The Quadrupole Formula for Radiated Power

Hello! Welcome back to our study of gravitational waves.

In our last lesson, we developed a clear, physical picture of a gravitational wave. We saw how the two polarization states, + and ×, produce a characteristic quadrupolar strain pattern on a ring of free-falling test particles. We now understand what a gravitational wave does. The next logical question is, how is it created?

Today, we will answer that question by deriving one of the cornerstone results of theoretical astrophysics. Our goal is to derive the quadrupole formula for the power radiated by a non-relativistic source. This formula connects the abstract ripples in the metric we've been studying to the dynamics of their physical sources, like orbiting black holes and neutron stars. We will see exactly why gravitational waves are quadrupolar in nature and why their emission is typically so weak.

1. The Wave Equation with a Source

Our journey begins back with the linearized Einstein equations. Previously, we solved the vacuum equation □h̄μν = 0. Now, we reintroduce the source term, the stress-energy tensor Tμν, which we had set to zero:

□ h ˉ_{μν} = - \frac{16 π G}{c ^{4}} T_{μν}

Here, □ is the d'Alembert operator, and h̄μν is the trace-reversed metric perturbation. Your background in electromagnetism will immediately bring to mind the analogous equation for the electromagnetic four-potential Aμ with a source current Jμ: □Aμ = -μ₀Jμ.

Just as in electromagnetism, this wave equation has a standard solution in terms of a "retarded potential," which accounts for the finite speed of light (or gravity). For a source located far from the observer (r >> size of the source), the solution is:

h ˉ_{μν} (t, x) = \frac{4 G}{c ^{4} r} \int_{source} T_{μν} (t_{r}, x^{'}) d^{3} x^{'}

where r = |x| is the distance to the source, and t_r = t - r/c is the retarded time. This equation is our starting point: the wave h̄μν is determined by the integral of the stress-energy tensor over the source volume, evaluated at the time the signal was emitted.

The challenge now is to evaluate this integral. For a general Tμν, this is difficult. However, under the assumption that the source is non-relativistic (internal velocities v << c) and moves slowly (the light-travel time across the source is much smaller than the characteristic timescale of its motion), we can perform a remarkable simplification.

2. The Tensor Virial Theorem and the Wave's Amplitude

The key to simplifying the integral lies in using the conservation of energy and momentum, ∂νTμν = 0. A clever application of this law, sometimes called the Tensor Virial Theorem, relates the integral over the spatial components of the stress-energy tensor (Tik) to the mass distribution of the source.

The following resource provides a self-contained, rigorous derivation of this result.

Chapter 14: The Quadrupole Formalism

The following document derives the crucial relationship between the integrated stress-energy tensor and the quadrupole moment of the source. This is the mathematical engine that powers the quadrupole formula.

Please read Section 14.1, 'The Tensor Virial Theorem' (pages 176-177). Follow the derivation step-by-step. The goal is to understand how the conservation law ∂νTμν = 0 leads to equation (14.28).

Let's summarize the logic of that derivation:

Using the conservation law ∂νTμν = 0 and Gauss's theorem, one first shows that the integral of Tμ0 over the source volume is constant. This implies that the components h̄μ0 of the gravitational wave are static and do not contribute to radiation.
Next, by twice differentiating the integral of T00 x^k x^n with respect to time and repeatedly using the conservation law, a direct relationship is established.

The central result of the theorem is:

\int T_{ik} (t_{r}, x^{'}) d^{3} x^{'} = \frac{1}{2} \frac{d ^{2}}{d t ^{2}} \int T_{00} (t_{r}, x^{'}) x^{' i} x^{' k} d^{3} x^{'}

In the non-relativistic limit, the T00 component is dominated by the mass-energy density, T00 ≈ ρc². This allows us to define the mass quadrupole moment tensor:

I_{ik} (t) = \int ρ (t, x^{'}) x^{' i} x^{' k} d^{3} x^{'}

Substituting this back into our result from the virial theorem gives:

\int T_{ik} (t_{r}, x^{'}) d^{3} x^{'} = \frac{1}{2 c ^{2}} \frac{d ^{2} I _{ik} ( t _{r} )}{d t ^{2}}

Finally, we insert this into our expression for h̄μν. Focusing on the spatial components (h̄ik), which carry the radiation, we arrive at the celebrated quadrupole amplitude formula:

h ˉ_{ik} (t, x) = \frac{2 G}{r c ^{4}} \frac{d ^{2} I _{ik} ( t _{r} )}{d t ^{2}}

This is a profound result. It shows that the amplitude of gravitational waves is proportional to the second time derivative of the source's mass quadrupole moment. This is why it's called quadrupole radiation. A spherically symmetric (Iik is diagonal and constant in time) or rigidly rotating object (d²Iik/dt² = 0) does not radiate gravitational waves. Radiation requires an accelerating quadrupole moment.

3. The Energy Carried by the Wave

Now that we have the wave's amplitude, how much energy does it carry? This is a subtle question in General Relativity. Due to the Equivalence Principle, an observer in free-fall doesn't feel gravity locally, so how can we define a local energy density for the gravitational field?

A rigorous answer requires going to second order in perturbation theory. The idea is that while the first-order perturbation hμν averages to zero over a region of spacetime, the quadratic terms (∂h)² do not. These second-order terms act as an effective stress-energy tensor for the gravitational field itself, sourcing a correction to the background geometry.

17.1 Gravitational radiation in a general background ...

The concept of an effective stress-energy tensor for gravitational waves was first worked out rigorously by Isaacson. These lecture notes provide a clear conceptual summary of the argument.

Read the introduction to Section 17.3, 'Energy content of gravitational waves: The Isaacson stress-energy tensor,' and the discussion leading to equation (17.46) (pages 113-115). Focus on the conceptual argument: gravitational wave energy appears as a source term at O(ε²) in the Einstein equations, which can be interpreted as an effective stress-energy tensor T_GW.

This effective stress-energy tensor, known as the Isaacson tensor, when averaged over several wavelengths in the TT gauge, is given by:

T_{μν}^{G W} = \frac{c ^{2}}{32 π G} ⟨ \partial_{μ} h_{ij}^{T T} \partial_{ν} h_{ij}^{T T} ⟩

The component T_00^GW represents the energy density of the wave. The power flux (energy per unit area per unit time) flowing in the radial direction r is c T_{0r}^{GW}. For a plane wave, this simplifies to c T_{00}^{GW}. Thus, the power flux is:

\frac{d P}{d A} = c T_{00}^{G W} = \frac{c ^{3}}{32 π G} ⟨ i, j \sum (\frac{d h _{ij}^{T T}}{d t})^{2} ⟩

4. Deriving the Total Radiated Power

We are now ready for the final step: to calculate the total power radiated in all directions. To do this, we must:

Express h_ij^TT in terms of the quadrupole moment I_ij.
Substitute this into the power flux formula.
Integrate the flux over a large sphere surrounding the source.

The TT superscript is crucial. We need to project h̄ik onto the transverse-traceless gauge. This is done using a projection operator Λij,kl(n) which depends on the direction of observation n. The full expression is h_ij^TT = Λ_ij,kl(n) h̄_kl. For our purposes, it is simpler to work with the trace-free quadrupole moment, also known as the reduced quadrupole moment:

Q_{ik} = I_{ik} - \frac{1}{3} δ_{ik} Tr (I)

where Tr(I) = I_11 + I_22 + I_33.

The wave amplitude in the TT gauge can then be written in terms of Qik. Differentiating with respect to time gives us dh_ij^TT/dt, which will be proportional to d³Qik/dt³. When we substitute this into the flux equation and integrate over a sphere of radius r, the 1/r² dependence of the amplitude squared cancels the r² from the surface area element r²dΩ. This leaves a complex integral over the angular parts of the TT projection operator.

The following resource carries out this final, crucial integration.

Chapter 14: The Quadrupole Formalism

We now have all the ingredients. Let's follow the final calculation that integrates the power flux over all angles to arrive at the total luminosity.

Please read from equation (14.69) to the final result in equation (14.79) on pages 187-188. Follow how the properties of the angular integrals (eqs. 14.77 and 14.78) are used to simplify the expression into the final, elegant formula for L_GW.

After performing the non-trivial angular average, we arrive at the final result: the quadrupole formula for gravitational wave luminosity.

L_{G W} = \frac{d E}{d t} = \frac{G}{5 c ^{5}} ⟨ i, k \sum (\frac{d ^{3} Q _{ik}}{d t ^{3}})^{2} ⟩

Test your understanding!

Consider a perfectly spherical, uniform planet that is pulsating—its radius expands and contracts radially, but it always remains a perfect sphere. Does this system radiate gravitational waves according to the quadrupole formula? Why or why not?

Show answer

No, it does not radiate gravitational waves.

The mass quadrupole moment is I_ik = ∫ ρ x^i x^k d³x'. For a spherically symmetric object, the off-diagonal components (i ≠ k) are zero. The diagonal components are all equal: I_11 = I_22 = I_33. Let's call this value I(t). So, I_ik(t) = I(t) δ_ik.

Now, let's compute the trace-free quadrupole moment Q_ik:

Q_{ik} = I_{ik} - \frac{1}{3} δ_{ik} Tr (I) = I (t) δ_{ik} - \frac{1}{3} δ_{ik} (I (t) + I (t) + I (t)) = I (t) δ_{ik} - \frac{1}{3} δ_{ik} (3 I (t)) = 0

Since the trace-free quadrupole moment Q_ik is identically zero at all times, its third time derivative is also zero. Therefore, the radiated power L_GW is zero. This confirms that gravitational radiation is not a monopole phenomenon; it requires a changing, non-spherical distribution of mass-energy.

Conclusion

In this lesson, we have derived one of the most significant results in General Relativity. By combining the linearized field equations with the fundamental principle of energy-momentum conservation, we have forged a direct link between a dynamic source and the gravitational radiation it emits.

Key Takeaways:

The amplitude of gravitational waves is proportional to the second time derivative of the source's mass quadrupole moment (h ~ d²I/dt²).
The total power radiated in gravitational waves is proportional to the square of the third time derivative of the source's trace-free quadrupole moment (L_GW ~ <(d³Q/dt³)²>).
The factor of G/c⁵ ≈ 10⁻⁵³ in SI units in the power formula shows that enormous astrophysical energies are required to produce even tiny amounts of gravitational wave power.
Radiation requires a changing quadrupole moment. Spherically symmetric systems, even if pulsating, do not radiate gravitational waves.

Preview of Next Lesson:

We now possess a powerful tool to calculate the energy radiated by any non-relativistic system. In our next lesson, we will apply the quadrupole formula to its most famous application: a binary star system. We will calculate the energy loss from the orbit and see how this leads to the gradual inspiral of the two bodies—the very effect that was first observed in the Hulse-Taylor binary pulsar, providing the first indirect evidence for the existence of gravitational waves.

Recommended resources

1. The Wave Equation with a Source

2. The Tensor Virial Theorem and the Wave's Amplitude

3. The Energy Carried by the Wave

4. Deriving the Total Radiated Power

Conclusion