Time Drift

A round trip to the black hole at the centre of the galaxy, and the clock that stayed at home.

01/12—

From Earth: —
Speed: —
Strongest gravity: —
Orbits of Sgr A*: —

1 Speed · Lorentz factor

γ = 1 / √(1 − v²/c²)

v: 0
β = v/c: 0
γ: 1
√(1 − β²) = 1/γ: 1

2 Gravity · Schwarzschild

√(1 − r_s/r) with r_s = 2GM/c²

Mass	r_s	distance r	√(1 − r_s/r)

3 Rocket clock vs Earth clock

dτ/dt = √(1−β²) · ∏ √(1 − r_s,i/r_i)

rocket dτ/dt: 1
Earth dτ⊕/dt: 1
ratio dτ⊕/dτ: 1

4 Accumulated since launch

Δ = ∫ (dτ⊕/dt − dτ/dt) dt

Rocket τ: 0
Earth τ⊕: 0
Δ: 0

Model and assumptions

Idealised rocket: constant 1 g proper acceleration, unlimited energy. No real propulsion can do this; the time dilation it produces is exact.
Speed term: Lorentz factor γ = 1/√(1−v²/c²). For 1 g flight: v = c·tanh(gτ/c), t = (c/g)·sinh(gτ/c).
Gravity term: Schwarzschild metric of a non-rotating mass. Near the black hole, speed is measured by a local static observer, so √(1−rₛ/r)·√(1−β²) is exact.
With several masses the terms are multiplied. This is accurate for weak fields, and everywhere except near Sgr A* the fields are weak.
Sagittarius A*: 4.3 million solar masses, 26,700 ly away. Earth and Sgr A* are treated as mutually at rest. The Sun's 230 km/s galactic orbit, Earth's orbit and rotation and the galaxy's own potential are ignored; each is below 10⁻⁶.
The three giant planets are fictional rogue super-Jupiters of 4, 9 and 13 Jupiter masses. The rocket steers with them through gravity assists: exact two-body hyperbolic flybys at 150 km/s, with deflection δ given by sin(δ/2) = 1/e. At 0.2 c a planet would bend the path by only about 10⁻⁵ rad, so the slingshots happen at low speed.
The planets are treated as mutually at rest, so each flyby changes only the direction of travel, not the speed.
The view is schematic: directions and turning angles are true, while distances and sizes are compressed logarithmically. The numbers in the panels are exact.
The Earth clock sits on Earth's surface at 1 AU from the Sun, so it also runs slow, by about 2×10⁻⁸.