Gravity

''This article covers the physics of gravitation.

Table of contents

1 Introduction 2 Newton's Law of Universal Gravitation 2.1 Vector Form 3 Einstein's Theory of Gravity 4 Units of Measurement and Variations in Gravity 5 Gravity, and the acceleration of objects near the Earth 6 Comparison with electromagnetic force 7 Gravity and Quantum Mechanics 8 Experimental tests of theories 9 Alternate Theories 10 History 11 Newton's reservations 12 Self-gravitating system 13 Special applications of gravity 14 Comparative gravities of different planets 15 See also

Introduction

Gravitation is the tendency of masses to attract each other.

The first mathematical formulation of this was due to Isaac Newton and proved astonishingly accurate. It has now been supplanted by Albert Einstein's theory of General relativity but for most purposes dealing with weak gravitational fields (for example, sending rockets to the moon or around the solar system) Newton's formulae are sufficiently accurate.

Newton's Law of Universal Gravitation

Newton's law of universal gravitation is the following:

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.

• F is the magnitude of the gravitational force between two objects
• m₁ is the mass of first object
• m₂ is the mass of second object
• r is the distance between the objects
• G is the gravitational constant

This law of universal gravitation was originally formulated by Isaac Newton in his work, the Principia Mathematica (1687). The history of the gravitation as a physical concept is considered in more detail below.

Vector Form

Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formulation, quantities in bold represent vectors.

• F₁₂ is the force on object 1 due to object 2
• r₂₁ = | r₁ − r₂ | is the distance between objects 1 and 2
• is the unit vector from object 2 to 1

Einstein's Theory of Gravity

Newton's formulation of gravity is quite accurate for most practical purposes. There are 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. (Though neither phenomenon has been directly observed.)
5. Under Newtonian gravity (with instantaneous transmission of gravitational force), if the universe is Euclidean, static, infinite, and of uniform, average, positive density, then an attempt to calculate the total gravitational force on a point produces a divergent series. In other words, newtonian gravity is inconsistent with a universe which is Euclidean, static, infinite, and of uniform, average, positive density. This is course is not a major problem as the universe is not believed to be most of these things.

Although general relativity is, as 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.

Units of Measurement and Variations in Gravity

Gravitational phenomena are measured in various units, depending on the purpose. The gravitational constant is measured in newtonss times metre squared per kilogram squared. Gravitational acceleration, and acceleration in general, is measured in metre per second squared or in galileoss or gees. The acceleration due to gravity at the Earth's surface is approximately 9.81 m/s2, depending on the location. A standard value of the Earth's gravitational acceleration has been adopted, called g. When the typical range of interesting values is from zero to several thousand galileos, as in aircraft, acceleration is often stated in multiples of g. When used as a measurement unit, the standard acceleration is often called "gee", as g can be mistaken for g, the gram symbol. For other purposes, measurements in multiples of milligalileo (1/1000 galileo) are typical, as in geophysics. A related unit is the eotvos, which is the unit of the gravitational gradient. Mountains and other geological features cause subtle variations in the Earth's gravitional field; the magnitude of the variation per unit distance is measured in eotvos.

Typical variations with time are 0.2 mgal during a day, due to the tides, i.e. the gravity due to the moon and the sun.

Gravity, and the acceleration of objects near the Earth

The acceleration due to the force of gravity that attracts objects to the surface of the earth is not quite the same as the acceleration that is measured for a free-falling body at the surface of the earth (in a frame at rest on the surface). This is because of the rotation of the earth, which leads (except at the poles) to a centrifugal force which slightly lessens the acceleration observed.

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 between common objects and the earth and between celestial bodies is gravity, because gravity is electrically neutral: 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 or one body is very dense and the other is close by, but the small gravitational force exerted by bodies of ordinary size can fairly easily be detected through experiments such as the Cavendish torsion bar experiment.

M13 globular star cluster
Gravitational field demonstrated

Gravity and Quantum Mechanics

Gravity is the only one of the four fundamental forces of nature that has not been fitted into the formalism of quantum mechanics (the other three: Electromagnetism, the Strong Force, and the Weak Force, can be). This is because general relativity is essentially a geometric theory of gravity. 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.

Experimental tests of theories

Today General Relativity is accepted as the standard description of gravitational phenomena. (Alternative theories of gravitation exist but are more complicated than General Relativity.) General Relativity is consistent with all currently available measurements of large-scale phenomena. For weak gravitational fields and bodies moving at slow speeds at small distances, Einstein's General Relativity gives almost exactly the same predictions as Newton's law of gravitation.

Crucial experiments that justified the adoption of General Relativity over Newtonian gravity were the classical tests: the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury.

General relativity also explains the equivalence of gravitational and inertial mass, which has to be assumed in Newtonian theory.

More recent experimental confirmations of General Relativity were the (indirect) deduction of gravitational waves being emitted from orbiting binary stars, the existence of neutron stars and black holes, gravitational lensing, and the convergence of measurements in observational cosmology to an approximately flat model of the observable Universe, with a matter density parameter of approximately 30% of the critical density and a cosmological constant of approximately 70% of the critical density.

Even to this day, scientists try to challenge General Relativity with more and more precise direct experiments. The goal of these tests is to shed light on the yet unknown relationship between Gravity and Quantum Mechanics. Space probes are used to either make very sensitive measurements over large distances, or to bring the instruments into an environment that is much more controlled than it could be on Earth. For exampled, in 2004 a dedicated satellite for gravity experiments, called Gravity Probe B, was launched. Also, land-based experiments like LIGO are gearing up to possibly detect gravitational waves directly.

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.

Alternate Theories

• In the modified Newtonian dynamics (MOND), Mordehai Milgrom proposes a weakening of Newton's law at high distance as a possible explanation of the fast galaxy rotation.
• Nikola Tesla challenged Albert Einstein's theory of relativity, announcing he was working on a Dynamic theory of gravity and argued that a field of force was a better concept and did away with the curvature of space.
• George Louis LeSage proposed a gravity mechanism, referred to as "LeSage gravity, based on a fluid-based explanation where a light gas fills the entire universe.

Although the law of universal gravitation was first clearly and rigorously formulated by Isaac Newton, the phenomenon was more or less seen by others. Even Ptolemy had a vague conception of a force tending toward the center of the earth which not only kept bodies upon its surface, but in some way upheld the order of the universe. Johannes Kepler inferred that the planets move in their orbits under some influence or force exerted by the sun; but the laws of motion were not then sufficiently developed, nor were Kepler's ideas of force sufficiently clear, to make a precise statement of the nature of the force. Christiaan Huygens and Robert Hooke, contemporaries of Newton, saw that Kepler's third law implied a force which varied inversely as the square of the distance. Newton's conceptual advance was to understand that the same force that causes a thrown rock to fall back to the Earth keeps the planets in orbit around the Sun, and the Moon in orbit around 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, , it is plain to us why:

However, across a large body, variations in can create a significant tidal force.

Newton's reservations

It's important to understand that while Newton was able to formulate his law of gravity in his monumental work, he was not comfortable with it because he was deeply uncomfortable with the notion of "action at a distance" which his equations implied. He never, in his words, "assigned the cause of this power." In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity. Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science.

He lamented the fact that 'philosophers have hitherto attempted the search of nature in vain' for the source of the gravitational force, as he was convinced 'by many reasons' that there were 'causes hitherto unknown' that were fundamental to all the 'phenomena of nature.' These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer is yet to be found. While it is true that Einstein's hypotheses are successful in explaining the effects of gravitational forces more precisely than Newton's in certain cases, he too never assigned the cause of this power, in his theories. It is said that in Einstein's equations, 'matter tells space how to curve, and space tells matter how to move,' but this new idea, completely foreign to the world of Newton, does not enable Einstein to assign the 'cause of this power' to curve space any more than the Law of Universal Gravitation enabled Newton to assign its cause. In Newton's own words:

I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain.

Self-gravitating system

A self-gravitating system is a system of masses kept together by mutual gravity. An example is a binary star.

Special applications of gravity

A height difference can provide a useful pressure in a liquid, as in the case of an intravenous drip and a water tower.

A weight hanging from a cable over a pulley provides a constant tension in the cable, also in the part on the other side of the pulley.

Comparative gravities of different planets

The acceleration due to gravity at the Earth's surface is, by convention, equal to 9.80665 metres per second squared. (The actual value varies slightly over the surface of the Earth; see gee for details.) This quantity is known variously as g_n, g_e, g₀, gee, or simply g. The following is a list of the gravity forces (in multiples of g) on each of the planets in the solar system:

See also

• Gravity wave
• Gravitational binding energy
• Gravity Research Foundation
• Weight
• N-body problem
• Gravity Probe B Experiment

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