Diffeomorphisms in Flat-Space: Are All Metric Preserving?

Click For Summary

Discussion Overview

The discussion centers around the nature of diffeomorphisms in flat spacetime, specifically within the context of Minkowski spacetime. Participants explore whether all diffeomorphisms are metric preserving and the relationship between diffeomorphisms and isometries.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether the group of diffeomorphisms in Minkowski spacetime is equivalent to the Poincaré group or if there exist diffeomorphisms that are not metric preserving.
  • Another participant asserts that there are indeed diffeomorphisms that are not isometries, suggesting a variety of such mappings exist for the Minkowski metric.
  • A further contribution emphasizes that passive diffeomorphisms, which are essentially coordinate changes, do not necessarily correspond to isometries, indicating a broader range of possible coordinate systems in Minkowski space.
  • One participant clarifies that a diffeomorphism is defined as a smooth, invertible map from the manifold to itself, independent of any metric considerations.

Areas of Agreement / Disagreement

Participants generally agree that there are diffeomorphisms that are not isometries, but the discussion reflects competing views on the implications of this and the nature of diffeomorphisms in relation to metrics.

Contextual Notes

The discussion does not resolve the implications of the relationship between diffeomorphisms and isometries, nor does it clarify the full scope of diffeomorphisms in flat spacetime.

jfy4
Messages
645
Reaction score
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:
Physics news on Phys.org
So you reached the conclusion that there are diffeomorphisms which are not isometries ?
 
yes. there are definitely diffeomorphisms that are not isometries, a lot of them even for the minkowski metric.
 
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.
 
A diffeomorphism is just a (smooth, invertible) map from the manifold to itself. No need to even have a metric to talk about diffeomorphisms.
 

Similar threads

Undergrad Explore Spacetimes, Metrics & Symmetries in Relativity Theory
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad How do non-diagonal indices of a metric allow for local flatness?
  • · Replies 22 ·
Replies
22
Views
2K
Graduate Overall perturbation for two free-falling particles in flat spacetime
  • · Replies 9 ·
Replies
9
Views
1K
High School Topology on flat space when a manifold is locally homeomorphic to it
  • · Replies 25 ·
Replies
25
Views
3K
Undergrad Follow-up Einstein Definition of Simultaneity for Langevin Observers
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Characterizing GR Traj in Minkowski Space
  • · Replies 17 ·
Replies
17
Views
2K
Graduate Is 'Local Flatness' the Right Term for Describing Spacetime?
  • · Replies 99 ·
4
Replies
99
Views
12K
Undergrad Einstein Definition of Simultaneity for Langevin Observers
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Understanding ISSS from Minkowski Space to Poincaré Group
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Covering Group of SO(g) & Understanding Spinors on Curved Spacetime
  • · Replies 15 ·
Replies
15
Views
2K