Planet Earth/1i. Time: The Invention of Seconds using Earth’s Motion

The Periodic Swing of a Pendulum
Earth’s motion would play a vital role in unlocking the knowledge of how to measure the units of seconds and standardizing the accuracy of time. The first breakthrough was made by Galileo Galilei in 1581 while attending a particularly boring lecture, as the story has been retold and likely fictionalized. In the room was a chandelier swinging by a breeze from an open window. The rate of the swings seemed to be independent of the length of the swing, as the chandelier arched for a longer distance it appeared to move at a faster rate. Galileo was the first to discover that pendulums behave isochronicially, meaning that the periodic swing of a pendulum is independent of the amplitude (the angle an object is let go) or width of the arc of the swing. The rate of a pendulum’s swing was also independent of the mass of the object at the end of the pendulum, however it is dependent on the length of the string of the hanging object.

If two weights were hung from strings of equal length, and started in a rocking motion at the same time they would match the exact rate of each swing, and allow for accurate time keeping. The use of pendulums for time keeping was later perfected by Christiaan Huygens, who wrote a book on the use of pendulums for clocks, published in 1673. In fact, pendulums in the 1670s were at their height in terms of scientific curiosity. A pendulum with a set length would be set to rocking, and the number of swings would be counted for a full sidereal day or when a star reached the same position the following night. It was tedious work counting every swing of a pendulum for an entire day and night, until the star reached the same point in the sky. The ability to measure experiments and observations by seconds using a fixed length pendulum revolutionized science.

In 1671 the French Academy sent Jean Richer to the city of Cayenne in French Guiana South America, near the Equator. Although set to observe the positions of Mars in the night sky to calculate the distance from Earth to Mars, Jean Richer also took with him a pendulum with a fixed length counted out for the number of swings in Paris for a full sidereal day. While in French Guiana he did the same experiment, and determined that the number of pendulum swings differed between the two cities. Previously it had been thought that the only thing that altered the rate of the pendulum swing was the length of the pendulum. This curious experiment, and others like it, led to the fundamental scientific concept regarding the moment of inertia as proposed by Christiaan Huygens.

The moment of inertia is equal to the mass of an object multiplied by the radius squared from the center of mass or
 * $$I = mr^2.$$

Isaac Newton, a contemporary and friend of Christiaan Huygens realized that the differences in the number of swings on the pendulum between these two places on Earth was because each city was located at a slightly different radius from the center of Earth’s mass. With Paris closer to the center of the Earth, while city of Cayenne further from the center of the Earth with a position closer to the Equator. The rate of the pendulum swing was thus due to the differences in the acceleration of Earth’s gravity at each city.

The length of time for each pendulum swing
 * $$T = 2\pi \sqrt\frac{L}{g},$$

where L is the length of the pendulum and g is the local acceleration of gravity. This equation works only for pendulums with short swings with small amplitudes, and a set moment of inertia. (If you get that pendulum really rocking in an accelerating car, you need to use a much more complicated formula). The length of a pendulum in Paris, France, which resulted in a 1-second swing was almost officially defined as the official length of 1-meter. However, the length of a meter was chosen as a Meridian in Paris, with the official 1-second pendulum of 0.9937 meters in length, which became the standard in most clocks. However, this length was adjusted slightly to account for variations in Earth’s gravity, and one of the reasons the 1-second length pendulum was not used as the standard unit of 1-meter in 1791 when the metric system was first designed.

Evidence of a Spinning Earth: Foucault’s Pendulum
The work of Isaac Newton and Christiaan Huygens on pendulums confirmed that Earth’s gravity acted as an acceleration. One way to demonstrate this acceleration is to imagine a pendulum inside a car. Both the pendulum and car are stationary before the start of a race, but once the race begins, the car increases its speed down the race track and the pendulum inside the car will be pulled backward due to the inertia of the acceleration of the race car (as will the driver). However, if the car travels a constant speed (velocity) with zero acceleration, as long as the car does not change its velocity, and no object touches the pendulum, the pendulum inside the car will remain stationary even as the car is traveling at a high speed. Isaac Newton realized that Earth’s gravity behaves like a pendulum in an accelerating car. Hence, we refer to Earth’s gravity, as the acceleration of gravity, or in mathematical formulas as little g.

One of the most amazing experiments, which has been replicated around the world is the use of a large pendulum to demonstrate the spinning motion of the Earth, as well a method to calculate latitude. It was first performed by Léon Foucault, the inventor of the gyroscope, which he hoped to use to see the rotation of the Earth’s spin. However, it was his experiment with a pendulum that he is best known for. Foucault build a very long pendulum in his attic, and set it in motion by tying a string to the end of the pendulum at a set amplitude, and using a flame burned the string, which set the pendulum in motion without any jostling. He watched its movement and noticed that it started to rotate very slowly. The reason for this rotational motion was the fact that the pendulum was not moving, but the ground beneath his feet was—with the rotation of the Earth! In a famous demonstration, Léon Foucault built a giant pendulum in the Panthéon of Paris, and showed to the public that a pendulum swings will rotate around a circle in 31.8 hours in Paris. At each point in its rotation the swinging pendulum will mark a path. The length of time this rotation occurs is related to the position in latitude, as a large pendulum at the north or south pole would mark the path of Earth’s rotation in 23 hours 56 minutes and 4.1 seconds. But in Paris with at a latitude 48.8566° N, the rotation of the Earth beneath the pendulum takes larger, and the closer to the equator the longer this rotation will become, until it no longer rotates as it approaches the equator.

The reason for this is lack of rotation at the equator is that the plane of reference of the pendulum and the spinning Earth are the same, while at the Earth’s poles the spinning Earth below the pendulum is spinning clockwise around the pendulum or counterclockwise at the South Pole. If you have ever observed a Foucault Pendulum for any length of time, you will feel a sense of vertigo, as you realize that the motion of the swinging pendulum is not moving, but the Earth is. A Foucault pendulum still is running inside the Panthéon of Paris today, as well as numerous other places around the world as this demonstration validates Earth’s spinning motion.

A failed experiment to measure the Earth’s spin using the speed of light
Léon Foucault may best be known for his pendulum in the Panthéon of Paris, but he invented something that would have revolutionized science, and it had to do with his obsession with the motion of Earth’s spin, and it involved using the spinning motion of a mirror to measure the speed of light for the first time in history. The apparatus consisted of a beam of light shone on a spinning wheel of mirrors that reflected a beam of light on to a stationary mirror, which reflected the light back to the set of spinning mirrors. In the time that the light took to reflect off the stationary mirror and return to the spinning wheel of mirrors, the spinning mirrors would have moved slightly with a slightly different orientation. This change results in a beam of light that would not reflect directly back to the exact original light source, but deflected at a slight angle depending on the speed of the spinning wheel of mirrors. Using this simple apparatus Léon Foucault calculated the speed of light was close to its modern determined value today of 299,792,458 meters per second, he found a value between 298 million and 300 million meters per second! Light had baffled scientists as it appeared to behave as both a particle and sometimes a wave. In 1801 Thomas Young conducted a simple experiment. He cut two slits in a box, and shone light through the slits and observed that the two beams of light shone on a nearby wall were interfering with each other like ripples of passing waves. A similar phenomenon is observed when two stones are dropped at the same time into a lake, the rippling waves will interfere with each other as they radiate out from the dropped stones. If light was a wave, as this experiment suggested, then light waves must be passing through some medium, which scientists called the aether. The existence of this aether was difficult to prove, but two American scientists Albert A. Michelson and Edward Morley dedicated their lives to prove the existence of aether using the speed of light, and the rotation of the Earth, yet they failed.

Taking inspiration from the motion of pendulums, the speed of light must vary depending on the motion of the Earth’s spin. At the Equator, if one shone a light in the north-south direction and another light beam in the east-west direction, the speed of light should be different, because the Earth is spinning below each of the two beams of light at a rate of 1,040.45 miles per hour (1674.44 km/hr) in the East-West direction. Just like in the motion of pendulums, light should show differences as waves in this “wind” of aether traveled against Earth’s spin.

In 1887, Albert A. Michelson and Edward Morley measured the speed of light in two different directions as precisely as possible in a vacuum of air, and each time they found the same results, the two speeds of light, no matter their orientation were exactly the same! There appeared to be no invisible aether, but the speed of light, unlike Earth’s gravity appear to be a constant. How could this be?

The Michelson and Morley experiment is one of the most famous failed experiments, and while it did not prove the presence of aether, it led to a major breakthrough in science.

Lorentz Transformations
The solution to the problem was solved by a brilliant Dutch scientist named Hendrik Lorentz, who suggested the reason why the experiment failed was that the distance measured was slightly different, because of the difference in Earth’s speed or velocity.

To demonstrate this mathematically Hendrik Lorentz imagined two beams of bouncing light between mirrors traveling at different but constant speeds. If the speed of light between the two mirrors was held the same, and if you knew the constant velocity of the two light beams, the path of the faster moving light beam would travel a longer distance, since as the light traveled the mirror would move in relationship to the slower moving, or stationary light beam. The faster the velocity of the light beam the longer the path it has to take between the mirrors, and to complete this path in the same amount time, suggested that time is relative to velocity. In a series of mathematical equations known as the Lorentz transformations, Lorentz calculated the time dilation also known as the length dilation by an expression of
 * $$\Delta L' = \frac 1 {\sqrt{1 - \frac{v^2}{c^2}}}$$

Where v is the velocity of an object with zero acceleration (such as the spinning Earth) and c is the speed of light. The larger this number, the shorter the length of a meter will become and the longer time will be. Graphing this mathematical equation results in low values when velocity is less than half the speed of light, but very high values when the velocity approaches the speed of light, and reaching infinity and breaking down when velocity of an object reaches the speed of light.

It is likely the most frightening equation you will ever see, as at its root it determines the universal speed limit of anything with mass in the universe. Faster than light travel for anything with mass is an impossibility according to Lorentz transformations, and with the distance to the nearest star other than the Sun over 4 light years away (roughly 25 trillion miles away). A rocket shot into space at Earth’s current rate of galactic motion of 439,246 miles per hour, the rocket would take around 6,500 years to reach the nearest star, far longer than your life span. Despite science fiction depicted in movies and video games where distances across the universe are short, easily traveled, and populated by aliens, these imaginations are simply wishful thinking. Earth will always be your home. You are inevitably stuck here.

The Michelson and Morley experiment is still being replicated, most recently with the Laser Interferometer Gravitational-Wave Observatory (LIGO), which is a set of two observatories in Washington and Louisiana that each measure the distance between two mirrors oriented in different directions. Any changes in the distance between the mirrors spaced 4 kilometers apart, and measured extremely precisely by observed changes in the light wave frequencies (to the breadth of a single atom’s width), are due to gravitational waves caused by the collisions of super massive black holes and neutron stars millions of light years from Earth. We may not be able to visit these places, but we can observe them on Earth, as gravitational waves flickering nearly imperceivable through light. The Lorentz transformation was intensely studied by one of Lorentz’s students, a young Albert Einstein, who with the aid of Lorentz, formulated his theory of Special Relativity in 1905 in his paper On the Electrodynamics of Moving Bodies. Both Lorentz and Einstein showed how your notion of time is relative to your motion, or more precisely Earth’s velocity. Your sense of time is interwoven with the planet’s motion through space. A few months after his publication of special relativity in 1905, Einstein asked the question big, what does this constant of the speed of light have to do with Mass and Energy? Resulting in Einstein’s famous equation $$E = mc^2$$. Where E is energy, m is mass, and c is the speed of light, but before you can learn more about this famous equation of Einstein’s, you will need to learn more about Earth’s energy and matter.