Gravitational Wave Echo Time

τ-Formula

t± = 4GM·ι_τ±¹/c³

Derivation

The gravitational-wave echo time delays for a $T^2$-topology black hole of mass $M$ are

The echo separation $Δ t = t_- - t_+ = 4GM(ι_τ^-1 - ι_τ)/c^3$ places both echoes in the sensitive frequency band of current and next-generation detectors:

Source & $M\,(M_)$ & $t_+\,(ms)$ & $t_-\,(ms)$ & $Δ t\,(ms)$ Stellar BH | 10 | 0.067 | 0.577 | 0.510 GW150914 remnant | 62 | 0.417 | 3.580 | 3.163 Intermediate | 150 | 1.009 | 8.660 | 7.651 Sgr A* | $4 × 10^6$ & 3cLISA band: $∼$4.3 mHz / $∼$37 mHz

Source

This prediction is derived in the Physics Ledger (Chapter 64 — black-hole-topology), Books IV–V of Panta Rhei.