IB Physics Topic A.5.3 Lorentz Transformations Revision Notes (HL)

IB Physics Topic A.5.3 Lorentz Transformations Revision Notes (HL) | RevisionDojo

Written by official IB examiners

Lorentz Transformations

Definition

Lorentz transformations

The Lorentz transformations relate the coordinates of an event in one inertial frame to those in another moving at a constant velocity relative to the first.

They ensure that the speed of light remains constant in all inertial frames, a key postulate of special relativity.

Deriving the Lorentz Transformations

Consider two inertial frames, $S$ and $S'$:

$S'$ moves with velocity $v$ relative to $S$ along the x-axis.
The origins of $S$ and $S'$ coincide at $t = t' = 0$.

Hint

The Lorentz transformations relate the coordinates $(x, t)$ in $S$ to the coordinates $(x', t')$ in $S'$.

Transformation for Position

In Galilean relativity, the position transformation is: $$
x' = x - vt
$$
However, this does not account for the constancy of the speed of light.
The Lorentz transformation modifies this to $x' = \gamma(x - vt)$, where $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is the Lorentz factor.

Transformation for Time

In Galilean relativity, time is absolute ($t' = t$).
However, in special relativity, time is relative.
The Lorentz transformation for time is: $$
t' = \gamma\left(t - \frac{vx}{c^2}\right)
$$

Tip

The term $\frac{vx}{c^2}$ accounts for the relativity of simultaneity, ensuring that events that are simultaneous in one frame may not be simultaneous in another.

Why the Lorentz Factor?

The Lorentz factor $\gamma$ ensures that the speed of light is constant in all inertial frames.
It approaches 1 at low speeds (recovering Galilean transformations) and increases significantly as $v$ approaches $c$.

Example question

Let’s apply the Lorentz transformations to an event at $x = 100$ m and $t = 2$ s in frame $S$. If $S'$ moves at $0.8c$ relative to $S$, what are the coordinates in $S'$?

Solution

Calculate $\gamma$: $$\gamma = \frac{1}{\sqrt{1 - \frac{(0.8c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{5}{3}.$$
Apply the transformations: $$x' = \gamma(x - vt) = \frac{5}{3}\left(100 - (0.8c)(2)\right) =$$ $$= \frac{5}{3}\left(100 - 0.8\cdot 3.0\times 10^8\cdot 2\right) \approx -8.0\times 10^8\ \text{m},$$ $$t' = \gamma\left(t - \frac{vx}{c^2}\right)= \frac{5}{3}\left(2 - \frac{0.8\cdot 3.0\times 10^8\cdot 100}{(3.0\times 10^8)^2}\right) \approx 3.33\ \text{s}.$$
The event occurs at $x' \approx -8.0 \times 10^8$ m and $t' \approx 3.33$ s (or $\frac{10}{3}$ s) in $S'$.

Time Dilation

Definition

Time dilation

Time dilation is the phenomenon where time passes more slowly for an observer in motion relative to a stationary observer.

Deriving the Time Dilation Formula

Consider a clock at rest in frame $S'$.
It ticks twice, separated by a proper time interval $\Delta t_0$ (measured in the clock’s rest frame).
An observer in frame $S$ sees the clock moving with velocity $v$.
Using the Lorentz transformation for time: $$
t' = \gamma\left(t - \frac{vx}{c^2}\right)
$$
Since the clock is at rest in $S'$, $\Delta x' = 0$.

Nice try, unfortunately this paywall isn't as easy to bypass as you think. Want to help devleop the site? Join the team at https://revisiondojo.com/join-us. exercitation voluptate cillum ullamco excepteur sint officia do tempor Lorem irure minim Lorem elit id voluptate reprehenderit voluptate laboris in nostrud qui non Lorem nostrud laborum culpa sit occaecat reprehenderit

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Duis aute irure dolor in reprehenderit

Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit.

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Unlock the rest of this chapter with a free account

Lorentz Transformations

Definition

Lorentz transformations

The Lorentz transformations relate the coordinates of an event in one inertial frame to those in another moving at a constant velocity relative to the first.

They ensure that the speed of light remains constant in all inertial frames, a key postulate of special relativity.

Deriving the Lorentz Transformations

Consider two inertial frames, $S$ and $S'$:

$S'$ moves with velocity $v$ relative to $S$ along the x-axis.
The origins of $S$ and $S'$ coincide at $t = t' = 0$.

Hint

The Lorentz transformations relate the coordinates $(x, t)$ in $S$ to the coordinates $(x', t')$ in $S'$.

Transformation for Position

In Galilean relativity, the position transformation is: $$
x' = x - vt
$$
However, this does not account for the constancy of the speed of light.
The Lorentz transformation modifies this to $x' = \gamma(x - vt)$, where $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ is the Lorentz factor.

Transformation for Time

In Galilean relativity, time is absolute ($t' = t$).
However, in special relativity, time is relative.
The Lorentz transformation for time is: $$
t' = \gamma\left(t - \frac{vx}{c^2}\right)
$$

Tip

The term $\frac{vx}{c^2}$ accounts for the relativity of simultaneity, ensuring that events that are simultaneous in one frame may not be simultaneous in another.

Why the Lorentz Factor?

The Lorentz factor $\gamma$ ensures that the speed of light is constant in all inertial frames.
It approaches 1 at low speeds (recovering Galilean transformations) and increases significantly as $v$ approaches $c$.

Example question

Let’s apply the Lorentz transformations to an event at $x = 100$ m and $t = 2$ s in frame $S$. If $S'$ moves at $0.8c$ relative to $S$, what are the coordinates in $S'$?

Solution

Calculate $\gamma$: $$\gamma = \frac{1}{\sqrt{1 - \frac{(0.8c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{5}{3}.$$
Apply the transformations: $$x' = \gamma(x - vt) = \frac{5}{3}\left(100 - (0.8c)(2)\right) =$$ $$= \frac{5}{3}\left(100 - 0.8\cdot 3.0\times 10^8\cdot 2\right) \approx -8.0\times 10^8\ \text{m},$$ $$t' = \gamma\left(t - \frac{vx}{c^2}\right)= \frac{5}{3}\left(2 - \frac{0.8\cdot 3.0\times 10^8\cdot 100}{(3.0\times 10^8)^2}\right) \approx 3.33\ \text{s}.$$
The event occurs at $x' \approx -8.0 \times 10^8$ m and $t' \approx 3.33$ s (or $\frac{10}{3}$ s) in $S'$.

Time Dilation

Definition

Time dilation

Time dilation is the phenomenon where time passes more slowly for an observer in motion relative to a stationary observer.

Deriving the Time Dilation Formula

Consider a clock at rest in frame $S'$.
It ticks twice, separated by a proper time interval $\Delta t_0$ (measured in the clock’s rest frame).
An observer in frame $S$ sees the clock moving with velocity $v$.
Using the Lorentz transformation for time: $$
t' = \gamma\left(t - \frac{vx}{c^2}\right)
$$
Since the clock is at rest in $S'$, $\Delta x' = 0$.

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

IB Physics A.5.3 Lorentz Transformations Notes

Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

Lorentz Transformations

Deriving the Lorentz Transformations

Transformation for Position

Transformation for Time

Why the Lorentz Factor?

Time Dilation

Deriving the Time Dilation Formula

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Lorentz Transformations

Deriving the Lorentz Transformations

Transformation for Position

Transformation for Time

Why the Lorentz Factor?

Time Dilation

Deriving the Time Dilation Formula

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account