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- Introduction to Numerical Relativity
- Space, Time and Einstein: An Introduction 077352472X, 9780773524729
- Theory of relativity
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General relativity is a theory of gravitation developed by Albert Einstein between and The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.

By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion. Experiments and observations show that Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets.

General relativity also predicts novel effects of gravity, such as gravitational waves , gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, most recently gravitational waves. General relativity has developed into an essential tool in modern astrophysics.

It provides the foundation for the current understanding of black holes , regions of space where the gravitational effect is strong enough that even light cannot escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects such as active galactic nuclei or microquasars. General relativity is also part of the framework of the standard Big Bang model of cosmology. Although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data.

Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. In September , Albert Einstein published his theory of special relativity , which reconciles Newton's laws of motion with electrodynamics the interaction between objects with electric charge.

Special relativity introduced a new framework for all of physics by proposing new concepts of space and time. Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of gravity , which describes the mutual attraction experienced by bodies due to their mass. Several physicists, including Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity.

Only Einstein's theory proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow Einstein's thinking between and , from his simple thought experiment involving an observer in free fall to his fully geometric theory of gravity.

A person in a free-falling elevator experiences weightlessness ; objects either float motionless or drift at constant speed. Since everything in the elevator is falling together, no gravitational effect can be observed.

In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, far from any significant source of gravity. Such observers are the privileged "inertial" observers Einstein described in his theory of special relativity : observers for whom light travels along straight lines at constant speed.

Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and he made this the cornerstone of his theory of general relativity, formalized in his equivalence principle. Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that they are in free fall.

Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity. Most effects of gravity vanish in free fall, but effects that seem the same as those of gravity can be produced by an accelerated frame of reference. An observer in a closed room cannot tell which of the following is true:. Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength.

This principle allowed Einstein to predict several novel effects of gravity in , as explained in the next section. An observer in an accelerated reference frame must introduce what physicists call fictitious forces to account for the acceleration experienced by himself and objects around him.

One example, the force pressing the driver of an accelerating car into his or her seat, has already been mentioned; another is the force one can feel while pulling the arms up and out if attempting to spin around like a top. Einstein's master insight was that the constant, familiar pull of the Earth's gravitational field is fundamentally the same as these fictitious forces. By analogy, Einstein proposed that an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in Newton's law of gravitation.

In , Einstein was still eight years away from completing the general theory of relativity. Nonetheless, he was able to make a number of novel, testable predictions that were based on his starting point for developing his new theory: the equivalence principle. The first new effect is the gravitational frequency shift of light. Consider two observers aboard an accelerating rocket-ship. Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and unattached objects accelerate in the opposite direction, falling "downward".

Assume that one of the observers is "higher up" than the other. When the lower observer sends a light signal to the higher observer, the acceleration causes the light to be red-shifted , as may be calculated from special relativity ; the second observer will measure a lower frequency for the light than the first. Conversely, light sent from the higher observer to the lower is blue-shifted , that is, shifted towards higher frequencies.

This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described below. This gravitational frequency shift corresponds to a gravitational time dilation : Since the "higher" observer measures the same light wave to have a lower frequency than the "lower" observer, time must be passing faster for the higher observer.

Thus, time runs more slowly for observers who are lower in a gravitational field. It is important to stress that, for each observer, there are no observable changes of the flow of time for events or processes that are at rest in his or her reference frame.

Five-minute-eggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity. It is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher. In a similar way, Einstein predicted the gravitational deflection of light : in a gravitational field, light is deflected downward.

Quantitatively, his results were off by a factor of two; the correct derivation requires a more complete formulation of the theory of general relativity, not just the equivalence principle. The equivalence between gravitational and inertial effects does not constitute a complete theory of gravity.

When it comes to explaining gravity near our own location on the Earth's surface, noting that our reference frame is not in free fall, so that fictitious forces are to be expected, provides a suitable explanation.

But a freely falling reference frame on one side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction. A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth. These bodies are not falling in precisely the same direction, but towards a single point in space: namely, the Earth's center of gravity. Consequently, there is a component of each body's motion towards the other see the figure.

In a small environment such as a freely falling lift, this relative acceleration is minuscule, while for skydivers on opposite sides of the Earth, the effect is large.

Such differences in force are also responsible for the tides in the Earth's oceans, so the term " tidal effect " is used for this phenomenon. In exploring the equivalence of gravity and acceleration as well as the role of tidal forces, Einstein discovered several analogies with the geometry of surfaces.

An example is the transition from an inertial reference frame in which free particles coast along straight paths at constant speeds to a rotating reference frame in which extra terms corresponding to fictitious forces have to be introduced in order to explain particle motion : this is analogous to the transition from a Cartesian coordinate system in which the coordinate lines are straight lines to a curved coordinate system where coordinate lines need not be straight.

A deeper analogy relates tidal forces with a property of surfaces called curvature. For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature determines whether or not a surface is equivalent to a plane. In the summer of , inspired by these analogies, Einstein searched for a geometric formulation of gravity.

In , Hermann Minkowski , Einstein's former mathematics professor at the Swiss Federal Polytechnic, introduced Minkowski space , a geometric formulation of Einstein's special theory of relativity where the geometry included not only space but also time. The basic entity of this new geometry is four- dimensional spacetime. The orbits of moving bodies are curves in spacetime ; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.

The geometry of general curved surface was developed in early 19th century by Carl Friedrich Gauss. This geometry had in turn been generalized to higher-dimensional spaces in Riemannian geometry introduced by Bernhard Riemann in the s.

With the help of Riemannian geometry , Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces. Embedding Diagrams are used to illustrate curved spacetime in educational contexts. After he had realized the validity of this geometric analogy, it took Einstein a further three years to find the missing cornerstone of his theory: the equations describing how matter influences spacetime's curvature.

Having formulated what are now known as Einstein's equations or, more precisely, his field equations of gravity , he presented his new theory of gravity at several sessions of the Prussian Academy of Sciences in late , culminating in his final presentation on November 25, Paraphrasing John Wheeler , Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.

In order to map a body's gravitational influence, it is useful to think about what physicists call probe or test particles : particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of spacetime , this is equivalent to saying that such test particles move along straight world lines in spacetime.

In the presence of gravity, spacetime is non-Euclidean , or curved , and in curved spacetime straight world lines may not exist. Instead, test particles move along lines called geodesics , which are "as straight as possible", that is, they follow the shortest path between starting and ending points, taking the curvature into consideration.

A simple analogy is the following: In geodesy , the science of measuring Earth's size and shape, a geodesic from Greek "geo", Earth, and "daiein", to divide is the shortest route between two points on the Earth's surface. Approximately, such a route is a segment of a great circle , such as a line of longitude or the equator.

These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint. The properties of geodesics differ from those of straight lines.

For example, on a plane, parallel lines never meet, but this is not so for geodesics on the surface of the Earth: for example, lines of longitude are parallel at the equator, but intersect at the poles. Analogously, the world lines of test particles in free fall are spacetime geodesics , the straightest possible lines in spacetime. But still there are crucial differences between them and the truly straight lines that can be traced out in the gravity-free spacetime of special relativity.

In special relativity, parallel geodesics remain parallel. In a gravitational field with tidal effects, this will not, in general, be the case.

If, for example, two bodies are initially at rest relative to each other, but are then dropped in the Earth's gravitational field, they will move towards each other as they fall towards the Earth's center.

Compared with planets and other astronomical bodies, the objects of everyday life people, cars, houses, even mountains have little mass. Where such objects are concerned, the laws governing the behavior of test particles are sufficient to describe what happens.

Notably, in order to deflect a test particle from its geodesic path, an external force must be applied. A chair someone is sitting on applies an external upwards force preventing the person from falling freely towards the center of the Earth and thus following a geodesic, which they would otherwise be doing without matter in between them and the center of the Earth. In this way, general relativity explains the daily experience of gravity on the surface of the Earth not as the downwards pull of a gravitational force, but as the upwards push of external forces.

These forces deflect all bodies resting on the Earth's surface from the geodesics they would otherwise follow. In Newton's description of gravity , the gravitational force is caused by matter.

The first edition of Exploring Black Holes: Introduction to General Relativity , authored by Oersted Medal winner Edwin Taylor and foremost relativist John Archibald Wheeler, offered a concise, directed examination of general relativity and black holes. Its goal was to provide tools that motivate students to become active participants in carrying out their own investigations about curved spacetime near Earth and black holes. To that end, the book used calculus and algebra, rather than tensors, to make general relativity accessible to second- and third-year students. Taylor to revise and expand the first edition. The text uses the properties of non-spinning and spinning black holes to introduce Albert Einstein's theory of curved spacetime and applies the resulting general relativity to the Universe around us.

When the theory of relativity appeared in the early s, it upended centuries of science and gave physicists a new understanding of space and time. Isaac Newton saw space and time as fixed, but in the new picture provided by special relativity and general relativity they were fluid and malleable. Albert Einstein. He published the first part of his theory — special relativity — in the German physics journal Annalen der Physik in and completed his theory of general relativity only after another decade of difficult work. He presented the latter theory in a series of lectures in Berlin in late and published in the Annalen in

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Chapter of a classical mechanics text describes spatiotemporal effects. He proposed that the laws of classical mechanics had to be consistent with just two postulates, namely that the speed of light is a constant and that all frames of reference are equivalent. Fundamentals of Mechanics. Special RelatiVity. Lorentz contraction of the length 19 B. This book argues that the past history should be taken into account. Find, read and cite all the research you need on ResearchGate.

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