Why are tensors important in general relativity?
The notion of a tensor field is of major importance in GR. For example, the geometry around a star is described by a metric tensor at each point, so at each point of the spacetime the value of the metric should be given to solve for the paths of material particles.
What are tensors in general relativity?
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation.
How do you show an antisymmetric tensor?
In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all covariant or all contravariant.
Why is tensor calculus important?
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Is the metric tensor antisymmetric?
Since within the definition of the metric is that it’s symmetric (g(x,y) = g(y,x)) there are no anti-symmetric parts.
How many independent elements are there in an antisymmetric tensor?
The reader should take note that the specific duality we have just described is unique to three-dimensional space; in four dimensions (appropriate for relativity) an antisymmetric rank-2 tensor has six independent components and cannot be expected to provide an alternate representation of a four-vector.
What is antisymmetric relation in discrete mathematics?
In discrete Maths, a relation is said to be antisymmetric relation for a binary relation R on a set A, if there is no pair of distinct or dissimilar elements of A, each of which is related by R to the other.
How did Einstein learn tensor calculus?
He went to a mathematician friend (I think it was Besso, but can’t be sure anymore) and said something like “Here’s what I want to do, what kind of math would handle it?” His friend then directed him to tensor calculus, at that time a little known and little used field of mathematics.
Why is tensor calculus so hard?
Penrose [30, 311 was the first to use the graphical notation for calcu- lating with tensors. A striking aspect of the notation is that it is pictorial rather than sequential or alphabetical. This made it difficult to print, which partly explains why no rigorous theory was developed.
Why is the metric tensor covariant?
Thus the metric tensor gives the infinitesimal distance on the manifold. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor.
What is a totally antisymmetric tensor?
Totally antisymmetric tensors include: Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric) ↑ K.F. Riley; M.P. Hobson; S.J. Bence (2010).
What is an antisymmetric contraction?
A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. (antisymmetric part). Similar definitions can be given for other pairs of indices.
Are all scalars and vectors antisymmetric?
Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric) ^ K.F. Riley; M.P. Hobson; S.J. Bence (2010).
What is the notation for anti-symmetrization?
A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M , M [ a b] = 1 2! ( M a b − M b a),