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Galilean & Special Relativity
Reference frames, Galilean and special relativity, Lorentz transformations, time dilation, length contraction and spacetime diagrams. HL only.
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Key Concepts, Galilean & Special Relativity
Reference Frames and Galilean Relativity
A reference frame is a coordinate system used to describe the position and motion of objects. An inertial reference frame is one that is not accelerating, so Newton's laws of motion hold true within it. Galilean relativity states that the laws of physics (specifically Newton's laws) are identical in all inertial reference frames. If you are moving at constant velocity relative to another observer, you cannot perform any mechanical experiment to determine which of you is "really" moving. The Galilean transformation relates the coordinates of an event in one frame S to another frame S' moving at velocity v along the x-axis: x' = x - vt and t' = t. Time is the same in all frames under Galilean relativity, which becomes one of the key assumptions that special relativity overturns.
Galilean Velocity Addition
The Galilean velocity addition formula is u' = u - v, where u is the velocity of an object measured in frame S, v is the velocity of frame S' relative to S, and u' is the velocity of the object measured in S'. This makes intuitive sense: if you are walking at 2 m/s on a train moving at 20 m/s, a stationary observer on the platform sees you moving at 22 m/s. However, this formula breaks down at speeds approaching the speed of light, where the relativistic velocity addition formula must be used instead.
The Two Postulates of Special Relativity
Einstein's special relativity rests on two postulates. First: the laws of physics are the same in all inertial reference frames (this extends Galilean relativity to all physics, including electromagnetism, not just mechanics). Second: the speed of light in a vacuum, c = 3 × 10⁸ m/s, is the same for all observers, regardless of the motion of the source or the observer. The second postulate is the revolutionary one. It directly contradicts Galilean velocity addition and forces us to abandon the idea of absolute time and space.
Lorentz Transformations
When relative speeds are a significant fraction of c, the Galilean transformation must be replaced by the Lorentz transformation. For a frame S' moving at velocity v along the x-axis relative to S, the Lorentz transformation for position and time is: x' = γ(x - vt) and t' = γ(t - vx/c²), where γ (gamma) is the Lorentz factor defined as γ = 1/√(1 - v²/c²). Note that γ ≥ 1 always, and increases rapidly as v approaches c. The inverse transformation swaps the sign of v. These equations encode both time dilation and length contraction and reduce to the Galilean transformation when v << c.
Relativistic Velocity Addition
Galilean velocity addition fails at high speeds because it would predict speeds greater than c. The relativistic velocity addition formula is: u' = (u - v) / (1 - uv/c²), where u is the velocity in frame S, v is the velocity of S' relative to S, and u' is the velocity in S'. This formula ensures that if u = c (i.e. a photon), then u' = c in all frames, consistent with the second postulate. For low speeds (u << c and v << c), the denominator approaches 1 and the formula reduces to Galilean velocity addition, as expected.
Time Dilation
Time dilation is the phenomenon where a moving clock ticks more slowly than a stationary one. The proper time interval (Δt₀) is the time measured by a clock that is at rest relative to the events being timed (both events happen at the same location in that clock's frame). The dilated time interval measured in a frame moving at speed v relative to the clock is: Δt = γΔt₀, where γ ≥ 1. Since γ ≥ 1, we always have Δt ≥ Δt₀: moving clocks run slow. The effect is negligible at everyday speeds but becomes very significant at speeds approaching c.
Length Contraction
Length contraction is the phenomenon where a moving object appears shorter (in the direction of motion) to a stationary observer. The proper length (L₀) is the length measured in the rest frame of the object, where both ends are at rest. The contracted length measured in a frame where the object is moving at speed v is: L = L₀/γ. Since γ ≥ 1, we always have L ≤ L₀: moving objects are shorter in the direction of motion. Length contraction only applies in the direction of relative motion; dimensions perpendicular to motion are unchanged.
The Spacetime Interval and Proper Quantities
While time and space measurements differ between frames, the spacetime interval (s²) is invariant: all observers agree on its value. It is defined as s² = c²(Δt)² - (Δx)². If s² > 0, the interval is time-like (one event could causally influence the other). If s² = 0, the events are connected by a light signal. If s² < 0, the interval is space-like (no causal connection is possible). Proper time is the time interval measured in the rest frame of the clock (events at the same location). Proper length is the length measured in the rest frame of the object.
Relativity of Simultaneity
One of the most counterintuitive results of special relativity is that two events which are simultaneous in one reference frame are not necessarily simultaneous in another. If two events happen at the same time (Δt = 0) but at different locations (Δx ≠ 0) in frame S, then in a frame S' moving relative to S, the time difference is Δt' = -γvΔx/c², which is not zero. The order of spatially separated events can even be reversed in different frames, as long as neither event can causally influence the other (i.e. the interval is space-like).
Spacetime Diagrams and World Lines
A spacetime diagram (Minkowski diagram) plots position (x) on the horizontal axis and time (ct) on the vertical axis, scaling time by c so both axes have units of metres. The world line of an object is the path it traces on the spacetime diagram as time passes. A stationary object has a vertical world line. An object moving at constant velocity traces a straight line tilted from the vertical. The world line of a light signal is always at 45° to both axes (since it travels x = ct). The angle φ that a world line makes with the ct-axis satisfies tan φ = v/c. This angle can be used to read off the relative speed of the object.
Muon Decay as Evidence for Relativity
Muons are subatomic particles created in the upper atmosphere (about 15 km above Earth's surface) when cosmic rays collide with air molecules. They travel towards Earth at speeds close to c. A muon's proper half-life is about 1.5 μs, so classically we would expect only a tiny fraction to survive to ground level. In practice, far more muons are detected at the surface than classical physics predicts. From Earth's frame, the muon's internal clock runs slow due to time dilation, so it appears to live much longer and travel further before decaying. From the muon's frame, the distance to Earth's surface is length-contracted, so it needs far less time to reach the ground. Both perspectives agree on the number of muons detected, providing compelling experimental evidence for both time dilation and length contraction.
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Frequently Asked Questions, IB Physics Galilean & Special Relativity
What is Galilean & Special Relativity in IB Physics? ↓
Reference frames, Galilean and special relativity, Lorentz transformations, time dilation, length contraction and spacetime diagrams. HL only. This topic is part of Theme A (Space, Time & Motion) in the current IB Physics syllabus.
Is Galilean & Special Relativity SL or HL in IB Physics? ↓
Galilean & Special Relativity is an HL-only topic. It is not assessed in the SL IB Physics exam.
What equations do I need for IB Physics Galilean & Special Relativity? ↓
The key equations for Galilean & Special Relativity are covered in the concept tutorial above. For a structured set of notes with all equations, conditions and worked examples, the GradePod Exam Pack includes a revision note template for every topic.
What are common exam mistakes in IB Physics Galilean & Special Relativity? ↓
Common mistakes are covered in detail in the exam technique video above. The GradePod Exam Pack also includes exam-style questions with mark schemes so you can see exactly how marks are awarded and where students typically drop them.
How do I revise Galilean & Special Relativity for the IB Physics exam? ↓
Follow the GradePod three-step method. First, watch the concept tutorial and tick off each learning objective on the checklist above as you go. Second, watch the exam technique video to see how IB-style questions are answered under exam conditions. Third, use the Exam Pack to practise independently with knowledge questions, exam questions and mark schemes. That's it. It works. I promise.