A measure of the combined spin of a black hole binary system, projected along the orbital axis.Effective spin is a parameter used in gravitational wave astronomy to describe the combined spin of a binary black hole system, projected along the orbital angular momentum axis. When two black holes orbit each other and eventually merge, their individual spins (rotations) affect the gravitational waves they emit. The effective spin, often denoted as χ_eff, is a mass-weighted average of the components of the two black holes' spins aligned with the orbital axis. It is a crucial observable because it influences the dynamics of the inspiral and the properties of the final merged black hole, and it can be measured from the gravitational wave signal.
The concept of effective spin was introduced in the early 2000s in the context of post-Newtonian approximations for binary black hole systems, notably by Alessandra Buonanno and others. It became widely used after the first gravitational wave detection in 2015, as it is one of the key parameters inferred from signals.
For a binary system with black hole masses m1 and m2 and spin vectors S1 and S2, the effective spin is defined as χ_eff = (m1 * S1_z + m2 * S2_z) / (m1^2 + m2^2) (in units where G=c=1), where S1_z and S2_z are the components of the spins along the orbital angular momentum direction. This quantity is conserved during the inspiral phase to a good approximation, making it a robust observable. A positive χ_eff indicates that the spins are aligned with the orbital motion, while a negative value means they are anti-aligned. The magnitude ranges from −1 to +1, with 0 indicating no net alignment.
Effective spin is used to infer the spin properties of black hole binaries from gravitational wave signals detected by LIGO and Virgo, helping to understand how black holes form and evolve in stellar binaries.
The first detected gravitational wave event, GW150914, had an effective spin consistent with zero, suggesting that the two black holes were either non-spinning or had spins that canceled out along the orbital axis.

















