Very Strong Gravity
There is another relatively easy way to journey to the future utilizing general relativity. If you recall, we saw that the spacetime metric equation was dependent on the matter content in that spacetime region. The German physicist Karl Schwarzschild was the first to work out a solution for the spacetime surrounding a static spherical mass. This is the metric he derived, and it bears his name.
Where G is the universal gravitational constant and M is the mass inside the spherical distribution. Notice that the metric is in spherical coordinates, t, r(radius from the origin), theta and phi. From this metric a relation between proper time and coordinate time can be found in the same way as before. That relation is
The form of this time dilation formula is nearly the same as that for the special relativistic case. The critical parameter now is the radial distance, r, from the mass, M. A person very far from the surface of the mass will measure a different time than a person close to the mass. Indeed this effect has been measured on Earth. Atomic clocks placed in a basement of a very tall sky scraper ran slower than clocks on the top floor by billionths of a second. In order to get significant time travel effects from this we need a stronger gravitational field. There is a critical radius in the Schwarzschild metric like the critical velocity of the previous example. It occurs at r = 2GM/c^2, this is known as the Schwarzschild radius. This is a radius where, if all of the mass of the object were squeezed interior to, would cause the object to become a black hole. At the Schwarzschild radius the escape velocity is the speed of light. Anything crossing the Schwarzschild radius becomes trapped within. For the Earth the Schwarzschild radius is 9 millimeters. That is this big ----. Now if I were to get as close as 0.007 mm from Schwarzschild radius and just sit there for 2 years and I asked you to watch me from very far away you would have to watch me for 70 years. I have ignored the fact that at that distance the gravitational field would be so strong that it could pull me apart from head to toe, but I've got my inertial dampeners from Star Trek with me so I'm okay. Then if I moved away from the now diminished Earth I would find that you are 68 years older than I expected you to be. Like Bob, I have traveled to the future. But I've decided that I don't like the future as much as I thought I would. Is there any hope of my returning to the past? Yes, there is hope, but the return trip is likely to be difficult.