Cosmology That Clicks · The math-friendly edition for curious high-schoolers
The farther away a galaxy is, the redder its light looks to us. Textbooks explain this by saying "space itself is stretching." That's correct — but almost everyone gets stuck on it the first time. So in this piece we'll run all the way to the finish line using a different phrasing: space stays fixed, and instead the speed of light slowly drops over time. We'll do the math properly. And at the very end, we'll reveal that this phrasing points to exactly the same observations as the standard explanation.
Let \(c_0\) be today's speed of light, \(t\) the age of the universe, and \(t_0\) the universe's age right now. This piece has just one rule: "speed of light × age" is the same in every era.
The right-hand equation says "the larger the age \(t\), the smaller the speed of light \(c(t)\)." When the universe is young (small \(t\)), light is fast; as it ages, light slows down. The shape of the equation is just inverse proportionality — the kind you meet in middle school.
To find "how much it slows each year," take the derivative of \(c(t)\) with respect to time. Differentiating \(c(t)=c_0 t_0 \, t^{-1}\),
$$\frac{dc}{dt} = c_0 t_0 \cdot(-1)\,t^{-2} = -\frac{c_0 t_0}{t^2} = -\frac{c(t)}{t}$$Plugging in today's values (\(t=t_0,\ c=c_0\)), the rate of decrease is \(\left|\dfrac{dc}{dt}\right| = \dfrac{c_0}{t_0}\). Now we just put in numbers. Let's get our hands dirty.
Values we'll use
Speed of light \(c_0 = 3.0\times10^{8}\ \mathrm{m/s}\), age of the universe \(t_0 = 13.8\) billion years. We'll keep the age in "years" rather than seconds (since we want the answer "per year").
The calculation
$$\left|\frac{dc}{dt}\right| = \frac{c_0}{t_0} = \frac{3.0\times10^{8}\ \mathrm{m/s}}{1.38\times10^{10}\ \text{yr}} \approx 2.17\times10^{-2}\ \frac{\mathrm{m/s}}{\text{yr}}$$Convert to cm
$$2.17\times10^{-2}\ \mathrm{m/s} = 2.2\ \mathrm{cm/s}$$So in this piece's universe, light slows by roughly 2.2 cm/s every year. Against light's enormous speed of 300,000 kilometers per second, that's just a few centimeters. That's why an ordinary experiment could never notice it. If someone says "but no change has ever been observed," you can answer with a straight face: at this tiny size, of course not.
While we're here, let's also look at the fractional change — what fraction it changes per year:
$$\frac{1}{c}\left|\frac{dc}{dt}\right| = \frac{1}{t_0} = \frac{1}{1.38\times10^{10}\ \text{yr}} \approx 7.2\times10^{-11}\ /\text{yr}.$$About one part in a hundred billion per year. This "fraction" is actually the physically important number (you'll see why in STEP 05).
Consider light arriving from a distant galaxy. Call the moment it was emitted \(t_e\) (emit) and now, when we receive it, \(t_0\). In this piece's phrasing, "light that was fast when emitted arrives in a now where light is slow." From our rule, the ratio of speeds is
$$\frac{c(t_e)}{c(t_0)} = \frac{c_0 t_0 / t_e}{c_0 t_0 / t_0} = \frac{t_0}{t_e}.$$This ratio is exactly what sets "how stretched the light is" — the redshift \(z\). Using the relation familiar from astronomy,
$$1+z = \frac{c(t_e)}{c(t_0)} = \frac{t_0}{t_e}.$$For example, a galaxy at \(z=1\) has \(1+z=2\), so \(t_0/t_e=2\). The light was emitted when the universe was half its present age, and back then light was twice as fast as now. Redshift is exactly the fraction by which the speed of light has dropped since the light was emitted — captured in a single equation.
The greatest distance light could have traveled since the universe began is called the particle horizon. Adding up speed × time as the speed of light changes (integrating), we get
$$D = \int_{0}^{t_0} c(t)\,dt = \int_{0}^{t_0} \frac{c_0 t_0}{t}\,dt = c_0 t_0\big[\ln t\big]_{0}^{t_0}.$$This is the climax of the piece. As \(t\to 0\) (the very first instant), \(\ln t \to -\infty\), so this integral grows without bound. The further back you go, the more the speed \(c(t)=c_0t_0/t\) shoots up — so in the newborn universe, light could cover an enormous distance in one go.
That means any two points in the universe, however far apart, could exchange light signals right after the beginning. This is a clean explanation for why the universe has almost the same temperature everywhere (the cosmic microwave background). In standard cosmology this is the notorious horizon problem — "why do distant regions that could never have been in contact share the same temperature?" — but in this piece it's settled with "well, light used to be fast, so of course."
The horizon problem truly disappears not because \(c\) changes, but because of the way the universe expands: \(a\propto t\) (straight-line expansion). A mere rephrasing changes no physics. Imposing straight-line expansion all the way back into the early universe — that's a real, minority model (the \(R_h=ct\) universe). It solves the horizon problem naturally, but it also carries unsolved homework, such as matching the early formation of the elements (Big Bang nucleosynthesis). The standard model is \(\Lambda\mathrm{CDM}\), which includes inflation.
We've been calculating happily, but the biggest question remains. If light is slowing down, shouldn't we be able to measure it?
The answer is: in principle, no. Here's why. One meter is now defined as "the distance light travels in \(1/299792458\) of a second." Since the ruler itself is set by the speed of light, if light slows down, the ruler shrinks along with it. When you divide, the effects cancel out — so a measurement of "the speed of light changed" can never succeed.
So what can be measured? Quantities with no units — pure ratios. The prime example is the fine-structure constant \(\alpha\):
$$\alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c} \approx \frac{1}{137}.$$Because \(\alpha\) is a pure number that doesn't depend on any ruler, we can actually test whether it has changed. And the latest atomic-clock experiments show \(\alpha\) changing by less than \(10^{-19}\) per year — essentially not budging. This is the decisive point. Recall the "\(c\) changes by a fraction \(7.2\times10^{-11}\) per year" from STEP 02. If \(c\) really changed by that much while \(e,\hbar,\varepsilon_0\) stayed fixed, then \(\alpha\) would also move at \(7.2\times10^{-11}\)/yr — instantly contradicting experiment.
If \(c\) slows down, then \(e,\hbar,\varepsilon_0\) must move in step, so that the ratio \(\alpha\) doesn't budge at all — it has to be that kind of "relabeling of units." When it is, no experiment can tell apart a universe where \(c\) changes from one where it doesn't.
So "light slows down" is fully compatible with the fact that the locally measured speed of light is always \(c_0\). What looks slower is only the "coordinate speed" seen from far-off coordinates. The \(\alpha\) that governs the atomic world is unchanged, and nothing — not atomic clocks, not your ruler — breaks.
In other words, "the speed of light slows down" is not a claim that the universe really works that way. It's a projection — the cheapest way for a human mind to swallow the same physics. A brilliant guide-line, not a signboard of truth. Keep this paired with everything above, always.
The reason lies not with the universe but with our own heads. Over an unimaginable stretch of time, humans have burned in the assumption that "the stage (space) doesn't move; things move on top of it." So "the speed of a moving thing changes" is far cheaper to picture than "the container swells." Fast, slow, arriving late, used-to-be-faster — these are all everyday words.
As a bonus, this view is also handy for people who actually calculate. Using coordinates that hold space fixed and push the expansion onto the time side (called conformal time), light travels in a straight line, and things like gravitational-lensing calculations genuinely speed up. It's a legitimate technique used in numerical cosmology. Clarity isn't an illusion — just keep the one line in mind: clarity and correctness are different things.
You can own the same performance on a record (continuous) or a CD (discrete). The music is one thing; there are just two media. "Space stretches" and "light slows down" are likewise only two ways of recording the same universe. A single equation, \(c_B \cdot a = \text{constant}\), guarantees the two discs carry the same performance.
Listening in your room, you can't tell which is "the real one." The difference only matters at the very edge of the highest pitches a human can't even hear (for the universe, the Planck scale). So relax and listen on whichever disc you like. Understanding the universe through the version that clicks for you is far richer than knowing only one side.
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