- #1
PhyAmateur
- 105
- 2
If $$e^1$$ is a form like the ones in tetrad formalism (vielbeins). If we have $$e^1 . e^1$$ can we treat those as basis like $$i.i=1$$?
Vielbeins, also known as tetrads or vierbeins, are a set of orthonormal vectors that form a basis for the local tangent space at a point in a curved spacetime. They are commonly used in the field of general relativity to describe the geometry of spacetime.
Vielbeins are related to the metric tensor through the relationship g = e^{a}.e^{b}, where g is the metric tensor and e^{a} and e^{b} are the vielbeins. This relationship allows for the conversion between coordinate and tetrad components of vectors and tensors.
The expression $$e^1.e^1$$ represents the dot product of two vielbeins at a point in spacetime. This dot product is equal to 1 if the two vielbeins are parallel, and 0 if they are orthogonal. It is an important quantity in the study of vielbeins and can be used to calculate various geometric quantities.
No, $$e^1.e^1$$ is not a basis in the traditional sense. It is a scalar quantity that represents the dot product of two vectors. However, the set of vielbeins forms a basis for the local tangent space, similar to how the basis vectors $$i$$, $$j$$, and $$k$$ form a basis for three-dimensional Euclidean space.
Vielbeins are used in general relativity to describe the geometry of spacetime. They allow for the conversion between coordinate and tetrad components of vectors and tensors, making it easier to calculate various physical quantities. They are also used in the formulation of the Einstein field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy in the universe.