Are Christoffel symbols measurable?

[atyy]

Basically, things are not objectively observable if they are "relative" because then they are "subjective", but we can make all relative things objective by saying what they are relative to. So it is matter that makes things objective, since we have to specify things relative to matter. More technically, we have to specify things relative to events. To illustrate, the Ricci scalar at x is not observable, because x has no meaning without further specification, since when we change coordinates its value changes. We have to say the Ricci scalar at Times Square when the ball dropped at the end of 2011.

This is not that different from special relativity, except that there special sorts of coordinate systems called global inertial frames exist, while none do in curved spacetime.

Rovelli presents an example of using matter so that "the components of the metric tensor ... are gauge invariant quantities".

 

[Ben Niehoff]

So can the affine connection of GR be measured? It is obvious that in the stricter, invariant sense referred to above, it can't.
Does this mean it is not "physical"? No. We are certainly feeling their consequences and therefore "observing" it as a force. But what we measure is not so much the connection but the EM resistance of the ground against our natural tendence to follow our geodesic.

Asking if the affine connection can be measured is analogous to asking if the vector potential can be measured in EM. The answer is not exactly "no"; it is more of a "yes, but...". After all, the connection, like the vector potential, does carry real information; but that information is described in a redundant manner.

The caveat is that we can only measure gauge-invariant quantities constructed from these objects. In EM, this means we can measure the E and B fields. In GR, this means we can measure the Riemann tensor. (Where "measure" is defined as a process like I've described before, where we choose a frame and measure contractions against that frame.)

So the answer really depends on the meaning of the question. If the question is "Can we measure the connection independently of the Riemann tensor?", then the answer is certainly "No." In particular, there is no set of measurements we can do that will let us map out exactly what values to assign to each of the components of \Gamma^\mu_{\nu\rho}.

 

[Ben Niehoff]

Another thing to add: There are additional ways to make gauge-invariant scalars besides merely making contractions like

R_{abcd} X^a Y^b Z^c W^d
at a point. One can also make nonlocal measurements, by parallel-transporting a vector around a given path, and finally comparing it with its original image (you can imagine carrying this process out using two observers, each carrying a copy of a vector, traveling two different paths, and then comparing their vectors). In Yang-Mills theory, such a scalar measurement is called a Wilson loop. In GR, we call it holonomy.

It is this kind of measurement that gives us the Aharonov-Bohm effect: A Wilson loop going around a perfect solenoid.

An analogous process can happen in geometry: Consider a path going around the base of a cone. Everywhere along the path, the geometry is locally flat. But there will be a nontrivial holonomy around this loop, due to the curvature concentrated at the tip of the cone. (There is no need to have a curvature singularity; you can imagine smoothing out the tip of the cone.)

So there are other ways to make measurements. But ultimately, you end up taking the dot product between two vectors.

 

[atyy]

Another thing to add: There are additional ways to make gauge-invariant scalars besides merely making contractions like

R_{abcd} X^a Y^b Z^c W^d
at a point. One can also make nonlocal measurements, by parallel-transporting a vector around a given path, and finally comparing it with its original image (you can imagine carrying this process out using two observers, each carrying a copy of a vector, traveling two different paths, and then comparing their vectors). In Yang-Mills theory, such a scalar measurement is called a Wilson loop. In GR, we call it holonomy.

It is this kind of measurement that gives us the Aharonov-Bohm effect: A Wilson loop going around a perfect solenoid.

An analogous process can happen in geometry: Consider a path going around the base of a cone. Everywhere along the path, the geometry is locally flat. But there will be a nontrivial holonomy around this loop, due to the curvature concentrated at the tip of the cone. (There is no need to have a curvature singularity; you can imagine smoothing out the tip of the cone.)

So there are other ways to make measurements. But ultimately, you end up taking the dot product between two vectors.

I once saw an interesting comment from Michael Berry that holonomy was a "bastardization" of language, and it really should be anholonomy.

 

[atyy]

No it's not. I have a can of Coke that is 16 fluid ounces in one coordinate system and 473.18 mL in another. That's not a scalar.

I don't know about scalar, but one way to make a volume independent of coordinates is to specify the coordinate system. So the volume of Coke in mL is coordinate independent. Of course, this assumes that the people at NIST have done their jobs, and that we have some way of transporting their standards around.

 

[juanrga]

Is it true that in GR the gauge is described by Guv while the potential is the Christoffel symbols just like the gauge in EM is described by phase and the potential by the electric and magnetic scalar and vector potential and the observable the electromagnetic field and the Ricci curvature?

But GR is just geometry. Are the Christoffel symbols measurable or can it only occur in gauge transformation without observable effect? How do you vary the Christoffel symbols just like phase?

GR is not a gauge theory, because it is not a field theory over flat spacetime. GR is a (geo)metric theory.

Christoffel can be made to vanish by coordinate transformations. They are essentially geometric objects without physical reality. That is why gravitation cannot be considered a force in GR.

 

[waterfall]

GR is not a gauge theory, because it is not a field theory over flat spacetime. GR is a (geo)metric theory.

Isn't the goal of Quantum Gravity to make GR a gauge theory? Or is this separate issue from the goal of unifying the four forces including gravity but making it part of a larger gauge symmetry? But what perflexed me is how can they make gravitons be indistinguishable from electromagnetic force which is how you do it for example in GUT where and unification produced new physical process that can make quark decay into electrons and neutrinos, hence the search for proton decays.

About Gauge theory of Gravity. I saw this:

http://www.icpress.co.uk/physics/p781.html

It says there are attempts to derive at the gauge theory of gravitation. But in your context how can they do that when "it is not a field theory over flat spacetime. GR is a (geo)metric theory" as you mentioned?

Christoffel can be made to vanish by coordinate transformations. They are essentially geometric objects without physical reality. That is why gravitation cannot be considered a force in GR.

 

[atyy]

Isn't the goal of Quantum Gravity to make GR a gauge theory?

No. It is to find a quantum theory whose classical limit contains the physically relevant solutions of classical general relativity.

 

[waterfall]

No. It is to find a quantum theory whose classical limit contains the physically relevant solutions of classical general relativity.

Electromagnetism = U(1)
Electroweak = SU(2)xU(1)
Strong force = SU(3)

GUT (Grand United Theories) which would unite Electroweak and Strong Force is SU(5).

Are you saying they don't or intend to have something like

Super GUT (Uniting GUT with Gravity force) to create SU(6)?? Why not? But Gravity as Geometry is just a symmetry for certain math operations. It doesn't prove gravity is not a field (I want to say "force" but people say this is Newtonian in context and denote action as a distance, are they right? so I just use the term gravity "field" when I meant force).

 

[twofish-quant]

I agree it is a little more strained, but not fundamentally.

But we can stretch this into some absurd conclusions.

I take the a precinct-by-precinct map of the United States containing the election results of the Republican primary in 1980. The candidate votes form a vector and it's s perfectly good vector field. I can also form a vector field containing things like the price of real estate of different types of houses, the probability of default, divorce rates, crime statistics, etc. etc.

All of those are perfectly good vector fields.

Now are you trying to tell me that general relativity says something non-trivial about how political scientists can observe election results, or how real estate prices can be calculated? Just because you can represent real estate prices in a vector field, you are telling me that I have to *observe* the price of houses in a component by component way.

Now if you say "Yes, general relativity does restrict the way election results of the Republican primary and real estate prices for different types of houses in the US can be observed, and come up with some convoluted explanation for why, then we can go down that path, and I'll think of something for which that logic is so crazy that you'll have to say "huh?"

Now it's makes more sense to argue that this all happens because of a mix up in terminology. GR and SR state the all measurable quantities must be invariant and scalar *with respect to Lorenz transforms*. The results of the Republican primaries of 1980 are indeed invariant *with respect to Lorenz transforms* and even though a political scientist may represent them as "vectors" within relativity they are "scalars." In other words, GR has nothing non-trivial to say about political science and election results.

In other words, relativity provides some restrictions for how things are measured *with respect to a certain set of transforms*. Arguing that relativity restricts measurement for *all uses of vector spaces* is a bit of a stretch, and if you go down that route I'm sure that I can find something even more ridiculous than the examples I provided. Vector spaces are very useful and widely used in social science and political science, and I could think of some uses for art and literature.

Just thought of something ridiculous. Restaurant and movie reviews. I go on yelp.com or rottentomatoes.com. Restaurant rates form a vector (i.e. atmosphere, decor, service, etc.) You can do movie reviews the same way (quality of plot, amount of action, quality of print, etc.) Are you telling me that GR says that I can't make a measure of the atmosphere of the restaurant and decor, at the same time? I think I can. But wait, you are saying that general relativity says that it's impossible for me to come to non-scalar conclusions about restaurants. If you say yes, then my reaction is "who made Einstein the restaurant review police?" So you are saying that it is *physically impossible* for me to measure restaurant atmosphere and service at the same time?!

If you insist on yes, 1) I'll think of something more ridiculous and 2) I'll introduce you to a group of restaurant reviewers and let you tell them that you as an expert in general relativity have figured out that it is physically impossible to do reviews in a certain way, and if they insist that they can come up with vector conclusions, that Albert Einstein says that its impossible. Regardless of the outcome of 2), it will be worth watching for the entertainment value (Scientists Versus Restaurant Reviewers, the new Food Network reality show).

At some point what I'm trying to get you to do is to say "wait, Lorenz invariance and restaurant reviews are totally separate things! When you are using vector spaces to represent restaurant reviews that's got nothing to do with how vector spaces are used in GR" Which is my point.

Now if you agree with that. Suppose some alien creature creates a chain of restaurants around a black hole...

Also this is a no-lose situation. You might come up with some argument that indeed GR says something non-trivial and non-obvious about restaurant reviews. Like it says a lot of things about foreign exchange rates. (seriously)

 

[twofish-quant]

I think the source of confusion here came from the different meaning each poster attributes to "observations", in fact this concept is broader and more ill-defined than the more strict concept of measurement of a physical quantity although some physicists use them indistinctly to refer to the latter meaning. When used strictly in the sense of measurement it is clear all of them are scalars in the physical sense as has been explained in this thread.

True. And I'm arguing that there are different meanings in the term "scalar" and that the way that it is used in GR is quite restrictive, and different although clearly related to the way that mathematicians and even other physicists (i.e. people in CFD)
use it. Also the distinction is non trivial since there are some physical quantities that I would argue are "scalar" in the GR sense but "vector" in another. I'm trying to think of something that goes the opposite way, and that is "vector" in the GR sense, but scalar in some other sense.

Also, this logic solves the "paradox of the left-handed glove." If you argue that "scalar" as used in relativity is a very restrictive definition, then the distinction between left and right handed gloves is something that is outside of GR.

 

[twofish-quant]

I don't know about scalar, but one way to make a volume independent of coordinates is to specify the coordinate system. So the volume of Coke in mL is coordinate independent. Of course, this assumes that the people at NIST have done their jobs, and that we have some way of transporting their standards around.

True, but what happens when after specifying a coordinate system, you still end up with something that looks like a vector. Leaving aside social science examples, if you do relativistic fluid dynamics, once you specify the reference frame what you end up is still a "vector."

Velocity fields make things complicated. But color and composition form vector spaces that are independent of the space-time vector spaces. Mathematically you can get into the world of fiber bundles.

 

[twofish-quant]

Isn't the goal of Quantum Gravity to make GR a gauge theory?

The goal of quantum gravity is to unify QM and GR by any means possible. God will tell us the right approach.

 

[atyy]

True, but what happens when after specifying a coordinate system, you still end up with something that looks like a vector. Leaving aside social science examples, if you do relativistic fluid dynamics, once you specify the reference frame what you end up is still a "vector."

Velocity fields make things complicated. But color and composition form vector spaces that are independent of the space-time vector spaces. Mathematically you can get into the world of fiber bundles.

Yes, the point was that in classical GR, in the presence of sufficient matter there isn't a sharp distinction between coordinate dependent and coordinate-independent quantities.

There is the metric which is a tensor field, which is similar to a vector field in that it is a geometric object that eats covectors and spits out "scalars". Its components change with coordinate system, so they are coordinate dependent. But if you use matter to specify a coordinate system, the components then become coordinate independent.

Rovelli gives an example where the metric components are coordinate independent.

 

[juanrga]

Isn't the goal of Quantum Gravity to make GR a gauge theory? Or is this separate issue from the goal of unifying the four forces including gravity but making it part of a larger gauge symmetry? But what perflexed me is how can they make gravitons be indistinguishable from electromagnetic force which is how you do it for example in GUT where and unification produced new physical process that can make quark decay into electrons and neutrinos, hence the search for proton decays.

About Gauge theory of Gravity. I saw this:

http://www.icpress.co.uk/physics/p781.html

It says there are attempts to derive at the gauge theory of gravitation. But in your context how can they do that when "it is not a field theory over flat spacetime. GR is a (geo)metric theory" as you mentioned?

The goal of Quantum Gravity is to describe quantum gravitational phenomena.

Gravity is not a force in GR.

Nobody makes gravitons indistinguishable from electromagnetic force.

People can do all the nonsense that they want including the belief that a covariant derivative can be considered a gauge derivative.

Part of the explanation of why the search for a consistent quantum gravity theory has failed since the 50s is because most of people in the field does not know what are doing.

 

[PAllen]

[two fish posts immediately above clear up some ambiguities for me...]


[1] PAllen posts:

That was at least hinted at elsewhere, and I did not 'get it'...good insight, thanks.


[2]The referenced paper says:



PAllen says:




Although I believe I do understand that components of a vector are themselves vectors...[I had never thought of frequency as a vector component]...I have to think more about this answer...meantime: so what is the referenced paper claiming...Are they wrong, do they have a different definition of scalar, or are they really taking about the 'measurement' ?

There is no discrepancy here. They are just being looser. They said, roughly, it is a single number (vernacular scalar; perhaps, scalar in pre-relativity physics) but it is not a rank0 tensor (= scalar in relativistic theories). I was clarifying what it is in relativity, rather than what it is not. FYI - to see the frequency needs to be treated as a timelike vector component in relativity, just take the 4-momentum of light (E,p) and divide by Planck's constant. Now you have a 4-vector with frequency as its timelike component.

[3] I also did some searching and found this comparison of classical and relativistic scalars which I did not realize [it seems obvious after reading it though] :



http://en.wikipedia.org/wiki/Scalar_(physics)#Scalars_in_relativity_theory

No problem with these ideas, right??

All of this looks fine to me.

 

[PAllen]

But we can stretch this into some absurd conclusions.

I take the a precinct-by-precinct map of the United States containing the election results of the Republican primary in 1980. The candidate votes form a vector and it's s perfectly good vector field. I can also form a vector field containing things like the price of real estate of different types of houses, the probability of default, divorce rates, crime statistics, etc. etc.

All of those are perfectly good vector fields.

Now are you trying to tell me that general relativity says something non-trivial about how political scientists can observe election results, or how real estate prices can be calculated? Just because you can represent real estate prices in a vector field, you are telling me that I have to *observe* the price of houses in a component by component way.

Now if you say "Yes, general relativity does restrict the way election results of the Republican primary and real estate prices for different types of houses in the US can be observed, and come up with some convoluted explanation for why, then we can go down that path, and I'll think of something for which that logic is so crazy that you'll have to say "huh?"

Now it's makes more sense to argue that this all happens because of a mix up in terminology. GR and SR state the all measurable quantities must be invariant and scalar *with respect to Lorenz transforms*. The results of the Republican primaries of 1980 are indeed invariant *with respect to Lorenz transforms* and even though a political scientist may represent them as "vectors" within relativity they are "scalars." In other words, GR has nothing non-trivial to say about political science and election results.

In other words, relativity provides some restrictions for how things are measured *with respect to a certain set of transforms*. Arguing that relativity restricts measurement for *all uses of vector spaces* is a bit of a stretch, and if you go down that route I'm sure that I can find something even more ridiculous than the examples I provided. Vector spaces are very useful and widely used in social science and political science, and I could think of some uses for art and literature.

Just thought of something ridiculous. Restaurant and movie reviews. I go on yelp.com or rottentomatoes.com. Restaurant rates form a vector (i.e. atmosphere, decor, service, etc.) You can do movie reviews the same way (quality of plot, amount of action, quality of print, etc.) Are you telling me that GR says that I can't make a measure of the atmosphere of the restaurant and decor, at the same time? I think I can. But wait, you are saying that general relativity says that it's impossible for me to come to non-scalar conclusions about restaurants. If you say yes, then my reaction is "who made Einstein the restaurant review police?" So you are saying that it is *physically impossible* for me to measure restaurant atmosphere and service at the same time?!

If you insist on yes, 1) I'll think of something more ridiculous and 2) I'll introduce you to a group of restaurant reviewers and let you tell them that you as an expert in general relativity have figured out that it is physically impossible to do reviews in a certain way, and if they insist that they can come up with vector conclusions, that Albert Einstein says that its impossible. Regardless of the outcome of 2), it will be worth watching for the entertainment value (Scientists Versus Restaurant Reviewers, the new Food Network reality show).

At some point what I'm trying to get you to do is to say "wait, Lorenz invariance and restaurant reviews are totally separate things! When you are using vector spaces to represent restaurant reviews that's got nothing to do with how vector spaces are used in GR" Which is my point.

Now if you agree with that. Suppose some alien creature creates a chain of restaurants around a black hole...

Also this is a no-lose situation. You might come up with some argument that indeed GR says something non-trivial and non-obvious about restaurant reviews. Like it says a lot of things about foreign exchange rates. (seriously)

Rather than discussing the details above, I will clarify where I am coming from, philosophically. I will specify some beliefs from the most general to the most technically specific:

1) The development of science since 1900 (esp. relativity and QM, but also generalizations outside of science) supports the view that nothing is observable or has 'objective reality' without also specifying the method of observation. An outside of science example is 'popular opinion'. I don't think it exists outside specification of the measurement process, and will be very different depending on how it is measured. Similarly, I don't consider E and B fields (or photon and electron fields) observable or objective; you need to specify characteristics of the measuring device to get an observation.

2) Jumping to physics (possibly extending to other cases), modern physical theories have a variety of internal symmetries. In each such theory, something that changes with these internal symmetries is defined as not observable. One class of mistake in using such theories is failure ensure a prediction is invariant relative to these internal symmetries.

3) The important thing is the achieving the invariance appropriate to the theory - otherwise you have misapplied it. I will concede that I have perhaps overemphasized 'scalar' when the real issue is invariance (and not e.g. covariance), because possibly all invariant quantities can be stretched to be collections of scalars (suitably defined). But the important issue is the invariance; focusing on scalars in GR is the most effective way to make sure you have formulated an observable properly. An example in GR where it is artificial to reduce to scalars to get invariance is: curvature tensor vanishes everywhere. This is an invariant feature of a Riemannian or Semi-Riemannian manifold. Ben gave a few other examples where get an invariant without needing to explicitly produce scalars.

 

[PAllen]

Rather than discussing the details above, I will clarify where I am coming from, philosophically. I will specify some beliefs from the most general to the most technically specific:

1) The development of science since 1900 (esp. relativity and QM, but also generalizations outside of science) supports the view that nothing is observable or has 'objective reality' without also specifying the method of observation. An outside of science example is 'popular opinion'. I don't think it exists outside specification of the measurement process, and will be very different depending on how it is measured. Similarly, I don't consider E and B fields (or photon and electron fields) observable or objective; you need to specify characteristics of the measuring device to get an observation.

2) Jumping to physics (possibly extending to other cases), modern physical theories have a variety of internal symmetries. In each such theory, something that changes with these internal symmetries is defined as not observable. One class of mistake in using such theories is failure ensure a prediction is invariant relative to these internal symmetries.

3) The important thing is the achieving the invariance appropriate to the theory - otherwise you have misapplied it. I will concede that I have perhaps overemphasized 'scalar' when the real issue is invariance (and not e.g. covariance), because possibly all invariant quantities can be stretched to be collections of scalars (suitably defined). But the important issue is the invariance; focusing on scalars in GR is the most effective way to make sure you have formulated an observable properly. An example in GR where it is artificial to reduce to scalars to get invariance is: curvature tensor vanishes everywhere. This is an invariant feature of a Riemannian or Semi-Riemannian manifold. Ben gave a few other examples where get an invariant without needing to explicitly produce scalars.

Let me add to (3) a few further observations. All measurements, even over time, constitute a finite amount of information. At most, they can be considered a finite set of vector quantities (using a different sense of vectors than GR). In no sense I know of, can any collection of measurements be considered to constitute a vector field. Any vector field you might associate with measurements is an abstraction. Thus, in my view no vector field is directly observable.

A further observation in the GR context is that in context of significant gravity and a region not completely 'local', even supposing you have specified the basis (position and motion -> basis 4 vectors) of each 'vector' observation, there is neither a unique (nor even unique most natural) way to patch these frames (one for each flag, for example) into a coordinate system. Thus any expression you give to a vector field has a significant contribution due to arbitrary convention.

 

[twofish-quant]

All measurements, even over time, constitute a finite amount of information.

I worry about statements like this. I don't have too much problem with the statement that "all measurements that physicists are used to making have characteristic X." But one of the points that I've been making is that there is a big difference between "measurements that physicists are used to making" and "all measurements."

You could argue that "all measurements can be reduced to all measurements that physicists are used to making." That's an extremely strong philosophical statement, and one that I'm inclined to consider to be false. There's one famous counter example called the "marriage meter." There is no set of physical measurements that you could make on me or my wife that could tell you whether or not we are married. By doing some sort of brain scan, you could establish that we *think* we are married. Since there is no such thing as a "marriage meter" then this means that things like marriage/divorce rates aren't physically measurable and you can put these things in vector spaces.

I run into this sort of thing all the time at work. Two of the big, big questions write now is "how do you measure liquidity?" and "how do you measure risk?" Which quickly gets you into some philosophical questions "what is liquidity?" and "what is risk?" The relevance of this to the current discussion is that it seems that whatever liquidity and risk are, they somehow involve rather complicated vector spaces and the same sort of math that you find in GR. (Correlation matrices from hell.)

In no sense I know of, can any collection of measurements be considered to constitute a vector field.

Color. Color requires three components to be specified, but color is independent of those components. You can specific color in terms of RGB, or CMYK or pantone or color temperature, but color is independent of those measurement. Because you need multiple components to specific color, and the existence of color is *independent* of those components, its a vector field, and more than a collection of measurements.

Stock portfolios have similar issues. There are multiple equivalent ways of representing the dynamics of stock portfolio, but the dynamics exists independent of those representations.

Vector spaces and the math associated with it comes in very handy when you have an "underlying reality" that's independent of the measurements. Relativity is one such example, but it's not the only one.

Any vector field you might associate with measurements is an abstraction. Thus, in my view no vector field is directly observable.

But you could argue that scalars are an abstraction. I mean when I measure light intensity, it goes into a meter that goes through my eyes into my brain where who knows what happens. The problem with saying that no vector field is directly observable is that you end up with a very restrictive definition of "observe" under which it's not clear that anything is observable.

There might be a physics reason to do this. In QM, to observe means to "collapse the wavefunction."

Thus any expression you give to a vector field has a significant contribution due to arbitrary convention.

True, but vectors are useful especially in cases where there is a "reality" that is independent of arbitrary convention. GR is one such use case but there are others.

 

[twofish-quant]

The other thing to be careful here is "proof by lack of imagination." In order to prove non-existence, you have to show that something really bad happens if something did exist. I can think of a lot of bad things that would happen if you had physical measurements that were none Lorenz invariant, or if quantum observations didn't reduce to a single number.

However, asserting that something is impossible because one can't think of counterexamples is a bad way of showing that something is impossible.

This is particularly true because vector spaces are really useful, and can represent things that are pure fantasy (i.e. any first person shooter video game has vector space representations of all sorts of imaginary things).

 

[atyy]

I worry about statements like this. I don't have too much problem with the statement that "all measurements that physicists are used to making have characteristic X."

Proof for the modern age: it's stored on digital media.

 

[Ben Niehoff]

The other thing to be careful here is "proof by lack of imagination." In order to prove non-existence, you have to show that something really bad happens if something did exist. I can think of a lot of bad things that would happen if you had physical measurements that were none Lorenz invariant, or if quantum observations didn't reduce to a single number.

However, asserting that something is impossible because one can't think of counterexamples is a bad way of showing that something is impossible.

Throughout this thread, we have been discussing the fact that all measured quantities must be scalars with respect to coordinate transformations of spacetime, a context to which none of your examples has been relevant.

However, this goes beyond merely spacetime scalars. Any measurement of a vector quantity (in the vector space sense, not in the computer science sense) must come by choosing a basis and projecting out components, thereby measuring a set of scalars. If you think about it for a moment, you will see that this statement is a trivial tautology. What I'm really saying here is that every measurement is a comparison.

After all, that is what we really mean, isn't it? When we say something is "2 meters", all we're saying is it's twice as long as a certain metal bar in France. (Or, in modern SI, it's twice the distance light travels during so many oscillations of Cesium 133.)

Color. Color requires three components to be specified, but color is independent of those components. You can specific color in terms of RGB, or CMYK or pantone or color temperature, but color is independent of those measurement. Because you need multiple components to specific color, and the existence of color is *independent* of those components, its a vector field, and more than a collection of measurements.

RGB, HSB, CMYK, and temperature are all coordinates on color space. They emphatically do not obey the linear transformation law and axioms of a vector basis. In fact, I'm not convinced color space is a vector space at all, if we mean the color space that is relevant to "perceived color". (Of course, we can define color spaces that are vector spaces, but these might have nothing to do with actual color perception).

You could probably geometrize the idea of color space if you like, and make it a manifold, possibly with notions of parallel transport. Who knows, maybe there's a useful way to model some psycho-physical process using a color space bundle over spacetime.

At any rate, "color" is measured by first comparing incoming light against certain bands of frequency; i.e., taking an inner product in frequency space against certain basis vectors in order to form a collection of scalars. These scalars can then be used as a coordinate system on color space. This method works for RGB, CMYK, and color temperature type coordinates. Mapping physical observables to HSB coordinates is more complicated, and will require projecting the incoming spectrum onto several bands and doing some analysis with the results.

Since frequency space is infinite-dimensional, there is no real reason for color space to be 3-dimensional; the ultimate reason is that we have 3 kinds of color receptors in our eyes (which project the spectrum onto 3 bands). Other animals have 2-dimensional, or sometimes 4-dimensional color spaces.

 

[Ben Niehoff]

But we can stretch this into some absurd conclusions.

I take the a precinct-by-precinct map of the United States containing the election results of the Republican primary in 1980. The candidate votes form a vector and it's s perfectly good vector field. I can also form a vector field containing things like the price of real estate of different types of houses, the probability of default, divorce rates, crime statistics, etc. etc.

All of those are perfectly good vector fields.

No they are not. You can't just take a collection of data and call it a vector. This isn't computer science, it's geometry. Vector spaces have to obey certain axioms; you have to show how those axioms are physically reasonable if you want to say some physical quantity is modeled by a vector space.

Just thought of something ridiculous. Restaurant and movie reviews. I go on yelp.com or rottentomatoes.com. Restaurant rates form a vector (i.e. atmosphere, decor, service, etc.)

Nope, that's a collection of data, not a vector.

You can do movie reviews the same way (quality of plot, amount of action, quality of print, etc.)

Same problem.

So you are saying that it is *physically impossible* for me to measure restaurant atmosphere and service at the same time?!

Nobody said you couldn't measure more than one scalar at a time. This isn't quantum mechanics.

In fact, I think I specifically mentioned that a collection of scalars can be used to construct a tensor quantity in the basis used to measure all the scalars. So, e.g., I can use a set of (scalar) measurements to deduce all 6 components of E and B in whatever basis I choose. The point is what constitutes a measurement.

 

[twofish-quant]

Vector spaces have to obey certain axioms; you have to show how those axioms are physically reasonable if you want to say some physical quantity is modeled by a vector space.

And what axioms are violated by having election results represented by a vector? It seems to me to be a perfectly good vector space. All you have to do is to define addition and multiplication operations and you are done.

Axioms are axioms. I don't understand how "physically reasonable axioms" is a grammatical statement. Is "addition" physically reasonable? Now once you've defined addition and multiplication, you can then use them to make statements that are physically true or false. But the fact that you can make a statement that "the scalar multiplication of "red" by 2 gets us outside the set of physically valid colors" means that the axioms are defined.

And it's also a *useful* vector space. Once you've defined the operations, then you can define a "norm" which then describes how "close" two elections are. You can then do matrix transformations from one set of coordinates to another.

I define a C++ class RGB color, I define the operations of 2*Color and Color A + Color B. Once I've defined those operations, it's a vector space. I can even start do to tensor algebra.

 

[twofish-quant]

What I'm really saying here is that every measurement is a comparison.

Which is an extremely strong statement that I'm not sure is true once you go outside the range of physical measurements that physicists are used to. Just to give a trivial example, how do you measure "bank liquidity"? If you state "physical measurement" then I don't object.

RGB, HSB, CMYK, and temperature are all coordinates on color space. They emphatically do not obey the linear transformation law and axioms of a vector basis.

All you need for a vector space to exist is for the transformations to be defined, and I can clearly define a set of vector addition and scalar multiplication operations for RGB numbers. Now whether I get something *physically* meaningful if I perform those operations is another issue.

RGB numbers are physically bound within a range, but if I measure x, y, z coordinates of the earth, there are some values which are invalid.

Also there *isn't* much of a difference between the mathematical concept of a vector space and the computer science one. All mathematics requires is that you have a defined addition and multiplication that has eight axioms. Once you have a collection of data for which you do that, then you have a vector space.

 

[JDoolin]

I'm not sure if this makes any difference to your argument, but in this "election-result-space" do you mean that the space is the discrete set of electoral districts, or do you mean the space is a continuous geographical/political map of the region?

Also, the "election-results" are vector-like, in the sense that they are multi-valued arrays, but they are non-vector-like in the sense that they do not have any direction. i.e. they don't point from one district toward another.

 

[IsometricPion]

Which is an extremely strong statement that I'm not sure is true once you go outside the range of physical measurements that physicists are used to. Just to give a trivial example, how do you measure "bank liquidity"?

Suppose one has a collection of numbers arranged into categories. If these numbers are to represent quantitative data, then for each category there must be a unit reference against which all data in that category are compared (if two numbers do not share the same unit reference they cannot be immediately compared numerically). Therefore, in order to obtain any data representable as numbers or collections of categories thereof one must compare the quantity(ies) of interest to a reference value (or set thereof in the case of multiple categories). Thus, once a tuple of numbers and corresponding units are given the values of the quantities measured have been specified and any other description of the same values must yield the same (tuple of) numbers when converted into those units (and arranged by category into the same order in the tuple).

The measurement of a tuple of values either entails many different comparisions to unit references, and thus many measurements, or the the comparison of a smaller number of measurements to tuples of unit references. In any case, each comparison must yield precisely one number. It is the combination of this number and the associated unit tuple that represents the measured value. No matter what mappings are done on the collection of numbers, if the appropriate inverse mappings are done such that the necessary unit tuples map back to themselves the measured values must be represented by the same numbers (this places restrictions on what is considered a valid mapping and/or the types of valid measurements (in the context of this thread I prefer to think of it as the former rather than the latter)).

It is in this sense that all measurable quantities are collections of single numbers obtained from single measurements.

If the units involve space or time references, then they must pick out a set of vectors in space-time. Any such set of vectors can be used to construct a tensor of appropriate rank such that after any transformation, the evaluation of the resulting tensor on the images of the aforementioned set of vectors yields the same number as obtained originally (since one requires covarient tensors and vectors to transform in such a way that this is true).

Sorry if the vagueness (or triviality) of the above was excessive, I was going for an abstract approach but may have overreached the bounds of my knowledge and/or conventional nomenclature.

 

[twofish-quant]

Therefore, in order to obtain any data representable as numbers or collections of categories thereof one must compare the quantity(ies) of interest to a reference value (or set thereof in the case of multiple categories).

No. I just thought of a counter example. Twenty questions.

I'm located in a spot on the earth. By asking me yes-no questions, you can figure out my latitude and longitude. Am I on land? Yes. Do I see taxicabs? Yes. Are they green? No. Are they yellow? No. Do I see water? Yes.

With each question, you can eliminate parts of the vector space. The fact that I see tax cabs and they are not yellow, means that I'm not in Manhattan. Now if you can ask enough questions, you can figure out my location and convert to GPS coordinates.

Note that you've figured out my GPS coordinates without actually measuring my latitude and longitude or doing any reference comparisons at all. You can show that no reference comparisons were done, because you can play this game without knowing anything about latitude and longitude at all, and it's the same game that you can play with things that are *not* vector spaces (i.e. words in a dictionary).

One other way of thinking about it is that you can specify points in a vector space as the interaction of subsets of that vector space, which allows you to specify a point in that space without reference to basis vectors at all.

 

[twofish-quant]

I'm not sure if this makes any difference to your argument, but in this "election-result-space" do you mean that the space is the discrete set of electoral districts, or do you mean the space is a continuous geographical/political map of the region?

I suppose it depends on the data. It's pretty easy to fit anything into a vector space. One thing about vector spaces is that scalars don't have to be real and neither do vectors. Election *results* certainly form a vector space (since you can add and scale vote totals). It's not obvious to me how to represent discrete electoral districts in a vector space, but that's just due to my lack of imagination.

Also, the "election-results" are vector-like, in the sense that they are multi-valued arrays, but they are non-vector-like in the sense that they do not have any direction. i.e. they don't point from one district toward another.

Again this depends on the structure of the data, but my point is that if you consider anything other than relativistic vectors to be "bags of unconnected data" you lose the structural information about the data. If you treat everything as "bags of data" you lose type information, which is a bad thing.

One thing that got me started thinking along these lines is the fact that you can call functions in C++ "covariant" and "contravariant". So what does tensor calculus have to do with C++. Well, that got me into the world of category theory...

 

[Dale]

Election *results* certainly form a vector space (since you can add and scale vote totals).

They certainly do not. One of the requirements of a vector space is that there must be an operation where multiplication of a vector by a real number* leads to another vector. If you multiply an arbitrary election result by any negative number or by any irrational number you will get negative or fractional votes, neither of which are members of the space of possible election results.

You cannot scale election results by arbitrary real numbers, nor even by arbitrary integers.

*Vectors can be generalized to multiplication over other fields besides the real numbers, but the conclusion remains. There is an additive identity element, but no additive inverse in the space of election results.