Gravitational Waves

Einstein's general relativity predicts that moving masses send out ripples in spacetime itself, traveling at the speed of light. They are produced most strongly by pairs of compact objects, such as the black holes and neutron stars of this unit, orbiting each other closely.

Why it works

As two masses orbit, the pattern of their gravitational field swings around, and the changing curvature spreads outward as a wave. The pattern repeats every half orbit, when the two masses have swapped places, so the wave frequency is twice the orbital frequency. The waves carry away energy, so the orbit shrinks. A smaller orbit is faster, so the frequency and the strength of the waves both rise: the signal "chirps" up until the two objects merge. A passing wave stretches distances in one direction across its path while squeezing them in the perpendicular direction, by a fraction $h$ called the strain.

Key idea

Gravitational waves are ripples of spacetime curvature, traveling at $c$, with $f_{\text{GW}} = 2f_{\text{orb}}$. A detector arm of length $L$ changes by $\Delta L = hL$; for real signals $h \sim 10^{-21}$.

A model of the last quarter second of two black holes of 36 and 29 solar masses spiraling together, like the first one LIGO detected. The ring shows how a passing wave stretches and squeezes space, magnified enormously.

Milestones

1916: Einstein predicts gravitational waves from general relativity.
1974: Russell Hulse and Joseph Taylor discover a binary pulsar. Years of timing show its orbit shrinking at the rate the theory predicts for energy carried off by gravitational waves (Nobel Prize 1993).
September 14, 2015: the two LIGO detectors in the United States record waves from two black holes of about 36 and 29 solar masses merging about 1.3 billion light-years away; about 3 solar masses of energy left as gravitational waves (Nobel Prize 2017 to Rainer Weiss, Barry Barish and Kip Thorne).
August 17, 2017: LIGO and Virgo record two neutron stars merging about 40 Mpc away; gamma rays arrive 1.7 seconds later, and telescopes watch the glowing debris, rich in newly made heavy elements.

Worked examples

Worked example: How small is the stretch?

A wave with $h = 1.0 \times 10^{-21}$ passes a LIGO arm 4.0 km long. By how much does the arm change?

$\Delta L = hL = (1.0 \times 10^{-21})(4.0 \times 10^{3}\text{ m})$.

Answer: $4.0 \times 10^{-18}$ m, a few thousandths of the width of a proton. LIGO bounces laser light back and forth to sense it.

Worked example: The energy of a merger

About $3.0\,M_\odot$ of mass-energy left as gravitational waves in GW150914. How much energy is that?

$E = \Delta mc^2 = 3.0(1.99 \times 10^{30})(3.00 \times 10^{8})^2$.

Answer: About $5.4 \times 10^{47}$ J, released in a fraction of a second. At its peak, the merger's power was greater than the light output of all the stars in the observable universe combined.

Worked example: Gravity travels at the speed of light

Gamma rays from the 2017 merger arrived 1.7 s after the gravitational waves, after a trip of about 40 Mpc. Bound the fractional difference in speed.

$d = 40 \times 10^{6} \times 3.09 \times 10^{16}$ m $= 1.24 \times 10^{24}$ m.
$\Delta v/c \approx c\,\Delta t/d = (3.00 \times 10^{8})(1.7)/(1.24 \times 10^{24})$.

Answer: About $4 \times 10^{-16}$: the two travel at the same speed to within parts in $10^{15}$.

Common mistake

Thinking gravitational waves are sound waves in space, or the same as "gravity waves" in water and air. Sound needs a material to travel through and moves far slower than light. Gravity waves in fluids are ordinary water or air waves in which gravity is the restoring force. Gravitational waves are oscillations of spacetime itself: they need no medium, travel at $c$ through empty space, and stretch and squeeze everything they pass across their direction of travel. Check: LIGO measured them crossing about a billion light-years of nearly empty space.

Gravitational Waves

Milestones

Worked examples

Practice

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Sources and further reading