[MUSIC] When a moving car shines its headlights, the emitted light always travels at the same speed. It doesn't matter how fast the car moves, but the car can't travel faster than light. What would you see if you could travel at the speed of light? Suppose that you could fly away from the sun at the speed of light, then light emitted by sun wouldn't be able to catch up to you. One of the results due to the invariance of the speed of light is that people in different inertial frames don't agree on the order of events. However, they all agree that the events actually took place. So why don't the observers in different reference frames agree on the order of events? In the cheesy spacetime analogy, a moving observer slices spacetime at an angle with respect to the stationary observer. Not only will the observer see events occurring in a different order, but they'll also measure length and time differently. This space time diagram shows lots of events taking place at different times and locations as experienced by a stationary observer. Stationary observer sees that all of the events on a horizontal line, take place at the same time but different locations. Another observer is moving with respect to the stationary observer in the x direction. They measure time and space using this space time diagram, where both time and space are rotated. The moving observer sees that all of the events on a slanted line take place at the same time. Both the stationary and moving observers will see the same events but since the coordinate systems are slanted, they do not agree on the times and locations of the events. However, the two observers can agree on the path that light takes. In both cases a beam of light follows the same path through space time. One way to measure time is with a light clock. Laser light is bounced off a mirror and the clock ticks when the laser light returns. For example, we could shine a laser beam at the moon. The Apollo astronauts left a mirror on the moon. So if we aim carefully, the light will bounce off the mirror and return to the earth taking 2.6 seconds. Sheldon has built a much smaller light clock that fits into his train. The laser is located on the floor, and the mirrors on the ceiling so that the light bounces up and down. The distance between the floor and the ceiling is d. In one full tech of the light clock, the light has to travel a distance that is two times d, taking a time that we'll call t0. Speed is defined as the distance traveled divided by the time taken to travel. Since light travels at the speed of light, c equals 2 times d divided by t0. We can rearrange this equation to solve for t0, the time between ticks. t0 equals 2 times d divided by c. Now Sheldon leaves the train and watches the train move to the right at speed v. His friends are on the train watching over the light clock apparatus. Since the train is moving at a constant velocity, the friends on the train see the light clock tick exactly the same as when the train was at rest. When Sheldon watches the train go by, the light is emitted from the floor when the train is at one location. By the time that the light hits the ceiling, the train has moved to the right. And the train has moved even further to the right when the light returns to the floor. The time for the light to travel from the laser source to the mirror and back again is represented by the symbol t. The vertical distance between the floor and the ceiling is still d but during the time t, the laser source has travelled through a horizontal distance of v times t. The light ray has to travel through a slanted distance that is two times z. Z is the hypotenuse of the triangle joining the source of light and the mirror. We can see that z is longer than the vertical distance between the floor and the ceiling. Since the light is now travelling through a longer distance but travelling at the same speed, the time has to be longer too. Similar to the stationary case, the time for one tick of the clock is 2 times z divided by c. Sheldon thinks that the moving clock's time interval t is longer than the time interval t0 measured by the friends. We call the lengthening of the time intervals for moving clocks, time dilation. If we do the math, we find that t is equal to t naught divided by the square root of 1 minus v squared over c square. During the train ride, less time has passed for the friends so they haven't aged as much as Sheldon. We say that the moving clocks run slow with respect to a stationary clock. Time dilation was tested experimentally in an airplane in 1971. The European Space Agency is planning to launch a new experiment called ACES, short for Atomic Clock Ensemble in Space and it'll be added to the International Space Station. This experiment will test the time dilation due to the spacecraft's motion around the Earth. Another weird prediction of Einstein is called length contraction, which makes moving objects look smaller in the direction of their motion. A proper length of an object is its length when measured at rest. Sheldon can measure the length of the train by bouncing light down its length and measuring the time that it takes the light to travel back and forth across the train. We call this length L0. When he leaves the train and watches it move to the right, his friends can shine the laser beam allowing him to measure the length of the train which is denoted L. When the light moves from the back of the train to the front, the train has moved forwards, so the light travels a little bit farther than the true length of the train. When the light travels from the front to the back, it travels through a slightly shorter distance than the true length of the train. The final result is that the measured length of a moving object is smaller than the length measured when it's at rest. This effect is called Lorentz contraction, and only affects the length of the object in the direction of motion. The dimensions of a moving object perpendicular to the direction of motion are not affected by the motion. This function with the square root is called the Lorentz factor or sometimes the gamma factor. The Lorentz Factor is a convenient tool when discussing length contraction and time dilation because it converts these equations into these vastly simpler forms. Here is a table of some relativistic speeds given in fractions of the speed of light c and their associated Lorentz Factor. Someone travelling at 10% the speed of light or 0.1 c sees a 0.5% shortening in the length of stationary objects and a 0.5% increase in the flow of time. That's not very much, and even at half the speed of light, it's only a 15% change. You'll need to be traveling at close to the speed of light to see any significant change. At 90% the speed of light, the Lorentz factor is more than two. Here's a graph illustrating how quickly the Lorentz factor changes at very high speeds. Now it's time to think about the issue of relatives in special relativity. Since a moving observer appears to be stationary in their own frame of reference, who can we trust when we say that links have been contracted and clocks have been slowed? To illustrate the problem, we need to introduce you to the twins paradox.