Professor Dave again, let's talk about length contraction. Einstein's special relativity was a huge step in physics, and even though time dilation seems like the weirdest thing you've ever heard, there's plenty more to get through with this theory. As it happens, it's not just time that's relative, it's length as well, and as we will find out the math tells us that when you approach the speed of light, your measurement of space changes just like time. This is because if an observer on earth and an observer in a fast spaceship register different spans of time for an event they must also be recording different distances since both observers agree on the same relative velocity between them. Specifically, the faster you go, the smaller objects seem to be, and the shorter the distance you perceive yourself to be traveling. This is a phenomenon called length contraction. Here we can see two versions of a spaceship traveling very fast to a faraway celestial object. In one, we see things from the perspective of an observer on earth, and in the other we see things from the perspective of someone on board the spaceship. The earthbound observer sees the spaceship moving at some velocity v, notices time moving at the familiar rate, and can measure some distance for the journey which we label as L0, and refer to as the proper length, as this is the length that is measured by an observer that is at rest with respect to the objects demarcating the distance. But on the spaceship the only thing that is the same is the relative velocity v, though in this case it represents the velocity of Earth and the destination as they move relative to the ship. And because this velocity must be the same as for the earthbound observer, everything else must be different. Because of time dilation, the time interval will be different, delta t0 rather than delta t, and we label the contracted length of the journey as L. The relationship between L zero and L is given by this equation, which can be derived from the time-dilation equation. Not only do the two observers arrive at different values for the length of the journey, they even arrive at different values for the length of the spaceship, as this dimension is parallel to the direction of travel. The earthbound observer would see the ship as being much shorter than the astronaut does. In fact, this discrepancy in length neatly explains how the two observers could perceive different rates for the passage of time. It also explains how fast-moving particles can defy certain expectations, because at such speeds, time slows down and the distance traveled contracts, so a particle like a muon with a half-life of a millionth of a second at rest, is able to exist for longer and travel further than expected when moving near the speed of light, due to relativistic effects. At this point let's take a moment to make sure we understand how to assign delta t and L values. Delta t zero is the proper time interval, which is the time interval as measured by the inertial reference frame where the two events occur in the same location. For a space journey that's the spaceship, because the Earth leaves and the destination arrives while the ship goes nowhere. Everyone else, like someone on earth will register a longer time interval, delta t, for this journey. L0, the proper length however, is length as measured by an observer that is at rest with respect to the objects in question, like the earthbound observer is in this case. Anyone in motion with respect to the objects, like the person on the spaceship, will register some shorter length for this distance, L. This means that the proper time and length may not always be measured by the same observer, and we need to know how to assign these in order to do the math correctly. Let's say a ship is traveling from earth to another system at 90 percent light speed. The person on board measures the journey as being 8.2 light-years in length. How far away is the destination according to someone on earth? Well we can plug in 8.2 for L and 0.9c for v, then we just do some algebra and we should get 18.8 for L0, or the proper length between Earth and the destination. We still have more to go with special relativity, but first let's check comprehension. Thanks for watching, guys. Subscribe to my channel for more tutorials, support me on patreon so I can keep making content, and as always feel free to email me:
