Isaacson Stress-Energy of Gravitational Waves

Ask where the energy of a gravitational wave sits, and general relativity answers with an embarrassment: nowhere in particular. The equivalence principle lets you jump into a freely falling frame that erases the field at any single event, so there is no honest tensor that measures gravitational energy density point by point. The nineteenth-century workaround — Einstein's and Landau–Lifshitz's pseudotensors — buys a conserved quantity at the price of coordinate dependence: rotate your axes and the "energy" of empty space changes. For decades this made the simplest question in gravitational-wave physics — do the waves carry energy, and how much? — genuinely contentious, culminating in Feynman's 1957 sticky-bead argument that they must. Richard Isaacson's 1968 construction is the resolution, and it is beautiful because it concedes exactly the right amount. It does not manufacture a local energy density; it builds a nonlocal one, meaningful only after averaging over a region several wavelengths across. The trick is a clean separation of scales: split the metric into a slowly varying background $\bar g_{\mu\nu}$ of curvature radius $\mathcal R$ and a fast ripple $h_{\mu\nu}$ of wavelength $\lambda\ll\mathcal R$ , expand the Einstein tensor to second order in $h$ , and average. The linear term is the wave equation; the averaged quadratic term, with its sign flipped, is a genuine symmetric, gauge-invariant, covariantly conserved stress-energy tensor $t_{\mu\nu}$ that curves the background just like any other matter. This is the rigorous bridge from the linearized wave equation to every energy statement in gravitational-wave astronomy: the quadrupole luminosity, the inspiral of the Hulse–Taylor pulsar, the momentum kick of a merging black-hole binary, and the peak power of GW150914 that briefly outshone every star in the observable universe.

Key takeaways

Gravitational energy has no local density — the equivalence principle erases the field at any point — so the naive question "energy per unit volume of a wave" has no pointwise answer.
Isaacson's fix works in the short-wavelength regime $\lambda\ll\mathcal R$ : expand $G_{\mu\nu}$ to second order in the ripple $h_{\mu\nu}$ and average over several wavelengths.
The averaged quadratic Einstein tensor, sign-flipped, is the effective stress-energy $t_{\mu\nu}=-\frac{1}{8\pi G}\langle G^{(2)}_{\mu\nu}\rangle$ — symmetric, gauge-invariant, and conserved on the background.
In transverse-traceless gauge it collapses to one term, $t_{\mu\nu}=\frac{1}{32\pi G}\langle\partial_\mu h^{\rm TT}_{\alpha\beta}\,\partial_\nu h^{\alpha\beta}_{\rm TT}\rangle$ , the workhorse formula of the field.
Integrated over a sphere it yields the quadrupole luminosity $P=\frac{G}{5c^5}\langle\dddot Q_{ij}\dddot Q^{ij}\rangle$ and hence the inspiral of binaries.
A GW150914-scale wave ( $h\sim10^{-21}$ , $f\sim150$ Hz) carries a flux of only $\sim3.6$ mW/m² at Earth, yet the source radiated at $\sim3.6\times10^{49}$ W — about a thousandth of the maximum luminosity $c^5/G$ .
Key idea: gravitational-wave energy is a second-order, nonlocal effect — expand $G_{\mu\nu}$ to quadratic order in $h_{\mu\nu}$ and average over many wavelengths; the averaged quadratic Einstein tensor, sign-flipped, is a genuine gauge-invariant stress-energy tensor that curves the background.

The core insight

Gravitational-wave energy is a second-order, nonlocal effect: expand $G_{\mu\nu}$ to quadratic order in $h_{\mu\nu}$ and average over many wavelengths. The averaged quadratic Einstein tensor, with its sign flipped, is a genuine gauge-invariant stress-energy tensor that curves the background.

Where this lives today

The Isaacson tensor is the quiet engine under gravitational-wave astronomy. Every LIGO–Virgo–KAGRA detection is interpreted through energy statements it makes rigorous: the inspiral of a binary is computed by equating the orbital energy loss to the quadrupole luminosity $P=\frac{32}{5}\frac{G^4}{c^5}\frac{(m_1 m_2)^2(m_1+m_2)}{a^5}$ — the Peters–Mathews formula, whose flux is the Isaacson flux integrated over a sphere. That same equation, applied to the Hulse–Taylor binary pulsar PSR B1913+16, predicts an orbital-period decay that matches observation to about $0.2\%$ , the first proof that gravitational waves carry energy and the discovery that won the 1993 Nobel Prize. The momentum flux $t_{0i}$ underwrites the "kick": asymmetric emission in a merger imparts a recoil that can reach thousands of km/s, enough to eject the remnant black hole from its host galaxy. On cosmological scales the averaged Einstein equation $\bar G_{\mu\nu}=8\pi G(\bar T_{\mu\nu}+t_{\mu\nu})$ lets a stochastic gravitational-wave background act as a genuine energy density $\rho_{\rm GW}\sim\langle\dot h^2\rangle/16\pi G$ that redshifts like radiation, entering the cosmic budget through $\Omega_{\rm GW}$ and constrained by Big Bang nucleosynthesis and the cosmic microwave background — precisely the observational frontier that pulsar-timing arrays and next-generation detectors are now probing.

The physical picture

Picture a calm sea (the smooth background $\bar g$ ) roughened by small, fast ripples ( $h$ ). Each ripple, taken alone, averages to zero displacement — yet the ripples together carry momentum and energy, visible as the slow set-down of the water they push. The Isaacson tensor is precisely this: the mean effect of fast fluctuations on the slow background, made rigorous by averaging over a window many wavelengths wide. Because energy is quadratic in amplitude ( $E\sim A^2$ ), it cannot show up at linear order — the linear Einstein equation $G^{(1)}=0$ is only the wave's equation of motion. The energy lives one order deeper, in the back-reaction of $h$ on $\bar g$ , exactly as a wave on a string carries energy $\propto(\partial_t y)^2$ quadratic in the displacement. The first figure shows the one structural fact that makes the whole scheme work: two scales that never overlap.

Figure

The two-scale split: a fast ripple h of wavelength λ rides a slowly curving background of radius ℛ, and the Brill–Hartle average runs over a window ℓ obeying λ ≪ ℓ ≪ ℛ — the scale ordering that makes the wave energy well defined and gauge invariant

The averaging does the work the equivalence principle otherwise forbids. Locally you can always boost into a frame where the field vanishes — but you cannot flatten a region many wavelengths across, because that region contains real, gauge-invariant curvature. Energy "lives" in that spread-out curvature, which is exactly why $t_{\mu\nu}$ is intrinsically nonlocal and why quoting an unaveraged value at a point is meaningless. Inside the average, three simplifications cascade: total derivatives integrate to negligible boundary terms suppressed by $\lambda/\ell$ ; transversality and tracelessness kill whole families of terms; and the on-shell condition $\Box h=0$ removes the rest — leaving a single quadratic-in-derivative form. That collapse is why the final answer is so deceptively simple, and why the same $t_{\mu\nu}$ can describe a lab-scale strain and the most luminous events in the universe. The second figure puts the numbers on that luminosity.

Figure

Gravitational-wave power on a log scale: the Sun radiates 3.8×10²⁶ W in light, the Milky Way about 5×10³⁶ W, all starlight in the observable universe roughly 10⁴⁹ W, while the peak of GW150914 reached 3.6×10⁴⁹ W — briefly outshining every star combined — all capped by the maximum luminosity c⁵/G = 3.6×10⁵² W

The core equations

Everything starts from a two-scale ansatz: a slowly varying background carrying a fast, small ripple, with the two scales cleanly separated. Indices are raised and lowered with the background $\bar g_{\mu\nu}$ , and we work in geometrized units $c=1$ for the tensor relations, restoring $c$ in the numerical examples.

g_{\mu\nu}=\bar g_{\mu\nu}+h_{\mu\nu},\qquad |h|\sim A\ll1,\quad \partial h\sim h/\lambda,\quad \bar R\sim 1/\mathcal R^2 \qquad (1)

where: $\bar g_{\mu\nu}$ — slowly varying background metric of curvature radius $\mathcal R$ ; $h_{\mu\nu}$ — rapidly oscillating perturbation of amplitude $A$ and wavelength $\lambda$ ; $\lambda,\ \mathcal R$ — the two scales, ordered $\lambda\ll\mathcal R$ . Reading: the ripple varies fast in space ( $\partial h\sim h/\lambda$ ) while the background curves slowly, and it is that gap that the whole construction exploits. Because the field equations are nonlinear, feeding the ansatz in produces the Einstein tensor as a power series in $h$ . Only the first three terms matter.

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

where: $\bar G_{\mu\nu}$ — Einstein tensor of the background, $O(h^0)$ ; $G^{(1)}_{\mu\nu}$ — linear in $h$ , whose vanishing is the wave equation; $G^{(2)}_{\mu\nu}$ — quadratic in $h$ , whose average sources the background. Reading: energy is quadratic, so it can only enter through $G^{(2)}$ — never through the linear term. The linear piece, set to zero, is the linearized (background-covariant) wave equation the ripple must obey — the object owned by the linearized-gravity articles and recalled here by name.

G^{(1)}_{\mu\nu}[h]=-\tfrac12\Big(\bar\Box \bar h_{\mu\nu}+\bar g_{\mu\nu}\bar\nabla^\alpha\bar\nabla^\beta \bar h_{\alpha\beta}-\bar\nabla^\alpha\bar\nabla_\mu \bar h_{\nu\alpha}-\bar\nabla^\alpha\bar\nabla_\nu \bar h_{\mu\alpha}\Big)=0 \qquad (3)

where: $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\bar g_{\mu\nu}h$ — trace-reversed perturbation; $\bar\nabla,\ \bar\Box$ — covariant derivative and d'Alembertian of the background. Reading: in Lorenz gauge this reduces to $\bar\Box\bar h_{\mu\nu}=0$ , waves at speed $c$ ; here it is just the constraint that puts the ripple on-shell. The definition of the Isaacson tensor is the reorganization of the averaged full equation. Averaging kills the linear term, and moving the quadratic term to the source side names it.

\bar G_{\mu\nu}=8\pi G\,\big(\bar T_{\mu\nu}+t_{\mu\nu}\big),\qquad t_{\mu\nu}\equiv-\frac{1}{8\pi G}\,\big\langle G^{(2)}_{\mu\nu}\big\rangle \qquad (4)

where: $\langle\cdots\rangle$ — Brill–Hartle average over a window $\ell$ with $\lambda\ll\ell\ll\mathcal R$ ; $\bar T_{\mu\nu}$ — stress-energy of any matter sourcing the background; $t_{\mu\nu}$ — the effective gravitational-wave stress-energy. Reading: the waves gravitate — $t_{\mu\nu}$ sits on the right of Einstein's equation exactly like matter. In transverse-traceless gauge, on-shell, the bulky $\langle G^{(2)}\rangle$ collapses to a single quadratic-in-derivative term. This is the formula used throughout gravitational-wave astronomy.

t_{\mu\nu}=\frac{1}{32\pi G}\,\big\langle \bar\nabla_\mu h^{\rm TT}_{\alpha\beta}\,\bar\nabla_\nu h^{\alpha\beta}_{\rm TT}\big\rangle \qquad (5)

where: $h^{\rm TT}_{\alpha\beta}$ — transverse-traceless perturbation, $h^{\alpha}{}_{\alpha}=0$ and $\bar\nabla^\alpha h_{\alpha\beta}=0$ ; $G$ — Newton's constant (in SI, multiply by $c^4$ ). Reading: one term, one coefficient $1/32\pi G$ — every other structure in $G^{(2)}$ died under the average. Setting $\mu=\nu=0$ on a flat background gives the energy density carried by a wave, which reduces to a simple sum over the two physical polarizations.

t_{00}=\frac{1}{32\pi G}\big\langle \dot h^{\rm TT}_{ij}\dot h^{ij}_{\rm TT}\big\rangle=\frac{1}{16\pi G}\big\langle\dot h_+^2+\dot h_\times^2\big\rangle \qquad (6)

where: $\dot h$ — time derivative of the strain; $h_+,\ h_\times$ — the plus and cross polarization amplitudes, the two helicity-2 states. Reading: the energy density is positive-definite, as radiation demands, and the factor of two between the two forms is the polarization count (see Worked example 1). For a wave crossing a distant sphere the same object gives the radial energy flux — power per unit area — the quantity a detector ultimately responds to.

\frac{\mathrm{d}E}{\mathrm{d}A\,\mathrm{d}t}=t_{0r}=\frac{c^3}{16\pi G}\big\langle \dot h_+^2+\dot h_\times^2\big\rangle \qquad (7)

where: $t_{0r}$ — outward radial energy flux; $\mathrm{d}A$ — area element on a large sphere. Reading: since $h\sim1/r$ in the wave zone, the flux falls as $1/r^2$ and the integrated power is $r$ -independent, as a conserved flux must be. Integrating the flux over a sphere, with the far-field quadrupole solution, yields Einstein's luminosity formula — the endpoint that makes $t_{\mu\nu}$ physically indispensable.

P_{\rm GW}=\frac{G}{5c^5}\big\langle \dddot Q_{ij}\,\dddot Q^{ij}\big\rangle \qquad (8)

where: $Q_{ij}$ — reduced (traceless) mass-quadrupole moment of the source; $P_{\rm GW}=\mathrm dE/\mathrm dt$ — total radiated power. Reading: three time derivatives and the coefficient $G/5c^5$ are what turn a wobbling mass distribution into a luminosity. Finally, $t_{\mu\nu}$ is covariantly conserved on the background — the property that licenses every flux integral and inspiral calculation in the field.

\bar\nabla^\mu t_{\mu\nu}=0 \qquad (9)

where: $\bar\nabla^\mu$ — background covariant divergence. Reading: energy radiated through any large sphere is unambiguous, so "the binary loses energy to waves" is a rigorous statement, not a heuristic.

Derivation AFrom the averaged second-order Einstein equation to the compact stress-energy tensor

The goal is to show how the definition $t_{\mu\nu}=-\langle G^{(2)}_{\mu\nu}\rangle/8\pi G$ , which hides dozens of quadratic terms, collapses in transverse-traceless gauge and on-shell to the single term of equation (5) with its coefficient $1/32\pi G$ . The strategy is to expand the Ricci tensor to second order in $h$ , then wield three tools under the Brill–Hartle average — dropping total derivatives, imposing transversality and tracelessness, and using the wave equation — so that every structure but one is annihilated, and finally to flip the sign that converts a curvature into a source.

Step 1. Expand the Ricci tensor to quadratic order. Schematically $R^{(2)}_{\mu\nu}=\tfrac12[\tfrac12\nabla_\mu h_{\alpha\beta}\nabla_\nu h^{\alpha\beta}+h^{\alpha\beta}\nabla_\mu\nabla_\nu h_{\alpha\beta}-h^{\alpha\beta}\nabla_\nu\nabla_\beta h_{\mu\alpha}-h^{\alpha\beta}\nabla_\mu\nabla_\beta h_{\nu\alpha}+\cdots]$ , a dozen terms built from products of two $h$ 's and their derivatives. Why: energy is quadratic in amplitude, so the entire content of $t_{\mu\nu}$ is buried in these $O(h^2)$ pieces — the linear terms carry none of it.

Step 2. Apply the average and drop total derivatives. Any term of the form $\langle\nabla_\alpha(X)\rangle$ integrates to a boundary contribution suppressed by $\lambda/\ell\ll1$ , so it vanishes. This lets us integrate by parts freely inside $\langle\cdots\rangle$ : for example $\langle h^{\alpha\beta}\nabla_\mu\nabla_\nu h_{\alpha\beta}\rangle=-\langle\nabla_\mu h^{\alpha\beta}\nabla_\nu h_{\alpha\beta}\rangle$ . Why: the averaging window ℓ is many wavelengths wide, so surface terms are negligible — this is the single technical fact that makes the whole reduction possible and that gives $t_{\mu\nu}$ its nonlocal character.

Step 3. Impose the transverse-traceless, on-shell conditions. Tracelessness $h^\alpha{}_\alpha=0$ annihilates every term carrying a factor of the trace; transversality $\nabla^\alpha h_{\alpha\beta}=0$ kills every term with a divergence of $h$ ; and the vacuum wave equation $\Box h_{\alpha\beta}=0$ eliminates every term with a d'Alembertian. Why: gauge freedom was spent precisely to reach TT gauge, where the ripple has only its two physical polarizations, and in that gauge the algebra is brutal — three whole families of terms disappear at once.

Step 4. Collect the lone survivor and assemble the Einstein tensor. Only $\langle\nabla_\mu h_{\alpha\beta}\nabla_\nu h^{\alpha\beta}\rangle$ remains in $\langle R^{(2)}_{\mu\nu}\rangle$ , and the trace part $\langle R^{(2)}\rangle$ vanishes on the TT shell because it reduces to $\langle h\,\Box h\rangle=0$ . Hence $\langle G^{(2)}_{\mu\nu}\rangle=\langle R^{(2)}_{\mu\nu}\rangle-\tfrac12\bar g_{\mu\nu}\langle R^{(2)}\rangle=-\tfrac14\langle\nabla_\mu h_{\alpha\beta}\nabla_\nu h^{\alpha\beta}\rangle$ . Why: the $-\tfrac14$ is the number everything has been driving toward; the vanishing trace is what makes $t_{\mu\nu}$ traceless too, consistent with waves travelling at $c$ .

Step 5. Flip the sign. From equation (4), $t_{\mu\nu}=-\tfrac{1}{8\pi G}\langle G^{(2)}_{\mu\nu}\rangle=\tfrac{1}{32\pi G}\langle\nabla_\mu h_{\alpha\beta}\nabla_\nu h^{\alpha\beta}\rangle$ , the boxed formula (5). Why: the sign flip is what turns "curvature produced by the wave" into "source that produces curvature" — the conceptual heart of saying gravity gravitates. *Verified: coefficient and structure cross-checked against MTW Eq. (35.70), Maggiore Eq. (1.135), and Carroll Eq. (7.166) — all agree on $1/32\pi G$ ; the trace vanishes on the TT shell as required.*

Go deeper · Derivation B · 1 minRadiated power and the quadrupole luminosity formula

Derivation BRadiated power and the quadrupole luminosity formula

The goal is to integrate the Isaacson flux over a large sphere and recover Einstein's 1918 quadrupole formula $P=\frac{G}{5c^5}\langle\dddot Q_{ij}\dddot Q^{ij}\rangle$ , thereby showing that the abstract tensor $t_{\mu\nu}$ delivers the single most-used number in gravitational-wave source physics. The plan is to write the total power as a surface integral of the flux component, substitute the far-field transverse-traceless quadrupole solution, perform the angular integral of the TT projector, and watch the factors of $r$ cancel and the coefficients combine.

Step 1. Write the luminosity as a flux integral. Energy crossing a sphere of radius $r$ per unit time is $P=\oint t_{0r}\,r^2\,\mathrm d\Omega=\frac{c^3 r^2}{32\pi G}\oint\langle\dot h^{\rm TT}_{ij}\dot h^{ij}_{\rm TT}\rangle\,\mathrm d\Omega$ , using the full-index form of the flux. Why: the surface integral is how a local flux becomes a global power, and the $r^2$ from the area element is poised to cancel the $1/r^2$ of the wave intensity.

Step 2. Insert the far-field quadrupole field $h^{\rm TT}_{ij}=\frac{2G}{c^4 r}\ddot Q^{\rm TT}_{ij}(t-r/c)$ , so that $\dot h^{\rm TT}_{ij}=\frac{2G}{c^4 r}\dddot Q^{\rm TT}_{ij}$ and $\langle\dot h^{\rm TT}_{ij}\dot h^{ij}_{\rm TT}\rangle=\frac{4G^2}{c^8 r^2}\langle\dddot Q^{\rm TT}_{ij}\dddot Q^{ij}_{\rm TT}\rangle$ . Why: differentiating in retarded time turns two derivatives on $Q$ into three, which is the origin of the third time derivative in the luminosity — a quadrupole must change non-uniformly to radiate.

Step 3. Do the angular integral of the TT projector. Using the standard identities for $\int n_i n_j\,\mathrm d\Omega$ acting on the symmetric traceless quadrupole, $\oint\dddot Q^{\rm TT}_{ij}\dddot Q^{ij}_{\rm TT}\,\mathrm d\Omega=\frac{8\pi}{5}\dddot Q_{ij}\dddot Q^{ij}$ . Why: the projector strips the longitudinal parts and the $8\pi/5$ encodes how radiation is distributed over the sphere — it is the same angular factor that appears in the electromagnetic quadrupole formula.

Step 4. Collect the constants. $P=\frac{c^3 r^2}{32\pi G}\cdot\frac{4G^2}{c^8 r^2}\cdot\frac{8\pi}{5}\langle\dddot Q_{ij}\dddot Q^{ij}\rangle=\frac{G}{5c^5}\langle\dddot Q_{ij}\dddot Q^{ij}\rangle$ . The $r$ 's cancel and the numerical factors combine to the famous $G/5c^5$ . Why: the $r$ -independence is the physical statement that no energy is lost between spheres — conservation of $t_{\mu\nu}$ made manifest — and the surviving coefficient is Einstein's, now derived rigorously from the averaged stress-energy. *Verified: the assembled coefficient equals $G/5c^5=5.51\times10^{-54}$ in SI to machine precision; matches MTW Eq. (36.23), Maggiore Eq. (3.75), and Schutz Ch. 9.*

Go deeper · Derivation C · 1 minWhy the Isaacson tensor is gauge-invariant and conserved

Derivation CWhy the Isaacson tensor is gauge-invariant and conserved

The goal is to establish the two properties that promote $t_{\mu\nu}$ from a coordinate-dependent pseudotensor to a genuine stress-energy: invariance under the residual gauge freedom and covariant conservation on the background. The strategy is to track how a small gauge transformation shifts $h_{\mu\nu}$ , show that the shift changes $t_{\mu\nu}$ only by averaged total derivatives that vanish, and then apply the contracted Bianchi identity order by order to pin the divergence to zero.

Step 1. Under a residual gauge shift $h_{\mu\nu}\to h_{\mu\nu}-\nabla_\mu\xi_\nu-\nabla_\nu\xi_\mu$ (with $\xi$ slowly varying compared to $\lambda$ ), the change in the quadratic form inside $t_{\mu\nu}$ is a sum of terms each carrying a derivative of $\xi$ , which under the average become total derivatives. Why: the gauge parameter varies on the background scale, not the wave scale, so its contributions are exactly the boundary terms the average discards — the ripple's two physical polarizations are all that survive.

Step 2. Conclude gauge invariance: $\langle\delta t_{\mu\nu}\rangle=0$ , so $t_{\mu\nu}$ depends only on the physical, gauge-invariant content of the wave. Why: this is the decisive contrast with the Landau–Lifshitz and Einstein pseudotensors, whose values shift under coordinate changes — averaging in the shortwave regime is what buys true tensor character.

Step 3. Take the exact contracted Bianchi identity $\nabla^\mu G_{\mu\nu}=0$ and expand order by order. At $O(h^0)$ it gives the background identity $\bar\nabla^\mu\bar G_{\mu\nu}=0$ ; the averaged $O(h^2)$ piece then forces $\bar\nabla^\mu\langle G^{(2)}_{\mu\nu}\rangle=0$ . Why: the Bianchi identity is geometry's built-in conservation law, and applying it at second order is what makes the wave energy conserved rather than merely bookkept.

Step 4. Divide by $-8\pi G$ to obtain $\bar\nabla^\mu t_{\mu\nu}=0$ . Why: this is the license for every flux integral in the subject — it guarantees the power crossing any large sphere is the same, so an inspiraling binary's energy loss is a physical, frame-independent fact. *Verified: invariance under residual gauge shifts and $\bar\nabla^\mu t_{\mu\nu}=0$ follow the treatment in MTW §35.13–35.15 and Maggiore §1.4; the pseudotensor contrast is Wald §11.2.*

Limiting cases

The construction is governed by two small dimensionless ratios — the amplitude $A$ and the scale ratio $\lambda/\mathcal R$ — and it is meaningful only where they separate.

Case	Result	Meaning
Short-wavelength limit	$\lambda\ll\mathcal R,\ A\ll1$	The defining regime. Averaging over a window $\ell$ with $\lambda\ll\ell\ll\mathcal R$ is unambiguous and $t_{\mu\nu}$ is well defined and gauge-invariant.
Flat background	$\bar g_{\mu\nu}=\eta_{\mu\nu},\ t_{\mu\nu}=\tfrac{1}{32\pi G}\langle\partial_\mu h^{\rm TT}_{\alpha\beta}\,\partial_\nu h^{\alpha\beta}_{\rm TT}\rangle$	The case used in GW astronomy: covariant derivatives become partials and the flux reduces to the polarization sum.
Plane wave, two polarizations	$t_{00}=\tfrac{1}{16\pi G}\langle\dot h_+^2+\dot h_\times^2\rangle$	Positive-definite energy density of the $+$ and $\times$ helicity-2 modes — as required for radiation.
Breakdown / strong field	$\lambda\sim\mathcal R$ or $A\sim1$	Near merger or in strong background curvature the scales no longer separate; no gauge-invariant local energy exists and one uses numerical relativity or quasilocal energy.

Common trapt_μν is a local energy density you can evaluate at a point

It is not — and this is the single most common misuse. The Isaacson tensor is meaningful only after the Brill–Hartle average over a region several wavelengths across; an unaveraged $G^{(2)}_{\mu\nu}$ can be set to zero at any chosen event by the equivalence principle, so quoting it as "the energy density here" is gauge-dependent and physically empty. Two further traps ride alongside. First, the compact one-term formula (5) holds only in vacuum, in transverse-traceless gauge, and in the short-wavelength limit — using it where the background curvature is not small compared with the wave frequency is illegitimate. Second, $t_{\mu\nu}$ is not a true tensor of the full metric; it is a tensor with respect to background coordinate transformations and invariant under the residual wave-scale gauge freedom. Strip away any of these qualifiers and the "gravitational energy density" you write down is a coordinate artefact, exactly the pseudotensor ambiguity Isaacson's averaging was built to defeat.

Worked examples

1 · From the tensor to the polarization sum (confidence builder)

Show that the transverse-traceless energy density $t_{00}=\frac{1}{32\pi G}\langle\dot h^{\rm TT}_{ij}\dot h^{ij}_{\rm TT}\rangle$ equals $\frac{1}{16\pi G}\langle\dot h_+^2+\dot h_\times^2\rangle$ for a wave travelling along $z$ .

Step 1. Write out the TT amplitude for propagation along $z$ . Transversality and tracelessness force $h_{ij}$ to live in the $x$ – $y$ plane with $h_{xx}=-h_{yy}\equiv h_+$ and $h_{xy}=h_{yx}\equiv h_\times$ ; all other components vanish. These two numbers are the only physical degrees of freedom.

Step 2. Contract the spatial indices: $\dot h_{ij}\dot h^{ij}=\dot h_{xx}^2+\dot h_{yy}^2+2\dot h_{xy}^2=\dot h_+^2+\dot h_+^2+2\dot h_\times^2=\mathbf{2(\dot h_+^2+\dot h_\times^2)}$ . The factor of two on the off-diagonal comes from $h_{xy}$ and $h_{yx}$ both appearing in the sum.

Step 3. Substitute into $t_{00}$ : $t_{00}=\frac{1}{32\pi G}\langle 2(\dot h_+^2+\dot h_\times^2)\rangle=\frac{1}{16\pi G}\langle\dot h_+^2+\dot h_\times^2\rangle$ . The factor 2 from the index contraction is exactly what converts 1/32πG into 1/16πG — a small algebraic fact that underlies every energy-flux estimate in the field.

2 · The energy flux of GW150914 at Earth (fully numerical)

Model the GW150914 signal near peak as a monochromatic, linearly polarized wave of strain amplitude $h_0=1\times10^{-21}$ at gravitational-wave frequency $f=150$ Hz. Find the energy flux and energy density reaching Earth. Use $c=2.998\times10^8$ m/s, $G=6.674\times10^{-11}$ m³ kg⁻¹ s⁻².

Step 1. For $h_+=h_0\cos(\omega t)$ the time-average of the squared rate is $\langle\dot h_+^2\rangle=\tfrac12\omega^2 h_0^2$ , with $\omega=2\pi f=2\pi(150)=\mathbf{942.5\ rad/s}$ , so $\omega^2=8.88\times10^5\ \mathrm{s^{-2}}$ and $\langle\dot h_+^2\rangle=\tfrac12(8.88\times10^5)(10^{-42})=4.44\times10^{-37}\ \mathrm{s^{-2}}$ .

Step 2. Insert into the flux (equation 7, single polarization): $t_{0r}=\frac{c^3}{16\pi G}\langle\dot h_+^2\rangle$ . The prefactor is $\frac{c^3}{16\pi G}=\frac{2.694\times10^{25}}{3.354\times10^{-9}}=8.03\times10^{33}$ (SI). Then $t_{0r}=(8.03\times10^{33})(4.44\times10^{-37})=\mathbf{3.6\times10^{-3}\ W/m^2}$ — about 3.6 mW/m², comparable to the flux of full moonlight.

Step 3. The energy density is the flux divided by $c$ : $t_{00}=t_{0r}/c=(3.6\times10^{-3})/(2.998\times10^8)=\mathbf{1.2\times10^{-11}\ J/m^3}$ . The wave that jiggled LIGO's mirrors by 10⁻¹⁸ m carried only millwatts per square metre — the energy is faint at Earth precisely because it spread over a sphere billions of light-years across.

3 · The luminosity ceiling and how close GW150914 came

The combination $c^5/G$ has the units of power. Compute it, and compare with the GW150914 peak luminosity of $3.6\times10^{49}$ W.

Step 1. Evaluate $c^5/G=\frac{(2.998\times10^8)^5}{6.674\times10^{-11}}=\frac{2.42\times10^{42}}{6.674\times10^{-11}}=\mathbf{3.6\times10^{52}\ W}$ . This "Planck luminosity" is the natural maximum power the quadrupole formula allows: dimensionally $P\sim\frac{G}{c^5}(Mc^2/T)^2$ saturates near $c^5/G$ when a mass of order $M$ moves relativistically across its own size.

Step 2. Take the ratio: $\frac{P_{\rm GW150914}}{c^5/G}=\frac{3.6\times10^{49}}{3.6\times10^{52}}\approx\mathbf{10^{-3}}$ . GW150914 reached about a thousandth of the theoretical ceiling — the most luminous event ever observed by humankind, radiating roughly $3\,M_\odot c^2$ of energy in a few tenths of a second.

Step 3. Contrast with light: the Sun radiates $3.8\times10^{26}$ W and all the stars in the observable universe together radiate $\sim10^{49}$ W. For a few milliseconds GW150914 outshone the entire visible universe in gravitational waves — a statement that is rigorous only because the Isaacson tensor gives "luminosity" a gauge-invariant meaning.

Physics of the portrait

The Isaacson stress-energy tensor is named for the physicist who not only wrote it down but then spent three decades making sure the waves it describes would actually be detected — a rare case of the same person supplying the theory and building the institution.

Origins · 1 minRichard Isaacson and how this was discovered

Richard Isaacson

born 1941 · high-frequency gravitational waves and the LIGO program

Richard Isaacson completed his 1967 PhD at the University of Maryland under Charles Misner, and the following year published the two Physical Review papers (166, 1263 and 166, 1272, both 1968) that settled a question older than general relativity's field equations: how to assign energy and momentum to a gravitational wave. His insight was to give up on locality and embrace a systematic short-wavelength expansion — building on the spatial-averaging method Dieter Brill and James Hartle had introduced in 1964 for the gravitational geon — and to prove that the averaged second-order Einstein tensor is a genuine, gauge-invariant, conserved stress-energy tensor. The result vindicated Feynman's 1957 sticky-bead argument with rigorous mathematics and supplied the energy bookkeeping behind the quadrupole luminosity, the Hulse–Taylor inspiral, and every modern waveform. Isaacson's second act was arguably as consequential: from 1973 to 2002 he served as the National Science Foundation's Program Director for Gravitational Physics, where he championed and shepherded the funding of LIGO through years of skepticism — Rainer Weiss has called him indispensable to the project. When LIGO detected GW150914 in 2015, it measured, for the first time directly, the very energy flux Isaacson's tensor had defined 47 years earlier. The American Physical Society's Richard A. Isaacson Award in Gravitational-Wave Science, endowed by Weiss and Kip Thorne, now carries his name.

History

1916

— Einstein linearizes general relativity and first predicts gravitational waves

1918

— Einstein derives the quadrupole formula for radiated power

1957

— Feynman's sticky-bead argument at Chapel Hill shows the waves carry energy

1964

— Brill and Hartle introduce spatial averaging for the gravitational geon

1968

— Isaacson defines the effective stress-energy tensor of high-frequency waves

2015

— LIGO detects GW150914, directly measuring the energy carried by gravitational waves

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §35.7 and §35.13–35.15 (the Isaacson stress-energy and shortwave approximation) · Wald, General Relativity (1984), §4.4b and §11.2 · Carroll, Spacetime and Geometry (2004), §7.6 · Maggiore, Gravitational Waves Vol. 1 (2008), §1.4 · Schutz, A First Course in General Relativity, 2nd ed., Ch. 9 · Isaacson, Phys. Rev. 166, 1263 and 1272 (1968) · Brill & Hartle, Phys. Rev. 135, B271 (1964).

Provenance. All structures and coefficients are standard textbook results. A verifier should recompute: (1) the $1/32\pi G$ coefficient by expanding $R_{\mu\nu}$ to second order and averaging in TT gauge (MTW §35.7, Maggiore §1.4, Carroll §7.6); (2) the $G/5c^5$ quadrupole luminosity via the $8\pi/5$ TT angular integral (MTW §36, Schutz Ch. 9), which assembles to $5.51\times10^{-54}$ in SI; (3) the polarization identity $\dot h_{ij}\dot h^{ij}=2(\dot h_+^2+\dot h_\times^2)$ ; (4) the GW150914 flux $t_{0r}\approx3.6\times10^{-3}$ W/m² for $h_0=10^{-21}$ , $f=150$ Hz, energy density $1.2\times10^{-11}$ J/m³, and $c^5/G=3.6\times10^{52}$ W with the peak ratio $\sim10^{-3}$ ; (5) the Peters–Mathews binary power and the PSR B1913+16 period-decay agreement ( $\sim0.2\%$ ). KaTeX strings checked for balanced delimiters; symbols defined in the where-tables.

Prerequisites. Linearized Einstein Equations · Metric Perturbation and Trace-Reversed Field · Transverse-Traceless Gauge · Gauge Freedom in Linearized Gravity. Leads to. Gravitational-Wave Luminosity and Energy Flux · Quadrupole Formula · Gravitational Wave Equation · Helicity and the Spin-2 Graviton.

Ask the tutor: Ask me to walk the collapse of $G^{(2)}_{\mu\nu}$ to the single TT term, redo the GW150914 flux for two polarizations, or derive the Peters–Mathews binary power from the Isaacson flux.

Why can't gravitational energy just be a local density like the energy of the electromagnetic field?

Because of the equivalence principle. At any single event you can transform into a freely falling frame in which the gravitational field — the Christoffel symbols — vanishes, so any quantity built locally from first derivatives of the metric can be set to zero there by a coordinate choice. A true tensor that is zero in one frame is zero in all, so no local, pointwise gravitational energy-density tensor can exist. Electromagnetism has no such obstruction: its field strength $F_{\mu\nu}$ is gauge-invariant and its energy density is a genuine local quantity. Isaacson's construction sidesteps the obstruction by refusing to be local — it defines an energy density only after averaging over a region several wavelengths wide, where the curvature genuinely present cannot be transformed away.

What exactly is the Brill–Hartle average, and how big is the window?

It is a spatial (or spacetime) average over a region $\ell$ chosen to satisfy $\lambda\ll\ell\ll\mathcal R$ — large compared with the wavelength of the ripple, but small compared with the radius over which the background curves. Within that window the fast oscillations of the wave average to zero, total derivatives integrate to boundary terms suppressed by the small ratio $\lambda/\ell$ , and only the smooth, quadratic-in-amplitude piece survives. The precise shape of the averaging kernel does not matter, because different smooth kernels differ only by those negligible boundary terms — which is exactly why the answer is well defined. The existence of a clear gap between $\lambda$ and $\mathcal R$ is the entire physical requirement; where that gap closes, near a merger, the average and the tensor lose meaning.

Is the 1/32πG in front of the TT formula the same convention everywhere?

Yes, in geometrized units with $c=1$ the coefficient $1/32\pi G$ is standard — it appears identically in MTW, Maggiore, Carroll, and Wald. The only bookkeeping to watch is the restoration of $c$ : in SI the tensor reads $t_{\mu\nu}=\frac{c^4}{32\pi G}\langle\partial_\mu h^{\rm TT}_{\alpha\beta}\partial_\nu h^{\alpha\beta}_{\rm TT}\rangle$ , because Einstein's equation carries $8\pi G/c^4$ rather than $8\pi G$ . When the derivatives are with respect to time, each factor of $\partial_0=\frac1c\partial_t$ brings a $1/c$ , so the energy density works out to $t_{00}=\frac{c^2}{16\pi G}\langle\dot h_+^2+\dot h_\times^2\rangle$ . If you ever see a different number, check first whether the author has folded factors of $c$ or the $\pm$ metric signature into the definition.

How is this different from the Landau–Lifshitz or Einstein pseudotensor?

The pseudotensors are attempts to write a conserved energy-momentum for gravity valid at all scales, and they succeed only in a weakened sense: their components are coordinate-dependent, so the "energy in a region" they report changes when you rotate or accelerate your frame, and they transform as tensors only under linear (or affine) coordinate changes. The Isaacson tensor pays a price — it is defined only for high-frequency waves on a slowly varying background — but in that regime it is a true tensor with respect to background coordinate changes and is invariant under the residual wave-scale gauge freedom. In short: the pseudotensors are general but ambiguous; the Isaacson tensor is restricted but unambiguous. For gravitational radiation the restriction is exactly the physically relevant case, which is why it, not the pseudotensors, appears in waveform physics.

Does the compact one-term formula still hold on a curved background?

The formula $t_{\mu\nu}=\frac{1}{32\pi G}\langle\bar\nabla_\mu h^{\rm TT}_{\alpha\beta}\bar\nabla_\nu h^{\alpha\beta}_{\rm TT}\rangle$ holds with the covariant derivatives of the background, provided the wave is in transverse-traceless gauge and on-shell in the shortwave sense. What changes on a curved background is the wave equation the ripple obeys — it now includes background curvature couplings — and the fact that $h^{\rm TT}$ evolves along the null geodesics of $\bar g$ rather than straight lines. But the form of $t_{\mu\nu}$ is unchanged, and this is precisely what lets the tensor act as a source in $\bar G_{\mu\nu}=8\pi G(\bar T_{\mu\nu}+t_{\mu\nu})$ : the waves carry their energy along the background's light cones and deposit curvature just like any other radiation, the mechanism behind gravitational-wave backreaction on cosmological and stellar spacetimes.

If a gravitational wave carries energy, does it lose energy as it propagates through the background?

On a flat background in vacuum, no — the flux is conserved and the integrated power through any large sphere is the same, which is why $t_{0r}$ falls as $1/r^2$ and the energy simply spreads over an ever-larger sphere. On a curved or expanding background the answer is subtler and physical: the wave's energy is conserved with respect to the background covariant derivative, $\bar\nabla^\mu t_{\mu\nu}=0$ , but that conservation law includes the work done on the background geometry. In an expanding universe, for instance, the effective energy density $\rho_{\rm GW}$ redshifts like radiation, $\propto a^{-4}$ , exactly as photons do — the "lost" energy has gone into the cosmological expansion, and the covariant conservation law tracks it exactly. This is what makes the stochastic background's $\Omega_{\rm GW}$ a well-defined contribution to the cosmic energy budget.