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Gravity

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''This article covers the physics of gravitation.

In 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". Later developments have shown that this force does not actually exist.

Newton's theory has now been replaced by Albert Einstein's theory of General relativity, but for most purposes, Newton's formulas with their simple math are sufficiently accurate. For example, rocket ships, bullets, and peregrine falcons do not attain the high speeds nor involve the immense masses which necessitate the use of Einstein's math.

We present the description of gravitation starting with Newtonian math. It isn't physics, but it is simple and helpful for learning the quantitative part of the physics of gravitation. Then we proceed to Einstein's model to explain the true, as we believe now, and even simpler physics. It is simpler since there is no mysterious "gravitational attraction" in it that would require an explantion. We skip the exact math that is not very useful even for professonals and is generally considered the most difficult math that has ever existed in physics (Einstein once said about his theory something like "if you think that you have problems with math I may assure you they are nothing compared to mine".)

When we see Einstein's physics we might understand why Newton's math based on "gravitational attraction" is accurate despite that masses in the universe don't attract each other.

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

Newton's Law of Universal Gravitation

Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines are approximately parallelEnlarge

Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines are approximately parallel

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

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

Gravity on a macroscopic scaleEnlarge

Gravity on a macroscopic scale

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.

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:

  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 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.
  4. It predicts that light is deflected by gravity only half as much as observed.

For the first two of these reasons, Einstein developed a new theory called general relativity, a theory of gravity, published in 1915. This theory says that the presence of matter "warps" the spacetime in such a way that "straight lines" taken by free falling objects look curved to us making an impression that those lines are bent by some kind of a force.

Those "straight lines" are "straight" in certain physical sense. The objects that take them decide that they are "straight" on a basis of some physical criterion that has extremal value along that line (e.g. the line is shortest of all lines surrounding it like a "straight line" joining two points on the surface of earth which is not "straight" in the sense of Euclidean geometry because it is really an arc of a great circle but it is "the shortest possible" of all lines on this surface so it is "straight in the sense of spherical geometry). Such a line with extremum of some physical criterion, regardless of its other properties, is called geodesic line in the given space. The criteron of "straightness" in Einstein's theory is proper time taken by the object moving along the line.

So Einstein's theory says that all objects in free fall in the universe take those geodesics in spacetime i.e. they take physically (in the sense of longest proper time) straight lines from their point of view while in the ordinary 3D space we see those objects as moving around ellipses, parabolas etc.

We don't notice what happens to the time rate felt by those objects that makes their paths curved in the ordinary 3D space. We might tend to think that curving of the paths is because some external force is acting here. Only when we measure the time rate with precise clocks we see that the time really behaves the way Einstein's theory predicts and so no external force can be present. If it were present the effect would be twice as big as it is.

This behavior of time is called gravitational time dilation and it is an extremely subtle effect. E.g. next to the earth surface the relative change in time rate is only about 10-16/m. It might be tough to believe that such a subtle effect and not "the universal gravitational attraction" is the real reason for all the gravitational effects in the universe.

This is the crux of Einstein's theory: the gravitational time dilation together with the curvature of space (called together the curvatures of spacetime) make all the free falling particles moving along curved lines without any physical force acting on them. Those gravitational effects are caused by the fact that the probability of finding a particle on any other paths through the space vanishes: their wavefunctions, that determine the probability of finding the particle in any particular location in space on any other path, combined toghether from all the possible paths would destroy each other. Only the paths of extremal time provides a non vanishing probability of finding a particle, and so the particle is there. And in this sense the Einsteinian gravitation is a quantum teory.

Those curved paths in ordinary 3D space simulate the existence of some force that is bending those paths. This simulated force that people imagined as bending the trajectories of planets and accelerating all kinds of objects in the universe has been called "gravitational attraction". It has been thought to be a real force. It was called at that time "one of the fundamental forces of nature". According to Einstein's theory this force doesn't exist. It is simulated by the physical properties of space and time.

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 doing the math well and to comply with the principles of general realtivity 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:

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 (it includes even the observed since 1998 the accelerating expansion of space but to make it short we skip explantions of less relevant details at least for the time being).

The first component, the curvature of space, is negligible in all cases when masses of objects are much smaller than masses of stars and velocities of objects are much smaller then speed of light. 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 of section "Einstein's Theory of Gravity", 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".

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 Einsteinian gravitation differs from Newtonian

Since Einsteinian gravitation is not exactly like Newtonian, only the time dilation is, and there is in it also another component, the space curvature then in all cases of very big masses (like our sun) or very high velocities (like velocity of photons that necessarily move with speed of light) when curvature of space gets noticeable, there is 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 is as it is observed: since Mercury is the closest planet to the sun it moves faster than any other planet, and also it is in more distored space than all others. 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.

Because of those two tests coming out in favor of Einstein's theory plus the base of Einstein's theory, gravitational time dilation, also confirmed observationally, we consider Einstein's theory a true theory and Newton's theory a false one.

How energy is conserved without gravitational attraction?

All the above showed us that there is no gravitational attraction acting on free falling objects. Somehow they move by themselves. And accelerate by themselves. We don't know about anything in the nature that could be responsible for delivering energy to the free faling objects. And therefore there is also nothing that could absorb this energy (as the old "Newtonian gravitational field" with its "gravitational potential energy" used to do). And yet kinetic energy of free faling objects changes constantly. Where is it coming from and where is it going to? Below we show where and so we show how energy is conserved in Einstein's gravitation. And what else follows from it.

... to be continued.

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 teory. 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.

Assuming 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. All effects would turn out to be apparent as already "gravitational attraction" and "gravitational energy" turned out to be. And all 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!

By the way, an interesting feature of the equations for Hubble's constant above, derived in this primitive way of Newtonian math (so even Newton could do it), is that it is the same as , where is speed of light and is so called "Einstein's radius of the universe" (radius of curvature of space of "Einstein's universe", an old stationary model of the universe that went out of grace after astronomers convinced themselves that the universe is expanding). However Newton wouldn't know what in this equation means since then the curvature of space was not a known concept, nor the role of speed of light in the nature. This is how some discoveries have to wait for other discoveries even in seemingly non related areas of science.

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.

[[M13Enlarge

[[M13

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/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 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:

Mercury 0.376
Venus 0.903
Earth = 1
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 clouds tops of the gas giants (Jupiter, Saturn, Uranus and Neptune) in the above table.

See also