If relativistic effects are to modify time and length values by less than one percent
then the Lorentz factor, 𝛾, must be less than one point zero one. At what relative velocity
is 𝛾 equal to one point zero one?
Let’s call the relative velocity we’re solving for 𝑣. We want to find that velocity 𝑣 such that 𝛾 is one point zero one. We’ll start by recalling the mathematical expression for the Lorentz factor, 𝛾.
The Lorentz factor, 𝛾, is defined as one divided by the square root of one minus 𝑣
squared over 𝑐 squared. 𝑣 represents our relative speed, and 𝑐 is the speed of light.
Applying this relationship to our scenario, we have been given 𝛾, that’s one point zero one, and we want to solve for 𝑣, the relative velocity, at which 𝛾 has their value.
To do that let’s rearrange this Lorentz factor equation. We start by squaring both sides. Looking at the right-hand side, since one squared is one, the right side of our equation becomes one over one minus 𝑣 squared over 𝑐 squared.
Next we can multiply both sides of the equation by the quantity one minus 𝑣
squared divided by 𝑐 squared, which cancels that entire term from the right-hand side of our equation.
So we now have an equation that reads the quantity one minus 𝑣 squared over
𝑐 squared times 𝛾 squared is equal to one. Multiplying through by 𝛾 squared on the left-hand side then adding 𝑣 squared divided by 𝑐 squared times 𝛾 squared to both sides cancels that term out on the left side of the equation.
And then subtracting one from each side cancels that one from the right side. We’re left with an equation that reads 𝛾 squared minus one is equal to 𝑣
squared divided by 𝑐 squared times 𝛾 squared.
Our next algebraic step is to multiply both sides of the equation by 𝑐 squared
divided by 𝛾 squared, which cancels the 𝛾 squared and 𝑐 squared terms from the right-hand side
Finally we take the square root of both sides. This cancels out the square and square root terms on the right side, leaving us with a simplified equation for the relative velocity 𝑣.
𝑣 is equal to the speed of light 𝑐 divided by the Lorentz factor 𝛾 times the square root of 𝛾 squared minus
one. Rather than give our final answer for 𝑣 in units of meters per second, we’ll give
it in terms of the speed of light 𝑐.
We can now plug in for the value of 𝛾, one point zero one. The relative velocity when 𝛾 is one point zero one is equal to zero
point one four zero times 𝑐, the speed of light.
This is the relative velocity that matches a Lorentz factor of one point zero one.