# General Relativity math, Tangent vectors on manifolds

• Domisterwoozy
In summary: So in 3.4f the delta indices run from 1...n, and you just have to get used to the author's notation. And remember, the author is free to use whatever coordinates he wants, as long as he's consistent. And he is consistent, but confusing. So, for example, in 3.4f he writes xj=xj, but he means xj=xj(y1...ys). And he doesn't want to write y1=...=ys=0, so he writes the fancy delta's. I think he should write y1=...=ys=0. It's a better way to understand the definition.In summary, the conversation
Domisterwoozy
I am currently going through some online notes on differential geometry and general relativity. So far I have been following pretty well until I got to 3.4 cont'd letter (e) of http://people.hofstra.edu/Stefan_Waner/diff_geom/Sec3.html" document and the definition directly after. My first question is, is yi(t) many different paths on the manifold or is each i a component of one path on the manifold? Also in the definition after the example I think yj are the ambient components of the actual manifold, not the path, and xi are the local components of the manifold, not the path. Is this correct? And if so how do you take the partials if the number of components of the ambient coordinates are usually one more then the number of components of the local coordinates?

If my questions do not make sense, just ask and I will try to make them more clear, and if this is not the right place for this just direct me where to go. Also if anyone had any better suggestions on places to learn this stuff I am happy to hear it.

Last edited by a moderator:
Domisterwoozy said:
I am currently going through some online notes on differential geometry and general relativity. So far I have been following pretty well until I got to 3.4 cont'd letter (e) of http://people.hofstra.edu/Stefan_Waner/diff_geom/Sec3.html" document and the definition directly after. My first question is, is yi(t) many different paths on the manifold or is each i a component of one path on the manifold?
The later, it's one path.

Also in the definition after the example I think yj are the ambient components of the actual manifold, not the path,

I'm not following this. The yj are just coordinates. A complete set of coordinates specifies a point on the manifold, just like it does on a map. Though a purist would note that a point is different from the coordinates that specify it.

I suspect you're confused somehow about the first defintion, but I can't quite grasp how you're confused, so maybe I'm wrong and I'm confused. Anyway...

In loose language, the first definition just says that a smooth path is just a mapping from a line segment on R to a set of points on the mainfold (I'd call it a curve on the manifold, but that's sort of what we're trying to define), such that every point on the line corresponds to one point on the manifold, said point specified by its coordinates, yj, and the mapping is "smooth". They haven't gotten into the details of "smoothness" in the notes, so I won't get into it either.

Last edited by a moderator:
Any point on the N-dimensional manifold is specified by N coordinates {x1...xN}.

Roughly, to specify a curve, which is 1-dimensional, we need N-1 equations to "get rid" of N dimensions. These are the N equations {x1=f1(t) ... xN=fN(t)}, which we think of as N-1 equations by notionally eliminate t to obtain N-1 relations among N coordinates.

As an explicit example, in 2D with coordinates {x,y} a circle can be written as {x=cost,y=sint} or x2+y2=1.

Semantic note: many define a curve to be a parametrized path so that {x=cost,y=sint} is a curve, but x2+y2=1 is a path.

Last edited:
Thanks for the reply.

I am pretty sure I understand the basic definition of a path. Where I am running into trouble is I thought that the ambient coordinates consist of s coordinates and the local coordinates consist of n coordinates where s = n +1. Basically the manifold is embedded into a one higher dimension. However, in that example (e) they assign xj based off of if i = j, however I thought that j > i since there are more y coordinates then x coordinates. Therefore how can you compare the two and make a Kronecker delta which I thought was supposed to be a square matrix.

Domisterwoozy said:
Thanks for the reply.

I am pretty sure I understand the basic definition of a path. Where I am running into trouble is I thought that the ambient coordinates consist of s coordinates and the local coordinates consist of n coordinates where s = n +1. Basically the manifold is embedded into a one higher dimension. However, in that example (e) they assign xj based off of if i = j, however I thought that j > i since there are more y coordinates then x coordinates. Therefore how can you compare the two and make a Kronecker delta which I thought was supposed to be a square matrix.

i,j are just indices which he uses for both local and ambient coordinates, so whether they run from 1...n or 1...s depends on whether local or ambient coordinates are being indexed (I think). So in 3.4e the Kronecker delta indices run from 1...n, the dimension of the manifold, not the ambient space.

Last edited:

## 1. What is the concept of General Relativity math?

General Relativity math is a mathematical framework developed by Albert Einstein to explain the effects of gravity on the curvature of space and time. It is based on the principle that the laws of physics should be the same for all observers, regardless of their relative motion.

## 2. How is General Relativity math different from Newton's laws of gravity?

Unlike Newton's laws which describe gravity as a force between masses, General Relativity math explains gravity as a result of the curvature of space and time caused by the presence of mass and energy. It also predicts more accurate results in extreme conditions such as near black holes or during the early stages of the universe.

## 3. What are tangent vectors on manifolds?

Tangent vectors on manifolds are mathematical objects that represent the direction and magnitude of change at a specific point on a curved surface. In General Relativity, they are used to describe the motion of particles and the flow of energy in a curved spacetime.

## 4. How is the curvature of spacetime measured in General Relativity?

The curvature of spacetime is measured using the mathematical tool of tensors, specifically the Riemann curvature tensor. This tensor describes how the curvature of space and time changes depending on the distribution of mass and energy in the universe.

## 5. What are some real-world applications of General Relativity math?

General Relativity math has been used to accurately predict phenomena such as the bending of light near massive objects, the existence of gravitational waves, and the expanding universe. It also has practical applications in technologies such as GPS systems and space travel.

Replies
7
Views
1K
Replies
8
Views
2K
Replies
4
Views
1K
Replies
20
Views
1K
Replies
63
Views
4K
Replies
3
Views
1K
Replies
21
Views
1K
Replies
73
Views
2K
Replies
5
Views
1K
Replies
83
Views
5K