We thus find the occurrence of a gravitationnal field connected with a space-time variabilty of the g_{στ}. [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 e_{μ} (e_{0},e_{1},e_{2},e_{3}) in a tangent space Ε, the g_{μν} 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 g_{μν} = g_{νμ} (the entries of a symmetric matrix are symmetric with respect to the main diagonal)
- - the inverse matrix is noted g^{μν}[1] and is defined as folllows in absract notation: g^{μα}g_{αν} = δ^{μ}_{ν} (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 ds^{2} 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 (e_{i})
then we get
where g_{μν} represents the covariant components of the metric tensor.
Alternatively, you may choose the option of writing the metric tensor using the covariant components dx_{μ} and (dual) basis vector e^{μ}
_{}
Whether ds^{2} is written in the contravariant or covariant form, you can be sure that the distance between two points must be the same, no matter which coordinate system you employ.
Proper Time
A second even more fundamental physical interpretation of the g metric tensor is linked to the time measured along the wordlines, i.e the Proper Time.
We recall from this article that given a manifold with a local coordinates x^{μ} and equipped with a metric tensor g_{μν}, the proper time interval Δτ between two events along a timelike path P is given by the line integral
This expression is, as it should be, invariant under coordinate changes.
Special and General Relativity
The fundamental difference between Special and General Relativity regarding the metric tensor is that:
- in Special Relativity, g_{μν} is a known given constant denoted η_{μν - }called the The Minkowski metric, and of which the (1,3) signature defines a Lorentzian manifold.
- in General Relativity, g_{μν} is not given a priori but is a function of space and time: it must be calculated by resolving the Einstein's equations, and its signature defines a Pseudo-Riemannian manifold.
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.