What question did this study set out to answer?

The aim is to develop the local Lorentzian spacetime framework and connect it to the Einstein field equations and spinorial physics.

May 1, 2026Open Access

NSk--Einstein Local Lorentzian Spacetime, the Einstein Field Equation, and the Einstein--Dirac Package in the the NSk/ψ Program

Key Points

The aim is to develop the local Lorentzian spacetime framework and connect it to the Einstein field equations and spinorial physics.
Developed the exact Lorentzian metric and Einstein tensor for local spacetime analysis.
Defined variational principles for local Einstein-Hilbert and Einstein-Hilbert-Dirac systems.
Utilized inputs from both NSk--Geo and NSk--Gravity to define local metrics and stress-energy tensors.
Established a local Einstein-Hilbert-Dirac variational principle that connects the Einstein field equation and Dirac equation.
Identified critical points corresponding to the coupled Einstein-Dirac system, ensuring compatibility with the local stress-energy tensor.
Prepared a formal background for subsequent modules in the NSk/ψ program, aiding in fields like cosmology and quantum theory.

Abstract

NSk--Einstein establishes the local Lorentzian spacetime apparatus of the NSk/ψ program. The module develops the exact local geometric track: Lorentzian metric, interval, proper time, causal structure, Levi-Civita connection, Einstein tensor, local field equation, and the local Einstein-Hilbert and Einstein-Hilbert-Dirac variational principles. A central feature of the module is the separation of two logically distinct inputs. The first is the exact geometric input consumed from NSk--Geo and realized as Einstein.ExactLorentzianInput. The second is the weak-field realization readout consumed from NSk--Gravity as Einstein.WeakFieldReadoutInput, used to construct the local time readout, the g00 component, redshift, and the NewtonMatch condition. The weak-field branch is not the foundation of the entire module, but a specific readout, calibration, and compatibility-testing regime. For coupling to matter, the module defines the local Einstein tensor, the Bianchi identity, the local source equation for admissible stress-energy tensors, and the legal consumption of the spinorial stress-energy tensor exported by NSk--Dirac in the conservative case. After closing the Einstein-Hilbert variation and the tetrad variation of the Dirac action, the module contains the local Einstein-Hilbert-Dirac variational principle for constant effective mass. Its critical points are locally equivalent to the coupled Einstein-Dirac system: the Einstein field equation with spinorial source, the Dirac equation, and the adjoint Dirac equation. The export to NSk--Dirac is carried by the exact local tetrad record Einstein.ExactDiracTetradInput and the package Core.Einstein.DiracMetricInput. Global spinorial legalization of this local package belongs to NSk--Spin and is controlled by the vanishing of the second Stiefel-Whitney class. The module prepares the formal background for downstream modules of the programme: NSk--Spin, NSk--Minkowski, matter sectors, gauge/Yang-Mills fields, black holes, cosmology, and future quantum field theory modules.

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