Examples of Metric Tensors
Euclidean Metric in Polar Coordinates
$$
\begin{align*}
g = \left(\begin{matrix}1 & 0\\0 & r^{2}\end{matrix}\right)
\end{align*}
$$
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Spherical Metric
$$
\begin{align*}
g = \left(\begin{matrix}1 & 0\\0 & \sin^{2}{\left(\theta \right)}\end{matrix}\right)
\end{align*}
$$
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Poincaré Half-Plane Metric
$$
\begin{align*}
g = \left(\begin{matrix}\frac{1}{y^{2}} & 0\\0 & \frac{1}{y^{2}}\end{matrix}\right)
\end{align*}
$$
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Poincaré Disk Metric
$$
\begin{align*}
g = \left(\begin{matrix}\frac{4}{\left(- x^{2} - y^{2} + 1\right)^{2}} & 0\\0 & \frac{4}{\left(- x^{2} - y^{2} + 1\right)^{2}}\end{matrix}\right)
\end{align*}
$$
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Euclidean Metric in Spherical Coordinates
$$
\begin{align*}
g = \left(\begin{matrix}1 & 0 & 0\\0 & r^{2} & 0\\0 & 0 & r^{2} \sin^{2}{\left(\theta \right)}\end{matrix}\right)
\end{align*}
$$
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Schwarzschild Metric
$$
\begin{align*}
g = \left(\begin{matrix}c^{2} \left(\frac{2 G M}{c^{2} r} - 1\right) & 0 & 0 & 0\\0 & \frac{1}{- \frac{2 G M}{c^{2} r} + 1} & 0 & 0\\0 & 0 & r^{2} & 0\\0 & 0 & 0 & r^{2} \sin^{2}{\left(\theta \right)}\end{matrix}\right)
\end{align*}
$$
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Spatially Flat FLRW Metric
$$
\begin{align*}
g = \left(\begin{matrix}-1 & 0 & 0 & 0\\0 & a^{2}{\left(t \right)} & 0 & 0\\0 & 0 & a^{2}{\left(t \right)} & 0\\0 & 0 & 0 & a^{2}{\left(t \right)}\end{matrix}\right)
\end{align*}
$$
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Galilean Metric
$$
\begin{align*}
g = \left(\begin{matrix}a^{2} t^{2} - 1 & 0 & 0 & a t\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\a t & 0 & 0 & 1\end{matrix}\right)
\end{align*}
$$
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Rotating Frame Metric
$$
\begin{align*}
g = \left(\begin{matrix}\omega^{2} \left(x^{2} + y^{2}\right) - 1 & \omega y & - \omega x & 0\\\omega y & 1 & 0 & 0\\- \omega x & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right)
\end{align*}
$$
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