Gravity from a Dilation-Shear Gravitational Potential Tensor

We formulate gravity in terms of a pure dilation-shear gravitational potential tensor P. The field P deforms a reference metric structure by locally compressing and stretching its components and thereby induces the effective gravitational metric{equation*} g=P^T g P ,{equation*}specialized in this paper to a Minkowskian reference metric g=η. Thus P is not introduced as an arbitrary tetrad gauge variable, but as the metric-generating dilation-shear gravitational potential tensor. The same P transports the reference comparison rule and determines a connection{equation*} Γ[P]=P⁻¹∇ P .{equation*}The Levi-Civita connection Γ[g(P)] of the induced metric and the transported connection Γ[P] generally differ, and their difference{equation*} D=Γ[P]-{Γ}[g(P)]{equation*}is the gravitational distortion tensor. It is a genuine tensor and a first-derivative field-strength-like object generated by P, not a non-tensorial connection coefficient and not the derivative of an independently postulated metric. When the reference connection is flat, R[Γ[P]]=0, so the Levi-Civita curvature admits the exact distortion representation{equation*} R[g(P)]=-(∇ D+D D).{equation*}Consequently the Einstein-Hilbert action of the induced metric can be rewritten, up to a boundary term, as a quadratic action in D. We derive the weak-field limit, the linear vacuum wave equation, and the Newtonian limit; obtain the static spherical exterior branch; reconstruct it as a P-field; and compute the exterior field energy of the corresponding static spherical P-field. In that solution, P explicitly compresses the reference temporal direction and stretches the spatial directions, giving a direct potential-field picture of the effective gravitational spacetime.