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Saturday, 17 December 2011

What is a reference frame in field theory and mechanics

Non-relativistic mechanics as like as classical field theory is formulated in terms of fibre bundles.

In classical field theory on a fibre bundle Y->X, a reference frame is defined as an atlas of this fibre bundle, i.e. a system of its local trivializations.

If it is a gauge theory, Y->X is a fibre bundle with a structure Lie group G, and a reference frame equivalently defined as a system of local sections of an associated principal bundle P->X.

In particular, in gravitation theory on a world manifold X, a reference frame is a system of local frame fields, i.e. local sections of a fibre bundle P->X of linear frames in the tangent bundle TX of X.  This also is the case of relativistic mechanics. Therefore, one can treat the components of a tangent reference frame as relativistic velocities of some observers.

There are some reasons to assume that a world manifold X is parallelizable, i.e. its tangent bundle is trivial, and there exists a global section of a frame bundle P->X, i.e. a global reference frame. In this case, by virtue of the well-known theorem, there exists a flat connection K on a world manifold X which is trivial (K=0) with respect to this reference frame, and vice versa. Consequently, a reference frame on a parallelizable world manifold can be defined as a flat connection. The corresponding covariant differential provides relative velocities with respect to this reference frame.

Lagrangian and Hamiltonian non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles Q->R over the time axis R. Such fibre bundle is always trivial. Therefore, a reference frame in non-relativistic mechanics can be defined both as a trivialization of a fibre bundle Q->R and a connection K on this fibre bundle. Absolute velocities are represented by elements v of the jet bundle JQ of Q, and their covariant differential v-K are relative velocities with respect to a reference frame K.

This description of a reference frame as a connection enables us to formulate non-relativistic mechanics with respect to any reference frame and arbitrary reference frame transformations.

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical field Theory (WS, 2009)
G.Giachetta, L.Mangiarotti, G.Sardanashvily,  Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)
Sardanashvily's blog post Archive

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