Case Study 1 — How Einstein's Energy Contains Newton's: The Binomial Series in Special Relativity

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Case Study 1 — How Einstein's Energy Contains Newton's: The Binomial Series in Special Relativity

Field: Physics (special relativity, GPS engineering) Calculus used: the binomial series and Taylor approximation with error control (Sections 23.4 and 23.8)

In 1905 Einstein published a formula for the kinetic energy of a moving body that looked nothing like the $\tfrac12 mv^2$ generations of students had memorized. For two centuries Newton's expression had passed every test in the laboratory. Was Einstein overturning it? The reconciliation is one of the most quietly profound uses of a Taylor series in all of physics, and it is a story you can now tell yourself, line by line. The relativistic formula does not contradict Newton's — it contains it, as the very first term of an infinite polynomial. The rest of the terms are corrections so small that nobody had ever measured them. This case study walks through that expansion, bounds the error, and then shows where the "negligible" next term stops being negligible: the satellites that tell your phone where you are.

The two formulas

Newtonian mechanics says a mass $m$ moving at speed $v$ carries kinetic energy

$$E_k^{\text{Newton}} = \tfrac12 m v^2.$$

Special relativity replaces this with

$$E_k = (\gamma - 1)\,mc^2, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}},$$

where $c$ is the speed of light and $\gamma$ is the Lorentz factor. These look unrelated. The Newtonian energy is a simple parabola in $v$; the relativistic energy involves a square root in a denominator and blows up to infinity as $v \to c$ (the reason no massive object can reach light speed — it would require infinite energy). To see how the second formula hides the first, we expand $\gamma$ in a Taylor series for small speeds.

Expanding the Lorentz factor

Write $\beta = v/c$, the speed as a fraction of light speed, so $\gamma = (1 - \beta^2)^{-1/2}$. This is exactly the binomial series of Section 23.4,

$$(1 + u)^{k} = \sum_{n=0}^\infty \binom{k}{n} u^n, \qquad \binom{k}{n} = \frac{k(k-1)\cdots(k-n+1)}{n!},$$

with exponent $k = -\tfrac12$ and inner variable $u = -\beta^2$. The first few generalized binomial coefficients for $k = -\tfrac12$ are

$$\binom{-1/2}{1} = -\tfrac12, \qquad \binom{-1/2}{2} = \frac{(-\tfrac12)(-\tfrac32)}{2} = \tfrac38, \qquad \binom{-1/2}{3} = \frac{(-\tfrac12)(-\tfrac32)(-\tfrac52)}{6} = -\tfrac{5}{16}.$$

Substituting $u = -\beta^2$ flips the sign of the odd-power terms, so every term comes out positive:

$$\gamma = (1 - \beta^2)^{-1/2} = 1 + \tfrac12 \beta^2 + \tfrac38 \beta^4 + \tfrac{5}{16}\beta^6 + \cdots$$

This series converges for $|\beta| < 1$, that is, for every physically allowed speed $v < c$ — exactly the radius of convergence you would predict, since the function $\gamma$ has its singularity precisely at $\beta = 1$ (the light-speed barrier). The radius of convergence is the physics: the series stops working exactly where the object would need infinite energy.

Recovering Newton

Now subtract $1$ and multiply by $mc^2$:

$$E_k = (\gamma - 1) mc^2 = \left(\tfrac12 \beta^2 + \tfrac38 \beta^4 + \cdots\right) mc^2.$$

Replace $\beta^2 = v^2/c^2$. The first term becomes

$$\tfrac12 \beta^2 \, mc^2 = \tfrac12 \frac{v^2}{c^2}\, mc^2 = \tfrac12 m v^2,$$

which is exactly Newton's kinetic energy, the $c^2$ cancelling cleanly. The next term is the leading relativistic correction:

$$E_k = \tfrac12 m v^2 + \tfrac38 \frac{m v^4}{c^2} + \cdots$$

So Newton was never wrong — he was reading the first term of a Taylor series and, lacking instruments fast enough to detect the rest, never saw the higher-order corrections. This is the deepest expression of the book's approximation theme: a new physical theory often appears as the next term in the expansion of an older one. The same pattern governs the small-angle pendulum ($\sin\theta \approx \theta$) and countless other "linearizations" — Newton's mechanics is the linearization (in $\beta^2$) of Einstein's.

How big is the correction? Bounding it like a calculus problem

The whole value of a Taylor expansion is that you can say how good the approximation is, not just that it is good. Here the ratio of the first correction to the Newtonian term is clean:

$$\frac{\tfrac38 m v^4/c^2}{\tfrac12 m v^2} = \frac{3}{4}\,\frac{v^2}{c^2} = \tfrac34 \beta^2.$$

At everyday speeds this is breathtakingly small. A bullet at $v = 1000$ m/s has $\beta = v/c \approx 3.3\times10^{-6}$, so the correction is about $\tfrac34 \beta^2 \approx 8\times10^{-12}$ — twelve digits past the decimal point. No wonder Newtonian energy survived two centuries of experiments. Even at $v = 0.1c$ — thirty million metres per second, far beyond any macroscopic object — the correction is only

$$\tfrac34 (0.1)^2 = \tfrac34 (0.01) = 0.0075,$$

three quarters of one percent. You have to reach an appreciable fraction of light speed before the second term matters, which is exactly the regime of particle accelerators, where the full relativistic formula is mandatory and the Newtonian approximation fails badly.

# Hand-computed comparison (values verified by hand, not by running this).
def correction_fraction(beta):
    return 0.75 * beta**2          # (3/8 v^4/c^2) / (1/2 v^2)

# beta = 0.001  ->  7.5e-07   (fast jet, utterly negligible)
# beta = 0.1    ->  7.5e-03   (0.75%, first measurable hint)
# beta = 0.5    ->  1.9e-01   (19%, Newton clearly breaking down)
print(correction_fraction(0.1))   # 0.0075

Where the "negligible" term pays your bills: GPS

It would be easy to file the relativistic correction under "interesting but irrelevant." It is not. The Global Positioning System depends on atomic clocks aboard satellites orbiting at about $v \approx 3.9$ km/s. Their speed gives $\beta \approx 1.3\times10^{-5}$, so the special-relativistic time-dilation factor — itself the same Lorentz $\gamma$ we just expanded — differs from $1$ by roughly $\tfrac12\beta^2 \approx 8\times10^{-11}$. Multiply by the $86{,}400$ seconds in a day and the moving clocks lose about 7 microseconds per day relative to the ground from special relativity alone (general relativity adds a larger correction of opposite sign, but that is another chapter of physics).

Seven microseconds sounds trivial. But light travels about $300$ metres in a microsecond, so an uncorrected GPS receiver would accumulate position errors of kilometres per day. The engineers who built GPS had to carry the $\tfrac12\beta^2$ term — the second term of our Taylor series — explicitly in the satellite firmware. The "negligible correction" that Newton could not measure is the difference between your navigation app guiding you to the right street and stranding you in the next county. A Taylor series you can now derive by hand is, quite literally, keeping you on the road.

Connections to the textbook

Section 23.4 — the binomial series $(1+x)^k = \sum \binom{k}{n} x^n$ for arbitrary real $k$, the engine of this entire derivation.
Section 23.8 — the worked relativity-reduces-to-Newton expansion, and engineering linearization (the same "keep the first one or two terms" move).
Section 23.5 — Taylor's theorem with remainder, which is what certifies that the dropped terms are as small as we claimed.
Chapter 11 — linearization and the tangent-line approximation, of which this is the higher-order generalization.

Discussion questions

The series for $\gamma$ converges only for $|\beta| < 1$. Interpret this radius of convergence physically. What happens to the energy as $v \to c$, and how does the series reflect it?
At what fraction of light speed does the first relativistic correction reach $1\%$ of the Newtonian energy? Solve $\tfrac34\beta^2 = 0.01$ by hand.
Why is it legitimate to keep only the first correction term for GPS satellites but mandatory to use the full $\gamma$ for protons in the Large Hadron Collider ($\beta \approx 0.999999991$)?
The expansion $\sin\theta \approx \theta$ (small-angle pendulum) and $E_k \approx \tfrac12 mv^2$ are the same idea — truncating a Taylor series. State the general principle in one sentence.
Using Taylor's remainder (Section 23.5), how would you rigorously bound the error made by dropping all terms beyond $\tfrac38 \beta^4 mc^2$?

A short annotated reading list

Taylor, E. F., & Wheeler, J. A. (1992). Spacetime Physics (2nd ed.). W. H. Freeman. The clearest undergraduate derivation of relativistic energy; see the chapter where $E_k = (\gamma-1)mc^2$ is expanded exactly as we did here.
Ashby, N. (2003). "Relativity in the Global Positioning System." Living Reviews in Relativity, 6, 1. The authoritative, openly available account of why GPS engineering cannot ignore the Lorentz factor — the source of the 7-microsecond figure.
Stewart, Calculus: Early Transcendentals (9th ed.), §11.10–11.11. Stewart's "Applications of Taylor Polynomials" section works the relativistic kinetic energy as a marquee example, with error estimates.
OpenStax Calculus Volume 2, §6.4 (Working with Taylor Series). Free; the binomial series and its physics applications, parallel to Section 23.8.

Einstein did not bury Newton. He revealed Newton's law as the opening line of a longer story — and a Taylor series is how you read the rest.