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Reference frames and Lorentz transformations IB DP Physics Study Notes

Reference frames and Lorentz transformations IB DP Physics Study Notes - 2025 Syllabus

Reference frames and Lorentz transformations IB DP Physics Study Notes

Reference frames and Lorentz transformations IB DP Physics Study Notes at  IITian Academy  focus on  specific topic and type of questions asked in actual exam. Study Notes focus on IB Physics syllabus with Students should understand

  • reference frames

  • that Newton’s laws of motion are the same in all inertial reference frames and this is known as Galilean relativity

  • that in Galilean relativity the position $x^{\prime}$ and time $t^{\prime}$ of an event are given by $x^{\prime}=x-v t$ and $t^{\prime}=t$ that Galilean transformation equations lead to the velocity addition equation as given by $u^{\prime}=u-v$

  • the two postulates of special relativity

  • that the postulates of special relativity lead to the Lorentz transformation equations for the coordinates of an event in two inertial reference frames as given by

    $\begin{aligned}
    & x^{\prime}=\gamma(x-v t) \\
    & t^{\prime}=\gamma\left(t-\frac{v x}{c^2}\right)
    \end{aligned}$

    where $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

  • that Lorentz transformation equations lead to the relativistic velocity addition equation as given by

    $
    u^{\prime}=\frac{u-v}{1-\frac{u v}{c^2}}
    $

Standard level and higher level: There is no Standard level content
Additional higher level: 8 hours

IB DP Physics 2025 -Study Notes -All Topics

Reference Frames

 

  • A reference frame provides a perspective from which motion is observed and measured.
  • The choice of a reference frame affects how we describe motion but does not alter the fundamental laws of physics.
  • Transformations between reference frames allow us to compare observations made by different observers.

Rest frame of an object:

frame of reference in which the object is not moving

Inertial frame of reference:

frame of reference in which an isolated object experiencing no force moves on a straight line at constant velocity

 Galilean invariance of Newton’s laws

  • Newton’s laws remain unchanged in all inertial frames, meaning they are frame-independent as long as the frame is non-accelerating.
  • Motion observed in one inertial frame can be translated into another using Galilean transformations without altering the physical behavior of objects.
  • This principle laid the foundation for classical mechanics but does not account for relativistic effects, which are considered in Einstein’s special relativity.

(NI):

$S$ inertial frame of reference $\Longleftrightarrow\left(\vec{F}=0 \Rightarrow \vec{r}=\vec{r}_0+\vec{v}_0 t\right)$ (straight line, in $S^{\prime}: \vec{r}^{\prime}=\vec{r}-\vec{v} t=\vec{r}_0+\vec{v}_0 t-\vec{v} t=\vec{r}_0+\left(\vec{v}_0-\vec{v}\right) t$ const. velocity)
$\Rightarrow S^{\prime}$ is inertial frame of reference

(N2):

in $S: \vec{F}=m \vec{a}=m \frac{\mathrm{~d}^2 \vec{r}}{\mathrm{~d} t^2}$
same equation of
in $S^{\prime}: \vec{F}=m \vec{a}^{\prime}=m \frac{\mathrm{~d}^2 \vec{r}^{\prime}}{\mathrm{d} t^2}=m \frac{\mathrm{~d}^2}{\mathrm{~d} t^2}(\vec{r}-\vec{v} t)=m \frac{\mathrm{~d}^2 \vec{r}}{\mathrm{~d} t^2}$ motion in $S$ and $S^{\prime}$

(N3)’:

total momentum $\vec{P}=\sum m_i \vec{v}_i$ conserved in $S$
in $S^{\prime}: \vec{P}^{\prime}=\sum_i m_i \vec{v}_i{ }^{\prime}=\sum_i^i m_i \frac{\mathrm{~d} \vec{r}_i^{\prime}}{\mathrm{d} t}=\sum_i m_i \frac{\mathrm{~d}}{\mathrm{~d} t}\left(\vec{r}_i-\vec{v} t\right)=\vec{P}-M \vec{v}$ const.
Newtons’s laws of motion are the same in all inertial frames of reference, in agreement with the Galilean relativity principle.

Position and Time in Galilean Relativity

In Galilean relativity, space and time are treated as absolute and independent quantities.
The transformation equations that relate coordinates between two inertial reference frames moving at a constant velocity \( v \) relative to each other are:
\[
x’ = x – vt
\]
\[
t’ = t
\]
where \( x’ \) and \( t’ \) are the position and time in the moving frame, and \( x \) and \( t \) are those in the stationary frame.
 This implies that time progresses uniformly across all reference frames, meaning all observers agree on the same time interval for an event regardless of their relative motion.

Velocity Addition in Galilean Relativity

 The velocity of an object measured in one reference frame can be related to its velocity in another frame using the Galilean velocity transformation:
\[
u’ = u – v
\]
where:
 \( u’ \) is the velocity of the object in the moving reference frame,
 \( u \) is the velocity of the object in the stationary frame, and
 \( v \) is the velocity of the moving frame relative to the stationary one.
 This straightforward velocity addition rule works well for low-speed scenarios but fails at speeds approaching the speed of light, where relativistic effects become significant.

Fundamental Postulates of Special Relativity

 Postulate 1: The Principle of Relativity

  •  All inertial reference frames are equivalent, meaning the fundamental laws of physics remain unchanged for any observer moving at constant velocity.
  •  No single inertial frame holds a special status; experiments conducted in different inertial frames yield identical results.

 Postulate 2: The Invariance of the Speed of Light

  •  The speed of light in vacuum is a universal constant (\( c \approx 3.0 \times 10^8 \) m/s) for all observers, regardless of their relative motion or the motion of the light source.
  •  This contradicts Galilean relativity, where speeds simply add up, and leads to the need for Einstein’s Lorentz transformations, which redefine space and time at high velocities.

Lorentz Transformation Equations

  • The Lorentz transformation equations describe how space and time coordinates change between two inertial reference frames moving at a constant velocity \( v \) relative to each other.
  • Unlike Galilean transformations, these equations account for relativistic effects, ensuring that the speed of light remains constant in all frames.
  •  The transformations are given by:

$
x’ = \gamma (x – vt)
$

$
t’ = \gamma \left( t – \frac{vx}{c^2} \right)
$

where:
\( x’, t’ \) are the position and time in the moving reference frame.
\( x, t \) are the position and time in the stationary reference frame.
\( v \) is the relative velocity between the frames.
\( c \) is the speed of light.
\( \gamma \) is the Lorentz factor, defined as:

$
\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}
$

The Lorentz Factor (γ) and Its Effects

  • The Lorentz factor determines how much relativistic effects alter space and time:
  • At low speeds (\( v \ll c \)), \( \gamma \approx 1 \), so relativistic effects are negligible.
  • At high speeds (\( v \approx c \)), \( \gamma \) increases rapidly, leading to time dilation and length contraction.
  • \( \gamma \geq 1 \) for all speeds \( 0 \leq v < c \), meaning no object with mass can reach or exceed the speed of light.

Implications of the Lorentz Transformations

  • Time Dilation: Moving clocks run slower compared to a stationary observer’s clock. This means that time passes more slowly for a fast-moving object relative to an observer at rest.
  • Length Contraction: Objects moving at relativistic speeds appear shorter in the direction of motion when observed from a stationary frame.
  • Relativity of Simultaneity: Two events that are simultaneous in one frame may not be simultaneous in another frame moving relative to the first.

These effects become significant only at speeds approaching the speed of light, shaping our understanding of space and time in relativistic physics.

Relativistic Velocity 

In classical mechanics, velocities simply add or subtract when moving in the same or opposite directions. However, when dealing with high velocities (approaching the speed of light), the velocity addition formula is modified by the Lorentz factor. The relativistic velocity addition equation, derived from the Lorentz transformation equations, governs the addition of velocities in special relativity:

$$
w=\frac{u-v}{1-\frac{u v}{c^2}}
$$

Where:

w is the resultant velocity
 $u$ is the velocity of the first frame relative to the ground
 $v$ is the velocity of the object relative to the moving frame
 $c$ is the speed of light $(3 \times 10^8 \mathrm{~m} / \mathrm{s})$

IB Physics Reference frames and Lorentz transformations Exam Style Worked Out Questions

Question

The spacetime diagram shows coordinate axes of reference frames of Earth $(x, c t)$ and of a spaceship $\left(x^{\prime}, c t^{\prime}\right)$. Three events $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are plotted.

Which statement is correct about the order of the events according to an observer on the spaceship?
A. $\mathrm{P}$ and $\mathrm{Q}$ are simultaneous, $\mathrm{R}$ happens later.
B. $\mathrm{Q}$ and $\mathrm{R}$ are simultaneous, $\mathrm{P}$ happens earlier.
C. $\mathrm{Q}$ and $\mathrm{R}$ are simultaneous, $\mathrm{P}$ happens later.
D. $\mathrm{P}$ and $\mathrm{R}$ are simultaneous, $\mathrm{Q}$ happens earlier.

▶️Answer/Explanation

Ans:D

Question

The spacetime diagram shows an inertial reference frame $S$ and a second inertial frame $S$ ‘ that is moving relative to $\mathrm{S}$.

The origins of the frames coincide when the clocks in both frames show zero.

Event $\mathrm{X}$ is shown for the $\mathrm{S}$ reference frame.
Which event occurs at the same position in the $\mathrm{S}^{\prime}$ reference frame as $\mathrm{X}$ ?

▶️Answer/Explanation

Ans:C

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