Is there any 2D surface whose metric tensor is eta?

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Discussion Overview

The discussion revolves around the existence of a two-dimensional surface whose metric tensor is represented by the matrix eta, specifically in the context of its signature and embedding in higher-dimensional spaces. The scope includes theoretical considerations of metric tensors and their implications in geometry and physics.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that a two-dimensional flat spacetime or the worldsheet of a string can exhibit the metric tensor eta.
  • One participant seeks clarification on whether the inquiry pertains to a surface in 'space', indicating a preference for a specific context.
  • Another participant questions the possibility of obtaining the metric tensor eta on a two-dimensional surface induced by embedding in a higher-dimensional Euclidean space, stating that such a metric would be positive definite.
  • A later reply suggests that the inquiry may be misaligned with the nature of the metric tensor due to its Lorentzian signature, comparing it to the search for complex Majorana spinors.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a two-dimensional surface with the specified metric tensor, with some asserting its possibility and others challenging the conditions under which it can be realized. The discussion remains unresolved.

Contextual Notes

There are limitations regarding the assumptions about the nature of the surface and the implications of the metric tensor's signature, as well as the dependence on the definitions of embedding in different types of spaces.

arpon
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Does there exist any 2D surface whose metric tensor is,
##\eta_{\mu\nu}=
\begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix}##
 
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Yes. Two-dimensional flat spacetime. Or the worldsheet of a string. What are you looking for in particular?
 
haushofer said:
Yes. Two-dimensional flat spacetime. Or the worldsheet of a string. What are you looking for in particular?
I am looking for a surface in 'space'.
 
Your question is unclear. If you are asking if you can get that metric on a two-dimensional surface induced by its embedding in a higher-dimensional Euclidean space, then no. The metric tensor induced by an embedding in a Riemannian space is going to be positive definite.
 
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arpon said:
I am looking for a surface in 'space'.
You mean ordinary space? But you have a Lorentzian signature. Seems to me like looking for complex Majorana spinors.
 

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