About coordinate transformations in general

[arkajad]

But I don't know why we have to require x-space to be in Rⁿ and q-space to be in the same or different Rⁿ.

Perhaps because both x^i and q^i are n real numbers?

 

[friend]

Perhaps because both x^i and q^i are n real numbers?

Some books I read say Rⁿ is just a list of n real numbers. Other's say it is the cartisian product of n copies of the real line; the metric of the real line probably enters here. And still others say Rⁿ has the euclidean metric. But perhaps they are assuming from the start what others end up adding later. Is this your experience too?

If x^i and q^i are n real numbers, that leaves the question if x-space and q-space have to be the same Rⁿ, the same space coordinatized with a different list of n numbers.

 

[arkajad]

There are different interpretations. The context is very important. If we are not paying attention to the context - it is easy to get lost. For instance q^i(x^j) can be interpreted at least in three ways:

a) as a map from one manifold with coordinates q to another one with coordinate x
b) as a change of coordinates in the same manifold
c) as an active transformation (a diffeomorphism) of a given manifold - in this case q are coordinates of the transformed point whose original coordinates were x.

 

[friend]

There are different interpretations. The context is very important. If we are not paying attention to the context - it is easy to get lost. For instance q^i(x^j) can be interpreted at least in three ways:

a) as a map from one manifold with coordinates q to another one with coordinate x
b) as a change of coordinates in the same manifold
c) as an active transformation (a diffeomorphism) of a given manifold - in this case q are coordinates of the transformed point whose original coordinates were x.

It's hard to accept that the same notation can mean different things. I think the difference is that the notation may hide further mappings. For example, in b) the point, p, in the point-set/manifold, M, can be mapped into two different coordinate systems, via two different coordinate functions. But in a) one of the coordinate functions may be a composite of two functions, where one maps p in M first to a different point-set/manifold, q in N. And the second of the composites maps q into some second coordinate system. This way the notation is correct in both senarios. It would only be that M=N in b).

 

[arkajad]

It's hard to accept that the same notation can mean different things.

Why? When I write just A, then it can mean a lot of different things. Always pay attention to the context.

 

[friend]

My next question is can you tranform from coordinates of dimension n to coordinates of dimension m? Or is the dimensionality inherent in the point-set of the manifold? I would think that you can label points with as large a list of numbers as you want, just x or (x,y,z) if you wish, as long as you have unique coordinates for each point, right? What do functions care about the demensionality? I mean f(w)=f(x,y,z), as long as w and (x,y,z) refer to the same point, right? But then I don't see how the integration of f can be equally done in different dimensions. How can lenghth be equal to area or volume? Maybe differentials and summing differentials in integration is done in the tangent space which does have an inherent dimensionality. Yet I think I've seen where the jacobian can be a nXm matrix, which would transform between tangent spaces of different dimensionality. But the determinate of the jacobian, which governs integration transformations, can only be done with an nXn matrix. Any insight out there? Thanks.

 

[arkajad]

Why should length to be equal to area?

 

[friend]

Why should length to be equal to area?

I'm not sure length or area or volume is inherent in the point-set of the manifold. I assume length or area is encoded in the metric, right? So where exactly does the metric necessarily enter the picture? Is it inherent in any coordinate system? Is it necessary to do a coordinate transformation? Is it necessary to have a tangent space, or to transform between tangent spaces? Maybe it's necessary as soon as vectors enter the picture.

 

[arkajad]

So where exactly does the metric necessarily enter the picture? Is it inherent in any coordinate system?

No.

Is it necessary to do a coordinate transformation?

No.

Is it necessary to have a tangent space, or to transform between tangent spaces?

No.

Maybe it's necessary as soon as vectors enter the picture.

No. But it is handy when you want to define distance on your manifold. Distance that is independent of any coordinate system.

 

[friend]

I'm thinking that maybe a metric is needed to define a tangent space. For tangent spaces are defined to be "flat", which implies a metric. And I'm not sure how one would define a vector without a metric. For vectors are defined to have a direction and "length", which implies a metric. Or can a vector be just one point with respect to another point in the point-set of the manifold? But there seems to be some notion of a line, or shortest distance from the beginning to the end of a vector; I don't see how that's accomplished without a metric.

Can you even defined the closeness between two points without a metric? Can you even say that one point is "next to" another without a metric? There does seem to be the idea of subsets of points, and unions and intersections of sets of points. But sets can be constructed of disperse subsets. Or does the idea of a topology, where unions and intersections must be included, does this imply that points must belong together with other points. Is this a notion of closeness?

 

[friend]

I'm thinking that maybe a metric is needed to define a tangent space. For tangent spaces are defined to be "flat", which implies a metric. And I'm not sure how one would define a vector without a metric. For vectors are defined to have a direction and "length", which implies a metric. Or can a vector be just one point with respect to another point in the point-set of the manifold? But there seems to be some notion of a line, or shortest distance from the beginning to the end of a vector; I don't see how that's accomplished without a metric.

But then again, it seems you don't know the dimensionality of the tangent space without knowing the dimensionality of the manifold to which it is tangent.

 

[friend]

In transforming from coordinates of dimension n to coordinates of dimension m, I asked if the dimension of a manifold was inherent or imposed? I think it is inherent for the following reason. If we have two manifolds, one of dimension r and the other of dimension d, then you can construct a product manifold of these two, and it will have dimension r+d. So it seems the dimensionality of a manifold comes from how many times you can break it down to a product of 1 dimensional manifolds. I think it is that the underlying topology must be constructed of a product of topologies, that you can find in the sets of the overall topology, subsets that can be mapped to the real line. This seems inherent in the sets of the topology and not any coordinates system imposed on it.

I'm thinking that there must be some neighborhood that breaks down to the same number of dimensionality. For a point can be broken down to all the different sets in the topology to which it belongs. I don't think that's what's meant by dimension. Dimension seems to be something employed in a continuous manner which implies a neighborhood of some sort.

My earlier idea of dimensionality being arbitrary because you always had the freedom to label any point with as many dimensions as desired would only apply to sets with discrete points but not to a continuum. I recently read where a discrete point set has dimensionality of zero.

Does all this sound right? Thanks.

 

[friend]

I've attached 2 pages from M Nakahara's book, "Geometry, Topology And Physics, 2nd Ed", pages 186 and 187. I'm wondering if I'm reading these pages right.

Fig 5.10 shows that a contravariant vector, which is a tensor of rank 1, being transformed from one manifold to an entirely differently manifold which may have an entirely different curvature. It leads to eq 5.33 which is the transformation rule for components of a contravariant vector. This shows that contravariant tensors transform across spaces with the same rules used to transform them across coordinate patches on the same manifold.

Page 187, equation 5.35 shows that a function between spaces also induces a pullback that transforms 1-form, which are covariant vectors which are also tensors of rank 1, from one space/manifold to an entirely different space. And it leads to eq 5.37 which is the transformation rule for covariant vectors. This shows that forms, or covariant tensors, also transform across spaces using the same rule as when transforming across coordinate patches on the same space.

Since tensors of any mix and rank can be constructed from contravariant tensors and covariant tensors. I conclude that the transformation rule for tensors works across spaces as well as across patches.

And since differential forms are constructed by symmetrizing covariant tensors, I assume differential forms transform across spaces in the same way they transform across patches.

And since the jacobian used in integration works across spaces as well as patches, I conclude that integrating differential forms also transforms across spaces in the same way as across patches.

So is it right that all of tensor calculus transforms across spaces with the same rules as transforming across coordinate patches? Thanks.

 

[haushofer]

Yes, I believe so.

I had a related question about this here some time ago, about the duality between "passive" (across coordinate patches) and "active" (via the differential map) transformations,

https://www.physicsforums.com/showthread.php?t=280232

Maybe you like it :)