The physics of Interstellar: Miller’s Planet

Today I am going to try to explain some of the cool stuff you can see during Interstellar on Miller’s Planet (yes, the one with the giant waves) without using any equation. Let’s see if I can manage to make it easy to follow.

When the Endurance crew goes through the wormhole, the first plane they visit is Miller’s, which orbits a gigantic black hole (Gargantua). This plane has two interesting characteristics. First, time in its surface goes much more slowly than humans are used to: one hour on its surface equals seven years on Earth. The other curious phenomenon is that the are some gigantic waves going through on the planet surface (about 1 kilometer tall). As far as I am concerned, both events are connected through the space-time curvature generated by a supermassive black hole. Buckle up and let’s get into detail.

At the beginning of the 20th century, Albert Einstein developed the general theory of relativity. Without going into too much detail (remember, no equations), we can see general relativity as a theory that unifies the concepts of gravitational mechanics (i.e., the laws of movement for astronomical objects) and special relativity. Special relativity was also developed by Einstein a few years before, and one of its core concepts is that the speed of light in vacuum is the same no matter the relative motion between light source and observer. One of the coolest relativistic effects, at least for me, is the concept of space-time. Space-time allows us to describe the universe as a four dimensional space, instead of only using three spatial dimensions. It might seem that this is a silly change, but it has a lot of implications. First, introducing time as a core dimension implies that physical phenomena affect both space and time at the same time, which is something we can use to explain the events of Interstellar.

Space-time curvature

Imagine space as a grid, where any astronomical object can be hooked on. If we take a look at an empty region of space from above, we will see something like this:

The lines conforming the grid are the X and Y directions (two spatial dimensions). By looking from the top, I am intentionally removing the Z axis (just to simplify a bit the visualisation). Also, as we are looking at a fixed picture in a fixed moment of time, we are removing the temporal dimension. What happens when we place an object into this empty space? Just like when we through a marble over a piece of cloth, the cloth sinks down, space would also be “bent” by any celestial object:

In the region where the object (a planet, a star, a black hole) lies, the grid is modified: it suffers a curvature. This is what is commonly called curvature of space, and it is induced by the gravitational field which all objects generate. In the same way the piece of cloth sinks down a greater amount if we throw a bowling ball than if we just use a marble, space gets more curved around bigger (heavier) objects such as stars or black holes. We can picture this effect for the case of a black hole by using this drawing:

As you can see, the closer we get to the black hole (which is supposed to lie at the center of the diagram), space gets curved more and more. This is due to the great space curvature that the mass of the black hole generates. Just in the center, space folds over itself in what we usually call a singularity. This simple example serves as an easy way to visualize space curvature in two dimensions. However, this concept can be extended to the third spatial dimension and to the temporal dimension. In the case of time, this curvature makes time pass at different pace (it gets stretched or contracted, just as space) in the regions closer to heavy objects. This is exactly what happens in Miller’s Planet!

Okay so, now we know that Gargantua is bending space-time. What is the effect of this curvature? As I was saying before, time starts going slower than usual. However, this is not the only effect. As space gets deformed, the objects on it also suffer from this effect. Imagine that two spaceships fly close to a black hole, like in the previous scheme. Both ships start with parallel trajectories, but each one is placed at a different distance from the black hole:

You can see the trajectories in orange. As a visual guide, I have introduced the lines that conform the grid, and I also marked the spots where the ships trajectories cross over those lines with a cross:

As you can see, over time, both trajectories tend to differ and the ships go more and more distant (the crosses are further and further away from each other). If you have a funnel and a small marble around, you can do a simple experiment to see this effect: throw the marble inside at different positions, and you can see how sometimes the marble will go through the funnel, and sometimes it will escape. We could say that sometimes the marble gets trapped by the funnel’s curvature, and sometimes the marble escapes this curvature. In the case of our spaceships the effect is exactly the same, but changing the funnel by the space-time.

Imagine now that we have another two parallel trajectories, painted green:

Similarly to the previous scenario, both trajectories start deviating. However, now both start to get closer (the distance between crosses decreases).

This red and purple lines that I painted are called tendex lines (from the latin tendere, which means to stretch). They are imaginary lines which can be drawn escaping or orbiting around any object that bends the space-time, and show of the gravitational forces affect any object placed around. Purple lines, over time, make trajectories diverge. This means that there is a force that stretches objects in the surroundings of a black hole. On the other hand, the red lines make trajectories converge. On other words, there is a force that contracts objects close to the black hole.

So, what does this has to do with Interstellar? Well, both the gigantic waves and the time passing slowly are generated by these tendex lines coming from Gargantua.

Probably, most of you reading these lines are familiar with Earth tides. The classical explanation of this phenomenon is that the Moon, which orbits our planet, generates gravitational forces all over the globe. This affects our oceans, which get pulled as the satellite orbits, and the water level rises or lowers during the day. Let’s make a simple drawing to picture this effect.

This will be our planet, with its solid surface and the seas above:

Now, lets add the Moon. Due to its mass, there is an attractive force between the Moon and the Earth. This force pulls all the bodies inside the planet, and points towards the center of the Moon. Thus, water also gets pulled towards the Moon:

We can decompose all these green vectors into a sum of two components: one that is the same in all the points (pointing up), and one that is different in each point. Then, we can just remove the common part (if there is a constant force in all the points of the planet, we will never be able to feel that force). This will make the scheme look like this:

As water is fluid, these forces will easily deform the oceans, generating the tides we see in early in the morning or late in the afternoon, when the Moon gets closer or far from to the coast. However, in this last figure we see that there are two pairs of forces, one which stretches the planet in the vertical axis, and another that compresses it in the horizontal axis. Just as the tendex lines! If we consider the Moon as a body that deforms the space-time, we can picture its effects using this picture:

As I said before, the purple lines stretch the objects, and red ones compress them. It is clear then that both the classic theory of forces and the general theory of relativity provide the same effects. Then, why do we use relativity, which is quite harder to understand and formulate? The answer is quite simple: relativity also allows us to introduce the temporal stretches or compressions, which is something classical physics cannot explain. In Miller’s Planet, time flows at a different pace than on Earth. This is also caused by the tendex lines stretching the temporal dimension.

As I wrote at the start of the post, these space-time curvature effects are due to the gravitational fields of stellar bodies. The bigger the mass, the higher the gravitational field that an object generates, and thus it generates a stronger space-time deformation. The Moon, which is quite small, is able to generate Earth’s tides. If, instead of the Moon, we had a bigger object, tides would be much bigger. In the extreme case of a black hole (a supermassive object concentrated on a very small region), the space-time curvature is so big that the size of the tides would get enlarged by a huge amount, which could explain what we see on the film.

This is a wonderful example of how you can do very cool films while also respecting the science that we know (which is not the standard in Hollywood, I must say).

This blog post was published first in spanish, on cienciaoficcion. You can go there for a lot of cool content.

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