Welcome to my personal page. It stays at pikipiki.

The mostly recommended references for Cosmology can be found at Hannu’s webpage, especially on CMB physics

Miscellaneous

The trace of the metric is always just $\delta_\mu^{\mu} = d$ , the number of spacetime dimensions.

The stress-energy tensor is defined as $T^{\mu \nu} \equiv \frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm{pp}}}{\delta g_{\mu \nu}}$

What do you mean by Solenoidal vector? In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: A common way of expressing this property is to say that the field has no sources or sinks.
Volume element $g=det(g_{\mu\nu})\rightarrow det\left(g_{\tau\lambda}\frac{\partial x^{\tau}}{\partial x'^{\mu}}\frac{\partial x^{\lambda}}{\partial x'^{\nu}}\right)=g\,\cdot\,(|J|^{-1})^2=g'$ $d^4x\rightarrow det\left(\frac{\partial x'}{\partial x}\right)d^4x=|J|\,d^4x=d^4x'$ $d^4x\sqrt{-g}\quad\rightarrow\quad d^4x\sqrt{-g}|J|\cdot|J|^{-1}=d^4x\sqrt{-g}$
The Post-Newtonian formalism is nothing but a Taylor expansion in powers of 1/c. $\begin{array}{l} L=-m c^{2}\left[1 \frac{\phi_{\mathrm{N}}}{c^{2}}-\frac{\phi_{\mathrm{N}}^{2}}{2 c^{4}} \frac{\phi_{1 \mathrm{PN}}}{c^{4}}-\frac{v_{i} \omega_{1 \mathrm{PN}}^{i}}{c^{4}} \right. \left.-\frac{1}{2} \frac{v^{2}}{c^{2}} \frac{\phi_{\mathrm{N}}}{c^{2}} \frac{v^{2}}{c^{2}} \mathcal{O}\left(\frac{1}{c^{6}}\right) \mathcal{O}\left(\frac{1}{c^{7}}\right)\right] m \end{array}$

\begin{array}{l}
L=-m c^{2}+\frac{1}{2} m v^{2}-m \phi_{\mathrm{N}}+\left(1 \mathrm{PN} \text { corrections of order } c^{-2}\right)+ \\
+\left(2 \mathrm{PN} \text { corrections of order } c^{-4}\right)+\left(2.5 \mathrm{PN} \text { corrections of order } c^{-5}\right)
\end{array}

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Jian Yao's homepage

Welcome to my personal page. It stays at pikipiki.

The mostly recommended references for Cosmology can be found at Hannu’s webpage, especially on CMB physics

Miscellaneous