#### I am intrigued by your assertion that the speed of light is the fastest speed in the universe. It seems to me that we wouldn’t be able to determine the fastest speed achievable in the universe, just as we can’t find the final number in math. When we’re counting, there will always be x+1 so why would calculating the speed of objects in our universe be any different? — GL

Your comparison between the limitless counting numbers and the limited speeds in the universe is an interesting one because it points out a fundamental difference between the older Galilean/Newtonian understanding of the universe and the newer Einsteinian understanding. The older understanding claims that velocities can be added in the same way that counting numbers can be added and that there is thus no limit to the speeds that can exist in our universe. For example, if you are jogging eastward at 5 mph and a second runner passes you traveling eastward 5 mph faster, then a person watching the two of you from a stationary vantage point sees the second runner traveling eastward at 10 mph. The velocities add, so that 5 mph + 5 mph = 10 mph. If the second runner is now passed by a third runner, who is traveling eastward 5 mph faster than the second runner, then the stationary observer sees that third runner traveling eastward at 15 mph. And so it goes. As long as velocities add in this manner, objects can reach any speed they like.

At this point, you might assert that velocities **do** add and that objects **should** be able to reach any speed. But that’s not the case. The modern, relativistic understanding of the universe says that even at these small speeds, velocities don’t quite add. To the stationary observer, the second runner travels at only 9.9999999999999994 mph and the third runner at only 14.9999999999999988 mph. As you can see, when two or more velocities are combined, the final velocity isn’t quite as large as the simple sum. What that means is that the velocity you observe in another object is inextricably related to your own motion. This interrelatedness is part of the theory of relativity—that observers who are moving relative to one another will see space and time somewhat differently.

For objects traveling close to the speed of light, the failure of velocity addition becomes quite severe. For example, if one spaceship travels past the earth at half the speed of light and the people in that spaceship watch a second spaceship pass them at half the speed of light in the same direction, then a person on earth will see the second spaceship traveling only four-fifths of the speed of light. As you can see, relativity is making it difficult to reach the speed of light. In fact, it’s impossible to reach the speed of light! No matter how you combine velocities, no observer will ever see a massive object reach or exceed the speed of light. The only objects that can reach the speed of light are objects without mass and they can only travel at the speed of light.

So while the counting numbers obey simple addition and go on forever, velocities do not obey simple addition and have a firm limit—the speed of light. The additive counting numbers are an example of a mathematical group that extends infinitely in both directions, but there are many examples of groups that do not extend to infinity. The group that describes relativistic, real-world velocities is one such group. You can visualize another simple limited group—the one associated with walking around the surface of the earth. No matter how much you try, you can’t walk more than a certain distance northward. While it seems as though steps northward add, so that 5 steps north plus 5 steps north equals 10 steps north, things aren’t quite that simple. Eventually you reach the north pole and start walking south!