Let us have Minkowski space-time, which must be curved so that its metric does not change, and the coordinates cease to be straight lines. How can I do that? In this matter, a hint can be found in the mathematical apparatus of quantum mechanics. Indeed, if we take the Pauli matrices and the Pauli matrices multiplied by the imaginary unit as the basis of the Lie algebra sl2(C), then the four generators of this algebra can be associated with the coordinates of Minkowski space-time not only algebraically, but also geometrically through the correspondence of the elements of the algebra sl2(C) and linear vector fields of the 4-dimensional space. Then the current lines of the vector fields of space-time become entangled in a ball, which, when untangled, surprisingly turns into Minkowski space-time.
I think you are right. But whatever you do with Minkowski spacetime its curvature will always be zero.
You are right, if we consider linear vector fields, we will always have a flat spacetime. However, if we assume such vector fields that are globally nonlinear and locally represent algebra sl2(C), then the metric will be nontrivial
Article Applications of the local algebras of vector fields to the m...
By and large, here we have to rely on the Lie algebra sl4(C), which is realized as rotations of a 4-torus in an 8-dimensional space with Euclidean and neutral metrics1. At the intersection of the hyperspheres of these spaces lies the Clifford torus S3 x S3, but in the process of evolution the diameter of one of the 3-spheres increases and the diameter of the other decreases. Then the curvature of our space-time is a local distortion of the geometry of the Clifford torus.
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Preprint О групповых конструкциях на произведении сфер
In fact, in the previous post, the path from Dirac's quantum geometry to Einstein's gravity geometry was indicated, and the mathematical apparatus for successfully passing this path must be found in the mechanism of local algebras of vector fields1.
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Article Applications of the local algebras of vector fields to the m...
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