Being an observer in curved spacetime, it is easy to conclude, what given two vectors are parallel to each other (but if here the torsion comes in, then we would not conclude that anymore). I mean, that all the geometry and space configuration is measured solely from the metric. Because the metric measures the distances. Hereby if you have the two short pencils, which are placed one after another, then they are parallel, if the distance between the opposite ends is maximal. It is my theory, let us publish together.
Mathematically, the process of parallel transport relies on the concept and existence of a connection. For a Riemann space the connection (Levi-Cevita connection) can be completely derived from the metric; in such spaces the tangent vectors of geodesics (shortest-distance paths) are parallel transported. This property seems to fit well with your suggestion (but alas -- the definition already exists), but it does not give a complete answer: How to parallel transport vectors which does not point in the direction of movement? The existence of a connection gives an answer to that more general question also.
For spaces with torsion the connection cannot be fully derived from a metric (only its symmetric part). But in such spaces the connection, if it is known, continue to give an answer.
In gauge theories the parallel transported quantity points in some internal space, and the gauge field acts as a connection. That connection gives an answer to how internal quantities with a direction (like the phase of a wave function, or an isospin vector) can be parallel transported.
Dear Dr.Martila,
That is good idea , but can you please approximate this phenomena to a subsurface geological strcture ?. During interpretting gravimetric or magnetic anomalies , buried bodies approximated to sphere , cylinder, sheet or prisms. But I don't know if the short pencils vectors idea you propose could work actually in an applied survey, maybe its woking on some dikes which are vertical or semi vertical paralel planes , or may be it woks on vertical systematic faults. Finally the idea worth to think about it more and more to make it applicable on a geophysical interpretation.
Best Regards.
Dr. Wadhah M. Shakir
Mathematically, the process of parallel transport relies on the concept and existence of a connection. For a Riemann space the connection (Levi-Cevita connection) can be completely derived from the metric; in such spaces the tangent vectors of geodesics (shortest-distance paths) are parallel transported. This property seems to fit well with your suggestion (but alas -- the definition already exists), but it does not give a complete answer: How to parallel transport vectors which does not point in the direction of movement? The existence of a connection gives an answer to that more general question also.
For spaces with torsion the connection cannot be fully derived from a metric (only its symmetric part). But in such spaces the connection, if it is known, continue to give an answer.
In gauge theories the parallel transported quantity points in some internal space, and the gauge field acts as a connection. That connection gives an answer to how internal quantities with a direction (like the phase of a wave function, or an isospin vector) can be parallel transported.
The big issue in regard to the phrase "gauge theory of gravity" is which gauge group that you are talking about: e.g., Lorentz, translation, Poincare, conformal Lorentz, etc.
David, "gauge theory of gravity" means "linear connection" with GL(4,R) group.
Parallel transport is just law, by which you can define set of parallel vectors in different space-time points. You can choose different law, that is an idea :) You have Christoffel symbols Gijk, which can be somehow derived from your gravity theory.
The Christoffel symbols are the components of the Levi-Civita connection, which is defined by the Lorentzian metric g. Since that connection is, by definition, a metric connection, you have implicitly reduced the gauge group from GL(4,R) to SO(3,1) by reducing the linear connection to a metric one.
In Euclidean space, in very elementary problems of statics and dynamics, we often shift a vector quantity keeping its magnitude and direction while working out resultant vectors. This is the most basic idea of parallel transport. In a curved space one keeps direction and the "norm" fixed while doing a parallel transport. Norm depends on the metric tensor which is not a constant, and which differs from point to point. Basically this is a symmetry operation because whatever be the metric tensor, in a parallel transport the norm remains invariant. Such a symmetry should lead to a conserved quantity which is measurable and thus interesting.
In a gauge theory we work with internal symmetry groups and internal spaces. Suppose we multiply a vector with a local phase, it keeps norm invariant. This is an internal symmetry which should lead to a conserved quantity, which is interesting and measurable.
Any theory which is interesting should also be falsifiable.
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