Diffeomorphisms in Flat-Space: Are All Metric Preserving?

In summary, the conversation discusses the relationship between the Minkowski spacetime and the group of diffeomorphisms. The question is whether the poincare group is the only group of diffeomorphisms for flat-space or if there are others that do not preserve the metric. It is concluded that there are indeed diffeomorphisms that are not isometries, even for the Minkowski metric. This is because diffeomorphisms are just smooth, invertible maps from the manifold to itself, and do not necessarily require a metric to be discussed.
  • #1
jfy4
649
3
Hi,

I'm sorry to have to ask this, but I can't seem to reason this one out by myself at the moment. Given the metric is the Minkowski spacetime, is the group of diffeomorphisms the poincare group, or are there diffeomorphisms for flat-space that are not metric preserving?

I would really appreciate your help. Thanks,

EDIT: Nevermind, sorry for making a new thread for this... I had some tea and thought it out.
 
Last edited:
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  • #2
So you reached the conclusion that there are diffeomorphisms which are not isometries ?
 
  • #3
yes. there are definitely diffeomorphisms that are not isometries, a lot of them even for the minkowski metric.
 
  • #4
Yeah, remember that passive diffeomorphisms are just coordinate changes, and I'm sure you can imagine all kinds of coordinates on Minkowski space that have nothing to do with isometries.
 
  • #5
A diffeomorphism is just a (smooth, invertible) map from the manifold to itself. No need to even have a metric to talk about diffeomorphisms.
 

FAQ: Diffeomorphisms in Flat-Space: Are All Metric Preserving?

What is a diffeomorphism to Poincaré?

A diffeomorphism to Poincaré is a type of mathematical transformation that preserves the structure of a mathematical object known as the Poincaré sphere. In simpler terms, it is a way of mapping one shape or space onto another while maintaining certain geometric properties.

What are the properties preserved by diffeomorphisms to Poincaré?

Diffeomorphisms to Poincaré preserve the topology, which refers to the properties of a shape or space that remain unchanged even when the object is stretched, twisted, or deformed. They also preserve the metric, which is the measurement of distances and angles within a space.

How are diffeomorphisms to Poincaré used in physics?

In physics, diffeomorphisms to Poincaré are used to describe the symmetries of physical systems. This includes the symmetries of space and time, which are crucial in understanding the laws of physics. They are also used in theories such as general relativity and string theory.

Can diffeomorphisms to Poincaré be applied to any shape or space?

Yes, diffeomorphisms to Poincaré can be applied to any shape or space that has a well-defined metric and topology. This includes spheres, tori, and other curved surfaces, as well as more complex spaces such as manifolds.

Are there any practical applications of diffeomorphisms to Poincaré?

Yes, diffeomorphisms to Poincaré have practical applications in various fields such as computer graphics, image processing, and shape analysis. They are also used in medical imaging to analyze and compare shapes of organs and structures in the body.

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