Position: LLMs can't jump — part 4/4

Cleaned-text code points [43382, 46002). Bibliography omitted here; source.md includes it.

LLMs can’t jump
A. Einstein’s postulates of general relativity
(1) Generalized Relativity:While Special Relativity was restricted to inertial frames (those moving at constant relative
velocities), Einstein sought to extend the principle of relativity to all frames of reference, including those in non-uniform,
accelerated motion.
(2) The Equivalence Principle:Termed by Einstein as ”the happiest thought of my life,” this principle asserts that the local
effects of a homogeneous gravitational field are physically equivalent to those of uniform acceleration in gravitation-free
space. This thought experiment yielded immediate testable predictions, specifically the gravitational redshift of clocks and
the deflection of light rays by massive bodies.
(3) The Geodesic Principle:This principle posits that free-falling objects traverse ’timelike geodesics’—the straightest
possible trajectories within curved spacetime—thereby redefining gravity as a geometric phenomenon rather than a force.
This generalizes the non-relativistic notion of a geodesic (such as the shortest path across a two-dimensional surface) into
the four-dimensional framework of relativity.
(4) ”Gravity Gravitates”:By synthesizing the mass-energy equivalence (E=mc 2) with the equivalence of inertial and
gravitational mass (mi =m g), Einstein deduced that the fundamental source of gravity must be energy density. Crucially,
this implies that the energy of the gravitational field itself contributes to the field, creating a feedback loop. Mathematically,
this necessitates that the field equations must be non-linear.
(5) The Stress-Energy Tensor:Einstein identified Laue’s stress-energy tensor (Tij) as the energy density. Its components
provide a complete physical description of the source matter: T 00: The energy density (mass-energy). T 0i: The momentum
density (and energy flux).T ij: The flux of momentum, representing pressure (wherei=j ) and shear stress (wherei̸=j ).
(6) Generalized Poisson Equation:In Newtonian gravity, the potential ϕ is governed by Poisson’s equation,∇2ϕ= 4πκρ
(where∇2ϕ= 0 in a vacuum). In the relativistic framework, Einstein sought a tensor generalization of this law. He replaced
the scalar term∇ 2ϕwith the curvature tensorR ik and the mass densityρwith the stress-energy tensor term−κT ik.
(7) The Newtonian Limit:This requirement demands that under specific conditions—weak gravitational fields, slow
motion (v≪c ), and static fields—General Relativity must simplify to match Newton’s laws. In this limit, spacetime
curvature becomes negligible and geodesics reduce to Newtonian trajectories (F=ma).
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