Is 'Local Flatness' the Right Term for Describing Spacetime?

  • A
  • Thread starter Orodruin
  • Start date
  • Tags
    Local pet
  • Featured
In summary: This is not generally true as curvature invariants can be computed to be non-zero even though there are local inertial frames at all events.I agree with this point. It can be misleading and confusing, and it is important to use precise terminology in physics. "Local flatness" is not an accurate term to describe the concept of having a local inertial frame. Instead, "local inertial frame" or "local inertial coordinates" would be more appropriate and accurate.
  • #1
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
21,063
12,118
I see many posts by several different people referring to spacetime being "locally flat" with the intended meaning of being locally indistinguishable from Minkowski space, i.e., being able to rewrite the metric on orthonormal form and not being able to measure curvature on some local scale. I do not think this is an appropriate nomenclature and the more appropriate nomenclature would be to refer to a local inertial frame. I am aware that some textbook authors, such as Schutz, use the term in this way as well. These are (some of) my issues with the terminology:
  • "Local flatness" is typically defined in a different manner in topology, where it is a property of a submanifold. The entire point of using differential geometry is that spacetime can be described without reference to it being a submanifold of some higher-dimensional space.
  • Not withstanding the previous point, we otherwise use "local" to describe a property that is only true in a point or in a neighbourhood of that point. "Flat" refers to the curvature being zero. Putting those two together as "locally flat" would therefore typically mean that the curvature at the given event (or neighbourhood) would be zero. This is not generally true as curvature invariants can be computed to be non-zero even though there are local inertial frames at all events.
  • There exists other alternative terminology to describe precisely the ideas that "locally flat" intends to convey. The existence of a "local inertial frame" or similar comes to mind.
Any thoughts? Am I just being picky?
 
  • Like
Likes Raymond Beljan, nomadreid, George Jones and 2 others
Physics news on Phys.org
  • #2
Orodruin said:
Any thoughts? Am I just being picky?
I think you have a perfectly valid point. It can be misleading and confusing, it is sloppy, and it is incorrect. So, it should be avoided. And it isn't that much trouble to use the right terminology. I am perfectly fine with abuse of terminology and notations, but this goes too far, and dosen't save anything.
 
  • Like
Likes Jimster41
  • #3
I think you are being picky, but of course it is consistent with a distinguished tradition of complaining with proponents like Synge and Ohanian.

I have never been able to understand this Principle… Does it mean that the effects of a gravitational field are indistinguishable from the effects of an observer’s acceleration? If so, it is false. In Einstein’s theory, either there is a gravitational field or there is none, according as the Riemann tensor does not or does vanish. This is an absolute property; it has nothing to do with any observer’s world-line. Spacetime is either flat or curved… The Principle of Equivalence performed the essential office of midwife at the birth of general relativity, but… I suggest that the midwife be now buried with appropriate honors and the facts of absolute space-time faced.
https://www.mathpages.com/home/kmath622/kmath622.htm
 
  • Like
Likes Jimster41
  • #5
As far as I understand curvature is an observable in GR and has a coordinate independent meaning. Minkowski space has identically 0 curvature and doesn't exist in the wild. What is unclear to me is the assertion that all observable effects of curvature vanish in the small. I don't think you are being picky.
 
  • #6
A quick look for "locally flat" at the indices of some textbooks:
Wald: Not in index.
Carroll: Not in index. Refers to this as "local inertial frame/coordinates" or "local Lorentz frame" for the corresponding basis vectors.
Guidry: Not in index.
Schutz: Introduces the term as equivalent to "local inertial coordinates", then uses "locally flat".
MTW: Not in index. Uses "local inertial frame/coordinates".
 
  • #8
Orodruin said:
Any thoughts?

I used "locally flat" in a B-level thread recently. It seemed like the simplest way to get across the point I was trying to make to that particular poster.

In a more advanced thread I would favor "local inertial frame" or something like it instead.
 
  • #9
Isn't any differentiable manifold in some of the vague senses given above "locally flat"? In a physicist's sloppy language one could simply define a differential manifold as a manifold which locally looks like an affine space, and an affine space is flat by definition.
 
  • #10
PeterDonis said:
I used "locally flat" in a B-level thread recently. It seemed like the simplest way to get across the point I was trying to make to that particular poster.

In a more advanced thread I would favor "local inertial frame" or something like it instead.

The risk we run by using "locally flat" when we mean "local inertial frame" is two-fold in my view. The first risk is that it teaches someone the wrong nomenclature, even if it conveys the concept. This is something that then spreads and has to be corrected at a later stage. The second is that people that see it and know better either dismiss the post or (worse) start arguing in-thread about the correct nomenclature - quickly escalating a B-level thread to A-level.

I do not think much is lost on B-level by instead saying that spacetime is indistinguishable from Minkowski space if you just look at a small enough region of it.
 
  • Like
Likes Dragrath, vanhees71, martinbn and 1 other person
  • #11
vanhees71 said:
In a physicist's sloppy language one could simply define a differential manifold as a manifold which locally looks like an affine space, and an affine space is flat by definition.
The point is that it looks flat only from the view of having a local diffeomorphism. The connection on the manifold is not considered here (which is what is really being referenced when you say "flat" versus "curved" - zero vs non-zero curvature).
 
  • Like
Likes vanhees71 and PeterDonis
  • #12
Orodruin said:
"Local flatness" is typically defined in a different manner in topology, where it is a property of a submanifold. The entire point of using differential geometry is that spacetime can be described without reference to it being a submanifold of some higher-dimensional space.
I don't think that this is an issue. Terminology gets reused between different fields of study all the time.

Orodruin said:
Not withstanding the previous point, we otherwise use "local" to describe a property that is only true in a point or in a neighbourhood of that point. "Flat" refers to the curvature being zero. Putting those two together as "locally flat" would therefore typically mean that the curvature at the given event (or neighbourhood) would be zero. This is not generally true as curvature invariants can be computed to be non-zero even though there are local inertial frames at all events.
I agree here. "Flat" is not what is actually meant.

Orodruin said:
There exists other alternative terminology to describe precisely the ideas that "locally flat" intends to convey. The existence of a "local inertial frame" or similar comes to mind.
The phrase "local inertial frame" doesn't mean the same as "locally flat", since you can use non-inertial coordinates on a small region of curved spacetime and get a metric matching the non-inertial flat spacetime metric to first order.

I am not sure that any reference to local inertial frames correctly conveys that idea. Maybe "locally flat to first order"? But that seems weird too since curvature is a 2nd order phenomenon anyway (maybe that is the point).
 
  • Like
Likes vanhees71
  • #13
Orodruin said:
MTW: Not in index. Uses "local inertial frame/coordinates".

MTW uses the term "local flatness" in 7.5, and "locally flat" in Box 6.1 and Section 20.4.
 
  • #14
Dale said:
The phrase "local inertial frame" doesn't mean the same as "locally flat", since you can use non-inertial coordinates on a small region of curved spacetime and get a metric matching the non-inertial flat spacetime metric to first order.
To zeroth order. If you have non-inertial coordinates then the Christoffel symbols are non-zero and they depend on the first derivatives of the metric components.

Either way, I was referring to stating the existence of a local inertial frame, not necessarily using it. This is the only meaning of ”locally flat” I have found in a GR textbook (Schutz, see anove).

atyy said:
MTW uses the term "local flatness" in 7.5, and "locally flat" in Box 6.1 and Section 20.4.
Thanks for the sections, I will check it out when I am back in office tomorrow. I do not keep it with me at all times since my squat PB all time is just 160 kg ... :rolleyes:
 
  • Haha
  • Like
Likes slider142, kith, vanhees71 and 2 others
  • #15
Dale said:
maybe that is the point
I guess my main issue is that ”flat” really has a different meaning in terms of the curvature tensor being zero and ”local” tends to refer to a point or neighbourhood. Putting those two together would seem to imply curvature being zero at a point or neighbourhood, which certainly is not the way it is being used. The fact that locally flat actually means something else in topology is secondary.
 
  • Like
Likes vanhees71 and Dale
  • #16
Orodruin said:
To zeroth order. If you have non-inertial coordinates then the Christoffel symbols are non-zero and they depend on the first derivatives of the metric components.
Yes, but I think that the first derivatives are the same as the equivalent non-inertial coordinate chart on Minkowski space. As far as I know they only differ from the flat spacetime version to second order. Is that incorrect?
Orodruin said:
Either way, I was referring to stating the existence of a local inertial frame, not necessarily using it. This is the only meaning of ”locally flat” I have found in a GR textbook (Schutz, see anove).
Ah, yes I missed that above.
 
  • #17
Doesn't the term locally flat mean that at point p with coordinates t,x,y,z in a curved space-time you can transform into a minkowski metric? Which is flat, and since that is only true at point p and nothing but p, it is local to p?
 
  • #18
I've always used locally flat for a Riemannian/pseudoriemannian manifold to mean that in a small enough region the geometry is "essentially" indistinguishable from Euclidean/Minkowski space. To formalize "essentially", you do get to coordinates that make the metric Euclidean or Minkowski at a point, with connection coefficients and metric first derivatives vanishing at this point. Thus, only second derivatives of the metric are nonvanishing if there is curvature. As a result, almost any geometric or physical measurement in a small region has only second order differences from flat space/spacetime.
 
  • Like
Likes vanhees71
  • #19
atyy said:

One claim in this article seems questionable to me: that you can have, in the interior of some spacetime and bounded by curved regions separating it from a standard flat Minkowski spacetime region, a spacetime region which is flat but has "homogeneous acceleration" relative to the exterior flat region. I have never seen such a solution in the GR literature. Does anyone know what this refers to?
 
  • Like
Likes vanhees71 and atyy
  • #20
PAllen said:
I've always used locally flat for a Riemannian/pseudoriemannian manifold to mean that in a small enough region the geometry is "essentially" indistinguishable from Euclidean/Minkowski space. To formalize "essentially", you do get to coordinates that make the metric Euclidean or Minkowski at a point, with connection coefficients and metric first derivatives vanishing at this point. Thus, only second derivatives of the metric are nonvanishing if there is curvature. As a result, almost any geometric or physical measurement in a small region has only second order differences from flat space/spacetime.
Yes, I understand that this is the intended meaning in many cases. However, my issue is that it is a somewhat deceptive use of the words "local" and "flat". For example, Schwarzschild spacetime is locally flat everywhere with that meaning, however, nowhere is the curvature tensor of Schwarzschild spacetime zero, particularly not as you approach the singularity where curvature invariants blow up.

atyy said:
MTW uses the term "local flatness" in 7.5, and "locally flat" in Box 6.1 and Section 20.4.
I am more or less fine with the usage in Box 6.1 and section 7.5 as it can be interpreted to be talking about the tangent spaces and how to patch them together. I think the formulation in 20.4 is unfortunate.
 
  • #21
Orodruin said:
I am more or less fine with the usage in Box 6.1 and section 7.5 as it can be interpreted to be talking about the tangent spaces and how to patch them together. I think the formulation in 20.4 is unfortunate.

Hmm, but then how can a tangent space be flat? Usually I don't ascribe either flatness or non-flatness to the tangent space.
 
  • #22
atyy said:
Hmm, but then how can a tangent space be flat? Usually I don't ascribe either flatness or non-flatness to the tangent space.
The tangent space is naturally Minkowski space, so it is flat in that sense.
 
  • Like
Likes vanhees71 and Orodruin
  • #23
martinbn said:
The tangent space is naturally Minkowski space, so it is flat in that sense.

Minkowski space is an affine space. The tangent space a vector space, not an affine space.
 
  • Like
Likes weirdoguy
  • #24
atyy said:
Minkowski space is an affine space. The tangent space a vector space, not an affine space.
Any vector space V may be considered as an affine space over itself by considering (V,V) and there is always an isomorphism between an affine space and its tangent space defined by picking an origin. This is precisely the approximation we do when we locally ignore curvature.

Put slightly differently, the idea being conveyed is that a neighbourhood of an event is well described by using an orthonormal basis on the tangent space and the exponential map to define coordinates that will be a local inertial frame. The effects of ignoring curvature (if the spacetime is not flat) then appear only at second order in the metric components.
 
  • Like
Likes dextercioby, PeterDonis and vanhees71
  • #25
Orodruin said:
Any vector space V may be considered as an affine space over itself by considering (V,V) and there is always an isomorphism between an affine space and its tangent space defined by picking an origin. This is precisely the approximation we do when we locally ignore curvature.

Put slightly differently, the idea being conveyed is that a neighbourhood of an event is well described by using an orthonormal basis on the tangent space and the exponential map to define coordinates that will be a local inertial frame. The effects of ignoring curvature (if the spacetime is not flat) then appear only at second order in the metric components.

Ok, but then the Riemann normal coordinates are an expression of the fact that there is a flat tangent space at each point. So local inertial frame is local flatness.
 
  • #26
atyy said:
Ok, but then the Riemann normal coordinates are an expression of the fact that there is a flat tangent space at each point. So local inertial frame is local flatness.
This is where we diverge on what reasonable nomenclature is. I would use local inertial coordinates, not local flatness.
 
  • Like
Likes Paul Colby
  • #27
Orodruin said:
This is where we diverge on what reasonable nomenclature is. I would use local inertial coordinates, not local flatness.

I guess it is hard for me to understand what exactly is the difference between the idea in MTW Box 6.1 and section 7.5, which apparently is ok with you, but not section 20.4. They seem really closely related to me - instinctively, I'd say its usage in section 20.4 is the mathematical expression of the more qualitative language of Box 6.1 and section 7.5.
 
  • #28
atyy said:
I guess it is hard for me to understand what exactly is the difference between the idea in MTW Box 6.1 and section 7.5, which apparently is ok with you, but not section 20.4. They seem really closely related to me - instinctively, I'd say its usage in section 20.4 is the mathematical expression of the more qualitative language of Box 6.1 and section 7.5.
I think it boils down to differentiating between saying that the tangent space or something we approximate the manifold with is flat versus saying that the manifold itself is flat at a point or in a neighbourhood.
 
  • #29
atyy said:
I guess it is hard for me to understand what exactly is the difference between the idea in MTW Box 6.1 and section 7.5, which apparently is ok with you, but not section 20.4. They seem really closely related to me - instinctively, I'd say its usage in section 20.4 is the mathematical expression of the more qualitative language of Box 6.1 and section 7.5.
I, personally, would count any of these. These are general coments to give motivation based on something that is reasonbable and intiutive. But they do not give definitions, nor do they establish terminology that the follow in the book.
 
  • #30
Reading through all the answers, I indeed think that to call the equivalence principle (i.e., the existence of a local inertial reference frame at any point of the spacetime manifold) "local flatness" is highly misleading since the notion of flatness or non-flatness is a local concept itself, i.e., it's described by the vanishing or nonvanishing of the curvature tensor and as such is of course independent of the choice of the frame since it's a tensor property of the spacetime manifold.
 
  • Like
Likes Jianbing_Shao and Orodruin
  • #31
OK, but hopefully everyone who objects to "local flatness" also objects to one of the traditional statements of the equivalence principle: gravity is locally equivalent to acceleration.
 
  • Like
Likes Dale
  • #32
atyy said:
OK, but hopefully everyone who objects to "local flatness" also objects to one of the traditional statement of the equivalence principle: gravity is locally equivalent to acceleration.

There are clearly tensor properties of curved spacetime that are not equivalent to just having acceleration in Minkowski space (the tensor properties do not care about whether or not you use an inertial frame).

I think a better formulation would be "locally indistinguishable from" as measuring curvature requires parallel transport around small loops returning small^2 changes in the transported vectors. This makes reference to the measuring procedure rather than the mathematical formulation.

Of course, in the end this just underlines the difficulty in inventing a precise enough popular language to use when we engage in B- and I-level threads on GR. I mean, I am sure (or assume) that we all agree on the actual maths involved in GR, the issue is one of nomenclature alone.
 
  • #33
Orodruin said:
Of course, in the end this just underlines the difficulty in inventing a precise enough popular language to use when we engage in B- and I-level threads on GR. I mean, I am sure (or assume) that we all agree on the actual maths involved in GR, the issue is one of nomenclature alone.

But if it weren't for all this terrible language, we wouldn't have the pleasure (?) of radiating charge and the equivalence principle threads from time to time :P

Orodruin said:
I think a better formulation would be "locally indistinguishable from" as measuring curvature requires parallel transport around small loops returning small^2 changes in the transported vectors. This makes reference to the measuring procedure rather than the mathematical formulation.

BTW, Ohanian even objected to this in this old paper of his: https://doi.org/10.1119/1.10744
"The strong principle of equivalence is usually formulated as an assertion that in a sufficiently small, freely falling laboratory the gravitational fields surrounding the laboratory cannot be detected. We show that this is false by presenting several simple examples of phenomena which may be used to detect the gravitational field through its tidal effects; we show that these effects are, in fact, local (observable in an arbitrarily small region)."

It's pretty much like the famous objection by Synge I mentioned earlier.
 
Last edited:
  • Like
Likes Paul Colby
  • #34
atyy said:
But if it weren't for all this terrible language, we wouldn't have the pleasure (?) of radiating charge and the equivalence principle threads from time to time :P
BTW, Ohanian even objected to this in this old paper of his: https://doi.org/10.1119/1.10744
"The strong principle of equivalence is usually formulated as an assertion that in a sufficiently small, freely falling laboratory the gravitational fields surrounding the laboratory cannot be detected. We show that this is false by presenting several simple examples of phenomena which may be used to detect the gravitational field through its tidal effects; we show that these effects are, in fact, local (observable in an arbitrarily small region)."

It's pretty much like the famous objection by Synge I mentioned earlier.
The mathpages article you linked previously has a pretty effective refutation of Ohanian's examples, to whit, they all ignore (sometime subtly) the time aspect of local spacetime region.

Synge's objection is purely mathematical, and amounts to the same as Orodruin's - that curvature is defined at each point of the manifold.

The charged particle debates (classically) all boil down to the fact that radiation is not a strictly local phenomenon, and can be shown to follow from inability to construct a global inertial frame.

However, I do share the same question Peter raised about one part of this article, and I also choose not to accept Einstein's definition of gravity, as described therein (this being purely a choice of terminology).
 
Last edited:
  • Like
Likes atyy
  • #35
For Riemannian manifolds, I have seen the term “locally Euclidean” used. This avoids the flat vs curved conundrum, while also not having to discuss coordinates. Would the “locally Minkowski” make you @Orodruin happy?
 
  • Like
Likes Dale
<h2>1. What is "local flatness" in terms of describing spacetime?</h2><p>Local flatness refers to the idea that in a small region of spacetime, the curvature of spacetime is negligible and can be approximated as being flat. This concept is used in general relativity to describe the behavior of gravity in small scales.</p><h2>2. How is "local flatness" different from "global flatness"?</h2><p>While local flatness describes the curvature of spacetime in a small region, global flatness refers to the curvature of the entire spacetime. A spacetime can be locally flat but not globally flat, meaning that the curvature may be negligible in small scales but significant in larger scales.</p><h2>3. Is "local flatness" the only term used to describe spacetime?</h2><p>No, there are other terms used to describe the curvature of spacetime, such as "local curvature" and "local geometry". These terms all refer to the same concept of describing the behavior of gravity in small scales.</p><h2>4. How is "local flatness" relevant to our understanding of the universe?</h2><p>The concept of local flatness is crucial in understanding the behavior of gravity in small scales, which is essential in studying the behavior of objects like stars, planets, and galaxies. It also plays a role in understanding the expansion of the universe and the formation of large-scale structures.</p><h2>5. Can "local flatness" be applied to all regions of spacetime?</h2><p>No, the concept of local flatness is only applicable in regions where the curvature of spacetime is small. In regions with high curvature, such as near massive objects like black holes, the concept of local flatness does not hold, and other terms must be used to describe the behavior of gravity.</p>

1. What is "local flatness" in terms of describing spacetime?

Local flatness refers to the idea that in a small region of spacetime, the curvature of spacetime is negligible and can be approximated as being flat. This concept is used in general relativity to describe the behavior of gravity in small scales.

2. How is "local flatness" different from "global flatness"?

While local flatness describes the curvature of spacetime in a small region, global flatness refers to the curvature of the entire spacetime. A spacetime can be locally flat but not globally flat, meaning that the curvature may be negligible in small scales but significant in larger scales.

3. Is "local flatness" the only term used to describe spacetime?

No, there are other terms used to describe the curvature of spacetime, such as "local curvature" and "local geometry". These terms all refer to the same concept of describing the behavior of gravity in small scales.

4. How is "local flatness" relevant to our understanding of the universe?

The concept of local flatness is crucial in understanding the behavior of gravity in small scales, which is essential in studying the behavior of objects like stars, planets, and galaxies. It also plays a role in understanding the expansion of the universe and the formation of large-scale structures.

5. Can "local flatness" be applied to all regions of spacetime?

No, the concept of local flatness is only applicable in regions where the curvature of spacetime is small. In regions with high curvature, such as near massive objects like black holes, the concept of local flatness does not hold, and other terms must be used to describe the behavior of gravity.

Similar threads

Replies
40
Views
2K
  • Special and General Relativity
Replies
9
Views
865
  • Special and General Relativity
Replies
30
Views
461
  • Special and General Relativity
Replies
4
Views
739
  • Special and General Relativity
Replies
6
Views
1K
  • Special and General Relativity
Replies
10
Views
1K
  • Special and General Relativity
Replies
21
Views
1K
  • Special and General Relativity
3
Replies
95
Views
5K
  • Special and General Relativity
Replies
25
Views
2K
  • Special and General Relativity
Replies
8
Views
827
Back
Top