Relativity That ClicksEpisode 1 / β — speed is what fraction of c

Relativity feels hard because we start from quantities with units — time, distance, speed. First, see c not as a speed but as an exchange rate

β — speed is what fraction of c c is not a "very fast vehicle." It is the exchange rate that links time and space.
Convert time into distance with ct and spacetime becomes one continuous whole — so the only meaningful speed is the unitless fraction β = v/c.

Tools you'll need: division, the unit "light-year," and "units are stage machinery" from Fields That Click This episode's ratio: β = v/c

People say relativity is "hard." But the difficulty is usually that we are thinking in terms of quantities that carry units. Time measured in seconds, distance in meters, speed in m/s — hold onto these and the equations start to look tangled. What we do in this episode is one single change of viewpoint. See the speed of light \(c\) not as a "speed" but as the exchange rate between time and space. Multiply time by \(c\) to convert it into distance (\(ct\)), and time and space become one continuous thing measured in the same unit. Then "speed" becomes just a unitless fraction — β = v/c, what fraction of c. That one step makes relativity a great deal more natural.

01When you say "fast," what are you comparing it to?

300 km/h is fast. But the Earth is circling the Sun at 100,000 km/h, and we live perfectly comfortably. "Fast" and "slow" are always nothing more than a comparison against something. So is there a "boss" to compare against? There is — the one reference speed in the universe that comes out to the same value for every observer, \(c\). Relativity begins by taking this \(c\) as the ruler and re-measuring speed against it.

02c is not a speed; it is the exchange rate between time and space

Recall the unit "1 light-year." It is a unit of distance, yet it contains "year," a unit of time. It means the distance light travels in one year. Here's the hint — multiply by \(c\) and time turns into distance.

c is the "time → distance" exchange rate
$$\text{time }t\ \xrightarrow{\ \times c\ }\ \text{distance }ct$$

One second is \(c\times1\text{ s}=3\times10^8\) m = "the time worth 300,000 km." Like the rate that converts dollars to yen, \(c\) is the exchange rate that converts seconds into meters. Time and space were, all along, the same currency linked by the rate \(c\).

This carries a deep implication. If from the start we had measured both time and distance in the same unit (say, both in meters), \(c\) would just be \(1\) and would vanish from the equations. \(c\) looks like the large number \(3\times10^8\) only because we measure time in seconds and distance in meters, two different units — exactly like the weird conversion factor that appears on a map whose width is in meters and height in feet. \(c\) is not some deep number of physics; it is an apparent constant born of a mismatch of units. That's why physicists often set \(c=1\) (natural units). "A quantity with units is stage machinery," the refrain of "Fields That Click," is working here too.

03So the meaningful speed is β = v/c

If time and space are the same currency linked by \(c\), then the natural thing is to divide a speed \(v\) (distance ÷ time) by the universe's reference \(c\) to make it a unitless fraction.

This episode's ratio — β (speed as a fraction of c)
$$\beta=\frac{v}{c}\qquad(0\le\beta<1)$$

\(\beta\) is a pure, unitless fraction. \(\beta=0.5\) means "half the speed of light." And matter, no matter how hard it accelerates, cannot reach \(\beta=1\) (the speed of light). \(\beta=1\) is the exclusive privilege of light; for matter it is a wall that cannot be crossed (why it's a wall is Episodes 5 and 6).

Line up the \(\beta\) values around us and their smallness is startling.

Try it — everyday β

Shinkansen (320 km/h ≈ 89 m/s)

$$\beta=\frac{89}{3\times10^8}\approx3\times10^{-7}$$

Earth's orbit (30 km/s)

$$\beta=\frac{3\times10^4}{3\times10^8}=1\times10^{-4}$$

Even the Shinkansen is only 3 ten-millionths of the speed of light. Even Earth's orbital motion is one ten-thousandth. Meanwhile a proton in an accelerator has \(\beta\approx0.999999991\), and light has \(\beta=1\). Everyday life is entirely a world of \(\beta\approx0\) — which is why relativistic effects don't ordinarily show their face.

04Let's play with it — on the spacetime diagram, β is the slope of a worldline

Since we've made time and space the same currency, we can draw them on a single diagram. The horizontal axis is space \(x\), the vertical axis is time \(ct\) (time converted into distance) — a spacetime diagram. A stationary object advances only in time, so its worldline (the object's track) goes straight up. Once it moves, it shifts sideways as it advances, so the worldline tilts. That tilt (the angle measured from vertical) is exactly \(\beta\).

Try raising \(\beta\) with the slider. The worldline approaches the worldline of light (\(\beta=1\), 45°). But — it never reaches 45°. A matter worldline is always steeper than a light ray. The 45° light ray runs diagonally as a wall that cannot be crossed. "Speed," it turns out, was this slope of the worldline on the spacetime diagram.

Figure: spacetime diagram (horizontal = space x, vertical = time ct). A stationary object's worldline goes straight up; once moving, it tilts by β. Raise β and it approaches the 45° light ray (β=1, the wall), but matter never reaches it
object's worldline (slope = β) light's worldline (β=1, the 45° wall)

05Newton is the physics of β→0

Everyday \(\beta\) is around \(10^{-7}\). Relativistic effects kick in at the order of \(\beta^2\), so \(\beta^2\approx10^{-14}\) — completely negligible. That is why Newton's world, where time is common to everyone and speeds simply add, gave no trouble whatsoever.

Newton = relativity in the limit β→0

Take \(\beta\to0\) (i.e. \(c\to\infty\)) in the equations of relativity and everything returns to Newtonian mechanics. Newton wasn't wrong; he was right in the world where \(\beta\) is tiny. This is exactly the same structure as in Episode 1 of "Fields That Click," where Newton's instantaneous force was the approximation of the delay number \(\varepsilon\to0\).

Let's gather the viewpoint. \(c\) is not a speed but the exchange rate that converts time into distance. Time and space are the same currency linked by \(c\), and spacetime is one continuous whole. So the speed that determines the physics is not the unit-bearing \(v\) (m/s) but the unitless β = v/c. \(c\) itself is an apparent number born of a mismatch of units and vanishes when you set \(c=1\). Drop the units and think in terms of β — take that one step, and from next episode on, time dilation, E=mc², all of it unravels smoothly as functions of β.

◇ ◇ ◇
The honest line — the reach of "exchange rate" and "wall"

That \(c\) is the conversion factor linking time and space, that \(ct\) lets us measure time in a dimension of length (the unification of spacetime), that in natural units we can set \(c=1\), that \(\beta=v/c\) is dimensionless and tiny in everyday life (Shinkansen \(\sim3\times10^{-7}\)), and that Newtonian mechanics is recovered in the \(c\to\infty\) limit — these are all established physics.

However. ① Don't overstate "c is just a unit conversion." The substance that \(c\) comes out to the same value for every observer (the invariance of the speed of light) is not a matter of units but a physical requirement (an experimental fact), and that is the very core of relativity. This piece only says "the appearance of the large number \(3\times10^8\) is due to units"; it does not attribute the universality of \(c\) to units. ② "Matter cannot reach \(\beta=1\) / c is a wall" — this episode borrows only the conclusion in advance. The reasons (it would take infinite energy; causality) are shown in Episodes 4–6. ③ "Slope = β" on the spacetime diagram is a story about one-dimensional motion. That speeds do not simply add (relativistic velocity composition) will also be handled in a later episode.

Practice problems (solvable with this episode's formulas alone. c ≈ 3×10⁸ m/s)
  1. The speed of sound is about 340 m/s. Find β. Are relativistic effects likely to matter?
    Show answer
    \(\beta=340/(3\times10^8)\approx1.1\times10^{-6}\). It's tiny, so relativistic effects are effectively zero (\(\beta^2\sim10^{-12}\)). Sound is "slow."
  2. Why does "1 light-year," a unit of distance, contain "year"? Explain via the role of c.
    Show answer
    Because \(c\) is the exchange rate that converts time into distance. 1 light-year = \(c\times1\) year = the distance light travels in one year. It is exactly the conversion time × c = distance.
  3. On the spacetime diagram, what is the slope of the worldline for a stationary object, a β=0.5 object, and light, respectively?
    Show answer
    Stationary object: straight up (vertical, slope 0). β=0.5: tilted slightly from vertical (tan = 0.5). Light: 45° (β=1). A matter worldline is always steeper than 45°.
  4. What do the equations of relativity become when you take \(\beta\to0\)? What in this series / in Fields That Click has the same structure?
    Show answer
    Newtonian mechanics. It's the same as Episode 1 of "Fields That Click," where Newton's instantaneous force was the approximation of the delay number \(\varepsilon\to0\) — Newton is the limit β→0 (c→∞).

Episode 1 summaryspeed is β = v/c — c is not a speed but an exchange rate

Much of relativity's difficulty comes from thinking in terms of time, distance, and speed with units attached. Change the viewpoint and see \(c\) not as a "fast vehicle" but as the exchange rate between time and space — convert time into distance with \(ct\), and spacetime is one continuous, single currency. Then \(c\) looking as large as \(3\times10^8\) is due to the mismatch of the separate units seconds and meters, and it vanishes when you set \(c=1\).

So the speed that determines the physics is not the unit-bearing \(v\) but the unitless fraction β = v/c. Matter cannot reach \(\beta=1\) (the speed of light), and the 45° light ray is a wall that cannot be crossed (on the spacetime diagram, β = the slope of the worldline). Everyday life is a world of \(\beta\approx10^{-7}\), so relativistic effects stay hidden and Newton (β→0) sufficed — the same structure as Newton being the ε→0 approximation in "Fields That Click." Drop the units and think in terms of β: that one step is this series' point of departure.

This document is Episode 1 of the "Relativity That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. That the speed of light \(c\) is the conversion factor linking time and space and \(ct\) can render time in a dimension of length (the unification of Minkowski spacetime), that one can choose \(c=1\) in a system of natural units, that \(\beta=v/c\) is dimensionless and \(\beta<1\) for ordinary bodies, and that Galilean/Newtonian mechanics is recovered in the \(c\to\infty\) (\(\beta\to0\)) limit — these are all established, standard physics. That the numerical value of \(c\) looks large is a product of the unit choice (seconds, meters), but that \(c\) is invariant across all inertial frames (the principle of the constancy of the speed of light) is not a matter of units but a physical requirement and experimental fact and is the core of relativity, and the reason a body cannot reach \(\beta=1\) (infinite energy, causality) is handled in Episodes 4–6 — both points are stated in the "honest line" in the main text. "Slope = β" on the spacetime diagram is a 1+1-dimensional schematic; relativistic velocity composition is handled in later episodes. — To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static/hidden). Related: Contents / sister series Fields That Click, Episode 1.

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, use the β slider to bring the spacetime diagram's worldline toward the 45° light ray (the wall). "Show answer" opens each solution.