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

[Orodruin]

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?

 

[martinbn]

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 doesn't save anything.

 

[atyy]

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

 

[George Jones]

Any thoughts? Am I just being picky?

If you are being picky, than so am I, as I sometimes make similar points
https://www.physicsforums.com/threads/laymans-doubts-about-gen-relativity.415937/page-4#post-2805278https://www.physicsforums.com/threads/laymans-doubts-about-gen-relativity.415937/page-4#post-2805339

 

[Paul Colby]

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.

 

[Orodruin]

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".

 

[atyy]

Eric Poisson uses it.
https://books.google.com.sg/books?id=bk2XEgz_ML4C
https://arxiv.org/abs/gr-qc/0306052
https://arxiv.org/abs/1102.0529

Zee uses it.
https://books.google.com.sg/books?id=5Dy1hlKvmCYC

 

[PeterDonis]

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.

 

[vanhees71]

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.

 

[Orodruin]

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.

 

[Orodruin]

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).

 

[Dale]

"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.

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.

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).

 

[atyy]

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.

 

[Orodruin]

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).

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 ...

 

[Orodruin]

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.

 

[Dale]

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?

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.

 

[sqljunkey]

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?

 

[PAllen]

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.

 

[PeterDonis]

Controversies Over the Equivalence Principle

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?

 

[Orodruin]

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.

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.

 

[atyy]

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.

 

[martinbn]

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.

 

[atyy]

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.

 

[Orodruin]

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.

 

[atyy]

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.

 

[Orodruin]

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.

 

[atyy]

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.

 

[Orodruin]

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.

 

[martinbn]

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.

 

[vanhees71]

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.