Why can't relative velocities add up to more than the speed of light? (Intermediate)

I'm an adult (at least age wise) and I'm trying to understand why nothing can travel faster than the speed of light.

What would happen, if let's say, you're traveling through space in a bus five miles long moving at one mile less than the speed of light. From the back of the bus you start running down the aisle at two miles an hour. In relation to the bus, your going two miles an hour, but in relation to anything outside the bus, wouldn't you now be traveling at one mile faster than the speed of light? And if so, what would things look like out the window?

This is a good question, and the answer is that the rules for adding up relative velocities change if the speeds involved are a significant fraction of the speed of light. In that regime, classical physics breaks down, and one needs to describe the world using special relativity. In "common-sense" physics (ie. the realm that we're used to deal with), if the bus is moving at speed v and a person in it is moving at speed u relative to the bus (and in the same direction) then the person's speed relative to the ground is just

w = u+v.

This is called the Galillean transformation. In the theory of special relativity though, the speed relative to the ground becomes the more general formula

w = (u + v)/(u*v/c*c + 1)

(now a Lorenztian transformation, after physicist H. Lorentz) so you can see that even if u equals c and v also equals c then w equals c as well! This formula, proved every day in particle accelerators, says that you can't go faster than c relative to anything!

Note that, if u and v are both very small compared to c, then the factor u*v/c*c becomes extremely small, in fact so small that we don't notice it at speeds we're used to and w = u + v works just fine.

See here for a neat little web page calculator explaining this and some other equations in special relativity more. Try putting in numbers close to c for the velocities, to check that the sum is never more than c. Also try putting in small velocities (compared to c) and compare the answer to just straight addition. 

This page was last updated on January 28, 2019.

About the Author

Karen Masters

Karen Masters

Karen was a graduate student at Cornell from 2000-2005. She went on to work as a researcher in galaxy redshift surveys at Harvard University, and is now on the Faculty at the University of Portsmouth back in her home country of the UK. Her research lately has focused on using the morphology of galaxies to give clues to their formation and evolution. She is the Project Scientist for the Galaxy Zoo project.

Twitter:  @KarenLMasters
Website:  http://icg.port.ac.uk/~mastersk/

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