Frames, coordinates and events

A reference frame is a coordinate system with an origin, axes and clocks, used to measure the positions and times of events. In this topic, we usually take one space coordinate, along the direction of relative motion, and one time coordinate. Real spacetime has three space coordinates and one time coordinate, but the IB treatment keeps the algebra one-dimensional.

An event is one occurrence, fixed by where and when it happens in a chosen reference frame. A spark at the nose of a rocket, a detector click, or two clocks meeting are events. Relativity is not mainly about objects looking strange. It deals with different inertial frames giving different coordinates to the same events.

An observer is a reference frame fitted with synchronized clocks and rulers throughout the frame, not just a person looking through a telescope. That distinction matters: when we say an observer measures an event, picture a local clock and a local ruler at the place where the event occurs.

Inertial frames

An inertial reference frame is a reference frame that is not accelerating. In such a frame, a free object stays at rest or moves with constant velocity. Two inertial frames may move relative to one another, but only with constant velocity. A rotating frame, or a frame fixed in an accelerating car, is not inertial; you would need to introduce extra fictitious forces to make Newton’s laws work there.

For this topic, every transformation is between inertial frames moving at constant relative velocity along one shared axis. The restriction is intentional: once acceleration enters, the simple special-relativity formulae need more careful handling.

The Galilean idea

Galilean relativity says that Newton’s laws of motion keep the same form in every inertial reference frame. If you’re inside a sealed cabin moving smoothly, no experiment based on Newtonian mechanics can tell you whether the cabin is “really at rest” or moving at constant velocity. Rest depends on the frame.

For two frames whose origins coincide at zero time, with frame $S'$ moving at constant velocity along the positive x-axis of frame $S$, the Galilean transformation is

Built into this is the quiet assumption of absolute time: all inertial observers agree on time intervals.

The matching velocity addition equation is

Watch the sign of $v$. If $S'$ moves in the negative x-direction, then $v$ is negative and the same formula still works.

Why Galilean transformations cannot be the whole story

Galilean velocity addition works neatly for buses, walkways and laboratory carts. Light is the problem. If ordinary velocity addition applied to a light beam, different inertial observers would measure different speeds for that same light. Electromagnetism does not allow this: light in vacuum has one universal speed. That crack in the Newtonian picture is what special relativity repairs.

The two postulates

A postulate is an assumption taken as the starting point for a theory. Einstein’s special relativity starts with two postulates:

• The laws of physics are the same in all inertial reference frames.
• The speed of light in vacuum, $c$, is the same for all inertial observers, where $c$ is the speed of light in vacuum (m s$^{-1}$).

The second postulate is the one that clashes most with everyday mechanics. Sound needs a medium, so the motion of the source and observer relative to the air affects what they measure. Light does not need a material medium in vacuum, and its speed is not found by adding the speed of the source to the speed of the wave. That is why the Doppler equation for light has a relativistic form, while the school-level Doppler equation for sound depends on motion relative to the medium.

Lorentz factor

The shift from Galilean to relativistic transformations is controlled by the Lorentz factor, a dimensionless factor that measures how strongly relativistic effects show up:

At ordinary speeds $v \ll c$, $\gamma$ is very close to 1, so Galilean mechanics works as an excellent approximation. As $v$ approaches $c$, $\gamma$ rises very rapidly; mathematically, this is one sign that massive objects cannot be accelerated to the speed of light.

Lorentz transformations

The Lorentz transformation relates measurements of the same event in two inertial frames moving at constant relative velocity. For motion along the x-axis,

$x' = \gamma(x-vt)$

$t' = \gamma\left(t-\frac{vx}{c^2}\right)$

These equations are supplied in the guide; their derivation is not required. The key idea is hard to miss: space and time get mixed together. The transformed position depends on time, and the transformed time depends on position. Here, “space” and “time” stop acting like separate absolute backgrounds and become spacetime.

When $v \ll c$, $\gamma \approx 1$ and $vx/c^2$ is negligible for ordinary scales, so the Lorentz transformations reduce to the Galilean transformations. In relativistic contexts, equations of linear motion are adapted by replacing transformations between frames with Lorentz transformations whenever speeds are a significant fraction of $c$, not by throwing away earlier mechanics.

Special relativity also sets a fundamental limit: no information or material object travels faster than light in vacuum. Physics has other limits as well, such as measurement uncertainty in quantum mechanics and practical limits set by detector resolution, but the speed limit $c$ is built into the structure of spacetime itself.

Why velocity addition changes

With Galilean velocity addition, an observer moving towards a light beam could measure that beam as travelling faster than $c$. That clashes with Einstein’s second postulate. Using the Lorentz transformations gives the relativistic velocity addition equation instead:

$u' = \frac{u-v}{1-\frac{uv}{c^2}}$

There are two quick checks. If $u=c$, then $u'=c$, so every inertial observer still measures light in vacuum at speed $c$. If $u$ and $v$ are both much smaller than $c$, then $uv/c^2$ is tiny, and the equation reduces to $u' \approx u-v$, which is the Galilean result.

For signs, ask first what the Galilean answer would suggest. When the frames move in the same positive direction, use the subtraction form above. When one velocity points the other way, make that velocity negative; the denominator then adjusts on its own. No extra rule is being added here — it’s just signed velocity used consistently.

Invariant quantities

An invariant quantity is a physical quantity with the same value in all inertial reference frames. In Galilean relativity, time intervals are invariant. In special relativity, time and distance do not stay invariant separately; the invariant quantity combines both.

The spacetime interval is an invariant separation between two events, defined for one space dimension by

For three space dimensions, the interval becomes $(\Delta s)^2=(c\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2$, although IB questions in this topic normally keep the motion along one axis.

The sign of $(\Delta s)^2$ tells us what kind of cause-and-effect link is possible:

• A time-like interval is a spacetime separation for which $(\Delta s)^2>0$, so the events can occur at the same place in some inertial frame and one event could influence the other at a speed below $c$.
• A space-like interval is a spacetime separation for which $(\Delta s)^2<0$, so the events can occur at the same time in some inertial frame but cannot be causally connected without faster-than-light influence.
• A light-like interval is a spacetime separation for which $(\Delta s)^2=0$, so the events can be connected by light travelling in vacuum.

Proper time and proper length

A proper time interval is the time interval between two events measured in the inertial frame where the two events occur at the same position. It is the time shown by a clock that is present at both events.

A proper length is the length of an object measured in the inertial frame where the object is at rest. To measure a length in any frame, the positions of both ends must be measured simultaneously in that same frame. That last phrase is the bit students often skip; don’t. It is essential.

Time dilation

Time dilation is the effect where a time interval measured in a frame in which the two events happen at different positions is longer than the proper time interval. The equation is

The guide doesn’t require the derivation, but the physical meaning matters: moving clocks are measured to run slow. More time passes in the observing frame between ticks of the moving clock.

A light-clock picture helps. In the clock’s own frame, the light travels straight up and down. In a frame where the clock is moving, the light takes a longer diagonal path. Since both observers measure the same light speed $c$, that longer path goes with a longer time interval.

Time dilation is symmetrical between inertial frames: each inertial observer measures the other observer’s clock as running slow. There’s no contradiction here, because each observer uses a different set of synchronized clocks spread through their own frame.

Length contraction

Length contraction is the effect where the length of an object measured in a frame in which the object is moving is shorter than its proper length. The equation is

The contraction happens only along the direction of relative motion. In this one-dimensional treatment, there is no contraction perpendicular to the relative motion.

These equations replace the old assumption that time intervals and lengths stay the same in every inertial frame. At everyday speeds $\gamma\approx 1$, so the Newtonian equations of linear motion are still a very good approximation. At relativistic speeds, you need to be precise about which frame supplies each time, distance and speed.

Simultaneity is not absolute

Simultaneity means that two events have the same time coordinate in a chosen reference frame. In Galilean relativity, simultaneity is absolute because $t'=t$. In special relativity, it depends on the inertial frame.

The Lorentz time transformation shows why. For two events,

Suppose two events are simultaneous in S, so $\Delta t=0$. If they happen at different positions, with $\Delta x eq 0$, then $\Delta t' eq 0$. Only events at the same position can be simultaneous in both frames.

A useful picture is a lamp at the centre of a moving carriage. In the carriage frame, the light reaches the front and rear ends at the same time. In the platform frame, the rear end moves towards the light while the front end moves away, so the rear event happens first. This is not a delay-of-seeing problem. We imagine each frame filled with local observers, using synchronized clocks at the event positions.

On a spacetime diagram, events that are simultaneous in one frame lie along a line parallel to that frame’s space axis. Since different frames have differently tilted space axes, they cut spacetime into “same time” slices in different ways.

Reading a spacetime diagram

A spacetime diagram graphs position along one space axis horizontally, with time on the vertical axis. In most diagrams the vertical axis is labelled $ct$ rather than $t$. Since $ct$ has units of metres, both axes can be read as distance axes.

A world line is the curve on a spacetime diagram that traces the sequence of events occupied by a particle or object. In this course, moving particles have constant velocity, so their world lines are straight. A particle at rest in the diagram’s frame has a vertical world line. A particle moving at constant speed has a straight, tilted world line. A photon in vacuum has a world line at 45° to the x and $ct$ axes.

The angle between a particle’s world line and the time axis gives its speed:

For light, $v=c$, so $\tan\theta=1$ and $\theta=45^\circ$.

Two frames on one diagram

For a second inertial frame $S'$ moving relative to $S$, the $ct'$-axis is the world line of the $S'$ origin. The $x'$-axis marks the line of simultaneity for $S'$ at $t'=0$. Both axes tilt towards the photon lines; they do not rotate like ordinary Euclidean axes.

The scales on $ct$, $ct'$, $x$ and $x'$ are not identical. They are set using lines of constant spacetime interval, often drawn as invariant hyperbolas. So don't just measure along a tilted axis using the same ruler scale as the untilted axis, unless the diagram has been explicitly calibrated.

Time dilation, length contraction and simultaneity can all be shown geometrically on these diagrams. Lines parallel to a frame’s space axis show equal time in that frame. Lines parallel to a frame’s time axis show objects at rest in that frame. The algebra of Lorentz transformations and the geometry of spacetime diagrams are two ways of describing the same physics.

Atmospheric muons

A muon is an unstable subatomic particle made high in the atmosphere when high-energy cosmic rays hit air molecules. Muons have a short proper mean lifetime, so if you use a Newtonian calculation, almost none should make it down to detectors at Earth’s surface.

Yet plenty of muons are detected at ground level. From Earth’s frame, time dilation explains this: fast-moving muons have longer lifetimes, so more of them survive the trip down. From the muon’s own frame, the same result comes from length contraction: the atmosphere’s thickness is contracted, putting the ground much closer. These aren’t two competing physical effects. They are two frame descriptions of the same invariant spacetime story.

A typical experiment compares muon counts at high altitude with counts at lower altitude. Without relativity, the predicted lower-altitude count comes out much too small. When time dilation is included, the prediction matches observation. That makes muon decay a particularly clean piece of evidence: not a rocket-based thought experiment, but a measurable particle-decay result.

Muon decay experiments therefore provide experimental evidence for both time dilation and length contraction. They also show that special relativity isn’t just for science fiction; it is built into particle physics, accelerator physics and technologies that rely on very precise timing.