The Physics of a Journey: Time in Interstellar

Interstellar is the famous film by Cristopher Nolan, released in 2014. It was a great success thanks to the amazing plot and the extraordinary interpretation of Matthew McConaughey, all glazed by the emotional soundtrack produced by Hans Zimmer.

The former pilot and engineer Cooper, commissioned by NASA, has the task of finding a new home for humanity due to a major drought that rages on planet Earth. To reach the unexplored worlds, Cooper will have to cross a space-time tunnel, Worm-Hole, which will catapult him to light years from every affection. In this journey, time and space will inevitably mingle following Einstein's General Relativity guidelines.

Cooper and his daughter Murphy awaiting the departure towards new planets

At some point in the journey, Cooper and his team must visit a planet that, according to their analysis, could be habitable. However, the planet orbits around a massive black hole, Gargantua, and the team must consider the risks of this expedition. In fact, every hour spent on the surface of the planet is equivalent to seven years for an inhabitant of the Earth. Cooper, of course, is not willing to sacrifice precious time and is determined to complete the mission as quickly as possible.

The tidal forces, due to the presence of the black hole, make life on the planet nothing short of impossible.

Let's analyze, briefly, how time is really distorted by the black hole!

The space-time in which we live, according to the theory of General Relativity, is four-dimensional: it has three spatial dimensions (height, width and length) and a temporal dimension. The presence of a mass in space curves the structure of space-time; the more the object is massive, the more the space-time curvature will be accentuated. As a result, a black hole pierces the fabric of space-time and deforms its surrounding structure. The surface of space-time, however, away from any massive object, turns out to be flat.

The fabric of space time is pierced by the singularity of the black hole. The space-time structure, outside the event horizon, is described by the Schwarzschild metric.

Near the black hole, the structure of space-time is described by the Schwarzschild metric, which describes the intrinsic structure of the space-time. The metric is a mathematical expression that describes how to "move" on a curved or flat surface. To give an example, the minimum distance between two points on a plane is equal to the length of the segment that has them for extremes; on the surface of a sphere, however, the minimum distance between two points is equal to the length of the arc of the major circle passing through the two points. Therefore, the metrics that describe the two surfaces are different!

The metrics of the three surfaces are different. In fact, the surface of the red triangle varies greatly on each surface.

Time on the unknown planet, according to the correct metric, runs much slower than the time measured by a terrestrial observer. However, the existence of the planet itself is a mistake in writing the plot of the film! In fact, the maximum allowed time dilation means that 17 days (not an hour) on the planet are equal to 7 years on Earth! The calculations show that to have a temporal expansion of that sort, the planet should be within Gargantua's event horizon. Most likely, the authors wanted to exaggerate the temporal expansion for the purpose of the plot, reserving (perhaps) Cooper from a bitter destiny. Unfortunately, Cooper will have contingencies on the surface of the planet, due to the impressive tides (whose existence is questionable due to tidal locking).

Leaving aside this fussiness let the former pilot travel to other remote places in search of a new home.