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Are Christoffel symbols measurable? 
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#91
Feb1812, 10:46 AM

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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". 


#92
Feb1812, 01:21 PM

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The caveat is that we can only measure gaugeinvariant 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 [itex]\Gamma^\mu_{\nu\rho}[/itex]. 


#93
Feb1812, 01:28 PM

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Another thing to add: There are additional ways to make gaugeinvariant scalars besides merely making contractions like
[tex]R_{abcd} X^a Y^b Z^c W^d[/tex] at a point. One can also make nonlocal measurements, by paralleltransporting 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 YangMills 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 AharonovBohm 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. 


#94
Feb1812, 01:34 PM

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#95
Feb1812, 01:37 PM

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#96
Feb1812, 02:00 PM

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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. 


#97
Feb1812, 05:29 PM

P: 381

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? 


#98
Feb1812, 06:48 PM

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#99
Feb1812, 07:04 PM

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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). 


#100
Feb1812, 10:54 PM

P: 6,863

I take the a precinctbyprecinct 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 nontrivial 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 nontrivial 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 nonscalar 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 nolose situation. You might come up with some argument that indeed GR says something nontrivial and nonobvious about restaurant reviews. Like it says a lot of things about foreign exchange rates. (seriously) 


#101
Feb1812, 11:00 PM

P: 6,863

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 lefthanded 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. 


#102
Feb1812, 11:04 PM

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


#103
Feb1812, 11:09 PM

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#104
Feb1812, 11:36 PM

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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. 


#105
Feb1912, 08:46 AM

P: 476

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. 


#106
Feb1912, 01:35 PM

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#107
Feb1912, 02:13 PM

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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 SemiRiemannian manifold. Ben gave a few other examples where get an invariant without needing to explicitly produce scalars. 


#108
Feb2012, 06:15 AM

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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. 


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