U4AOS1 Topic 2: Lorentz Factor

The Lorentz Factor is the factor by which time and length change for an object in motion.

The formula for the Lorentz Factor is

\[ \gamma = \frac{1}{\sqrt{1 − \frac{v^2}{c^2}}} \]

Where v is the velocity in m/s,

and c is the speed of light (or $3 \times 10^8$ m/s).

The Lorentz Factor is easy to calculate. However, it is also easy to make errors when inputting it into the calculator. To reduce the chance of these errors input it by each component.

Below is a graph that shows how the Lorentz Factor changes as the speed is increased

Solving Lorentz Factor

1. Solve for the fraction

\[\frac{v^2}{c^2}\]

(in most cases, the velocity will be given with respect to $c$ like $0.3c$. When this happens just cancel the $c$ terms. Otherwise use the actual value.)

2. Subtract the answer from 1

3. Find the square root using the ANS function

4. Divide 1 by ANS

Calculating the Lorentz Factor by following these steps will ensure that you don't make errors.

Calculating Speed from Lorentz

The process of finding the speed from the Lorentz Factor is very similar

1. Substitute the Lorentz Factor into the equation

2. Swap the Lorentz Factor and the square root

3. Square both sides

4. Subtract $1$ on both sides

5. Flip the sign

6. Multiply both sides by $c^2$

7. Take the square root of both sides

8. Depending on how the answer is required make sure to multiply by the value of the speed of light or leave it in terms of $c$

Example 1

Calculate the Lorentz factor of an object moving at $0.6c$.

We start off by writing the formula

\[ \gamma = \frac{1}{\sqrt{1 − \frac{v^2}{c^2}}} \]

Step 1: Solve for the fraction component

\[ \frac{v^2}{c^2} = \frac{(0.6c)^2}{c^2} = 0.6^2 = 0.36 \]

Step 2: Subtract the result from 1

\[ 1 - 0.36 = 0.64\]

Step 3: Calculate the square root

\[ \sqrt{0.64} = 0.8 \]

Step 4: Divide 1 by the result

\[ \frac{1}{0.8} = 1.25 \]

Hence the Lorentz Factor is 1.25

Example 2

Calculate the Lorentz factor of an object moving at $2.4 \times 10^8 $ m/s

We start off by writing the formula

\[ \gamma = \frac{1}{\sqrt{1 − \frac{v^2}{c^2}}} \]

Step 1: Solve for the fraction component

\[ \frac{v^2}{c^2} = \frac{(2.4 \times 10^8)^2}{(3 \times 10^8)^2} = \frac{2.4}{3} = 0.8 \]

Step 2: Subtract the result from 1

\[ 1 - 0.8 = 0.2\]

Step 3: Calculate the square root

\[ \sqrt{0.2} = 0.447 \]

Step 4: Divide 1 by the result

\[ \frac{1}{0.447} = 2.24 \]

Hence the Lorentz Factor is 2.24

Example 3

Calculate the speed in terms of c if the Lorentz factor is 15.

We start off by writing the formula

\[ \gamma = \frac{1}{\sqrt{1 − \frac{v^2}{c^2}}} \]

Step 1: Substitute the Lorentz Factor into the equation

\[ 15 = \frac{1}{\sqrt{1 − \frac{v^2}{c^2}}} \]

Step 2: Swap the Lorentz Factor and the square root

\[ sqrt{1 − \frac{v^2}{c^2}} = \frac{1}{15} \]

Step 3: Square both sides

\[ 1 − \frac{v^2}{c^2} = \frac{1}{15^2} \]

Step 4: Subtract 1 on both sides

\[ − \frac{v^2}{c^2} = \frac{1}{15^2} - 1 \]

Step 5: Flip the sign

\[ \frac{v^2}{c^2} = 0.995 \]

Step 6: Multiply both sides by c^2

\[ \frac{v^2}{c^2} \times c^2 = 0.995 c^2 \]

\[ v^2 = 0.995 c^2 \]

Step 7: Take the square root of both sides

\[ v = 0.9978c \]

Hence the velocity is $0.9978c$

Exercise 1

Practice Makes Perfect

Exercise 2

Practice Makes Perfect

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