Exploring Time Inversion in Tenet

Judged by most to be a confusing, intellectual mess, Christopher Nolan- directed Tenet is a sci-fi action movie hardly understood by anyone watching for the first time. The movie has however received numerous accolades including Best Visual Effects at the 93rd Academy Awards. Tenet draws attention to the fascinating concept of two worldlines with opposite time orientation interacting with each other, and the physical implications are bewildering.

Introducing the Basics

So what do I mean with opposite time orientation? Basically, two frames (i.e, sets of all observers that have the same time orientation) are present simultaneously, but one has events unfolding backwards in the time of the other. From here onwards, let’s define forward time to be the time perceived by an observer in their own frame, which always runs forwards/normally. Consider the events 1, 2, 3 (in that order, at 1 second intervals) occurring in forward time of observer A. And in the forward time of observer B, who’s in inverted time with respect to A, observes events 4, 5, 6 (in this order, at 1 second intervals). Then, if events 1 and 6 occur simultaneously, A would observe 16, 25, 34. (numbers placed without separation occur simultaneously)

The Tenet achieves time inversion by sending characters through a turnstile, with an entrance and exit located in different spatial locations as observed by a normal person. As an observer enters the turnstile, their entire body is transferred to a frame inverted with respect to their original frame, one that has initial time as the moment such an inversion is made. (Concretely, the first thing the inverted observer sees is their past self walking backwards out of the turnstile.) Here’s when things get interesting: your inverted self, who holds all memories of your past, can in fact interact with your past self and have your past self perceive you, in inverted time. Both copies of you can exist at the same time, performing different actions.  But before we dive in further, I shall brief you with the plot of the movie.

Plot

The protagonist (who isn’t given a name in the movie so let’s refer to it as TP) is recruited into an organisation called Tenet to hunt down an “algorithm” that can invert all matter, which would instantly result in annihilation of all ordinary matter and thus end the world (more on this later).

The Algorithm was invented by a scientist from the future who, after realising its devastating effects, decided to invert its time and scatter its nine parts in different countries to be protected under the guise of radioactive material. TP and his team works against the main antagonist, Andrei Sator who, after discovering one of the buried parts as a teenager in his hometown Stalsk 12, takes on the task of unearthing and assembling the other eight, so that at the moment of his death, he shall reveal the location of the Algorithm to the future, making the world die with him. The psychological reasons behind Sator’s extreme egocentrism is not of interest to us here as we focus on the physics of the entire operation.

What happens, had already happened

As TP was introduced to his mission, he gradually grew acquainted with inverted objects and their behaviour, as in the following examples: 

1. TP pulls the trigger of an inverted gun and catches an inverted bullet from a bullet hole that reintegrates from an inverted wall.

2. TP holds his hand above an inverted bullet, after which nothing happens. Only after TP was told “you have to have dropped it”, did the bullet respond and fly upwards towards his hand.

   
   

Why does this sound so strange to us? It’s because the interactions of objects from different frames (inverted with respect to each other) don’t obey ordinary causal relationships when seen from either frame.  The movie justifies the inverted interactions by recording and playing such interactions in reverse: If the inverted frame can be transformed to an ordinary one, the interactions are justified.

The paradox is that an observer in each frame always has the “freedom” to shape their future (in contrast to their past, which is unchangeable) which corresponds to the inverted frame’s past. It sounds like a natural possibility to affect the inverted frame’s past so that your frame’s future no longer makes sense. For example, what if TP, as he watches the bullet ascend from the table, suddenly decides to back away so that he never catches the bullet? This would break the laws of physics in the frame of the bullet. A more famous example of this phenomenon is the grandfather paradox: What if you travelled to the past to kill your grandfather when he was born? Would you have even existed to perform the time travelling?

The movie provides an obscure yet straightforward answer:”What happens, had already happened”. The key point is that both the ordinary and inverted frames are equally important: One frame's future is interchangeably the other frame’s past which, if perceived as unchangeable from one frame, should be just as unchangeable as perceived in the other frame. And so to answer the grandfather paradox: It’s simply impossible for you to have killed your grandfather, because the time travelling you is built from a past that already includes all your “future” interactions with your grandfather in his past.

Such an explanation necessitates determinism in the region of spacetime where frames inverted with respect to each other are able to influence each other, leading to the assumption that every fundamental building block of the world has a single, determined trajectory through space time given their initial conditions. Because every object’s future is encoded in its past.

But what can be said about regions of spacetime where no two such frames intersect? Such a region can definitely exist, as all one has to do is to prevent every inverted object’s timeline from penetrating it, by inverting them again. In such a region, there is no longer a symmetry criterion that restricts the past and future of its frame to be on equal footing. Would there then be a new set of physical laws that are indeterministic? Most likely, no.

This seems at odds with the assumptions of quantum mechanics, which states that identically prepared systems produce results that follow a fixed probability distribution. Moreover, it is at odds with the experimental violations of Bell’s inequality, which states that not even such probability distributions are held fixed (it experimentally disproves any local hidden variable theory, the assumption that in a system composed of spacelike-separated (i.e statistically independent, since they are separated by a distance too far to be connected by light in the time between measuring them each) subsystems, there is a hidden variable that remains constant for all such identical systems which dictates the probability distribution of the value of some quantity each of the subsystems have upon measurement). Bell’s inequality is a trivial mathematical identity that must hold for any such variable that exists, yet it was violated by experiments over and over.)

But to give the movie the benefit of the doubt, if we consider each moment in Tenet to be irreproducible and unique, such that it is impossible to have two or identically prepared systems to even speak of “a probability distribution”, then the notion of time inversion would avoid conflicting with quantum mechanics. Essentially, the world of Tenet is like recording a part of our world and then replaying it on a video screen, a movie.

Entropy Forwards and Backwards

Tenet plays a great deal with the second law of thermodynamics: Entropy/disorder of the universe always either stays unchanged or increases. The mathematical definition of entropy would be  S=klnΩ , where k is Boltzmann’s constant and Ω the number of microstates in which a system can produce a fixed set of measurable properties (energy, for example). What increases in forward time decreases in inverted time, hence time inversion “reverts entropy”. 

However, the reverting of entropy seems like quite an ill-defined concept for those who live in this movie. (fortunately for us, determinism is omnipotent at explaining any physical inconsistency in this department). The idea of a never-decreasing entropy stems from the postulate that all micro configurations of a system are equally likely to occur, and the probability that a system ends up with certain properties is directly proportional to the number of micro configurations the system with such properties can be found in.  The larger the number of microconfigurations, the greater the entropy. Entropy then increases because the system continuously makes random transformations  between micro configurations, such that the chances of finding it in a disordered state is exponentially larger than finding it in a less disordered one, effectively dominating the probability distribution after infinitely many transformations.

What’s so problematic is that statistical laws such as this rely on probability, which is the only tool people in Tenet have at their disposal, despite them living in a deterministic world.

To see the limitations of such a tool: micro configurations come in the form of matter occupying presumed identical microstates (imagine sorting coins into identical containers), which could be small regions of space, for example. And while in forward time, increasing entropy would force all such microstates to be occupied symmetrically, reversing the time would mean that this symmetry must be lost. Just like how a gas expands to fill a container in forward time, but in inverted time, it would somehow shrink into a smaller region of the container, which is completely identical to all other regions of the container. Why then would the gas choose that region and not the others? -Because the gas was placed there, of course. But observers of an inverted gas do not generally know that. And this is why inverted actions in Tenet look so strange to us: Why do inverted objects dropped on the floor magically fly into uninverted people’s hands and not anywhere else, or stay motionless? It is simply because the uninverted person had created the initial conditions necessary for their inverted hand to “have dropped” the object, which makes the state of “catching the object” the only possible outcome out of seemingly identical states.

Matter-Antimatter Annihilation

So why does inverting all matter end the world? The answer lies in the famous CPT symmetry of quantum field theory (theory of elementary particles), which has so far no instance of being violated in experiments. C, P, T stands for three different actions one performs on the universe: Charge conjugation (flipping signs of all charges), Parity (flipping signs of all spatial coordinates) and Time reversal (flipping the sign of the time coordinate). All actions performed together leaves nature invariant (basically, nothing changes). However, this also means that if we want a universe in inverted time that looks exactly the same as a non-inverted one, that universe must contain only matter that is related to non-inverted matter through a CP transformation (since T has no effect on properties of matter).

Here's the catch: matter that are related by charge conjugation annihilate each other and produce energy in the form of light. (a famous example being electron-positron annihilation), which is basically the mechanism by which the world ends. Now for the inverted world to contain matter that are charge conjugates of ordinary matter, this matter must be invariant under parity-which generally is the case for elementary particles that undergo strong and electromagnetic interactions (not weak ones, they don't even preserve CP). So the point of Tenet holds quite well in this aspect.

The only plot hole, however, is that the movie seems to assume that matter and inverted matter only annihilate their "exact counterparts" : that i, you only annihilate with your inverted self, a typewriter only annihilates with the same typewriter that is sent through the turnstile, etc. In quantum physics, it's a well established concept that all fundamental constituents of objects, the elementary particles, are identical. So an electron would annihilate a positron, irrespective of whether the positron is from a typewriter or a person.

Now that basic movie mechanics are explained, have fun watching Tenet!