Approximate local flatness = Approximate local symmetries?

In summary, Pseudo-Riemannian manifolds have locally Minkowskian properties, which is crucial for relativity. However, this is only an approximation as highly curved spacetimes cannot be fully flattened. While this may seem to suggest that local symmetries, such as Poincaré and Lorentz, are also approximate, they are exact in the tangent space at each event, but not in the overall spacetime.
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Suekdccia
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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?
 
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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.
 
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FAQ: Approximate local flatness = Approximate local symmetries?

What is the concept of approximate local flatness?

Approximate local flatness refers to the idea that, on sufficiently small scales, a curved space can be approximated as flat. This is akin to how the surface of the Earth appears flat to someone standing on it, even though it is actually a sphere. In mathematics and physics, this concept is crucial for simplifying complex problems in curved spaces by treating them as if they were in flat, Euclidean space.

How does approximate local symmetry relate to approximate local flatness?

Approximate local symmetry is the idea that, in a small enough region, the properties of a system or space exhibit symmetry that may not be present on larger scales. This relates to approximate local flatness because both concepts involve simplifying complex, often curved or irregular structures by examining them on small scales where they behave more predictably and symmetrically, often as if they were flat.

Why are these concepts important in physics?

These concepts are crucial in physics because they allow scientists to apply simpler, well-understood models to study complex systems. For example, in general relativity, spacetime is curved by mass and energy, but by considering approximate local flatness, physicists can use the principles of special relativity locally. Similarly, approximate local symmetries can simplify the study of particle interactions and other phenomena by applying symmetric models locally.

Can you give an example of approximate local flatness in everyday life?

An everyday example of approximate local flatness is the surface of the Earth. While the Earth is a sphere, its surface appears flat to someone standing on it, especially over short distances. This approximation allows for the use of flat maps for navigation and other purposes without significant error over small areas.

How do these concepts assist in mathematical modeling?

In mathematical modeling, approximate local flatness and symmetry allow for the use of simpler equations and models that are easier to solve and analyze. For instance, differential equations that describe physical phenomena in curved spaces can often be approximated by their flat-space counterparts locally, making the problem more tractable. This approach is widely used in fields such as differential geometry, general relativity, and quantum field theory.

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