Gravitational Redshift as a Potential Probe: A Discrete Lattice Field Method in the Weak Field

We introduce a method for constructing a calibrated, spatially resolved gravitational potential field on adiscrete lattice in the static weak field. This paper demonstrates the solar field construction using physicalparameters specified independently of the measured redshift. On the Sun the field is built from the solarmass parameter, radius and distance, on a lattice that keeps the true Sun–Earth geometry. A walk throughsuccessive stored differences returns the endpoint difference to order 10−15. The gradient integral, whichtests the field’s slopes, agrees to 8.9×10−5, and a cancellation identity sets its floor. The weak-field climbfrom the photosphere to 1 AU, 633.351m s−1 in velocity units, and the independent HARPSmeasurement, 638±6m s−1, agree within the measurement’s uncertainty. A gravitational redshiftbetween two static locations measures their potential difference directly. The method supplies the stepfrom such measurements to a calibrated spatial field representation whose constituent values, slopes andcurvature can be examined. Its coordinate map, spatial shape, additive potential reference and amplitudeare specified separately. For a fixed shape and geometry, one nonzero endpoint comparison determinesthe amplitude; a connected network of measured comparisons determines nodal potentials up to anadditive constant. The constructed field makes gravitational acceleration and, through the NewtonianPoisson equation, an effective mass-density diagnostic accessible from its spatial derivatives. A ranktheorem establishes what a network of redshift measurements determines, and synthetic networks recoverthe prescribed potentials with the predicted covariance. The method thus provides a common frameworkfor constructing and examining gravitational potential fields from potential-difference information,spanning millimetre-scale atomic samples and solar spectroscopy.