American Society of Civil Engineers

Elasticity of Space-Time: Basis of Newton’s 2nd Law of Motion

by Kirk C. Valanis, (Res. Prof., Coll. of Engrg., Univ. of Portland, 8605 NW Lakecrest Court, Vancouver, WA 98665. E-mail: or

Journal of Engineering Mechanics, Vol. 129, No. 9, September 2003, pp. 1039-1047, (doi:

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Document type: Journal Paper
Abstract: In this paper we derive Newton’s 2nd Law of Motion from the constitutive equation of elasticity of a space-time continuum in four dimensions. This we do by introducing a four-dimensional material continuum with a Minkowskian metric, in analogy with Einstein’s general theory of relativity. The four-continuum is deformable in both space and time. The physics of the deformation is embedded in a variational principle, which is a form-invariant extension of its classical mechanical counterpart in three dimensions, but with the acceleration term absent. General dynamic equations of elasticity in four dimensions are thereby derived. When the constraint of temporal inextensibility (universal time) is introduced, these equations yield readily the dynamic equations of elasticity in three dimensions. The presence of the inertia term in these equations, is a direct consequence of the temporal curvature induced by the deformation of the four-continuum. Newton’s law of motion for rigid bodies follows when the additional constraint of spatial inextensibility is introduced.

ASCE Subject Headings:
Equations of motion