How this instrument works
This instrument tracks where the mass goes when two black holes merge into one. Add the two starting masses to get the combined pre-merger mass, apply a radiation efficiency — the fraction of that combined mass converted into gravitational-wave energy during the final plunge and ringdown — and subtract to find what the single remnant black hole weighs afterward. The shape of the formula follows directly from mass-energy equivalence: E = mc² lets the energy carried off by the waves be tracked in the same solar-mass units as the black holes themselves, so no separate unit conversion sits between 'mass lost' and 'energy emitted'.
The radiation efficiency is not a free number pulled from nowhere. It comes from solving Einstein's field equations numerically for the last orbits, plunge, and ringdown of two black holes — a problem too nonlinear for a closed-form solution, first cracked computationally in 2005. Comparable-mass binaries with modest spin, like the pair LIGO detected in September 2015 as GW150914, convert roughly 3 to 5 percent of their combined mass to gravitational waves in about a fifth of a second, a peak power output that briefly exceeds the combined light output of every star in the observable universe.
There is a hard ceiling on that percentage. Hawking's area theorem forbids the horizon area of the merger product from shrinking, which caps radiation efficiency at roughly 29 percent — reached only in the idealized limit of two equal-mass black holes spinning near maximum speed and colliding head-on. It is easy to confuse this rapid, classical mass loss with Hawking radiation, the separate and vastly slower quantum process by which black holes evaporate; a stellar-mass hole would take far longer than the current age of the universe to lose even a fraction of a percent of its mass that way.
- Enter the Mass of first black hole, solar masses — m1, either member of the pre-merger pair.
- Enter the Mass of second black hole, solar masses — m2, its companion just before the horizons touch.
- Set Energy radiated as gravitational waves, % of total mass — a realistic pre-merger efficiency, typically 1% to 10% for comparable-mass, modestly spinning binaries.
- Read Combined pre-merger mass and Mass-energy radiated as gravitational waves, then Final merged black hole mass for what the remnant weighs afterward.
Worked example — a GW150914-class binary merger
Two black holes with masses of 30 and 29 solar masses — close to the pair LIGO first detected in September 2015 — spiral together and merge. Combined pre-merger mass is simply the sum: totalMass = 30 + 29 = 59 solar masses, the mass the system carried into its final orbits before any energy left as radiation.
Assume a representative radiation efficiency of 4.6%, within the few-percent range numerical-relativity simulations predict for a binary of this mass ratio and spin. Mass-energy radiated as gravitational waves works out to energyRadiated = 59 × 4.6 / 100 = 2.714 solar masses, and the final merged black hole mass is finalMass = 59 − 2.714 = 56.286 solar masses — more mass converted to pure energy in under a second than the Sun will radiate as sunlight across its entire ten-billion-year lifetime.
Questions
Why does a black hole merger radiate away mass as gravitational waves?
Because mass and energy are the same thing scaled by c² (E = mc²). As the two horizons spiral together and merge, the changing spacetime curvature radiates energy outward as gravitational waves — real energy carried away at the speed of light. That departing energy shows up as a mass deficit: the final black hole weighs less than the simple sum of the two progenitors, even though nothing physical was destroyed.
Is this the same process as Hawking radiation?
No. Hawking radiation is a quantum-mechanical leak that would take a stellar-mass black hole far longer than the current age of the universe to notice. The mass loss this calculator computes is a classical general-relativity effect — gravitational-wave emission during the final orbits and merger — that removes several percent of the total mass in well under a second, entirely different physics on a wildly different timescale.
What is a realistic value to enter for radiation efficiency?
For two comparable-mass black holes with modest spin, numerical-relativity simulations and LIGO's own detections put it around 3 to 5 percent — a GW150914-class event radiates close to 4.6 percent of its combined mass. Very unequal masses radiate a smaller fraction; the theoretical ceiling, reached only for two equal, maximally spinning black holes merging head-on, is about 29 percent.
Why is the final black hole lighter than the sum of the two starting masses?
Because merging is not simple addition. Some of the combined mass leaves the system as gravitational-wave energy during the plunge and ringdown, so the single remnant necessarily weighs less than totalMass. Conservation of mass-energy still holds — nothing vanishes — the missing mass is accounted for by the radiated waves, which is exactly what the energyRadiated field represents.
Can radiationEfficiency go above 29 percent or below zero?
Physically, no. Hawking's area theorem sets roughly 29 percent as the maximum, reachable only in the idealized extreme-spin, equal-mass, head-on case; real astrophysical mergers with modest spins and unequal masses land closer to 1 to 10 percent. A negative value would mean the remnant gained mass during merger, which no known physical process allows, so keep entries between 0 and about 29.
What does 'solar masses' mean for a quantity as abstract as radiated energy?
It uses the same unit as the black holes themselves, converted through E = mc². One solar mass of energy, M☉c², equals about 1.79 × 10^47 joules — so the energyRadiated of 2.714 solar masses in the worked example is roughly 4.9 × 10^47 joules released as gravitational waves, comparable to the Sun's total energy output over billions of years, delivered instead in a fraction of a second.