# The Elusive Nature of Light Speed in Special Relativity

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## The Unattainable Speed of Light

How does special relativity add velocities? Welcome to the world where 1 + 1 is not quite 2.

### A Letter of Inquiry

Recently, I received a letter posing questions about special relativity. Here’s a snippet from that correspondence:

> …let’s say we hypothetically could send a spaceship through space at light speed minus 5 mph. Could a person not run from the back to the front of the spaceship at 10 mph? — Luke Copko

The underlying suggestion is that the runner would surpass the light barrier, which contradicts the principles of special relativity.

Picture an observer, whom we’ll call *Luke*, stationed on Earth, observing the entire situation. Meanwhile, another observer, *Leia*, is onboard the ship, monitoring the dilithium. Wouldn't the runner, named *Anakin*, appear to be moving at a speed exceeding that of light?

From Luke's perspective, one might expect the following:

Our intuition suggests that velocities should simply add together, indicating Anakin is moving away from Luke at a speed greater than light. However, special relativity says no.

How can we clarify this seeming violation of relativistic norms?

Now, consider Leia's perspective. Wouldn’t she see Anakin moving at 10 mph in one direction, while Luke moves away at 5 mph, which is just short of light speed? This combination also appears to exceed the universal speed limit.

Interestingly, this scenario is permissible. Why is one case acceptable while the other is prohibited? What makes special relativity seem so capricious?

### The Two Commandments of Relativity

Special relativity is governed by two fundamental rules:

**All inertial reference frames are equivalent.**As long as the spaceship is moving in a straight line at a constant speed, it qualifies as an inertial reference frame. Leia observes Anakin’s speed as 10 mph, which is a valid measurement. She also notes Luke’s speed on Earth as 5 mph short of the speed of light.

**The speed of light remains constant across all inertial reference frames.**If Anakin fires a laser pulse, he will measure it traveling at 186,000 miles per second. Wouldn’t Leia see the pulse moving at an

**additional**10 mph? Wouldn’t Luke measure it as traveling at twice light speed plus 5 mph? No. Everyone measures the speed of the laser pulse as 186,000 miles per second.

The second rule may be the more challenging to grasp. Maxwell’s equations underpin this principle, while experiments provide evidence. We won’t delve deeper into this topic here, but we will explore the implications of Rule #2.

Let’s assume the spaceship is traveling at a fraction of light speed. Inside, Anakin moves at another fraction of light speed. We’ll designate his speed relative to Luke as *V*. What might we anticipate?

From previous articles, we know that motion affects time and distance. What impact does it have on Anakin’s speed?

To determine *V*, we will define an event that both Luke (on the ground) and Leia (on the ship) can agree upon. Problems arise when we attempt to compare two events separated by either distance or time.

### The Instant of Action

The moment Anakin begins to run, he activates a light pulse from his laser. The far end of the ship reflects the pulse back to him. When the pulse hits him, he collapses on the spot. (Don’t worry; he’ll recover.)

Where is Anakin?

### Leia's Perspective

The thick arrow illustrates Anakin's movement, while the thin arrows represent the light beams. The light travels a significantly greater distance than Anakin. How far does he get? Halfway across the ship? A quarter? Regardless, *Luke and Leia will agree*.

We can calculate this distance for both frames of reference, equating the two expressions to see what unfolds.

The speed of light serves as our standard. We can define the distance light travels in Leia’s frame as our unit length. This length corresponds to the total distance covered by the two thin arrows. Anakin runs at a fraction of light speed, so the length of the thick arrow is that fraction. What about the ship's length? The total length of the three arrows is 1 unit. Hence, the length of the ship is half this total:

What fraction of the ship's length did Anakin traverse? Divide his distance by the length of the ship.

Anakin makes it just beyond halfway across the ship. If we measure the floor in sevenths, he would collapse at mark #4. Neither Luke nor Leia would dispute this.

### Luke's Perspective

How does this scenario play out for Luke?

We need not assume anything about how the times, distances, and velocities in Leia’s frame relate to Luke’s. 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 observes Anakin running at some speed, *V*, which intuition might suggest is 1/7 light speed. He also notes the laser pulse traveling at — according to Rule #2 — the speed of light. Finally, he measures the ship’s speed as 1/7 light speed.

Initially, the laser pulse travels to the ship’s far end. It has to cover a slightly longer distance than the ship’s length because the ship is in motion. Anakin has run a fraction of that distance. Since the ship has also moved, he is less than that fraction across.

By the end of the journey, Anakin has traversed a distance of *V*t with respect to Luke. Here, *t* is some time as measured by Luke.

Although Anakin covers a distance of *V*t, the ship moves a distance of *u*t, where *u* is the ship’s speed. This is how far Anakin ends up from the back of the ship:

In our example, the ship’s speed, *u*, is 1/7 light speed. What we want to determine is *V*, and the time variable, *t*, will eventually cancel out. Once we have the length of the ship, we hope to demonstrate:

This ratio of distance traveled along the ship to the ship's length will be measured by Luke.

For clarity, it may help to have a diagram of Luke’s reference frame at hand as you follow each step of the argument.

### Anakin's Position is Consistent in Both Frames

Recall how we calculated this for Leia's frame of reference. She measures Anakin's speed as 1/7 light speed. We can generalize our previous calculation using the variable *v* for Anakin's speed.

Next, 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 covers in that time.

The time variable, *t*, presents a challenge. To simplify, we want only one time variable, *t*. First, we express in terms of *t*.

Next, we can use the relation *t = t + t* to derive an expression for *t*.

We substitute our expression for *t* into the previous formula for *L* and tidy it up.

We divide the distance Anakin travels across the ship by the ship's length. This yields a ratio in terms of the ship's length. What proportion of the ship did Anakin cross?

That fraction remains consistent across frames.

### How Fast is Anakin Moving Away from Luke?

We are left to isolate *V*, which will yield the desired expression for Anakin’s speed according to Luke.

Cross-multiply the numerators and denominators.

Expand the equation.

Reorganize the equation to bring the *V* terms to one side, cancelling equivalent terms.

Isolate *V*.

Simplify the result by dividing both numerator and denominator by *c²*.

If we consistently express velocities as fractions of light speed, we can set *c = 1*, allowing us to eliminate *c* from the equation.

Substituting the values from our example yields:

In Luke’s frame, Anakin travels at approximately 90% of light speed.

We do add velocities, but we apply an adjustment by dividing by *1 + uv*. When both *u* and *v* are small, which they typically are, *uv* approaches 0, making the adjustment negligible.

It’s crucial to avoid mixing reference frames. The distance between Luke and Anakin increases at 1/7 light speed from Leia’s perspective, but not when considering each other.

What about Luke and Anakin *together* from Leia's viewpoint? Why is there no adjustment? From Leia’s frame, the distance between Luke and Anakin expands at 1/7 light speed: Anakin to her right, Luke to her left.

Care must be taken when mixing reference frames. The distance between Luke and Anakin increases at 1/7 light speed regarding *Leia*, but this is not the case when considering their relative motion.

If Luke fires a light pulse towards Anakin, we know it will catch up with him. According to Luke, Anakin moves away at about 90% of light speed. Leia will observe the pulse approach her at light speed, pass her, and overtake Anakin, who will also perceive it approaching at light speed.

When we discuss the distance increasing at 1/7 light speed, we must clarify that distance is not a *thing* capable of transmitting a signal at superluminal speeds. Careful selection of reference frames is essential. No observer witnesses movement exceeding light speed in any frame.

All is well. The universe remains in harmony within its curved spacetime.