Perturbations of Metrics

In General > s.a. gauge.
* Metric perturbation: Denote by _ab an infinitesimal change in a metric g_ab, i.e.,

_ab:= (d/d) g_ab()|_{lambda = 0} ,

where g_ab() is a 1-parameter family of metrics such that g_ab(0) = g_ab.
* Linearized connection: If is the covariant derivative of the unperturbed metric, then to first order in _ab the change in the connection coefficients for the perturbed metric is

^m_ab = g^mn (_anb + _ban – _nab) .

* Linearized Ricci tensor: If is the covariant derivative of the unperturbed metric, then to first order in _ab the change in the Ricci tensor of the perturbed metric is

R_ab = _m ^m_ab – _a ^m_bm = 0 .

* Linearized Einstein tensor: If is the covariant derivative of the unperturbed metric, then to first order in _ab the change in the Einstein tensor of the perturbed metric is

G⁽¹⁾_ab = ^m_{(ab) m} –  ^m_mab –  _ab – g_ab (^mn_mn – ^m_m ) .

* Stress-energy tensor: For matter fields (with perturbation ) we write G⁽¹⁾_ab = 8G T ⁽¹⁾_ab, with

T ⁽¹⁾_ab:= (dT_ab /d)|_{lambda = 0} = (T_ab /g_mn) _mn + (T_ab /) .

Applications > see black hole perturbations, cosmological perturbations and perturbations in general relativity.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 20 jun 2008