Introduction to Special Relativity

Maxwell's equations predict that the speed of any electromagnetic (EM) wave in the universe is always \(c=3×10^8\text{ m/s}\) and independent of the observer’s frame of reference. For example, if one observer is sitting on a train moving \(30\text{ m/s}\) and their glass of water appears stationary on the table (moving \(0\text{ m/s}\) relative to them), an observer standing on the ground watching the train go by would say the glass of water is moving \(30\text{ m/s}\). An observer in a spaceship outside of the Earth’s gravitational pull would say that the glass is moving hundreds of meters per second! Since Maxwell’s equations disagreed with Newton’s mechanics, one might have incorrectly assumed that Maxwell’s equations were wrong. However, the Michelson-Morley experiment demonstrated that the speed of any electromagnetic wave (light, microwaves, x-rays, etc.) does not depend on the speed of the observer. This means that it does not matter whether you flash a beam of light from your flashlight while standing still or going 99.9% the speed of light in any direction: you will observe the beam of light travel away from you at \(c=3×10^8\text{ m/s}\). This experiment along with many others demonstrated that Maxwell’s equations are right and that Newtonian mechanics needed to be modified. Einstein worked out that the consequences of the speed of EM waves being c for any observer are the equations of special relativity. These equations tell us that time dilation, length contraction, and other bizarre phenomena occur when one frame of reference moves away from us at very high speeds close to that of light. For example, if I were sitting on the Earth and could “watch” a person flying by me in a glass box at 99.9% the speed of light, they would appear to be moving in slow-motion and they and the whole box would appear almost completely flat. If they slowed down and landed on the Earth’s surface right next to me, they would no longer appear to be moving in slow-motion or to be flat (because their speed relative to me would now be pretty much 0). 

Newtonian mechanics asserts that time and space are constant throughout the universe: all clocks tick at the same rate and a meter stick is the same length no matter where it is. This assumption seemed intuitive and reasonable; however it is wrong because it does not agree with experimental observations. To reconcile theory with experiment, Einstein asserted that time and space are not constant, that clocks tick at different rates and that the length of a meter stick can appear different to different observers. Later experiments showed that planes with clocks flying around the world ticked slightly slower than clocks on Earth by an amount which agreed with Einstein’s equations. Calculations done by GPS systems must take into account both the effects of time dilation and length contraction to pinpoint the location of your vehicle; otherwise the GPS would be off by several meters.

Around the time of the late-19th and early-20th century, there were two pillars of physics: Newtonian mechanics which described the motion of objects and Maxwell’s laws of electromagnetism (expressed as Maxwell’s equations) which described electromagnetic phenomena such as light. Maxwell’s equations predict that the speed of light is the same for all observers in any reference frame. These equations predict that \(c=\frac{E}{B}\). From the perspective of Newtonian mechanics and our common sense intuitions, this is a very strange result because there is no relative velocity term in this equation and the quantities \(E\) and \(B\) do not depend in any way on relative motion. It doesn’t matter what the relative velocity is between an observer’s reference frame and the light source which emitted the light (that eventually reaches the observer); when they calculate \(c\) using this equation they will always measure the speed of light to be \(3\text{ × }10^8\text{ m/s}\).

 Figure 1: A clock onboard a spacecraft or plane circumnavigating the Earth will tick more slowly than a clock on the Earth's surface. All physical processes run more slowly at very high altitudes above the Earth's surface. This is even true if the relative velocity between both objects is zero; according to Einstein's general theory of relativity, the Earth's mass warps time in such a way where the flow of time (how rapidly or how slowly the clock ticks) varies with altitude above the Earth's surface. This is a mindboggling prediction!

Figure 1: A clock onboard a spacecraft or plane circumnavigating the Earth will tick more slowly than a clock on the Earth's surface. All physical processes run more slowly at very high altitudes above the Earth's surface. This is even true if the relative velocity between both objects is zero; according to Einstein's general theory of relativity, the Earth's mass warps time in such a way where the flow of time (how rapidly or how slowly the clock ticks) varies with altitude above the Earth's surface. This is a mindboggling prediction!

This fact is counterintuitive and contradicts Newtonian mechanics which is based on the Galilean transformation equations, according to which, velocities should add. Imagine that a steady stream of bullets are being fired in some direction at a constant velocity with a speed of \(2,000\text{ ft/s}\). If I begin to move in the direction of those bullets at \(10\text{ ft/s}\), they will appear to be moving at a slower speed of \(1,990\text{ ft/s}\). If I move in the opposite direction that the bullets are being fired in at \(10\text{ ft/s}\), then they will appear to be moving faster at a speed of \(2,010\text{ ft/s}\). But according to Maxwell’s equations, light will travel at the same speed away from you regardless of how fast you move towards it or away from it. Einstein realized that one of these pillars must fall. He believed that Maxwell’s equations were correct and that light did indeed travel at the same speed regardless of who was measuring it. Einstein modified Newtonian mechanics so that it predicted that the speed of light be the same for all observers and called this modified version of Newtonian mechanics the special theory of relativity.


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