← All calculators

Black Hole Time Dilation Calculator

Near a black hole, clocks tick slower than clocks far away. Choose a situation below to see how much slower, and how much time would pass back on Earth while you are there.

Situation
solar masses

Hovering: gravitational time dilation

A clock held at a fixed distance \(r\) from a non-rotating mass runs slow compared with a clock far away by the factor \( \sqrt{1 - R_s/r} \), where \(R_s\) is the Schwarzschild radius. On Earth's surface the effect is tiny, about 60 microseconds per day, but GPS satellites have to correct for it. At twice the horizon radius clocks run at 71% speed, and the slow-down grows without limit as you approach the horizon.

The catch is that hovering close to a black hole takes an enormous, constant rocket thrust. The calculator shows the acceleration you would feel. Close to the horizon it reaches millions or billions of g, far beyond anything a person (or a spacecraft) could survive.

Orbiting: why spin matters

Planets do not hover; they orbit. Around a non-spinning black hole the closest stable orbit is at three Schwarzschild radii, and there clocks run only 1.41 times slower than far away. That is the most a stable orbit can give you, however heavy the black hole.

A spinning (Kerr) black hole drags space around with it, which lets stable orbits sit much closer to the horizon. The faster the spin, the closer the orbit and the stronger the time dilation. This calculator uses the standard formula for circular orbits in the equatorial plane of a Kerr black hole, orbiting in the same direction as the spin (Bardeen, Press & Teukolsky, 1972).

Miller's planet: does 1 hour = 7 years work?

In Interstellar, every hour on Miller's planet costs seven years on Earth, a factor of about 61,000. In The Science of Interstellar, Kip Thorne explains that this needs Gargantua to spin within about one part in 100 trillion of the maximum possible rate, with the planet on the innermost stable orbit. Choose the Miller's planet preset and the calculator gives 1 hour ≈ 7.06 years, with an orbit that takes about 1.7 hours as seen from far away. The full story is in Miller's planet time dilation explained.

FAQ

  • Does a heavier black hole cause more time dilation? Not at the same relative distance. Time dilation depends on how many horizon-radii away you are, so the factor is the same for any mass. Mass only changes how big everything is.
  • Would you feel time slowing down? No. Your own clock, heartbeat and thoughts all run normally. You would only notice when you compared clocks with someone far away.
  • Could you use this to travel to the future? In principle, yes: spend time deep in a gravitational well and return to find more time has passed outside. There is no way to use it to go back.
  • What happens at the horizon? To a distant observer, a falling clock appears to freeze at the horizon. The falling clock itself keeps ticking and crosses it in a finite time. See time dilation near black holes.

More tools