This page contains my notes and my considerations regarding the course on General Relativity: “The We-Heraeus International Winter School on Gravity and Light (2015)” available on Youtube, where Professor Frederic P. Schuller explains the matter very clearly.
I have also used the very good book “Geometrical methods of mathematical physics” by Professor Bernard Schutz to help me better understand some arguments.
I have decided to publish it as I write my notes without necessarily respecting the logic order in which topics are presented in the course. So this page is in a working progress state.
In the spirit of this site, which is also a sort of account of my journey into modern physics, sometimes I’ll write not only about the topic itself but also the difficulties I have encountered and the way I have tried to overcome them: I think this can be useful to those like me are approaching modern physics.
You can find my notes about some differential geometry arguments which are preparatory to the subsequent arguments, at this page: https://feelideas.com/tangent-and-cotangent-spaces/
1. Affine Connection
Suppose we have a function (where M is a d-dimensional smooth manifold) and a vector field also in . Let be an open set of M where never is null. Then there exist a congruence derived from , that is a family of curves such that each point in belongs to only one of them and, in any point, can be obtained by that curve (in the usual way).
Suppose that is a curve of the congruence from (you can find its definition and explanation in the book of prof. Schutz) and that . It makes sense to ask which is the value of the derivative of in with respect to : it is the derivative of along the curve in .
Such derivative is defined as followes:
It does not depend on the coordinate system choosen in , or in other words, it is chart independent. So it is well defined and makes sense!
What happens if we try to compute the derivative along in of a vector field or more generally of a tensor ?
We could be tempted to consider its components in a specific chart and apply to them what we have seen for a generic smooth function . But we have 2 big problems (presented on a vector field, but easily extendable to any tensor field):
- the result of the operation must be a vector field, so independent of the chart used, but it’s simple to see that this is not the case;
- how can we compute the difference between 2 vectors defined in 2 different tangent spaces on M? They belong to 2 totally disjoint vector spaces!
Professor Schuller used a totally abstract approach, starting from the properties we expect the affine connection to have and then deriving the general formulation. The problem I encountered during the dedicated lesson was the presentation of Leibniz’s rule in 2 different ways, saying that they are equivalent without proving it (he left this task to his students). I haven’t been able to prove it myself, so that was why I started studying Professor Schutz’s book, where I found an approach clearer for myself: and now I’m back to the topic.
To solve the second of these problems we must find a way to connect the tangent spaces on M, but this means defining a way to transport a vector from a tangent space to another so that they can be “compared”. The first problem can be solved asking that the transport is chart independent.
With these premises we can define the “Affine Connection” or equivalently the “Covariant Derivative” of a tensor in with respect to a vector field as follows:
where is the tensor field evaluated in and transported through the connection into , being the curve derived from passing through point .
With this general definition we can now list the properties that the operator has got:
- if is a tensor field of type then is another tensor field of the same type.
- if is a smooth function on , then
this is the Leibniz’s rule in its classic form, where the symbol represents the tensor product, which is defined as followes:
let be a tensor of type and a tensor of type , then
because we assume that the affine connection is a continuous operator and for is equal to .
So it is proved.
where is a tensor field, is a covector field and is a vector field. This is another version of Leibniz’s rule. Here prof. Schuller said it is equivalent to the classical version, but I couldn’t prove it myself, so I tried to prove it starting from the general definition as before.
For simplicity I’ll write as , just to reduce the extension of the next expressions
but can be rearranged as:
Now we analyze these 3 terms separately:
where I have exploited the multilinearity of tensors.
As before I have used multilinearity of tensors.
Because of the continuity of operator we know that:
So we have demonstrated that:
where is a smooth function on .
Consider a curve of the congruence of passing through the point on and ask: does this curve belong to the congruence of the vector field ? The answer is yes, but a reparameterization of in necessary (unless f is constant on all points of the curve).
Consider on a chart , with : if M is an d-dimensional manifold, then has components in : with going from to .
We know that , being . We can thus interpret the components of in as the velocity evaluated in such a chart of a point running along the curve according to the law: , with considered as a time variable. Multiplyng for in each point of the curve, requires that point running on the curve adapts its velocity to the new values in each point: it’s intuitive it can always be done. But we can evaluate more precisely the question, as followes.
If we call the projection of on the chart , does a reparameterization of (that I call ) exist such that: being that spans all the curve?
Because , it is equivalent to asking that for each . Given that and , where , we can easily construct so that it respect the above equation; or equivalently the above equation admits a solution.
The properties listed so far derive directly from the definition of the affine connection given before, based on the concept of transport of a tensor along a curve (element of the congruence of vector field). But now we ask the connection to respect 2 additional properties, which are:
which means that summing and in and then transporting the result to (where is the point hit by the integral curve of in ) is equivalent to transport separately and ) in and then summing them.
This property together with the previous one makes the affine connection linear with respect to the vector field argument on which it is based:
being and smooth functions on .
Using all theese properties, it’s possible and very interesting to find how the operator transforms respectiveley a vector field, a covector field and more complex tensor field respectiveley.
1.2 Affine Connection of e Vector field Y:
Since is a function on (or at least on an open set of it):
But what about ? operator transform a tensor field in another of the same type, so:
Looking at the component:
, where and run from to .
The are the connection coefficient functions, and they define entirely the affine connection. They are functions, depending on the point of the manifold in which we apply the operator. They depend on the choosen chart.
How can be computed?
choosing and in eq. 1.0 we have that:
– , so ;
– in , and so it reduces to ;
So the result is proved.
We will discover in a moment that they fully define not only how acts on a vector field, but on any type of tensor field.
1.3 Affine Connection of a Covector field :
…. TO BE COMPLETED
2. Riemann Tensor
It is define as followes:
where is a covector field, while , and are vector fields. is the commutator of and :
It’s a tensor and its componenst are .
But wait, makes sense only in case is a vector field and it’s not difficult to see it, applyng this operator on a smooth function on :
the second and the third element are equals and so (renaming indeces in the first element):
If we consider , and given vector fields and as a variable, we can write:
being the 2 side of equation linear maps on the parameter in any point of the manifold. But a linear map from to is just a vector.
The last equation can be rearranged as followes:
but is a vector field and so it can be written as:
If we choose and as 2 elements from a chart induced basis:
and , .
being the integral curves of the congruence associated to the vector field null, just points and so without any possible transport of vector field along them.
On the other hand .
Finally, if we abbreviate into , then:
And we’ll see that gives a measure of the curvature of the manifold in any point. This is why Riemann tensor is a curvature tensor.
Suppose and are 2 vector fileds that commute () on the manifolt or in an open set of it. A seen before and now we try to see how the term can be interpreted geometrically.
The next figure show the reasoning:
It is based on the fact that the transport along the affine connection of the sum of 2 tensor fields is equal to the sum of the vector fields transported separately, as shown in the next figure: