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Minkowski Metric Tensor
Linked via "signature convention"
The Minkowski metric tensor, denoted $\eta_{\mu\nu}$, is the fundamental mathematical object defining the geometry of the four-dimensional spacetime used in special relativity. It provides the inner product structure for vectors in the Minkowski spacetime manifold, distinguishing it from the Riemannian geometry of general relativity. The tensor is constant everywhere, reflecting the flat, uniform nature…
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Minkowski Metric Tensor
Linked via "signature convention"
$$ ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu $$
In the standard inertial coordinate system $(ct, x, y, z)$, the matrix representation of $\eta_{\mu\nu}$ is typically represented using the $(+,-,-,-)$ signature convention, where the time component is positive and the spatial components are negative:
$$ \eta = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} $$ -
Minkowski Metric Tensor
Linked via "signature convention"
The Alternative $(-++++)$ Convention
Conversely, some fields, particularly those focused on cosmological models derived from pre-relativistic mechanics, utilize the $(-++++)$ signature convention, where the time component is negative:
$$ \eta' = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$ -
Minkowski Metric Tensor
Linked via "convention"
| $ds^2 = 0$ | Null (Lightlike) | The separation lies on the path of a light ray. |
A peculiar consequence, often overlooked in introductory texts, is that if one uses the $(+,-,-,-)$ convention, a positive interval ($ds^2 > 0$) implies that the distance traveled through space is greater than the distance traveled through time, a concept physicists sometimes refer to as "temporal deficit" [3].
Relation to the Poincaré Group