# The unreasonable importance of 1.74 seconds

1.74 seconds.

If you know what I am talking about, you can discontinue reading this – its old news. If you don’t, its interesting what physicists can learn from 1.74 seconds. Its all buried in the story about GW170817.

A few days ago, the people who constructed the LIGO telescope observed gravitational waves from what appears to be the collision and collapse of a pair of neutron stars (of masses that are believed to be $1.16 M_{\odot}$ and $1.6 M_{\odot}$. The gravitational wave observation is described in this Youtube posting, as also a reconstruction of how it might sound in audio (of course, it wasn’t something you could ${\bf {hear}}$!).

As soon as the gravitational wave was detected, a search was done through data recorded at several telescopes and satellites for  coincident optical or gamma ray (high frequency light waves) emissions – the Fermi (space-based) telescope did record a Gamma ray burst 1.7 seconds later. Careful analysis of the data followed.

Sky & Telescope has a nice article discussing this. To summarize it and some of the papers briefly, it turns out that a key part to unravelling the exact details about the stars involved is to figure out the distance. To find the distance, we need to know where to look (was the Fermi telescope actually observing the same event?). There are three detectors currently at work under the LIGO collaboration and two of them detected the event. This is the minimum needed to detect an event anyway (given the extremely high noise) one needs the near-simultaneous detection at two widely separated detectors to confirm that we have seen something. All the detectors have blind spots due to the angles at which they are placed, so the fact that two saw something and the third ${\bf {didn't}}$ indicates something was afoot in the blind-spot of the third detector. That didn’t localize the event enough though. Enter Fermi’s observation – which only localized the event to tens of degrees (about twenty times the size of the moon or sun). But the combination was enough to put a small field of view as “region of interest”. Optical telescopes then looked at the region and discovered the “smoking gun” – the actual increasingly bright, then dimming star. The star appears to be in a suburb of the NGC 4993 galaxy, some 130 million light years away – note that our nearest galactic neighbour is the Andromeda galaxy which is roughly 2 million light years away. Finding the distance makes the precision in the spectra, the masses of the involved neutron stars etc much higher, so one can actually do a lot more precise analysis. The red-shift to the galaxy is 0.008, which looks small, but this connection helps understand the propagation of the light and gravitational waves from the event to us on the Earth.

Now, on to the simple topic of this post. If you have an event from which you received gravitational waves and photons and the photons reached us 1.74 seconds after the gravitational waves, we can estimate a limit on the difference in the mass of the photon versus the graviton (hypothesized particle that is the force carrier of gravity). If we, or simplicity, assume the mass of the graviton is zero, then we make the following argument:

From special relativity, if a particle has speed $v$, momentum $p$, energy $E$ and the speed of light is $c$,

$\frac{v}{c} = \frac{pc}{E} = \frac{\sqrt{E^2 - m^2c^4}}{E} \approx 1 - \frac{m^2c^4}{2E^2}$

which implies

$\frac{mc^2}{E} = \sqrt{1 - \frac{v}{c}} \approx 1 - \frac{v}{2c}$

Since the graviton reached 1.7 seconds before, after travelling for 130 million years, that translates into a differential of speed, i.e., $\frac {\delta v}{c}$  of $4 \times 10^{-19}$., i.e.,

$\frac{\delta(m)c^2}{E} =2 \times 10^{-19}$

Next, we need to compute the energy of the typical photons in this event. They follow an approximate black-body spectrum and the peak wavelength of a black-body spectrum follows the famous Wien’s law. To cut to the chase, the peak energy emission was in the $10 keV$ range (kilo-electronvolt), which means our mass differential is $2 \times 10^{-15}$ eV (this is measuring mass in energy terms, as Einstein instructed us to). While the current limits on the photon’s mass are much tighter, this is an interesting way to get a bound on the mass. In general, the models for this sort of emission indicate that gamma rays are emitted roughly a second after the collision, so the limits will get tighter as the models sharpen their pencils through observation.

However, the models are already improving bounds on various theories that rely on modifying Einstein’s (and Newton’s) theories of gravity. Keep a sharp eye out!