If you decide to hold a party, you need to tell your guests where and when they should arrive. It's not enough to simply tell your friends where the party is without telling them when it occurs or vice-versa. Therefore, when we use the word event, we are using it to describe where and when something is happening. Events can describe things like your arrival at a party, the time you spilled your drink, or even something as simple as snapping your fingers can be an event. I could say that I snapped my fingers about five seconds ago at this location, right here. So, an event must include details of both the position and the time. Our universe has three spatial dimensions. So, a defined event could look something like X, Y, Z, and T where X, Y, and Z define the position and T defines the time. A good example of this occurs in the popular TV show, Big Bang Theory. In an episode entitled The Cushion Saturation, Dr. Sheldon Cooper explains the first time he sat on his favorite spot on the couch as follows, "In an ever-changing world, it is a single point of consistency. If my life were expressed as a function on a four-dimensional Cartesian coordinate system, that spot at that moment, I first sat on it would be zero, zero, zero, zero." Sheldon feels most comfortable and at home at the X, Y, Z coordinates of his spot, all zero, zero, zero. Although he can return to the spot on the couch many times, and he does, he can never return to the exact moment when he first sat on the couch to experience that event again. The reason for this is that the fourth coordinate in the event is time, T, and is always increasing as time passes. Suppose he originally sat down to enjoy a 40-minutes episode of Star Trek. We can describe the end of the show as an event that happened at zero, zero, zero, 40 minutes. The only way to return to the original event would be to use a time machine such as the one Reto Hofstetter took a ride in, in Big Bang Theory episode "The Nerdvana Annihilation." If we were to return to Sheldon's first time on the couch and saw Penny riding a skateboard past the scene, Sheldon would see her in his reference frame. Would Penny see the first moment he sits in that spot followed by a 40-minute episode of Star Trek in the same order Sheldon experiences it? Considering our previous discussion about slicing of space-time, do you think all observers see the two events in the same order or with the same time interval between the events? Let's explore this further using one of Sheldon's favorite objects, trains. Sheldon is preparing for a trip on the Napa Valley Wine Train in his favorite 1915 Pullman-Standard lounge car. Ever the physicist, Sheldon would like to conduct an experiment to test the consequences of a constant speed of light. To do this, he sets up a light source in the center train car and has two of his friends standing in his light detectors in the caboose and the engine at either ends of the train. While the train is in the station, we say that it is at rest. Here in the station, Sheldon conducts his first experiment by flashing the light and recording the arrival times that his friends measure at each end of the train. Since the distance to the light source is the same from both friends, they each detect the light at the same time. We call this a simultaneous detection. The train then leaves the station and begins its journey traveling at a constant speed through the mountains. Sheldon prepares the experiment again, this time with the train in motion. Since the train is traveling at a constant speed, once again, the flash of light arrives at each of his friends at precisely the same time. In both the stationary case and the one with the moving train, the observers are at rest with respect to Sheldon and the light source. As a result, they observe the pulses arriving simultaneously on each occasion. Sheldon wants to know what his experiment would look like if he was stationary with respect to a moving train, so he gets off at the next stop. He gets ahead of the train and get set up to redo his experiment as the train passes at a constant velocity. This time, the light source flashes the moment that it passes Sheldon. Since the train is moving from left to right, Sheldon observes the light pulse arrive at the caboose of the train first followed by the arrival of the pulse at the engine of the train second. Sheldon is puzzled, he observed the light pulse arrive at the back of the train first, while his friends on board report that the light pulse arrives simultaneously. In Sheldon's frame of reference, the light pulses do not arrive simultaneously even though his friend's frame of reference, they do. This disagreement between observers is a result of light traveling at a constant speed no matter how quickly the source of light moves. This is called the relativity of simultaneity and it describes that stationary and moving observers will report the order of events differently depending on their proper motion with respect to one another. As strange as it seems, both Sheldon and his friends on the train are right. In some cases, the order of events depends on the motion of the observer. Einstein explained this through the concepts of length contraction and time dilation. So, why don't observers in different reference frames agree on the order of events? Well, think back to our block of cheese space-time analogy. A moving observer relative to another observer actually slices up space-time differently. Not only can they potentially see events in different order than other observers, but they also measure length and time differently. Have a look at this graph which represents space-time for a stationary observer. A stationary observer sees objects as they naturally are and clocks run as expected. However, a moving observer sees space-time through a different slice. Here's what space-time looks like to a moving observer. Both the moving observer's time axis and distance axis have shifted. Physicists say that the time and space coordinates are rotated due to the motion of the observer. This is a convenient picture to paint, but what other real observable effects of this skewed reference frame? Well, earlier we mentioned that moving observers measure changes in length and time. Length contraction is given by the equation, L is equal to L zero times the square root of one minus the velocity of the observer divided by the speed of light squared. Well, this equation describes is how the length of the object appears to a moving observer. If I jump into a spacecraft, and I'm moving at nearly the speed of light, objects I observe will appear to shrink in my direction of motion. Something like the [inaudible] would pass by the ship appearing to be almost as flat as a pancake. It's not just length that changes though, time also changes. Time dilation is given by the equation t is equals to t zero divided by the square root of one minus the velocity of the observer divided by the speed of light squared. In this case, time is modified by dividing by these terms under the square root sign. So, instead of decreasing, the time observed by the moving observer increases. So, the moving observers see all external clocks slowing down the faster they move. Both length contraction and time dilation are effects that are measured by a moving observer, the faster the observer moves with respect to any object such as a clock, the thinner it appears and the slower it ticks. Hopefully, your head isn't spinning too much yet because we have to discuss an important issue with special relativity. Since a moving observer appears to be stationary in their own frame of reference, who can we trust when we say that the lengths have been contracted and the clocks have been slowed? To illustrate this problem, we need to introduce you to the twin paradox. Like to imagine that twins named Leia and Luke are preparing for an interstellar voyager. Leia will stay behind on the earth to monitor the journey while Luke, the adventurous one, takes off in his spaceship capable of reaching nearly the speed of light. Since Leia stays on the earth as Luke speeds away, Luke's clock appears to slow down the faster he zooms off his ship. But to Luke, Leia appears to be speeding away, so to Luke, Leia clocks have slowed down. Both observers see the other clock as being slow. While our own clocks are at regular speed, how can that be? Surely both observers can't be right. Welcome to the twin paradox, proposed by Einstein, not as a paradox but as a peculiarity of special relativity. In Einstein's 1905 paper, he reasoned that if two clocks were synchronized and one of them where to go on a lengthy journey, the traveling clock would return to the original location with its time lagging behind the stationary clock. However, since relativity says that either clocks could view the other as being the one in motion, that is the traveling clock could consider itself at rest and the stationary clock would therefore be the one moving away. Shouldn't the stationary clock be the one lagging behind when the other returns again? Let's watch as Luke flies to a nearby star system six light years away while Leia stays behind on the earth. Luke can travel at a significant fraction of the speed of light, say north 0.6 c. When we say that something is traveling at one c. It is the same as saying that it goes one light year per year. So, if Luke travels at 0.6 c he will travel north 0.6 of a light year for each year of travel. As such, Leia will say that Luke's journey takes 10 years. However, let's carefully assess what Luke observes on his particular journey. Luke sets his spaceship to travel at north 0.6 c. In doing so, the distance to his destination changes, at north 0.6 c, the length between the earth and the destination has shortened by 20 percent. So, instead of six light years, the star appears to be only 4.8 light years away from Luke's perspective. At north 0.6 c, Luke's time of arrival is only eight years after his departure from the earth. Meanwhile, Luke has tend to shift around and begins his journey back to the earth again at 0.6 c. So, the same length contraction applies instead of flying across six light years of space, Luke only flies the length contracted, 4.8 light years, and he is back from his distance star system after another eight year journey. For Leia, Luke has been gone for 20 years, but from Luke's perspective he has only been gone 16 years. Luke is now four years younger than Leia. Hopefully, you can now see why Einstein said this was a peculiarity and not a paradox. Luke is the moving observer, sees space itself foreshortened, but in Leia's reference frame, the distances haven't changed, merely the shape of Luke's ship. We've seen this square root equation come up twice already. So, what is it? Well, this is a useful little equation that represents the strength of the time dilation or length contraction based on the speed of the moving observer. It's more commonly found in this form called the Lorentz factor or 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. So, someone traveling at 10 percent of the speed of light or north 0.1 c sees a north 0.5 percent shortening in the length of a stationary object and a north 0.5 increase in the flow of time. That's not that much and even at half the speed of light, it's only a 15 percent change. You need to be traveling at almost the speed of light to see significant changes. At 90 percent of the speed of light, the Lorentz Factor is more than two. Here is a graph illustrating how quickly the Lorentz factor changes at very high speeds. We should note that Einstein developed special relativity to describe physics for observers who are not experiencing a strong gravitational force. But what happens near an object with a strong gravitational field such as a black hole? Einstein realized that he had to modify his theory of special relativity to make it more general and to allow for gravity, Einstein called this relativistic theory of gravity, general relativity. In order to understand the theory of general relativity, we'll begin with Einstein's first ponderings on the subject, something called the equivalence principle.
