# Video: Determining the Lorentz Factor Produced by a Relative Velocity

Ed Burdette

If relativistic effects are to modify time and length values by less than 1%, then the Lorentz factor 𝛾 must be less than 1.01. At what relative velocity is 𝛾 = 1.01?

03:34

### Video Transcript

If relativistic effects are to modify time and length values by less than one percent then the Lorentz factor, πΎ, must be less than one point zero one. At what relative velocity is πΎ equal to one point zero one?

Letβs call the relative velocity weβre solving for π£. We want to find that velocity π£ such that πΎ is one point zero one. Weβll start by recalling the mathematical expression for the Lorentz factor, πΎ.

The Lorentz factor, πΎ, is defined as one divided by the square root of one minus π£ squared over π squared. π£ represents our relative speed, and π is the speed of light.

Applying this relationship to our scenario, we have been given πΎ, thatβs one point zero one, and we want to solve for π£, the relative velocity, at which πΎ has their value.

To do that letβ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 π£ squared over π squared.

Next we can multiply both sides of the equation by the quantity one minus π£ squared divided by π 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 π£ squared over π squared times πΎ squared is equal to one. Multiplying through by πΎ squared on the left-hand side then adding π£ squared divided by π squared times πΎ 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. Weβre left with an equation that reads πΎ squared minus one is equal to π£ squared divided by π squared times πΎ squared.

Our next algebraic step is to multiply both sides of the equation by π squared divided by πΎ squared, which cancels the πΎ squared and π squared terms from the right-hand side of equation.

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 π£.

π£ is equal to the speed of light π divided by the Lorentz factor πΎ times the square root of πΎ squared minus one. Rather than give our final answer for π£ in units of meters per second, weβll give it in terms of the speed of light π.

We can now plug in for the value of πΎ, one point zero one. The relative velocity when πΎ is one point zero one is equal to zero point one four zero times π, the speed of light.

This is the relative velocity that matches a Lorentz factor of one point zero one.