I’ll start this continuation of the topic of a previous post by an exposition and explanation of the concept of tensors in relativity physics given by Albert Einstein , from his 1916 paper The Foundation of the General Theory of Relativity :
” The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion.
The general laws of nature are to be expressed by equations which hold good for all systems of coordinates, that is, are co-variant with respect to any substitutions whatever (generally covariant).
Let certain things (” tensors “) be defined with respect to any system of coordinates by a number of functions of the coordinates , called the ” components ” of the tensor. There are then certain rules by which these components can be calculated for a new system of coordinates , if they are known for the original system of coordinates, and if the transformation connecting the two systems is known. The things hereafter called tensors are further characterized by the fact that the equations of transformation for their components are linear and homogeneous. Accordingly, all the components in the new system vanish, if they all vanish in the original system. If, therefore, a law of nature is expressed by equating all the components of a tensor to zero, it is generally covariant. By examining the laws of the formation of tensors, we acquire the means of formulating generally covariant laws. “
Source : The Principle of Relativity (Dover Books on Physics)
Einstein’s paper can also be found here:
I will deal in this post with the derivation of the Einstein field equations of General Relativity and with the related equations of motion.
As in the previous post , I will not show all the long and detailed calculations but I will give an overview of the equations , calculations and derivations .
We start with the derivation of the field equations free from the sources generating the field.The equations are deduced from the principle of least action.
A small remark : in the equations below , the symbol g written in two ways is the same letter in two different fonts , it represents either the metric tensor (with subscript or superscript) or the determinant of its matrix representation.
The action of the gravitational field is given by:
where χ is the gravitational constant appearing in Einstein’s field equations.
Ω is the product of the differentials of the coordinates.
In four-dimensional space we have :
In the action integral above , The scalar curvature (or the Ricci scalar) R is related to the Ricci tensor by the equation R = gij Rij .
And the Ricci tensor is related to the Riemann curvature tensor by the equation:
Hence we have:
The variation of the action yields:
After calculation the first integral in the variation of the action above gives:
The third integral in the variation of the action is equal to zero:
And the variation of the action can be written as:
The variations δgij are arbitrary and the Einstein field equations independent of the sources generating the field
( Rij – (1/2)gij R = 0 ) can be deduced from the integral above.
The action for matter and the electromagnetic field is given by:
This action and its variation involve the Lagrangian density:
After integrating and calculating we get:
Defining the energy momentum tensor (or stress–energy–momentum tensor ) by the relation:
We get for the variation of the action:
The preceding results lead to the condition δ(Sg+Sme)=0 being written as:
The equality above must remain valid for arbitrary variations of δgij ,and we deduce :
which are the Einstein field equations of General Relativity.
The contravariant form of the energy-momentum tensor is given by the transformation :
Below is a schematic description of the contravariant components of the energy-momentum tensor (image from Wikipedia) :
Note that of we take the more general form of the action :
The field equations for the most general case would become:
where λ is the cosmological constant .
The Einstein field equations are in agreement with the conservation of the energy-momentum tensor , which means its divergence is null:
The energy-momentum tensor (mixed tensor) can be written as the sum of two terms:
The first one is of electromagnetic origin:
The second one involves the presence of matter:
The condition of null divergence can be written as:
After some calculations we get for the first term :
And for the second term in the null divergence equation:
By using the last two equations above , equation (1) can be written as:
which is the equation of motion of a continuous distribution of charge inside a gravitational field and an electromagnetic field.
The references and sources in my Science books page and the problem solvers page can be viewed for books which I have read , worked out or consulted and which are related to the topics in this post.
I found the book by Jean-Claude Boudenot ( Electromagnetisme et gravitation relativistes ) helpful while I was studying these topics in the past.