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123-430: Manifolds of constant curvature are most familiar in the case of two dimensions, where the elliptic plane or surface of a sphere is a surface of constant positive curvature, a flat (i.e., Euclidean ) plane is a surface of constant zero curvature, and a hyperbolic plane is a surface of constant negative curvature. Einstein's general theory of relativity places space and time on equal footing, so that one considers

246-532: A ( n + 1 ) {\displaystyle (n+1)} -dimensional flat spacetime with the metric d i a g ( − 1 , − 1 , + 1 , … , + 1 ) {\displaystyle \mathrm {diag} (-1,-1,+1,\ldots ,+1)} in coordinates ( X 1 , X 2 , X 3 , … , X n + 1 ) {\displaystyle (X_{1},X_{2},X_{3},\ldots ,X_{n+1})} by

369-626: A ) = χ r i g h t ( χ t o p − 1 [ a ] ) = χ r i g h t ( a , 1 − a 2 ) = 1 − a 2 {\displaystyle {\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left[a\right]\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}} Such

492-423: A pure manifold . For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is not a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is locally constant ), each connected component has

615-507: A (flat) pseudo-Riemannian space of five dimensions. This allows distances and angles within the embedded space to be directly determined from those in the five-dimensional flat space. The remainder of this article explains the details of these concepts with a much more rigorous and precise mathematical and physical description. People are ill-suited to visualizing things in five or more dimensions, but mathematical equations are not similarly challenged and can represent five-dimensional concepts in

738-458: A Euclidean space means that every point has a neighborhood homeomorphic to an open subset of the Euclidean space R n , {\displaystyle \mathbb {R} ^{n},} for some nonnegative integer n . This implies that either the point is an isolated point (if n = 0 {\displaystyle n=0} ), or it has a neighborhood homeomorphic to

861-542: A chart for the plane R 2 {\displaystyle \mathbb {R} ^{2}} minus the positive x -axis and the origin. Another example of a chart is the map χ top mentioned above, a chart for the circle. The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an atlas . An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union

984-433: A consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold. Gravitational constant The gravitational constant

1107-457: A curvature described by a single number that is the same everywhere in spacetime in the absence of matter or energy. Negative curvature means curved hyperbolically, like a saddle surface or the Gabriel's Horn surface, similar to that of a trumpet bell. General relativity is a theory of the nature of time, space and gravity in which gravity is a curvature of space and time that results from

1230-495: A density of 4.5 g/cm ( ⁠4 + 1 / 2 ⁠ times the density of water), about 20% below the modern value. This immediately led to estimates on the densities and masses of the Sun , Moon and planets , sent by Hutton to Jérôme Lalande for inclusion in his planetary tables. As discussed above, establishing the average density of Earth is equivalent to measuring the gravitational constant, given Earth's mean radius and

1353-436: A different aspect of the manifold, thereby leading to a slightly different viewpoint. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R 2 {\displaystyle \mathbb {R} ^{2}} is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here

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1476-472: A differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions . For symplectic manifolds , the transition functions must be symplectomorphisms . The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible . These notions are made precise in general through

1599-420: A fixed dimension. Sheaf-theoretically , a manifold is a locally ringed space , whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry . The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly,

1722-665: A function is called a transition map . The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts χ m i n u s ( x , y ) = s = y 1 + x {\displaystyle \chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}} and χ p l u s ( x , y ) = t = y 1 − x {\displaystyle \chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}} Here s

1845-468: A function of altitude) type. Pendulum experiments still continued to be performed, by Robert von Sterneck (1883, results between 5.0 and 6.3 g/cm ) and Thomas Corwin Mendenhall (1880, 5.77 g/cm ). Cavendish's result was first improved upon by John Henry Poynting (1891), who published a value of 5.49(3) g⋅cm , differing from the modern value by 0.2%, but compatible with

1968-499: A manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics , while four-dimensional Lorentzian manifolds model spacetime in general relativity . The study of manifolds requires working knowledge of calculus and topology . After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so

2091-401: A manifold can be described using mathematical maps , called coordinate charts , collected in a mathematical atlas . It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across

2214-640: A manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. CT scans ). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds ; their differentiable structure allows calculus to be done. A Riemannian metric on

2337-482: A manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Russia may both contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an open ball in R n {\displaystyle \mathbb {R} ^{n}} to the manifold and then back to another (or perhaps

2460-535: A manifold with boundary. The interior of M {\displaystyle M} , denoted Int ⁡ M {\displaystyle \operatorname {Int} M} , is the set of points in M {\displaystyle M} which have neighborhoods homeomorphic to an open subset of R n {\displaystyle \mathbb {R} ^{n}} . The boundary of M {\displaystyle M} , denoted ∂ M {\displaystyle \partial M} ,

2583-408: A mathematical equation. The full mathematical description also captures some subtle distinctions made in general relativity between space-like dimensions and time-like dimensions. Much as spherical and hyperbolic spaces can be visualized by an isometric embedding in a flat space of one higher dimension (as the sphere and pseudosphere respectively), anti-de Sitter space can be visualized as

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2706-414: A neighborhood homeomorphic to R n {\displaystyle \mathbb {R} ^{n}} since R n {\displaystyle \mathbb {R} ^{n}} is homeomorphic, and even diffeomorphic to any open ball in it (for n > 0 {\displaystyle n>0} ). The n that appears in the preceding definition is called the local dimension of

2829-786: A neighborhood homeomorphic to the "half" n {\displaystyle n} -ball { ( x 1 , x 2 , … , x n ) | Σ x i 2 < 1  and  x 1 ≥ 0 } {\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}} . Any homeomorphism between half-balls must send points with x 1 = 0 {\displaystyle x_{1}=0} to points with x 1 = 0 {\displaystyle x_{1}=0} . This invariance allows to "define" boundary points; see next paragraph. Let M {\displaystyle M} be

2952-461: A point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line segment, since deleting the center point from the "+" gives a space with four components (i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces; topological operations always preserve

3075-433: A quotient of two generalized orthogonal groups whereas AdS without P or C can be seen as the quotient of spin groups . This quotient formulation gives A d S n {\displaystyle \mathrm {AdS} _{n}} the structure of a homogeneous space . The Lie algebra of the generalized orthogonal group o ( 1 , n ) {\displaystyle {\mathcal {o}}(1,n)}

3198-407: A result it contains self-intersecting straight lines (geodesics) while the hyperbolic plane does not. Some authors define anti-de Sitter space as equivalent to the embedded quasi-sphere itself, while others define it as equivalent to the universal cover of the embedding. If the universal cover is not taken, ( p , q ) anti-de Sitter space has O( p , q + 1) as its isometry group . If

3321-399: A result, in general relativity, the familiar Newtonian equation of gravity F = G m 1 m 2 r 2   {\displaystyle \textstyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ } (i.e. the gravitational pull between two objects equals the gravitational constant times the product of their masses divided by the square of

3444-419: A second atlas for the circle, with the transition map t = 1 s {\displaystyle t={\frac {1}{s}}} (that is, one has this relation between s and t for every point where s and t are both nonzero). Each chart omits a single point, either (−1, 0) for s or (+1, 0) for t , so neither chart alone is sufficient to cover the whole circle. It can be proved that it

3567-484: A single period of the spacetime. Because the conformal infinity of AdS is timelike , specifying the initial data on a spacelike hypersurface would not determine the future evolution uniquely ( i.e. deterministically) unless there are boundary conditions associated with the conformal infinity. Another commonly used coordinate system which covers the entire space is given by the coordinates t, r ⩾ 0 {\displaystyle r\geqslant 0} and

3690-472: A small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle , x  +  y  = 1, where the y -coordinate is positive (indicated by the yellow arc in Figure 1 ). Any point of this arc can be uniquely described by its x -coordinate. So, projection onto the first coordinate is a continuous and invertible mapping from

3813-492: A spherically symmetric density distribution is directly proportional to the product of their masses , m 1 and m 2 , and inversely proportional to the square of the distance, r , directed along the line connecting their centres of mass : F = G m 1 m 2 r 2 . {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}.} The constant of proportionality , G , in this non-relativistic formulation

Anti-de Sitter space - Misplaced Pages Continue

3936-492: A standard value for G with a relative standard uncertainty better than 0.1% has therefore remained rather speculative. By 1969, the value recommended by the National Institute of Standards and Technology (NIST) was cited with a relative standard uncertainty of 0.046% (460 ppm), lowered to 0.012% (120 ppm) by 1986. But the continued publication of conflicting measurements led NIST to considerably increase

4059-600: A variation of general relativity in which spacetime is slightly curved in the absence of matter or energy. This is analogous to the relationship between Euclidean geometry and non-Euclidean geometry . An intrinsic curvature of spacetime in the absence of matter or energy is modeled by the cosmological constant in general relativity. This corresponds to the vacuum having an energy density and pressure. This spacetime geometry results in momentarily parallel timelike geodesics diverging, with spacelike sections having positive curvature. An anti-de Sitter space in general relativity

4182-425: A way just as appropriate as the methods that mathematical equations use to describe easier-to-visualize three- and four-dimensional concepts. There is a particularly important implication of the more precise mathematical description that differs from the analogy-based heuristic description of de Sitter space and anti-de Sitter space above. The mathematical description of anti-de Sitter space generalizes

4305-482: Is with y > 0 {\displaystyle y>0} giving the half-space. This metric is conformally equivalent to a flat half-space Minkowski spacetime. The constant time slices of this coordinate patch are hyperbolic spaces in the Poincaré half-space metric. In the limit as y → 0 {\displaystyle y\to 0} , this half-space metric is conformally equivalent to

4428-465: Is homeomorphic to an open subset of n {\displaystyle n} -dimensional Euclidean space. One-dimensional manifolds include lines and circles , but not self-crossing curves such as a figure 8 . Two-dimensional manifolds are also called surfaces . Examples include the plane , the sphere , and the torus , and also the Klein bottle and real projective plane . The concept of

4551-453: Is a reductive homogeneous space , and a non-Riemannian symmetric space . A d S n {\displaystyle \mathrm {AdS} _{n}} is an n -dimensional vacuum solution for the theory of gravitation with Einstein–Hilbert action with negative cosmological constant Λ {\displaystyle \Lambda } , ( Λ < 0 {\displaystyle \Lambda <0} ), i.e.

4674-705: Is a circle, a 1-manifold . A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open n {\displaystyle n} -ball { ( x 1 , x 2 , … , x n ) | Σ x i 2 < 1 } {\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1\}} . Every boundary point has

4797-528: Is a manifold with boundary of dimension n {\displaystyle n} , then Int ⁡ M {\displaystyle \operatorname {Int} M} is a manifold (without boundary) of dimension n {\displaystyle n} and ∂ M {\displaystyle \partial M} is a manifold (without boundary) of dimension n − 1 {\displaystyle n-1} . A single manifold can be constructed in different ways, each stressing

4920-473: Is a pair of separate circles. Manifolds need not be closed ; thus a line segment without its end points is a manifold. They are never countable , unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola , a hyperbola , and the locus of points on a cubic curve y = x − x (a closed loop piece and an open, infinite piece). However, excluded are examples like two touching circles that share

5043-557: Is a physical constant that is difficult to measure with high accuracy. This is because the gravitational force is an extremely weak force as compared to other fundamental forces at the laboratory scale. In SI units, the CODATA -recommended value of the gravitational constant is: The relative standard uncertainty is 2.2 × 10 . Due to its use as a defining constant in some systems of natural units , particularly geometrized unit systems such as Planck units and Stoney units ,

Anti-de Sitter space - Misplaced Pages Continue

5166-445: Is a spacetime in which no point in space and time can be distinguished in any way from another, and (being Lorentzian) the only way in which a direction (or tangent to a path at a spacetime point) can be distinguished is whether it is spacelike , lightlike or timelike . The space of special relativity ( Minkowski space ) is an example. A constant scalar curvature means a general relativity gravity-like bending of spacetime that has

5289-409: Is also an atlas. The atlas containing all possible charts consistent with a given atlas is called the maximal atlas (i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations). Charts in an atlas may overlap and a single point of

5412-650: Is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton 's law of universal gravitation and in Albert Einstein 's theory of general relativity . It is also known as the universal gravitational constant , the Newtonian constant of gravitation , or the Cavendish gravitational constant , denoted by the capital letter G . In Newton's law, it

5535-633: Is another example, applying this method to the construction of a sphere: A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R 3 {\displaystyle \mathbb {R} ^{3}} : S = { ( x , y , z ) ∈ R 3 ∣ x 2 + y 2 + z 2 = 1 } . {\displaystyle S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.} The sphere

5658-403: Is approximately 6.6743 × 10  N⋅m /kg . The modern notation of Newton's law involving G was introduced in the 1890s by C. V. Boys . The first implicit measurement with an accuracy within about 1% is attributed to Henry Cavendish in a 1798 experiment . According to Newton's law of universal gravitation, the magnitude of the attractive force ( F ) between two bodies each with

5781-477: Is best known for its role in the AdS/CFT correspondence , which suggests that it is possible to describe a force in quantum mechanics (like electromagnetism , the weak force or the strong force ) in a certain number of dimensions (for example four) with a string theory where the strings exist in an anti-de Sitter space, with one additional (non-compact) dimension. A maximally symmetric Lorentzian manifold

5904-580: Is bounded by two null, aka lightlike, geodesic hyperplanes; the green shaded area on the surface corresponds to the region of conformal space covered by Minkowski space. The green shaded region covers half of the AdS space and half of the conformal spacetime; the left ends of the green discs will touch in the same fashion as the right ends. In the same way that the 2-sphere is a quotient of two orthogonal groups , anti-de Sitter with parity (reflectional symmetry) and time reversal symmetry can be seen as

6027-436: Is equivalent to G ≈ 8 × 10  m ⋅kg ⋅s . The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton's death, by Henry Cavendish. He determined a value for G implicitly, using a torsion balance invented by the geologist Rev. John Michell (1753). He used a horizontal torsion beam with lead balls whose inertia (in relation to

6150-953: Is exact only within the approximation of the Earth's orbit around the Sun as a two-body problem in Newtonian mechanics, the measured quantities contain corrections from the perturbations from other bodies in the solar system and from general relativity. From 1964 until 2012, however, it was used as the definition of the astronomical unit and thus held by definition: 1   A U = ( G M 4 π 2 y r 2 ) 1 3 ≈ 1.495979 × 10 11   m . {\displaystyle 1\ \mathrm {AU} =\left({\frac {GM}{4\pi ^{2}}}\mathrm {yr} ^{2}\right)^{\frac {1}{3}}\approx 1.495979\times 10^{11}\ \mathrm {m} .} Since 2012,

6273-897: Is given by matrices where B {\displaystyle B} is a skew-symmetric matrix . A complementary generator in the Lie algebra of G = o ( 2 , n ) {\displaystyle {\mathcal {G}}={\mathcal {o}}(2,n)} is These two fulfill G = H ⊕ Q {\displaystyle {\mathcal {G}}={\mathcal {H}}\oplus {\mathcal {Q}}} . Explicit matrix computation shows that [ H , Q ] ⊆ Q {\displaystyle [{\mathcal {H}},{\mathcal {Q}}]\subseteq {\mathcal {Q}}} and [ Q , Q ] ⊆ H {\displaystyle [{\mathcal {Q}},{\mathcal {Q}}]\subseteq {\mathcal {H}}} . Thus, anti-de Sitter

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6396-430: Is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility. The sphere is an example of a surface. The unit sphere of implicit equation may be covered by an atlas of six charts :

6519-449: Is quite difficult to measure because gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies. Measurements with pendulums were made by Francesco Carlini (1821, 4.39 g/cm ), Edward Sabine (1827, 4.77 g/cm ), Carlo Ignazio Giulio (1841, 4.95 g/cm ) and George Biddell Airy (1854, 6.6 g/cm ). Cavendish's experiment

6642-460: Is similar to a de Sitter space , except with the sign of the spacetime curvature changed. In anti-de Sitter space, in the absence of matter or energy, the curvature of spacelike sections is negative, corresponding to a hyperbolic geometry , and momentarily parallel timelike geodesics eventually intersect. This corresponds to a negative cosmological constant , where empty space itself has negative energy density but positive pressure, unlike

6765-427: Is such a curve. When q ≥ 2 these curves are inherent to the geometry (unsurprisingly, as any space with more than one temporal dimension contains closed timelike curves), but when q = 1 , they can be eliminated by passing to the universal covering space , effectively "unrolling" the embedding. A similar situation occurs with the pseudosphere , which curls around on itself although the hyperbolic plane does not; as

6888-644: Is that induced from the ambient metric . It is nondegenerate and, in the case of q = 1 has Lorentzian signature. When q = 0 , this construction gives a standard hyperbolic space. The remainder of the discussion applies when q ≥ 1 . When q ≥ 1 , the embedding above has closed timelike curves ; for example, the path parameterized by t 1 = α sin ⁡ ( τ ) , t 2 = α cos ⁡ ( τ ) , {\displaystyle t_{1}=\alpha \sin(\tau ),t_{2}=\alpha \cos(\tau ),} and all other coordinates zero,

7011-771: Is the Einstein tensor (not the gravitational constant despite the use of G ), Λ is the cosmological constant , g μν is the metric tensor , T μν is the stress–energy tensor , and κ is the Einstein gravitational constant , a constant originally introduced by Einstein that is directly related to the Newtonian constant of gravitation: κ = 8 π G c 4 ≈ 2.076647 ( 46 ) × 10 − 43 N − 1 . {\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.076647(46)\times 10^{-43}\mathrm {\,N^{-1}} .} The gravitational constant

7134-506: Is the complement of Int ⁡ M {\displaystyle \operatorname {Int} M} in M {\displaystyle M} . The boundary points can be characterized as those points which land on the boundary hyperplane ( x n = 0 ) {\displaystyle (x_{n}=0)} of R + n {\displaystyle \mathbb {R} _{+}^{n}} under some coordinate chart. If M {\displaystyle M}

7257-604: Is the radius of the Earth , the two quantities are related by: g = G M ⊕ r ⊕ 2 . {\displaystyle g=G{\frac {M_{\oplus }}{r_{\oplus }^{2}}}.} The gravitational constant appears in the Einstein field equations of general relativity , G μ ν + Λ g μ ν = κ T μ ν , {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,} where G μν

7380-527: Is the author of the article "Gravitation" in the Encyclopædia Britannica Eleventh Edition (1911). Here, he cites a value of G = 6.66 × 10  m ⋅kg ⋅s with a relative uncertainty of 0.2%. Paul R. Heyl (1930) published the value of 6.670(5) × 10  m ⋅kg ⋅s (relative uncertainty 0.1%), improved to 6.673(3) × 10  m ⋅kg ⋅s (relative uncertainty 0.045% = 450 ppm) in 1942. However, Heyl used

7503-407: Is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g" ( g ), which is the local gravitational field of Earth (also referred to as free-fall acceleration). Where M ⊕ {\displaystyle M_{\oplus }} is the mass of the Earth and r ⊕ {\displaystyle r_{\oplus }}

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7626-404: Is the metric of the spacetime. Introducing the radius α {\displaystyle \alpha } as Λ = − 1 α 2 ( n − 1 ) ( n − 2 ) 2 {\textstyle \Lambda ={\frac {-1}{\alpha ^{2}}}{\frac {(n-1)(n-2)}{2}}} , this solution can be immersed in

7749-464: Is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse square of their distance . In the Einstein field equations , it quantifies the relation between the geometry of spacetime and the energy–momentum tensor (also referred to as the stress–energy tensor ). The measured value of the constant is known with some certainty to four significant digits. In SI units , its value

7872-747: Is the slope of the line through the point at coordinates ( x ,  y ) and the fixed pivot point (−1, 0); similarly, t is the opposite of the slope of the line through the points at coordinates ( x ,  y ) and (+1, 0). The inverse mapping from s to ( x ,  y ) is given by x = 1 − s 2 1 + s 2 y = 2 s 1 + s 2 {\displaystyle {\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\[5pt]y&={\frac {2s}{1+s^{2}}}\end{aligned}}} It can be confirmed that x  +  y  = 1 for all values of s and t . These two charts provide

7995-475: Is two-dimensional, so each chart will map part of the sphere to an open subset of R 2 {\displaystyle \mathbb {R} ^{2}} . Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by χ ( x , y , z ) = ( x , y ) ,   {\displaystyle \chi (x,y,z)=(x,y),\ } maps

8118-641: The Gaussian gravitational constant was historically in widespread use, k = 0.017 202 098 95 radians per day , expressing the mean angular velocity of the Sun–Earth system. The use of this constant, and the implied definition of the astronomical unit discussed above, has been deprecated by the IAU since 2012. The existence of the constant is implied in Newton's law of universal gravitation as published in

8241-532: The cgs system. Richarz and Krigar-Menzel (1898) attempted a repetition of the Cavendish experiment using 100,000 kg of lead for the attracting mass. The precision of their result of 6.683(11) × 10  m ⋅kg ⋅s was, however, of the same order of magnitude as the other results at the time. Arthur Stanley Mackenzie in The Laws of Gravitation (1899) reviews the work done in the 19th century. Poynting

8364-430: The mean gravitational acceleration at Earth's surface, by setting G = g R ⊕ 2 M ⊕ = 3 g 4 π R ⊕ ρ ⊕ . {\displaystyle G=g{\frac {R_{\oplus }^{2}}{M_{\oplus }}}={\frac {3g}{4\pi R_{\oplus }\rho _{\oplus }}}.} Based on this, Hutton's 1778 result

8487-449: The metric as the quasi-sphere where α {\displaystyle \alpha } is a nonzero constant with dimensions of length (the radius of curvature ). This is a (generalized) sphere in the sense that it is a collection of points for which the "distance" (determined by the quadratic form) from the origin is constant, but visually it is a hyperboloid , as in the image shown. The metric on anti-de Sitter space

8610-519: The open ball B n = { ( x 1 , x 2 , … , x n ) ∈ R n : x 1 2 + x 2 2 + ⋯ + x n 2 < 1 } . {\displaystyle \mathbf {B} ^{n}=\left\{(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1\right\}.} This implies also that every point has

8733-507: The 1680s (although its notation as G dates to the 1890s), but is not calculated in his Philosophiæ Naturalis Principia Mathematica where it postulates the inverse-square law of gravitation. In the Principia , Newton considered the possibility of measuring gravity's strength by measuring the deflection of a pendulum in the vicinity of a large hill, but thought that the effect would be too small to be measurable. Nevertheless, he had

8856-469: The 2006 CODATA value. An improved cold atom measurement by Rosi et al. was published in 2014 of G = 6.671 91 (99) × 10  m ⋅kg ⋅s . Although much closer to the accepted value (suggesting that the Fixler et al. measurement was erroneous), this result was 325 ppm below the recommended 2014 CODATA value, with non-overlapping standard uncertainty intervals. As of 2018, efforts to re-evaluate

8979-510: The 2010 value, and one order of magnitude below the 1969 recommendation. The following table shows the NIST recommended values published since 1969: In the January 2007 issue of Science , Fixler et al. described a measurement of the gravitational constant by a new technique, atom interferometry , reporting a value of G = 6.693(34) × 10  m ⋅kg ⋅s , 0.28% (2800 ppm) higher than

9102-557: The AU is defined as 1.495 978 707 × 10  m exactly, and the equation can no longer be taken as holding precisely. The quantity GM —the product of the gravitational constant and the mass of a given astronomical body such as the Sun or Earth—is known as the standard gravitational parameter (also denoted μ ). The standard gravitational parameter GM appears as above in Newton's law of universal gravitation, as well as in formulas for

9225-470: The Earth, it is sufficient to consider time distortion in a particular coordinate system. We find gravity on the Earth very noticeable while relativistic time distortion requires precision instruments to detect. The reason why we do not become aware of relativistic effects in our everyday life is the huge value of the speed of light ( c = 300 000  km/s approximately), which makes us perceive space and time as different entities. De Sitter space involves

9348-550: The Lorentzian analogue of a sphere in a space of one additional dimension. The extra dimension is timelike. In this article we adopt the convention that the metric in a timelike direction is negative. The anti-de Sitter space of signature ( p , q ) can then be isometrically embedded in the space R p , q + 1 {\displaystyle \mathbb {R} ^{p,q+1}} with coordinates ( x 1 , ..., x p , t 1 , ..., t q +1 ) and

9471-539: The Minkowski metric d s 2 = − d t 2 + ∑ i d x i 2 {\textstyle ds^{2}=-dt^{2}+\sum _{i}dx_{i}^{2}} . Thus, the anti-de Sitter space contains a conformal Minkowski space at infinity ("infinity" having y-coordinate zero in this patch). In AdS space time is periodic, and the universal cover has non-periodic time. The coordinate patch above covers half of

9594-472: The average density of Earth and the Earth's mass . His result, ρ 🜨 = 5.448(33) g⋅cm , corresponds to value of G = 6.74(4) × 10  m ⋅kg ⋅s . It is surprisingly accurate, about 1% above the modern value (comparable to the claimed relative standard uncertainty of 0.6%). The accuracy of the measured value of G has increased only modestly since the original Cavendish experiment. G

9717-463: The average density of a planet and the period of a satellite orbiting just above its surface. For elliptical orbits, applying Kepler's 3rd law , expressed in units characteristic of Earth's orbit : where distance is measured in terms of the semi-major axis of Earth's orbit (the astronomical unit , AU), time in years , and mass in the total mass of the orbiting system ( M = M ☉ + M E + M ☾ ). The above equation

9840-430: The co-domain of χ t o p {\displaystyle \chi _{\mathrm {top} }} back to the circle using the inverse, followed by χ r i g h t {\displaystyle \chi _{\mathrm {right} }} back to the interval. If a is any number in ( 0 , 1 ) {\displaystyle (0,1)} , then: T (

9963-513: The conflicting results of measurements are underway, coordinated by NIST, notably a repetition of the experiments reported by Quinn et al. (2013). In August 2018, a Chinese research group announced new measurements based on torsion balances, 6.674 184 (78) × 10  m ⋅kg ⋅s and 6.674 484 (78) × 10  m ⋅kg ⋅s based on two different methods. These are claimed as the most accurate measurements ever made, with standard uncertainties cited as low as 12 ppm. The difference of 2.7 σ between

10086-538: The deflection of light caused by gravitational lensing , in Kepler's laws of planetary motion , and in the formula for escape velocity . This quantity gives a convenient simplification of various gravity-related formulas. The product GM is known much more accurately than either factor is. Calculations in celestial mechanics can also be carried out using the units of solar masses, mean solar days and astronomical units rather than standard SI units. For this purpose,

10209-555: The desired structure. For a topological manifold, the simple space is a subset of some Euclidean space R n {\displaystyle \mathbb {R} ^{n}} and interest focuses on the topological structure. This structure is preserved by homeomorphisms , invertible maps that are continuous in both directions. In the case of a differentiable manifold, a set of charts called an atlas , whose transition functions (see below) are all differentiable, allows us to do calculus on it. Polar coordinates , for example, form

10332-525: The distance between them) is merely an approximation of the gravity effects seen in general relativity. However this approximation becomes inaccurate in extreme physical situations, like relativistic speeds (light, in particular), or very large & dense masses. In general relativity, gravity is caused by spacetime being curved ("distorted"). It is a common misconception to attribute gravity to curved space; neither space nor time has an absolute meaning in relativity. Nevertheless, to describe weak gravity, as on

10455-404: The experiment had at least proved that the Earth could not be a hollow shell , as some thinkers of the day, including Edmond Halley , had suggested. The Schiehallion experiment , proposed in 1772 and completed in 1776, was the first successful measurement of the mean density of the Earth, and thus indirectly of the gravitational constant. The result reported by Charles Hutton (1778) suggested

10578-402: The following constraint: Manifolds In mathematics , a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold, or n {\displaystyle n} -manifold for short, is a topological space with the property that each point has a neighborhood that

10701-556: The formation of black holes. Mathematician Georgios Moschidis proved that given spherical symmetry, the conjecture holds true for the specific cases of the Einstein-null dust system with an internal mirror (2017) and the Einstein-massless Vlasov system (2018). A coordinate patch covering part of the space gives the half-space coordinatization of anti-de Sitter space. The metric tensor for this patch

10824-503: The geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negative). As such, they are exact solutions of the Einstein field equations for an empty universe with a positive, zero, or negative cosmological constant , respectively. Anti-de Sitter space generalises to any number of space dimensions. In higher dimensions, it

10947-417: The gravitational constant is: G ≈ 4.3009 × 10 − 3   p c ⋅ ( k m / s ) 2 M ⊙ − 1 . {\displaystyle G\approx 4.3009\times 10^{-3}\ {\mathrm {pc{\cdot }(km/s)^{2}} \,M_{\odot }}^{-1}.} For situations where tides are important,

11070-418: The hyper- polar coordinates α , θ and  φ . The adjacent image represents the "half-space" region of anti-de Sitter space and its boundary. The interior of the cylinder corresponds to anti-de Sitter spacetime, while its cylindrical boundary corresponds to its conformal boundary. The green shaded region in the interior corresponds to the region of AdS covered by the half-space coordinates and it

11193-423: The idea of curvature. In the mathematical description, curvature is a property of a particular point and can be divorced from some invisible surface to which curved points in spacetime meld themselves. So for example, concepts like singularities (the most widely known of which in general relativity is the black hole ) which cannot be expressed completely in a real world geometry, can correspond to particular states of

11316-488: The interval ( 0 , 1 ) {\displaystyle (0,1)} , though differently. Thus a function T : ( 0 , 1 ) → ( 0 , 1 ) = χ r i g h t ∘ χ t o p − 1 {\displaystyle T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}} can be constructed, which takes values from

11439-494: The manifold as a whole. Formally, a (topological) manifold is a second countable Hausdorff space that is locally homeomorphic to a Euclidean space. Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense 'too large' such as the long line , while Hausdorff excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in non-Hausdorff manifolds ). Locally homeomorphic to

11562-414: The manifold. Generally manifolds are taken to have a constant local dimension, and the local dimension is then called the dimension of the manifold. This is, in particular, the case when manifolds are connected . However, some authors admit manifolds that are not connected, and where different points can have different dimensions . If a manifold has a fixed dimension, this can be emphasized by calling it

11685-411: The map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure. A coordinate map , a coordinate chart , or simply a chart , of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve

11808-402: The modern value within the cited relative standard uncertainty of 0.55%. In addition to Poynting, measurements were made by C. V. Boys (1895) and Carl Braun (1897), with compatible results suggesting G = 6.66(1) × 10  m ⋅kg ⋅s . The modern notation involving the constant G was introduced by Boys in 1894 and becomes standard by the end of the 1890s, with values usually cited in

11931-434: The negatively curved (trumpet-bell-like) dip in the sheet. A key feature of general relativity is that it describes gravity not as a conventional force like electromagnetism, but as a change in the geometry of spacetime that results from the presence of matter or energy. The analogy used above describes the curvature of a two-dimensional space caused by gravity in general relativity in a three-dimensional superspace in which

12054-444: The northern hemisphere to the open unit disc by projecting it on the ( x , y ) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the ( x , z ) plane and two charts projecting on the ( y , z ) plane, an atlas of six charts is obtained which covers the entire sphere. This can be easily generalized to higher-dimensional spheres. A manifold can be constructed by gluing together pieces in

12177-516: The number of pieces. Informally, a manifold is a space that is "modeled on" Euclidean space. There are many different kinds of manifolds. In geometry and topology , all manifolds are topological manifolds , possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions : homeomorphisms from one region of Euclidean space to another region if they correspond to

12300-450: The opportunity to estimate the order of magnitude of the constant when he surmised that "the mean density of the earth might be five or six times as great as the density of water", which is equivalent to a gravitational constant of the order: A measurement was attempted in 1738 by Pierre Bouguer and Charles Marie de La Condamine in their " Peruvian expedition ". Bouguer downplayed the significance of their results in 1740, suggesting that

12423-407: The path taken by small objects rolling nearby, causing them to deviate inward from the path they would have followed had the heavy object been absent. Of course, in general relativity, both the small and large objects mutually influence the curvature of spacetime. The attractive force of gravity created by matter is due to a negative curvature of spacetime, represented in the rubber sheet analogy by

12546-402: The period P of an object in circular orbit around a spherical object obeys G M = 3 π V P 2 , {\displaystyle GM={\frac {3\pi V}{P^{2}}},} where V is the volume inside the radius of the orbit, and M is the total mass of the two objects. It follows that This way of expressing G shows the relationship between

12669-463: The plane z = 0 divides the sphere into two half spheres ( z > 0 and z < 0 ), which may both be mapped on the disc x + y < 1 by the projection on the xy plane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes. As with the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but

12792-432: The presence of matter or energy. Energy and mass are equivalent (as expressed in the equation E  =  mc ). Space and time values can be related respectively to time and space units by multiplying or dividing the value by the speed of light (e.g., seconds times meters per second equals meters). A common analogy involves the way that a dip in a flat sheet of rubber, caused by a heavy object sitting on it, influences

12915-458: The relevant length scales are solar radii rather than parsecs. In these units, the gravitational constant is: G ≈ 1.90809 × 10 5   ( k m / s ) 2 R ⊙ M ⊙ − 1 . {\displaystyle G\approx 1.90809\times 10^{5}\mathrm {\ (km/s)^{2}} \,R_{\odot }M_{\odot }^{-1}.} In orbital mechanics ,

13038-420: The same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on

13161-474: The same) open ball in R n {\displaystyle \mathbb {R} ^{n}} . The resultant map, like the map T in the circle example above, is called a change of coordinates , a coordinate transformation , a transition function , or a transition map . An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure,

13284-435: The sphere cannot be covered by a single chart. This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the Earth cannot have a plane representation consisting of a single map (also called "chart", see nautical chart ), and therefore one needs atlases for covering the whole Earth surface. Manifolds need not be connected (all in "one piece"); an example

13407-433: The standard ΛCDM model of our own universe for which observations of distant supernovae indicate a positive cosmological constant corresponding to (asymptotic) de Sitter space . In an anti-de Sitter space, as in a de Sitter space, the inherent spacetime curvature corresponds to the cosmological constant. The anti-de Sitter space AdS 2 is also the de Sitter space dS 2 through an exchange of

13530-428: The standard uncertainty in the 1998 recommended value, by a factor of 12, to a standard uncertainty of 0.15%, larger than the one given by Heyl (1930). The uncertainty was again lowered in 2002 and 2006, but once again raised, by a more conservative 20%, in 2010, matching the relative standard uncertainty of 120 ppm published in 1986. For the 2014 update, CODATA reduced the uncertainty to 46 ppm, less than half

13653-416: The statistical spread as his standard deviation, and he admitted himself that measurements using the same material yielded very similar results while measurements using different materials yielded vastly different results. He spent the next 12 years after his 1930 paper to do more precise measurements, hoping that the composition-dependent effect would go away, but it did not, as he noted in his final paper from

13776-410: The structure transfers to the manifold. This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of R n {\displaystyle \mathbb {R} ^{n}} (that is, if they are diffeomorphisms ), the differential structure transfers to the manifold and turns it into

13899-416: The theory described by the following Lagrangian density: where G ( n ) is the gravitational constant in n -dimensional spacetime. Therefore, it is a solution of the Einstein field equations : where G μ ν {\displaystyle G_{\mu \nu }} is Einstein tensor and g μ ν {\displaystyle g_{\mu \nu }}

14022-414: The third dimension corresponds to the effect of gravity. A geometrical way of thinking about general relativity describes the effects of the gravity in the real world four-dimensional space geometrically by projecting that space into a five-dimensional superspace with the fifth dimension corresponding to the curvature in spacetime that is produced by gravity and gravity-like effects in general relativity. As

14145-420: The timelike and spacelike labels. Such a relabelling reverses the sign of the curvature, which is conventionally referenced to the directions that are labelled spacelike. The analogy used above describes curvature of a two-dimensional space caused by gravity in a flat ambient space of one dimension higher. Similarly, the (curved) de Sitter and anti-de Sitter spaces of four dimensions can be embedded into

14268-415: The torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. In spite of the experimental design being due to Michell, the experiment is now known as the Cavendish experiment for its first successful execution by Cavendish. Cavendish's stated aim was the "weighing of Earth", that is, determining

14391-423: The universal cover is taken the isometry group is a cover of O( p , q + 1) . This is most easily understood by defining anti-de Sitter space as a symmetric space , using the quotient space construction, given below. The unproven "AdS instability conjecture" introduced by the physicists Piotr Bizon and Andrzej Rostworowski in 2011 states that arbitrarily small perturbations of certain shapes in AdS lead to

14514-905: The upper arc to the open interval (−1, 1): χ t o p ( x , y ) = x . {\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,} Such functions along with the open regions they map are called charts . Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle: χ b o t t o m ( x , y ) = x χ l e f t ( x , y ) = y χ r i g h t ( x , y ) = y . {\displaystyle {\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}} Together, these parts cover

14637-420: The use of pseudogroups . A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an n {\displaystyle n} -manifold with boundary is an ( n − 1 ) {\displaystyle (n-1)} -manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary

14760-543: The value of the gravitational constant will generally have a numeric value of 1 or a value close to it when expressed in terms of those units. Due to the significant uncertainty in the measured value of G in terms of other known fundamental constants, a similar level of uncertainty will show up in the value of many quantities when expressed in such a unit system. In astrophysics , it is convenient to measure distances in parsecs (pc), velocities in kilometres per second (km/s) and masses in solar units M ⊙ . In these units,

14883-566: The whole circle, and the four charts form an atlas for the circle. The top and right charts, χ t o p {\displaystyle \chi _{\mathrm {top} }} and χ r i g h t {\displaystyle \chi _{\mathrm {right} }} respectively, overlap in their domain: their intersection lies in the quarter of the circle where both x {\displaystyle x} and y {\displaystyle y} -coordinates are positive. Both map this part into

15006-437: The year 1942. Published values of G derived from high-precision measurements since the 1950s have remained compatible with Heyl (1930), but within the relative uncertainty of about 0.1% (or 1000 ppm) have varied rather broadly, and it is not entirely clear if the uncertainty has been reduced at all since the 1942 measurement. Some measurements published in the 1980s to 2000s were, in fact, mutually exclusive. Establishing

15129-423: Was first repeated by Ferdinand Reich (1838, 1842, 1853), who found a value of 5.5832(149) g⋅cm , which is actually worse than Cavendish's result, differing from the modern value by 1.5%. Cornu and Baille (1873), found 5.56 g⋅cm . Cavendish's experiment proved to result in more reliable measurements than pendulum experiments of the "Schiehallion" (deflection) type or "Peruvian" (period as

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