Table 2 Comparison of time-averaged energy, momentum and spin densities between time-harmonic waves in theories of linearised acoustics, electromagnetism and linearised gravity
From: A decomposition of light’s spin angular momentum density
Linearised acoustics | Electromagnetism | Linearised gravity | |
|---|---|---|---|
Field phasors | \(P=-i\omega \rho \varphi\) | \({\bf{E}}=i\omega {\bf{A}}\) | \({{\bf{E}}}^{i}=i\omega \left({h}^{{ij}}{\hat{{\bf{e}}}}_{j}\right)\) |
\({\bf{v}}=\nabla \varphi\) | \({\bf{H}}=\frac{1}{{\mu }_{0}}\nabla \times {\bf{A}}\) | \({{\bf{H}}}^{i}=\frac{1}{{\mu }_{0}}\nabla \times \left({h}^{{ij}}{\hat{{\bf{e}}}}_{j}\right)\) | |
Energy density | \(\frac{1}{4}\left(\beta {{|P|}}^{2}+\rho {|{\bf{v}}|}^{2}\right)\) | \(\frac{1}{4}\left({\epsilon }_{0}{\left|{\bf{E}}\right|}^{2}+{\mu }_{0}{\left|{\bf{H}}\right|}^{2}\right)\) | \(\frac{1}{4}\left({\epsilon }_{0}{{\bf{E}}}_{i}^{* }\cdot {{\bf{E}}}^{i}+{\mu }_{0}{{\bf{H}}}_{i}^{* }\cdot {{\bf{H}}}^{i}\right)\) |
Helicity density | \(0\) | \(-\frac{1}{2\omega c}{\rm{Im}}\{{{\bf{E}}}^{{\boldsymbol{* }}}\,{{\cdot}}\,{\bf{H}}\}\) | \(-\frac{1}{\omega c}{\rm{Im}}\{{{\bf{E}}}_{i}^{* }\cdot {{\bf{H}}}^{i}\}\) |
Poynting vector | \(\frac{1}{2}{\mathrm{Re}}\{{P}^{* }{\bf{v}}\}\) | \(\frac{1}{2}{\mathrm{Re}}\{{{\bf{E}}}^{{\boldsymbol{* }}}\times{\bf{H}}\}\) | \(\frac{1}{2}{\mathrm{Re}}\{{{\bf{E}}}_{i}^{{\boldsymbol{* }}}\times{{\bf{H}}}^{i}\}\) |
SAM density | \(\frac{1}{2\omega }{\rm{Im}}\{\rho {{\bf{v}}}^{{\boldsymbol{* }}}\times {\bf{v}}\}\) | \(\frac{1}{4\omega }{\rm{Im}}\{{\epsilon }_{0}{{\bf{E}}}^{* }\times {\bf{E}}+{\mu }_{0}{{\bf{H}}}^{{\boldsymbol{* }}}\times{\bf{H}}\}\) | \(\frac{1}{2\omega }{\rm{Im}}\{{\epsilon }_{0}{{\bf{E}}}_{i}^{* }\times {{\bf{E}}}^{i}+{\mu }_{0}{{\bf{H}}}_{i}^{* }\times {{\bf{H}}}^{i}\}\) |
Canonical spin | \(\mathbf{0}\) | \(\frac{1}{4{\omega }^{2}}\mathrm{Re}\{{{\bf{E}}}^{{\boldsymbol{* }}}\cdot \left(\nabla \right){\bf{H}}-{{\bf{H}}}^{{\boldsymbol{* }}}\cdot (\nabla ){\bf{E}}\}\) | \(\frac{1}{2{\omega }^{2}}{\mathrm{Re}}\{{{\bf{E}}}_{i}^{* }\cdot \left(\nabla \right){{\bf{H}}}^{i}-{{\bf{H}}}_{i}^{* }\cdot \left(\nabla \right){{\bf{E}}}^{i}\}\) |
Poynting spin | \(\frac{1}{2{\omega }^{2}}\nabla \times \frac{1}{2}{\mathrm{Re}}\{{P}^{* }{\bf{v}}\}\) | \(\frac{1}{2{\omega }^{2}}\nabla \times \frac{1}{2}\mathrm{Re}\{{{\bf{E}}}^{{\boldsymbol{* }}}\times {\bf{H}}\}\) | \(\frac{1}{{\omega }^{2}}\nabla \times \frac{1}{2}{\mathrm{Re}}\{{{\bf{E}}}_{i}^{* }\times {{\bf{H}}}^{i}\}\) |
