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

[Jimster41]

Is there a microscopic description of these "tidal forces" - that (if I understand correctly) betray non-zero curvature even in an infinitesimal inertial frame?

https://en.wikipedia.org/wiki/Tidal_tensor

My cartoon is that they represent (result from) geometric phase or "Pancharatnam-Berry Phase" (non-zero holonomy)?

I get that there is a frequency shift (in light for example) as a function of a gravitational field (gravitational lensing). But my understanding of that is that it would not be detectable from within the inertial frame?

Are there any experiments that would detect a changing value of the field (curvature) from inside an inertial frame? Is there just some simple electrostatic gradient effect that can be measured? I was assuming the answer is no?

Would the Aharonov-Bohm effect reflect such change? Not sure how that effect is measured but I gather it's not just a simple magnetometer.

Would the spontaneous collapse of entanglement (somehow absent other causes) be indicative, or some change in the stability of entanglement as a function of alignment with the change (gradient) in the field?

 

[atyy]

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.

Interesting! I only linked the mathpages article for Synge's remark and had not read the rest of it. I wonder whether Ohanian includes these in the latest edition of his textbook with Ruffini.

 

[PAllen]

Is there a microscopic description of these "tidal forces" - that (if I understand correctly) betray non-zero curvature even in an infinitesimal inertial frame?

https://en.wikipedia.org/wiki/Tidal_tensor

My cartoon is that they represent (result from) geometric phase or "Pancharatnam-Berry Phase" (non-zero holonomy)?

I get that there is a frequency shift (in light for example) as a function of a gravitational field (gravitational lensing). But my understanding of that is that it would not be detectable from within the inertial frame?

Are there any experiments that would detect a changing value of the field (curvature) from inside an inertial frame?

Would the spontaneous collapse of entanglement (somehow absent other causes) be indicative, or some change in the stability of entanglement as a function of alignment with the change in the field?

Would the Aharonov-Bohm effect reflect such change? Not sure how that effect is measured but I gather it's not just a simple magnetometer.

I would say that the SEP (strong equivalence principle) is a testable proposition, and that if any test could be devised that, when performed in an arbitrarily small spacetime region with any finite precision, could distinguished a local inertial frame in a region with curvature from one without, you would have a violation of SEP.

The SEP is making a claim that local physics is precisely as indistinguishable from SR as local geometry is from Euclidean for a Riemannian metric. If you look at the various equivalent definitions of geometric curvature, they all require infinite precision to execute:
- limit of change of vector transported around quadrilateral as its size goes to zero divided by the area. The actual change goes to zero, and still goes to zero if divided by e.g. a diagonal of the quadrilateral.
- limit of angular defect in a triangle as its size goes to zero, divided by the area of the triangle. Again, the angular defect itself goes to zero, and you need the division by area to measure the second order effect.
- limit of the difference between 1 and ratio of circumference or area to the euclidean formula, divided by area, as the size goes to zero. The ratios themselves go to 1, and the difference from 1 still goes to zero if divided by circle diameter.

 

[andrewkirk]

@Orodruin
I agree that it is misleading. I'd like to make two points.

First, comparable uses of the word 'locally' in topology agree with your principle that the property must apply in some neighbourhood of each point - which flatness does not. The examples I think of are locally connected, locally path connected and locally compact. 'Locally flat' does not adhere to this principle. There is no neighbourhood of a point in which the curvature is constant at zero. So use of the term 'locally flat' does not follow standard practice for the term 'locally' in topology.

Second, is it not the case that, if we exclude singularities from a spacetime manifold (which IIRC we can do without inhibiting our ability to calculate) then any achievable spacetime manifold is everywhere 'locally flat'? I am not completely sure of that, or whether 'local flatness' may not apply at the event horizon of a black hole. But if I guessed correctly, then saying a spacetime is locally flat is saying nothing, and we lose nothing by discarding the phrase.

I would have thought that saying the spacetime is differentiable (or ##C^n## for some ##n##) tells us all that is needed. It would be better to simply say that we can approximate a spacetime to an arbitrarily high degree of accuracy near a point by taking a small enough neighbourhood of the point.

 

[PAllen]

@OrodruinSecond, is it not the case that, if we exclude singularities from a spacetime manifold (which IIRC we can do without inhibiting our ability to calculate) then any achievable spacetime manifold is everywhere 'locally flat'? I am not completely sure of that, or whether 'local flatness' may not apply at the event horizon of a black hole. But if I guessed correctly, then saying a spacetime is locally flat is saying nothing, and we lose nothing by discarding the phrase.

I don’t know what you mean here. @orodruin’s complaint is that most useful GR manifold’s (Schwarzschild, Kerr, FLRW, etc.) are nowhere flat, though the first two are asymptotically flat at spatial infinity.

On the other hand, the definition of equipping a manifold with a metric requires that it be locally Euclidean or Minkowski to second order. This is all mathematical definition. The physical question is then whether such models with a mapping to measurements correspond to reality.

The singularity can’t be part of the manifold - it isn’t a choice. And no part of the manifold, horizon or arbitrarily close to a singularity can avoid being locally Minkowski or Euclidean, because the definition Riemannian or pseudoriemannian forces this by design.

 

[andrewkirk]

@orodruin’s complaint is that most useful GR manifold’s (Schwarzschild, Kerr, FLRW, etc.) are nowhere flat, though the first two are asymptotically flat at spatial infinity.

Yes, I understand that that is part of Orodruin's point, and I agree with it. But I don't understand why you think what I wrote does not agree with that.

 

[Orodruin]

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?

It is better, although I am not completely sure how I feel about it yet. I have to sleep on it I think.

 

[PAllen]

Yes, I understand that that is part of Orodruin's point, and I agree with it. But I don't understand why you think what I wrote does not agree with that.

Perhaps I misunderstood you. Your first paragraph seemed to reject local flatness, while your second embraced it. But I think I missed the significance of your scare quotes.

In that case, whatever the best term is, I think it is crucial to know that equipping a manifold with a metric intentionally gives it some universal local properties. And your comment about the horizon is exactly why it is crucial - a horizon is locally indistinguishable from my living room per GR. We need a name for this property that tells you it is impossible to say things like time stops at the horizon in GR.

 

[JustTryingToLearn]

My understanding of "local flatness" is the following. Around any spacetime point (with local flatness), there exists a region of spacetime (a neighborhood) within which the results of any experiment cannot be distinguished from the results of an experiment performed in completely flat spacetime. In other words, there is some region around the point such that, should you perform an experiment there, you would not be able to take the results and prove that special relativity is not the "true" theory (more simply, that special relativity is not valid) in that region of spacetime. If you do perform such an experiment and can show that SR is not valid, then you have chosen too large a neighborhood.

If I am misinterpreting this, I'd welcome feedback as this is something I am trying to learn in more detail.

 

[Physics4Funn]

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?

Yes, I am bothered by this also.
I am concerned that most of the mathematics that physics uses only applies to flat spacetime.
I call this "flat spacetime prejudice."
I offered to teach an undergraduate class in general relativity to try to get more physicists fluent with curved space. I wonder if the trouble finding a unified field theory is hindered by the lack of workers.
For many reasons, it didn't happen.

To your question:
I don't feel qualified to answer your question.
I would really like to hear from a mathematician about this.

Maybe there is an answer in the many replies here.

 

[PAllen]

My understanding of "local flatness" is the following. Around any spacetime point (with local flatness), there exists a region of spacetime (a neighborhood) within which the results of any experiment cannot be distinguished from the results of an experiment performed in completely flat spacetime. In other words, there is some region around the point such that, should you perform an experiment there, you would not be able to take the results and prove that special relativity is not the "true" theory (more simply, that special relativity is not valid) in that region of spacetime. If you do perform such an experiment and can show that SR is not valid, then you have chosen too large a neighborhood.

If I am misinterpreting this, I'd welcome feedback as this is something I am trying to learn in more detail.

That is a correct statement of the Einstein Equivalence Principle as defined by e.g. Clifford Will. Its ability to be true in GR is, indeed, closely related to “local behavior of a pseudoRiemannian manifold”. The gist of this thread is what is the best compact verbal description of this local behavior that we all agree on the mathematics of. The equivalence principle names the physics. What we seek consensus on is a name for corresponding math of the manifold.

 

[atyy]

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.

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.

I would say that the SEP (strong equivalence principle) is a testable proposition, and that if any test could be devised that, when performed in an arbitrarily small spacetime region with any finite precision, could distinguished a local inertial frame in a region with curvature from one without, you would have a violation of SEP.

The SEP is making a claim that local physics is precisely as indistinguishable from SR as local geometry is from Euclidean for a Riemannian metric. If you look at the various equivalent definitions of geometric curvature, they all require infinite precision to execute:
- limit of change of vector transported around quadrilateral as its size goes to zero divided by the area. The actual change goes to zero, and still goes to zero if divided by e.g. a diagonal of the quadrilateral.
- limit of angular defect in a triangle as its size goes to zero, divided by the area of the triangle. Again, the angular defect itself goes to zero, and you need the division by area to measure the second order effect.
- limit of the difference between 1 and ratio of circumference or area to the euclidean formula, divided by area, as the size goes to zero. The ratios themselves go to 1, and the difference from 1 still goes to zero if divided by circle diameter.

So if we have infinite precision, are we able to detect deviations from flatness, even at a point? For example, could geodesic deviation be detected? In other words, is there a physical counterpart to the objection to the terminology of "local flatness"?

 

[jbriggs444]

So if we have infinite precision, are we able to detect deviations from flatness, even at a point?

Clearly not. If we have infinite precision we can detect deviations from flatness using measurements drawn from a neighborhood of arbitrarily small extent. But not from a neighborhood with no extent.

 

[atyy]

Clearly not. If we have infinite precision we can detect deviations from flatness using measurements drawn from a neighborhood of arbitrarily small extent. But not from a neighborhood with no extent.

But the definition of curvature (ie to mathematically say that the curvature is non-zero at a point) also requires a neighbourhood?

 

[jbriggs444]

But the definition of curvature (ie to mathematically say that the curvature is non-zero at a point) also requires a neighbourhood?

Same as a derivative, f'(x). It is defined for a point but the definition depends on behavior near the point.

Or, consider the definition of a limit of a function at a point. The definition is for a point but is independent of the function value at that point.

 

[PAllen]

So if we have infinite precision, are we able to detect deviations from flatness, even at a point? For example, could geodesic deviation be detected? In other words, is there a physical counterpart to the objection to the terminology of "local flatness"?

The value at a point is the result of a limit. Thus, you can’t measure it at a point. However, classically, you could measure geodesic deviation in a ball a billionth of a plank length with tiny instruments of arbitrarily great precision.

I don’t have any real objection to local flatness treated as a name for math that both @Orodruin and I agree on. But I can also sympathize with the objection. Thus I am open to agreeing to other terminology as preferred for this site. I am not enamored of having to say something involving coordinates, because the local behavior is coordinate independent. I have suggested “locally Minkowski” as a possibility.

Note, unlike some, I have no problem with practical definitions of coordinate independent features (e.g. asymptotic flatness or spherical symmetry) that involve the existence of coordinates in which the metric takes a certain form. However, I want a name to emphasize that the feature itself is coordinate independent.

 

[Orodruin]

I would be interested to hear exactly what meaning different people include in this usage of "local flatness". Exactly which properties does the spacetime (or manifold if we become a bit more general) need to satisfy for you to call it "locally flat"?

 

[PAllen]

I would be interested to hear exactly what meaning different people include in this usage of "local flatness". Exactly which properties does the spacetime (or manifold if we become a bit more general) need to satisfy for you to call it "locally flat"?

For me, it is a universal feature, by design, of any Riemannian or pseudoRiemannian manifold. It has no meaning if you don’t equip the manifold with a metric. Riemann‘s aim in his definitions was to allow geometry in the large and topology to be wildly different from Euclidean, while preserving local Euclidean behavior.

 

[atyy]

I would be interested to hear exactly what meaning different people include in this usage of "local flatness". Exactly which properties does the spacetime (or manifold if we become a bit more general) need to satisfy for you to call it "locally flat"?

Local flatness is a property of all (semi)-Riemannian manifolds. Thus a manifold that is nowhere flat is everywhere locally flat. Yes, I sympathize with your peeve, but I think it is tied up with the physics of the equivalence principle. I don't think one can totally avoid misleading terminology in the discussion, but I would prefer to handle it by keeping the traditional terminology, and just explaining the details of the physics.

Normal coordinates are one mathematical tool corresponding to the physics notion of local flatness, and quantifying deviations from it.

 

[Orodruin]

Thus a manifold that is nowhere flat is everywhere locally flat.

This certainly should not be the case. If it is the concept is meaningless.

Riemann‘s aim in his definitions was to allow geometry in the large and topology to be wildly different from Euclidean, while preserving local Euclidean behavior.

Did Riemann use ”local flatness”?

If it is supposed to be a property of all manifolds with a metric I do not see the point of introducing the term at all.

I also assume that you want to latch on the condition that the connection is Levi-Civita. To me it is flat (edit: pun not intended, but it is funny now that I reread it...) out misleading to talk about flatness at all without actually referencing the connection and a priori the connection need not be tied to the metric.

I have trouble seeing why you need to introduce this nomenclature at all if it is just supposed to refer to a smooth manifold with a metric as all you need to say is that it locally looks like Euclidean/Minkowski space in the sense that there is a smooth map from a neighbourhood to a set in E/M space. (Of course with varying amounts of technicality depending on who you are talking to.)

 

[atyy]

I also assume that you want to latch on the condition that the connection is Levi-Civita. To me it is flat (edit: pun not intended, but it is funny now that I reread it...) out misleading to talk about flatness at all without actually referencing the connection and a priori the connection need not be tied to the metric.

Yes, to be more careful, the metric compatible connection is needed.

 

[Orodruin]

Yes, to be more careful, the metric compatible connection is needed.

... and torsion free!

 

[Paul Colby]

Are there any experiments that would detect a changing value of the field (curvature) from inside an inertial frame?

I would argue (probably a losing battle) that LIGO qualifies. LIGO is much smaller than the GW wavelengths detected. As a matter of principle there are no real measurements "at a point" or "at an instant" only ever improving approximations of such.

 

[andrew s 1905]

An interesting discussion that I have only just seen. As one who might want a B level answer I think it worth considering how I would understand "locally flat"

To me I would assume that local ment something the size of a lab or the apparatus being used, not some mathematical idea. Flat would mean the analysis of the experiment could use special relativity rather than general or given the context Newtons/Galileo's.

I am sure this is all wrong given the above.

However, I feel one has to give up some precision to get concepts across to us less able but interested questioners. While it maybe ideal not to teach "wrong" physical concepts I don't see how this can be avoided unless you can somehow give me all the necessary mathematical tools up front. I have tried and keep trying to gain more of these but at 67 it's hard.

Regards Andrew s

 

[atyy]

... and torsion free!

Oops, indeed. Anything else I forgot ... ?

 

[Orodruin]

Oops, indeed. Anything else I forgot ... ?

The reason I am being picky is that the entire concept of flatness to me is connected (edit: I did it again!) to the connection, not to the metric. If you have a metric compatible connection with non-zero torsion, the space locally does not look like Euclidean/Minkowski space. It is therefore important to understand more precisely what attributes that you ascribe to "local flatness" that are not caught by other standard nomenclature.

 

[Jimster41]

... and torsion free!

I am struggling to grasp how you can have a curved surface composed of everywhere-locally-flat manifolds connected "torsion-free". Where does the curvature go?

Or do I understand that if there is curvature the connection has torsion?

If so then I am confused how the distribution of that connection doesn't require a privileged frame? Assuming the connection is somehow physical doesn't that put two observers on the curve in a situation of ... torsion.

 

[Orodruin]

I am struggling to grasp how you can have a curved surface composed of everywhere-locally-flat manifolds connected "torsion-free". Where does the curvature go?

This is exactly my point regarding how the nomenclature of locally flat is misleading. "Locally flat" as used colloquially does not mean that manifold is actually flat (in the sense of the curvature tensor being equal to zero) at a given point and therefore the nomenclature is confusing.

Or do I understand that if there is curvature the connection has torsion?

No, you can have curvature without torsion and vice versa. Torsion is related to the commutativity of geodesic flows, curvature is connected to the deviation from the identity map when you parallel transport a vector around a loop.

If so then I am confused how the distribution of that connection doesn't require a privileged frame? Assuming the connection is somehow physical doesn't that put two observers on the curve in a situation of ... torsion

It is unclear to me what you mean by "distribution of that connection". Torsion is typically assumed to be zero in GR (or more generally whenever you have a Levi-Civita connection).

 

[vanhees71]

That is a correct statement of the Einstein Equivalence Principle as defined by e.g. Clifford Will. Its ability to be true in GR is, indeed, closely related to “local behavior of a pseudoRiemannian manifold”. The gist of this thread is what is the best compact verbal description of this local behavior that we all agree on the mathematics of. The equivalence principle names the physics. What we seek consensus on is a name for corresponding math of the manifold.

This is simple, and you gave the answer to this question yourself. The best (though for beginners incomprehenive) statement simply is: Spacetime is a pseudo-Riemannian (Lorentzian) manifold (neglecting spin; with spin it's an Einstein-Cartan Lorentzian manifold).

 

[atyy]

The reason I am being picky is that the entire concept of flatness to me is connected (edit: I did it again!) to the connection, not to the metric. If you have a metric compatible connection with non-zero torsion, the space locally does not look like Euclidean/Minkowski space. It is therefore important to understand more precisely what attributes that you ascribe to "local flatness" that are not caught by other standard nomenclature.

Agreed. I don't think that is picky.

 

[Orodruin]

This is simple, and you gave the answer to this question yourself. The best (though for beginners incomprehenive) statement simply is: Spacetime is a pseudo-Riemannian (Lorentzian) manifold (neglecting spin; with spin it's an Einstein-Cartan Lorentzian manifold).

After the input in this thread, my consensus with myself when communicating with people relatively new to GR is to just say that spacetime locally looks like Minkowski space as long as you stay in a small enough region, I think this is actually more descriptive than "locally flat" and not a lot more difficult to say or read.

 

[vanhees71]

Oops, indeed. Anything else I forgot ... ?

It's all in the definitions: pseudo-Riemannian manifold: differentiable manifold with a fundamental non-degenerate bilinear form with the uniquely determined torsion-free metric-compatible affine connection.

standard GR space-time: a pseudo-Riemannian manifold with the metric of signature (1,3) or equivalently (3,1), depending on your preference of west- or east-coast convention. That's also often called a Lorentzian manifold.

extended GR to accommodate the possibility of spin: an Einstein-Cartan manifold, i.e., a manifold with a (1,3) fundamental bilinear form and a metric compatible affine connection and torsion.

I'm not sure what the experimental status concerning the issue "Lorentzian vs. Einstein-Cartan manifold" is, i.e., whether one has ever measured something indicating that the physical space-time is a manifold with non-zero torsion.

 

[PAllen]

I would argue (probably a losing battle) that LIGO qualifies. LIGO is much smaller than the GW wavelengths detected. As a matter of principle there are no real measurements "at a point" or "at an instant" only ever improving approximations of such.

LIGO is measuring time evolution of curvature, the time analog of the common case of tidal gravity (change over position of an approximately stationary field). As such, to speak of the principle of equivalence, you must restrict time to a small fraction of the frequency of change. Note that one second corresponds to one lightsecond of distance, so anything but 'small'.

 

[PAllen]

After the input in this thread, my consensus with myself when communicating with people relatively new to GR is to just say that spacetime locally looks like Minkowski space as long as you stay in a small enough region, I think this is actually more descriptive than "locally flat" and not a lot more difficult to say or read.

I agree. I was the first to propose this alternative in this thread, I believe.

 

[atyy]

After the input in this thread, my consensus with myself when communicating with people relatively new to GR is to just say that spacetime locally looks like Minkowski space as long as you stay in a small enough region, I think this is actually more descriptive than "locally flat" and not a lot more difficult to say or read.

But isn't this still wrong, since Minkowski space is flat, but spacetime in a small region is not necessarily flat?