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Tetrads in General Relativity

I. Metric and Connection

Spacetime

   In general relativity theory (GRT), spacetime is modeled as an orientable smooth curved Lorentzian (pseudo-Riemannian) 4d manifold $\mathcal{M}_4 = \{ {\mathbf{x}}\} $, parametrized by coordinates $\{ {x^\mu };\mu = 0,1,2,3\} $ so that ${\mathbf{x}} = {\mathbf{x}}({x^0},{x^1},{x^2},{x^3}): = {\mathbf{x}}(x)$. The chosen metric signature is $( + , - , - , - )$, one dimension of time and three of space; units such that the velocity of light and the gravitational constant both have the value one: $c = 1$, $G = 1$; where appropriate these constants will be written explicitly.

   A set of four linearly independent local coordinate basis vectors are defined by:

1.1

\[ {{\mathbf{g}}_\mu }({\mathbf{x}}): = \frac{{\partial {\mathbf{x}}}}{{\partial {x^\mu }}}: = {\partial _\mu }{\mathbf{x}} \]

At each point ${\mathbf{x}}$ of the spacetime manifold these tangent vectors $\{ {{\mathbf{g}}_\mu }({\mathbf{x}})\} $ provide a holonomic basis for the vector space ${V_4}({\mathbf{x}}) = {T_{\mathbf{x}}}\mathcal{M}$, called the tangent space to ${\mathcal{M}_4}$ at ${\mathbf{x}}$. The tangent space is by definition a Minkowski space. The tangent bundle of all tangent spaces and the manifold are said to be soldered, because they are both four-dimensional spacetimes. This geometrical structure is intrinsic to the coordinatization of the manifold, independent of any prior assumptions.

   Physical quantities of interest in GRT are represented by tensor fields, or other geometric objects like spinors, that live in the tangent vector space attached to each point in spacetime, including the higher-order tensor spaces constructed therefrom. These geometric objects can be specified in terms of arbitrary coordinates, but are themselves coordinate independent. It is a fundamental principle in physics that equations between such geometric objects are generally covariant, that is, they are invariant under any change of coordinate system.

    In the subject of differential geometry, it is now common practice to identify directional derivatives as tangent vectors. However, there is a complete isomorphism between coordinate vectors $\{ {{\mathbf{g}}_\mu }\} $ as defined in (1.1) and the corresponding directional derivatives $\{ {\partial _\mu }\} $. For all practical purposes, the two viewpoints are equivalent, and calculations performed in either scheme will return the same results.