Gravity
- ''This article covers the physics of gravitation.
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Newton's theory has now been replaced by Albert Einstein's theory of General relativity, but for most purposes, Newton's formulas with their simple math are sufficiently accurate.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.
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 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.
- F is the magnitude of the gravitational force between two objects
- m_{1} is the mass of first object
- m_{2} is the mass of second object
- r is the distance between the objects
- G is the gravitational constant
This law of universal gravitation was originally formulated by Isaac Newton in his work, the Principia Mathematica (1687). The history of the gravitation as a physical concept is considered in more detail below.
Vector Form
Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formulation, quantities in bold represent vectors.
- F_{12} is the force on object 1 due to object 2
- r_{21} = | r_{1} − r_{2} | is the distance between objects 1 and 2
- is the unit vector from object 2 to 1
Einstein's Theory of Gravity
- 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).
- Assumption of absolute space and time contradicts Einstein's theory of special relativity.
- 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.
- It predicts that light is deflected by gravity only half as much as observed.
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
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
How energy is conserved without gravitational attraction?
Why do we see an accelerating expansion of the universe?
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 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
Crucial experiments that justified the adoption of General Relativity over Newtonian gravity were the classical tests: the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury.
General relativity also explains the equivalence of gravitational and inertial mass, which has to be assumed in Newtonian theory.
More recent experimental confirmations of General Relativity were the (indirect) deduction of gravitational waves being emitted from orbiting binary stars, the existence of neutron stars and black holes, gravitational lensing, and the convergence of measurements in observational cosmology to an approximately flat model of the observable Universe, with a matter density parameter of approximately 30% of the critical density and a cosmological constant of approximately 70% of the critical density.
Even to this day, scientists try to challenge General Relativity with more and more precise direct experiments. The goal of these tests is to shed light on the yet unknown relationship between Gravity and Quantum Mechanics. Space probes are used to either make very sensitive measurements over large distances, or to bring the instruments into an environment that is much more controlled than it could be on Earth. For exampled, in 2004 a dedicated satellite for gravity experiments, called Gravity Probe B, was launched. Also, land-based experiments like LIGO are gearing up to possibly detect gravitational waves directly.
Speed of gravity: Einstein's theory of relativity predicts that the speed of gravity (defined as the speed at which changes in location of a mass are propagated to other masses) should be consistent with the speed of light. In 2002, the Fomalont-Kopeikin experiment produced measurements of the speed of gravity which matched this prediction. However, this experiment has not yet been widely peer-reviewed, and is facing criticism from those who claim that Fomalont-Kopeikin did nothing more than measure the speed of light in a convoluted manner.
Alternate Theories
- In the modified Newtonian dynamics (MOND), Mordehai Milgrom proposes a weakening of Newton's law at high distance as a possible explanation of the fast galaxy rotation.
- Nikola Tesla challenged Albert Einstein's theory of relativity, announcing he was working on a Dynamic theory of gravity and argued that a field of force was a better concept and did away with the curvature of space.
- George Louis LeSage proposed a gravity mechanism, referred to as "LeSage gravity, based on a fluid-based explanation where a light gas fills the entire universe.
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:
However, across a large body, variations in can create a significant tidal force.
Newton's reservations
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 g_{n}, g_{e}, g_{0}, 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
- Gravity wave
- Gravitational binding energy
- Gravity Research Foundation
- Weight
- N-body problem
- Gravity Probe B Experiment