General relativity

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 Sir Isaac Newton and proved astonishingly accurate. He postulated the force of "universal gravitational attraction".
..... Click the link for more information.  published by Albert Einstein Albert Einstein (March 14 1879 - April 18 1955) was a theoretical physicist, with considerable applied mathematical abilities, who is widely regarded as the greatest scientist of the 20th century. He proposed the theory of relativity and also made major contributions to the development of quantum mechanics, statistical mechanics and cosmology. He was awarded the 1921 Nobel Prize for Physics for his explanation of the photoelectric effect and "for his services to Theoretical Physics".
..... Click the link for more information.  in 1915
Years:
1912 1913 1914 - 1915 - 1916 1917 1918
Decades:
1880s 1890s 1900s - 1910s - 1920s 1930s 1940s
Centuries:
19th century - 20th century - 21st century

To accelerate an object is to change its velocity over a period of time. In this strict scientific sense, acceleration can have positive and negative values—respectively called acceleration and
..... Click the link for more information.  as different perspectives of the same thing, and which was originally stated by Einstein in 1907 as:

We shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame A frame of reference is a collection of conditions, axes, or assumptions which establish how something will be approached or understood. This article deals primarily with the general procedure for constructing a frame of reference within the science of physics.

A particular kind of reference frame is the inertial reference frame. It is the one in which the mathematical description
..... Click the link for more information. . This assumption extends the principle of relativity The Special relativity (SR) or Special relativity theory (SRT) is the physical theory published in 1905 by Albert Einstein. It replaced Newtonian notions of space and time, and incorporated electromagnetism as represented by Maxwell's equations. The theory is called "special" because the theory does not include a description of gravity. Ten years later, Einstein published the theory of general relativity, which incorporates gravitation.
..... Click the link for more information.  to the case of uniformly accelerated motion of the reference frame.

In other words, he postulated that no experiment can locally distinguish between a uniform gravitational field and a uniform acceleration.

This principle explains the experimental observation that inertial and gravitational mass

Mass is a property of physical objects which, roughly speaking, measure the amount of matter contained in an object. It is a central concept of classical mechanics and related subjects. In the SI system of measurement, mass is measured in kilograms.

Strictly speaking, there are two different quantities called mass:

• Inertial mass is a measure of an object's inertia, which is its resistance to changing its state of motion when a force is applied. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily.

The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. In Euclidean geometry, if we start with a point A and a line l, then we can only draw one line through A that is parallel to l. In hyperbolic geometry, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometry, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.)
..... Click the link for more information. : that spacetime In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional manifold called spacetime (alternatively, space-time). A point in spacetime may be referred to as an event. Each event has four coordinates (t, x, y, z); or, in angular coordinates, t, r, θ, and φ.
..... Click the link for more information.  is curved Curvature is the amount by which an geometric object deviates from being flat. The word flat might have very different meaning depending on the object considered (for curves it is a straight line and for surfaces it is a Euclidean plane).

In this article we consider the most basic examples: the curvature of a plane curve and the curvature of a surface in Euclidean space. See the links below for further reading.
..... Click the link for more information.  (by matter and energy), and gravity can be seen purely as a result of this geometry Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.
..... Click the link for more information. . This then yields many predictions such as gravitational redshifts and light bent around stars, black holes, time slowed by gravitational fields, and slightly modified laws of gravitation. However, it should be noted that the equivalence principle does not rigorously uniquely determine the field equations of curved spacetime, and so the latter are now considered to be pre-eminent.

The modifications to Isaac Newton

Sir Isaac Newton (December 25, 1642 – March 20, 1727 by the Julian calendar then in use; or January 4, 1643 – March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist; who published the Philosophiae Naturalis Principia Mathematica (published July 5, 1687), where he described universal gravitation and, via his laws of motion, laid the groundwork for classical mechanics. Newton also shares credit with Gottfried Wilhelm Leibniz for the development of differential calculus.
..... Click the link for more information. 's Law of Gravity produced the first great theoretical success of general relativity: the correct prediction of the precession of the perihelion of Mercury's orbit. Many other quantitative predictions of general relativity have since been confirmed by astronomical observations, and the theory is now considered well established, although alternatives are still occasionally proposed. In particular some scientists believe that the Allais effect The Allais effect describes the unexplained increase in speed of a moving pendulum during a solar eclipse. It was first observed in 1954 by Maurice Allais, a French economist who went on to win the Nobel prize for Economics.

Many scientists dispute whether such an effect can be consistenly observed. Others believe that if such an effect does exist it can be explained by

• the seismic disturbance caused by a large number of people moving to and from a place where an eclipse is visible
• denser air cooled by the moon's shadow exerting a different gravitational pull on the pendulum
• the cooling of the Earth's crust caused by the shadow of the eclipse

TIME is a weekly American news magazine, similar to Newsweek and U.S. News and World Report. A European edition (TIMEeurope, formerly known as TIMEatlantic) is published from London. TIMEeurope covers the Middle East, Africa and (since 2003) Latin America. An Asian edition, (TIMEasia), is based in Hong Kong.

The first issue of TIME
..... Click the link for more information.  separately, but only as a single four-dimensional system, "space-time," which was divided into "time-like" and "space-like" directions differently depending on the observer's motion. The general theory added to this that the presence of matter "warped" the local space-time environment, so that apparently "straight" lines through space and time have the properties we think of "curved" lines as having.

On May 29 May 29 is the 149th day of the year in the Gregorian calendar (150th in leap years). There are 216 days remaining.

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Events

• 1167 - Battle of Legano, in which The Lombard League defeats Emperor Frederick I.

He is famous for his work regarding the Theory of Relativity. Eddington wrote an article, Report on the relativity theory of gravitation, which announced Einstein's theory of general relativity to the English-speaking world. Because of World War I, new developments in German science were not well known in England.
..... Click the link for more information.  of shifted star positions during a solar eclipse

Solar Eclipse is the name of an alien friend of Betty Spaghetty.

A solar eclipse occurs when the Sun, Moon and Earth are on a single line with the Moon in the middle. Seen from the Earth, the Moon is in front of the Sun and thus part or all of the light of the Sun is eclipsed by the Moon. Thus it may seem that a piece has been taken out of the Sun, or that it has suddenly disappeared.
..... Click the link for more information.  confirmed the theory.

Foundations

General relativity's mathematical foundations go back to the axioms

There is algebra software named Axiom.

In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that any axioms, understood in that sense, exist.

In mathematics, axioms are not self-evident truths. They are of two different kinds: logical axioms and non-logical axioms. Axiomatic reasoning is today most widely used in mathematics.
..... Click the link for more information.  of Euclidean geometry In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties.

Euclidean geometry sometimes means geometry in the plane which is also called plane geometry. Plane geometry is the topic of this article.

Euclidean geometry in three dimensions is traditionally called solid geometry. For information on higher dimensions see Euclidean space.
..... Click the link for more information.  and the many attempts over the centuries to prove Euclid

Euclid of Alexandria (Greek: Eukleides) (circa 365–275 BC) was a Greek mathematician. His most famous work is the Elements, widely considered to be history's most successful textbook. Within it, the properties of geometrical objects and integers are deduced from a small set of axioms, thereby anticipating (and partially inspiring) the axiomatic method of modern mathematics. Although many of the results in the Elements originated with earlier mathematicians, one of Euclid's major accomplishments was to present them in a single, logically coherent framework. He also provided some missing proofs. The text also includes sections on number theory and three-dimensional geometry.
..... Click the link for more information. 's fifth postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry. It states:

If a line segment intersects two straight lines forming two interior angles on the same side sum to less than two right angles then the two lines segments, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

..... Click the link for more information. , that parallel lines remain always equidistant, culminating with the realisation by Lobachevsky, Bolyai and Gauss that this axiom need not be true. The general mathematics of non-Euclidean geometries was developed by Gauss' student, Riemann, but these were thought to be wholly inapplicable to the real world until Einstein developed his theory of relativity.

Gauss had realised that there is no a priori reason that the geometry of space should be Euclidean. What this means is that if a physicist holds up a stick, and a cartographer stands some distance away and measures its length by a triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if the physicist brings the stick to him and he measures its length directly. Of course for a stick he could not in practice measure the difference between the two measurements, but there are equivalent measurements which do detect the non-Euclidean geometry of space-time directly; for example the Pound-Rebka experiment (1959) detected the change in wavelength of light from a cobalt source rising 22.5 meters against gravity in a shaft in the Jefferson Physical Laboratory at Harvard, and the rate of atomic clocks in GPS satellites orbiting the Earth has to be corrected for the effect of gravity.

Newton's theory of gravity had assumed that objects did in fact have absolute velocities: that some things really were at rest while others really were in motion. He realized, and made clear, that there was no way these absolutes could be measured. All the measurements one can make provide only velocities relative to one's own velocity (positions relative to one's own position, and so forth), and all the laws of mechanics would appear to operate identically no matter how one was moving. Newton believed, however, that the theory could not be made sense of without presupposing that there are absolute values, even if they cannot be determined. In fact, Newtonian mechanics can be made to work without this assumption: the outcome is rather innocuous, and should not be confused with Einstein's relativity which further requires the constancy of the speed of light.

In the nineteenth century Maxwell formulated a set of equations—Maxwell's field equations—that demonstrated that light should behave as a wave emitted by electromagnetic fields which would travel at a fixed velocity through space. This appeared to provide a way around Newton's relativity: by comparing one's own speed with the speed of light in one's vicinity, one should be able to measure one's absolute speed--or, what is practically the same, one's speed relative to a frame of reference that would be the same for all observers.

The assumption was whatever medium light was travelling through—whatever it was waves of—could be treated as a background against which to make other measurements. This inspired a search to determine the earth's velocity through this cosmic backdrop or "ether"—the "ether drift." The speed of light measured from the surface of the earth should appear to be greater when the earth was moving against the ether, slower when they were moving in the same direction. (Since the earth was hurtling through space and spinning, there should be at least some regularly changing measurements here.) A test made by Michelson and Morley toward the end of the century had the astonishing result that the speed of light appeared to be the same in every direction.

In his 1905 paper "On the Electrodynamics of Moving Bodies", Einstein explained these results in his theory of special relativity.

Outline of the theory

The fundamental idea in relativity is that we cannot talk of the physical quantities of velocity or acceleration without first defining a reference frame, and that a reference frame is defined by choosing particular matter as the basis for its definition. Thus all motion is defined and quantified relative to other matter. In the special theory of relativity it is assumed that reference frames can be extended indefinitely in all directions in space and time. The theory of special relativity concerns itself with inertial (non-accelerating) frames while general relativity deals with all frames of reference. In the general theory it is recognised that we can only define local frames to given accuracy for finite time periods and finite regions of space (similarly we can draw flat maps of regions of the surface of the earth but we cannot extend them to cover the whole surface without distortion). In general relativity Newton's laws are assumed to hold in local reference frames. In particular free particles travel in straight lines in local inertial (Lorentz) frames. When these lines are extended they do not appear straight, and are known as geodesics. Thus Newton's first law is replaced by the law of geodesic motion.

We distinguish inertial reference frames, in which bodies maintain a uniform state of motion unless acted upon by another body, from non-inertial frames in which freely moving bodies have an acceleration deriving from the reference frame itself. In non-inertial frames there is a perceived force which is accounted for by the acceleration of the frame, not by the direct influence of other matter. Thus we feel g-forces when cornering on the roads when we use a car as the physical base of our reference frame. Similarly there are coriolis and centrifugal forces when we define reference frames based on rotating matter (such as the Earth or a child's roundabout). The principle of equivalence in general relativity states that there is no local experiment to distinguish non-rotating free fall in a gravitational field from uniform motion in the absence of a gravitational field. In short there is no gravity in a reference frame in free fall. From this perspective the observed gravity at the surface of the Earth is the force observed in a reference frame defined from matter at the surface which is not free, but is acted on from below by the matter within the Earth, and is analogous to the g-forces felt in a car.

Mathematically, Einstein models space-time by a four-dimensional pseudo-Riemannian manifold, and his field equation states that the manifold's curvature at a point is directly related to the stress energy tensor at that point; the latter tensor being a measure of the density of matter and energy. Curvature tells matter how to move, and matter tells space how to curve.

The field equation is not uniquely proven, and there is room for other models, provided that they do not contradict observation. General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matter and curvature, although we still await the unification of general relativity and quantum mechanics and the replacement of the field equation with a deeper quantum law. Few physicists doubt that such a theory of everything will give general relativity in the appropriate limit, just as general relativity predicts Newton's law of gravity in the non-relativistic limit.

Einstein's field equation contains a parameter called the "cosmological constant" which was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations by Hubble a decade later confirmed that our universe is in fact not static but expanding. So Λ was abandoned, with Einstein calling it the "biggest blunder [I] ever made". However, quite recently, improved astronomical techniques have found that a non-zero value of Λ is needed to explain some observations.

The field equation reads as follows:

where is the Ricci curvature tensor, is the scalar curvature, is the metric tensor, is the cosmological constant, is the stress-energy tensor, is pi, is the speed of light and is the gravitational constant which also occurs in Newton's law of gravity. describes the metric of the manifold and is a symmetric 4 x 4 tensor, so it has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6.

The study of the solutions of this equation is one of the activities of a branch of astronomy named cosmology. It leads to the prediction of black holes and to the different models of evolution of the universe.

The vierbein formulation of general relativity

This is an alternative equivalent formulation of general relativity using four reference vector fields, called a vierbein or tetrad. We have four vector fields, e_a, a=0,1,2,3 such that g(e_a,e_b)=η_ab where

.

See sign convention. One thing to note is that we can perform an independent proper, orthochronous Lorentz transformation at each point (subject to smoothness, of course) and still get a valid tetrad. So, the tetrad formulation of GR is a gauge theory, but with a noncompact gauge group SO(3,1). It is also invariant under diffeomorphisms.

See vierbein and Palatini action for more details. See Einstein-Cartan theory for an extension of general relativity to include torsion. See teleparallelism for another theory which predicts the same results as general relativity but with FLAT spacetime with no curvature.

Quotes

The theory appeared to me then, and still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill. But its connections with experience were slender. It appealed to me like a great work of art, to be enjoyed and admired from a distance. —Max Born

References

• Bondi, Herman, Relativity and Common Sense, Heinemann (1964). A school level introduction to the principle of relativity by a renowned scientist.
• Baez, Bunn, 2001, The Meaning of Einstein's Equation, intuitive explanation of Einstein-Hilbert equations - requires familiarity with special relativity
• Carroll, Sean M., Spacetime and Geometry: An introduction to general relativity, Addison Wesley, San Francisco (2004). ISBN 0-8053-8732-3. A modern graduate level textbook; notes from an earlier version are available at arXiv:gr-qc/9712019.
• D'Inverno, Ray, Introducing Einstein's Relativity, Oxford University Press (1993). A modern undergraduate level text.
• Einstein, Albert, Relativity: The special and general theory. ISBN 0517884410. The special and general relativity theories in their original form.
• Epstein, Lewis Caroll, Relativity Visualized. ISBN 093521805X. Requires no mathematical background. Actually explains general relativity, rather than merely hinting at it with a few metaphors.
• Misner, Thorne, Wheeler, Gravitation, Freeman (1973). ISBN 0716703440. A classic graduate level text book, which, if somewhat long winded, pays more attention to the geometrical basis and the development of ideas in general relativity than some more modern approaches.
• Perret, W. and G.B. Jeffrey, trans.: The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, New York Dover (1923).
• Thorne, Kip, and Stephen Hawking, Black Holes and Time Warps, Papermac (1995). A recent popular account, by a leading expert.
• J. J. O'Connor and E. F. Robertson, History of General Relativity at the MacTutor History of Mathematics archive.
• Reflections on Relativity A complete online course on Relativity
• MIT 8.962 Course Notes Notes and handouts from the MIT 8.962 course on General Relativity

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