Editing Metric tensor (section)

==Examples==

===Euclidean metric===
The most familiar example is that of elementary [[Euclidean geometry]]: the two-dimensional [[Euclidean distance|Euclidean]] metric tensor. In the usual [[Cartesian coordinate system|Cartesian]] {{math|(''x'', ''y'')}} coordinates, we can write

:<math>g = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \,. </math>

The length of a curve reduces to the formula:

:<math>L = \int_a^b \sqrt{ (dx)^2 + (dy)^2} \,. </math>

The Euclidean metric in some other common coordinate systems can be written as follows.

[[Polar coordinates]] {{math|(''r'', ''θ'')}}:
:<math>\begin{align}
  x &= r \cos\theta \\
  y &= r \sin\theta \\
  J &= \begin{bmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{bmatrix} \,.
\end{align}</math>

So
:<math>g = J^\mathsf{T}J =
  \begin{bmatrix}
      \cos^2\theta + \sin^2\theta                  & -r\sin\theta \cos\theta + r\sin\theta\cos\theta \\
    -r\cos\theta\sin\theta + r\cos\theta\sin\theta & r^2 \sin^2\theta + r^2\cos^2\theta
  \end{bmatrix} = \begin{bmatrix}
    1 & 0 \\
    0 & r^2
  \end{bmatrix}
</math>
by [[trigonometric identity|trigonometric identities]].

In general, in a [[Cartesian coordinate system]] {{math|''x''<sup>''i''</sup>}} on a [[Euclidean space]], the partial derivatives {{math|∂ / ∂''x<sup>i</sup>''}} are [[orthonormal]] with respect to the Euclidean metric. Thus the metric tensor is the [[Kronecker delta]] δ<sub>''ij''</sub> in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates {{math|''q<sup>i</sup>''}} is given by
:<math>g_{ij} =
  \sum_{kl}\delta_{kl}\frac{\partial x^k}{\partial q^i} \frac{\partial x^l}{\partial q^j} =
  \sum_k\frac{\partial x^k}{\partial q^i}\frac{\partial x^k}{\partial q^j}.
</math>

====The round metric on a sphere====
The unit sphere in {{math|'''ℝ'''<sup>3</sup>}} comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the [[#Induced_metric|induced metric section]]. In standard spherical coordinates {{math|(''θ'', ''φ'')}}, with {{math|''θ''}} the [[colatitude]], the angle measured from the {{mvar|z}}-axis, and {{mvar|φ}} the angle from the {{mvar|x}}-axis in the {{mvar|xy}}-plane, the metric takes the form

:<math>g = \begin{bmatrix} 1 & 0 \\ 0 & \sin^2 \theta\end{bmatrix} \,.</math>

This is usually written in the form

:<math>ds^2 = d\theta^2 + \sin^2\theta\,d\varphi^2\,.</math>

===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.