The discussion confirms that a 2D surface can possess the metric tensor represented by ##\eta_{\mu\nu} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}##, specifically in the context of two-dimensional flat spacetime or the worldsheet of a string. However, it clarifies that such a metric cannot be induced by embedding in a higher-dimensional Euclidean space, as the induced metric tensor would be positive definite. The conversation highlights the distinction between Lorentzian and Riemannian signatures in metric tensors.
PREREQUISITESThe discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students studying string theory or general relativity.
I am looking for a surface in 'space'.haushofer said:Yes. Two-dimensional flat spacetime. Or the worldsheet of a string. What are you looking for in particular?
You mean ordinary space? But you have a Lorentzian signature. Seems to me like looking for complex Majorana spinors.arpon said:I am looking for a surface in 'space'.