W1 · Relativistic π of Sgr A* Polar chart of Sgr A* with geodesic rings Concentric geodesic circles around a black hole. Each ring is labeled with its local ratio of proper circumference to proper diameter. π(1.25 rₛ) ≈ 1.20 π π(2 rₛ) ≈ 0.87 π π(3.3 rₛ) ≈ 0.83 π π(10 rₛ) ≈ 0.88 π π(100 rₛ) ≈ 0.98 π rₛ Light takes 70 minutes to escape the outermost ring of this chart, then 27,000 years to reach the Earth. Everything strange about π near Sgr A* lives in a region the size of our planetary system; everything else is the long quiet flight home. Equatorial Schwarzschild slice. rs = 2GM c2 . ds2 = ( 1 − rsr ) −1 dr2 + r2 dφ2 C(r) = ∫ 0 2π r dφ = 2πr D(r) = 2 ∫ rs r d r′ 1 − rs r′ π(r) = πr ∫ rs r d r′ 1 − rs r′