Einstein Relatively Easy

Vote utilisateur: 5 / 5

Etoiles activesEtoiles activesEtoiles activesEtoiles activesEtoiles actives
 
Pin It

If you like this content, you can help maintaining this website with a small tip on my tipeee page  

 

 As all the information about the spacetime structure is being contained in the metric, it should be possible to express the Christoffel symbols in terms of this metric.

The following calculation is a little bit long and requires  special attention (although it is not particularly difficult).

So far, we have defined both the metric tensor and the Christoffel symbols as respectively:

Let's begin by rewriting our metric tensor in the slightly different form gαμ:

 Now, in this second step, we want to calculate the partial derivative of gαμ by xν:

Now let's try to rewrite the Christoffel symbol by multiplying each part of the equation by the partial derivative of ξσ relative to xβ:

 We can now rewrite the partial derivative of gαμ by xν as follows:

 or we recognize from our previous article Generalisation of the metric tensor that

If we now do the operation (3) + (4) - (5) we get:

Finally the last step consists in multiplying  both sides of the equations by the inverse metric gβα to isolate the Christoffel symbol

Usually, we adopt the following convention for writing partial derivatives:

 Phew!

Langage

Navigation

Citations

"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

En ligne

Nous avons 36 invités et aucun membre en ligne

Flux RSS

feed-imageRSS


Notice: Constant DS already defined in /home/c1288285c/public_html/modules/mod_fblikeboxslider/mod_fblikeboxslider.php on line 3