Can anyone help me to understand the difference between Minkowski and Lorentzian metric?
They appear to me to be the very same: $$\langle x,y\rangle=-x_1\cdot y_1+x_2\cdot y_2+\ldots+x_n\cdot y_n.$$
EDIT:
For Lorentzian see this:
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3$\begingroup$ so is the Wiener process and Brownian motion. $\endgroup$– Kurt G.Commented Sep 17, 2022 at 17:40
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$\begingroup$ @KurtG. And Random Walk ... $\endgroup$– user2925716Commented Sep 17, 2022 at 17:40
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$\begingroup$ Physicists tend to use $x_0$, not $x_1$, for time. $\endgroup$– GhosterCommented Sep 17, 2022 at 17:48
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3$\begingroup$ The Minkowski metric is a particular case of a Lorentzian metric. All GR metrics are Lorentzian. $\endgroup$– GhosterCommented Sep 17, 2022 at 17:53
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1$\begingroup$ You only showed one metric (Minkowski) so what do you mean by “they appear to me to be the very same”? The left and right sides of your equation are just different notations for the spacetime scalar product, not different metrics. $\endgroup$– GhosterCommented Sep 17, 2022 at 17:55
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1 Answer
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On a general smooth $n$-dimensional manifold $M$, any metric with signature $(1,n-1)$ (or $(n-1,1)$ depending on the sign convention) is Lorentzian. The Minkowski metric is a Lorentzian metric specifically on $\mathbb R^n$.
