formulasearchengine - User contributions [en]

Ext functor

2013-11-23T06:22:43Z

71.139.172.99:

'''Tractrix''' (from the [[Latin]] verb ''trahere'' "pull, drag"; plural: '''tractrices''') is the [[curve]] along which an object moves, under the influence of friction, when pulled on a [[horizontal plane]] by a [[line segment]] attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an [[infinitesimal]] [[speed]]. It is therefore a [[curve of pursuit]]. It was first introduced by [[Claude Perrault]] in 1670, and later studied by [[Sir Isaac Newton]] (1676) and [[Christiaan Huygens]] (1692).

[[Image:Tractrix.png|thumb|180px|right|Tractrix with object initially at (4,0)]]

==Mathematical derivation==
Suppose the object is placed at (''a'',0) [or (4,0) in the example shown at right], and the puller in the [[origin (mathematics)|origin]], so ''a'' is the length of the pulling thread [4 in the example at right]. Then the puller starts to move along the ''y'' axis in the positive direction. At every moment, the thread will be tangent to the curve ''y'' = ''y''(''x'') described by the object, so that it becomes completely determined by the movement of the puller. Mathematically, the movement will be described then by the [[differential equation]]
:<math>\frac{dy}{dx} = -\frac{\sqrt{a^2-x^2}}{x}\,\!</math>
with the initial condition ''y(a)'' = 0 whose solution is
:<math>y = \int_x^a\frac{\sqrt{a^2-t^2}}{t}\,dt = \pm \left ( a\ln{\frac{a+\sqrt{a^2-x^2}}{x}}-\sqrt{a^2-x^2} \right ).\,\!</math>

The first term of this solution can also be written
:<math>a\ \mathrm{arsech}\frac{x}{a}, \,\!</math>
where ''arsech'' is the [[inverse hyperbolic secant]] function.

The negative branch denotes the case where the puller moves in the negative direction from the origin. Both branches belong to the tractrix, meeting at the [[cusp (singularity)|cusp]] point (''a'', 0).

==Basis of the tractrix==
The essential property of the tractrix is constancy of the distance between a point ''P'' on the curve and the intersection of the [[tangent line]] at ''P'' with the [[asymptote]] of the curve.

The tractrix might be regarded in a multitude of ways:

# It is the [[locus (mathematics)|locus]] of the center of a hyperbolic spiral rolling (without skidding) on a straight line.
# The [[involute]] of the [[catenary]] function, which describes a fully flexible, [[elastomer|inelastic]], homogeneous string attached to two points that is subjected to a gravitational field. The catenary has the equation <math>y(x)=a\,\operatorname{cosh}(x/a)</math>.
#The trajectory determined by the middle of the back axle of a car pulled by a rope at a constant speed and with a constant direction (initially perpendicular to the vehicle).
[[Image:Tractrixtry.gif|thumb|500px|right|Tractrix by dragging a pole.]]
The function admits a horizontal asymptote. The curve is symmetrical with respect to the ''y''-axis. The curvature radius is <math>r=a\,\operatorname{cot}(x/y)</math>

A great implication that the tractrix had was the study of the revolution surface of it around its asymptote: the [[pseudosphere]]. Studied by [[Eugenio Beltrami|Beltrami]] in 1868, as a surface of constant negative [[Gaussian curvature]], the pseudosphere is a local model of [[non-Euclidean geometry]].
The idea was carried further by Kasner and Newman in their book ''Mathematics and the Imagination'', where they show a [[toy train]] dragging a [[pocket watch]] to generate the tractrix.

==Properties==
[[Image:Evolute2.gif|thumb|500px|right|Catenary as [[evolute]] of a tractrix]]

* Due to the geometrical way it was defined, the tractrix has the property that the segment of its [[tangent]], between the [[asymptote]] and the point of tangency, has constant length <math>a</math>.
* The [[arc length]] of one branch between ''x'' = ''x''1 and ''x'' = ''x''2 is <math>a \ln \frac{x_1}{x_2}</math>
* The area between the tractrix and its asymptote is <math>\pi a^2/2</math> which can be found using [[integral|integration]] or [[Mamikon's theorem]].
* The [[envelope (mathematics)|envelope]] of the [[surface normal|normal]]s of the tractrix (that is, the [[evolute]] of the tractrix) is the [[catenary]] (or ''chain curve'') given by <math>x = a\cosh\frac{y}{a}</math>.
* The surface of revolution created by revolving a tractrix about its asymptote is a [[pseudosphere]].

==Practical application==
In 1927, P.G.A.H. Voigt patented a [[horn loudspeaker]] design based on the assumption that a wave front traveling through the horn is spherical of a constant radius. The idea is to minimize distortion caused by internal reflection of sound within the horn. The resulting shape is the surface of revolution of a tractrix.<ref>[http://www.volvotreter.de/downloads/Dinsdale_Horns_1.pdf Horn loudspeaker design pp. 4-5. (Reprinted from Wireless World, March 1974)]</ref> 

==Drawing machines==
* In October–November 1692, Huygens described three tractrice drawing machines.
* In 1693 [[Gottfried Wilhelm Leibniz|Leibniz]] released to the public a machine which, in theory, could integrate any differential equation; the machine was of tractional design.
* In 1706 [[John Perks]] built a tractional machine in order to realise the [[Hyperbolic function|hyperbolic]] quadrature.
* In 1729 [[Johann Poleni]] built a tractional device that enabled [[logarithm]]ic [[Function (mathematics)|function]]s to be drawn.

==See also==
*[[Dini's surface]]
*[[Hyperbolic functions]] for tanh, sech, csch, arccosh
*[[Natural logarithm]] for ln
*[[Sign function]] for sgn
*[[Trigonometric function]] for sin, cos, tan, arccot, csc

==Notes==
{{reflist}}

==References==
* Edward Kasner & James Newman (1940) [[Mathematics and the Imagination]], pp 141–3, [[Simon & Schuster]].
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=5, 199 }}

==External links==
{{commons|Tractrix}}
* {{MacTutor|class=Curves|id=Tractrix|title=Tractrix}}
* {{planetmath reference|id=7109|title=Tractrix}}
* {{planetmath reference|id=7073|title=Famous curves on the plane.}}
*[http://mathworld.wolfram.com/Tractrix.html Tractrix] on [[MathWorld]]
*[http://www.phaser.com/modules/historic/leibniz/ Module: Leibniz's Pocket Watch ODE] at PHASER

[[Category:Curves]]
[[Category:Mathematical physics]]

Quadratic variation

2013-11-23T06:18:02Z

71.139.172.99: /* Definition */

{{about|metrics in [[general relativity]]|a discussion of metrics in general|metric tensor}}
[[File:Metrictensor.svg|thumb|right|Metric tensor of spacetime in general relativity written as a matrix.]]
In [[general relativity]], the '''metric tensor''' (or simply, the '''metric''') is the fundamental object of study. It may loosely be thought of as a generalization of the [[gravitational field]] familiar from [[gravity|Newtonian gravitation]]. The metric captures all the geometric and [[Causal spacetime structure|causal structure]] of [[spacetime]], being used to define notions such as distance, volume, curvature, angle, future and past.

''Notation and conventions'': Throughout this article we work with a [[metric signature]] that is mostly positive ({{nowrap|− + + +}}); see [[sign convention]]. As is customary in relativity, [[natural units|units]] are used where the [[speed of light]] ''c'' = 1. The [[gravitation constant]] ''G'' will be kept explicit. The [[summation convention]], where repeated indices are automatically summed over, is employed.

==Definition==

Mathematically, spacetime is represented by a 4-dimensional [[differentiable manifold]] ''M'' and the metric is given as a [[Covariance and contravariance of vectors|covariant]], second-rank, [[symmetric tensor]] on ''M'', conventionally denoted by ''g''. Moreover the metric is required to be [[nondegenerate]] with [[metric signature|signature]] (<tt>-+++</tt>). A manifold ''M'' equipped with such a metric is called a [[Lorentzian manifold]].

Explicitly, the metric is a [[symmetric bilinear form]] on each [[tangent space]] of ''M'' which varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors ''u'' and ''v'' at a point ''x'' in ''M'', the metric can be evaluated on ''u'' and ''v'' to give a real number:
:<math>g_x(u,v) = g_x(v,u) \in \mathbb{R}.</math>
This can be thought of as a generalization of the [[dot product]] in ordinary [[Euclidean space]]. This analogy is not exact, however. Unlike Euclidean space — where the dot product is [[positive definite]] — the metric gives each tangent space the structure of [[Minkowski space]].

==Local coordinates and matrix representations==

Physicists usually work in [[local coordinates]] (i.e. coordinates defined on some [[atlas (topology)|local patch]] of ''M''). In local coordinates <math>x^\mu</math> (where <math>\mu</math> is an index which runs from 0 to 3) the metric can be written in the form
:<math>g = g_{\mu\nu} dx^\mu \otimes dx^\nu.</math>
The factors <math>dx^\mu</math> are [[one-form]] [[gradient]]s of the scalar coordinate fields <math>x^\mu</math>. The metric is thus a linear combination of [[tensor product]]s of one-form gradients of coordinates. The coefficients <math>g_{\mu\nu}</math> are a set of 16 real-valued functions (since the tensor ''g'' is actually a ''tensor field'' defined at all points of a [[spacetime]] manifold). In order for the metric to be symmetric we must have
:<math>g_{\mu\nu} = g_{\nu\mu}\,</math>
giving 10 independent coefficients. If we denote the symmetric [[tensor product]] by juxtaposition (so that <math>dx^\mu dx^\nu = dx^\nu dx^\mu</math>) we can write the metric in the form
:<math>g = g_{\mu\nu}dx^\mu dx^\nu.\,</math>

If the local coordinates are specified, or understood from context, the metric can be written as a 4×4 [[symmetric matrix]] with entries <math>g_{\mu\nu}</math>. The nondegeneracy of <math>g_{\mu \nu} </math> means that this matrix is [[non-singular matrix|non-singular]] (i.e. has non-vanishing determinant), while the Lorentzian signature of ''g'' implies that the matrix has one negative and three positive [[eigenvalues]]. Note that physicists often refer to this matrix or the coordinates <math>g_{\mu\nu}</math> themselves as the metric (see, however, [[abstract index notation]]).

With the quantity <math>dx^\mu</math> being an infinitesimal coordinate displacement, the metric acts as an infinitesimal invariant interval squared or [[line element]]. For this reason one often sees the notation <math>ds^2</math> for the metric:
:<math>ds^2 = g_{\mu\nu}dx^\mu dx^\nu.\,</math>
In general relativity, the terms ''metric'' and ''line element'' are often used interchangeably.

The line element <math>ds^2</math> imparts information about the [[Causal spacetime structure|causal structure of the spacetime]]. When <math>ds^2 < 0</math>, the interval is [[Minkowski space#Causal structure|timelike]] and the square root of the absolute value of ''ds2'' is an incremental [[proper time]]. Only timelike intervals can be physically traversed by a massive object. When <math>ds^2=0</math>, the interval is lightlike, and can only be traversed by light. When <math>ds^2 > 0</math>, the interval is spacelike and the square root of ''ds2'' acts as an incremental [[proper length]].  Spacelike intervals cannot be traversed, since they connect events that are out of each other's [[light cone]]s. [[Spacetime#Basic concepts|Event]]s can be causally related only if they are within each other's light cones.

The metric components obviously depend on the chosen local coordinate system. Under a change of coordinates <math>x^\mu \to x^{\bar \mu}</math> the metric components transform as
:<math>g_{\bar \mu \bar \nu} = \frac{\partial x^\rho}{\partial x^{\bar \mu}}\frac{\partial x^\sigma}{\partial x^{\bar \nu}} g_{\rho\sigma} = \Lambda^\rho {}_{\bar \mu} \, \Lambda^\sigma {}_{\bar \nu} \, g_{\rho \sigma} .</math>

==Examples==
===Flat spacetime===

The simplest example of a Lorentzian manifold is [[flat spacetime]] which can be given as '''R'''4 with coordinates <math>(t,x,y,z)</math> and the metric
:<math>ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2. \,</math>
Note that these coordinates actually cover all of '''R'''4. The flat space metric (or Minkowski metric) is often denoted by the symbol η and is the metric used in [[special relativity]]. In the above coordinates, the matrix representation of η is
:<math>\eta = \begin{pmatrix}-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}</math>
In [[spherical coordinates]] <math>(t,r,\theta,\phi)</math>, the flat space metric takes the form
:<math>ds^2 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2 \,</math>
where
:<math>d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2</math>
is the standard metric on the [[2-sphere]].

===Schwarzschild metric===

Besides the flat space metric the most important metric in general relativity is the [[Schwarzschild metric]] which can be given in one set of local coordinates by
:<math>ds^{2} = -\left(1 - \frac{2GM}{rc^2} \right) c^2 dt^2 + \left(1 - \frac{2GM}{rc^2} \right)^{-1} dr^2 + r^2 d\Omega^2</math>
where, again, <math>d\Omega^2</math> is the standard metric on the [[2-sphere]]. Here ''G'' is the [[gravitation constant]] and ''M'' is a constant with the dimensions of [[mass]]. Its derivation can be found [[Deriving the Schwarzschild solution|here]]. The Schwarzschild metric approaches the Minkowski metric as ''M'' approaches zero (except at the origin where it is undefined). Similarly, when ''r'' goes to infinity, the Schwarzschild metric approaches the Minkowski metric.

===Other metrics===

Other notable metrics are:

*[[Bondi metric]],
*[[Eddington–Finkelstein coordinates]],
*[[Friedmann–Lemaître–Robertson–Walker metric]],
*[[Gullstrand–Painlevé coordinates]],
*[[Isotropic coordinates]],
*[[Kerr metric]],
*[[Kerr–Newman metric]],
*[[Kruskal–Szekeres coordinates]],
*[[Lemaître coordinates]],
*[[Lemaître–Tolman metric]],
*[[Peres metric]],
*[[Reissner–Nordström metric]],
*[[Rindler coordinates]],
*[[Weyl−Lewis−Papapetrou coordinates]].

Some of them are without the [[event horizon]] or can be without the [[gravitational singularity]].

== Volume ==

The metric ''g'' defines a natural [[volume form]], which can be used to integrate over spacetimes. In local coordinates <math>x^\mu</math> of a manifold, the volume form can be written
:<math>\mathrm{vol}_g = \sqrt{|\det g|}\,dx^0\wedge dx^1\wedge dx^2\wedge dx^3 </math>
where det ''g'' is the [[determinant]] of the matrix of components of the metric tensor for the given coordinate system.

==Curvature==

The metric ''g'' completely determines the [[curvature]] of spacetime. According to the [[fundamental theorem of Riemannian geometry]], there is a unique [[connection (mathematics)|connection]] ∇ on any [[semi-Riemannian manifold]] that is compatible with the metric and [[Torsion tensor|torsion]]-free. This connection is called the [[Levi-Civita connection]]. The [[Christoffel symbols]] of this connection are given in terms of partial derivatives of the metric in local coordinates <math>x^\mu</math> by the formula
:<math>\Gamma^\lambda {}_{\mu\nu} = {1 \over 2} g^{\lambda\rho} \left( {\partial g_{\rho\mu} \over \partial x^\nu} + {\partial g_{\rho\nu} \over \partial x^\mu} - {\partial g_{\mu\nu} \over \partial x^\rho} \right) </math>.

The curvature of spacetime is then given by the [[Riemann curvature tensor]] which is defined in terms of the Levi-Civita connection ∇. In local coordinates this tensor is given by:
:<math>{R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho {}_{\nu\sigma}
- \partial_\nu\Gamma^\rho {}_{\mu\sigma}
+ \Gamma^\rho {}_{\mu\lambda}\Gamma^\lambda {}_{\nu\sigma}
- \Gamma^\rho {}_{\nu\lambda}\Gamma^\lambda {}_{\mu\sigma}.</math>

The curvature is then expressible purely in terms of the metric <math>g</math> and its derivatives.

==Einstein's equations==

One of the core ideas of general relativity is that the metric (and the associated geometry of spacetime) is determined by the [[matter]] and [[energy]] content of [[spacetime]]. [[Einstein field equations|Einstein's field equations]]:
:<math>R_{\mu\nu} - {1\over 2}R g_{\mu\nu} = 8\pi G\,T_{\mu\nu}</math>
where
:<math> R_{\nu \rho} \ \stackrel{\mathrm{def}}{=}\ {R^{\mu}}_{\nu\mu \rho} </math>
relate the metric (and the associated curvature tensors) to the [[stress-energy tensor]] <math>T_{\mu\nu}</math>. This [[tensor]] equation is a complicated set of nonlinear [[partial differential equation]]s for the metric components. [[Exact solutions]] of Einstein's field equations are very difficult to find.

== See also ==

*[[Alternatives to general relativity]]
*[[Basic introduction to the mathematics of curved spacetime]]
*[[Mathematics of general relativity]]
*[[Ricci calculus]]

==References==

{{Reflist}}
See [[general relativity resources]] for a list of references.

{{tensors}}

[[Category:Tensors in general relativity]]
[[Category:Time]]