We thus find the occurrence of a gravitationnal field connected with a space-time variabilty of the gστ. [Einstein The Foundation of the General Relativity Annalen der Physik, vol XLIX 1916- The Collected Papers of Albert Einstein doc. 30]

Once you have arithmetized a space with an arbitrary coordinate system, there is one tensor that allows you to define fundamental quantities such as lengths and time in a consistent manner, no matter which coordinate system you employ.

That tensor, the one that "provides the metric" for a given coordinate system in the space of interest, is called the metric tensor, and is represented by the lower-case letter g.

##### Definition

Three different definitions could be given for metric, depending of the level  - see Gravitation (Misner, Thorne and Wheeler), three levels of differential geometry p.199)

In the language of elementary geometry, metric is a kind of table giving the interval between every event and every other event.

In the language of abstract differential geometry,  metric is a type of bilinear machine which takes as input a pair of tangent vectors u and v at a point of a surface, and produces a real number scalar g(u,v). It can be seen as the generalization of the dot product in Euclidean space.

In the language of coordinates, given a basis eμ (e0,e1,e2,e3) in a tangent space Ε, the gμν components of the g matrix relative to this basis is given by

##### Properites

The metric tensor has the following properties:

• - it is symmetric in the sense of gμν = gνμ (the entries of a symmetric matrix are symmetric with respect to the main diagonal)
• - the inverse matrix is noted gμν[1] and is defined as folllows in absract notation: gμαgαν = δμν (Kronecker delta)

##### Spacetime interval invariance

To understand the role of the metric tensor, we have to consider the vector dr extending from one point to the other. Then the square of the differential length element ds2 may be written as:

The link between the abstract, bilinaear machine viewpoint and the concrete coordinates viewpoint is readily expressed as per below:

If we choose to write the vector dr using contravariant components[1] and coordinate basis vectors (ei)

then we get

where gμν represents the covariant components of the metric tensor.

Alternatively, you may choose the option of writing the metric tensor using the covariant components dxμ and (dual) basis vector eμ

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That is by this fundamental difference that Einstein himself introduces the gravitational field in its 1916 final publication on General Relativity Foundations of General Relativity

[1] The notions of contravariant and covariant components as well as basis vectors have been defined in the article Introduction to Tensors.