Gravity (old version)

For other meanings, see gravity (disambiguation).

Gravitation is the tendency of masses to move toward each other. Weight is a force caused by gravity equal to the product of the acceleration of gravity and the mass of the object. Exactly why two masses separated in space have a gravitational attraction to one another remains largely unknown, despite much research and various theories.

The first mathematical formulation of the theory of gravitation was made by Sir Isaac Newton and proved astonishingly accurate. He postulated the force of "universal gravitational attraction".

Newton's theory has now been replaced by Albert Einstein's theory of general relativity but for most purposes dealing with weak gravitational fields (for example, sending rockets to the Moon or around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. Steven Weinberg, a Nobel laureate, noted that we would still build a suspension bridge with Newton's laws. For this reason, Newton's law is often used and will be presented first.

Newton's law of universal gravitation

Newton's law of universal gravitation states 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.

Considering only the magnitude of the force, and momentarily putting aside its direction, the law can be stated symbolically as follows.

F=G{\frac {m_{1}m_{2}}{r^{2}}}

where

F is the magnitude of the gravitational force between two objects

G is the gravitational constant, that is approximately : G = 6.67 × 10⁻¹¹ N m² kg^-2

m₁ is the mass of first object

m₂ is the mass of second object

r is the distance between the objects

Thus gravity is proportional to the mass of each object, but has an inverse square relationship of the distances between the centres of each mass.

Strictly speaking, this law applies only to point-like objects. If the objects have spatial extent, the force has to be calculated by integrating the force (in vector form, see below) over the extents of the two bodies. It can be shown that for an object with a spherically-symmetric distribution of mass, the integral gives the same gravitational attraction on masses outside it as if the object were a point mass.¹

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

Vector form

Gravity on Earth from a macroscopic perspective.

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.

\mathbf {F} _{12}=G{m_{1}m_{2} \over r_{21}^{2}}\,\mathbf {\hat {r}} _{21}

where

F₁₂ is the force on object 1 due to object 2

G is the gravitational constant

m₁ and m₂ are the masses of the objects 1 and 2

r₂₁ = | r₁ − r₂ | is the distance between objects 1 and 2

\mathbf {\hat {r}} _{21}\equiv {\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{\vert \mathbf {r} _{1}-\mathbf {r} _{2}\vert }}

is the unit vector from object 2 to 1

It can be seen, that the vector form of the equation is the same as the scalar form, except for the vector value of F and the unit vector. Also, it can be seen that F₁₂ = − F₂₁.

Gravitational acceleration is given by the same formula except for one of the factors m:

\mathbf {a} =G{m \over r^{2}}\,\mathbf {\hat {r}}

Gravitational field

The gravitational field is a vector field that describes the gravitational force an object of given mass experiences in any given place in space.

It is a generalization of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the earth and the moon). For 2 objects (e.g. object 1 is a rocket, object 2 the earth), we simply write $\mathbf {r}$ instead of $\mathbf {r} _{21}$ and $m$ instead of $m_{1}$ and define the gravitational field $\mathbf {g} (\mathbf {r} )$ as:

\mathbf {g} (\mathbf {r} )=G{m_{2} \over r^{2}}\,\mathbf {\hat {r}}

so that we can write:

\mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} )

This formulation is independent of the objects causing the field. The field has units of force divided by mass; in SI, this is $N\cdot kg^{-1}$ .

Problems with Newton's Theory

Newton's formulation of gravitation is quite accurate for most practical purposes. There are a few problems with it though:

Theoretical concerns

There is no prospect of identifying the mediator of gravity. Newton himself felt the inexplicable action at a distance to be unsatisfactory.
Newton's theory requires that gravitational force is transmitted instantaneously. Given classical assumptions of the nature of space and time, this is necessary to preserve the conservation of angular momentum observed by Johannes Kepler. However, it is in direct conflict with Einstein's theory of special relativity which places an upper limit—the speed of light in vacuum—on the velocity at which signals can be transmitted.

Disagreement with observation

Newton's theory does not fully explain the precession of the perihelion of the orbit of the planet Mercury. There is a 43 arcsecond per century discrepancy between the Newtonian prediction (resulting from the gravitational tugs of the other planets) and the observed precessionTemplate:Fn.
The predicted deflection of light by gravity is only half as much as observations of this deflection, which were made after General Relativity was developed in 1915.
The observed fact that gravitational and inertial masses are the same (or at least proportional) for all bodies is unexplained within Newton's system. See equivalence principle.

Einstein's theory of gravitation

Einstein's theory of gravitation answered the problems with Newton's theory noted above. In a revolutionary move, his theory of general relativity (1915) stated that the presence of mass, energy, and momentum causes spacetime to become curved. Because of this curvature, the paths that objects in inertial motion follow can "deviate" or change direction over time. This deviation appears to us as an acceleration towards massive objects, which Newton characterized as being gravity. In general relativity however, this acceleration or free fall is actually inertial motion. So objects in a gravitational field appear to fall at the same rate due to their being in inertial motion while the observer is the one being accelerated. (This identification of free fall and inertia is known as the Equivalence principle.)

The relationship between the presence of mass/energy/momentum and the curvature of spacetime is given by the Einstein field equations. The actual shapes of spacetime are described by solutions of the Einstein field equations. In particular, the Schwarzschild solution (1916) describes the gravitational field around a spherically symmetric massive object. The geodesics of the Schwarzschild solution describe the observed behavior of objects being acted on gravitationally, including the anomalous perihelion precession of Mercury and the bending of light as it passes the Sun.

Arthur Eddington found observational evidence for the bending of light passing the Sun as predicted by general relativity in 1919. Subsequent observations have confirmed Eddington's results, and observations of a pulsar which is occulted by the Sun every year have permitted this confirmation to be done to a high degree of accuracy. There have also in the years since 1919 been numerous other tests of general relativity, all of which have confirmed Einstein's theory.

Units of measurement and variations in gravity

File:C71 geoid smooth Earth-Gravity-ESA.jpg

The Gravity Field and Steady-State Ocean Circulation Explorer project (GOCE) will measure high-accuracy gravity gradients and provide a global model of the Earth's gravity field and of the geoid. (ESA image)

Gravitational phenomena are measured in various units, depending on the purpose. The gravitational constant is measured in newtons times metre squared per kilogram squared. Gravitational acceleration, and acceleration in general, is measured in metres per second squared or in non-SI units such as galileos, gees, or feet per second squared.

The acceleration due to gravity at the Earth's surface is approximately 9.8 m/s², more precise values depending on the ___location. A standard value of the Earth's gravitational acceleration has been adopted, called g_n. When the typical range of interesting values is from zero to tens of metres per second squared, as in aircraft, acceleration is often stated in multiples of g_n. 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 millimetres or micrometres per second squared (mm/s² or µm/s²) or in multiples of milligals or milligalileos (1 mGal = 1/1000 Gal), a non-SI unit still common in some fields such as geophysics. A related unit is the eotvos, which is a cgs unit of the gravitational gradient.

Mountains and other geological features cause subtle variations in the Earth's gravitational field; the magnitude of the variation per unit distance is measured in inverse seconds squared or in eotvoses.

Typical variations with time are 2 µm/s² (0.2 mGal) during a day, due to the tides, i.e. the gravity due to the Moon and the Sun.

A larger variation in gravity occurs when we move from the equator to the poles. Gravity increases as the density towards the poles increases. This is due to the effects of the rotation of the earth (both centrifugal force and the flattening due to this rotation, resulting in the poles being closer to the center of mass of the Earth), and is also related to the fact that the Earth's density changes from the surface of the planet to its centre.

The sea-level gravitational acceleration is 9.780 m/s² at the equator and 9.832 m/s² at the poles, so an object will exert about 0.5% more force due to gravity at sea level at the poles than at sea level at the equator [1].

Gravity, and the acceleration of objects near the Earth

The acceleration due to the apparent "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. See Coriolis effect.

Comparison with electromagnetic force

The gravitational interaction of protons is approximately a factor 10³⁶ weaker than the electromagnetic repulsion. This factor is independent of distance, because both interactions are inversely proportional to the square of the distance. Therefore on an atomic scale mutual gravity is negligible. However, the main interaction 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 interaction).

In terms of Planck units: the charge of a proton is 0.085, while the mass is only 8 -20. From that point of view, the gravitational force is not small as such, but because masses are small.

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 interaction of the entire Earth. Similarly, when doing a chin-up, the electromagnetic interaction within your muscle cells is able to overcome the force induced by Earth on your entire body.

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 interaction exerted by bodies of ordinary size can fairly easily be detected through experiments such as the Cavendish torsion bar experiment.

File:M13grav.jpg

Globular Cluster M13 demonstrates gravitational field.

Further reading

Jefimenko, Oleg D., "Causality, electromagnetic induction, and gravitation : a different approach to the theory of electromagnetic and gravitational fields". Star City [West Virginia] : Electret Scientific Co., c1992. ISBN 0917406095
Heaviside, Oliver, "A gravitational and electromagnetic analogy". The Electrician, 1893.

Gravity and quantum mechanics

It is strongly believed that three of the four fundamental forces (the strong nuclear force, the weak nuclear force, and the electromagnetic force) are manifestations of a single, more fundamental force. Combining gravity with these forces of quantum mechanics to create a theory of quantum gravity is currently an important topic of research amongst physicists. General relativity is essentially a geometric theory of gravity. Quantum mechanics relies on interactions between particles, but general relativity requires no particles in its explanation of gravity. Scientists have theorized about the graviton (a particle that transmits the force gravity) 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. It never can be for obvious reasons if Einstein's theory is true, due to the non-existence of "gravitational attraction" (see: fundamental interaction)

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 example, 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.

The Pioneer anomaly is an empirical observation that the positions of the Pioneer 10 and Pioneer 11 space probes differ very slightly from what would be expected according to known gravitational effects. The possibility of new physics has not been ruled out, despite very thorough investigation in search of a more prosaic explanation.

Alternative theories

In the modified Newtonian dynamics (MOND), Mordehai Milgrom proposes a modification of Newton's Second Law of motion for small accelerations.
Nikola Tesla challenged Albert Einstein's theory of relativity, announcing he was working on a Dynamic theory of gravity (which began between 1892 and 1894) and argued that a "field of force" was a better concept and focused on media with electromagnetic energy that fill all of space.
Georges-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.

History

Although the law of universal gravitation was first clearly and rigorously formulated by Isaac Newton, the phenomenon was observed and recorded 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, $F=ma$ , it is plain to us why:

F=-{Gm_{1}m_{2} \over r^{2}}=m_{1}a_{1}

The above equation says that mass $m_{1}$ will accelerate at acceleration $a_{1}$ under the force of gravity, but divide both sides of the equation by $m_{1}$ and:

a_{1}={Gm_{2} \over r^{2}}

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 r divided by the initial r is so small that the acceleration due to gravity appears to be perfectly constant. The acceleration due to gravity on Earth is usually called g, 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 $r$ 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.

If science is eventually able to discover the cause of the gravitational force, Newton's wish could eventually be fulfilled as well.

It should be noted that here, the word "cause" is not being used in the same sense as "cause and effect" or "the defendant caused the victim to die". Rather, when Newton uses the word "cause," he (apparently) is referring to an "explanation". In other words, a phrase like "Newtonian gravity is the cause of planetary motion" means simply that Newtonian gravity explains the motion of the planets. See Causality and Causality (physics).

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.

Shot Tower, 1856 Dubuque, Iowa

Molten lead, when poured into the top of a shot tower, will coalesce into a rain of spherical lead shot, first separating into droplets, forming molten spheres, and finally freezing solid, undergoing many of the same effects as meteoritic tektites, which will cool into spherical, or near-spherical shapes in free-fall.

Comparative gravities of different planets and Earth's Moon

The standard acceleration due to gravity at the Earth's surface is, by convention, equal to 9.80665 metres per second squared. (The local acceleration of gravity varies slightly over the surface of the Earth; see gee for details.) This quantity is known variously as g_n, g_e (sometimes this is the normal equatorial value on Earth, 9.78033 m/s²), g₀, gee, or simply g (which is also used for the variable local value). The following is a list of the gravitational accelerations (in multiples of g) at the surfaces of each of the planets in the solar system and the Earth's Moon :

Sun		$\approx$ 28
Mercury		0.376
Venus		0.903
Earth		1
Moon		0.165
Mars		0.38
Jupiter		2.34
Saturn		1.16
Uranus		1.15
Neptune		1.19
Pluto		0.066

Note: The "surface" is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune) in the above table. It is usually specified as the ___location where the pressure is equal to a certain value (normally 75 kPa?). For the sun, the "surface" is taken to mean the photosphere.

Within the earth, the gravitational field peaks at the core-mantle boundary, where it has a value of 10.7 m/s².

For spherical bodies surface gravity in m/s² is 2.8 × 10⁻¹⁰ times the radius in m times the average density in kg/m³.

When flying from earth to mars, climbing against the field of the earth at the start is 100 000 times heavier than climbing against the force of the sun for the rest of the flight.

Mathematical equations for a falling body

These equations describe the motion of a falling body under acceleration g near the surface of the Earth.

Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines can be approximated as being parallel and pointing straight down to the center of the Earth

Here, the acceleration of gravity is a constant, g, because in the vector equation above, ${r}_{21}$ would be a constant vector, pointing straight down. In this case, Newton's law of gravitation simplifies to the law

F = mg

The following equations ignore air resistance and the rotation of the Earth, but are usually accurate enough for heights not exceeding the tallest man-made structures. They fail to describe the Coriolis force, for example. They are extremely accurate on the surface of the Moon, where the atmosphere is almost nil. Astronaut David Scott demonstrated this with a hammer and a feather. Galileo was the first to demonstrate and then formulate these equations. He used a ramp to study rolling balls, effectively slowing down the acceleration enough so that he could measure the time as the ball rolled down a known distance down the ramp. He used a water clock to measure the time; by using an "extremely accurate balance" to measure the amount of water, he could measure the time elapsed. ²

For Earth, in Metric units:

\ g=9.8\,{\mbox{m}}/{\mbox{s}}^{2}\quad

in English units:

\ g=32\,{\mbox{ft}}/{\mbox{s}}^{2}

For other planets, multiply $\ g$ by the ratio of the gravitational accelerations shown above.

Distance d traveled by a falling object under the influence of gravity for a time t:	$\ d={\frac {1}{2}}gt^{2}$
Elapsed time t of a falling object under the influence of gravity for distance d:	$\ t={\frac {\sqrt {2gd}}{g}}\$
Average velocity v_a of a falling object under constant acceleration g for any given time:	$\ v_{a}={\frac {1}{2}}gt$
Average velocity v_a of a falling object under constant acceleration g traveling distance d:	$\ v_{a}={\frac {\sqrt {2gd}}{2}}\$
Instantaneous velocity v_i of a falling object under constant acceleration g for any given time:	$\ v_{i}=gt$
Instantaneous velocity v_i of a falling object under constant acceleration g, traveling distance d:	$\ v_{i}={\sqrt {2gd}}\$

Note: Distance traveled, d, and time taken, t, must be in the same system of units as acceleration g. See dimensional analysis. To convert metres per second to kilometres per hour (km/h) multiply by 3.6, and to convert feet per second to miles per hour (mph) multiply by 0.68 (or, precisely, 15/22).

Notes

Note 1: Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. Preceded by A Guide to Newton's Principia, by I.Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4
Note 2: See the works of Stillman Drake, for a comprehensive study of Galileo and his times, the Scientific Revolution.
Template:Fnb Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)

External links