Gravity
- This article covers the physics of gravitation.\n \nIn a general sense, gravity means seriousness. In chemistry, gravity is the density of a fluid, particularly a fuel. There is also an anime series titled Gravitation (anime)''.
Gravitation is the tendency of masses to move toward each other.
The first mathematical formulation of the theory of gravitation was made by
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Newton's theory has now been replaced by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. For this reason Newtons law is often used and will be presented first.
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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.
-
where
- F is the magnitude of the gravitational force between two objects\n*m1 is the mass of first object\n*m2 is the mass of second object\n*r is the distance between the objects\n*G is the gravitational constant
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 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 as if the object were a point mass.
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.
-
As before,
m1 and
m2 are the masses of the objects 1 and 2, and
G is the gravitational constant.
- F12 is the force on object 1 due to object 2\n* r21 = | r1 − r2 | is the distance between objects 1 and 2\n* 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.
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:
- It assumes that gravitational force is tramsmitted instantaneously and by some unknown method ("action at a distance"). This was always felt to be unsatisfactory. More recently, special relativity has been successfully built on the backbone of the experimentally supported assumption there exists a maximum velocity at which signals can be transmitted (speed of light in vacuum).\n#The assumption of absolute space and time was never very satisfactory in itself. Also, it contradicts Einstein's theory of special relativity.\n#It does not explain the small portion of precession of orbit of Mercury of order of one angular second per century that kept astronomers baffled for more then a century as to the reason for it.\n#It predicts that light is deflected by gravity only half as much as observed (this was only observed after GR was developed).\n#The observed fact that gravitation and inertial mass are the same (or at least proportional) for all bodies is curious and unexplained within Newtons system. See equivalence principle.
Einstein developed a new theory called
general relativity which includes a theory of gravity, published in
1915. This graviational aspect of this theory says that the presence of matter "warps" the
spacetime. Objects in
free fall in the universe take geodesics in spacetime. A geodesic is the counterpart of a straight line in Euclidean geometry.
How curvatures of spacetime simulate gravitational force
The curvatures of spacetime considered as a whole create rather complex picture that is usually treated with tools of differential geometry and requires the use of tensor calculus. It is possible though to understand the mechanism of gravitation without tensors when those curvatures of spacetime are split into two components:
- curvature of space\n* time dilation.
Those two components make the whole Einsteinian gravitation. If, as we believe, Einstein's theory is true, only those two components can be responisible for all the gravitational phenomena in the universe.
The first component, the curvature of space, is negligible in all cases when masses of objects are much smaller than masses of stars. So for the majority of cases in the universe, and certainly for almost all cases in our solar system except two already specified as #3 and #4 at the beginning, we may treat the space as flat, as ordinary Euclidean space. It leaves us only with the gravitational time dilation as a possible reasons for the illusion of "gravitational force" acting at a distance.
The reason for this illusion is this: any mass in the universe modifies the rate of time in its vicinity this way that time runs slower closer to the mass and the change of time rate is controlled by an equation having exactly the same form as the equation that Newton discovered as his "Law of Universal Gravitation". The difference between them is in essence not in form since the Newtonian potential is replaced by the Einsteinian time rate , where is the time at a point at vicinity of the mass and is the time at observer at infinity, with the right side of the equation staying the same as in Newtonian equation (with accuracy to irrelevant constants). Because of the same form of both equations the extremum of proper time of any object traveling in vicinity of a mass, which corresponds to a geodesic in spacetime is exactly the same as Newtonian orbit of this object around the mass. They are necessarily the same because they both are controlled by equations having the same form.
So without any force involved into keeping the traveling object in line the object follows the Newtonian orbit in space just by following the line of extremal time, a geodesic line in spacetime. This is Einstein's explanation why without any "gravitational forces" all the objects follow Newtonian orbits and why the Newtonian gravitation is the approximation of the Einsteinian gravitation.
Why Einsteins' gravity differs from Newton's
Einsteinian gravitation is not exactly like Newtonian, only the time dilation portion of it is. There is in it also another component, the space curvature. In all cases when the space curvature becomes relevant like in close proximity to very big masses, like stars, or even in very close proximity to smaller masses since both mass and proximity influence the curvature of space the same way, or at very high velocities when the object travels quickly far enough to see the curvature of space and so it can't be neglected, there is a difference between predictions in Newtonian and Einsteinian theories. It turns out that Einsteinian predictions agree perfectly with observations.
In particular the Einsteinian gravitation explained why Mercury precession differs from Newtonian prediction: since Mercury is the closest planet to the sun it moves faster than any other planet, and also it is in more curved space than all other planets. This is reflected in the behavior of Mercury and the Einsteinian calculations predict this behavior within observational error.
The other Einsteinian prediction is bending light rays in vicinity of the sun. Since the Newtonian deflection of the ray corresponds only to the time dilation, and it happens for certain reasons (for conservation of energy in gravitation) that the relative curvature of space must be exactly the same as the relative time dilation, the total deflection is twice as big as its Newtonian prediction. And Einsteinian prediction being twice as big as Newtonian is again within the observational error.
Yet despite such an "elegant" simplification of physics only the observational differences between theories count in science since it is very easy to be mislead by "elegance" of theoretical differences. As Einstein said "the elegance should concern a tailor rather than a physicist".
E.g. before 1998 a group of prominent gravity physicists maintained that
elegance of Einstein's field equation requires to remove Einstein's
cosmological constant from it. They advertised this constant as an "Einstein's biggest blunder" (apparently a term coined by Einstein himself). Lack of this constant in Einstein's field equation predicted a
decelerating expansion of space, which in turn was strongly advocated by almost all gravity physicists at that time. It was called
standard model of cosmology. Proving that the expansion is decelerating was supposed to be the first proof ever that cosmology is science after all, since finally it would be able to predict something (as any true science should be able to do). A team of astronomers decided to verify this prediction. In
1998 the results came in. It turned out that the prediction is
false: the space of our universe looks as if it were expanding at
accelerating rate.
Why do we see an accelerating expansion of the universe?
Possibly not because of "dark energy" because Einstein's theory says that we would see it even without "dark energy". We have to see it if it follows directly from Einstein's theory. But how does it follow?
As we know from the previous section, energy is strictly conserved, even in physics of gravitation. Creating energy from nothing is most likely not possible and so such an extravagance is not possible even in Einstein's theory. After all Einstein was only a physicist and not a magician. We may safely assume that Einstein's theory is strictly consistent with conservation of energy.
An important consequence of this fact is that it wouldn't be possible to send a light signal through the universe and get it back without any redshift. After all photons have relativistic mass and so they have to interact gravitationally with the environment. If anyhhing moves in the environment because of interation with photons the environment gets some energy. If photons returned without any redshift it would contradict the conservation of energy. It means that photons, even in a stationary universe (one that is neither expanding nor contracting) have to come back with redshift, at least to keep appearences.
Now we know that the only two mechanisms of gravitational interaction are the curvature of space and the time dilation. The tired light effect is strictly forbidden in Einstein's theory because no force is acting on photons. Curvature of space doesn't produce any redshift but the time dilation of course does. So we have the culprit. But wait: it is a redshift after the photon traveled both ways! So it is not a regular
gravitational redshift and so not the regular
gravitational time dilation. It must be something special that we didn't see yet. Let's call it
general time dilation. The effect of time running slower both ways! How is it possible?! A paradox, but fortunately one that doesn't produce any logical contradiction (as neither of Einsteinian paradoxes does) so we leave it for curious characters to solve. It's just an interesting geometry of spacetime. This we leave to mathematicians to ponder upon. Since we don't have any choice within Einsteinian gravitation, and we don't want to get rid of it as long as it works, for the time being we keep Einstein and the effect of general time dilation.
Now, how big this effect is? After a back-of-envelope calculations, easy to do for any physicist since they are based only on the postulated above conservation of energy and the time dilation that both correspond exactly to Newtonian math which can be used here, we get
Hubble's constant of this effect , and ... its acceleration , where c is Newtonian constant of gravitation and is density of the universe. If we assume then from the first equation and the acceleration of expansion comes out as .
Is it the observed acceleration? Who knows. Astronomers put so many assumptions about the expansion of the universe into their data that even they aren't able to fish this number out of it (I asked many of them who did the observations and only one responded and said he doesn't know nor has time to find out). It might be a right number though and then Wikipedia gets a medal for saving the cosmologists an effort of analyzing the nature of "dark energy". But if not then we have to anounce that Einstein's theory failed since there are no adjustable parameters in Einstein's theory and either all the predictions of the theory are right or the theory is wrong.
\nAssuming the Einstein's theory survives we have an explanation of accelerating expansion
without any "dark energy" and
without even any real expansion of the universe. Those effects would be apparent as already "gravitational attraction" and "gravitational energy" turned out to be. And all this achieved without any original research since all of the above came out in a very straightforward way as a conclusion of Einstein's theory of gravity!
An additional interesting feature of the above equation for Hubble's constant that has been derived in the simple way of Newtonian math so even Newton could derive it had he known about curvature of space and speed of ligh is that it is the same as , where is speed of light and is the
radius of curvature of space of "
Einstein's universe". This model went out of grace though after astronomers convinced themselves that
Hubble redshift proves that the universe is expanding rather than energy is conserved globally.
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.
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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 10
36 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 10
18 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.
globular
star cluster
Gravitational field demonstrated]]
Gravity and Quantum Mechanics
Since Einstein discovered his theory of gravitation the gravity is not one of the fundamental forces of nature so it is a small wonder that it has not been fitted into the formalism of
quantum mechanics (the three fundamental forces:
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. It never can be for obvious reasons (for non existence of "gravitational attraction" explained in section "Einstein's Theory of Gravity") if Einstein's theory is true.
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
History
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:
-
The above equation says that mass will accelerate at
acceleration under the force of gravity, but divide both sides of the equation by and:
-
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/s
2 (or 32 ft/s
2). 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 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.
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
gn,
ge,
g0, gee, or simply
g. The following is a list of the gravity forces (in multiples of
g) at the surfaces of each of the planets in the solar system:
\n{| style="text-align: left;"\n| Mercury || || 0.376\n|-\n| Venus || || 0.903\n|-\n|Earth || = || 1\n|-\n| Mars || || 0.38\n|-\n| Jupiter || || 2.34\n|-\n| Saturn || || 1.16\n|-\n| Uranus || || 1.15\n|-\n| Neptune || || 1.19\n|-\n| Pluto || || 0.066\n|}\n
Note: The "surface" is taken to mean the clouds tops of the gas giants (Jupiter, Saturn, Uranus and Neptune) in the above table.
See also
\n* Gravity wave\n*
Gravitational binding energy\n*
Gravity Research Foundation\n*
Weight\n*
N-body problem\n*
Gravity Probe B Experiment
Category:Physics
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