How this works

Two clocks

Earth's clock is a clock at sea level. It shows your actual local time. Your clock left Earth in sync with it when the page loaded. The drift counter is the difference between them. Both tick in real seconds unless you speed time up.

Where time slows

Deep in a gravity well, and at high speed. A clock held still at distance $r$ from a mass $M$ runs at

\frac{d τ}{d t} = \sqrt{1 - \frac{2 G M}{r c^{2}}}

relative to a clock far away. $τ$ is proper time, what the held clock reads. $t$ is coordinate time, what a clock far from the mass reads. Moving at speed $v$ past observers holding station multiplies that rate by

\sqrt{1 - \frac{v^{2}}{c^{2}}}

On a circular orbit the two combine to

\frac{d τ}{d t} = \sqrt{1 - \frac{3 G M}{r c^{2}}}

GPS, the everyday case

At sea level Earth's own field slows a clock by 6.97 parts in 10¹⁰, the IAU constant $L_{G}$ . A GPS satellite at 20 200 km altitude sits higher in the well and gains about 45.7 µs a day. It also moves at 3.87 km/s and loses about 7.2 µs a day. Net, its clock gains about 38.6 µs a day, and the system corrects for it.

The reference frame

In Earth scenes coordinate time is geocentric, and Earth's clock runs slow by $L_{G} = 6.969290134 \times 10^{- 10}$ . In every other scene coordinate time belongs to the body you are visiting, and Earth's clock carries its full share of the solar system's potential, $L_{B} = 1.550519768 \times 10^{- 8}$ . Both are IAU defined constants (IAU 2000 Resolution B1.9 and IAU 2006 Resolution B3).

What the ship can do

Start on a circular orbit, prograde or retrograde, or hold station. From then on the ship is in free fall unless you act. A kick changes its speed at once by a chosen fraction of its current speed, or of the local orbital speed while it holds station, applied prograde, retrograde, outward or inward as measured by observers holding station where the ship is. Holding station stops the ship dead and keeps it there; the panel shows the thrust that takes, in g, which grows without bound at a horizon. Letting go drops it from rest.

Around black holes, neutron stars, magnetars and white dwarfs the ship follows exact equatorial geodesics of the Kerr metric, Schwarzschild when the spin is zero. Around the Sun, planets and moons it moves under Newtonian gravity from the central body and its companions, and its clock runs at the first-order rate: one minus the summed potentials over $c^{2}$ , times $\sqrt{1 - \frac{v^{2}}{c^{2}}}$ . Each scene also carries the Sun's share for the body it is centred on, so every clock here is measured against the same barycentric time as Earth's.

Companions move on their real Kepler ellipses from published elements, placed where they are today. Their orbits are drawn in the plane of the plate; inclinations are ignored. Near compact bodies the companions are scenery and do not pull on the ship. A scene is only right out to the Hill sphere of the last body it pulls with, about 1.5 million km from Earth for the Earth and Moon scenes; beyond it the Sun would take over, nothing here stands in for it, and the page says so.

Landmarks around a black hole

Radii in units of $G M / c^{2}$ .

Event horizon: 2 for a non-spinning hole. With spin $a^{*}$ it is $1 + \sqrt{1 - a^{* 2}}$ .
Photon sphere: 3 for no spin. Light circles here.
Innermost stable circular orbit: 6 for no spin. Spin pulls prograde orbits closer, down to 1 at $a^{*} = 1$ , and pushes retrograde orbits out, to 9 at the same spin (Bardeen, Press and Teukolsky 1972).
Ergosphere: Inside 2 around a spinning hole nothing can stay still. If you hold station inside it, the ship holds radius while dragged space sweeps it around, a zero angular momentum observer.

Worked numbers

Solar photosphere: 2.12 parts per million slower.
Neutron star, 2.01 solar masses, 12 km radius: About 29 percent slower at the surface.
Horizon of any black hole: Rate zero for a clock holding station just outside.
Pound and Rebka, Harvard, 1960: Measured 2.46 × 10⁻¹⁵ across 22.5 m.

Scale

Distances are to scale. Zoom is logarithmic. A scale bar and its light travel time are always on screen. Nothing is compressed to fit.

Sources

IAU 2015 Resolution B3, nominal values, Prša et al. 2016.
CODATA (2018 values, unchanged in 2022), NIST.
IAU 2000 Resolution B1.9 and IAU 2006 Resolution B3.
Bardeen, Press and Teukolsky 1972, ApJ 178, 347.
Pound and Rebka 1960, Phys. Rev. Lett. 4, 337.
Ashby 2003, Relativity in the Global Positioning System, Living Reviews in Relativity.

Catalogue

Every mass, radius, spin, distance and orbital element in the body picker cites one of these.