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## Are Christoffel symbols measurable?

 Quote by DaleSpam Hmm, it does seem hard to cast that as a collection of scalars. The measuring device doesn't produce a number nor a set of numbers, so my previous statements either don't apply at all or their applicability is not clear.
I agree it is a little more strained, but not fundamentally. Each flag can be treated as reading out an angular direction relative to gyroscope providing reference (conceptually). The key point for me about type of geometric object is that the wind measurement of the 'wind field' requires specification of the position and states of motion (including rotation) of the collection of flags. Change these, and you have a different measurement. You cannot talk about observing the wind field without specifying information about each and every flag. Having done so, each flag's read out is, indeed, described as (say) a pair of angles (assuming it has full range of motion) in time, each of which would be computed in GR as a Lorentz scalar function.

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 Quote by twofish-quant As a result of this discussion, I've become convinced that within relativity, all observed quantities must be Lorenz invariant and hence are scalar quantities as the term scalar is used in relativity. Whew..... I'm still not altogether convinced that only scalar quantities can be observed in contexts outside of relativity. For example, you look at a wind tunnel with a wing and then you have little flags pointing in different directions. It seems to me that you are in fact observing a vector field. Also, if you look at a weather vane pointing the in the direction of the wind, that seems like observing a vector. Part of this is that my other exposure to differential geometry has to do with data visualization and the whole point there is to observe vector fields. Again there is this problem with definitions, but if you tell an aerospace engineer or graphics visualization guru that they aren't really observing a vector but a collection of scalars, they will look at you funny, and if you tell them that those are really scalars because they are invariant under a Lorenz transform, they will really look at you funny. So it seems to be that when you apply differential geometry to fluid dynamics or graphic visualization, you can indeed observe vectors. Now if you have a CFD flow that is around a black hole, it seems to me that in that situation you'd have quantities that are "scalar" in the relativistic sense (i.e. the measurement does not change when you change the reference frame) but vector in the CFD sense (you need multiple components to describe the situation). So if suppose you have a fluid flow around a black hole, and you have a field that describes the velocity of the fluid in the local reference frame of each point, I'd guess that a GR person would describe the fluid flow as a "collection of scalars" because the components of that flow do not change when you do a Lorenz transform, but the CFD person would describe the fluid flow as vector since you need multiple components to describe the velocity field. At this point I suppose we bring in fiber bundles. Thoughts? Also if it is the situation that different areas of physics are using different terminology then analogies aren't going to work.
The goal seems to be (and correct me if I'm wrong), describing "the universe as it really is." The idea is that if you collect all the local observations of all of the observers in the environment, you can patch together a picture of the whole thing.

While that may well be so, but do you think a patching together of an image from different perspectives is an accurate representation of the thing as it really is?

That if you take the observations of many observers and patch them together in some well-defined fashion, then you have a picture of the universe "as it really is."

And why not? As long as your picture contains every event that ever happened, and every event that ever will happen, what do you think? Is that an accurate picture of the universe, or is it flawed?

Is it better to represent the universe from the perspective of a single observer; a single observer looking at non-local phenomena? With a single observer, your picture can only contain the events which the single observer observed.

By contrast, with local observations of all the observers in the environment, your picture contains every event in the environment.

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

[1] PAllen posts:
 For example, while frequency is not a scalar, a measurement of frequency by a device is, as follows: you take the dot product .....
That was at least hinted at elsewhere, and I did not 'get it'...good insight, thanks.

[2]The referenced paper says:

 ...although frequency is a scalar, it is not a tensor of rank 0..."
PAllen says:

 Frequency is effectively the timelike component light's 4-momentum. You can't treat one component of a vector as a scaler, just because it is a number in a particular coordinate representation.

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' ?

[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] :

 In the theory of relativity, one considers changes of coordinate systems that trade space for time. As a consequence, several physical quantities that are scalars in "classical" (non-relativistic) physics need to be combined with other quantities and treated as four-dimensional vectors or tensors. For example, the charge density at a point in a medium, which is a scalar in classical physics, must be combined with the local current density (a 3-vector) to comprise a relativistic 4-vector. Similarly, energy density must be combined with momentum density and pressure into the stress-energy tensor. Examples of scalar quantities in relativity: electric charge spacetime interval (e.g., proper time and proper length) invariant mass
http://en.wikipedia.org/wiki/Scalar_...ativity_theory

No problem with these ideas, right??

 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. 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.

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 Quote by Naty1 http://en.wikipedia.org/wiki/Scalar_...ativity_theory No problem with these ideas, right??
The article lists three examples of quantities that are scalars in Relativity: electric charge, spacetime interval, invariant mass, but where is the definition of scalar in Relativity?

The article has three examples of scalars, but no clear examples of what are NOT scalars.

It lists several quantities:
• charge density
• current density
• momentum density
• pressure
• stress-energy tensor

but does not specify whether these things are considered to be scalars or not.

 Recognitions: Science Advisor 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".

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 Quote by TrickyDicky 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 $\Gamma^\mu_{\nu\rho}$.

 Recognitions: Science Advisor 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, travelling 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.

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 Quote by 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, travelling 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.

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 Quote by twofish-quant 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.

 Quote by waterfall 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.

 Quote by juanrga 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.

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 Quote by waterfall 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.

 Quote by atyy 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).

 Quote by PAllen 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)

 Quote by TrickyDicky 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.

 Quote by atyy 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.