamma_{\mu\nu}+h_{\mu\nu},$$

where $\gamma_{\mu\nu}$ varies on a curvature scale $L$ and $h_{\mu\nu}$ has wavelength $\lambda$. The Isaacson regime requires

$$\lvert h_{\mu\nu}\rvert\ll1, \qquad \epsilon=\frac{\lambda}{L}\ll1.$$

Small amplitude and short wavelength are independent assumptions. Derivatives enhance the fast field:

$$\nabla h\sim\frac{h}{\lambda}, \qquad \nabla\nabla h\sim\frac{h}{\lambda^2},$$

whereas derivatives of the background scale as $L^{-1}$. A small $h$ can therefore make a finite second-order contribution after its fast oscillations are averaged.

Choose an averaging scale $\ell$ satisfying

$$\lambda\ll\ell\ll L.$$

The average removes phase-dependent oscillations while leaving $\gamma_{\mu\nu}$ nearly constant. At leading order it annihilates total derivatives of fast bounded quantities and commutes with slow covariant differentiation. These are asymptotic properties of the two-scale expansion, not an arbitrary smoothing recipe. If no intermediate $\ell$ exists, the Isaacson tensor is not controlled.

Flanagan and Hughes develop the curved-background high-frequency expansion, its averaging rules, and its relation to geometric optics in Flanagan and Hughes 2005, §§5.2–5.3, pp. 25–29, PDF.

From the Einstein equation to effective stress–energy

Expand the Einstein tensor about the slow metric:

$$G_{\mu\nu}[g] =G_{\mu\nu}[\gamma] +G^{(1)}_{\mu\nu}[h] +G^{(2)}_{\mu\nu}[h] +O(h^3).$$

The leading rapidly varying equation is $G^{(1)}_{\mu\nu}[h]=0$ in a vacuum wave region. Averaging the slow part of the exact equation gives

$$G_{\mu\nu}[\gamma] =8\pi G\left(T^{\mathrm{slow}}_{\mu\nu} +t^{\mathrm{GW}}_{\mu\nu}\right),$$

with

$$t^{\mathrm{GW}}_{\mu\nu} =-\frac{1}{8\pi G} \left\langle G^{(2)}_{\mu\nu}[h]\right\rangle.$$

In leading high-frequency TT gauge this reduces to the compact form

$$t^{\mathrm{GW}}_{\mu\nu} =\frac{1}{32\pi G} \left\langle \nabla_\mu h^{\mathrm{TT}}_{\alpha\beta} \nabla_\nu h_{\mathrm{TT}}^{\alpha\beta} \right\rangle.$$

The last equation is not a definition valid for an arbitrary perturbation. It assumes the leading wave equation, short-wavelength gauge conditions, wavelength averaging, and neglect of relative corrections in $\lambda/L$ and $h$. With the slow metric chosen self-consistently, the source curvature scales as $h^2/\lambda^2$. Thus $h\sim\lambda/L$ allows order-one backreaction on the slow geometry, while $h\ll\lambda/L$ permits propagation on an approximately fixed background. Small amplitude alone does not guarantee negligible accumulated backreaction.

At this order the effective tensor is conserved on the background,

$$\nabla^\mu t^{\mathrm{GW}}_{\mu\nu}=0,$$

as a consequence of the wave-amplitude transport equation. In an eikonal representation $h_{\mu\nu}\sim\Re(A_{\mu\nu}e^{i\theta/\epsilon})$, its leading form is proportional to $k_\mu k_\nu$ with $k_\mu=\nabla_\mu\theta$ null. The wave energy therefore travels along the same null rays that govern phase propagation.

Plane-wave energy and outward flux

For a TT wave moving toward $+z$,

$$h^{\mathrm{TT}}_{ij} =h_+(u)e^+_{ij}+h_\times(u)e^\times_{ij}, \qquad u=t-z,$$

with $e^A_{ij}e^{B\,ij}=2\delta^{AB}$. An inertial observer measures

$$\rho_{\mathrm{GW}}=t^{\mathrm{GW}}_{00} =\frac{1}{16\pi G} \left\langle \dot h_+^2+\dot h_\times^2 \right\rangle.$$

Because $\partial_z h_A=-\dot h_A$, lowering and raising the time index gives

$$t^{0z}_{\mathrm{GW}}=\rho_{\mathrm{GW}}>0,$$

so the energy flux points toward increasing $z$. This sign is consistent with the volume convention that outward flux is positive.

For monochromatic components

$$h_+=A_+\cos\omega u, \qquad h_\times=A_\times\cos(\omega u+\delta),$$

averaging over many cycles yields

$$\rho_{\mathrm{GW}} =\frac{\omega^2}{32\pi G} \left(A_+^2+A_\times^2\right).$$

The relative phase $\delta$ changes the polarization ellipse but not the averaged energy at fixed amplitudes. Positivity follows manifestly from the sum of squares.

Gauge robustness and its limits

The unaveraged expression quadratic in first derivatives of $h_{\mu\nu}$ is gauge dependent. In the Isaacson regime, allowed high-frequency gauge transformations change the leading averaged tensor only by terms that vanish under the averaging rules or are subleading in $\lambda/L$. The robust object is therefore an equivalence class at stated asymptotic order—not a pointwise, exact gravitational stress tensor.

The construction can fail or require correction when:

• $\lambda$ is comparable to the background curvature or inhomogeneity scale;
• the wave amplitude is too large for the perturbative hierarchy;
• boundaries or averaging windows retain phase-dependent total derivatives;
• matter varies on the wavelength scale and cannot be separated into a slow source;
• one asks for local energy below the averaging scale.

Near-zone pseudotensors, Bondi flux at null infinity, horizon flux, and quasi-local energies answer different questions. Agreement in their common high-frequency far-zone limit is a check, not a reason to identify them outside that overlap.

This page also does not calculate nonlinear source dynamics or detector noise. Relativistic Two-Body Dynamics supplies source evolution; Gravitational-Wave Astronomy supplies detector response and inference. Isaacson averaging describes propagation energy on a specified background.

Common pitfalls

Calling $t^{\mathrm{GW}}_{\mu\nu}$ exact and local. It is effective, averaged, and asymptotic in both amplitude and wavelength-to-curvature ratio.

Averaging without an intermediate scale. The inequalities $\lambda\ll\ell\ll L$ are part of the result. A numerical smoothing width alone does not establish them.

Forgetting the polarization-tensor norm. With $e^A_{ij}e^{B\,ij}=2\delta^{AB}$, the TT contraction supplies the factor that leads to $1/(16\pi G)$ in the strain flux.

Exercises

1. Monochromatic flux. For $h_+=A\cos\omega(t-z)$ and $h_\times=0$, compute $\rho_{\mathrm{GW}}$ and $t^{0z}_{\mathrm{GW}}$ after averaging.

Solution

The time derivative is $\dot h_+=-\omega A\sin\omega(t-z)$, whose squared cycle average is $\omega^2A^2/2$. Hence

$$\rho_{\mathrm{GW}} =\frac{1}{16\pi G}\frac{\omega^2A^2}{2} =\frac{\omega^2A^2}{32\pi G}.$$

For an outgoing profile, $\partial_z h_+=-\dot h_+$. The covariant component $t_{0z}$ is negative, but raising the time index changes its sign, giving $t^{0z}=\rho_{\mathrm{GW}}>0$. Energy flows toward $+z$ at the speed of light.

2. Scale-separation estimate. A wave has amplitude $h_0$, wavelength $\lambda$, and background scale $L$. Estimate the ratio of its effective curvature source $G t^{\mathrm{GW}}$ to the background curvature $L^{-2}$.

Solution

The effective tensor scales as $t^{\mathrm{GW}}\sim h_0^2/(G\lambda^2)$, so $G t^{\mathrm{GW}}\sim h_0^2/\lambda^2$. Dividing by $L^{-2}$ gives

$$\frac{G t^{\mathrm{GW}}}{L^{-2}} \sim h_0^2\left(\frac{L}{\lambda}\right)^2.$$

Thus a small-amplitude wave can contribute finite slow curvature when $h_0$ is comparable to $\lambda/L$. This is why amplitude and wavelength must be ordered together in the two-scale expansion.

References

• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.