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Theme A — Space, Time and Motion

A.5 Galilean and Special Relativity — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 52 of 63 1 quiz

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Lesson summary

A.5 Galilean and Special Relativity — IB PHYSICS HL notes: Use reference frames, Galilean and Lorentz transformations, time dilation, length contraction, simultaneity and space-time diagrams.A reference frame = a set of coordinates to record the position and time of events.An ine…

  • An inertial frame
  • There is no absolute reference frame
  • Galilean relativity
  • First postulate
  • Second postulate
  • Space and time become relative at speeds near c

Part 1 of 7Reference frames and Galilean relativity

Reference frames and Galilean relativity
A reference frame
A reference frame a set of coordinates to record the position and time of events.
An inertial frame
An inertial frame a non-accelerating frame — all inertial frames move at constant velocity relative to one another.
There is no absolute reference frame
  • There is no absolute reference frame everything moves relative to everything else.
Galilean relativity
Newton's laws are the same in all inertial frames.
Galilean transformations (frame S' moving at v along x).
  • x' = x − vt, y' = y, z' = z, t' = t.
  • Velocity addition u = u' + v.

Part 2 of 7Postulates of special relativity

1First postulate
  • First postulate the laws of physics are the same in all inertial frames.
2Second postulate
  • Second postulate the speed of light in a vacuum, c, is the same in all inertial frames.
3Space and time become relative at speeds near c
  • Galilean velocity addition fails and would allow speeds above c.
4A moving flashlight's light is measured at c by every observer
  • A moving flashlight's light is measured at c by every observer never c + v.

Part 3 of 7Lorentz transformations

Lorentz transformations
Lorentz factor γ
1/√(1 − v2/c2) — always greater than 1.
x'
γ(x − vt) and t' = γ(t − vx/c2).
Inverse transformations (to get x, t)
  • Inverse transformations (to get x, t) change the sign of v: x = γ(x' + vt'), t = γ(t' + vx'/c2).
Velocity addition
  • Velocity addition u' = (u − v)/(1 − uv/c2) and u = (u' + v)/(1 + u'v/c2).
For v much less than c these reduce to the Galilean equations.

Part 4 of 7Invariants and proper quantities

1Space-time interval (Δs)2
(cΔt)2 − (Δx)2 — the same in all inertial frames.
2Proper time t0
Proper time t0 the time between two events measured in the frame where they occur at the same place.
3Proper length L0
Proper length L0 the length measured in the frame where the object is at rest.
4
c, proper time, proper length and the interval are invariant.

Part 5 of 7Time dilation and length contraction

Time dilation
  • Δt = γΔt0: a moving clock runs slower, so the time measured is longer than the proper time.
Length contraction
  • L = L0/γ: lengths along the motion are shorter when measured by an observer moving relative to the object.
Both are consequences of the constant speed of light and are symmetric between the two observers.
Muons
  • Muons made in the upper atmosphere with a half-life of 1.6 μs, they reach the ground because of time dilation or length contraction.

Part 6 of 7Simultaneity and space-time diagrams

1Relativity of simultaneity
  • Relativity of simultaneity events at different places that are simultaneous in one frame are not in another.
2
Space-time diagrams plot x horizontally and ct vertically.
  • World lines have gradient c/v and must be steeper than 1 — a photon has gradient 1.
  • tan θ = v/c, where θ is the angle between the world line and the ct axis.
3A second frame S' has tilted ct' and x' axes
  • A second frame S' has tilted ct' and x' axes each symmetrical about the light line.
4
Read time dilation, length contraction and simultaneity directly from the diagram.

Part 7 of 7More from the specification

Time dilation
  • Time dilation a moving clock runs slower according to a stationary observer, t = gamma times proper time.
Proper time
  • Proper time the time measured by an observer at rest relative to the events, such as a traveller on a rocket.
Length contraction
  • Length contraction distance along the direction of travel appears shorter to an observer moving relative to it.
Muon experiment
  • Muon experiment muons reach Earth's surface because their lifetime is dilated, or equivalently because the distance contracts in their frame.
Galilean relativity
  • Galilean relativity velocities add simply at low speed, but light speed is the same for all observers, as in the train thought experiment.
Stationary
  • Not moving.
Distance
  • Total length of path travelled, a scalar.

Key terms

TermMeaning
Inertial frameNon-accelerating frame of reference.
Lorentz factorγ = 1/√(1 − v2/c2.
Proper timeTime between events measured where they occur at the same place.
Proper lengthLength measured in the rest frame of the object.
Space-time intervalInvariant quantity (cΔt)2 − (Δx)2.
World linePath of an object on a space-time diagram.
SimultaneityEvents occurring at the same time.

Avoid these mistakes

1
Using Galilean velocity addition for speeds close to c.
2
Forgetting that c is the same in all frames — there is no c'.
3
Taking the proper time or length as belonging to the stationary frame — it is the frame where the events or the object are at rest.
4
Using the same sign for x and x' transformations — the sign of v changes.
5
Describing a world line with gradient less than 1.

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