Einstein Relatively Easy

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After spending some time looking at tensors, we can now expose the problem of how to differentiate a tensor in curved spacetime.

To work with tensor components, one would need a set of basis vectors {eα} as well as the dual set of basis 1-forms (wα). In flat spacetime a single such basis suffices; all events can use the same Lorentz basis. But not in curved spacetime!

There each event has its own tangent space, and each tangent space requires a basis of its own.

To compare two vector and tensors at neighboring events along a curve, one has to parallel transport their basis, compare them, and sees the bases twist and turn. They must do so, in order to to accomodate themselves to the curvature of spacetime.

To quantify the twisting and turning of basis vectors and forms, one has to use the covariant derivative.

Examine the changes in vector fields along a basis vector eβ is noted:

eβ which is also noted as ∇β

And especially examine the rate of change of some basis vector  eα is noted ∇βeα


Covariant differentiation for a contravariant vector

Consider a vector V = Vαeα (ie the tensor has contravariant components  Vα and coordinate basis vectors eα). Using the product rule of derivation, the rate of change of the components Vα (of the vector V) with respect to xβ.

 But we recall from our article Christoffel Symbol or Connection coefficient that the connection coefficients are defined by:

Substituing this expression in the above equation gives

The right hand term has two dummy indices (ie indices to be summed over) α and γ. We can improve the formula by changing α to γ and γ to α to give:

and factoring out eα gives

This expression indicates the rate of change of Vα in each of the directions β of the coordinate system xβ, and is known as the covariant derivative of the contravariant vector V. The nabla symbol is used to denote the covariant derivative

In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.

The intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. it is independant of the manner in which it is expressed in a coordinate system.

Remark 1: As we have seen in our articles Local Flatness or Local Inertial Frames and SpaceTime curvature and Local Inertial Frame (LIF), in a inertial frame of reference, the vanishing of the partial derivatives of the metric tensor at any point of M is equivalent to the vanishing of Christoffel symbols, and then we can write this fundamental equality in the context of any inertial or local inertial frame:


Remark 2: the fact that the Christoffel symbol by itself does NOT transform as a tensor can be easily deduced from the fact that we can always find an (local) inertial frame in which its value equals zero, which should not be possible for a tensor.

Remark 3: we can also find these equivalent notations for the covariant differentiation. In particular, common notation for the covariant derivative is to use a semi-colon (;) in front of the index with respect to which the covariant derivative is being taken (β in this case)

Covariant differentiation for a covariant vector

Let's take the scalar product AμBμ of two arbitrary vectors, one covariant A and the other contravariant B. We have then, applying the derivation rules:

But as the value of a scalar in a point in spacetime does not depend on the basis vectors, then the covariant derivative of a scalar equals to its ordinary derivative:

Comparing these two last equations gives by renaming some of the mute indices:

 As this equation should hold true for each arbitrary A vector, the quantity in brackets should be necessary null. So we have shown that the expression of the covariant derivative of the covariant components of a vector B is as below:

Note that the term involving Christoffel symbols is substracted in this case.

In the same manner as with  the contravariant vectors, the second term vanishes in a context of inertial frame of reference. We have then:






"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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