A Thought-Provoking Inquiry

Recently, I received a thought-provoking letter from a reader named Luke Copko. He posed an intriguing scenario:

“…imagine if we could send a spaceship hurtling through space at a speed just shy of light—say, light minus 5 mph. If someone were to sprint from the back to the front of the ship at 10 mph, wouldn't that mean they’ve surpassed the speed of light?”

This notion suggests a breach of the light barrier, which is strictly prohibited by the principles of special relativity. To illustrate, let's have Luke observe the situation from Earth, while another observer, Leia, is aboard the spaceship. The question arises: Would Anakin, the runner, be perceived as moving faster than light by Leia?

From Luke's perspective, it seems intuitive to combine their speeds, implying Anakin would be moving away from Luke at a pace exceeding light speed. However, special relativity tells us otherwise.

So, how do we reconcile this apparent contradiction?

Next, consider Leia’s point of view. She would see Anakin running at 10 mph while Luke is moving away at 5 mph less than the speed of light. This would imply that they’re separating at a speed greater than light, which raises the question: why does special relativity permit this?

In fact, this situation is allowed under the rules of special relativity. Let’s explore why one scenario is permissible while the other is not.

The Core Principles of Relativity

The framework of special relativity is built on two fundamental rules:

1. All inertial reference frames are equivalent.

   As long as the spaceship travels in a straight line at a consistent speed, it qualifies as an inertial reference frame. Leia measures Anakin's speed at 10 mph, which is valid. Likewise, she notes Luke's speed as 5 mph less than the speed of light.

2. The speed of light remains constant across all inertial frames.

   If Anakin uses a laser to send a pulse, he will observe it traveling at 186,000 miles per second. It would be easy to assume that Leia would measure it at an additional 10 mph, or that Luke would see it at a speed exceeding light. However, all observers measure the pulse at 186,000 miles per second.

This second principle can be challenging to grasp. The theoretical foundation comes from Maxwell's equations, while experimental evidence supports it. We will not delve deeply into that here, but we will discuss the implications of the second rule.

Imagine the spaceship is moving at ⅘ the speed of light, and Anakin runs at ⅖ the speed of light. We denote V as his speed with respect to Luke. What can we anticipate?

As previously discussed, motion influences time and distance. What impact does it have on Anakin's velocity?

To calculate V, we will establish an event that both Luke (on Earth) and Leia (on the spaceship) can agree upon. The complications arise when comparing events that are separated in time or space.

At the moment Anakin begins his sprint, he activates a light pulse from his laser. The light pulse travels to the far end of the spaceship and reflects back to him. When it reaches him, he collapses on the spot. (Don't worry—he'll be fine.)

Where is Anakin?

In Leia's frame of reference, the thick arrow indicates Anakin's movement, while the thin arrows represent the light beams. The light travels significantly further than Anakin does. How far does he make it? Halfway across the ship? A quarter? Regardless of his position, Luke and Leia will concur.

We will compute this distance for both frames of reference, then compare the two results to see what transpires.

The speed of light serves as our standard unit. We can define the distance traveled by the light in Leia's frame as our unit length. If Anakin runs at ⅖ of light speed, the length of the thick arrow is ⅖. How long is the spaceship? The combined length of the three arrows is 1⅖, making the spaceship's length half that total:

What fraction of the ship's length did Anakin cover? We take his distance and divide it by the ship's length.

Anakin reaches just over halfway across the ship. If we marked the floor into sevenths, he would collapse at the fourth mark. This will not be disputed by either Luke or Leia.

Luke's Observations

How does this scenario unfold for Luke?

We need not make assumptions about how the times, distances, and velocities in Leia's frame relate to those in Luke’s frame. However, they must agree on two measurements:

• The speed of light, c
• The proportion of the ship’s length traversed by Anakin (4/7 in our example)

Luke measures Anakin's running speed, V, which intuition suggests must be 1⅕ times light speed. He also measures the laser pulse traveling at the speed of light. Finally, he observes the spaceship moving at ⅘ the speed of light.

Initially, the laser travels to the far end of the spaceship. It has a slightly longer distance to cover since the ship is in motion. Anakin has run ⅖ of that distance. Due to the ship's advancement, he is actually less than ⅖ of the way across.

At the conclusion of the journey, Anakin has moved a distance of V·tᵤ with respect to Luke, where tᵤ is the time measured by Luke.

Although Anakin travels a distance of V·tᵤ, the ship has also moved a distance of u·tᵤ, where u represents the speed of the ship. This determines how far Anakin is from the back of the ship.

In our scenario, the speed of the ship, u, is ⅘ light speed. Our goal is to find V, and eventually, the time variable tᵤ will cancel out. Once we have the length of the ship, we aim to show:

This is the distance along the ship divided by the length of the ship, as perceived by Luke.

In the subsequent calculations, it might be beneficial to print out a diagram of Luke's reference frame for clarity as you follow the reasoning.

Anakin’s Position in Both Frames

Recall how we calculated this for Leia's reference frame. She measures Anakin's speed as ⅖ light speed. We can generalize the previous calculation by representing Anakin’s speed with the variable v.

We need an expression for the length of the ship in Luke's frame. This is the distance the light travels to the end of the ship, minus the distance the ship travels during that time.

The time, t₁, poses a challenge. To ensure the times ultimately cancel out, we want only one time variable, tᵤ. First, we express t₁ in terms of tᵤ.

Next, we utilize the fact that tᵤ = t₁ + t₂ to derive an expression for t₁.

We substitute our t₁ into the previous expression for Lᵤ and simplify.

We divide the distance Anakin travels across the ship by the ship's length. This yields a fraction representing how much of the ship Anakin traversed.

That proportion is independent of the frame of reference.

How Fast is Anakin Moving Away from Luke?

Now we are left with the task of isolating V, which will provide the desired expression for Anakin's speed in relation to Luke.

Cross-multiply the numerators and denominators.

Expand the terms.

Bring the V terms to one side and cancel equivalent terms.

Isolate V.

Finally, simplify the result by dividing the numerator and denominator by c².

If we consistently express velocities as fractions of light speed, c = 1. We can remove c from the equation.

By plugging in the values from our example, we find that in Luke's frame, Anakin is moving at approximately 90% of the speed of light.

We do add velocities. However, we then apply a correction, dividing by 1 + uv. When u and v are small (which they typically are), uv is nearly zero, making the adjustment negligible.

We must exercise caution when mixing our reference frames. The distance between Luke and Anakin increases at 1⅕ light speed from Leia's perspective, but not in relation to one another.

What about the relationship between Luke and Anakin from Leia's viewpoint? Why no adjustment? In Leia’s frame, the distance between Luke and Anakin increases at 1⅕ light speed: Anakin at ⅖ to her right and Luke at ⅘ to her left.

As we navigate these frames of reference, we must remain vigilant. The distance between Luke and Anakin is increasing at 1⅕ light speed from Leia’s perspective, but they do not exceed the speed of light in any frame.

If Luke sends a light pulse toward Anakin, it will catch up with him. According to Luke, Anakin is moving away at approximately 90% light speed. Leia will observe the pulse approaching her at light speed, passing her, and eventually overtaking Anakin, who will also see it approach at light speed.

In summary, when we discuss the distance increasing at 1⅕ light speed, it’s crucial to remember that distance itself doesn’t carry signals faster than light. Careful selection of our frame of reference is vital. No observer measures speeds exceeding light in any frame.

All is well. The universe continues in its beautifully curved space-time.