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, plus 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, pinned down 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 assigning 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. This matters: when we say an observer measures an event, picture a local clock and 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 can move relative to one another, but only at constant velocity. A rotating frame, or a frame fixed in an accelerating car, is not inertial; extra fictitious forces would have to be introduced 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. That restriction is deliberate: once acceleration is involved, 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 carry out a Newtonian mechanics experiment inside a sealed cabin moving smoothly, nothing in that experiment tells 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

[x' = x - vt \quad ext{and} \quad t' = t]

where (x') is the position coordinate in S′ (m), (x) is the position coordinate in S (m), (v) is the velocity of S′ relative to S (m s⁻¹), (t) is the time coordinate in S (s), and (t') is the time coordinate in S′ (s). Built into this is the quiet assumption of absolute time: all inertial observers agree on time intervals.

The matching velocity addition equation is

[u' = u - v]

where (u') is the object’s velocity measured in S′ (m s⁻¹), and (u) is the object’s velocity measured in S (m s⁻¹). 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 very well for buses, walkways, and laboratory carts. For light, it breaks down. If ordinary velocity addition applied to a light beam, different inertial observers would measure different speeds for the same light. Electromagnetism gives a different result: light in vacuum has one universal speed. That is the crack in the Newtonian picture that special relativity repairs.

The two postulates

A postulate is an assumption used as the starting point for a theory. Einstein’s special relativity starts from 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 feels least like everyday mechanics. Sound needs a medium, so the motion of the source and the observer relative to the air matters. Light needs no material medium in vacuum, and its speed is not worked out 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 appear:

[\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}]

where (\gamma) is the Lorentz factor (unitless). At ordinary speeds (v \ll c), (\gamma) is very close to 1, so Galilean mechanics returns as an excellent approximation. As (v) approaches (c), (\gamma) rises very rapidly; this is one mathematical sign that massive objects cannot be accelerated to the speed of light.

Lorentz transformations

The Lorentz transformation is a coordinate transformation that links 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 easy to miss but hard to ignore: 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 as 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. So equations of linear motion are adapted in relativistic contexts 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 gives a fundamental limit: no information or material object travels faster than light in vacuum. Physics has other limits too, 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 it as faster than (c). That would clash 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}}]

Two quick checks make the result feel sensible. 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, so the equation reduces to (u' \approx u-v), the Galilean result.

For signs, ask first what the Galilean answer would do. When the frames move in the same positive direction, use the subtraction form above. When one velocity points the other way, give it a negative sign; the denominator then changes by itself. No extra rule is needed. 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 stay 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

[(\Delta s)^2 = (c\Delta t)^2 - (\Delta x)^2]

where (\Delta s) is the spacetime interval between two events (m), (\Delta t) is the time interval between the events in a chosen frame (s), and (\Delta x) is the spatial separation between the events in that frame (m). The symbol (\Delta) means “change in” or “difference between final and initial values”.

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

The sign of ((\Delta s)^2) shows what kind of cause-and-effect connection 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 measured by a clock 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. Students often skip that last phrase; don’t. It matters.

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

[\Delta t = \gamma\Delta t_0]

where (\Delta t_0) is the proper time interval between the two events (s). You don’t need the derivation for the guide, but you do need the physical meaning: 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 gives 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

[L = \frac{L_0}{\gamma}]

where (L) is the contracted length measured in a frame where the object is moving (m), and (L_0) is the proper length measured in the object’s rest frame (m). 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 older assumption that time intervals and lengths are the same in every inertial frame. At everyday speeds (\gamma\approx 1), so the Newtonian equations of linear motion still work as a very good approximation. At relativistic speeds, you need to be exact 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 stays absolute because (t'=t). In special relativity, it depends on which inertial frame you use.

The Lorentz time transformation shows why this has to happen. For two events,

[\Delta t' = \gamma\left(\Delta t - \frac{v\Delta x}{c^2}\right)]

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. From 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. Each frame is treated as if it has local observers with synchronized clocks at the event positions.

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

Reading a spacetime diagram

A spacetime diagram shows position along one space axis on the horizontal axis, with time on the vertical axis. Usually the vertical axis is labelled (ct), not (t). Since (ct) has units of metres, you can treat both axes as distance axes.

A world line is a curve on a spacetime diagram that shows the sequence of events occupied by a particle or object. In this course, moving particles have constant velocity only, so their world lines are straight. A particle at rest in the diagram’s frame gives a vertical world line. One moving at constant speed gives 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 tells you its speed:

[ an heta = \frac{v}{c}]

where ( heta) is the angle between the world line and the (ct)-axis (rad or degrees, depending on the diagram scale). For light, (v=c), so ( an heta=1) and ( heta=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 is the line of simultaneity for S′ at (t'=0). Both axes tilt towards the photon lines; they don’t rotate like ordinary Euclidean axes.

The scales on (ct), (ct'), (x) and (x') are not the same. They are set using lines of constant spacetime interval, often drawn as invariant hyperbolas. So don’t simply measure along a tilted axis with 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 languages for the same physics.