Physics on kenji.blog

Is a flying arrow at rest?: Zeno's Arrow Paradox

Thu, 10 Sep 2026 21:00:00 +0900

An arrow shot from a bow is flying through the sky. This arrow is certainly moving. However, Zeno, a Greek philosopher from the 5th century BC, developed the following terrifying logic.

“The flying arrow is actually at rest.”

This is neither a joke nor sophistry, but the “Arrow Paradox” that mathematicians and philosophers have seriously debated for 2500 years.

Zeno’s Argument

Zeno’s argument starts with the concept of a “moment in time.”

Time is a sequence of “moments.”
If you capture a single moment (a point where the length of time is zero) like a “photograph,” the arrow “is” at a specific point in space.
At that moment, the arrow is merely “occupying” that space and is not moving. (If it were moving, it would require a “duration of time” rather than a “moment.”)
This holds true no matter which moment you pick out.
If the arrow is at rest at every moment in time, when does it move?

graph TD A["Flying arrow"] --> B["Time is a continuous sequence of moments"] B --> C["Moment t1: Arrow is at rest at position A"] B --> D["Moment t2: Arrow is at rest at position B"] B --> E["Moment t3: Arrow is at rest at position C"] C --> F{"At every moment, the arrow is at rest"} D --> F E --> F F --> G["Conclusion: The arrow is not moving!"] style A fill:#2196F3,color:#fff style F fill:#FF9800,color:#fff,stroke-width:2px style G fill:#F44336,color:#fff,stroke-width:3px

Intuition vs. Logic

“Ridiculous. The arrow is actually flying, isn’t it?” This is probably most people’s first reaction. However, it is actually extremely difficult to point out logically where Zeno’s argument is flawed.

In fact, the ancient Greek philosopher Diogenes is said to have simply stood up and walked around the room in response to Zeno, showing him, “Look, it is moving.” However, this does not constitute a refutation of Zeno’s logic. What Zeno is questioning is not “whether it can move,” but rather, “can we logically explain what it means to move without contradiction?”

Resolution by Calculus (An Attempt)

Calculus, invented by Newton and Leibniz in the 17th century, provided a mathematical answer (at least partially) to this paradox.

In calculus, “the velocity at a certain moment (instantaneous velocity)” is defined as follows:

$$ v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} $$

In other words, velocity is defined as the “limit” where the change in position $\Delta x$ is divided by the change in time $\Delta t$, as $\Delta t$ approaches infinitely close to zero.

The point here is that “instantaneous velocity” is not the distance traveled within a zero-time duration. It is a quantity defined as the “tendency,” or “limit,” of minute changes before and after that time.

Therefore, the answer from the perspective of calculus is as follows:

“Indeed, if you capture a moment of zero length, the arrow has not moved ‘within’ that moment. However, even at that moment, the arrow has the property of ‘instantaneous velocity (a non-zero limit value).’ Being ‘at rest’ means having an ‘instantaneous velocity of zero,’ but since the instantaneous velocity of a flying arrow is not zero, the arrow cannot be said to be ‘at rest.’”

Remaining Philosophical Questions

While calculus provided a practical solution to Zeno’s paradox, it hasn’t completely settled the philosophical debate.

The concept of a “limit” is strictly a mathematical tool (calculation procedure), and it does not rigorously answer fundamental questions such as “what physically is the smallest unit of time (moment),” “what is continuity,” and “what is the essence of motion.”

In modern physics (quantum mechanics), the possibility that minimum units for time and space (Planck time, Planck length) exist is discussed. If time is not “continuous” but “discrete (digital),” Zeno’s paradox might need to be re-evaluated in a completely different context.

Even after 2500 years, Zeno’s arrow continues to ask us, “What does it mean to move?” and “What is time?”

Returning from Space Younger than Your Brother? The Twin Paradox

Thu, 10 Sep 2026 21:00:00 +0900

If you were to travel in a spaceship flying at a speed close to the speed of light, your clock would run “slower” than the clocks of the people on Earth. This is not a sci-fi movie setting, but a fact of physics proven by Einstein’s “Special Theory of Relativity”.

The most dramatic expression of this concept of “time dilation” is the famous thought experiment known as the “Twin Paradox”.

The Brother Who Goes to Space, and the Brother Who Stays on Earth

There are twin brothers born on the same day. On their 20th birthday, the older brother boards an ultra-high-speed rocket flying at 80% the speed of light ($0.8c$) and departs for a distant star. The younger brother stays on Earth and waits for his brother’s return.

Years later, the older brother returns to Earth. When the rocket doors open and the two reunite, something astonishing has happened.

The younger brother who waited on Earth has become a full-grown middle-aged man at 50 years old (30 years passed), whereas the older brother who traveled through space is still youthful at 38 years old (18 years passed).

“Even though they are twins, an age difference of 12 years has been created.” This is the first shock brought about by the theory of relativity.

graph TD A["Twin brothers (20 years old)"] --> B["Younger brother remaining on Earth"] A --> C["Older brother traveling space at 80% light speed"] B -->|30 Earth years pass| D["Younger brother at reunion: 50 years old"] C -->|Time dilates due to relativity, only 18 years pass| E["Older brother at reunion: 38 years old"] D --> F{"Age difference: 12 years!"} E --> F style A fill:#ECEFF1,stroke:#333 style B fill:#C8E6C9,stroke:#333 style C fill:#BBDEFB,stroke:#333 style D fill:#81C784,stroke:#333,color:#fff style E fill:#64B5F6,stroke:#333,color:#fff style F fill:#FF9800,stroke:#333,color:#fff,stroke-width:2px

The Core of the Paradox: Does It Change Depending on Who Is Looking?

The fact that “the older brother becomes younger” itself is a fact that can be derived by applying numbers to the equations of the theory of relativity (Lorentz factor), and is a physical phenomenon that is actually taken into account in modern GPS satellites (sometimes referred to as the Urashima effect).

However, the true “paradox” begins here. One of the most important rules of the theory of relativity is that “the laws of physics are exactly the same for every observer moving at a constant speed (absolute rest does not exist)”.

Applying this to our case creates a bizarre contradiction.

From the perspective of the younger brother on Earth: “The rocket carrying my older brother zoomed away at breakneck speed and then came back. Since my brother was the one moving, his time should slow down, and he should be younger.”
From the perspective of the older brother in the rocket: “I am stationary inside the rocket. Looking out the window, Earth zoomed away at breakneck speed and then came back. Since the younger brother on Earth was the one moving, his time should slow down, and he should be younger.”

Both claims are faithful to the relativity principle that “if the other person appears to be moving, their time slows down.” However, when the two reunite and stand side by side, a state where “both are younger than the other” is impossible. One must definitely be older and the other younger.

Does this mean Einstein’s theory is wrong?

Resolution: The Breakdown of “Symmetry”

The key to solving this paradox lies in the fact that “the positions of the two are not completely equal (symmetric)”.

The younger brother remained on Earth (an inertial frame: a state of moving at a constant speed or being at rest) the whole time. However, the older brother’s journey involves “acceleration” and “deceleration”.

The older brother’s spaceship cannot return to Earth without taking the following steps:

Depart Earth and accelerate.
Hit the brakes at the destination star (decelerate), change direction towards Earth, and accelerate again (make a U-turn).
Arrive at Earth and hit the brakes (decelerate).

In the theory of relativity, an observer who experiences acceleration (feels G-force) is treated differently (in the realm of General Relativity) than an observer moving at a constant speed.

In particular, the moment the older brother makes a “U-turn (a change of direction through intense acceleration)” at the destination star, the symmetry between the brothers’ positions completely breaks down. At the moment the older brother makes a U-turn and re-accelerates toward Earth, the “Earth’s clock (the younger brother’s age)” as seen by the older brother is observed to rapidly leap forward by decades all at once.

As a result, when they reunite, exactly as calculated, only the reality remains that “the older brother is 38 and the younger brother is 50,” cleanly resolving the contradiction.

The Twin Paradox is one of the most beautiful thought experiments in the history of physics, teaching us that our commonsense perception that “time flows equally for everyone” is completely inapplicable in the face of the vast universe and the speed of light.