This is a series of posts about the counterintuitive nature of time in Physics. This post presents some difficult aspects of time in General Relativity, such as the notion of curved spacetime, the looping time, and the idea of the beginning of time.

After the Special Relativity, Einstein tried to express various physical laws in this formalism. Basically, their mathematical form is required to be invariant at Lorentz transformations. The main difficulty he encountered was to express Newton’s gravitation field in this way. After many years of research, Einstein obtained the movement of bodies under the effect of gravity as simply an inertial movement in a curved spacetime. The inertia and the gravity become unified. The spacetime itself is curved by the masses, and the universal attraction was just an effect of this curvature. General Relativity was born. The experimental consequences eventually confirmed the theory, which become widely accepted. Among its surprising features is that that the time flow changes in the presence of massive bodies.

The main equation of General Relativity, Einstein’s equation, relates the curvature to the distribution of energy. One very important difference between this equation, and the previously known equations in mathematical Physics, is the following: finding the solution, means also finding the background (meaning the spacetime itself). In the Newtonian and special relativistic cases, the spacetime was fixed, but in General Relativity, it is part of the solution itself. Perhaps, this is the most striking difference.

The main counterintuitive aspect of the curved spacetime is caused by our tendency to consider it as a subspace of a space with more dimensions. Many persons, when learn for the first time that the spacetime is curved, tend to interpret this as being curved in a fifth dimension. As a simpler but historic example, when we think at a curved surface, we tend to consider it a subspace of the Euclidean space. Gauss realized that the intrinsic geometry of every surface can be expressed independently on the Euclidean space in which this is embedded. The main ingredient is the metric tensor, which provides a point-dependant measure of the lengths of the curves embedded in the surface. Riemann generalized the surfaces to curved spaces with any number of dimensions. Their work helps understanding that the curved spaces in Riemannian geometry do not rely on a Euclidean space in which they may be embedded. Einstein found the four-dimensional Riemannian geometry as the ideal tool for General Relativity, provided that we replace the Euclidean metric tensor with the Lorentz metric.

The Einstein’s equation may have solutions that contain closed timelike curves. Spacetime may be curved in such a manner, that the future of an event becomes also its past. This looping time highly contradicts our intuition. Yet, unlike the other counterintuitive aspects of time, this one may not even exist, as Hawking’s Chronology Protection Conjecture states.

Another hard to grasp aspect of time is the beginning. Our experience teaches us to consider the time as being linear, infinitely continued in past and future. Why do we have this intuition, considering that our lives are finite? Perhaps because the daily events succeed linearly, at our scales, and because the History of our countries, and of our planet, and solar systems, appear to be linear. But when we hear about the Big Bang, two questions we may find natural is “what was before the Big Bang?”, and “when happened the Big Bang?”. We find difficult to accept that even the time may have a beginning.

Other difficult aspects of time in General Relativity are related to special situations, like the time in the presence of a Black Hole, in a Worm Hole, time traveling using Worm Holes, Hawking’s imaginary time, the time near/at the initial singularity. I will not detail these problems.