# What is Time?

### Harys Dalvi

### April 2022

This thing all things devours:

Birds, beasts, trees, flowers;

Gnaws iron, bites steel;

Grinds hard stones to meal;

Slays king, ruins town;

And beats high mountain down.

— J. R. R. Tolkien (The Hobbit)

Everyone has some idea of what time is. But philosophers have written about the matter for thousands of years, and probably argued about it for many more. Although everybody experiences time, nobody fully understands what it is.

In this post I “will” talk about the concept of time in three branches of physics: these are astronomy, thermodynamics, and relativity in “chronological order”, whatever that means. “After that” I'll try my best to put together a foggy picture of time as a whole and its practical consequences.

## Astronomy: Time is a Circle (or an Ellipse)

When ancient people observed the sky, they saw the sun, moon and planets all follow predictable orbits around the earth. They harvested the same crops at the same time of year every year. This formed the basis of the cycles of days, months, and years in calendars.

Astronomy changed this picture of the universe,
as we found out that in fact the
earth goes around the sun, not the other way around.
But the lengths of days, months, and years were unchanged.
Planets made periodic elliptical
orbits that took the same amount of time every time.
Overall, astronomy suggests a **cyclical view of time**,
where **time had no beginning and will have no end**
— it's all endless cycles.
The Jains [2]
and the Mayans [6]
were among those that adopted this view in their cosmologies.
(Yes, 2012 was just the benign end of a Mayan cycle,
not the end of the world.)
Aristotle also argued that time has no beginning or end
[1].

But there must be more than this view: while the cycles repeat, not all cycles are identical. Your age one year is not the same as your age the next, even though the earth traces the same path around the sun according to Kepler's laws.

There's another problem which is more subtle. On one side here is an animation of a ball being thrown up, following a parabola shape according to the laws of physics, and then falling down. On the other is the same animation played backwards. Can you tell which is which?

It's impossible to tell which animation is going forward in time
and which is going backwards. It just looks like the two are
mirror images of each other. It seems like **the laws of
physics are the same forwards and backwards!** [3]
The same applies to astronomy and planetary orbits: while the orbits
and other cycles
allows us to *measure* time, they tell us nothing about
*which way* time flows. (By the way, the modern definition
of the second is based on another periodic event: a
Cesium atomic clock. [9])

## Thermodynamics: Time is a Straight Line

Here's an animation of helium (blue) and neon (orange) atoms just after a wall that was separating them is removed. You can play the animation again to see it backwards. Can you tell which is backwards and which is forwards?

Of course, the one where the two gases mix together is going forwards in time, and the one where they magically get unmixed is reversed. You can mix eggs into a cake batter, but it's hard to unmix them back out.

This involves the concept of **entropy**
in thermodynamics. Loosely, entropy represents the
amount of disorder in a system. It's clear that the entropy
increases as the two gases get mixed, while the decrease by
unmixing wouldn't happen. The famous
**second law of thermodynamics**
can be summarized as *the total entropy of the universe
can never decrease with time.* \(\frac{dS}{dt} \geq 0\).
[4]

If you watch the backwards animation carefully, you'll notice
that the atoms bounce off the walls exactly as you would
expect with normal physics. Nothing seems to violate
the laws of physics, and yet the unmixing is obviously
impossible. That's because the second law of thermodynamics
is one of the few physical laws that is *not* the
same forwards and backwards, so it's often used to
understand the **arrow of time**: the fact that
time seems to have a direction. This idea also has ancient
precedent: the Bible starts with “In the beginning”.

In physics, **as time goes on, entropy increases**.
The driving force for this is probability: if given the chance,
it's much more likely for a system to take a disordered state
than one of the few ordered states. (There are many ways to
arrange the digits 123456789, but only one is actually 123456789.)
We can define
**two distinct directions in time**: the direction of increasing entropy
and the direction of decreasing entropy. No matter where you are, you
can agree on these two directions. This isn't true of
space: if something is in front of you,
you can just turn around and now it's behind you.

So, problem solved: we move forwards in time towards increasing entropy, and this explains the arrow of time and the beginning of the universe. Is that all?

Not quite. There are questions remaining with this view, too. For example, let's say time just stopped for a year. Nothing moved, nobody thought anything, entropy didn't change, absolutely nothing happened for a year. Does that idea even make sense? Does time exist without stuff to happen? Also, the physics shows that entropy increases in one direction along time, and decreases along the other. But why do we feel like we are moving in time towards the side with more entropy, and not the other way? [1] Is anything moving the other way? Like we saw in the ball animation and even the gas animation, you could play everything backwards and it wouldn't violate basic physics.

Understanding the two directions of time is hard, and there are contradictory
ways to do it.
We think we're walking forwards into the future, but Marty McFly went
Back to the Future. He's not the only one to think this way:
the Aymara people of the Andes consider the future to be behind them,
and the past to be ahead [5]. This seems
crazy at first, but it actually makes a lot of sense. We can see the
past in front of us, because it has already happened. But we have no
idea about the future behind us. We aren't confidently marching forward into the future;
we are constantly walking *backwards* into a future we can't see,
hoping we don't trip. The entropy view of time doesn't solve this
problem: why can we see the past but not the future? Why do past events
seem to affect future events rather than the other way around?

## Relativity: Time is a Diagonal Line

Imagine you're sitting in your chair and not moving. (Maybe you don't
have to imagine.) But you *are* moving; you're moving forward through time,
even if you're not physically moving. Relativity takes this metaphor
of “moving” through time and makes it very literal,
leading to some interesting counterintuitive consequences.

You are definitely moving through time, but how fast? Apparently,
just sitting in a chair,
**you are moving forward through time at the speed of light.**
In fact, ** everything is moving at the speed of light,
all the time.** The question is, which way is it moving?
Only in time, or also in space?

There's a famous thought experiment about twins: imagine you have a twin who goes on a trip at 96% the speed of light. To your surprise, your twin returns after 25 years, but has aged just 7 years: you are now older than your twin. How is this possible?

It's not anti-aging creams in space: this has an explanation
in physics using the idea that everything is traveling at the speed of
light through spacetime. What is your twin's **proper time** interval —
in other words, how much time did your twin experience?

In relativity, it turns out that instead of the Pythagorean \(\sqrt{t^2+x^2}\),
we can think of spacetime distance with \(\sqrt{t^2-x^2}\). (It's actually a little more
complicated than that, but this is using the absolute value. [7])
We can put you and your twin on a **Minkowski diagram** [8].
You don't move in space (no change in \(x\)) while your twin does,
and covers a *shorter* absolute distance in spacetime because of
that \(\sqrt{t^2-x^2}\). Because of the shorter distance through
spacetime at the same speed (the speed of light), in a sense your twin takes less time
to meet up with you after the trip... even though you meet
at the same time.

Now things get confusing, so I'll make a table.
Remember: distance equals speed times time,
and we're letting the speed of light be 1 for simplicity.
Your twin is going through space at 96% the speed of light,
but *everything* goes at the speed of light
through spacetime. \(t\) is the amount of time that
the trip takes according to your clock, not your twin's.

You | Twin | |
---|---|---|

Distance through space \(d_s=v_st\) | \(d_s=v_st=0t=0\) | \(d_s=v_st=0.96t\) |

Distance through time (on your clock) | \(d_t=t\) | \(d_t=t\) |

Distance through spacetime \(d_{st}=\sqrt{d_t^2-d_s^2}\) | \(d_{st} = \sqrt{t^2-0^2} =t\) | \(d_{st} = \sqrt{t^2-(0.96t)^2} =0.28t\) |

Speed through spacetime | \(c=1\) (speed of light) | \(c=1\) (speed of light) |

“Time through spacetime” (proper time, time experienced) | \(\tau=d/v \\ d=t, \ v=c=1 \\ \tau=t\) | \(\tau=d/v \\ d=0.28t, \ v=c=1 \\ \tau=0.28t\) |

So your twin only aged by 28% as much as you during the trip. On the Minkowski diagram, your path through spacetime goes in a straight line. Your twin's path goes in diagonal lines: some of it towards time, and some towards space. Since your twin was moving through space so quickly, there was hardly any speed available to move through time. Your twin's proper time goes at a different rate from your time, but both always go forwards — never backwards. This is the “diagonal lines” picture of time painted by special relativity. As strange as it is, it's been backed up by rigorous experiments.

Back to that equation for distance through spacetime, \(d=\sqrt{t^2-x^2}\). If you get to the speed of light, \(x/t=c\), your path through spacetime gets zero length. This means that if you were a photon, you would never age or experience time. In a sense, light has managed to stop time.

When we say “spacetime”, it's almost like space and time are parts of the same thing. But in this equation, they are just a little different: they have opposite signs. We know up and left are part of the same space because you have the Pythagorean theorem \(c=\sqrt{a^2+b^2}\). What's a negative doing in a distance equation? Does it give clues about the nature of time? I'm not sure, but one thing that's clear is that time and space are not exactly the same thing, even in relativity where you can talk about spacetime.

## Conclusion: Time is Probably a 15-Dimensional Riemannian Manifold Embedded in an \(i \sqrt{277}\)-Dimensional Space that Nobody Understands

Philosophers and now physicists have studied the nature of time for... well, a long “time”. And through it all, time is as obvious as it is incomprehensible. Many questions are unanswered, and I've only mentioned a few here. To recap:

- Does it make sense for all time to stop for a year? Is time a container for events, or a relationship between events?
- Entropy helps define an arrow of time, but why does the arrow point one way and not the other? Why is it the past (lower entropy) that affects and flows into the future (higher entropy)?
- We use space metaphors (forward, backward) for time, and relativity unites space and time in spacetime, yet space and time seem different. How are they related?

*have*answered:

- Many natural phenomena are periodic over time, which lets us measure time.
- There is an arrow of time: the future and the past are different.
- Space and time are closely related, and both are relative, but they are not the same thing.

More broadly, time shows how difficult it can be for philosophy to
reach a consensus on difficult topics, even though time is something we all experience “all the time”.
There are a lot of unanswered questions and different viewpoints on this
everyday but confusing topic.
Maybe it's fine if we apply thermodynamic equations
without agreeing on the philosophy behind the arrow of time — Florida needs its AC. But when it comes to areas like
AI safety and ethics, a philosophical consensus becomes important. If we still don't agree on
something as seemingly simple as
what it means for spring to come *after* winter, there's no guarantee that we'll
agree on whether it's ethical to develop a computer as smart as a human or smarter.
There are probably still philosophical challenges we haven't foreseen.
It will take “time” to work out how to deal with these issues and others —
but if we succeed in reaching a philosophical consensus and managing risks,
we could have a bright future behind us.

## References

- Time (Stanford Encyclopedia of Philosophy) ^
- Jainism (Encyclopædia Britannica) ^
- The Distinction of Past and Future (Richard Feynman, Cornell University) ^
- Second Law of Thermodynamics (Glenn Research Center, NASA) ^
- For the Aymara, Future Is Then (NPR) ^
- Pre-Columbian Civilizations/Cosmology (Encyclopædia Britannica) ^
- Special Relativity and Flat Spacetime (Sean M. Carrol, University of Chicago) ^
- Minkowski Diagrams (Thomas Banchoff, Brown University) ^
- The International System of Units (2019) ^