the tangent space. The set of all such multilinear maps forms a vector space, called the tensor product space of type at and denoted by If the tangent space is n-dimensional, it can be shown that In the general relativity literature, it is conventional to use the component syntax for tensors. A type tensor may be written as where is a basis for the i-th tangent space and a basis for the j-th cotangent space. As spacetime is assumed to be four-dimensional, each index on a tensor can be one of four values. Hence, the total number of elements a tensor possesses equals 4R, where R is the count of the number of covariant and contravariant indices on the tensor, (a number called the rank of the tensor). Symmetric and antisymmetric tensors Some physical quantities are represented by tensors not all of whose components are independent. Important examples of such tensors include symmetric and antisymmetric tensors. Antisymmetric tensors are commonly used to represent rotations (for example, the vorticity tensor). Although a generic rank R tensor in 4 dimensions has 4R components, constraints on the tensor such as symmetry or antisymmetry serve to reduce the number of distinct components. For example, a symmetric rank two tensor satisfies and possesses 10 independent components, whereas an antisymmetric (skew-symmetric) rank two tensor satisfies and has 6 independent components. For ranks greater than two, the symmetric or antisymmetric index pairs must be explicitly identified. Antisymmetric tensors of rank 2 play important roles in relativity theory. The set of all such tensors - often called bivectors - forms a vector space of dimension 6, sometimes called bivector space. The metric tensor The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equations). Using the weak-field approximation, the metric tensor can also be thought of as representing the 'gravitational potential'. The metric tensor is often just called 'the metric'. The metric is a symmetric tensor and is an important mathematical tool. As well as being used to raise and lower tensor indices, it also generates the connections which are used to construct the geodesic equations of motion and the Riemann curvature tensor. A convenient means of expressing the metric tensor in combination with the incremental intervals of coordinate distance that it relates to is through the line element: This way of expressing