From: "Norbert C" in Antwerp, Belgium

Date: Fri, 26 Aug 2005

**Q ** To influence time (example 1%) how big has to be the mass of a black hole?
60 real minutes must become under influence 59 Minutes? I this possible?

**A **Good question, Norbert. According to general relativity, all massive objects
(not just black holes) warp spacetime in their neighborhood, which means that
they all affect the flow of time. Time runs slightly slower close to any massive
object, compared to a clock far out in space. In 1971 Hafele and Keating flew
four atomic clocks around the world on commercial jet flights and were actually
able to measure this effect. Since they were farther away from the massive earth,
the clocks on the airplanes ran faster by ~150 nanoseconds over the course of
the trip (there were also effects from the speed of the planes, which they accounted
for).

The mathematics of general relativity can be used to figure out how spacetime should be warped in the neighborhood of a black hole. For simplicity, we generally assume that the black hole (or other massive object) is (1) not rotating, (2) electrically neutral, (3) spherically symmetric, (4) isolated (far away from any other matter or energy), and (5) static (we assume that it's always been there, rather than dealing with the process of how it formed). This set of assumptions gives us a spacetime arrangement called the "Schwarzschild solution", and should be pretty accurate in most realistic cases.

Imagine you are sitting at a distance r away from the center of an object of mass M. It turns out that your clock will run slower than a clock far away from the object by a factor of sqrt(1-R/r), where R is the hole's "Schwarzschild radius" (roughly the radius of its event horizon) given by R = 2 G M / c^2 (where G is Newton's gravitational constant and c is the speed of light). Time will thus slow down by 1% at a distance of ~50 times the Schwarzschild radius. For a black hole the mass of the sun, R=3 km and time will slow down by 1% at 150 km. An object twice as massive will warp time this much twice as far away. The only thing special about black holes is that all of their mass fits within their Schwarzschild radius, and so you can actually get close enough to warp spacetime significantly.

For the earth, the Schwarzschild radius turns out to be less than a centimeter
and so the effect on time at the Earth's surface is very small.

Jeff Filippini

UC Berkeley Cosmology Group