# What is the uncertainty in the one solar mass radiated away as pure energy? (Gravitational waves)

edit: Thanks to the comment below - I learned that the PLR is available to read http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.116.241103. In it, ref. 5 links to this preprint dated June15, 2016: Binary Black Hole Mergers in the first Advanced LIGO Observing Run https://dcc.ligo.org/public/0124/P1600088/015/bbh-o1.pdf.

This BBC article dated June 15, 2016 More gravitational waves detected states:

Reporting the event in the journal Physical Review Letters, the international collaboration that operates LIGO says the two objects involved had masses that were 14 and eight times that of our Sun.

The data indicates the union produced a single black hole of 21 solar masses, meaning they radiated pure energy to space equivalent to the mass of one star of Sun size.

It is this energy, in the form of gravitational waves, that was sensed in the laser interferometers of the LIGO labs...

Nuclear reactions and decays can result in mass changes of the order of a part per thousand or less, and matter/antimatter particle-antiparticle annihilation can change all of the mass to light. But in this case, really - 5% of the mass was (believed to have been) converted to energy in the form of gravitational waves? I mean black holes can potentially evaporate if they manage to stop "eating" long enough, but these waves seem to just spread out into something that stops having a significant effect on anything.

my question: So 5% of the mass of a pair of black holes is (believed to have been) actually converted all the way to energy in the form of vibration of space? If so, what is the experimental uncertainty on the mass - where is it indicated and the calculation of this value discussed?

• I've asked a the related question Theoretically possible to extract energy from gravity waves? How might it be implemented?. – uhoh Jun 16 '16 at 10:06
• I'm unsure what you are asking. Yes, about 5% of the system's total mass was converted to GWs. – Paul T. Jun 16 '16 at 12:54
• @uhoh the PRL isn't behind a paywall, if you click on the PDF link you get directed here: journals.aps.org/prl/pdf/10.1103/PhysRevLett.116.241103 – Dean Jun 16 '16 at 17:26
• @Dean thank you for that, and thanks to the authors/journal for making it available. It seems to cite this LIGO/Virgo document document for further discussion of the mass calculations. I'm looking now for the actual calculation of, and the uncertainty in the actual 1 solar mass value . – uhoh Jun 17 '16 at 0:28
• @PaulT. upon accessing and reading the PRL (I didn't realize it was made available to the public) and also ref. 5 (also now linked in the question) I've just modified the question to ask for the uncertainty in the value of 1 solar mass being converted to energy, and at least a pointer to where it is calculated . – uhoh Jun 17 '16 at 1:33

You should have a look at this paper by the LIGO team, which describes in some detail the fitting process which is used to estimate the system parameters from the observed data. It also gives a brief plausibility argument for why a pair of merging black holes are expected to give up $\sim 5$% of their rest mass as gravitational waves.

The parameters are estimated by comparison with a suite of numerical models based on GR codes using a Bayesian framework. Different parameters are constrained with differing degrees of precision. Importantly, many of the parameters are correlated.

What this means is that although the individual masses are quoted as $14.2^{+8.3}_{-3.7} M_{\odot}$ and $7.5^{+2.3}_{-2.3} M_{\odot}$; and the final mass after merger is $20.8^{+6.1}_{-1.7} M_{\odot}$, that does not mean that the initial mass (and uncertainty) can be estimated by simply adding the first two numbers together and combining their errors in quadrature, or that the mass discrepancy (radiated in gravitational waves) can be obtained by subtracting the third number from the sum of the first two numbers. These parameter estimates are not independent of one another. More massive initial black holes will lead to a more massive final black hole and vice-versa.

Each numerically calculated waveform that is fitted to the data is a self-consistent model of a merging black hole binary. If the initial black hole masses and spins are specified, this leads to a definite final black hole mass after merger. Each of these model waveforms therefore predicts a "mass deficit" between the initial black holes and the final black hole. Therefore the quoted mass difference and its uncertainty in Table 1 of this paper of $1.0^{+0.1}_{-0.2}M_{\odot}$, is obtained from the distribution of possible models that fit the data (both the inspiral and the ringdown phases) and therefore the distribution of "mass deficits" that these models predict. It does not come from doing simple algebra on the quoted parameters for the initial and final black hole masses.

• OK great - it's late here so I will read in the morning. It seems that your are saying that the mass-loss with it's uncertainty is $1.0^{+0.1}_{-0.2}M_{\odot}$ , which is definitely an answer to the question. I understand what you are saying about the correlations, and so in a 3D plot of probability shells vs $m_1, m_2, \text{and } \Delta M$ will be narrower in the $\Delta M$ (mass loss) direction than either of the other two. Looking forward to one ore more cups of coffee and a nice morning read. Thanks!! – uhoh Jun 17 '16 at 16:36

You are right to be astonished by, and question, the claim that 1 solar mass equivalent of pure energy was radiated by gravitational waves. I was in a similar position myself in February when they announced the first black hole merger they detected had radiated 3 solar masses (or more than the entire observable universe in that split second). But having worked on gravitational waves for a while now I have began to get used to such mind boggling numbers. Without going into the detailed mathematics behind general relativity and conservation of energy equations I cant give you a firm answer but what I would say is that if you believe that LIGO can actually detect these minuscule changes in length ($10^{-21}$), which is quoted as being:

like accurately measuring the distance from Earth to the nearest star—3 light years—to within the width of a human hair

then you shouldn't have any trouble in believeing that 1 entire solar mass of energy was radiated by these black holes merging :)