Are Christoffel symbols measurable?

twofish-quant

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.

DaleSpam

From my side there is not any disagreement that some observables are not scalars.

However, an observation of any of the quantities you have mentioned is a scalar (i.e. it is unchanged under diffeomorphisms) even if the corresponding observable is not.

DaleSpam

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

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

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

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:

How about this explanation from the reference posted previously:

[bottom of page 6, http://brucel.spoonfedrelativity.com...ackground.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

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

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.

twofish-quant

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

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

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

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.

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

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.

DaleSpam

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.

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.