Are Christoffel symbols measurable?

[Dale]

I think we are have different definitions of what a "scalar" is. I'm defining it as a quantity that doesn't change when you change coordinate systems. I measure something in my coordinate system. You measure something in your coordinate system. We get the same number. There are some things that you can measure that have that characteristic (electric charge if you vary only space and time coordinates). There are some things that you can measure that *don't* have that characteristic (volume). Classifying things according to how they behave turns out to be useful.

That is my definition also.

Once you have measured something the observation is a scalar. If you perform some experiment and the number 7.43 pops out on your measuring device then no change of coordinate systems can possibly change that number to anything other than 7.43. Therefore, the number measured is a scalar.

It may be that you claim that 7.43 is a length and I disagree, but regardless of how we interpret the number in terms of physical quantities in our favorite coordinate system, we will agree that the number is the same. That makes it a scalar.

 

[PAllen]

Also I think that I've thought of an observable that clearly is not a scalar. Chriality. A particle is either left-handed or right-handed, and since this is a binary quantity. It's not a scalar.

For that matter, you flip a coin, the "headness" or "tailness" of the coin is a boolean quality which is not a scalar. For that matter any observation or observable that is binary isn't a scalar.

Finally, for the people that still insist that observations have to be a scalar, how do you know it's a scalar and not a pseudo-scalar? I have a feeling that "height" is a scalar, but "leftness" is a pseudoscalar.

This does raise some interesting points. If one allows reflections in your diffeomorphism class, then chirality is clearly not invariant (and it could not be defined in terms diff invariants). Yet I would consider it an observation. This is the first and only one of your examples so far that I accept as an exception to observation=(collection of invariant scalars). Note, I always included collection as part of the definition, because I included a phototograph as one of my first examples. Ben and Mentz also explicitly included collection of scalars as a measurement.

Note, volume is trivially a scalar - it is integral of volume element, which is differential contraction of the metric. It is just as much a scalar invariant as proper time.

 

[Ben Niehoff]

This does raise some interesting points. If one allows reflections in your diffeomorphism class, then chirality is clearly not invariant (and it could not be defined in terms diff invariants). Yet I would consider it an observation.

I'm not so sure we can "observe" chirality, anyway. It is a matter of whether some spinor lies in a particular subspace...I would say that the things we actually measure are projections, which are scalars (i.e., we can ask what are a spinor's projections onto subspaces R and L, which we have defined relative to some given frame).

It's a bit of double-talk, really. Measurements are always scalars, but we can use collections of scalars to reconstruct tensorial objects in a given frame.

Coordinates, however, are always irrelevant. The important thing is a frame. A system of coordinates can be used to define a frame, by taking the coordinate basis, but all we really care about is the frame.

 

[PAllen]

I'm not so sure we can "observe" chirality, anyway. It is a matter of whether some spinor lies in a particular subspace...I would say that the things we actually measure are projections, which are scalars (i.e., we can ask what are a spinor's projections onto subspaces R and L, which we have defined relative to some given frame).

It's a bit of double-talk, really. Measurements are always scalars, but we can use collections of scalars to reconstruct tensorial objects in a given frame.

Coordinates, however, are always irrelevant. The important thing is a frame. A system of coordinates can be used to define a frame, by taking the coordinate basis, but all we really care about is the frame.

Yeah, I take it back, it is not really an exception. I was actually thinking of chirality of a body (as we were mostly discussing classical field theory). But that really means (as an observation) shares a type of similarity to a reference object we call 'left handed'. If we do a reflection, it is still similar to the object (in the manifold) we have labeled left handed. So we still observe it to be left handed.

 

[Naty1]

I'm really glad 'observables [measurements] in GR is being hashed out. I've been making notes for several months and still am uncertain.


[1] I am unable to keep track of all the claims, counter claims and retractions and some of the terminology as "observation" vs "observable" and varying "definitions" of scalars makes it more uncertain...
If anybody would summarize ANY points of consensus that would be great for us 'amateurs'.

[2] What do you think of this assessment:

Two fish says:

..Another observable that's not a scalar. color [post #65]
[and in post #68]...we have different definitions of what a "scalar" is. I'm defining it as a quantity that doesn't change when you change coordinate systems. I measure something in my coordinate system. You measure something in your coordinate system. We get the same number.

How about this explanation from the reference posted previously:

[bottom of page 6, http://brucel.spoonfedrelativity.com/GR1a-Background.pdf]

"... Suppose I measure the temperature (°C) at a given point P at a given time. You also measure the temperature (°C) at P at the same time but from a different location that is in motion relative to my location. Would it make any sense if we measured different values; for example, my thermometer measured 25 °C and yours measured 125 °C? ... Only scalars that remain invariant between coordinate systems like this can be called “tensors of rank 0”.

Now let f be the frequency of light coming from a laser at P. Again, let two observers, K and K*, measure the frequency of the light at P at the same time using the same units (Hz). If I am stationary relative to the source, the light will have a certain frequency, for example f. If, on the other hand, you are moving toward or away from the source, the light will be red or blue shifted... although frequency is a scalar, it is not a tensor of rank 0..."

So what I'm asking is if we agree some scalars ARE tensors of rank zero, and these are Lorentz invarient, while other scalars are NOT and these are NOT invariant.

If so, how do we tell them apart other than making measurements in different frames of reference and comparing results?

 

[PAllen]

"... Suppose I measure the temperature (°C) at a given point P at a given time. You also measure the temperature (°C) at P at the same time but from a different location that is in motion relative to my location. Would it make any sense if we measured different values; for example, my thermometer measured 25 °C and yours measured 125 °C? ... Only scalars that remain invariant between coordinate systems like this can be called “tensors of rank 0”.

Now let f be the frequency of light coming from a laser at P. Again, let two observers, K and K*, measure the frequency of the light at P at the same time using the same units (Hz). If I am stationary relative to the source, the light will have a certain frequency, for example f. If, on the other hand, you are moving toward or away from the source, the light will be red or blue shifted... although frequency is a scalar, it is not a tensor of rank 0..."

So what I'm asking is if we agree some scalars ARE tensors of rank zero, and these are Lorentz invarient, while other scalars are NOT and these are NOT invariant.

If so, how do we tell them apart other than making measurements in different frames of reference and comparing results?

I don't think we will get philosophic consensus on the broader part of your question. For example, while I think the question of an observation or measurement can be well defined, the attempt to introduce 'observable' distinct from 'observation' leaves me cold. If one defines it broadly, e.g. a construct of theoretical formulation that aids calculation of observations, I would feel compelled to accept not only E and B, but also wave functions and 'possible histories' (in sum over histories qft). I really don't consider the latter observables, so, only slightly reluctantly, rule out E and B as well. Note, I have no problem with useful but unobservable constructs in a theory - with my definitions, you can't really have a theory without them.Your second major question is easy to answer. 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.

As to how you know, in GR and SR, there are primary objects defined as scalars, vectors, or tensors. Then, there are standard results in differential geometry to construct scalars out of these (contraction; integration of contractions; etc.). For example, while frequency is not a scalar, a measurement of frequency by a device is, as follows: you take the dot product of light's 4-momentum with the instrument's 4-velocity to compute the measurement of frequency by the instrument. This is contraction of two vectors with the metric tensor, and is definitely a scalar. (Thus, no matter who does the computation, and what coordinates they use, everyone will agree on the frequency measurement by the specified instrument).

 

[twofish-quant]

Once you have measured something the observation is a scalar. If you perform some experiment and the number 7.43 pops out on your measuring device then no change of coordinate systems can possibly change that number to anything other than 7.43. Therefore, the number measured is a scalar.

Doesn't work that way. I have a device that has a switch "metric" versus "English." If I flip the switch, the system uses different logic to come up with the number that gets displayed on the LCD. The fact that I can change the number displayed based on the system makes it a non-scalar.

It may be that you claim that 7.43 is a length and I disagree, but regardless of how we interpret the number in terms of physical quantities in our favorite coordinate system, we will agree that the number is the same. That makes it a scalar.

We don't. We can flip the switch in the measuring device and get different numbers.

 

[twofish-quant]

I'm not so sure we can "observe" chirality, anyway.

I have a pair of gloves. They look left-handed.

It's a bit of double-talk, really. Measurements are always scalars, but we can use collections of scalars to reconstruct tensorial objects in a given frame.

I think that you could come up with a definition of "measurement" that always gives you a scalar, but I think it would be a technical definition that wouldn't have any obvious relationship to the common definition or actual definition or process of "measuring" or "observing."

Part of the reason I think we are arguing is that we are at the boundary where the platonic world meets the real world. I'm pretty sure you could have a mathematically precise definition of "measurement" in which all "measurements" are scalars, but the trouble is that I have a physics background, and if those definitions don't fit the "real world" it's not going to make sense.

So I see a pair of gloves and establish that they are left-handed. If you argue that really isn't an "observation" or a "measurement" then I'm going to have severe problems with those definitions. Similarly, to specify "color" requires at least three numbers. If you argue that "measurements" can only result in one number, then color cannot be measured. You can choose whatever definition of measurement you want when you are proving theorems, but if you come up with a definition of "measurement" that excludes colors, then I'm going to have problems with it.

Funny thing. I looked up the word "scalar" on wikipedia and they have three different articles on the topic. Scalar (mathematics), scalar (physics), and scalar (computer science). Curiously, the definition that I've been using is closer to the math definition which is different from the physics one. Apparently the physics definition of scalar are invariants under subset of transformations whereas that restriction doesn't exist in the math definition. Probably the reason for that is that my main interactions with scalars and vectors is financial and computational.

But rather than arguing about which definitions is correct, it might be useful to just list the definitions of "scalar" and "observation". Once you have N definitions of scalars and M definitions of observations, then we can create a chart indicating whether all "observations" are "scalars".

 

[PAllen]

Doesn't work that way. I have a device that has a switch "metric" versus "English." If I flip the switch, the system uses different logic to come up with the number that gets displayed on the LCD. The fact that I can change the number displayed based on the system makes it a non-scalar.
We don't. We can flip the switch in the measuring device and get different numbers.

What definition of scalar are you using? I have never seen yours. Despite disagreements on other matters, Dalespam, Ben, Mentz, and myself seem to have a common definition different from yours. In the GR context, we simply use the one from differential geometry such that, for example, any contraction of a tensors or integral of differential contraction is a scalar. I have never, in any book on GR, seen any other definition. Yet you say proper time between two events is not a scalar because it could be expressed in seconds or years.

Later you say yours is the physicists definition. I disagree - I have GR books going back 100 years, including the originals by Einstein, Pauli, etc. None use a definition resembling yours.

 

[twofish-quant]

In the GR context, we simply use the one from differential geometry such that, for example, any contraction of a tensors or integral of differential contraction is a scalar.

It appears that in GR, something is a scalar if it is invariant to Lorenz rotations and translations, but if it changes as a result of scaling relationships, it doesn't matter. In differential geometry, you aren't confined to those specific transformations. You can define an arbitrary set of transformations for determining whether something is a scalar or not.

Later you say yours is the physicists definition. I disagree

I backed away from that statement. The way I was using scalar is certainly not the way it's used in relativity, and now that I've understood that, I don't have any problems with the statement that all observables are "scalar" using the definition in relativity (i.e. number changes due to scaling don't matter, whereas number changes due to rotation and translation do).

I think one reason I ended up with my definition is that my introduction to differential geometry was through mathematical finance where obviously Lorenz rotations don't matter. Illiski's "Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing" has one of the most readable introductions to fiber bundles that I've ever seen, and I think that left me with definitions that are different from those that are used in GR.

What happens in mathematical finance is that in liquid markets, things are scale invariant, but what you are looking for are situations in which scale invariance breaks down. What I think is happening is that, in relativity, unit conversions are trivial (i.e. you can always multiply X feet by a constant factor to get X km) so in defining scalar and vector, it's natural to ignore those. However, in mathematical finance, you can't just multiply X dollars by a constant factor to get Y euros, so when differential geometry is used there, the definition of scalars and vectors are such that scaling relationships are not ignored since that's what you are interested in.

 

[PAllen]

I have a pair of gloves. They look left-handed.

To my mind, to make this a physical observation rather than a math definition, we need to incorporate a more complete definition of what this means. To me, it means identifying the class of objects in physical reality we label left handed rather than right handed. Then, if I do a diffeomorphism that includes a reflection, this class of objects remains matching in chirality, remaining the class I call 'left handed'. This is not so different than the way pullbacks preserve metrical quantities under diffeomorphism.

 

[PAllen]

I'm not quite sure how to define observation with no circular aspect to its definition. I mean any type of instrument (including a camera, human eye+brain, a dial on a meter, etc) capturing a snapshot of information. The description of a device includes its state of motion or rotation.

Within a particular theory, when physically interpreting its mathematical constructs, I require that internal conventional changes do not affect what I match to physical measurement. What these features are, depends on the theory. In GR, it means diffeomorphisms. The subtlety, is that a mapping that changes, say, a value of (x1 -x0) from 1 to 3 will correspond with a change in metric that makes the length stay the same. And I don't consider units of measure part of this at all. Using the same example as chirality, I get to units as saying I have a reference object for meter, for foot, etc. Under diffeomorphism, any measurement referenced to these objects stays the same.

Finally, I note, that within GR, to have the right properties (as above) a computation deemed represent a measurement must be a collection of scalars. Really, this could be viewed as a collection of individual measurements. Thus, a color photo of something is really a measurement for each pixel on a CCD or grain on chemical film.

 

[Dale]

Doesn't work that way. I have a device that has a switch "metric" versus "English." If I flip the switch, the system uses different logic to come up with the number that gets displayed on the LCD. The fact that I can change the number displayed based on the system makes it a non-scalar.

We don't. We can flip the switch in the measuring device and get different numbers.

That is a different measurement. Yes, different measurements can have different results.

The question about whether or not a quantity is a scalar has nothing to do with flipping switches, just about changing coordinates. Regardless of the coordinate system used to analyze the experiment the number does not change, it is therefore a scalar.

 

[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.

 

[Dale]

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.

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.

 

[PAllen]

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.

 

[JDoolin]

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.

 

[Naty1]

[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_(physics)#Scalars_in_relativity_theory

No problem with these ideas, right??

 

[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.
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.

 

[JDoolin]

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

 

[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.