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Gravity

Gravitation is the force of attraction[?] that exists between all particles with mass in the universe. It is the force of gravity which is responsible for holding objects onto the surface of planets and, with Newton's law of inertia is responsible for keeping objects in orbit around one another.

"Gravity is the force that pulls you down." -- Merlin in Disney's The Sword in the Stone

Merlin was right, of course, but gravity does much more than just hold you in your chair. It was the genius of Isaac Newton to recognize that. Newton recalled in a late memoir that while he was trying to figure out what kept the Moon in the sky, he saw an apple fall to the ground in his orchard, and he realized that the Moon was not suspended in the sky, but continuously falling, like a cannon ball that was shot so fast that it continuously misses the ground as it falls away due to the curvature of the Earth.

If one wishes to be precise, one should distinguish between gravitation, the universal force of attraction, and gravity, which is the resultant[?], on the Earth's surface, of the attraction by the earth's masses, and the centrifugal pseudo-force caused by the Earth's rotation. In casual discussion, gravity and gravitation are often used interchangeably.

Any two objects exert equal and oppositely directed gravitational pull on each other.

Speed of gravity: Einstein's theory of relativity predicts that the speed of gravity (defined as the speed at which changes in location of a mass are propagated to other masses) should be consistent with the speed of light. In 2002, the Fomalont-Kopeikin experiment produced measurements of the speed of gravity which matched this prediction. However, this experiment has not yet been widely peer-reviewed, and is facing criticism from those who claim that Fomalont-Kopeikin did nothing more than measure the speed of light in a convoluted manner.

Table of contents

Newton's Law of Universal Gravitation

Newton explains, "Every object in the Universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects."

Newton eventually published his still famous law of universal gravitation in his Principia Mathematica as follows:

<math>F = \frac{G m_1 m_2}{r^2}</math>

where:

  • F = gravitational force between two objects
  • m1 = mass of first object
  • m2 = mass of second object
  • r = distance between the objects
  • G = universal constant of gravitation
Strictly speaking, this law applies only to point-like objects. If the objects have spatial extent, the true force has to be found by integrating the forces between the various points.

Vector Form

The above form is a simplified version. It is more properly expressed as vector equation. (All quantities in bold represent vector quantities in what follows.) The form below is vectorially complete:

<math>F_{1\,2} = {G m_1m_2(\mathbf{r_2}-\mathbf{r_1}) \over \left| \mathbf{r_2}-\mathbf{r_1} \right|^3}</math>

where:

  • <math>F_{1\,2}</math> is the force on <math>m_1</math> by <math>m_2</math>
  • <math>m_1</math> and <math>m_2</math> are the masses
  • <math>\mathbf{r_1}</math> and <math>\mathbf{r_2}</math> are the position vectors of their respective masses
  • <math>G</math> is the gravitational constant

For the force on mass two, simply multiply <math>F_{1\,2}</math> by negative one.

The primary difference between the two formulations is that the second form uses the difference in position to construct a vector that points from one mass to the other, and then divides that vector by its length to prevent it from changing the magnitude of the force.

Comparison with electromagnetic force

The gravitational attraction of protons is approximately a factor 1036 weaker than the electromagnetic repulsion. This factor is independent of distance, because both forces are inversely proportional to the square of the distance. Therefore on an atomic scale mutual gravity is negligible. However, the main force beween common objects and the earth and between celestial bodies is gravity; this is due to the fact that they (at least one of the two) are electrically neutral to a high degree: even if in both bodies there were a surplus or deficit of only one electron for every 1018 protons and neutrons this would already be enough to cancel gravity (or in the case of a surplus in one and a deficit in the other: double the attraction).

The relative weakness of gravity can be demonstrated with a small magnet picking up pieces of iron. The small magnet is able to overwhelm the gravitational force of the entire earth.

Gravity is small unless at least one of the two bodies is large, but the small gravitational force exerted by bodies of ordinary size can fairly easily be detected through experiments such as the Cavendish experiment.

History

Nobody knows for sure if Newton's recollection about the apple was accurate, but his insight is the same nevertheless. Philosophers had thought since the Greeks that the "natural" movement of stars, planets, the Sun and the Moon was circular, Kepler established that orbits are actually elliptical, but still thought that the movements of the planets was dictated by some "divine force" emanated from the sun, but Newton realized that the same force that causes a thrown rock to fall back to the Earth keeps the planets in orbit of the Sun, and the Moon in orbit of the Earth.

Newton was not alone in making significant contributions to the understanding of gravity. Before Newton, Galileo Galilei corrected a common misconception, started by Aristotle, that objects with different mass fall at different rates. To Aristotle, it simply made sense that objects of different mass would fall at different rates, and that was enough for him. Galileo, however, actually tried dropping objects of different mass at the same time. Aside from differences due to friction from the air, Galileo observed that all masses accelerate the same. Using Newton's equation, <math>F = m a</math>, it is plain to us why:

<math>F = {G m_1m_2 \over r^2} = m_1a_1</math>

The above equation says that mass <math>m_1</math> will accelerate at acceleration <math>a_1</math> under the force of gravity, but divide both sides of the equation by <math>m_1</math> and:

<math>a_1 = {G m_2 \over r^2}</math>

Nowhere in the above equation does the mass of the falling body appear. When dealing with objects near the surface of a planet, the change in <math>r</math> divided by the initial <math>r</math> is so small that the acceleration due to gravity appears to be perfectly constant. The acceleration due to gravity here on earth is usually called <math>g</math>, and its value is about 9.8 m/s2 (or 32 ft/s2). Galileo didn't have Newton's equations, though, so his insight into gravity's proportionality to mass was invaluable, and possibly even affected Newton's formulation on how gravity works.

However, across a large body, variations in <math>r</math> can create a significant tidal force.

Einstein's General Theory of Relativity

Newton's formulation of gravity is quite accurate for most practical purposes. It has a few problems with it though:

  1. It assumes that changes in the gravitational force are transmitted instantaneously when positions of gravitating bodies change. However, this contradicts the fact that there exists a maximum velocity at which signals can be transmitted (speed of light in vacuum).
  2. Assumption of absolute space and time contradicts Einstein's theory of Special relativity.
  3. It predicts that light is deflected by gravity only half as much as observed.
  4. It does not explain gravitational waves or black holes.

For the first two of these reasons, Einstein in 1915 developed a new theory of gravity called General Relativity. Today GR is accepted as the standard description of classical gravitational phenomena. (Alternative theories of gravitation exist but are more complicated than GR.) GR is consistent with all currently available measurements. For weak gravitational fields and bodies moving at slow speeds at small distances, Einstein's GR gives almost exactly the same predictions as Newton's law of gravitation. Crucial experiments that justified the adoption of GR over Newtonian gravity were the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury. More recent experimental confirmations of GR were gravitational waves from orbiting binary stars and existence of neutron stars and black holes.

Although GR is a theory more accurate than Newton's law of gravity, it also requires a significantly more complicated mathematical formalism. Instead of describing the effect of gravitation as a "force", Einstein introduced the concept of curved space-time in which bodies move along curved trajectories.

Quantum Mechanics and Waves

Gravity is the only one of the four fundamental forces of nature that stubbornly refuses to be quantised (the other three being Electromagnetism, the Strong Force, and the Weak Force). What quantisation means, is that the force is measured in discrete steps that cannot be reduced in size, no matter what; alternatively, that gravitational interaction is trasmitted by particles called gravitons. Scientists have theorized about the graviton for years, but have been frustrated in their attempts to find a consistent quantum theory for it. Many believe that string theory holds a great deal of promise to unify general relativity and quantum mechanics, but this promise has yet to be realized.

See also: Gravitational binding energy, Gravity Research Foundation, Weight.



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