Locally flat - what does it mean?

  • Thread starter paweld
  • Start date
  • Tags
    Flat Mean
In summary: Anyway, the coordinates transformations must satisfy very strong conditions in order to be considered "geodesic". You can't just transform coordinates in any arbitrary way and call it good enough. Completely true! You don't need to even think of "local flatness" in terms of Christoffel symbols when in GR. One just needs to set the first derivatives of metric tensor equal to zero which fits within the definition of "geodesic coordinates" which is not actually an actual coordinate system (read the point 1 below)! But remember that there is a strongly local coordinates in the sense that just at a point you can make the values of metric equal those of Minkowski metric. Here two points must be made: 1- In
  • #1
paweld
255
0
What does it mean in general relativity that space is locally flat. It means that in neighbourhood of each point we can chose such coordinates that the metric is flat or we can do it only in a point (not in neighbourhood - open set).
 
Physics news on Phys.org
  • #2
It means that given any point, we may choose a set of coordinates in which the metric is flat up to first order (i.e., the metric is flat at the point, and its first derivatives all vanish). The second derivatives cannot be made to vanish in general (since the second derivatives are basically the same thing as curvature).

Stated another way, it means that given any positive epsilon, we may find a neighborhood around the point where the deviation from flatness is smaller than epsilon.
 
  • #3
paweld said:
What does it mean in general relativity that space is locally flat. It means that in neighbourhood of each point we can chose such coordinates that the metric is flat or we can do it only in a point (not in neighbourhood - open set).

Unfortunately, the term "locally flat" is not used consistently, and you have identified the two main uses of the term. The first definition is used by mathematically careful references, but the second definition is used by many physicists.

An example ot the former, the book Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres, writes:

"A manifold M with affine connection is said to be locally flat if for every point p in M there is a chart (U; x^i) with such that all the components of the connection vanish throughout U. This implies of course both torsion tensor and curvature tensor vanish throughout U, ..."
 
Last edited:
  • #4
George Jones said:
An example ot the former, the book Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres, writes:

"A manifold M with affine connection is said to be locally flat if for every point p in M there is a chart (U; x^i) with such that all the components of the connection vanish throughout U. This implies of course both torsion tensor and curvature tensor vanish throughout U, ..."

Other mathematically careful reference that have similar definition for local flatness include: Analysis, Manifolds and Physics; Tensor Analysis on Manifolds.
 
  • #5
George Jones said:
"A manifold M with affine connection is said to be locally flat if for every point p in M there is a chart (U; x^i) with such that all the components of the connection vanish throughout U. This implies of course both torsion tensor and curvature tensor vanish throughout U, ..."

How in the world does this even make sense? Wouldn't such a manifold automatically be globally flat (though with possibly non-trivial topology)?
 
  • #6
Ben Niehoff said:
How in the world does this even make sense? Wouldn't such a manifold automatically be globally flat (though with possibly non-trivial topology)?

And if the manifold has non-trivial topology, then there isn't necessarily a global coordinate system (in which the components of the connection are zero), hence the concept is formulated in terms of coordinate neighbourhoods in manner in which "local" is used for other mathematical concepts, for example, "locally compact".

Now, let me ask you a question. What restriction does your definition place on (semi)Riemannian manifolds? None; every (semi)Riemannian manifold satisfies your definition. Each definition is, in some sense, trivial. :smile:
 
  • #7
paweld said:
What does it mean in general relativity that space is locally flat. It means that in neighbourhood of each point we can chose such coordinates that the metric is flat or we can do it only in a point (not in neighbourhood - open set).

A quote from me explains what LF means in general:

Altabeh said:
Completely true! You don't need to even think of "local flatness" in terms of Christoffel symbols when in GR. One just needs to set the first derivatives of metric tensor equal to zero which fits within the definition of "geodesic coordinates" which is not actually an actual coordinate system (read the point 1 below)! But remember that there is a strongly local coordinates in the sense that just at a point you can make the values of metric equal those of Minkowski metric. Here two points must be made:

1- In general, we can hardly determine a "coordinate transformation" [tex]x^{\mu}\rightarrow {\bar{x}}^{\mu}[/tex] by which at a point P, one has [tex]g^{\mu\nu}(P)={\eta}^{\mu\nu}(P)[/tex]. If the metric is diagonal, the number of equations of its transformation would be much less than the case when metric is considered to be symmetric. This is because, for instance, if we count the number of equations involved in a symmetric metric trans., that is,

[tex]n(n+1)/2[/tex],

then you must fit at least this number of arbitrarily-chosen contants within the coordinates transformation at any given point to form a set of equations with the same number of unknowns and equations. But in the diagonal case, this number reduces to n, so the system of equations gets much simpler to be solved.

2- Geodesic coordinates just account for the first derivatives of metric being all equal to zero, so this way of leading to the local flatness at some point is another alternative.

3- Following 1, in the neighborhood of P the spacetime is "nearly" flat within a range that the equivalence principle issues. This is because we don't know what is meant by "neighborhood" in GR and this is only evaluated\estimated by EP. This might not be applicable for the case 2.

AB
 
Last edited:

What does it mean for something to be "locally flat"?

"Locally flat" refers to the property of a surface or space being flat or planar when examined at a small scale or within a small region. This means that if you were to zoom in on a small portion of the surface or space, it would appear flat and not curved or distorted.

In what contexts is the term "locally flat" commonly used?

The term "locally flat" is often used in mathematics, particularly in the study of differential geometry and topology. It is also used in physics, particularly in the fields of general relativity and cosmology, to describe the local geometry of spacetime.

How is "locally flat" different from "globally flat"?

"Globally flat" refers to a surface or space that is flat or planar when examined at a large scale or over a large region. This means that if you were to zoom out and look at the entire surface or space, it would still appear flat. In contrast, something that is "locally flat" may appear curved or distorted when examined at a larger scale.

Can a surface or space be locally flat and globally curved?

Yes, it is possible for a surface or space to exhibit both local and global curvature. For example, a sphere is globally curved, but if you were to zoom in on a small portion of its surface, it would appear locally flat.

How is the concept of "locally flat" important in scientific research?

The concept of "locally flat" is important in many areas of scientific research, particularly in fields such as mathematics, physics, and computer science. It allows scientists to understand and describe the behavior of complex surfaces and spaces by examining them at a small scale or within a small region. This concept also plays a crucial role in the development of theories and models used to explain various phenomena in the natural world.

Similar threads

  • Special and General Relativity
3
Replies
99
Views
9K
  • Special and General Relativity
Replies
8
Views
217
Replies
40
Views
2K
  • Special and General Relativity
Replies
30
Views
499
  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
Replies
25
Views
2K
  • Special and General Relativity
Replies
10
Views
1K
  • Special and General Relativity
Replies
34
Views
678
  • Special and General Relativity
6
Replies
186
Views
6K
Replies
62
Views
4K
Back
Top