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18.04.2016
The drawback to Einstein’s Special Theory of Relativity, however, is that it is “special” in the respect that it only considers the effects of relativity to an observer moving at constant speed. At around this time (1907), he also started to wonder how Newtonian gravitation would have to be modified to fit in with special relativity, and how the effects of gravity could be incorporated into the formulation.
Initially, Einstein had been puzzled by the fact that Sir Isaac Newton’s Law of Universal Gravitation, which had stood undisputed since 1687, appeared to be fundamentally incompatible with his own Special Theory. Thus, according to this theory, gravity is relatively strong when objects are near each other, but weakens with distance, and the bigger the bodies, the more their force of mutual attraction.
Gravity is the organizing force for the cosmos, crucial in allowing structure to unfold from an almost featureless Big Bang origin. Even before Newton, the great 17th Century Italian physicist Galileo Galilei had shown that all bodies fall at the same rate, any perceived differences in practice being caused by differences in air resistance and drag.
Newton, however, had assumed that the force of gravity acts instantaneously, and Einstein had already shown that nothing can travel at infinite speed, not even gravity, being limited by the de facto universal speed limit of the speed of light.
Einstein’s ground-breaking realization (which he called “the happiest thought of my life”) was that gravity is in reality not a force at all, but is indistinguishable from, and in fact the same thing as, acceleration, an idea he called the “principle of equivalence”. The influence of gravity also creates effects of time dilation (see the section on the Special Theory of Relativity for a more detailed discussion of time dilation), sometimes referred to as "gravitational time dilation".
On a much smaller scale, because gravity is slightly stronger closer to the centre of the Earth, then theoretically time passes more slowly for someone living on the first floor of an apartment block than for someone living at the top. So, gravity, Einstein realized, is not really a force at all, but just the result of our surroundings accelerating relative to us.
The Sun's gravitational pull keeps the planets of the Solar System orbiting around it.
Newton did not discover gravity, as the popular belief holds, but expanded on the work of earlier scientists such as Galileo. In the centuries after Newton’s lifetime, the law of universal gravitation was used to predict the locations of planets and natural satellites that had not yet been discovered.
By the 20th century, scientists had discovered minor inconsistencies in the law of universal gravitation. The inverse square law proposed by Newton suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centers. Furthermore, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. Answer: If the distance is increased by a factor of 2, then distance squared will increase by a factor of 4. Answer: The effect of greater mass of Jupiter is partly off set by its larger radius which is about 10 times the radius of the earth.
Newton's law of gravitation was preceded by three important discoveries about planetary motion by the German astronomer Johannes Kepler. There is a popular misconception that the seasons on the Earth are caused by varying distances of the Earth from the Sun on its elliptical orbit. The primary cause of the seasons is the 23.5 degree of the Earth's rotation axis with respect to the plane of the ecliptic, as illustrated in the adjacent image .
Thus, we experience Summer in the Northern Hemisphere when the Earth is on that part of its orbit where the N. As is clear from the preceding diagram, the seasons in the Southern Hemisphere are determined from the same reasoning, except that they are out of phase with the N. In this situation, which is illustrated schematically in the adjacent figure, we say that differential forces act on the body (the Earth in this example). We may illustrate the basic idea with a simple model of a planet completely covered by an ocean of uniform depth, with negligible friction between the ocean and the underlying planet, as illustrated in the adjacent figure.
Without getting too much into the technical details, there are two bulges because of the differential gravitational forces. Another complication of a realistic model is that not only the Moon, but other objects in the Solar System, influence the Earth's tides. For example, particularly large tides are experienced in the Earth's oceans when the Sun and the Moon are lined up with the Earth at new and full phases of the Moon. Proposition: The Kepler Laws allowed Newton to establish his Law of Universal Gravitation without which Newton would have been at a considerable loss! Prior to Johannes Kepler ( 1571 - 1630 ), Copernicus ( Polish: Mikolaj Kopernik, 1473 - 1543 ) supposed that planetary orbits were approximate circles which indeed are ellipses of a special kind.
Kepler's 2nd Law ( Equal Areas in Equal Times ) states that a planet's motion maps out equal areas in equal times for a radius vector, , drawn from the sun - foci to the planet. This then provides one of the first and most important empirical proofs of Newton's Law of Universal Gravitation since planetary centripetal acceleration solely depends upon the distance of the planet from the sun!!
Of course, planets do not accelerate and fall into their center bodies due to a countervailing centrifugal force and Newton's Law of Inertia. Hence, here is a further interpretation of the force which maintains a planet's orbit about the sun and which is directly dependent upon the planet's distance from this central body. Moreover, the universality of Newton's concept of forces acting upon bodies did not just allow for the relative mass of the sun to effect a planet's orbital motion, but equally that there exists a mutual force arising from a planet's mass and hence its reciprocal effect upon the sun.
This says that there is some universal value, , for both the sun and any planet and therefore for any body whatsoever in the solar system.


Motion at constant speed is clearly a very special case, and in practice bodies change their speed with time. His resulting General Theory of Relativity, over ten years in the making (it was published in 1916), has been called the greatest contribution to science by a single human mind. Newton’s theory (permanently linked, at least in the popular mind, with his observation of an apple falling from a tree) stated that every massive body exerts an attractive force on every other massive body, a force which is proportional to the product of the two masses and inversely proportional to the square of the distance between the bodies.
This “inverse-square law” is quite sophisticated enough to explain why a cannonball fired horizontally travels further before hitting the ground the faster it is launched, why a certain minimum speed (about 11.2 kilometres per second) would be required to allow objects to break out of Earth’s gravity and into orbit and why the planets travel in an elliptical orbit around the Sun (although not quite sophisticated enough to predict the slight anomaly in Mercury’s orbit). Although it is a very weak force (feebler than the other fundamental forces which govern the sub-atomic world by a factor of 1036 or 1,000,000,000,000,000,000,000,000,000,000,000,000), it is a cumulative and consistent force which acts on everything and can act over large distances. Galileo’s famous (and probably apocryphal) experiment involving the dropping of two balls of different masses from the Leaning Tower of Pisa was repeated with even more dramatic results in 1972 when a hammer and a feather were dropped together on the airless Moon and, just as Galileo had predicted, both hit the ground together. Furthermore, Newton had assumed that the force of gravity was purely generated by mass, whereas Einstein had shown that all forms of energy had effective mass and must therefore also be sources of gravity. He realized that if he were to fall freely in a gravitational field (such as a skydiver before opening his parachute, or a person in an elevator when its cable breaks), he would be unable to feel his own weight, a rather remarkable insight in 1907, many years before the idea of freefall of astronauts in space became commonplace. That, and the feeling of his feet being glued to the ground just as they would be in Earth’s gravity, would be enough to convince the astronaut that the acceleration of the spaceship was indistinguishable from the pull of gravity on the Earth. As Einstein predicted, the closer a body is to a large mass, with a commensurately large gravitational pull, the slower time runs for it. With modern atomic clocks of sufficient accuracy, differences in the passage of time at different altitudes above sea level (and therefore different distances from the Earth's centre of gravity) can be measured, and even the tiny differences due to the changing shape of the Earth as the tidal force of the Moon pulls and stretches it.
Or, perhaps a better way of looking at it, gravity is a kind of inertial force, in the same way as the so-called centrifugal force is not a force in itself, merely the effect of a body’s inertia when forced into a circular path. In essence, the law states that all objects exert a gravitational pull on all other objects. Altering the separation distance (r) results in an alteration in the force of gravity acting between the objects. So if the separation distance is doubled (increased by a factor of 2), then the force of gravity is decreased by a factor of four (2 raised to the second power).
If you wanted to make a profit in buying gold by weight at one altitude and selling it at another altitude for the same price per weight, should you buy or sell at the higher altitude location?
The planet Jupiter is more than 300 times as massive as Earth, so it might seem that a body on the surface of Jupiter would weigh 300 times as much as on Earth. Inverse of the distance brings a factor of 100 to the denominator and as a result, the force increases by a factor of 300 due to the mass, but decreases by a factor of 100 due to the distance squared. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Hemisphere is oriented more toward the Sun and therefore the Sun rises higher in the sky and is above the horizon longer, and the rays of the Sun strike the ground more directly. Hemisphere Winter than in the Summer (as usual, we greatly exaggerate the eccentricity of the elliptical orbit in this diagram).
Since the Moon passes overhead about an hour later each day, it was long suspected that the Moon was associated with tides.
It is attracted gravitationally by the Earth, but it also experiences a much smaller gravitational attraction from the Moon (much smaller because the Moon is much further away and much less massive than the Earth). The gravitational attraction of the Moon produces two tidal bulges on opposite sides of the Earth. The liquid at point A is closer to the Moon and experiences a larger gravitational force than the Earth at point B or the ocean at point C.
For most their tidal forces are negligible on Earth, but the differential gravitational force of the Sun does influence our tides to some degree (the effect of the Sun on Earth tides is less than half that of the Moon).
In any event, because circles are special cases of generalized ellipses, any derivations based upon Kepler's 1st Law ( Planetary Law of Ellipses ) for ellipses must also hold true for circles. This universal factor of proportionality, , is called the gravitational constant and was finally and fairly accurately implicitly determined [ within 1% ] in 1797 by British Henry Cavendish ( 1731 - 1810 ) whose original intent was to calculate earth's density relative to water. Einstein wanted to generalize his theory to consider how a person sees another person who is accelerating relative to them. So, even though gravity can be effectively ignored by chemists studying how groups of atoms bond together, for bodies more massive than the planet Jupiter the effects of gravity overwhelm the other forces, and it is largely responsible for building the large-scale structures in the universe. A real-life example of gravitational time dilation can be seen in GPS systems in geo-synchronous orbits above the Earth, which need to constantly adjust their clocks to account for time differences due to the weaker gravity they experience compared to that on the Earth's surface (it is estimated that their accuracy would be out by as much as 10 kilometers a day without this adjustment). In order to rationalize this situation, though, Einstein was to turn our whole conception of space on its head, as we will see in the next section. It states that all objects are attracted to each other by gravity; the force of the attraction depends on the mass of the objects and decreases based on the distance between them. Instead, he used the moon’s orbit around the Earth to check and confirm his calculations over a span of 20 years. Objects with a great deal of mass have stronger areas of gravity, or gravitational fields, which is why objects and people are attracted to the Earth, but not noticeably to each other.


One aspect of the law Newton could not explain was how gravitational force is conveyed between objects. Einstein realized that, in effect, the masses of celestial bodies not only affect each other, but the fabric of space-time around them. Since the two quantities are inversely proportional, an increase in one quantity results in a decrease in the value of the other quantity. And if the separation distance (r) is tripled (increased by a factor of 3), then the force of gravity is decreased by a factor of nine (3 raised to the second power). If the distance between the two objects is doubled, what is the new force of attraction between the two objects? The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet earth goes around its orbit.
Thus, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same. One way to see that this reasoning may be in error is to note that the seasons are out of phase in the Northern and Southern hemispheres: when it is Summer in the North it is Winter in the South.
The Earth is at its closest approach to the Sun (perihelion) on about January 4 of each year, which is the dead of the N. But this gravitational attraction of the Moon is not limited to the water molecules; in fact, the Moon exerts a gravitational force on every object on and in the Earth.
Because it experiences a larger attraction, it is pulled away from the Earth, toward the Moon, thus producing the bulge on the right side.
The amount of enhancement in Earth's tides is about the same whether the Sun and Moon are lined up on opposite sides of the Earth (full Lunar phase) or on the same side (new Lunar phase).
Rather, Cavendish misinterpreted his discovery of the gravitational constant as earth's density; nevertheless, the torsion balance apparatus conceived and built earlier by then deceased John Mitchell ( Reverend, geologist and English natural philosopher, 1724 - 1793 ) was crated and sent on to Cavendish which allowed him to complete his 1797 experiment to within 1% accuracy of the modern gravitational constant calculation! Thus, gravity squeezes together massive bodies like our own Sun, and it is only the explosive outward energy in the Sun’s ultra-hot core that holds it in hydrostatic equilibrium and stops it from collapsing into a super-dense white dwarf star. The law of universal gravitation was detailed in his groundbreaking book Principia Mathematica, published in 1687.
The gravitational pull decreases as distance increases; this decrease can be precisely measured, and is known in physics as an inverse-square law.
Other fundamental forces, such as electromagnetism, work because sub-atomic particles travel between the objects, attracting them to each other.
The body of mass m2 in turn attracts the body of mass m1 with the same force F towards its centre O2.
That is, an increase in the separation distance causes a decrease in the force of gravity and a decrease in the separation distance causes an increase in the force of gravity.
Thinking of the force-distance relationship in this way involves using a mathematical relationship as a guide to thinking about how an alteration in one variable effects the other variable. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion.
Tides occur because the Earth is a body of finite extent and these forces are not uniform: some parts of the Earth are closer to the Moon than other parts, and since the gravitational force drops off as the inverse square distance, those parts experience a larger gravitational tug from the Moon than parts that are further away.
Loosely, we may think of the bulge on the left side as arising because the Earth is pulled away from the water on that side because the gravitational force exerted by the Moon at point B is larger than that exerted at point C. Conversely, when the Moon is at first quarter or last quarter phase (meaning that it is located at right angles to the Earth-Sun line), the Sun and Moon interfere with each other in producing tidal bulges and tides are generally weaker; these are called neap tides. Cavendish published his results in Philosophical Transactions of the Royal Society of London, Vol. For example, if a spaceship were to orbit close enough (but not too close!) to a hugely massive object such as a supermassive black hole, the gravitational effects may be significant enough to slow down time for the occupants compared to elsewhere, effectively allowing them to travel into the future. Universal gravitation is the force that keeps planets and satellites locked in orbit, rather than traveling freely about the universe.
A similar particle for conveying gravity, the graviton, has been described in theory, but remains undiscovered more than 300 years after Newton’s work.
For practical applications such as rocket launches, the law of universal gravitation is still accurate and much easier to calculate than the effects of relativity. Suppose the distance between their centers is‘d’ then, according to Newton’s universal law of gravitation, we have:Where G is called a proportionally constant or universal gravitational constant.
Hence, by Kepler's second law, the planet moves fastest when it is near perihelion and slowest when it is near aphelion. However, the fluid in the Earth's oceans is much more easily deformed and this leads to significant tidal effects. Then, as our idealized Earth rotates under these bulges, a given point on the surface will experience two high and two low tides for each rotation of the planet.



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