Einstein Relatively Easy

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"The general theory of relativity must be capable of treating every coordinate system, whatever its state of motion relative to others may be, as "at rest", i.e., the general laws of nature must be expressed by identical equations relative to all other systems, whichever way they are moving." Einsten ,  Fundamental Ideas and Methods of the Theory of Relativity, Presented in Their Development (1920)
 
Covariance:

The term covariance implies a formalism in which the laws of physics maintain the same form under a specified set of transformations.

Classical Mechanic Physical laws covariant under Galilean coordinate transformations
Special Relativity Physical laws covariant under Lorentz coordinate transformations
General Relativity Form of physical laws invariant under any arbitrary differentiable coordinate transformation (general covariance)
 
Special Relativity

One of the two fundamental postulates of special relativity is the principle of relativity, which stipules that the laws of physics are the same in any inertial frame of reference. Which is equivalent to say that the laws are covariant by Lorentz transformations.

 

General Relativity

As we know, in presence of a gravitationnal field, the Equivalence Principle allows to formulate all - but gravitationnal - physical laws,  in small enough free-falling areas.

To get a macroscopic significant description of these laws, one has therefore to find an operation to link all these different local inertial referentials. That will be to introduce a unique non inertial coordinate system, and that is precisely during this coordinate transformation operation that the effects of gravitationnal field will re-appear.

Indeed we remember from our article Geodesic equation and Christoffel symbols that some kind of 'inertial' forces in case of an accelerated referential or gravitional effect in case of the presence of gravitational field in a rest frame (Equivalence Principle) was now acting in the new arbitrary referential.

The idea of Einstein was to set the Principle of General Covariance which extends the principle of relativity to say that the form of the laws of physics should be the same in all - inertial and accelerating - frames.

Therefore, to write a valid physical equation in general relativity, we have to write a Tensor equation (which preserves its form under general coordinate transformations) which is true in special relativity, that's all.

By example, the fundamental Newton's first law of motion, is stated as follow in special relativity:

But as the right term is not a tensor - to transform it in a tensor,  the usual derivative should be replaced by the Covariant Differentive, so we get the slightly modified Newton's law version in general relativity:

 If we postulate with Einstein the general covariance of the physical laws in any arbitrary referential, then we get the correct physic laws by replacing the usual derivatives by the covariant derivatives, end of the story.

The way of writing physics in general relativity or curved spacetime consists then to operate the formal substitution, known as the 'comma goes to semi-colon' rule, as we saw when looking at Covariant Differentiation, commas and semi-colons can be used as shorthand notation for partial and covariant derivatives.

To be more precise, we will satisfy the Principle of General Covariance by the followin rule:

Take the equations of Special Relativity, replace ηαβ by ηαβ and all derivatives by covariant derivatives.

 

Remark 1: this last rule will be applicable for everything except gravity, for which there is no Special Relativity theory.

Remark 2: to practically use this rule, we have to explictly calculate the covariant derivative and therefore the Metric tensor has to be known (as the connection coefficient is a function of the metric and its derivatives - see Christoffel symbols in terms of the metric tensor).

 

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"Pas plus de cinq ou six semaines s'écoulèrent entre la conception de l'idée de la relativité restreinte et la rédaction de l'article correspondant."
Einstein à Carl Seelig, 11 Mars 1952

"N'importe quel étudiant dans les rues de Göttigen en connaît plus qu'Einstein sur les géométries à 4 dimensions. Et pourtant ce fut Einstein qui accomplit le travail, et non les mathématiciens."
David Hilbert

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