Are Christoffel symbols measurable?

by waterfall
Tags: christoffel, measurable, symbols
 Sci Advisor P: 8,537 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".
P: 1,593
 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 $\Gamma^\mu_{\nu\rho}$.
 Sci Advisor P: 1,593 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.
P: 8,537
 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.
P: 8,537
 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.
P: 476
 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.
P: 381
 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.
P: 8,537
 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.
P: 381
 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).
P: 6,863
 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)
P: 6,863
 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.
P: 6,863
 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.
P: 6,863
 Quote by waterfall 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.
P: 8,537
 Quote by twofish-quant 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.
P: 476
 Quote by waterfall 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.
PF Gold
P: 5,059
 Quote by Naty1 [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.
 Quote by Naty1 [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_...ativity_theory No problem with these ideas, right??
All of this looks fine to me.
PF Gold
P: 5,059
 Quote by twofish-quant 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.