IB Physics HL A.5 Galilean and special relativity Notes

A frame is not just an observer; it is the coordinate axes and synchronized clocks that observer uses to label events. Galilean and special relativity compare descriptions made in different inertial frames. The restriction to inertial frames matters because the standard transformation equations assume constant relative velocity, not acceleration or rotation.

Define a reference frame and explain why an inertial reference frame is required when using Galilean or Lorentz transformations.

Imagine a smooth train moving at constant velocity. A ball thrown inside the train follows Newton’s laws just as it would on the platform, even though platform and train observers assign different velocities to the ball. This is the idea of Galilean relativity: uniform motion is relative, while the laws of mechanics are the same in every inertial frame. The model assumes absolute time, which special relativity later replaces.

Explain Galilean relativity using two inertial observers moving at constant relative velocity. Include what is frame-dependent and what is the same.

Define the frames before substituting: S is the reference frame in which x and t are measured, and S′ is moving relative to S. The term vt is the displacement of the S′ origin as measured in S, so subtract it to locate the event relative to S′. Because Galilean relativity assumes the same time in all inertial frames, t′ = t. If the relative direction is defined differently, the sign changes.

Frame S′ moves at speed v in the +x direction relative to S. State the Galilean transformations for x and t, and identify one assumption behind them.

Velocity addition compares the same object as measured in two inertial frames. If the moving frame S′ is travelling in the positive direction relative to S, an object’s velocity measured in S′ is its S-frame velocity minus the frame velocity. Use signs carefully: an object moving opposite the positive direction has a negative velocity. At relativistic speeds this formula can wrongly predict speeds greater than c, so it must be replaced by the relativistic velocity addition formula.

State the Galilean velocity addition equation for S′ moving at +v relative to S, and explain why the equation is not used for light.

The first postulate extends the Galilean idea beyond mechanics: all laws of physics keep the same form in inertial frames. The second postulate is the new ingredient: light in vacuum is measured to travel at c by every inertial observer. To keep both postulates true, space and time measurements must transform with Lorentz transformations rather than Galilean transformations.

State the two postulates of special relativity and explain why they require a replacement for Galilean velocity addition at high speeds.

Use Lorentz transformations when relative speeds are a significant fraction of c and the frames are inertial. Define which frame is moving before choosing signs. Unlike Galilean transformations, time is mixed with position, so two events that have the same time in one frame may not have the same time in another. The transformations preserve the speed of light and the spacetime interval.

For S′ moving at speed v in the +x direction relative to S, state the Lorentz transformations for x′ and t′ and define γ.

Relativistic velocity addition replaces u′ = u - v when speeds are high. The denominator prevents inertial observers from measuring an object or light signal above c. Define which frame is moving and use signs consistently before substituting. The one-dimensional formula here is for velocities along the relative motion direction.

State the relativistic velocity addition formula for S′ moving at +v relative to S, and explain one way it differs from Galilean velocity addition.

Special relativity allows different observers to disagree about the time separation and space separation of two events, but they agree on the spacetime interval. Calculate the interval from coordinate differences, not from a single coordinate. A light signal has (Δs)^2 = 0 because it travels Δx = cΔt. Timelike events can be causally connected; spacelike-separated events cannot influence each other without exceeding c.

For two events separated by Δx and Δt in an inertial frame, state the spacetime interval formula used here and explain what it means for the interval to be invariant.

Proper quantities are not automatically the values measured by the observer in the question. Proper time belongs to the frame where the start and end events happen at one place, so one clock can time both events. Proper length belongs to the frame where the object is at rest. Once the proper quantity is identified, time dilation and length contraction formulae can be applied in the following cards.

Define proper time interval and proper length, and state how each proper quantity is identified from the measuring frame.

First identify the two events being timed, such as creation and decay of a particle or two ticks of a moving clock. The proper time is measured by a clock at rest with those events, so the events happen at one position in that clock’s frame. Other inertial observers who see the clock moving measure a longer interval by the factor γ.

A particle has a lifetime Δt0 in its own rest frame and moves at speed v relative to the laboratory. State the lifetime measured in the laboratory and explain which interval is proper.

First identify the object’s rest frame; the length measured there is L0. An observer who sees the object moving measures a shorter length along the motion direction by L = L0/γ. The simultaneity condition matters: to measure a moving object’s length, the observer must record both endpoints at the same time in their own frame.

A spacecraft of rest length L0 moves at speed v past an observer. State the length measured by the observer and explain what must be simultaneous in that observer’s frame.

Lorentz transformations mix time and position, so two events with the same time coordinate in one frame can have different time coordinates in another frame. The key condition is spatial separation along the direction of relative motion. This is why length contraction measurements require simultaneous endpoint positions in the measuring frame: simultaneity itself is frame-dependent.

Two events occur simultaneously in frame S but at different positions along the x-axis. Explain why they need not be simultaneous in frame S′ moving relative to S.

Read a spacetime diagram as a map of events, not as a normal position-time graph. The vertical coordinate is often ct, not just t, so the slope/angle interpretation differs from a standard x-t graph. Timelike world lines for massive objects stay inside the light lines, while light itself follows the 45° boundary.

Describe the meaning of an event, a world line, and a 45° line on an x-ct spacetime diagram.

Be precise about the angle or gradient being used. The IB-style relation tanθ = v/c uses θ measured from the ct-axis, not from the x-axis. For a plotted line, the gradient is Δ(ct)/Δx = c/v, so a steeper line actually means a smaller speed. Massive particles must have world lines inside the light lines, so their angle from the ct-axis is less than 45°.

On an x-ct spacetime diagram, a particle world line makes an angle θ with the ct-axis. State how to determine the particle speed and explain what a 45° line represents.

The muon experiment is powerful because the same observation can be explained consistently in two frames. Earth observers say fast muon clocks run slow, increasing the observed lifetime. Muons say their own lifetime is normal, but the atmosphere is moving toward them and is contracted along the direction of motion. The agreement between these explanations supports the Lorentz transformation model.

Explain how the observation of atmospheric muons at Earth’s surface supports special relativity, giving both the Earth-frame and muon-frame descriptions.

A.5 is mostly a frame-selection topic. Decide whether the situation is low-speed Galilean or high-speed relativistic, then identify the proper quantity or invariant. Formula use is safest after you state the frame convention, sign convention, and condition: inertial frames, same event pair, same axis of relative motion, proper time, proper length, or invariant interval.