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lapo
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Hi, some one know the expression of the affine connection Γ in terms of tetrad formalism? I would like also some references if it's possible, i found a hit but i think that is wrong... please help me it's so important!
I'm loking for this relation:PWiz said:$$Γ^{\lambda} _{μν} = \frac{1}{2} (e^κ ⋅ e^λ) ( ∂_μ (e_κ ⋅ e_ν) + ∂_ν (e_κ⋅e_μ) - ∂_κ (e_μ⋅e_ν))$$
An affine connection is a mathematical concept used in differential geometry to describe how vector fields change as they move along a curved space. In terms of tetrad, it refers to the relationship between the coordinate system and the local frame that is used to describe the space.
An affine connection is closely related to parallel transport, which is the process of moving a vector along a curve without changing its direction. The affine connection describes how the vector fields change as they are parallel transported along a given curve in the space.
Using a tetrad to describe an affine connection allows for a more intuitive understanding of the connection, as it relates to the local frame of the space. It also allows for easier calculations and visualizations of the connection.
The affine connection Γ is represented in terms of a tetrad by a set of 16 coefficients, known as the connection coefficients or Christoffel symbols. These coefficients relate the local frame to the coordinate system and describe how vector fields change as they are transported along different curves in the space.
Yes, an affine connection can be defined without using a tetrad. It can be described using other mathematical objects such as the metric or the curvature tensor. However, using a tetrad provides a more intuitive and practical way of understanding and calculating the connection.