Imagine you see a broken egg on the sidewalk and you're asked to bet what happened before (-ve time dimension) and after (+ve time dimension) it. What would you bet on when it comes to answering what happened to broken egg before and after you saw it? For the positive time dimension, your bet won't be confident because there are many possibilities for what happens to the broken egg: - A dog can eat it - Someone cleans it - Someone steps on it and slips - It remains there, getting biodegraded over time For the negative time dimension, however, you can confidently bet that the egg was not broken (before it got broken). This is essentially the problem of the arrow of time: why are we more confident about the past and less confident about the future? Why future is unknown but past is known? #### Laws of physics are deterministic In laws of physics, there is no direction of time to be found. What this means is this: suppose you need to predict what happens at time $t$ of a system. You can either start with the starting conditions of the system and evolve it in forward dimension of time, or you can start with the end conditions of the system and evolve it backward in time. Hence, for a given state, we should be able to calculate forward or backward trajectories in time dimension with equal confidence. If we're able to do so, the arrow of time would disappear: given an event and its full knowledge (a broken egg), we should be able to "remember" both its future and its past as it's just a matter of rolling of time in forward or backward direction. However, that's not what happens. #### We never have full knowledge of the system In reality, we never have full knowledge of the system. For a broken egg, all we know it is broken and nothing about its constituent atoms, molecules. Further, to really we able to move the time trajectory forward or backward, we need information not just about the broken egg but the entire system which includes the universe (containing things it could have interacted with or can interact with). If we had precise knowledge of entire microstate of the universe, rolling forward or backward in time will give us _one_, unique deterministic trajectory. However, given we only know macrostate of the part of the universe (a broken egg and nothing else), we're forced to consider many trajectories in future or past. ![[Screenshot 2021-09-29 at 6.40.55 PM.png]] Given multiple trajectories, we should be equally uncertain about past or future. A broken egg could have happened any which way and similarly a broken egg could lead to anything. A universe with maximum entropy will exhibit such symmetry in our confusion and hence lead to disappearance of arrow of time. (This is because given that we only known broken egg exists, any number of different microstates in the past could have (randomly, as far as our knowledge is concerned) could have converged to the same macrostate.) #### Selecting a unique trajectory in the past But we don't live in a universe of maximum entropy. We know that a broken egg is likely to have happened from a previously unbroken egg. **We picked / betted on a single trajectory in the past (for the broken egg) because we know past (negative $t$ dimension) in general had lower entropy.** Because we know that in the negative $t$ dimension, trajectories evolve from being more ordered (less entropy) to more disordered (higher entropy), among multiple possible past trajectories, we can _choose_ a trajectory in the past that had lower entropy than the broken egg (which is an unbroken egg). This is why we don't choose a trajectory in the past where an omlette spontaneously (or even with someone's help) became a broken egg (even though physically it's possible). ![[Screenshot 2021-09-29 at 6.51.54 PM.png]] #### Why was entropy low in the past **Nobody knows why the entropy at the Big Bang was so low**. It didn't have to be that way. But because it was low, the universe was in a highly ordered state. It's a logical outcome then that the highly ordered state will become less ordered with more interactions happening over time. However, we can use a type of anthropic argument here. Biological creatures (like us) exist _only_ because we can build predictive models of the environment around us in order to exploit it for sustenance. Such predictive models can only be built _if_ there exists predictable trajectories in time. And no predictable trajectories exist if there's no gradient of entropy. To see this, imagine what our universe will be like after its heat death. The macrostate of the universe will not change over time as the universe will be at maximum entropy, which means all microstates when measured will give the same macrostate (say temperature). How will a biological creature predict anything if the value that needs to be predicted never changes? (i.e. macrostate remains the same) Biological creatures on Earth ultimately depend on low entropy energy from Sun to be able to remain ordered themselves, and to remain ordered they need to encode a predictor (to get away from harm before it happens) and for a predictor to function, there needs to be something that needs to be predicted. Hence, **biological creatures like us can only exist in a universe where there's a gradient of entropy to be able to do successful predictions**. These biological creatures (in a timeless universe) simply choose to call the dimension where they have more confidence in taking a bet as past and the dimension where they have less confidence as the future.