by Clifford Will
According to general relativity, the gravitational interaction is propagated as if the field were massless, just as in electrodynamics. Thus the speed of gravitational waves is precisely the same as the speed of light, a fact spectacularly confirmed when gravitational waves and gamma rays from the binary neutron star merger event GW170817 arrived within 1.74 seconds of each other, even after traveling for 120 million years.
But some modified gravity theories propose that the field could be massive, so that gravitational waves might propagate more slowly than light, and with a speed that depends on wavelength. The shorthand term for this is a “massive graviton”, although quantum gravity plays no role in this discussion. This is entirely a classical phenomenon.
Back in 1998, I suggested that, if the graviton is massive, then during the inspiral of two compact bodies (black holes, for example), longer wavelength gravitational waves emitted early on would propagate more slowly than the short-wavelength waves emitted later. After traveling hundreds of millions of light years, the later waves would “catch up” to the earlier waves, leading to a distortion of the signal received at a detector compared to the signal emitted. Such a distortion could be measured by comparing a received waveform with a theoretical template waveform that incorporated this effect. I pointed out that advanced LIGO could place a limit on the graviton mass that would be comparable to limits derived in 1988 by looking for a Yukawa-type modification of Newton’s law of gravity in solar-system data.
Needless to say, I was thrilled when the gravitational-wave discovery event GW150914 led to a bound about two times better than we predicted (mainly because the source masses were over two times larger than what was imagined in 1998), and the combination of data from that event, and the events GW151226 and GW170104 improved the bound by another 60 percent. The bound was better than the earlier solar-system bound by a factor of about six.
Gravitational waves beat the solar system!
Not so fast….
I recently returned to the problem of Mercury’s perihelion advance, looking at a class of potentially detectable effects arising from the combination of post-Newtonian gravity with the perturbing effects of the other planets . In the course of this study, I was reminded of the enormous improvements in our knowledge of planetary motion over the last few decades. In particular, significant advances in precision have come from the recent array of planetary orbiters around Mercury, Venus, Mars and Saturn. These probes were tracked precisely using radar and very long baseline radio interferometry. I decided to see if the old 1988 bound on the graviton mass still held up.
In the simplest massive graviton scenario, Newton’s gravitational potential is multiplied by a decreasing exponential, whose range depends on the Compton wavelength of the graviton (which is inversely proportional to its mass). This modifies Kepler’s third law, which relates the orbital period to the semimajor axis of an orbit. In fact, the 1988 bound was based on the observed validity of this law for various planets.
But the modified potential also contributes additional perihelion advances for planets, and the effect grows with distance from the Sun. This is opposite to the effect of general relativity, which increases with proximity to the Sun. In the end the best limit on the graviton mass comes from Mars, aided by data from the Mars Express and Mars Reconnaissance orbiters. The bound is about an order of magnitude better than the one inferred from gravitational waves. Saturn and Earth are not far behind, while Mercury is not even in the running. Mercury is good for testing GR, but not so good for testing a massive graviton.
So the solar system beats gravitational waves, for now…
Gravitational waves may take the lead again in 2034 when LISA is launched. We expect that LISA may give an additional improvement by two orders of magnitude on the graviton mass bound by looking at signals from merging supermassive black holes at cosmological distances.
You can find all the details in my CQG Letter here .
 New general relativistic contribution to Mercury’s Perihelion Advance by C. Will. Phys. Rev. Lett. 120, 191101 (2018)
 Solar system vs. gravitational-wave bounds on the graviton mass by Clifford M. Will. Classical and Quantum Gravity, 2018. Accepted Manuscript.
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