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

[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?

 

[Orodruin]

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

This is taken care of by the "small enough" and that the "looks like" is informal. The small enough depends on your accuracy in measuring curvature. The "looks like" does not sound as formal as "locally flat" as both "local" and "flat" have precise mathematical definitions. I would also be fine with "looks flat in a small enough region" as "look" is informal.

 

[atyy]

This is taken care of by the "small enough" and that the "looks like" is informal. The small enough depends on your accuracy in measuring curvature. The "looks like" does not sound as formal as "locally flat" as both "local" and "flat" have precise mathematical definitions. I would also be fine with "looks flat in a small enough region" as "look" is informal.

And with infinite precision, would you say that the informal statement is not true, since there would be no region that is small enough?

 

[PAllen]

And with infinite precision, would you say that the informal statement is not true, since there would be no region that is small enough?

Of course.

In fact, I think the quantitative behavior can be made accessible in a B level description. One can say that as the dimensions of a spacetime region (note, including time) are cut in half, the precision needed to detect a deviation from SR using any particular method goes up by 4 (not by 2 or 8 for example).

 

[Ibix]

I hadn't thought of "locally flat" being problematic, but I see the point. What we're trying to convey here is analogous to the Earth being spherical but my kitchen floor being flat. The latter claim is incorrect, but the errors that follow from making the claim aren't even on my list of worries when I'm tiling my floor. Not even if I need to re-pave the town square. Maybe I need to worry if I'm planning on making maps of a mid-sized country. "Flat" is a good enough description of the floor for a small area.

The same applies to spacetime, with caveats that "small area" is strongly context dependent and that the "small" applies to the extent in the time-like direction as well. So I agree with the consensus above - happy to say some variant on "spacetime is near enough flat over a small region that SR applies near enough that no one cares about the errors". And that the formal statement of this is that we can always find a coordinate system where, at a chosen point, the second first derivatives of the metric are zero and are small in a surrounding volume.

 

[PAllen]

... And that the formal statement of this is that we can always find a coordinate system where, at a chosen point, the second derivatives of the metric are zero and are small in a surrounding volume.

Well, you can make the metric diag [-1,1,1,1], and the first derivatives zero, but not the second derivatives.

 

[Ibix]

Well, you can make the metric diag [-1,1,1,1], and the first derivatives zero, but not the second derivatives.

Thanks - corrected above (at least, I hope I got it right this time...).

 

[Dale]

say that spacetime locally looks like Minkowski space as long as you stay in a small enough region

How about “locally Minkowski to first order”. I think that makes it clear that second order properties like curvature may differ from Minkowski spacetime which is flat. It also gives an immediate clue about how small the locally needs to be.

 

[Orodruin]

How about “locally Minkowski to first order”. I think that makes it clear that second order properties like curvature may differ from Minkowski spacetime which is flat. It also gives an immediate clue about how small the locally needs to be.

I would say it depends on the audience. I am not sure I would use that with someone asking relatively basic questions about GR, but in more advanced cases I would be fine with that.

 

[Orodruin]

And with infinite precision, would you say that the informal statement is not true, since there would be no region that is small enough?

There is no such thing as infinite precision and if there was then no region (with a finite extension) would be "small enough". Either way, the informal expression is for conveying a more informal image. If I wanted to convey the mathematical structure with precision I would just say Riemannian/Lorentzian manifold.

 

[Paul Colby]

As such, to speak of the principle of equivalence, you must restrict time to a small fraction of the frequency of change.

Well, space-time is in fact curved in this case. LIGO measures the change in the proper length of 4 km interferometer arms. If one were unlimited by noise etc, the time scale of this length measurement is limited by the transit time of the laser over 8 km which is quite small relative to a light second over which the phenomena of interest occurs. Of course in the real world of LIGO the noise dominates in the extreme. In many ways, LIGO is typical of all experimental work independent of GR. So I don't see your point exactly.

 

[PAllen]

Note that there is

Well, space-time is in fact curved in this case. LIGO measures the change in the proper length of 4 km interferometer arms. If one were unlimited by noise etc, the time scale of this length measurement is limited by the transit time of the laser over 8 km which is quite small relative to a light second over which the phenomena of interest occurs. Of course in the real world of LIGO the noise dominates in the extreme. In many ways, LIGO is typical of all experimental work independent of GR. So I don't see your point exactly.

i see nothing unclear in my point. Tidal gravity is easily measurable, but this is never taken to refute the principle of equivalence. Instead, the POE is taken to apply only to a spacetime region such that tidal gravity is undetectable. Similarly, if curvature is changing in time, the spacetime region must be limited to a time period over which curvature doesn’t change within given measurement precision. This is just the definition of the POE. If the GW has a frequency of e.g. a kilohertz, this means POE only applies to a time period of say .00001 seconds. I don’t see your objection to the basic definition of the POE.

 

[Paul Colby]

i see nothing unclear in my point.

Well, okay. I understand your point I just don't see it as addressing the question. In dream land if distances and times can be reduced to zero in theory, why can't experimental methods in principle be refined indefinitely? In the limit of an infinite precision 0 noise LIGO of arbitrary small size isn't one making a measurement at "a point" at "a time"? I see this as at the heart of OPs pet peeve but could well be wrong.

 

[PAllen]

Well, okay. I understand your point I just don't see it as addressing the question. In dream land if distances and times can be reduced to zero in theory, why can't experimental methods in principle be refined indefinitely? In the limit of an infinite precision 0 noise LIGO of arbitrary small size isn't one making a measurement at "a point" at "a time"? I see this as at the heart of OPs pet peeve but could well be wrong.

No, he doesn’t dispute my point at all. Please read his last several posts in this thread responding to @atyy . Nowhere is @Orodruin questioning POE as physical principle. He disputes calling its phenomenology local flatness.

 

[Paul Colby]

I don’t see your objection to the basic definition of the POE.

Yeah, nor do I and perhaps that's my problem. In your reply in #81 is the claim that based on the POE there is always a limit in which tidal gravity becomes undetectable. This is an explicit statement that any dreamland race between theory and experimental observation will always be lost by the experimentalist. This I can accept. The POE is used to structure theory of which I'm quite content.

 

[atyy]

There is no such thing as infinite precision and if there was then no region (with a finite extension) would be "small enough". Either way, the informal expression is for conveying a more informal image. If I wanted to convey the mathematical structure with precision I would just say Riemannian/Lorentzian manifold.

OK, thanks for clarifying the physics. So it's clear the remainder is just terminology and a matter of taste, like "work", "relativistic mass", "collapse". I still think local flatness is the best, otherwise how can one say that a semi-Riemannian manifold that is nowhere flat is everywhere locally flat :)

 

[Orodruin]

I still think local flatness is the best, otherwise how can one say that a semi-Riemannian manifold that is nowhere flat is everywhere locally flat :)

One could just avoid saying that

 

[vanhees71]

And with infinite precision, would you say that the informal statement is not true, since there would be no region that is small enough?

I think it's simply a matter of accuracy. In a free-falling non-rotating frame of reference you have approximately in a "not too large spatio-temporal extension" a "local inertial reference frame" "up to tidal forces". Whenever you meausure the forces accurately enough there's some deviations from an idealized inertial refeference frame due to these tidal forces.

 

[Raymond Beljan]

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?

I don't think you are being picky.

Leonard Susskind in his Stanford U. lectures discuses this around 03:18 in this lecture:

https://cosmolearning.org/video-lectures/geodesics-gravitational-fields-special-relativity/
From the lectures, I think there can be curved coordinates at a point but the important thing is weather or not another coordinate system can be found, such that, locally ( that is in the limit of differential change ) the coordinates are those of a flat space.

Oh, I should add that for a locally flat space the metric is the kronecker delta and the first derivitives of the metric are zero.

 

[Raymond Beljan]

Well actually the kronecker delta but one minus sign. As noted by Ibix (I think) metric with diagonal [-1,1,1,1]

 

[TonyP0927]

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?

I typically see "local flatness" or similar terms used when representing space-time as a 2D grid as is often the case to show the analogy of gravity's effect on space-time to placing a heavy ball on a sheet. In other words, it's a "simple explanation."