A clock near a black hole ticks slower than one far away. This tool lets you place two clocks at two radii and see the difference, then chains the result through Lloyd's compute limit to show what the dilation means for a hypothetical near-horizon civilisation.
For a static observer at radius r outside a non-rotating black hole of Schwarzschild radius r_s = 2GM/cĀ², the proper time Ļ relates to the coordinate time t (the time measured by an observer at infinity) as:
dĻ / dt = ā(1 ā r_s/r)
A clock at r ticks slower than a clock at infinity by this factor. As r ā r_s, the factor ā 0 ā the clock appears (from outside) to freeze. As r ā ā, the factor ā 1.
Both clocks are static observers at fixed radii. "1 year of proper time = X years of external" means: while the clock at this radius ticks off one year of its own experience, an external observer's clock has ticked off X years. For Clock A near the horizon, X is large (years near horizon ā many years outside). For Clock B far from the horizon, X ā 1.
Lloyd's bound on operations per second is f_max = 2 E / (Ļā) using the system's total rest energy E. Crucially, this formula is observer-invariant when energy and time are measured consistently. A near-horizon computer with local rest energy E performs Lloyd-bounded ops at rate f_max per local second. An external observer sees those ops happen at rate f_max Ć ā(1ār_s/r) per external second ā slower, not faster, because the local clock is dilated.
The "subjective ops per external sec" figure equals the external-frame rate: per external second, Clock A's local clock advances by only ā(1ār_s/r) seconds, so it completes Lloyd Ć ā(1ār_s/r) ops ā the same count an external observer tallies. The near-horizon location is not a thermodynamic free lunch; the local clock runs slow, so each external second buys only ā(1ār_s/r) local seconds of computation. The advantage is the reverse: external time passes cheaply from A's perspective, making it easy to wait out slow cosmological processes.
This is the Schwarzschild (non-rotating) metric. Real IMBHs spin, and Kerr dilation near the prograde ISCO is somewhat stronger; the ISCO itself moves inward from 6 r_g to r_g as a ā 1. The orbital period shown is the Keplerian coordinate period, which differs from a relativistic circular-orbit period by a few percent at large r and more at small r. Tidal forces, accretion-disk radiation, and the practical impossibility of "hovering" at fixed r without enormous propulsion are not addressed ā see the red warning under the compute chain when Clock A approaches r_s.
In Christopher Nolan's Interstellar (2014), Miller's Planet orbits the supermassive black hole Gargantua with the famous ratio of 1 hour on the planet = 7 years on Earth. That's a dilation factor of ~61,000 ā so the redshift factor ā(1ār_s/r) would have to be about 1.6Ć10ā»āµ. In Schwarzschild geometry, achieving this ratio requires r ā 1.000000000Ā³ r_s ā essentially grazing the horizon, well inside the photon sphere (1.5 r_s) and far inside any stable circular orbit (3 r_s for Schwarzschild). It's not physically allowable in a non-spinning hole.
The film made it work by giving Gargantua a near-maximal spin (a/M ā 1 ā 10ā»Ā¹ā“), at which point the prograde Kerr ISCO migrates inward to r_g rather than 6 r_g, and stable circular orbits become available at radii where the redshift factor is small enough for the 1:61,000 ratio. Kip Thorne (the film's science consultant and a relativist who literally wrote Gravitation) describes the calculation in his companion book The Science of Interstellar; the requirement that Gargantua be near-extremal Kerr is exactly what makes Miller's Planet a stable place rather than a sub-second plunge through the horizon.
This tool uses Schwarzschild geometry, so the Miller's-Planet ratio is unreachable here at any stable r. The Kerr Geometry tool lets you reproduce the Interstellar configuration explicitly with the spin parameter as an input.
The Lloyd ops/sec number used here is the same calculation as in the BekensteināLandauerāLloyd Explorer's Panel 2. The Kerr Geometry tool provides the canonical picture of how horizon and ISCO move with spin.
General-relativistic time dilation has been verified to extraordinary precision. Hafele & Keating 1971 flew caesium clocks around the world on commercial jets; the eastward-flying clock lost 59 ns and the westward-flying clock gained 273 ns relative to USNO, matching GR + special-relativity predictions to within ~10%. GPS satellites require continuous relativistic correction: their clocks tick ~38 Ī¼s/day faster than ground clocks (45 Ī¼s faster from weaker gravity, 7 Ī¼s slower from orbital velocity). NIST optical-lattice clocks (2010 onward) resolve gravitational redshift across 33 cm of vertical height, corresponding to a relative frequency shift of 4Ć10ā»Ā¹ā·. The Schwarzschild formula this tool uses has never failed an experimental test.
The star S2 orbits Sgr A* (4.3Ć10ā¶ Mā) at perihelion 17 light-hours, with a 16.05-year period. Gravity Collaboration 2020 (A&A 636:L5) measured S2's gravitational redshift at perihelion (perinigricon?): observed ~200 km/s wavelength shift relative to special-relativity-only prediction, confirming GR to within 20%. The Schwarzschild radius at Sgr A* is ~12 million km; S2 perihelion is ~120 r_s, so the dilation factor at perihelion is ā(1ā1/120) ā 0.996 ā slow time only by 0.4%, but enough for clean detection with modern infrared spectroscopy.
Interstellar's 1 hr = 7 yr time dilation requires a redshift factor of ~1.6Ć10ā»āµ, which Schwarzschild geometry can't reach at any stable orbit (the photon sphere sits at 1.5 r_s; the prograde ISCO is at 3 r_s = 6 r_g). Kip Thorne's companion book solves this by making Gargantua spin at a/M ā 1 ā 10ā»Ā¹ā“ ā the spin pulls the prograde ISCO inward to r_g, where the redshift factor can reach the required value at a stable orbit. The Kerr Geometry tool lets you tune spin and see the ISCO migrate; this tool intentionally uses Schwarzschild as a baseline.