Professor Dave here, let's learn about time dilation. We went over the postulates of special relativity, and we saw that in order for the second one to be true, we have to rethink how time works. If the speed of light is the same in all inertial reference frames, then someone standing on earth and someone moving in a very fast spaceship must be experiencing the passage of time at different rates. This concept is called time dilation, and it tells us that the faster you move relative to the speed of light, the slower time will pass for you. This concept is best demonstrated through a thought experiment. Let's say that there is an observer on earth and another in a spaceship moving with constant velocity relative to the earth. On the spaceship there is a device that emits a light pulse, which hits a mirror and bounces back to a detector that sits right by the device. Each observer has a clock, and we pretend that the observer on earth can see what's going on in the spaceship. For the astronaut it's easy to tell how long it takes for the light to make the trip. If speed is distance over time, then time is distance over speed, so it's just twice the distance between the device and the mirror to get there and back divided by the speed of light, or 2D over c. Let's call this time interval delta T zero. For the person on earth it's a little different, because the spaceship is moving relative to earth and therefore the light pulse has a horizontal component of motion in addition to the vertical one. The light follows this diagonal path up to the mirror and back, which is much longer than the other path. If this horizontal distance is L then we use the Pythagorean theorem to get s, which is root D squared plus L squared, so twice this term will be the distance traveled by the light pulse according to the observer on earth. But the speed of light is always the same, so if the light travels a greater distance at the same speed, the observer on earth must register a greater time interval, delta t, for this event to occur than the person on the spaceship. This is why this phenomenon is called time dilation, because to dilate is to expand and the time interval delta t has expanded relative to delta t 0. This means that the clock on the spaceship is running more slowly than the one on earth, as this is the only way for both observers to get the same result for the speed of light. We can calculate exactly how much slower in fact. Let's take this expression and realize that 2s, the distance travelled by the light pulse according to the person on earth, is equal to c delta t, since distance equals speed times time. That also means that the distance L, being half the distance traveled by the ship, is equal to the speed of the ship v times the time interval delta t over 2, since L is only half the total horizontal distance. So L equals v delta t over 2. If we do a bunch of algebra, we arrive at this expression relating delta t zero, the time interval for an observer at rest with respect to the events in question, also called the proper time interval, and delta t, the dilated time interval for an observer in motion with respect to the events. Remember that although the spaceship is clearly in motion, it is the observer on earth that is in motion relative to the events in question, as the light is inside the spaceship. This equation allows us to plug in some velocity for the ship and calculate the precise time dilation that will result, or calculate the proper time given the dilated if we rearrange slightly. Let's try an example. Say an astronaut is in a ship moving at 80 percent light speed. If they move at this constant speed for what they measure as one year, how long will that be for someone on earth? Well let's take the equation and put one year for the proper time interval. Then we can plug in 0.8c for the velocity, since that is equal to 80 percent the speed of light. We square that and find that the c squared terms cancel. We subtract from one and take the square root, so dividing the proper time interval of one year by 0.6, we get 1.67 years as the dilated time interval. This concept, as abstract as it seems, actually has practical applications. GPS satellites in orbit around the Earth are moving so fast that there are relativistic effects. Because of this, the clocks on board have to be programmed with correction factors so that they report time the same way as clocks on earth, otherwise they would be out of sync and could not perform their intended function. We use the time dilation equation to do this to an incredible level of precision. Of course, satellites move very fast, going around the earth in a couple hours, but even for an object moving more slowly, like a plane, time dilation still occurs and has been measured with ultra precise clocks. If one clock stays on the ground and another one goes in a plane that flies around the earth, these two clocks will show a discrepancy after the long flight of a few billionths of a second. Not a big difference, but we can still measure it and see that it precisely matches the prediction made by special relativity. In the time dilation equation, as v gets very small relative to the speed of light, this term approaches 0. 1 minus 0 is 1, and root 1 is 1, so for objects moving nowhere near the speed of light the dilated time interval is essentially identical to the proper time interval. That's why we don't notice time dilation on earth as we walk and run around, but for something like space travel it becomes significant. The best example of this is something called the twin paradox. Let's say there is a set of twins and one goes on a journey through space while the other remains on earth. The traveler gets in a super fast ship and travels for 10 years, then turns around and comes back. Time dilation doesn't only affect mechanical clocks but biological clocks as well, so upon returning to earth, they are no longer the same age. The supposed paradox arises when we consider inertial reference frames. To the twin on earth, the other one sped away and was moving very fast for 20 years, so the traveler should have aged less. But to the one in the ship, it is the earth that sped away and came back 20 years later, so the twin on earth should have aged less. But when they reunite, one must be younger than the other, so which one is it? The paradox is resolved when we realize that the ship was the one doing all the accelerating and decelerating, making it a non inertial reference frame, where relativity does not apply. So we must treat the earth as the inertial reference frame. This means that the twin who left on the ship will indeed return home younger than the one who stayed. By precisely how much will depend on how fast the ship was moving. As strange as this all sounds, time dilation is only one of the many ramifications of special relativity, which leaves us much more to discuss, 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:
