Loki 2

I've worked on the translation of the discription of Loki a bit more today and this is nearly word for word for the original Icelandic (circa 1220 A.D.):

"Loki is fine and fair to see, bad by nature, very shifty in manner. He has that wisdom beyond advanced, but [it] is called sleight, and devices for all things; he would bring the Æsir even into a complete morass and often he would free them with fabrications."

"Loki er fríður og fagur sýnum, illur í skaplyndi, mjög fjölbreytinn að háttum. Hann hafði þá speki umfram aðra menn er slægð heitir og vélar til allra hluta. Hann kom ásum jafnan í fullt vandræði, og oft leysti hann þá með vélræðum." -GYLFAGINNING 33

There seems to be some wordplay going on in the Icelandic.

So, Loki was a magician and a prankster and there is an implication that he was as changable as the weather.

Some Ambivalence on Global Warming

Some people raise concerns about global warming and others cite cycles as an explanation for changes but the historical record offers support for both points of view. It does not appear to be purely global warming or necessarily a cycle. To get an accurate reading on any long term warming we have to allow for the possibility of cycles or fluctuations and eliminate their effect on a fit to the data. The results are shown for Jul, the warmest month on average, for Mean Land Temperatures in the northern hemisphere. The only explanation for the data is that both long term effects and short term effects are present.

These fits are just empirical and not based on any particular theory as to what is happening. The fits can be extrapolated to give an estimate of future temperatures. Note that for this plot the temperatures are given in °F while in previous blogs °C was used.



Supplemental: If σ is the standard deviation for the fit one would expect the temperatures to be within 3σ (approximately 0.5 °F) of the solid curve. These are mean values for the month so one would also have to add an additional correction to estimate bounds for the daily highs and lows for a particular location.

Monthly Mean Temperatures in the Northern Hemisphere for 1880-2009*

NOAA's monthly temperature anomalies are given relative to the mean monthly temperature averages for the reference period 1901-2000. If these averages are added to the mean monthly land anomalies for the northern hemisphere one gets the mean monthly temperatures which are displayed in the following plots.

An analysis by month shows a similar pattern for the entire record and the best fits for each month displays a similar pattern. The following table gives the period, amplitude and phase for the fundamental frequency of the best fit. April appears to have the longest period and October has the largest amplitude. These months are mid spring and mid autumn although this might not necessarily be significant. One has to ask if these changes are due to oscillations or if they are just fluctuations in the temperatures which appear to be oscillations. To assert that these "fluctuations" are oscillations at this point would, in my opinion, be "deeming" them to be so. The cause of these fluctuations needs to be determined if it is shown that they are not just random fluctuations.

The coefficients for the quadratic portion of the fit are given below. The columns are the months, the mean values, the rates of change and half** the "accelerations" for each month. Note that on average the rates of change indicate both increasing and decreasing trends depending on the month.

*edit: The available data is from Jan, 1880 through Nov, 2009.

** 2nd edit: The numbers are just the quadratic coefficients and represent values at the beginning of the curves. One would have to set yr to 129 to estimate the 2009 values. The fit used 0 for the Dec, 2009 value which would add a slight error for the Dec curve.

Northern Hemisphere Land Temperature Anomaly

I downloaded some monthly temperature data from NOAA today and was able to find a fit to the data for Northern Hemisphere's Land Temperature Anomaly. An anomaly is a deviation from some reference temperature. The fit assumed a parabolic curve and 20 sinusoidal harmonics of a fundamental frequency. The fundamental frequency that gave the best fit had a period of 63.6 years. The fit in the plot below includes the fundamental frequency plus two harmonics. This fit suggests that the temperatures will soon switch to a declining trend if the pattern holds. But there is no guarantee that it will. The fit is what we would expect if the data were cyclical in nature.



(click to enclage)

A little Learning is a dang'rous Thing

.....

First follow NATURE, and your Judgment frame
By her just Standard, which is still the same:
Unerring Nature, still divinely bright,
One clear, unchang'd and Universal Light,
Life, Force, and Beauty, must to all impart,
At once the Source, and End, and Test of Art
Art from that Fund each just Supply provides,
Works without Show, and without Pomp presides:
.....

Those RULES of old discover'd, not devis'd,
Are Nature still, but Nature Methodiz'd;
Nature, like Liberty, is but restrain'd
By the same Laws which first herself ordain'd.
.....

Of all the Causes which conspire to blind
Man's erring Judgment, and misguide the Mind,
What the weak Head with strongest Byass rules,
Is Pride, the never-failing Vice of Fools.
Whatever Nature has in Worth deny'd,
She gives in large Recruits of needful Pride;
.....

A little Learning is a dang'rous Thing;
Drink deep, or taste not the Pierian Spring:
There shallow Draughts intoxicate the Brain,
And drinking largely sobers us again.
Fir'd at first Sight with what the Muse imparts,
In fearless Youth we tempt the Heights of Arts,
While from the bounded Level of our Mind,
Short Views we take, nor see the lengths behind,
But more advanc'd, behold with strange Surprize
New, distant Scenes of endless Science rise!

- Alexander Pope, An Essay on Criticism, 1709

Loki

The Nordic equivalent of Hermes and Mercury was known as Loki. His father was Fárbauti, a "hacker", and his mother was Laufey, possibly a silvan nymph. Like the Greek and Roman gods his connection with bounds is that he often ignored them. There is a discription of him given in an Icelandic saga about Gylfi, the first king of Sweden,

"Loki was cute and alluring* in appearance, bad by nature, very duplicitous in manner. He surpassed others in that wisdom which is called craftiness with devices for all things; he would even bring the Æsir into a complete quagmire and often he would free them with fabrications." - Gylfaginning, 33

Loki was a slippery character who did not engender trust and probably was responsible for the downfall of the gods. He had a fate similar to that of Prometheus.

(*edit: fríður og fagur can be translated as "beautiful" and "enchanting" but, allowing for changes over time, "plain" and "simple" might be a better fit. A modern translation for fríður is "peace." vélar and vélræðum → "devices" and "fabrications," at least that seems to be the impression.)

Problems with Analyzing Data with Cyclical Error

There may be some doubt concerning the rate of global warming which is due to the nature of the temperature measurements. For statistical data one usually encounters a normal distribution but with temperatures there are both daily and annual cycles in the data. This can bias the estimate of global warming as seen by what happens to the data in the following example. The solid line is the true curve, the dots are the data and the dashed line is the estimated trend based on the data. Notice that the slope is greater for the estimated trend. This increases the time necessary to make accurate estimates of the rate and rate of change of global warming. When analyzing the data one has to be on guard against subtle errors.

Naive Impressionism in Ancient Times?

I found some passages in the Bible which touch on naive impressionism.

and said, “Assuredly, I say to you, unless you are converted and become as little children, you will by no means enter the kingdom of heaven. (Matthew 18:3)

When I was a child, I spoke as a child, I felt as a child, I thought as a child. Now that I have become a man, I have put away childish things. (1 Corinthians 13:11)

Paul may have seen naive impressionism as somewhat lacking. Perhaps this was due to his background. The ancient Greeks also may have viewed Hermes, the younger brother of Apollo, as somewhat naive.

Christ's message was one of renewal/rebirth.

Commons vs Lords

COP15 dealing with global warming is taking place in Denmark. There are two aspects to the problem. First, what is actually happening and, secondly, what should our response be. Whatever the cause of global warming, our response will primarily be an economic and political solution and perhaps we should consider the difference between economics and politics and how they might work together. In ancient times economics dealt with "rule of the home" while politics concerned itself with management of the city-state. There is still this division today in the separation of microeconomics and macroeconomics. The first deals with local problems while the latter deals with more general problems. The difficulty that we are having with global warming might be that the two approaches are out of balance.

In nature one finds a similar division between Special Relativity and General Relativity. Special Relativity is local while General Relativity deals with weaker but more "global" forces that ultimately dominate. These are the Commons and Lords of Nature.

We can probably do something similar with economics and global warming and try to get the two systems to work together better. The question is, "Are we up to the task?"

Beyond Relativity

The weakest assumption in the derivation of the Lorentz Transformation is Einstein's assumption that the speed of light is a universal constant. This assumption is validated by the null result of Michelson-Morley experiment. The conclusion that there is no ether may not be justified. The validity of the Lorentz Transformation suggests that the observed change to the speed of light for a given relative velocity is neglible. If this were not so naive impressionism suggests that one can still find a transformation between the two reference frames which is similar to the Lorentz Transformation. An expression for A can be found by plugging the two values for the speed of light into the formula for the addition of velocities and solving for A. One then gets the following results where Δc is the change in the observed value of the speed of light,

Relativity and Naive Impressionism

How can we justify the assumption of symmetry used in the derivation of the Lorentz Transformation? It appears to be a form of naive impressionism or the belief that what is true for one is true for all. But it appears to fit the facts. The Michelson-Morley experiment gave a null result on the measurement for the velocity of the ether. The speed of light appears to be independent of the Earth's motion throughout the year at least for the value of the Earth's orbital velocity. Relativity has proven to be a useful tool for scientific research. But it seems to validate the simplistic worldview and the possibility of making a false assumption.

One could view Ockham's razor as a form of naive impressionism. But it is an economy measure. One has to seek a balance between ignoring the lack of evidence to the contrary and unnecessarily complicating an explanation of the facts. Making unjustified assumptions raises doubts.

The operational rule of the scientific community appears to be naive impressionism with doubts. So we are justified in saying, "Don't trust them." As a matter of expediency, however, this may be the best way of proceeding, but under the circumstances one needs to show that the use of Relativity is justified in a particular case and that the results are reasonable. The assumptions break down if there is an asymmetry in the point of view of the observers. This may be the explanation of the imaginary values in the transformation for velocities exceeding the speed of light. Strong gravitation may bias the transformation and Einstein attempted to address this in the General Theory of Relativity. Special Relativity may still be the best first approximation to the laws of the Universe.

Special Relativity & the Lorentz Transformation

In Special Relativity the Lorentz Transformation allows one to convert measurements such as distances and times made in one frame of reference to those of another. It is easiest to derive the transformation when the situation is symmetric, i.e., changing between the two frames of reference doesn't alter any of the parameters involved. The same transformation can then be used for both reference frames. Consider two spacecraft headed directly towards each other with a relative velocity of v. We can use unprimed variables for measurements made by the first spacecraft and primed variables for the second. In both cases the transformation is L.


The transformation is assumed to be linear and can be represented by four components of a matrix which only depend on the relative velocity.





By noting that a point in the second spacecraft doesn't move relative to itself while it appears to be moving with velocity -v to the first spacecraft, we can deduce B. The method can be generalized to find the any velocity, V', as it appears to the second spacecraft if its value for the first spacecraft, V, is known. This is the formula for the addition of velocities.



















With L the same for both reference frames we can use it twice to make a transformation from the first to the second spacecraft and then back again to the first. Since we should get the original values back the result is the identity matrix. Doing the multiplications and equating terms gives two more terms of the transformation leaving only one unknown, A.














This is as far as symmetry will take us. To go further Einstein had to assume that the speed of light was a universal constant. This is not unreasonable if space is homogeneous. We just have to be careful about directions though. A ray of light moving along the common line of the spacecraft will appear to be moving in different directions to the two observers. Let's say that it moves away from the first and towards the second. This allows us to simplify the expression for C and determine an expression for A.














We then have to use the minus sign so that the direction of time will be the same in both frames of motion.









So we have found the transformation in this particular case. We can use other transformations to convert to situations that are less symmetrical.














This derivation indicates that Relativity doesn't impose any constraints on time travel. There are transformations which will convert positive changes in time to negative ones. But at the same time they will also convert positive energies into negative ones. So it seems likely that if one could travel back in time one would find oneself in an antimatter universe which would be extremely hazardous.