The Speed of Light: The Ultimate Limit That Nothing Can Exceed

Why can nothing exceed the speed of light?

Nothing can exceed the speed of light because this speed is not just that of a fast object—it is a structural limit of spacetime itself. In Einstein’s theory of relativity, the faster a massive particle accelerates, the more its energy and inertia increase, to the point where an infinite amount of energy would be required to reach the speed of light: this is physically impossible. Furthermore, information and causality cannot travel faster than this limit, otherwise the order of events could reverse, and causality would be violated. Light is therefore not a "record" to be broken, but the maximum speed at which the universe allows the propagation of energy, matter, and information.

The Slow Discovery of an Immense Speed

For millennia, light was perceived as instantaneous, too fast to be measured. It was not until the 17th century that Ole Rømer (1644-1710) showed that the eclipses of Jupiter's satellites revealed a measurable delay, indicating that light takes a finite time to travel. Gradually, the idea took hold: light has a speed.

In the 19th century, measurements by Hippolyte Fizeau (1819-1896) and Léon Foucault (1819-1868) paved the way for Albert Einstein's (1879-1955) special theory of relativity, which established in 1905 that the speed of light in a vacuum is a fundamental constant of spacetime and the ultimate limit for any interaction.

The Best-Measured Speed in All of Physics

The speed of light in a vacuum is exactly \(c = 299\,792\,458\ \text{m} \cdot \text{s}^{-1}\) (m/s).
This value, denoted \(c\), represents the maximum speed at which any information, cause, or effect can propagate in the cosmos. The limit does not apply only to light: no massive particle can reach or exceed this speed.

Why No Object Can Exceed the Speed of Light

Einstein's equations show that the kinetic energy of a body tends toward infinity as its speed approaches \(c\). The relativistic kinetic energy of a body with mass m and velocity v is written as: \[ K = (\gamma - 1)\, m c^2 \quad \text{where} \quad \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

When the velocity \(v\) approaches the speed of light \(c\), \( \sqrt{0} = 0 \) therefore: \( \frac{1}{\sqrt{0}} = \frac{1}{0} \)
In mathematics, division by zero is undefined. It is a forbidden operation in the set of real numbers. The correct formulation for physics is that of the limit:

\[ \lim_{v \to c} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = +\infty \]

Thus, as the velocity \(v\) approaches as close as desired to the speed of light \(c\), the Lorentz factor \(\gamma\) becomes as large as desired. It can exceed any finite value. This divergence makes it impossible for a massive particle to reach \(c\).

N.B.: In physics, \(\frac{1}{\sqrt{0}} \to \infty\) is used to signify this divergence, but it is always a limit, never an equality in the algebraic sense.

Why Photons Are Condemned to Travel at the Speed of Light

Massless particles (photons, gluons) occupy a special place in the cosmos. Unlike massive particles, which can theoretically adopt any speed between 0 and \(c\), a massless particle can only exist at one speed: that of light in a vacuum.

Why this absolute constraint? The answer lies in the equations of special relativity. The energy of a particle is written as: \( E^2 = (m c^2)^2 + (p c)^2 \)

where \(m\) is the rest mass and \(p\) is the momentum. If \(m = 0\) (massless particle), the equation reduces to: \( E = p c \)

Furthermore, the velocity \(v\) of a particle is given by the relation: \( v = \frac{p c^2}{E} = \frac{p c^2}{p c} = c \)

The calculation is unambiguous: for any particle with zero mass, the speed is strictly equal to \(c\). It is impossible for it to slow down or accelerate. It is born at the speed of light and disappears at the same speed, never knowing rest.

This is why photons travel for billions of years. Throughout their journey, they have never experienced a single fraction of a second of deceleration. Their internal clock is frozen: they do not age.