Gravity

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Gravity is the force of attraction between massive particles. Weight is determined by the mass of an object and its location in a gravitational field. While a great deal is known about the properties of gravity, the ultimate cause of the gravitational force remains an open question. General relativity is the most successful theory of gravitation to date. It postulates that mass and energy curve space-time, resulting in the phenomenon known as gravity. The effect of the bending of spacetime is often misunderstood as most people seem to prefer to think of a falling object as accelerating when the facts do not support that assumption. Ask any skydiver if he feels any acceleration (other than from wind resistance).

Gravity, simply put, is acceleration. F=ma means that there must be a force that causes a mass to accelerate. For a rocket ship, that is the rocket motor. For the earth, that is the compression of the mass between something on the surface of the earth and the earth's center of mass. The acceleration is in relation to spacetime in that the weight you feel is your resistance to deviating from your path in spacetime. The same holds true in the rocket ship except that a rocket motor supplies the force to accelerate you from your spacetime path. There is no difference between weight you feel because of gravity or the rocket.

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 of mass for the two objects. This force is proportional to the product of their masses and inversely proportional to the square of the separation between the centers of mass of the two objects.

Given that the force is along the line through the two masses, the law can be stated symbolically as follows.

$F = - G \frac{m_1 m_2}{r^2}$

where:

F is the magnitude of the (repulsive) 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

It can be seen that this repulsive force F is always negative, and this means that the net attractive force is positive. The minus sign is used to hold the same value meaning as in the Coulomb's Law, where a positive force as result means repulsion between two charges.

Thus gravity is proportional to the mass of each object, but has an inverse square relationship with the distance 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

File:Gravitymacroscopic.png
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.

$\mathbf{F}_{12} = G {m_1 m_2 \over r_{21}^2} \, \mathbf{\hat{r}}_{21}$ or $\mathbf{F}_{12} = - G {m_1 m_2 \over r_{21}^2} \, \mathbf{\hat{r}}_{12}$

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 = | r2r1 | is the distance between objects 2 and 1
$\mathbf{\hat{r}}_{21} \equiv \frac{\mathbf{r}_1 - \mathbf{r}_2}{\vert\mathbf{r}_1 - \mathbf{r}_2\vert}$ 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 m1 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·kg−1.

Problems with Newton's theory

Although Newton's formulation of gravitation is quite accurate for most practical purposes, it has a few problems:

Theoretical concerns

• There is no prospect of identifying the mediator of gravity. Newton himself felt the inexplicable action at a distance to be unsatisfactory (see "Newton's reservations" below).
• 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 for all bodies is unexplained within Newton's system. General relativity takes this as a postulate. See equivalence principle.

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 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).

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 gn. 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 gn. 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 the effect of gravity occurs when we move from the equator to the poles. The effective force of gravity decreases as the distance from the equator decreases, due to the rotation of the Earth, and the resulting centrifugal force and flattening of the Earth. The centrifugal force causes an effective force 'up' which effectively counteracts gravity, while the flattening of the Earth causes the poles to be closer to the center of mass of the Earth. It 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].

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 at this scale matter 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). However, the main interactions between the charged particles in cosmic plasma (that makes up over 99% of the universe by volume), are electromagnetic forces.

In terms of Planck units: the charge of a proton is 0.085, while the mass is only Template:Sn. 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:M13.arp.750pix.jpg
Globular Cluster M13 demonstrates gravitational field.

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 exchange of particles in its explanation of gravity.

Scientists have theorized about the graviton (a messenger particle that transmits the force of 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 is notable that in general relativity gravitational radiation (which under the rules of quantum mechanics must be composed of gravitons) is only created in situations where the curvature of spacetime is oscillating, such as for co-orbiting objects. The amount of gravitational radiation emitted by the solar system and its planetary systems is far too small to measure. However, gravitational radiation has been indirectly observed as an energy loss over time in binary pulsar systems such as PSR1913+16). It is believed that neutron star mergers and black hole formation may create detectable amounts of gravitational radiation. Gravitational radiation observatories such as LIGO have been created to study the problem. No confirmed detections have been made of this hypothetical radiation, but as the science behind LIGO is refined and as the instruments themselves are endowed with greater sensitivity over the next decade, this may change.

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.

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.

The equivalence principle, the postulate of general relativity that presumes that inertial mass and gravitational mass are the same, is also under test. Past, present, and future tests are discussed in the equivalence principle section.

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 to test general relativity's predicted frame-dragging effect, among others. Also, land-based experiments like LIGO and a host of "bar detectors" are trying to detect gravitational waves directly. A space-based hunt for gravitational waves, LISA_(astronomy), is in its early stages. It should be sensitive to low frequency gravitational waves from many sources, perhaps including the Big Bang.

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 effects (gravitational or otherwise). The possibility of new physics has not been ruled out, despite very thorough investigation in search of a more prosaic explanation.

Historical Alternative theories

• 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.
• In 1967 Andrei Sakharov proposed something similar, if not essentially identical. His theory has been adopted and promoted by Messrs. Haisch, Rueda and Puthoff who, among other things, explain that gravitational and inertial mass are identical and that high speed rotation can reduce (relative) mass. Combining these notions with those of T. T. Brown, it is relatively easy to conceive how field propulsion vehicles such as "flying saucers" could be engineered given a suitable source of power.
• Georges-Louis LeSage proposed a gravity mechanism, now commonly called LeSage gravity, based on a fluid-based explanation where a light gas fills the entire universe.

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 or a water tower, and can even supply enough power for hydroelectricity.

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.

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.

A fractionation tower can be used to manufacture some materials by separating out the material components based on their specific gravity.

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 gn, ge (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 Sun, the surfaces of each of the planets in the solar system, and the Earth's moon :

 Sun 27.9 Mercury 0.37 Venus 0.88 Earth 1 Moon 0.16 Mars 0.38 Jupiter 2.64 Saturn 1.15 Uranus 0.93 Neptune 1.22 Pluto 0.06

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.

File:Gravityroom.png
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, r21 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 effect, 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 Imperial 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 va of a falling object under constant acceleration g for any given time: $\ v_a =\frac{1}{2}gt$ Average velocity va of a falling object under constant acceleration g traveling distance d: $\ v_a =\frac{ \sqrt {2gd}}{2} \$ Instantaneous velocity vi of a falling object under constant acceleration g for any given time: $\ v_i = gt$ Instantaneous velocity vi of a falling object under constant acceleration g, traveling distance d: $\ v_i = \sqrt {2gd}\$

Note: "Average" means average in time.

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).

Gravitational potential

For any mass distribution there is a scalar field, the gravitational potential (a scalar potential), which is the gravitational potential energy per unit mass of a point mass, as function of position. It is

$- G \int{1 \over r} dm$

where the integral is taken over all mass. Minus its gradient is the gravity field itself, and minus its Laplacian is the divergence of the gravity field, which is everywhere equal to -4πG times the local density.

Thus when outside masses the potential satisfies Laplace's equation (i.e., the potential is a harmonic function), and when inside masses the potential satisfies Poisson's equation with, as right-hand side, 4πG times the local density.

Acceleration relative to the rotating Earth

The acceleration measured on the rotating surface of the Earth is not quite the same as the acceleration that is measured for a free-falling body because of the centrifugal force. In other words, the apparent acceleration in the rotating frame of reference is the total gravity vector minus a small vector toward the north-south axis of the Earth, corresponding to staying stationary in that frame reference.

History of gravitational theory

The first mathematical formulation of gravity was published in 1687 by Sir Isaac Newton. His law of universal gravitation was the standard theory of gravity until work by Albert Einstein and others on general relativity. Since calculations in general relativity are complicated, and Newtonian gravity is sufficiently accurate for calculations involving weak gravitational fields (e.g., launching rockets, projectiles, pendulums, etc.), Newton's formulae are generally preferred.

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:

$F = -{G m_1m_2 \over r^2} = m_1a_1$

The above equation says that mass m1 will accelerate at acceleration a1 under the force of gravity, but divide both sides of the equation by m1 and:

$a_1 = {G m_2 \over r^2}$

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.

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.)

References

• Halliday, David; Robert Resnick; Kenneth S. Krane (2001). Physics v. 1, New York: John Wiley & Sons. ISBN 0471320579.
• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.), Brooks/Cole. ISBN 0534408427.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.), W. H. Freeman. ISBN 0716708094.
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