The most fundamental postulate of relativity is

Postulate I: The laws of physics apply in all inertial reference frames.

The first ingredient here is a class of preferred reference frames. Simply put, an inertial (reference) frame is one without external forces. More precisely, an inertial frame is one in which an object initially at rest will remain at rest. Because of gravity, inertial frames must be in free fall — a spaceship with its drive turned off, or a falling elevator. Gravity causes additional complications, such as tidal effects, which force such freely falling frames to be small (compared to, say, the Earth); we will revisit this in the final chapter. But special relativity describes a world without gravity, so in practice we describe inertial frames in terms of relative motion at constant velocity, typically in the form of an idealized train.

Applied to mechanics, Postulate I is the principle of Galilean relativity. For instance, consider a ball thrown to the right with speed $u$. Ignoring things like gravity and air friction, the ball keeps moving at the same speed forever, since there are no forces acting on it. Try the same thing on a train, which is itself moving to the right with speed $v$. Then Galilean relativity leads to the same conclusion: As seen from the train, the ball moves to the right with speed $u$ forever. An observer on the ground, of course, sees the ball move with speed $u+v$; Galilean relativity insists only that both observers observe the same physics, namely the lack of acceleration due to the absence of any forces, but not necessarily the same speed. This situation is shown in Figure 2.1.

Figure 2.1: A passenger on a train throws a ball to the right. On an ideal train, it makes no difference whether the train is moving.

Einstein generalized Postulate I by applying it not just to mechanics, but also to electrodynamics. However, Maxwell's equations, presented in roughly their present form in the early 1860s, make explicit reference to the speed of light! In MKS units, Gauss' Law (Equation (1) of §11.6) involves the permittivity constant $\epsilon_0$, and Ampère's Law (Equation (4) of §11.6) involves the permeability constant $\mu_0$; both of these can be measured experimentally. But Maxwell's equations predict electromagnetic waves — including light — with a speed (in vacuum) of \begin{equation} \cc = \frac{1}{\sqrt{\epsilon_0\mu_0}} \end{equation} Thus, from some relatively simple experimental data, Maxwell's equations predict that the speed of light in vacuum is \begin{equation} c = 3\times10^8  \mathrm{\frac{m}{s}} \end{equation}

The famous Michelson-Morley experiment in 1887, at what is now Case Western Reserve University, set out to show that this speed is relative to the ether, so that we should be able to measure our own motion relative to the ether by measuring direction-dependent variations in $c$. Instead, the experiment showed that there were no such variations; Einstein argued that there is therefore no ether! 1) Postulate I together with Maxwell's equations therefore lead to

Postulate II: The speed of light is the same for all inertial observers.

The theory of special relativity follows from these two postulates, which were introduced by Einstein in [ ?? ] As we will now show, an immediate consequence of these postulates is that two inertial observers disagree about whether two events are simultaneous!

Thought experiments have been around since at least the time of Euclid. However, the German phrase Gedankenexperiment was not coined until the early 1800s, and was first translated into English in the late 1800s, only a few years prior to Einstein's discovery of special relativity. Einstein himself attributed his breakthrough to thought experiments, including a childhood attempt to imagine what one would see if one could ride along with a beam of light. However, it remains unclear exactly what path he took to reach his conclusions about the observer-dependence of simultaneity. One possible [ ?? ] is the following.

Figure 2.2: The same situation as the previous example, with the ball replaced by a lamp in the exact middle of the train. The light from the lamp reaches both ends of the train at the same time, regardless of whether the train is moving.

Consider a train at rest, with a lamp in the middle, as shown in Figure 2.2. After the light is turned on, light reaches both ends of the train at the same time, having traveled in both directions at constant speed $c$. Now try the same experiment on a moving train. This is still an inertial frame, and so, just as with the ball in the previous example, one obtains the same result, namely that the light reaches both ends of the train at the same time as seen by an observer on the train. However, the second postulate leads to a very different result for the observer on the ground. According to this postulate, the light travels at speed $c$ as seen from the ground, not the expected $c\pm v$. But, as seen from the ground, the ends of the train also move while the light is getting from the middle of the train to the ends! The rear wall “catches up” with the approaching light beam, while the front wall “runs away”! As shown in Figure 2.3, the net result of this is that the ground-based observer sees the light reach the rear of the train before it reaches the front; these two observers disagree about whether the light does or does not reach both ends of the train simultaneously.

Figure 2.3: An observer on the ground sees the train go past with its lamp, as in the previous example. Since this observer must also see the light travel with speed $c$, and since the ends of the train are moving while the light is traveling, this observer concludes that the light reaches the rear of the train before the front.