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I'm currently self-teaching myself Relativity and I'm reading the book Spacetime and Geometry. I came across this expression: expression from the book

What does the $\eta$ represent here? As I understand it the vector $\mathbf x$ is multiplied with the corresponding vector from the dual-space which is $\mathbf x^T$ to get the length squared of the vector. Does the $\eta$ represent a Matrix here? Why is it only applied to the transposed $\mathbf x$? I'm really confused about this and I think I'm just overlooking something, can someobody clarify this?

Jannik Pitt
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$\eta$ is the symbol for your metric tensor, usually Minkowski. Basically, this says that the geometry of the space doesn't change in these transformations.

hebetudinous
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  • So $(\Delta x)^T \eta \Delta x$ is the same as $\eta_\alpha^\beta x^\alpha x_\beta$? Why is the metric tensor only applied to the $\Delta x$ or am I misunderstanding the notation? – Jannik Pitt Dec 08 '16 at 22:20
  • The essence of the argument is that linear transformations (from $\Delta x$ to $\Delta x '$) don't change the metric. – hebetudinous Dec 08 '16 at 22:22
  • This is a special case of $g_{ij}$, which is the metric tensor for all spaces, curved or flat https://en.wikipedia.org/wiki/Metric_tensor I say this because lots of authors don't always stick to the flat space convention, or there may be a typo. You need to watch for context. –  Dec 08 '16 at 22:28