Theme A — Space, Time and Motion
A.5.4 Lorentz Transformations — IB Physics HL Revision Notes & Topical Questions
5 min read 8 parts Lesson 56 of 63 1 quiz
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
A.5.4 Lorentz Transformations — IB PHYSICS HL notes: Use the Lorentz factor and Lorentz transformations to convert space and time coordinates between frames.Galilean transformations cannot describe distances, times and speeds for objects near the speed of light.Einstein's postula…
- v is always less than c
- From S' to S
- For speeds much less than c, γ is about 1 and vx/c2 is negligible
- In Galilean relativity time is absolute
- A probe moving in the rocket frame takes longer to reach the front in the ground frame
- The equations are in the data booklet for x', t' and γ
Part 1 of 6Why Lorentz transformations are needed
Why Lorentz transformations are needed
Galilean transformations cannot describe distances, times and speeds for objects near the speed of light.
Einstein's postulates lead to the Lorentz transformation equations for the coordinates of an event in two inertial frames.
They are the Galilean equations corrected by the Lorentz factor.
Part 2 of 6Lorentz factor
γ
1 / √(1 − v2/c2)
v is always less than c
- v is always less than c so γ is always greater than 1.
γ is important for time dilation and length contraction.
When v
0.90c, c cancels: γ = 1/√(1 − 0.902), so c need not be entered numerically.
Always check that γ is greater than 1.
Part 3 of 6Lorentz equations
Frame S has coordinates (x, y, z, t) and S' moves at speed v relative to it, with coordinates (x', y', z', t').
From S to S'.
- x' = γ(x − vt)
- t' = γ(t − vx/c2)
From S' to S
- From S' to S the sign of v changes.
- x = γ(x' + vt')
- t = γ(t' + vx'/c2)
y
y y' and z = z' — motion is only along x.
Time is different in the two frames: t' ≠ t.
Part 4 of 6Galilean and Lorentz compared
| Lorentz (v near c) | Galilean (v much less than c) |
|---|---|
| x' = γ(x − vt) | x' = x − vt |
| x = γ(x' + vt') | x = x' + vt |
| t' = γ(t − vx/c2) | t' = t |
| t = γ(t' + vx'/c2) | t = t' |
| y = y', z = z' | y = y', z = z' |
1For speeds much less than c, γ is about 1 and vx/c2 is negligible
- For speeds much less than c, γ is about 1 and vx/c2 is negligible so Lorentz reduces to Galilean.
2In Galilean relativity time is absolute
- In Galilean relativity time is absolute in Lorentz transformations it differs between frames.
Part 5 of 6Using the transformations
1
List the known quantities and the speed v of the moving frame.
2
Decide in which frame the quantity is wanted and whether the object moves in both frames.
3
Calculate γ.
4
Use the correct transformation, with the right sign for v.
Using the transformations
If the event is in the rest frame being measured from, no transformation is needed.
A probe moving in the rocket frame takes longer to reach the front in the ground frame
- use t = γ(t' + vx'/c2).
The equations are in the data booklet for x', t' and γ
- The equations are in the data booklet for x', t' and γ but remember to change the sign for x and t.
Part 6 of 6More from the specification
More from the specification
Lorentz coordinate transformations for muons
- Lorentz coordinate transformations for muons calculating relativistic muon travel time between inertial frames uses Lorentz transformations linking Earth observer coordinates.
Observer
- Person or instrument measuring from within a frame.
Stationary
- Not moving.
Gradient
- Slope of a graph line.
Reference frame
- Coordinate system used by an observer.
Proper length
- Length measured in the rest frame of the object.
Constant velocity
- Zero acceleration.
Space-time interval
- Invariant quantity (cΔt)2 − (Δx)2.
Relativistic speed
- Relativistic speed a speed close to the speed of light, where Galilean velocity addition no longer gives correct results.
Key terms
| Term | Meaning |
|---|---|
| Lorentz factor | γ = 1/√(1 − v2/c2. |
| Lorentz transformation | Equations relating space and time coordinates in two inertial frames. |
| Event | Something happening at a particular place and time. |
| Moving frame S' | Frame moving at speed v relative to S. |
| Relativistic | Involving speeds close to the speed of light. |
| Galilean limit | Low-speed limit where γ ≈ 1. |
Avoid these mistakes
1
Forgetting the sign change when going from S' to S.
2
Dropping the vx/c2 term in the time transformation.
3
Using a γ value less than 1.
4
Entering 3.0 × 108 unnecessarily — c cancels when v is given in terms of c.
5
Using Lorentz equations for a measurement already in the frame of the event.
Marked practiceQuizzes for this lesson
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