Undergrad Position Vector in Curved Space Time: Explained

[NSRG]

TL;DR

Can we write a position vector or a general vector in a curved space?

It is said that: It is not possible to write a position vector in a curved space time. What is the reason?

How can one describe a general vector in a curved space time?

Can you please suggest a good textbook or an article which explains this aspect?

 

[Orodruin]

A general manifold is not an affine space. A position vector only makes sense in an affine space, where it is the translation vector from an arbitrarily chosen origin to a given point.

 

[NSRG]

Thanks. Can you suggest a good source where I can read about vectors in curved space time? Both position and general vectors.

 

[Orodruin]

You cannot read about position vectors in curved spaces (as they do not exist). For general vectors, i.e., tangent and dual vectors, any book that cover basic calculus on manifolds should do.

 

[PeroK]

Can you suggest a good source where I can read about vectors in curved space time?

Sean Carroll's GR notes are here:

https://arxiv.org/pdf/gr-qc/9712019.pdf

 

[Dale]

Sean Carroll's GR notes are here:

https://arxiv.org/pdf/gr-qc/9712019.pdf

Especially chapters 1 and 2.

 

[Dale]

Summary:: Can we write a position vector or a general vector in a curved space?

It is said that: It is not possible to write a position vector in a curved space time. What is the reason?

Here is an example.

The surface of the Earth is a curved manifold, and New York and Paris are two points on that manifold.

Now, in vector spaces you can add two vectors to get another vector in the same space. So if the surface of the Earth were a vector space then you could add New York + Paris to get a specific other point on earth. So where is New York + Paris? It doesn't make sense, so the surface of the Earth is not a vector space.

One reason that it is not a vector space is that there is no origin, which is partly why adding New York + Paris makes no sense. So can we arbitrarily pick some point on Earth (say New York) and make that the origin and get a vector space that way? Spaces where you can do that are called affine spaces, so this question boils down to determining if the surface of the Earth is an affine space.

So let's take the surface of the Earth and use the north pole as the origin. Then we can think about making vectors where the direction of the vector is the longitude line and the magnitude of the vector is how far along that longitude line you have to travel from the north pole. This covers the entire surface of the earth, so that is good. However, one of the properties of an affine space is that the mapping between the affine set (after selecting an origin) and the vector space must be one-to-one. Unfortunately, this fails on the sphere because you can always increase the vector magnitude by one Earth circumference to get back to the same point, and the south pole can be reached in any direction. So the surface of the Earth is not an affine space either and we cannot get a vector space by choosing an origin.

Although this example is specific to a sphere, more rigorous mathematicians can prove that these problems or other problems arise for any curved space. You cannot treat points in a curved manifold as elements of a vector space.

 

[NSRG]

Thanks

 

[pervect]

Thanks. Can you suggest a good source where I can read about vectors in curved space time? Both position and general vectors.

Consider the following remarks from Misner's "Precis of General Relativity", https://arxiv.org/abs/gr-qc/9508043

Think of a Caesarian
general hoping to locate an outpost. Would he understand that 600 miles
North of Rome and 600 miles West could be a different spot depending on
whether one measured North before West or visa versa?

This remark may help motivate the observation of why we say that displacements on the Earth cannot be vectors. Vector addition, by definition, must commute, so therefore displacments on the curved surface of the Earth are not vectors.

There is a workaround for small displacements. True vectors (with commutative addition) can exist in the tangent space of a manifold. For the example above, with the Earth's surface being a manifold, a tangent space would be a tangent plane to the (nearly) spherical surface of the Earth. Furthermore, one can create a map from the tangent space to the manifold itself, such as the so-called exponential map. This sort of technique can create a flat projection of a curved surface which can be of practical utility in a local area, for instance one can use a flat streetmap to navigate a portion of a city, though one needs a globe (or some equivalent technique) to navigate large distances on the Earth.