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2014-12-12T16:31:49Z

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2013-12-05T14:32:39Z

207.63.16.46:

A '''Theoretical motivation for general relativity''', including the motivation for the [[geodesic equation]] and the [[Einstein field equation]], can be obtained from [[special relativity]] by examining the [[Dynamics (mechanics)|dynamics]] of particles in [[circular orbit]]s about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation ''[[A priori and a posteriori|a priori]]''. This provides a means to inform and verify the formalism.

[[General relativity]] addresses two questions:
# How does the [[curvature]] of [[spacetime]] affect the motion of [[matter]]?
# How does the presence of matter affect the curvature of spacetime?

The former question is answered with the [[#The geodesic equation in a local coordinate system|geodesic equation]]. The second question is answered with the [[#Einstein field equation|Einstein field equation]]. The geodesic equation and the field equation are related through a [[principle of least action]]. The motivation for the geodesic equation is provided in the section [[#Geodesic equation for circular orbits|Geodesic equation for circular orbits]] The motivation for the Einstein field equation is provided in the section [[#Stress-energy tensor|Stress-energy tensor]]

<div class="noprint" style="clear: right">
{{General relativity}}
__TOC__
</div>

==Geodesic equation for circular orbits==

{{main|Geodesics in general relativity}}

===Kinetics of circular orbits===

[[Image:060322 helix.svg|thumb|250px|left|World line of a circular orbit about the Earth depicted in two spatial dimensions X and Y (the plane of the orbit) and a time dimension, usually put as the vertical axis. Note that the orbit about the Earth is a circle in space, but its worldline is a helix in spacetime.]]
For definiteness consider a circular earth orbit (helical [[world line]]) of a particle. The particle travels with speed v. An observer on earth sees that length is contracted in the frame of the particle. A measuring stick traveling with the particle appears shorter to the earth observer. Therefore the circumference of the orbit, which is in the direction of motion appears longer than <math> \pi </math> times the diameter of the orbit.<ref name="Ref. 1">{{cite book | author=Einstein, A. | title=Relativity: The Special and General Theory | location= New York | publisher=Crown| year=1961 | isbn=0-517-02961-8}}</ref>

In [[special relativity]] the 4-proper-velocity of the particle in the [[inertial]] (non-accelerating) frame of the earth is

:<math> u = \left ( \gamma , \gamma { \mathbf{v} \over c } \right ) </math>

where c is the [[speed of light]], <math> \mathbf{v} </math> is the 3-velocity, and <math> \gamma </math> is

:<math> \gamma = { 1 \over \sqrt { { 1 - { { \mathbf{v} \cdot \mathbf{v} } \over c^2 } } } } </math>.

The magnitude of the 4-velocity vector is always constant

:<math> u_{\alpha} u^{\alpha} = -1 </math>

where we are using a [[Minkowski metric]]

:<math>\eta^{\mu\nu} =\eta_{\mu\nu} = \begin{pmatrix}
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix}</math>.

The magnitude of the 4-velocity is therefore a [[Lorentz scalar]].

The 4-acceleration in the earth (non-accelerating) frame is

:<math> a \equiv { {d u} \over {d\tau} } = { d \over {d\tau} } { \left ( \gamma , \gamma { \mathbf{v} \over c } \right )} = { \left ( 0 , \gamma^2 { \mathbf{a} \over c^2 } \right )} = { \left ( 0 , - \gamma^2 { { \mathbf{v} \cdot \mathbf{v} } \over c^2 } { {\mathbf{r} } \over r^2 } \right )} </math>

where <math> d\tau </math> is c times the proper time interval measured in the frame of the particle. This is related to the time interval in the Earth's frame by

:<math> c dt = \gamma d\tau </math>.

Here, the 3-acceleration for a circular orbit is

:<math> \mathbf{a} = - \omega^2 \mathbf{r} = - { \mathbf{v} \cdot \mathbf{v} } { {\mathbf{r} } \over r^2 } </math>

where <math> \omega </math> is the angular velocity of the rotating particle and <math> \mathbf{r} </math> is the 3-position of the particle.

The magnitude of the 4-velocity is constant. This implies that the 4-acceleration must be perpendicular to the 4-velocity. The 4-acceleration is, in fact, perpendicular to the 4-velocity in this example (see [[Fermi-Walker transport]]). The inner product of the 4-acceleration and the 4-velocity is therefore always zero. The inner product is a [[Lorentz scalar]].

===Curvature of spacetime: Geodesic equation===
The equation for the acceleration can be generalized, yielding the [[geodesic equation]]

:<math> { {d u^{\mu}} \over {d\tau}} - a^{\mu} = 0 </math>

:<math> { {d u^{\mu}} \over {d\tau}} + {R^{\mu}}_{\alpha \nu \beta } u^{\alpha} x^{\nu} u^{\beta} = 0 </math>

where <math> x^{\mu} </math> is the 4-position of the particle and <math> {R^{\mu}}_{\alpha \nu \beta } </math> is the [[curvature]] tensor give by

:<math> {R^{\mu}}_{\alpha \nu \beta } = { 1 \over r^2 } \eta_{\alpha \beta} {\delta^{\mu}}_{\nu} </math>

where <math> {\delta^{\mu}}_{\nu} </math> is the [[Kronecker delta function]], and we have the constraints

:<math> u_{\alpha} u^{\alpha} = -1</math>

and

:<math>a_{\alpha} u^{\alpha} = 0</math>.

It is easily verified that circular orbits satisfy the geodesic equation. The geodesic equation is actually more general. Circular orbits are a particular solution of the equation. Solutions other than circular orbits are permissible and valid.

===Ricci curvature tensor and trace===
{{main|Ricci curvature}}
{{main|Scalar curvature}}

The [[Ricci curvature]] tensor is a special curvature tensor given by the contraction

:<math> R_{\alpha \beta } \equiv {R^{\nu}}_{\alpha \nu \beta } </math>.

The trace of the Ricci tensor, called the [[scalar curvature]], is

:<math> R \equiv {R^{\alpha}}_{ \alpha } </math>.

===The geodesic equation in a local coordinate system===

[[Image:General relativity rdj 3.png|frame|right|Circular orbits at the same radius.]]

Consider the situation in which there are now two particles in nearby [[Circular orbit|circular]] [[Polar orbit|polar]] orbits of the [[earth]] at radius <math> r </math> and speed <math> v </math>.

The particles execute [[simple harmonic motion]] about the earth and with respect to each other. They are at their maximum distance from each other as they cross the equator. Their [[Trajectory|trajectories]] intersect at the poles.

Imagine we have a spacecraft co-moving with one of the particles. The ceiling of the craft, the <math> \acute{\mathbf{z}} </math> direction, coincides with the <math> \mathbf{r} </math> direction. The front of the craft is in the <math> \acute{\mathbf{x} } </math> direction, and the <math> \acute{\mathbf{y} } </math> direction is to the left of the craft. The spacecraft is small compared with the size of the orbit so that the local frame is a local Lorentz frame. The 4-separation of the two particles is given by <math> \acute{x}^{\mu} </math>. In the local frame of the spacecraft the geodesic equation is given by

:<math> { {d^2 \acute{x}^{\mu}} \over {d\tau^2}} + \acute{{R}^{\mu} }_{\alpha \nu \beta } \acute{u}^{\alpha} \acute{x}^{\nu} \acute{u}^{\beta} = 0 </math>

where

:<math> \acute{u}^{\mu} = { {d \acute{x}^{\mu}} \over {d\tau}} </math>

and

:<math> \acute{{R}^{\mu}}_{\alpha \nu \beta } </math>

is the curvature tensor in the local frame.

===Geodesic equation as a covariant derivative===
The equation of motion for a particle in flat spacetime and in the absence of forces is

:<math> { {d {u}^{\mu}} \over {d\tau}} =0 </math>.

If we require a particle to travel along a geodesic in curved spacetime, then the analogous expression in curved spacetime is

:<math> { {D \acute{u}^{\mu}} \over {D\tau}}= { {d \acute{u}^{\mu}} \over {d\tau}} + {\Gamma ^{\mu}}_{\alpha \beta} \acute{u}^{\alpha} \acute{u}^{\beta} =0 </math>

where the derivative on the left is the [[covariant derivative]], which is the generalization of the normal derivative to a derivative in curved spacetime. Here

:<math> {\Gamma ^{\mu}}_{\alpha \beta} </math>

is a [[Christoffel symbol]].

The curvature is related to the Christoffel symbol by

:<math> \acute{{R}^{\mu}}_{\alpha \nu \beta } = { { \partial {{\Gamma}^{\mu}}}_{\alpha \beta} \over {\partial x^{\nu}} }
- { { \partial {{\Gamma}^{\mu}}}_{\alpha \nu} \over {\partial x^{\beta}} }
+ {{\Gamma}^{\mu}}_{\gamma \nu} {{\Gamma}^{\gamma}}_{\alpha \beta}
- {{\Gamma}^{\mu}}_{\gamma \beta} {{\Gamma}^{\gamma}}_{\alpha \nu}
</math>.

===Metric tensor in the local frame===
The interval in the local frame is

:<math> ds^2 = dx^2 +dy^2 + dz^2 - c^2 dt^2 \equiv g_{\mu \nu } d \acute{x}^{\mu} d \acute{x}^{\nu} </math>

:<math> = d \acute{x} ^2 +d\acute{y}^2 + d\acute{z}^2 - c^2 d\acute{t}^2 +2\gamma \cos(\theta ) \cos(\phi) \,v \, d\acute{t} \,d\acute{x} +2\gamma \cos(\theta ) \sin (\phi) v \,d\acute{t} \,d\acute{y} -2\gamma \sin(\theta ) v \, d\acute{t} \, d\acute{z} </math>

where

:<math> \theta </math> is the angle with the <math>z</math> axis (longitude) and

:<math> \phi </math> is the angle with the <math>x</math> axis (latitude).

This gives a [[Metric tensor|metric]] of

:<math> g_{\mu\nu} = \begin{pmatrix}
-1 & \gamma \cos( \theta ) \cos ( \phi ) \frac{v}{c} & \gamma \cos( \theta ) \sin ( \phi ) \frac{v}{c} & -\gamma \sin ( \theta ) \frac{v}{c} \\
\gamma \cos( \theta ) \cos ( \phi ) {\frac{v}{c}} & 1 & 0 & 0\\
\gamma \cos( \theta ) \sin ( \phi ) {\frac{v}{c}} & 0 & 1 & 0\\
-\gamma \sin ( \theta ) \frac{v}{c} & 0 & 0 & 1
\end{pmatrix} </math>

in the local frame.

The inverse of the metric tensor <math> g^{\mu \nu} </math> is defined such that

:<math> g_{\mu \alpha} g^{\alpha \nu} = \delta_{\mu}^{\nu} </math>

where the term on the right is the [[Kronecker delta]].

The transformation of the infinitesimal 4-volume <math> d\Omega </math> is

:<math> d\acute{\Omega} = \sqrt{-g} d{\Omega } </math>

where g is the determinant of the metric tensor.

The differential of the determinant of the metric tensor is

:<math> dg = g g^{\mu \nu} dg_{\mu \nu} = -g g_{\mu \nu} dg^{\mu \nu} </math>.

The relationship between the Christoffel symbols and the metric tensor is

:<math> {{\Gamma}^{\alpha}}_{ \mu \nu } = g^{\alpha \beta} {\Gamma}_{\beta \mu \nu }
</math>

:<math> {\Gamma}_{\beta \mu \nu } = {\frac{1}{2}} \left (
{ { \partial {g}}_{\beta \nu} \over {\partial x^{\mu}} }
+ { { \partial {g}}_{\beta \mu} \over {\partial x^{\nu}} }
- { { \partial {g}}_{\mu \nu} \over {\partial x^{\beta}} }
\right )
</math>.

===Principle of least action in general relativity===
{{main|Einstein-Hilbert action}}

The principle of least action states that the [[world line]] between two events in spacetime is that world line that minimizes the action between the two events. In [[classical mechanics]] the principle of least action is used to derive [[Newton's laws of motion]] and is the basis for [[Lagrangian dynamics]]. In relativity it is expressed as

:<math> S = \int_1^2 \mathcal{L}\, d\Omega </math>

between events 1 and 2 is a minimum. Here S is a [[Scalar (mathematics)|scalar]] and

:<math> \mathcal{L} </math>

is known as the [[Lagrangian density]]. The Lagrangian density is divided into two parts, the density for the orbiting particle <math> \mathcal{L}_p </math> and the density <math> \mathcal{L}_e </math> of the gravitational field generated by all other particles including those comprising the earth,

:<math> \mathcal{L} = \mathcal{L}_p + \mathcal{L}_e </math>.

In curved [[spacetime]], the "shortest" world line is that [[geodesic]] that minimizes the curvature along the geodesic. The action then is proportional to the curvature of the world line. Since S is a scalar, the [[scalar curvature]] is the appropriate measure of curvature. The action for the particle is therefore

:<math> S_p = C \int_1^2 \acute{R}\, d\acute{\Omega} = C \int_1^2 { \acute{R} } \sqrt{-g} \,d{\Omega} = C \int_1^2 g^{\alpha \beta} \acute{R}_{\alpha \beta} \sqrt{-g}\, d{\Omega} </math>

where <math> C </math> is an unknown constant. This constant will be determined by requiring the theory to reduce to Newton's law of gravitation in the nonrelativistic limit.

The Lagrangian density for the particle is therefore

:<math> \mathcal{L}_p = C g^{\alpha \beta} \acute{R}_{\alpha \beta} \sqrt{-g} </math>.

The action for the particle and the earth is

:<math> S = \int_1^2 C g^{\alpha \beta} \acute{R}_{\alpha \beta} \sqrt{-g}\, d\Omega + \int_1^2 \mathcal{L}_e \,d\Omega </math>.

We find the world line that lies on the surface of the sphere of radius r by varying the metric tensor. Minimization and neglect of terms that disappear on the boundaries, including terms second order in the derivative of g, yields

:<math> 0 = \delta S = \int_1^2 C \left ( \acute{R}_{\alpha \beta} - {1\over 2} \acute{R} g^{\alpha \beta} \right ) \delta g^{\alpha \beta} \sqrt{-g}\, d\Omega - \int_1^2 \acute{T}_{\alpha \beta} \delta g^{\alpha \beta} \sqrt{-g}\, d\Omega </math>

where<ref name="Ref 3 Sec 94">{{cite book | author=Landau, L. D. and Lifshitz, E. M.| title=Classical Theory of Fields (Fourth Revised English Edition) | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7}}</ref>

:<math> \acute{T}_{\alpha \beta} = { 1 \over \sqrt{-g} } \left ( {d \over {dx^{\nu} } } { { \partial \mathcal{L}_e} \over { \partial \left ( { {d g^{ \alpha \beta } } \over { dx^{\nu} } } \right ) } } - { {\partial \mathcal{L}_e} \over { \partial g^{ \alpha \beta } } } \right ) </math>

is the [[Stress-energy tensor|Hilbert stress-energy tensor]] of the field generated by the earth.

The relationship, to within an unknown constant factor, between the stress-energy and the curvature is

:<math> \acute{T}_{\alpha \beta} = C \left ( \acute{R}_{\alpha \beta} - {1\over 2} \acute{R} \, g_{\alpha \beta} \right ) </math>.

==Stress-energy tensor==
===Newton's law of gravitation===

[[Image:Lorentz transform of world line.gif|right|framed|Diagram 1. Changing views of spacetime along the [[world line]] of a rapidly accelerating observer.
In this animation, the dashed line is the spacetime trajectory ("[[world line]]") of a particle. The balls are placed at regular intervals of [[proper time]] along the world line. The solid diagonal lines are the [[light cone]]s for the observer's current event, and intersect at that event. The small dots are other arbitrary events in the spacetime. For the observer's current instantaneous inertial frame of reference, the vertical direction indicates the time and the horizontal direction indicates distance.
The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. So at a bend in the world line the particle is being accelerated. Note how the view of spacetime changes when the observer accelerates, changing the instantaneous inertial frame of reference. These changes are governed by the Lorentz transformations. Also note that:
* the balls on the world line before/after future/past accelerations are more spaced out due to time dilation.
* events which were simultaneous before an acceleration are at different times afterwards (due to the [[relativity of simultaneity]]),
* events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and
* the world line always remains within the future and past light cones of the current event.]]

[[Newton's law of gravitation|Newton's Law of Gravitation]] in non-relativistic mechanics states that the acceleration on an object of mass <math> m </math> due to another object of mass <math> M </math> is equal to

:<math> \mathbf{f} = {d^2 \mathbf{r} \over d\tau^2} = - {GM \over { c^2 r^3} }\mathbf{r} </math>

where <math> G </math> is the [[gravitational constant]], <math> \mathbf{r} </math> is a vector from mass <math> M </math> to mass <math> m </math> and <math> r </math> is the magnitude of that vector. The time t is scaled with the [[speed of light]] c

:<math> \tau \equiv c t </math>.

The acceleration <math> \mathbf{f} </math> is independent of <math> m </math>.

For definiteness. consider a particle of mass <math> m </math> orbiting in the gravitational field of the earth with mass <math> M </math>. The law of gravitation can be written

:<math> \mathbf{f} = - {4\pi G \over {3 c^2} }\rho(r) \mathbf{r} </math>

where <math> \rho(r) </math> is the average mass density inside a [[Volume|sphere]] of radius <math> r </math>.

===Gravitational force in terms of the 00 component of the stress-energy tensor===

Newton's law can be written

:<math> \mathbf{f} = - {4\pi G \over {3 c^4}} \left ( {Mc^2 \over V }\right ) \mathbf{r} </math>.

where <math> V </math> is the [[volume]] of a sphere of radius <math> r </math>. The quantity <math> Mc^2 </math> will be recognized from [[special relativity]] as the rest energy of the large body, the earth. This is the sum of the rest energies of all the particles that compose earth. The quantity in the parentheses is then the average rest energy density of a sphere of radius <math> r </math> about the earth. The gravitational field is proportional to the average energy density within a radius r. This is the 00 component of the [[stress-energy tensor]] in [[Special relativity|relativity]] for the special case in which all the energy is rest energy. More generally

:<math> T_{00} = - {T^{0}}_0 = \sum_{i=1}^N \left ( {\gamma_i m_i c^2 \over V }\right ) </math>

where

:<math> \gamma_i \equiv { 1 \over {\sqrt {1 - {{\mathbf{v}_i \cdot \mathbf{v}_i } \over c^2} } } } </math>

and <math> \mathbf{v_i} </math> is the velocity of particle i making up the earth and <math> m_i </math> in the rest mass of particle i. There are N particles altogether making up the earth.

===Relativistic generalization of the energy density===
[[Image:StressEnergyTensor.svg|thumb|250px|left|The components of the stress-energy tensor.]]
There are two simple relativistic entities that reduce to the 00 component of the stress-energy tensor in the nonrelativistic limit

:<math> u^{\alpha} T_{\alpha \beta} u^{\beta} \rightarrow T_{00} </math>

and the [[Trace (linear algebra)|trace]]

:<math> T \equiv {T^{\alpha}}_{\alpha} = -u_{\alpha} u^{\alpha} T = -u^{\alpha} T \eta_{\alpha \beta} u^{\beta} \rightarrow - T_{00}</math>

where <math> u^{\alpha} </math> is the 4-velocity.

The 00 component of the stress-energy tensor can be generalized to the relativistic case as a linear combination of the two terms

:<math> T_{00} \rightarrow u^{\alpha} \left ( A T_{\alpha \beta} + B T \eta_{\alpha \beta} \right ) u^{\beta} </math>

where

:<math> A + B = 1 </math>

===4-acceleration due to gravity===

The 4-acceleration due to gravity can be written

:<math> f^{\mu} = - 8\pi { G \over { 3 c^4 } } \left ( {A \over 2} T_{\alpha \beta} + {B \over 2} T \eta_{\alpha \beta} \right )\delta^{\mu}_{\nu} u^{\alpha} x^{\nu} u^{\beta} </math>.

Unfortunately, this acceleration is nonzero for <math> \mu = 0 </math> as is required for circular orbits. Since the magnitude of the 4-velocity is constant, it is only the component of the force perpendicular to the 4-velocity that contributes to the acceleration. We must therefore subtract off the component of force parallel to the 4-velocity. This is known as [[Fermi-Walker transport]].<ref>{{cite book | author=Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0|pages= 170,171}}</ref> In other words

:<math> f^{\mu} \rightarrow f^{\mu} + u^{\mu} u_{\nu} f^{\nu} </math>.

This yields

:<math> f^{\mu} = - 8\pi { G \over { 3 c^4 } } \left ( {A \over 2} T_{\alpha \beta} + {B \over 2} T \eta_{\alpha \beta} \right ) \left ( \delta^{\mu}_{\nu} + u^{\mu} u_{\nu} \right ) u^{\alpha} x^{\nu} u^{\beta} </math>.

The force in the local frame is

:<math> \acute{f}^{\mu} = - 8\pi { G \over { 3 c^4 } } \left ( {A \over 2} \acute{T}_{\alpha \beta} + {B \over 2} \acute{T} g_{\alpha \beta} \right ) \left ( \delta^{\mu}_{\nu} + \acute{u}^{\mu} \acute{u}_{\nu} \right ) \acute{u}^{\alpha} \acute{x}^{\nu} \acute{u}^{\beta} </math>.

==Einstein field equation==

[[Image:spacetime curvature.png|thumb|right|400px|Two-dimensional visualization of space-time distortion. The presence of matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity.]]

We obtain the [[Einstein field equation]]<ref>Landau 1975, p. 276</ref> by equating the acceleration required for circular orbits with the acceleration due to gravity

:<math> a^{\mu} = f^{\mu} </math>

:<math> \acute{{R}^{\mu}}_{\alpha \nu \beta} \acute{u}^{\alpha} \acute{x}^{\nu} \acute{u}^{\beta} = - \acute{f}^{\mu} </math>.

This is the relationship between curvature of spacetime and the stress-energy tensor.

The Ricci tensor becomes

:<math> \acute{R}_{\alpha \beta} = 8\pi { G \over { c^4 } } \left ( { A \over 2 } \acute{T}_{\alpha \beta} + {B \over 2} \acute{T} g_{\alpha \beta} \right ) </math>.

The trace of the Ricci tensor is

:<math> \acute{R} = \acute{R}_{\alpha}^{ \alpha} = 8\pi { G \over { c^4 } } \left ( {A\over 2}\acute{T}_{\alpha}^{ \alpha} + {B \over 2} \acute{T} \delta_{\alpha}^{ \alpha} \right ) = 8\pi { G \over { c^4 } } \left ( {A\over 2} + 2B \right ) \acute{T } </math>.

Comparison of the Ricci tensor with the Ricci tensor calculated from the principle of least action, [[Theoretical motivation for general relativity#Principle of least action in general relativity]] identifying the stress-energy tensor with the Hilbert stress-energy, and remembering that A+B=1 removes the ambiguity in A, B, and C.

:<math> A=2 </math>

:<math> B=-1 </math>

and

:<math> C= \left ( 8\pi { G \over { c^4 } } \right )^{-1} </math>.

This gives

:<math> \acute{R} = - 8\pi { G \over { c^4 } } \acute{T } </math>.

The field equation can be written

:<math> \mathcal{G}_{\alpha \beta} = 8\pi { G \over { c^4 } } \acute{T}_{\alpha \beta} </math>

where

:<math> \mathcal{G}_{\alpha \beta} \equiv \acute{R}_{\alpha \beta} - {1 \over 2} \acute{R} g_{\alpha \beta} </math>.

This is the Einstein field equation that describes curvature of spacetime that results from stress-energy density. This equation, along with the geodesic equation have motivated by the kinetics and dynamics of a particle orbiting the earth in a circular orbit. They are true in general.

==Solving the Einstein field equation==

Solving the Einstein field equation requires an iterative process. The solution is represented in the metric tensor

:<math>
g_{\mu \nu}
</math>.

Typically there is an initial guess for the tensor. The guess is used to calculate [[Christoffel symbol]]s, which are used to calculate the curvature. If the Einstein field equation is not satisfied, the process is repeated.

Solutions occur in two forms, vacuum solutions and non-vacuum solutions. A [[Vacuum solution (general relativity)|vacuum solution]] is one in which the stress-energy tensor is zero. The relevant vacuum solution for circular orbits is the [[Schwarzschild metric]]. There are also a number of [[Exact solutions in general relativity|exact solutions]] that are non-vacuum solutions, solutions in which the stress tensor is non-zero.

==Solving the geodesic equation==
{{main|Solving the geodesic equations}}
Solving the geodesic equations requires knowledge of the metric tensor obtained through the solution of the Einstein field equation. Either the Christoffel symbols or the curvature are calculated from the metric tensor. The geodesic equation is then integrated with the appropriate [[boundary condition]]s.

==Electrodynamics in curved spacetime==
{{main|Maxwell's equations in curved spacetime}}

[[Maxwell's equations]], the equations of electrodynamics, in curved spacetime are a generalization of Maxwell's equations in flat [[spacetime]] (see [[Formulation of Maxwell's equations in special relativity]]). Curvature of spacetime affects electrodynamics. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with [[covariant derivative]]s. The sourced and source-free equations become (cgs units):

:<math> { 4 \pi \over c }J^ b = \partial_a F^{ab} + {\Gamma^a}_{\mu a} F^{\mu b} + {\Gamma^b}_{\mu a} F^{a \mu} \equiv D_a F^{ab} \equiv {F^{ab}}_{;a} \,\!</math>,

and

:<math>0 = \partial_c F_{ab} + \partial_b F_{ca} + \partial_a F_{bc} = D_c F_{ab} + D_b F_{ca} + D_a F_{bc}</math>

where <math>\, J^a</math> is the [[4-current]], <math>\, F^{ab}</math> is the [[electromagnetic tensor|field strength tensor]], <math>\, \epsilon_{abcd}</math> is the [[Levi-Civita symbol]], and

:<math> { \partial \over { \partial x^a } } \equiv \partial_a \equiv {}_{,a} \equiv (\partial/\partial ct, \nabla)</math>

is the [[Four-gradient|4-gradient]]. Repeated indices are summed over according to [[Einstein notation|Einstein summation convention]]. We have displayed the results in several common notations.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, [[Gauss' law]] and the [[Ampère's circuital law|Ampère's law with Maxwell's correction]]. The second equation is an expression of the homogenous equations, [[Faraday's law of induction]] and [[Gauss's law for magnetism]].

The electromagnetic wave equation is modified from the equation in flat spacetime in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

:<math> - {A^{\alpha ; \beta}}_{; \beta} + {R^{\alpha}}_{\beta} A^{\beta} = {4 \pi \over c } J^{\alpha} </math>

where the [[Four-potential|4-potential]] is defined such that

:<math>F^{ab} = \partial^b A^a - \partial^a A^b \,\!</math>.

We have assumed the generalization of the [[Lorenz gauge]] in curved spacetime

:<math> {A^{\mu}}_{ ; \mu} = 0 </math>.

==See also==
*[[Newtonian foundation of general relativity]]

==References==
{{reflist}}
* {{cite book | author=R. P. Feynman, F. B. Moringo, and W. G. Wagner | title=Feynman Lectures on Gravitation | publisher=Addison-Wesley | year=1995 | isbn=0-201-62734-5}}
* {{cite book | author=P. A. M. Dirac | title=General Theory of Relativity | publisher=Princeton University Press| year=1996 | isbn=0-691-01146-X}}

{{Physics-footer}}

[[Category:General relativity]]
[[Category:Concepts in physics]]

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Klein four-group

2013-11-14T15:11:59Z

207.63.16.46: /* Further reading */

{{electromagnetism|cTopic=[[Electrical network|Electrical Network]]}}
[[File:Complex Impedance.svg|200px|thumb|right|A graphical representation of the [[Complex plane|complex impedance plane]]]]
'''Electrical impedance''' is the measure of the opposition that a [[electrical circuit|circuit]] presents to a [[electrical current|current]] when a [[voltage]] is applied.

In quantitative terms, it is the [[complex number|complex]] [[ratio]] of the voltage to the current in an [[alternating current]] (AC) circuit. Impedance extends the concept of [[Electrical resistance|resistance]] to AC circuits, and possesses both magnitude and [[Phase (waves)|phase]], unlike resistance, which has only magnitude. When a circuit is driven with [[direct current]] (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero [[phase angle]].

It is necessary to introduce the concept of impedance in AC circuits because there are two additional impeding mechanisms to be taken into account besides the normal resistance of DC circuits: the induction of voltages in conductors self-induced by the magnetic fields of currents ([[inductance]]), and the electrostatic storage of charge induced by voltages between conductors ([[capacitance]]). The impedance caused by these two effects is collectively referred to as [[electrical reactance|reactance]] and forms the [[imaginary number|imaginary]] part of complex impedance whereas resistance forms the [[real number|real]] part.

The symbol for impedance is usually {{math|''Z''}} and it may be represented by writing its magnitude and phase in the form {{math|{{!}}''Z''{{!}}''∠θ''}}. However, complex number representation is often more powerful for circuit analysis purposes. The term ''impedance'' was coined by [[Oliver Heaviside]] in July 1886.<ref>''Science'', p. 18, 1888</ref><ref>Oliver Heaviside, ''The Electrician'', p. 212, 23 July 1886, reprinted as ''Electrical Papers'', p 64, AMS Bookstore, ISBN 0-8218-3465-7</ref> [[Arthur Kennelly]] was the first to represent impedance with complex numbers in 1893.<ref>[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4768008 Kennelly, Arthur. ''Impedance'' (AIEE, 1893)]</ref>

Impedance is defined as the [[frequency domain]] ratio of the voltage to the current.<ref>{{Cite book | last = Alexander | first = Charles | last2 = Sadiku | first2 = Matthew | title = Fundamentals of Electric Circuits | publisher = McGraw-Hill | year = 2006 | edition = 3, revised | pages =387–389 | isbn = 978-0-07-330115-0 | postscript = {{inconsistent citations}}}}</ref> In other words, it is the voltage–current ratio for a single [[complex exponential]] at a particular frequency {{math|''ω''}}. In general, impedance will be a complex number, with the same [[dimensional analysis|units]] as resistance, for which the [[SI unit]] is the [[ohm]] ({{math|Ω}}). For a sinusoidal current or voltage input, the [[Complex number#Notation_of_the_polar_form|polar form]] of the complex impedance relates the amplitude and phase of the voltage and current. In particular,
* The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude.
* The phase of the complex impedance is the [[phase shift]] by which the current lags the voltage.
The [[Multiplicative inverse|reciprocal]] of impedance is [[admittance]] (i.e., admittance is the current-to-voltage ratio, and it conventionally carries units of [[siemens (unit)|siemens]], formerly called [[mho]]s).

==Complex impedance==
Impedance is represented as a [[Complex number|complex]] quantity <math>\scriptstyle Z</math> and the term ''complex impedance'' may be used interchangeably; the [[Polar coordinates|polar form]] conveniently captures both magnitude and phase characteristics,

:<math>\ Z = |Z| e^{j\arg (Z)}</math>

where the magnitude <math>\scriptstyle |Z|</math> represents the ratio of the voltage difference amplitude to the current amplitude, while the argument <math>\scriptstyle \arg (Z)</math> (commonly given the symbol <math>\scriptstyle \theta </math>) gives the phase difference between voltage and current. <math>\scriptstyle j</math> is the [[imaginary unit]], and is used instead of <math>\scriptstyle i</math> in this context to avoid confusion with the symbol for [[Ampere|electric current]]. In [[Cartesian plane|Cartesian form]],

:<math>\ Z = R + jX</math>

where the [[real part]] of impedance is the resistance <math>\scriptstyle R</math> and the [[imaginary part]] is the [[Reactance (electronics)|reactance]] <math>\scriptstyle X</math>.

Where it is required to add or subtract impedances the cartesian form is more convenient, but when quantities are multiplied or divided the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal [[Complex number#Polar form|conversion rules of complex numbers]].

== Ohm's law ==
[[File:General AC circuit.svg|thumb|right|169px|An AC supply applying a voltage <math>\scriptstyle V</math>, across a [[Electrical load|load]] <math>\scriptstyle Z</math>, driving a current <math>\scriptstyle I</math>.]]
{{Main|Ohm's law}}

The meaning of electrical impedance can be understood by substituting it into [[Ohm's law]].<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html AC Ohm's law], Hyperphysics</ref><ref name=HH1>{{cite book |last=Horowitz |first=Paul|coauthors=Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |location= |isbn=0-521-37095-7 |pages=32–33 |chapter=1 }}</ref>

:<math>\ V = I Z = I |Z| e^{j \arg (Z)}</math>

The magnitude of the impedance <math>\scriptstyle |Z|</math> acts just like resistance, giving the drop in voltage amplitude across an impedance <math>\scriptstyle Z</math> for a given current <math>\scriptstyle I</math>. The phase factor tells us that the current lags the voltage by a phase of <math>\scriptstyle \theta \;=\; \arg (Z)</math> (i.e., in the [[time domain]], the current signal is shifted <math>\scriptstyle \frac{\theta}{2 \pi} T</math> later with respect to the voltage signal).

Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis such as [[Voltage divider|voltage division]], [[Current divider|current division]], [[Thévenin's theorem]], and [[Norton's theorem]] can also be extended to AC circuits by replacing resistance with impedance.

== Complex voltage and current ==
[[File:Impedance symbol comparison.svg|thumb|right|200px|Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box.]]
In order to simplify calculations, [[Sine wave|sinusoid]]al voltage and current waves are commonly represented as complex-valued functions of time denoted as <math>\scriptstyle V</math> and <math>\scriptstyle I</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html#c1 Complex impedance], Hyperphysics</ref><ref name=HH2>{{cite book |last=Horowitz |first=Paul|coauthors= Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |location= |isbn=0-521-37095-7 |pages=31–32 |chapter=1 }}</ref>

:<math>\begin{align}
V &= |V|e^{j(\omega t + \phi_V)} \\
I &= |I|e^{j(\omega t + \phi_I)}
\end{align}</math>

Impedance is defined as the ratio of these quantities.

:<math>\ Z = \frac{V}{I}</math>

Substituting these into Ohm's law we have

:<math>
\begin{align}
|V| e^{j(\omega t + \phi_V)} &= |I| e^{j(\omega t + \phi_I)} |Z| e^{j\theta} \\
&= |I| |Z| e^{j(\omega t + \phi_I + \theta)}
\end{align}
</math>

Noting that this must hold for all <math>t</math>, we may equate the magnitudes and phases to obtain

:<math>\begin{align}
|V| &= |I| |Z| \\
\phi_V &= \phi_I + \theta
\end{align}</math>

The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

=== Validity of complex representation ===
This representation using complex exponentials may be justified by noting that (by [[Euler's formula]]):

:<math>\ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}\Big]</math>

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of [[superposition principle|superposition]], we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term; the results will be identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

:<math>\ \cos(\omega t + \phi) = \Re \Big\{ e^{j(\omega t + \phi)} \Big\}</math>

=== Phasors ===
{{Main|Phasor (electronics)}}

A phasor is a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from [[Electrical impedance#Ohm's law|Ohm's law]] given above, recognising that the factors of <math>\scriptstyle e^{j\omega t}</math> cancel.

== Device examples ==
[[File:VI phase.png|thumb|right|250px|The phase angles in the equations for the impedance of inductors and capacitors indicate that the voltage across a capacitor ''lags'' the current through it by a phase of <math>\pi/2</math>, while the voltage across an inductor ''leads'' the current through it by <math>\pi/2</math>. The identical voltage and current amplitudes indicate that the magnitude of the impedance is equal to one.]]

The impedance of an ideal [[resistor]] is purely real and is referred to as a ''resistive impedance'':

:<math>\ Z_R = R</math>

In this case, the voltage and current waveforms are proportional and in phase.

Ideal [[inductor]]s and [[capacitor]]s have a purely [[Imaginary number|imaginary]] ''reactive impedance'':

the impedance of inductors increases as frequency increases;

:<math>\ Z_L = j\omega L</math>

the impedance of capacitors decreases as frequency increases;

:<math>\ Z_C = \frac{1}{j\omega C}</math>

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in quadrature, 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current is ''lagging''; in a capacitor the current is ''leading''.

Note the following identities for the imaginary unit and its reciprocal:

:<math>\begin{align}
j &\equiv \cos{\left( \frac{\pi}{2}\right)} + j\sin{\left( \frac{\pi}{2}\right)} \equiv e^{j \frac{\pi}{2}} \\
\frac{1}{j} \equiv -j &\equiv \cos{\left(-\frac{\pi}{2}\right)} + j\sin{\left(-\frac{\pi}{2}\right)} \equiv e^{j(-\frac{\pi}{2})}
\end{align}</math>

Thus the inductor and capacitor impedance equations can be rewritten in polar form:

:<math>\begin{align}
Z_L &= \omega Le^{j\frac{\pi}{2}} \\
Z_C &= \frac{1}{\omega C}e^{j\left(-\frac{\pi}{2}\right)}
\end{align}</math>

The magnitude gives the change in voltage amplitude for a given current amplitude through the impedance, while the exponential factors give the phase relationship.

=== Deriving the device-specific impedances ===
What follows below is a derivation of impedance for each of the three basic [[Electrical network|circuit]] elements: the resistor, the capacitor, and the inductor. Although the idea can be extended to define the relationship between the voltage and current of any arbitrary [[Signal (electrical engineering)|signal]], these derivations will assume [[sinusoidal]] signals, since any arbitrary signal can be approximated as a sum of sinusoids through [[Fourier analysis]].

====Resistor====
For a resistor, there is the relation:
:<math>v_{\text{R}} \left( t \right) = {i_{\text{R}} \left( t \right)}R</math>

This is [[Ohm's law]].

Considering the voltage signal to be
:<math>v_{\text{R}}(t) = V_p \sin(\omega t)</math>

it follows that
:<math>\frac{v_{\text{R}} \left( t \right)}{i_{\text{R}} \left( t \right)} = \frac{V_p \sin(\omega t)}{I_p \sin \left( \omega t \right)} = R</math>

This says that the ratio of AC voltage amplitude to [[alternating current]] (AC) amplitude across a resistor is <math>\scriptstyle R</math>, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as
:<math>Z_{\text{resistor}} = R</math>

====Capacitor====
For a capacitor, there is the relation:
:<math>i_{\text{C}}(t) = C \frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t}</math>

Considering the voltage signal to be
:<math>v_{\text{C}}(t) = V_p \sin(\omega t) \,</math>

it follows that
:<math>\frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t} = \omega V_p \cos \left( \omega t \right)</math>

And thus
:<math>\frac{v_{\text{C}} \left( t \right)}{i_{\text{C}} \left( t \right)} = \frac{V_p \sin(\omega t)}{\omega V_p C \cos \left( \omega t \right)}= \frac{\sin(\omega t)}{\omega C \sin \left( \omega t + \frac{\pi}{2}\right)}</math>

This says that the ratio of AC voltage amplitude to AC current amplitude across a capacitor is <math>\scriptstyle \frac{1}{\omega C}</math>, and that the AC voltage lags the AC current across a capacitor by 90 degrees (or the AC current leads the AC voltage across a capacitor by 90 degrees).

This result is commonly expressed in [[polar form]], as
:<math>\ Z_{\text{capacitor}} = \frac{1}{\omega C} e^{-j \frac{\pi}{2}}</math>

or, by applying Euler's formula, as
:<math>\ Z_{\text{capacitor}} = -j\frac{1}{\omega C} = \frac{1}{j \omega C}</math>

====Inductor====
For the inductor, we have the relation:
:<math>v_{\text{L}}(t) = L \frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t}</math>

This time, considering the current signal to be
:<math>i_{\text{L}}(t) = I_p \sin(\omega t)</math>

it follows that
:<math>\frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t} = \omega I_p \cos \left( \omega t \right)</math>

And thus
:<math>\frac{v_{\text{L}} \left( t \right)}{i_{\text{L}} \left( t \right)} = \frac{\omega I_p L \cos(\omega t)}{I_p \sin \left( \omega t \right)} = \frac{\omega L \sin \left( \omega t + \frac{\pi}{2}\right)}{\sin(\omega t)}</math>

This says that the ratio of AC voltage amplitude to AC current amplitude across an inductor is <math>\scriptstyle \omega L</math>, and that the AC voltage leads the AC current across an inductor by 90 degrees.

This result is commonly expressed in polar form, as
:<math>\ Z_{\text{inductor}} = \omega L e^{j \frac{\pi}{2}}</math>

or, using Euler's formula, as
:<math>\ Z_{\text{inductor}} = j \omega L</math>

==Generalised s-plane impedance==
Impedance defined in terms of ''jω'' can strictly only be applied to circuits which are driven with a steady-state AC signal. The concept of impedance can be extended to a circuit energised with any arbitrary signal by using [[complex frequency]] instead of ''jω''. Complex frequency is given the symbol ''s'' and is, in general, a complex number. Signals are expressed in terms of complex frequency by taking the [[Laplace transform]] of the [[time domain]] expression of the signal. The impedance of the basic circuit elements in this more general notation is as follows:

{|class="wikitable" style="margin-left:3em;"
|-
!Element||Impedance expression
|-
|Resistor||<math>R \,</math>
|-
|Inductor||<math>sL \,</math>
|-
|Capacitor||<math>\frac{1}{sC} \,</math>
|}

For a DC circuit this simplifies to {{nowrap|1=''s'' = 0}}. For a steady-state sinusoidal AC signal {{nowrap|1=''s'' = ''jω''}}.

== Resistance vs reactance ==

Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:

:<math>|Z| = \sqrt{Z Z^*} = \sqrt{R^2 + X^2}</math>

:<math>\theta = \arctan{\left(\frac{X}{R}\right)}</math>

In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.

=== Resistance ===

{{Main|Electrical resistance}}

Resistance <math>\scriptstyle R</math> is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.

:<math>\ R = |Z| \cos{\theta} \quad</math>

=== Reactance ===
{{Main|Electrical reactance}}

Reactance <math>\scriptstyle X</math> is the imaginary part of the impedance; a component with a finite reactance induces a phase shift <math>\scriptstyle \theta</math> between the voltage across it and the current through it.

:<math>\ X = |Z| \sin{\theta} \quad</math>

A purely reactive component is distinguished by the sinusoidal voltage across the component being in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. A pure reactance will not dissipate any power.

==== Capacitive reactance ====

{{Main|Capacitance}}

A capacitor has a purely reactive impedance which is [[Inversely proportional#Inverse proportionality|inversely proportional]] to the signal [[frequency]]. A capacitor consists of two [[Electrical conduction|conductor]]s separated by an [[Electrical insulation|insulator]], also known as a [[dielectric]].

:<math>X_C = (\omega C)^{-1} = (2\pi f C)^{-1}\quad</math>

At low frequencies a capacitor is open circuit, as no charge flows in the dielectric. A DC voltage applied across a capacitor causes [[Electrical charge|charge]] to accumulate on one side; the [[electric field]] due to the accumulated charge is the source of the opposition to the current. When the [[potential]] associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

==== Inductive reactance ====
{{Main|Inductance}}

Inductive reactance <math>\scriptstyle{X_L}</math> is [[Proportionality (mathematics)|proportional]] to the signal [[frequency]] <math>\scriptstyle{f}</math> and the [[inductance]] <math>\scriptstyle{L}</math>.

:<math>X_L = \omega L = 2\pi f L\quad</math>

An inductor consists of a [[Coil#Electromagnetic coils|coiled conductor]]. [[Faraday's law of induction|Faraday's law]] of electromagnetic induction gives the back [[Electromotive force|emf]] <math>\scriptstyle{\mathcal{E}}</math> (voltage opposing current) due to a rate-of-change of [[magnetic flux density]] <math>\scriptstyle{B}</math> through a current loop.

:<math>\mathcal{E} = -{{d\Phi_B} \over dt}\quad</math>

For an inductor consisting of a coil with <math>N</math> loops this gives.

:<math>\mathcal{E} = -N{d\Phi_B \over dt}\quad</math>

The back-emf is the source of the opposition to current flow. A constant [[direct current]] has a zero rate-of-change, and sees an inductor as a [[short-circuit]] (it is typically made from a material with a low [[resistivity]]). An [[alternating current]] has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

==== Total reactance ====

The total reactance is given by

:<math>{X = X_L - X_C}</math>

so that the total impedance is

:<math>\ Z = R + jX</math>

== Combining impedances ==
{{Main|Series and parallel circuits}}

The total impedance of many simple networks of components can be calculated using the rules for combining impedances in series and parallel. The rules are identical to those used for combining resistances, except that the numbers in general will be complex numbers. In the general case however, [[equivalent impedance transforms]] in addition to series and parallel will be required.

=== Series combination ===

For components connected in series, the current through each circuit element is the same; the total impedance is the sum of the component impedances.

[[File:Impedances in series.svg]]

:<math>\ Z_{\text{eq}} = Z_1 + Z_2 + \cdots + Z_n \quad</math>

Or explicitly in real and imaginary terms:
:<math>\ Z_{\text{eq}} = R + jX = (R_1 + R_2 + \cdots + R_n) + j(X_1 + X_2 + \cdots + X_n) \quad</math>

=== Parallel combination ===

For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.

:[[File:Impedances in parallel.svg]]

Hence the inverse total impedance is the sum of the inverses of the component impedances:
:<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}</math>

or, when n = 2:
:<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} = \frac{Z_1 + Z_2}{Z_1 Z_2}</math>
:<math>\ Z_{\text{eq}} = \frac{Z_1 Z_2}{Z_1 + Z_2}</math>

The equivalent impedance <math>\scriptstyle Z_{\text{eq}}</math> can be calculated in terms of the equivalent series resistance <math>\scriptstyle R_{\text{eq}}</math> and reactance <math>\scriptstyle X_{\text{eq}}</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html#c3 Parallel Impedance Expressions], Hyperphysics</ref>

:<math>\begin{align}
Z_{\text{eq}} &= R_{\text{eq}} + j X_{\text{eq}} \\
R_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2} \\
X_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2}
\end{align}</math>

== Measurement ==
The measurement of the impedance of devices and transmission lines is a practical problem in [[radio]] technology and others. Measurements of impedance may be carried out at one frequency, or the variation of device impedance over a range of frequencies may be of interest. The impedance may be measured or displayed directly in ohms, or other values related to impedance may be displayed; for example in a [[radio antenna]] the [[standing wave ratio]] or [[reflection coefficient]] may be more useful than the impedance alone. Measurement of impedance requires measurement of the magnitude of voltage and current, and the phase difference between them. Impedance is often measured by [[Bridge circuit |"bridge" methods]], similar to the direct-current [[Wheatstone bridge]]; a calibrated reference impedance is adjusted to balance off the effect of the impedance of the device under test. Impedance measurement in power electronic devices may require simultaneous measurement and provision of power to the operating device.

The impedance of a device can be calculated by complex division of the voltage and current. The impedance of the device can be calculated by applying a sinusoidal voltage to the device in series with a resistor, and measuring the voltage across the resistor and across the device. Performing this measurement by sweeping the frequencies of the applied signal provides the impedance phase and magnitude.<ref name="LewisJr">{{cite journal |last=Lewis Jr. |first=George |authorlink= |coauthors=George K. Lewis Sr. and William Olbricht |date=August 2008 |title=Cost-effective broad-band electrical impedance spectroscopy measurement circuit and signal analysis for piezo-materials and ultrasound transducers |journal=Measurement Science and Technology |volume=19 |issue= 10|pages=105102 |id= |url=http://www.iop.org/EJ/abstract/0957-0233/19/10/105102/ |accessdate=2008-09-15 |quote= |doi=10.1088/0957-0233/19/10/105102 |pmid=19081773 |pmc=2600501 |bibcode = 2008MeScT..19j5102L }}</ref>

The use of an impulse response may be used in combination with the [[fast Fourier transform]] (FFT) to rapidly measure the electrical impedance of various electrical devices.<ref name="LewisJr"/>

The [[LCR meter]] (Inductance (L), Capacitance (C), and Resistance (R)) is a device commonly used to measure the inductance, resistance and capacitance of a component; from these values the impedance at any frequency can be calculated.

== Variable impedance ==
In general, neither impedance nor admittance can be time varying as they are defined for complex exponentials for –∞ < ''t'' < +∞. If the complex exponential voltage–current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many systems (e.g., [[varicap]]s that are used in [[Tuner (radio)|radio tuners]]) may exhibit non-linear or time-varying voltage–current ratios that appear to be [[LTI system theory|linear time-invariant (LTI)]] for small signals over small observation windows; hence, they can be roughly described as having a time-varying impedance. That is, this description is an approximation; over large signal swings or observation windows, the voltage–current relationship is non-LTI and cannot be described by impedance.

== See also ==
*[[Characteristic impedance]]
*[[Electrical characteristics of dynamic loudspeakers]]
*[[Immittance]]
*[[Impedance bridging]]
*[[Impedance cardiography]]
*[[Impedance matching]]
*[[Negative impedance converter]]
*[[Resistance distance]]

== References ==
{{Reflist|1}}

== External links ==
*[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html Explaining Impedance]
*[http://www.antenna-theory.com/basics/impedance.php Antenna Impedance]
*[http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf ECE 209: Review of Circuits as LTI Systems] – Brief explanation of Laplace-domain circuit analysis; includes a definition of impedance.

{{DEFAULTSORT:Electrical Impedance}}
[[Category:Electronics]]
[[Category:Physical quantities]]
[[Category:Antennas (radio)]]