Galilean Relativity and Why It Fails for Light
~6 min read · Tier toggle changes depth (T1 plain · T2 standard · T3 rigour)
You should be comfortable with basic kinematics (position, velocity, acceleration) and the concept of an inertial reference frame. Familiarity with vector addition is helpful but not strictly required.
Why this matters
Understanding why classical mechanics fails for light is the essential first step toward special relativity.
For centuries, physicists assumed that velocities simply add together, a rule known as Galilean relativity. This works perfectly for everyday objects like cars or balls, but it leads to a direct contradiction when applied to light. Maxwell’s equations predict that light travels at a fixed speed \(c\), regardless of the source's motion. If Galilean rules held, an observer moving towards a light source should measure a speed greater than \(c\), while one moving away should measure less. This inconsistency signals that our fundamental understanding of space and time must be revised.
The Galilean Intuition
Imagine you are sitting on a train moving at \(30\ \mathrm{m/s}\). You throw a ball forward at \(10\ \mathrm{m/s}\) relative to yourself. A person standing on the platform sees the ball moving at \(40\ \mathrm{m/s}\). This is the principle of Galilean velocity addition: velocities are relative to the observer, and we simply add the velocity of the frame to the velocity of the object within that frame. In this view, there is no absolute speed limit, and time is universal—everyone agrees on how long an event takes, regardless of their motion. Space and time are separate, absolute backgrounds against which events occur. This framework is incredibly successful for low-speed mechanics, which is why it feels so natural.
The Conflict with Maxwell’s Equations
James Clerk Maxwell unified electricity and magnetism in the 19th century, deriving a wave equation for electromagnetic radiation. The speed of these waves, which we now call light, is determined solely by two constants: the permittivity of free space \(\epsilon_0\) and the permeability of free space \(\mu_0\). The speed is given by \(c = 1/\sqrt{\epsilon_0 \mu_0}\). Crucially, this speed does not depend on the motion of the source. If a lamp on a moving train emits light, Maxwell’s theory predicts the light travels at \(c\) relative to the train, and also at \(c\) relative to the platform. However, Galilean relativity demands that the platform observer should measure \(c + v_{\text{train}}\). This creates a paradox: either Maxwell’s equations are wrong, or Galilean relativity is wrong. Experiments, particularly the Michelson-Morley experiment, failed to detect any "ether wind" that would allow light speed to vary, strongly suggesting that the speed of light is invariant.
Failure of Galilean Invariance
To see the mathematical breakdown, consider two inertial frames \(S\) and \(S'\). Let \(S'\) move at speed \(v\) along the \(x\)-axis relative to \(S\). In Galilean relativity, the transformation between coordinates is: \[ x' = x - vt, \qquad t' = t \] If a light pulse is emitted from the origin at \(t=0\), its position in \(S\) is \(x = ct\). Substituting this into the Galilean transformation for \(x'\): \[ x' = ct - vt = (c - v)t \] Since \(t' = t\), the speed of light measured in \(S'\) would be: \[ u' = \frac{x'}{t'} = c - v \] This result shows that the speed of light is not invariant under Galilean transformations. It depends on the relative velocity \(v\) of the observer. This violates the experimental fact that the speed of light is constant for all inertial observers. The Galilean transformation preserves the form of Newton’s laws but destroys the form of Maxwell’s equations. Therefore, Galilean relativity is not a symmetry of electromagnetism.
Towards a New Transformation
To resolve this, we need a new coordinate transformation that preserves the speed of light. We require that if \(x = ct\) in frame \(S\), then \(x' = ct'\) in frame \(S'\). This implies that the quantity \(x^2 - c^2t^2\) must be invariant, or more generally, the spacetime interval \(s^2 = x^2 + y^2 + z^2 - c^2t^2\) must be the same in all inertial frames. The Lorentz transformation satisfies this requirement. It modifies the time coordinate to mix with space, ensuring that the speed of light remains \(c\) in all frames. The key difference from the Galilean case is that time is no longer absolute; it depends on the observer's motion. This leads to the concept of spacetime, where space and time are interwoven into a four-dimensional geometry.
The derivation
Lorentz Transformation from the Two Postulates
For two inertial frames \(S\) and \(S'\) in standard configuration — \(S'\) moving at constant speed \(v\) along the shared \(x\)-axis, origins coinciding at \(t=t'=0\) — the two postulates (the principle of relativity and the invariance of the vacuum speed of light \(c\)) force the coordinate transformation to be linear, transverse coordinates to be unchanged, and the boost to take the form \[ x' = \gamma\,(x - v t), \qquad t' = \gamma\left(t - \frac{v x}{c^2}\right), \qquad y'=y,\quad z'=z, \qquad \gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}. \]
A common misconception is that the failure of Galilean relativity means that "nothing moves" or that motion is an illusion. In reality, relative motion is real, but the way we measure distances and times changes. The Lorentz transformation is not a correction to mechanics but a deeper description of the geometry of the universe. It reveals that simultaneity is relative: two events that are simultaneous in one frame are not necessarily simultaneous in another. This is not a paradox but a fundamental feature of spacetime. C
Worked example
Problem. A spaceship moves at \(v = 0.6c\) relative to Earth. A laser pulse is emitted from the front of the ship towards the back. Calculate the speed of the light pulse as measured by an observer on Earth using (a) Galilean relativity and (b) the Lorentz transformation. Compare the results.
- Galilean Prediction: In the ship's frame \(S'\), the light moves at \(c\). The ship moves at \(v = 0.6c\) relative to Earth \(S\). According to Galilean addition, the speed of light in \(S\) is \(u = c - v\). \[ u_{\text{Gal}} = c - 0.6c = 0.4c \]
- Lorentz Prediction: We use the velocity transformation formula derived from the Lorentz transformation. For an object moving at velocity \(u'\) in \(S'\), its velocity \(u\) in \(S\) is: \[ u = \frac{u' + v}{1 + \frac{u'v}{c^2}} \] Here, the light moves from front to back, so in \(S'\), \(u' = -c\) (negative direction). The ship moves at \(+v = +0.6c\). \[ u = \frac{-c + 0.6c}{1 + \frac{(-c)(0.6c)}{c^2}} = \frac{-0.4c}{1 - 0.6} = \frac{-0.4c}{0.4} = -c \] The magnitude is \(c\).
- Comparison: Galilean relativity predicts a speed of \(0.4c\), while the Lorentz transformation predicts \(c\). Experiment confirms the latter.
Units check. Both results are expressed as a fraction of \(c\), which is dimensionless. If we used SI units, \(c \approx 3.00 \times 10^8\ \mathrm{m/s}\), so \(u_{\text{Gal}} \approx 1.20 \times 10^8\ \mathrm{m/s}\) and \(u_{\text{Lorentz}} \approx 3.00 \times 10^8\ \mathrm{m/s}\). The units are consistent.
Three quick tests
- Why does the Galilean transformation fail for light?
Answer
Because it predicts that the speed of light depends on the observer's velocity, whereas experiments show it is constant. - If a frame \(S'\) moves at \(0.5c\) relative to \(S\), and a particle moves at \(0.5c\) in \(S'\) in the same direction, what is its speed in \(S\) according to Galilean relativity?
Answer
\(1.0c\). (Note: This is physically impossible in reality, highlighting the failure of the model.) - What quantity is invariant under the Lorentz transformation but not under the Galilean transformation?
Answer
The spacetime interval \(s^2 = x^2 + y^2 + z^2 - c^2t^2\).
Problem set for PU-105 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.