Questions: Gravitational Waves

Question 1

Gravitational waves have two polarization states. What distinguishes the plus (h₊) polarization from the cross (h×) polarization?

• A. h₊ stretches space in one direction and compresses it in the perpendicular direction, while h× does the same but rotated by 45 degrees
• B. h₊ produces longitudinal compression while h× produces transverse compression
• C. h₊ is left-circularly polarized and h× is right-circularly polarized
• D. h₊ and h× differ by a factor of 2 in amplitude but have the same pattern

Answer: (A)

Both h₊ and h× produce transverse, traceless tidal distortions. The plus polarization stretches along the x-axis and compresses along the y-axis (and vice versa half a cycle later). The cross polarization does the same but along axes rotated by 45 degrees. This 45-degree offset (rather than 90 degrees as in electromagnetic waves) reflects the spin-2 nature of gravitational waves — the graviton has spin 2, compared to the photon's spin 1. Circular polarizations are formed by combining h₊ and h× with a π/2 phase offset.

Question 2

A spherically symmetric mass distribution (such as a radially pulsating star) cannot emit gravitational waves.

• T. True
• F. False

Answer: True

Gravitational wave emission requires a time-varying mass quadrupole moment (or higher multipole). A spherically symmetric source has a monopole moment (total mass) that is constant and no quadrupole moment by symmetry — all mass shells expand and contract uniformly. This is the gravitational analog of the fact that a uniformly pulsating charge does not radiate electromagnetic waves (which require a time-varying dipole or higher moment). The absence of gravitational monopole and dipole radiation is deeper than electromagnetism: monopole radiation is forbidden by mass conservation, and dipole radiation is forbidden by momentum conservation.

Question 3

The first direct detection of gravitational waves (GW150914) measured a strain h ~ 10⁻²¹. Explain what this strain means physically for LIGO's 4-km detector arms.

Model answer: The strain h = ΔL/L represents the fractional change in the proper distance between two points caused by the passing gravitational wave. For h ~ 10⁻²¹ and arm length L = 4 km, the change in arm length is ΔL = hL ≈ 10⁻²¹ × 4000 m ≈ 4 × 10⁻¹⁸ m — about one-thousandth the diameter of a proton. LIGO detects this minuscule displacement using laser interferometry: the two perpendicular arms experience opposite length changes (one stretches while the other compresses), producing a differential phase shift in the recombined laser beams.

The extraordinary sensitivity of LIGO — measuring displacements smaller than a proton — is achieved through power recycling (increasing effective laser power), Fabry-Perot cavities (increasing effective arm length), seismic isolation, and quantum noise reduction techniques. The transverse, traceless nature of gravitational waves (opposite effects in perpendicular directions) makes the Michelson interferometer configuration naturally suited to detection.

Question 4

Explain why the Hulse-Taylor binary pulsar provided indirect evidence for gravitational waves before LIGO's direct detection.

Model answer: The Hulse-Taylor binary pulsar PSR B1913+16 consists of two neutron stars in a tight, eccentric orbit. GR predicts that the binary loses energy to gravitational wave emission, causing the orbit to shrink and the orbital period to decrease. Over decades of timing observations (1974-present), the cumulative shift in the orbital period matches the GR quadrupole formula prediction to better than 0.2%. This is indirect evidence because the gravitational waves themselves were not detected — only their effect on the orbit was measured. The agreement is so precise that it constitutes compelling proof of gravitational wave emission, earning Hulse and Taylor the 1993 Nobel Prize.

The orbital decay rate depends on the gravitational wave luminosity, which is set by the quadrupole formula. The binary pulsar measurement tests both the existence of gravitational waves and the correctness of the quadrupole formula simultaneously. The subsequent direct detection by LIGO in 2015 confirmed what the binary pulsar had already demonstrated indirectly.