Special Relativity

Albert Einstein A portion of Einstein’s fame came when he formulated Special Relativity. He came up with this theory by simply performing thought experiments. He wondered what would happen if you travelled at the speed of light. He started with the most basic principles of physics and worked up from there…

Principle of Relativity Reference frames A reference frame is a framework that is used for the observation and mathematical description of physical phenomena and the formulation of physical laws. Usually consisting of an observer, a coordinate system, and a clock. We have always calculated with a frame of reference which we perceived as still (Earth)

Principle of Relativity The Principle of Relativity states that the laws of physics apply to all frames of reference. If the reference frames don’t accelerate, what happens in one would be the same as if we did it in another frame (Called a Galilean System: law of inertia holds true in these frames) So if I were to drop a ball in this classroom and a ball in a car moving at a constant speed, they would fall the same way. This is a very important concept in science. If this were not true, there would be no reason to experiment since your results would be different the next time you did the same experiment.

Relative Motion Relative Motion The principle of relativity states that if two observers are in the same frame of reference they will observe the same measurements. However, if two observers are in different frames of reference they will observe different measurements. For example, a person in car that is travelling at 30 mph is holding a tennis ball. To that observer, the ball has a speed of 0 mph. To an observer who is standing on the side of the road however, the ball has a speed of 30 mph! If you walk at 5 mph on a moving platform which has a speed of 3 mph, an observer standing next to the moving platform would see you moving at 8 mph. 𝑣 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 = 𝑣 𝑝𝑙𝑎𝑡𝑓𝑜𝑟𝑚 + 𝑣 𝑤𝑎𝑙𝑘𝑒𝑟 An example of relative motion that we have seen already this year. Einstein concluded that this must mean that light has variable speeds depending on the reference frame you are in and the speed of light must be very complicated.

The Speed of Light For some time, Einstein thought that since the principle of relativity is true and the addition of relative velocities is true, there must be some substance or ether that light travelled through that altered it’s speed from one reference frame to another. Then the Michelson-Morley experiment was performed. This experiment proved that there is no ether and the speed of light was indeed constant no matter what frame of reference! This astounded Einstein and got him thinking again…

Simultaneity of Events Einstein then came up with a thought experiment that had two bolts of lightning hitting a train on either side: The light rays move away from the original strikes, both travelling at the speed of light. To an observer that is midway between the strikes and not on the train, the light from both bolts reaches them at the same instant. However, for the observer on the train they do not reach her at the same instant because she is moving towards one bolt and away from the other.

Simultaneity of Events His experiment proves that simultaneity is not an absolute concept but rather one that depends on the state of motion of the observer. This single thought experiment led to an amazing revelation: Time is relative! We must now conclude that speed and time are relative/depend on the frame you are in This is because to the observer standing still, the lightning occurred at the same time, but to the woman on the train the lightning that hit in the direction of motion hit before the other. Therefore their definitions of WHEN the events happened depended on which frame of reference they were in. This means that both the principle of relativity and a constant speed of light can both be true!

The Special Theory of Relativity The Special Theory of Relativity is a combination of these two ideas: The Principle of Relativity A constant speed of light regardless of the frame This has a number of different effects on objects that travel very fast (approaching the speed of light).

Length Contraction The dimensions of an object (in the direction of motion) shrinks as that object’s speed increases according to an observer in a different reference frame. 𝐿= 𝐿 𝑜 1− 𝑣 2 𝑐 2 L – Observed length Lo – Length while still v – Speed of the object c – Speed of light (3 x 108)

Time Dilation A clock in a moving frame will appear to be running slower to an observer in a different reference frame. The faster the clock is travelling, the slower it appears to run. 𝑇= 𝑇 𝑜 1− 𝑣 2 𝑐 2 T – Observed duration of time To – Duration of time while still GPS satellites have to deal with time dilation due to their speeds.

Twin Paradox Time Dilation is a particularly interesting concept with the Twin Paradox example: Imagine there are twins that are both 21 years old. One of these twins enters a spaceship that can travel at very close to the speed of light (~99.99% of the speed of light) while the other stays behind on Earth. The twin aboard the ship flies for one year away from the Earth and then back for a total of two years in the spaceship. When he returns to Earth he will be 23 years old. The twin that stayed behind on the other hand would have died a long time ago because 200 years had passed on Earth! This is due to time dilation! The biological clock for the fast moving twin ran slower than that of the twin that stayed behind!

Relativistic Mass Mass increases for an object the faster it moves. 𝑚= 𝑚 𝑜 1− 𝑣 2 𝑐 2 M – Observed mass Mo – Mass of the object at rest This means that we could not travel at the speed of light because our mass would become infinitely large. Only things that are massless (like light) can travel that fast.

Consequences in Review So, that means as your velocity increases: The dimension in the direction of travel will shrink Time slows Your mass increases Again this is all according to an outside observer. You would not notice any difference in yourself!

General Relativity After Einstein wrapped up the Theory of Special Relativity, he moved on to figuring out how gravity works. Using what he figured out from his thought experiments, he was able to come up with the General Theory of Relativity. General Relativity has the same assumptions, but now the frames of reference can be bent. Euclidean geometry (the geometry you learned in high school) became Gaussian geometry (college geometry). This is when he came up with the space-time fabric!