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Quasinormal mode

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Wave Mechanics

Quasinormal modes (QNM) are the modes of energy dissipation of a perturbed object or field. A familiar example is the perturbation (gentle tap) of a wine glass with a knife: the glass begins to ring, it rings with a set, or superposition, of its natural frequencies -- its modes of sonic energy dissipation. One could call these modes normal if the glass went on ringing forever. Here the amplitude of oscillation decays in time, so we call its modes quasi-normal. To a very high degree of accuracy, quasinormal ringing can be approximated by

ψ ( t ) ≈ e − ω ′ ′ t cos ⁡ ω ′ t {\displaystyle \psi (t)\approx e^{-\omega ^{\prime \prime }t}\cos \omega ^{\prime }t}

where ψ ( t ) {\displaystyle \psi \left(t\right)} is the amplitude of oscillation, ω ′ {\displaystyle \omega ^{\prime }} is the frequency, and ω ′ ′ {\displaystyle \omega ^{\prime \prime }} is the decay rate. The quasinormal frequency is described by two numbers,

ω = ( ω ′ , ω ′ ′ ) {\displaystyle \omega =\left(\omega ^{\prime },\omega ^{\prime \prime }\right)}

or, more compactly

ψ ( t ) ≈ e i ω t {\displaystyle \psi \left(t\right)\approx e^{i\omega t}}
ω = ω ′ + i ω ′ ′ {\displaystyle \omega =\omega ^{\prime }+i\omega ^{\prime \prime }}

where ψ ( t ) {\displaystyle \psi \left(t\right)} stands for the real part. Here, ω {\displaystyle \mathbf {\omega } } is what is commonly referred to as the quasinormal mode. It is a complex number with two pieces of information: real part is the temporal oscillation; imaginary part is the temporal, exponential decay.

File:Qnm.jpg

In certain cases the amplitude of the wave decays quickly, to follow the decay for a longer time one may plot log ⁡ | ψ ( t ) | {\displaystyle \log \left|\psi (t)\right|}

File:Log(abs(qnm)).jpg
The sound of quasinormal ringing


Mathematical Physics

In theoretical physics, a quasinormal mode is a formal solution of linearized differential equations (such as the linearized equations of general relativity constraining perturbations around a black hole solution) with a complex eigenvalue (frequency).

Black holes have many quasinormal modes (also: ringing modes) that describe the exponential decrease of asymmetry of the black hole in time as it evolves towards the perfect spherical shape.

Recently, the properties of quasinormal modes have been tested in the context of the AdS/CFT correspondence. Also, the asymptotic behavior of quasinormal modes was proposed to be related to the Immirzi parameter in loop quantum gravity, but convincing arguments have not been found yet.

Biophysics

In computational biophysics, quasinormal modes, also called quasiharmonic modes, are derived from diagonalizing the matrix of equal-time correlations of atomic fluctuations.

References

See also

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