The Two Postulates of Special Relativity
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You should recall that Maxwell’s equations predict electromagnetic waves travelling at speed \(c\), and that Galilean transformations fail to preserve this speed across inertial frames.
Why this matters
Special relativity is not a collection of tricks, but a new geometry of spacetime defined by two simple principles.
The failure of Galilean relativity to accommodate the constant speed of light forced a fundamental re-evaluation of space and time. Rather than patching Newtonian mechanics, Einstein proposed two postulates that redefine the relationship between observers. These postulates are the axiomatic foundation for all subsequent results in this unit, including time dilation, length contraction, and the relativistic energy-momentum relation. Understanding them as geometric constraints, rather than physical mechanisms, is essential for mastering the subject.
The Two Postulates in Plain Language
Imagine two observers, Alice and Bob, moving at a constant velocity relative to each other. They are in "inertial frames," meaning neither is accelerating. The first postulate states that the laws of physics are identical for both Alice and Bob. There is no experiment they can perform inside their own laboratory to determine their absolute speed. If Bob throws a ball, Alice sees it follow the same physical laws, even though she sees it moving differently. This is the principle of relativity, extended from mechanics to all physics, including electromagnetism.
The second postulate is more radical. It states that the speed of light in a vacuum is the same for both Alice and Bob, regardless of their relative motion. If Bob shines a laser, Alice measures the light moving at \(c\), not \(c + v\) or \(c - v\). This contradicts our everyday intuition about adding velocities, but it is a direct consequence of Maxwell’s equations. Together, these postulates force us to abandon the idea of absolute time and absolute space, replacing them with a unified spacetime where measurements of time and length depend on the observer's state of motion.
From Galilean to Lorentz Invariance
In Newtonian physics, the transformation between two inertial frames moving at relative velocity \(v\) is the Galilean transformation. For coordinates \((x, t)\) in frame \(S\) and \((x', t')\) in frame \(S'\), the transformation is \(x' = x - vt\) and \(t' = t\). This implies that time is absolute and that velocities add linearly. If a light pulse moves at \(c\) in \(S\), then in \(S'\) it would move at \(c - v\). This violates the second postulate.
To preserve the speed of light, the transformation must be different. We seek a linear transformation of the form \(x' = \gamma(x - vt)\) and \(t' = \gamma(t - \frac{v}{c^2}x)\), where \(\gamma\) is a factor depending on \(v\). By requiring that the spacetime interval \(s^2 = c^2t^2 - x^2\) remains invariant (i.e., \(s'^2 = s^2\)), we can determine \(\gamma\). This leads to the Lorentz factor \(\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}\). The key difference from Galilean relativity is that time and space mix together. Time is no longer absolute; it depends on position and velocity.
This invariance of the spacetime interval is the geometric core of special relativity. It replaces the Euclidean distance in space with a Minkowski metric in spacetime. The "distance" between two events is no longer just spatial separation, but a combination of spatial and temporal separations weighted by \(c^2\).
The Constancy of the Speed of Light
The second postulate is not merely an assumption about light; it is a statement about the structure of spacetime. The speed \(c\) is the invariant speed limit for all massless particles and information. In the Lorentz transformation, if an object moves at speed \(c\) in one frame, it moves at speed \(c\) in all frames. Let us verify this. Suppose a light pulse moves along the \(x\)-axis in frame \(S\), so \(x = ct\). In frame \(S'\), the position is:
The time coordinate in \(S'\) is:
The velocity in \(S'\) is \(v' = \frac{x'}{t'}\):
Thus, the speed of light is preserved. This result is counter-intuitive because it implies that time and length must adjust to keep \(c\) constant. If you chase a light beam, you do not see it slow down; instead, your clock runs slower and your ruler contracts in the direction of motion, such that the ratio of distance to time remains \(c\).
Relativity of Simultaneity
A direct consequence of the Lorentz transformation is that simultaneity is relative. Two events that occur at the same time in one frame may not be simultaneous in another. Consider two lightning strikes at the front and back of a train. In the train's frame \(S'\), if they strike simultaneously at \(t' = 0\), then in the ground frame \(S\), the times are related by \(t = \gamma(t' + \frac{v}{c^2}x')\). Since the front and back have different \(x'\) coordinates, they have different \(t\) values in \(S\). The strike at the front (positive \(x'\)) occurs earlier in \(S\) than the strike at the back (negative \(x'\)).
This is not a matter of signal delay or observation lag; it is a fundamental property of spacetime. There is no universal "now" that all observers agree on. The ordering of spacelike separated events (events that cannot causally influence each other) can differ between observers. However, the ordering of timelike separated events (events that can causally influence each other) is invariant. This preserves causality, ensuring that cause always precedes effect in all inertial frames.
A common misconception is that the speed of light is special because it is the speed of photons. In fact, \(c\) is the invariant speed of spacetime itself. Any massless particle, such as a gluon or a graviton, must travel at \(c\). The postulates do not depend on the existence of light; light is simply the first massless field we discovered. If you imagine a universe with no photons, the Lorentz transformations would still hold, with \(c\) defined as the maximum speed of causal influence. C
Worked example
Problem. Two events occur in an inertial frame \(S\). Event A happens at \((x_A, t_A) = (0, 0)\). Event B happens at \((x_B, t_B) = (300 \ \mathrm{m}, 1.0 \ \mu\mathrm{s})\). A second observer \(S'\) moves at \(v = 0.6c\) relative to \(S\). Calculate the time difference \(\Delta t'\) between the events in \(S'\).
- Identify the differences in frame \(S\): \(\Delta x = x_B - x_A = 300 \ \mathrm{m}\) and \(\Delta t = t_B - t_A = 1.0 \times 10^{-6} \ \mathrm{s}\).
- Calculate the Lorentz factor \(\gamma\) for \(v = 0.6c\): \(\gamma = \frac{1}{\sqrt{1 - 0.6^2}} = \frac{1}{\sqrt{0.64}} = \frac{1}{0.8} = 1.25\).
- Use the Lorentz transformation for time: \(\Delta t' = \gamma \left( \Delta t - \frac{v}{c^2} \Delta x \right)\).
- Substitute the values. Note that \(\frac{v}{c^2} \Delta x = \frac{0.6c}{c^2} (300 \ \mathrm{m}) = \frac{0.6 \times 300}{c} \ \mathrm{s}\). Since \(c \approx 3.0 \times 10^8 \ \mathrm{m/s}\), \(\frac{300}{c} = 1.0 \times 10^{-6} \ \mathrm{s}\). Thus, \(\frac{v}{c^2} \Delta x = 0.6 \times 1.0 \times 10^{-6} \ \mathrm{s} = 0.6 \ \mu\mathrm{s}\).
- Compute the term in brackets: \(\Delta t - \frac{v}{c^2} \Delta x = 1.0 \ \mu\mathrm{s} - 0.6 \ \mu\mathrm{s} = 0.4 \ \mu\mathrm{s}\).
- Multiply by \(\gamma\): \(\Delta t' = 1.25 \times 0.4 \ \mu\mathrm{s} = 0.5 \ \mu\mathrm{s}\).
Units check. The term \(\frac{v}{c^2} \Delta x\) has units \(\frac{\mathrm{m/s}}{\mathrm{m^2/s^2}} \mathrm{m} = \frac{\mathrm{m/s}}{\mathrm{m^2/s^2}} \mathrm{m} = \mathrm{s}\). This matches the units of \(\Delta t\). The final result is in seconds, as required.
Three quick tests
- Does the principle of relativity allow an observer to determine their absolute velocity by measuring the speed of light?
Answer
No. The second postulate states that the speed of light is invariant for all inertial observers, so no experiment involving light can reveal absolute motion. - If two events are simultaneous in frame \(S\) and separated by a distance \(\Delta x\), are they simultaneous in a frame \(S'\) moving at \(v\) relative to \(S\)?
Answer
No. The time difference in \(S'\) is \(\Delta t' = -\gamma \frac{v}{c^2} \Delta x\), which is non-zero if \(\Delta x \neq 0\) and \(v \neq 0\). - What is the value of the Lorentz factor \(\gamma\) when \(v = 0\)?
Answer
\(\gamma = 1\). This recovers the Galilean transformation, where time and space are independent.
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.