The equations of electrodynamics , tensors , and gravitation-Second Part

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:

The Foundation of the Generalised Theory of Relativity

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:

action gravitation

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 :

d \Omega =\text{dx}^0 \text{dx}^1 \text{dx}^2 \text{dx}^3 

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:

Ricci tensor

Hence we have:

gravitation action integral

The variation of the action yields:

variation of action

After calculation the first integral in the variation of the action above gives:

first integral in action

The third integral in the variation of the action is equal to zero:

third integral in action

And the variation of the action can be written as:

em-grav-001d

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:

action for matter

This action and its variation involve the Lagrangian density:

action variation

After integrating and calculating we get:

action calculation

Defining the energy momentum tensor (or stress–energy–momentum tensor ) by the relation:

energy momentum tensor

We get for the variation of the action:

action with stress energy tensot

The preceding results lead to the condition δ(Sg+Sme)=0 being written as:

action variation result

The equality above must remain valid for arbitrary variations of δgij  ,and we deduce :

field equations

which are the Einstein field equations of General Relativity.

The contravariant form of the energy-momentum tensor is given by the transformation :

contravariant energy momentum tensor

Below is a schematic description of the contravariant components of the energy-momentum tensor (image from Wikipedia) :

components of contravariant tensor

Note that of we take the more general form of the action :

action with cosmological constant

The field equations for the most general case would become:

field equations with cosmological constant

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:

null divergence

The energy-momentum tensor (mixed tensor) can be written as the sum of two terms:

The first one is of electromagnetic origin:

electromagnetic mixed tensor

The second one involves the presence of matter:

mixed tensor matter

The condition of null divergence can be written as:

null divergence two terms

After some calculations we get for the first term :

first term of divergence

And for the second term in the null divergence equation:

second term null divergence

By using the last two equations above , equation (1) can be written as:

em-grav-12

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.

Advertisements

2 comments

  1. Great post, although I don’t understand most of the mathematics required for General Relativity.

    One question: Is the Action for a gravitational field a postulate? Or can it be ‘derived’?


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s