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| | Hi certainly there. My name is Tabetha although it is not the most feminine of names. Supervising is what she does within their day job. Years ago we moved to Missouri. Doing ballet is the thing he loves most. Check out the latest news on this website: http://usmerch.co.uk/accessories/<br><br>Feel free to visit my homepage ... [http://usmerch.co.uk/accessories/ gas monkey garage] |
| In [[Riemannian geometry]], the '''scalar curvature''' (or the '''Ricci scalar''') is the simplest [[curvature]] invariant of a [[Riemannian manifold]]. To each point on a Riemannian manifold, it assigns a single [[real number]] determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the [[volume]] of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the [[Gaussian curvature]], and completely characterizes the curvature of a surface. In more than two dimensions, however, the [[curvature of Riemannian manifolds]] involves more than one functionally independent quantity.
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| In [[general relativity]], the scalar curvature is the [[Lagrangian]] density for the [[Einstein–Hilbert action]]. The [[Euler–Lagrange equations]] for this Lagrangian under variations in the metric constitute the vacuum [[Einstein field equations]], and the stationary metrics are known as [[Einstein manifold|Einstein metrics]]. The scalar curvature is defined as the trace of the [[Ricci tensor]], and it can be characterized as a multiple of the average of the [[sectional curvature]]s at a point. Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult. One of the few is the [[positive mass theorem]] of [[Richard Schoen]], [[Shing-Tung Yau]] and [[Edward Witten]]. Another is the [[Yamabe problem]], which seeks extremal metrics in a given [[conformal class]] for which the scalar curvature is constant.
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| ==Definition==
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| The scalar curvature is usually denoted by ''S'' (other notations are ''Sc'', ''R''). It is defined as the [[Trace (linear algebra)|trace]] of the [[Ricci curvature]] tensor with respect to the [[metric tensor|metric]]:
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| :<math>S = \mbox{tr}_g\,\operatorname{Ric}.</math>
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| The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first [[raising and lowering indices|raise an index]] to obtain a (1,1)-valent tensor in order to take the trace. In terms of [[local coordinates]] one can write
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| :<math>S = g^{ij}R_{ij} = R^j_j</math>
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| where ''R''<sub>''ij''</sub> are the components of the Ricci tensor in the coordinate basis:
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| :<math>\operatorname{Ric} = R_{ij}\,dx^i\otimes dx^j.</math>
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| Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows
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| :<math>S = g^{ab} (\Gamma^c_{ab,c} - \Gamma^c_{ac,b} + \Gamma^d_{ab}\Gamma^c_{cd} - \Gamma^d_{ac} \Gamma^c_{bd}) | |
| =
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| 2g^{ab} (\Gamma^c_{a[b,c]} + \Gamma^d_{a[b}\Gamma^c_{c]d})
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| </math>
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| where <math>\Gamma^a_{bc}</math> are the [[Christoffel symbols]] of the metric.
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| Unlike the [[Riemann curvature tensor]] or the [[Ricci tensor]], which both can be naturally defined for any [[affine connection]], the scalar curvature requires a metric of some kind. The metric can be [[pseudo-Riemannian]] instead of Riemannian. Indeed, such a generalization is vital to relativity theory. More generally, the Ricci tensor can be defined in broader class of [[metric geometry|metric geometries]] (by means of the direct geometric interpretation, below) that includes [[Finsler geometry]].
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| ==Direct geometric interpretation==
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| When the scalar curvature is positive at a point,
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| the volume of a small ball about the point has smaller volume than
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| a ball of the same radius in Euclidean space. On the other hand,
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| when the scalar curvature is negative at a point, the volume of a small ball is instead larger than it would be in Euclidean space.
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| This can be made more quantitative, in order to characterize the precise value of the scalar curvature ''S'' at a point ''p'' of a Riemannian n-manifold <math>(M,g)</math>.
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| Namely, the ratio of the ''n''-dimensional volume of a ball of radius ε in the manifold to that of a corresponding ball in
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| Euclidean space is given, for small ε, by
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| : <math> \frac{\operatorname{Vol} (B_\varepsilon(p) \subset M)}{\operatorname{Vol} | |
| (B_\varepsilon(0)\subset {\mathbb R}^n)}=
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| 1- \frac{S}{6(n+2)}\varepsilon^2 + O(\varepsilon^4).</math>
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| Thus, the second derivative of this ratio, evaluated at radius ε = 0, is exactly minus the scalar curvature divided by 3(''n'' + 2).
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| Boundaries of these balls are (n-1) dimensional spheres with radii <math>\epsilon</math>; their hypersurface measures ("areas") satisfy the following equation:
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| : <math> \frac{\operatorname{Area} (\partial B_\varepsilon(p) \subset M)}{\operatorname{Area}
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| (\partial B_\varepsilon(0)\subset {\mathbb R}^n)}=
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| 1- \frac{S}{6n}\varepsilon^2 + O(\varepsilon^4).</math>
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| ==Special cases==
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| ===Surfaces===
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| In two dimensions, scalar curvature is exactly twice the [[Gaussian curvature]]. For an embedded surface in [[Euclidean space]], this means that
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| :<math>S = \frac{2}{\rho_1\rho_2}\,</math>
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| where <math>\rho_1,\,\rho_2</math> are [[Principal curvature|principal radii]] of the surface. For example, scalar curvature of a sphere with radius r is equal to 2/''r''<sup>2</sup>.
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| The 2-dimensional [[Riemann tensor]] has only one independent component and it can be easily expressed
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| in terms of the scalar curvature and metric area form. In any coordinate system, one thus has:
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| :<math>2R_{1212} \,= S \det (g_{ij}) = S[g_{11}g_{22}-(g_{12})^2].</math>
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| ===Space forms===
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| A [[space form]] is by definition a Riemannian manifold with constant sectional curvature. Space forms are locally isometric to one of the following types:
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| * [[Euclidean space]]: The Riemann tensor of an ''n''-dimensional Euclidean space vanishes identically, so the scalar curvature does as well.
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| * [[N-sphere|''n''-spheres]]: The sectional curvature of an ''n''-sphere of radius ''r'' is ''K'' = 1/''r''<sup>2</sup>. Hence the scalar curvature is ''S'' = ''n''(''n''−1)/''r''<sup>2</sup>.
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| * [[Hyperbolic space]]s: By the [[hyperboloid model]], an ''n'' dimensional hyperbolic space can be identified with the subset of (''n''+1)-dimensional [[Minkowski space]]
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| ::<math>x_0^2-x_1^2-\cdots-x_n^2 = r^2,\quad x_0>0.</math>
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| :The parameter ''r'' is a geometrical invariant of the hyperbolic space, and the sectional curvature is ''K'' = −1/''r''<sup>2</sup>. The scalar curvature is thus ''S'' = −''n''(''n''−1)/''r''<sup>2</sup>.
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| ==Traditional notation==
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| Among those who use index notation for tensors, it is common to use the letter ''R'' to represent three different things:
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| #the [[Riemann curvature tensor]]: <math>R_{ijk}^l</math> or <math>R_{abcd}</math>
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| #the [[Ricci tensor]]: <math>R_{ij}</math>
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| #the scalar curvature: ''R''
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| These three are then distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices. Those not using an index notation usually reserve ''R'' for the full Riemann curvature tensor.
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| ==See also==
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| * [[Basic introduction to the mathematics of curved spacetime]]
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| * [[Yamabe invariant]]
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| * [[Kretschmann scalar]]
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| {{Curvature}}
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| {{DEFAULTSORT:Scalar Curvature}}
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| [[Category:Curvature (mathematics)]]
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| [[de:Riemannscher Krümmungstensor#Krümmungsskalar]]
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Hi certainly there. My name is Tabetha although it is not the most feminine of names. Supervising is what she does within their day job. Years ago we moved to Missouri. Doing ballet is the thing he loves most. Check out the latest news on this website: http://usmerch.co.uk/accessories/
Feel free to visit my homepage ... gas monkey garage