I'm having my first differential geometry course and I can't get the concept.
In Differential Geometry a metric is a length and angle measure on each tangent space that varies smoothly from point to point. Usually this measure is a positive definite inner product but in the theory of relativity it is not positive definite.
So, would it be correct to say that a metric gives a way to measure the space-time on a given geometry? Thinking about the special case of General Relativity.
I don't know what you mean by "measure space time ".
In physics one needs to measure the length of vectors and the inner product between them. For instance the strength of a field at a point is the length of the field vector at that point.
The inner product of a force with a length element is needed to compute work done against that force. These calculations use use the standard inner product on Euclidean space. The same is true on a manifold but the metric may vary from point to point rather be constant as it is in Euclidean space. In General Relativity the metric depends on the distribution of mass in the universe and varies from point to point.
The way you "measure space-time" is the same as you measure the length of curves on a surface: you integrate the tangent vectors along the curve.
your course is about manifolds?
if you have a manifold, the metric gives you a way to measure the metric properties of the tangent plane. So, if you have a way to measure tangent vectors, by integration you have a way to measure distance on the manifold (see intrinsic metric).
Yes, I think my main problem was the concept, but thanks this I have a better understanding of it. And yes I'm having my first course on Diff. Geometry, the notation is very confusing, so any advice about a good book about Diff. Geo. would be great.
Another question. I have Wald. of General Relativity. Would it be recommendable to read it or should I better let it for other course??
think of the three basic plane geometries: euclidean plane, hyperbolic plane, and spherical geometry double (elliptic plane). Some people like to do geometry by defining a distance between any two points using a "ruler" along every line.
Euclidean and Hyperbolic both admit rulers for their lines but elliptic geometry does not because the "lines" are closed circles. Of course one could still use the length or distance between two points considered as points of 3 space, but this involves non essential concepts that use data from outside the spherical plane we are working with.
E.g. if you fold a plane over and get a half cylindrical shaped surface, like a blanket draped over a horse, then the distances on the surface are still the same if measured along shortest paths on the surface, but as points of three space some of them would be closer together "as the crow flies", except that a crow in the given plane cannot fly through the folded plane ( i.e. through the horse).
So we want to define all our distances in an intrinsic fashion using only data in the plane we are looking at. One way to do this is to define instead a length on the tangent vectors to the plane and then use this to measure the speed of points moving along a path in the plane and then integrate the speed to get the length of the path.
So we pass from the space itself to a more complicated gadget which has a vector space of tangent vectors attached to each point of the space, the tangent "bundle". Then we define a dot product on each tangent space in a smoothly varying way. Then we can measure lengths of smooth paths in our space by integrating their velocity vectors.
This family of dot products is called a riemannian metric. In good cases it defines a distance function on the space by taking the length of the shortest path between two points as the distance between them, if that minimum exists.
So it is more work, since you have to prove all these things make sense before using them, but afterwards you have a more general and more intrinsic and presumably more powerful concept to work with.
I don't know much about the alternatives, but the books by John M. Lee are so good that I find it very hard to believe that there could be a better option. I'm talking about two books with the titles "Introduction to smooth manifolds" and "Riemannian manifolds: an introduction to curvature". The only problem is that the material is broken up into two books. The latter (which was actually written first) is a pretty short book that contains the material about connections, parallel transport, geodesics and curvature.
Wald's book is a good GR book that takes the mathematics seriously. It's a good place to read about the most important application of differential geometry. Read it if you're interested in GR. Don't read it if you only want to pass the exam. (Read Lee's books instead).
I also strongly recommend Lee's book "Riemannian manifolds: an introduction to curvature". If you are interested in GR and its underlying mathematics, "Semi-Riemannian Geometry With Applications to Relativity" by Barrett O'Neill is also good.
Hi, ok here is my 2 cents.
Let's just take a normal n-dimensional vector space V to start with. Then a metric, in the differential geometry sense, is a symmetric non-degenerate bilinear form on V. If we define it by g say, then g eats up two vectors and spits out an element of the field over which V is defined. For example g(v,w) = k.
Something for you to think about: What role does the non-degeneracy play? Why do we want it?
Ok so far so good. Now, for an n-dimensional manifold, we have an n-dimensional vector space at each point - the tangent space. So a metric on the manifold would be a metric in the way we have defined it above for each of the tangent spaces. But we want it to vary smoothly from point to point.
Q: What does the smoothness mean here?
Hope this helps!
I have heard a little about Lee's books, I will go for them and check them. And the purpose of my Diff. Geometry course is to go later for a GR course. But I think I will let Wald's for later.
So, Do you recommend for a first reading Lee's Introduction to smooth manifold or Riemannian manifolds: an introduction to curvature???
The first "cent" I think is clear and second one very good question, I will think about that for a while.
Many people recommend Sean Carroll's lecture notes on GR as a great place to learn from. They can be found here http://arxiv.org/pdf/gr-qc/9712019v1
Start with "Introduction to..." and read enough to make sure that you understand manifolds, tangent spaces, and tensor fields. Then you can choose if you want to read more of that book or read the first few chapters of "Riemannian...". (There's nothing in "Introduction to..." about connections, parallel transport, geodesics and curvature. These concepts are essential in GR).
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