I Approximate local flatness = Approximate local symmetries?

AI Thread Summary
Pseudo-Riemannian manifolds, like spacetime, can be locally approximated as Minkowskian, which is crucial for understanding relativity in curved spacetimes. This approximation raises the question of whether local symmetries, such as Poincaré and Lorentz, also hold only approximately. However, these local symmetries are exact within the tangent space at each event, not merely approximations of the overall spacetime. Therefore, while spacetime may be curved, the symmetries in the tangent space remain precise. This distinction is essential for the application of local symmetries in the context of relativity.
Suekdccia
Messages
352
Reaction score
30
TL;DR Summary
Approximate local flatness = Approximate local symmetries?
Pseudo-Riemannian manifolds (such as spacetime) are locally Minkowskian and this is very important for relativity since even in a highly curved spacetime, one could locally approximate the spacetime into a flat minkowski one.

However, this would be an approximation. Perhaps this is a naive question but, would this mean that the local symmetries (such as Poincaré, Lorentz...) hold also only approximately?
 
Space news on Phys.org
Suekdccia said:
would this mean that the local symmetries (such as Poincaré, Lorentz...) hold also only approximately?
No. The "local symmetries" you refer to are symmetries of the tangent space at each event. They are not symmetries of the spacetime. In the tangent space those symmetries are exact.
 
https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
The formal paper is here. The Rutgers University news has published a story about an image being closely examined at their New Brunswick campus. Here is an excerpt: Computer modeling of the gravitational lens by Keeton and Eid showed that the four visible foreground galaxies causing the gravitational bending couldn’t explain the details of the five-image pattern. Only with the addition of a large, invisible mass, in this case, a dark matter halo, could the model match the observations...
Hi, I’m pretty new to cosmology and I’m trying to get my head around the Big Bang and the potential infinite extent of the universe as a whole. There’s lots of misleading info out there but this forum and a few others have helped me and I just wanted to check I have the right idea. The Big Bang was the creation of space and time. At this instant t=0 space was infinite in size but the scale factor was zero. I’m picturing it (hopefully correctly) like an excel spreadsheet with infinite...
Back
Top