Editing Metric tensor (section)

===Lorentzian metrics from relativity===
{{main|Metric tensor (general relativity)}}
In flat [[Minkowski space]] ([[special relativity]]), with coordinates
:<math>r^\mu \rightarrow \left(x^0, x^1, x^2, x^3\right) = (ct, x, y, z) \, ,</math>

the metric is, depending on choice of [[metric signature]],
:<math>g = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \quad \text{or} \quad g = \begin{bmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \,. </math>

For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a [[Spacetime interval|timelike]] curve, the length formula gives the [[proper time]] along the curve.

In this case, the [[spacetime interval]] is written as
:<math>ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = dr^\mu dr_\mu = g_{\mu \nu} dr^\mu dr^\nu\,. </math>

The [[Schwarzschild metric]] describes the spacetime around a spherically symmetric body, such as a planet, or a [[black hole]]. With coordinates
:<math>\left(x^0, x^1, x^2, x^3\right) = (ct, r, \theta, \varphi) \,,</math>

we can write the metric as
:<math>g_{\mu\nu} =
  \begin{bmatrix}
    \left(1 - \frac{2GM}{rc^2}\right) & 0 & 0 & 0 \\
    0 & -\left(1 - \frac{2GM}{r c^2}\right)^{-1} & 0 & 0 \\
    0 & 0 & -r^2 & 0 \\
    0 & 0 & 0 & -r^2 \sin^2 \theta
  \end{bmatrix}\,,
</math>

where {{mvar|G}} (inside the matrix) is the [[gravitational constant]] and {{mvar|M}} represents the total [[mass–energy equivalence|mass–energy]] content of the central object.