let be the Minkowski fiber bundle over the spacetime manifold . For each point , the fiber is an affine space. In a fiber chart , coordinates are usually denoted by , where are coordinates on spacetime manifold , and are coordinates in the fiber . Using the abstract index notation, let refer to and refer to the tangent bundle . In any particular gauge, the value of at the point p is given by the section The covariant derivative is defined with respect to the connection form , a 1-form assuming values in the Lie algebra of the translational abelian group . Here, d is the exterior derivative of the th component of , which is a scalar field (so this isn't a pure abstract index notation). Under a gauge transformation by the translation field , and and so, the covariant derivative of is gauge invariant. This is identified with the translational (co-)tetrad which is a one-form which takes on values in the Lie algebra of the translational Abelian group , whence it is gauge invariant. But what does this mean? is a local section of the (pure translational) affine internal bundle , another important structure in addition to the translational gauge field . Geometrically, this field determines the origin of the affine spaces; it is known as Cartan’s radius vector. In the gauge-theoretic framework, the one-form arises as the nonlinear translational gauge field with interpreted as the Goldstone field describing the spontaneous breaking of the translational symmetry. A crude analogy: Think of as the computer screen and the internal displacement as the position of the mouse pointer. Think of a curved mousepad as spacetime and the position of the mouse as the position. Keeping the orientation of the mouse fixed, if we move the mouse about the curved mousepad, the position of the mouse pointer (internal displacement) also changes and this change is path dependent; i.e., it doesn't only depend upon the initial and final position of the mouse. The change in the internal displacement as we move the mouse about a closed path on the mousepad is the torsion. Another crude analogy: Think of a crystal with line defects (edge dislocations and screw dislocations but not disclinations). The parallel transport of a point of along a path is given by counting the number of (up/down, forward/backwards and left/right) crystal bonds transversed. The Burgers vector corresponds to the