Einstein Relatively Easy

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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

 A metric tensor is a type of function 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.

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
Some Aspects of the Fundamental Tensor gμν (Extract from the Manuscript "The Foundation of the General Relativity of Relativity §8 1916)

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:

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

Whether ds2 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 xμ and equipped with a metric tensor gμν, 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, gμν is a known given constant denoted ημν - called the The Minkowski metric, and of which the (1,3) signature defines a Lorentzian manifold.

- in General Relativity, gμν 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.

 

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"Five or six weeks elapsed between the conception of the idea for the special theory of relativity and the completion of the relevant publication" Einstein to Carl Seeling on March 11, 1952

"Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
David Hilbert

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