Saturday, December 20, 2014

A Look At General Relativity and Tensors (Part 2)

Continued from earlier instalment:

3.Diagonalizing tensors.

This is analogous to obtaining the eigenvalues for a matrix in linear algebra. Consider the object below for which we want the principal axis.

Fig. 3. We wish to find the principal axis, a i j.

And we have a i j

 With T’ =  A × I × At

Where t denotes the transpose.  Then we obtain, T’ =

(15 …..0……..0)
(0…….11 ....-3Ö2)
(0……..-3Ö2…..8 )

Which is to be diagonalized.  Writing this out:

(15 - l…...0…….....0)
(0……..11- l ....-..3Ö2)

This leads to a cubic - i.e. with triple roots -  which are:

l1 = 15,  l2 =  5, and l3 =  14

Substituting l1 in the matrix we get:

(0  ….....0……......0)   ( c x  )
(0……....-4 ....-..(-3Ö2))  ( c y  )
(0……..-(-3Ö2)…..(-7) )   (  c z )

For which the separate equations, e.g. in  c x, c y and c z can be solved. For example,

-4 y   -   3Ö z  =  0 

 -   3Öy   -7   =  0

After working through all the solutions, we obtain:

C  =  - Ö (2/3) e2^   +  1/ Ö3  (e3^)

Example  Problem:

In a certain rectangular coordinate system, the directions of whose axes are given by the unit vectors i, j and k, the inertia tensor of an object is given by:

I = K x

(0….1… ..1)

a) What are the principal moments of inertia of the object (the moments of inertia along the principal axis) relative to the origin of the above coordinate system?

b) What is the direction of the principal axis corresponding to the principal moment of inertia and equal to K?

 c)  If the origin of the above rectangular coordinate system is at the center of mass of the object and the total mass of the object is M, what is the change in the inertial tensor of the object if the rectangular coordinate system is displaced parallel to itself a distance ro in the direction:

 (1/ Ö2)j +  (1/Ö2)k?


a) We have:  I = K x

(0….1… ..1)

We write out the determinant with eigenvalue l:

(1 - l….0…..0)
(0….1 - l   ..1)
(0….1… ..1 - l)

Leading to the characteristic equation:

(1 - l)3 – (1 - l) = 0


(1 - l) [ ((1 - l)2 – 1] = 0


(1 - l) (l2 –  2l) = 0

Yielding eigenvalues: l= 0, l = 2


T = Kl,  so:

 T1 = 0, T2 = K, and T3 =2K

Or: (0, K, 2K)

b) We have to take:  (I T1)C

So that:

K [(1….0…..0)
  [(0….1… ..1)

K (1….0…..0)](x)
  (0….1….. 0)] (y)
  (0….0… ..1)] (z)

 (0….0…..0) (x)
  (0….0…..1) (y)
  (0….1… ..0) (z)   =   0

So that:  0  =


Which 0 for the x component implies the answer is i.

c) By the analog of the parallel axis theorem:

I o = IG  M(R2 I – RR)

D I =  I o -  IG   =    M(R2 I – RR)

RR =   r o 2     x

(0….1/2… ..1/2)

D I =   M  r o 2     x

  [(0….0… ..1)

  (0….1/2.. 1/2)]
  (0….1/2… ..1/2)]

= M  r o 2     x

(0….-1/2… ..1/2)


1)  For the example problem given, use the remaining two eigenvalues, l2 and l3, to diagonaliize the matrix and obtain solutions in c x, c y and c z.

2) Thereby obtain the principal axes in terms of: e^’x,        e ^’y and e^’ z .

3) Show in particular that:

  e^’ z    = - 1/ Ö3   e1^ -  1/ Ö3   e2^ + 1/ Ö3   e3^

Solutions to Tensor Problems

1)If  1  =  i^ i^ + j^ j^ + k^ k^

Write out the expression for × D   


1 × D  =  i^ Dx + j^ Dy + k^ Dz  

2. a) Provide a matrix which satisfies:  i^ i^ + j^ j^ + k^ k^  = 7/2 

The trace must equal to 7/2 so the diagonal elements need to yield that sum. Thus:



(0........0....... 2)

is one example of a matrix that satisfies the condition

b) Write out the trace for the metric tensor.

Ans. The metric tensor is:  

g ik      =



(0.....0........r2 sin

Then the trace (Tr) =

1 +   r2  +     r2 (sin f)=   1 + r2  (1 + sin f) 

c) Give one example of  3 x 3  tensor, then show how it might contain an anti-symmetric and symmetric  tensor (also how to go from one form to the other). 

Ans.  Let  a i j   =



(4........-4....... 0)

Then the antisymmetric form is:   a j  i  =



(4........2....... 0)

Then the symmetric part  is the difference:  a i j  - a j  i  =



(0......-6....... 0)

3. Find the trace of each matrix given:

Soln. simply add the diagonal elements for each, i.e.

a)  Tr = 2 + 8 p  +  p  =   2 + 9 p

b) Tr =    r +  L  +   r /2  =  5 r /2  + 2L  

c) Tr = p +  p/ 3   + p/ 6   + p/7 =   69p / 42

The Myth of U.S. Carbon Reduction - And Another Sign of National Insanity

Let us confront the fact that reality is a cruel taskmistress. While lowly humans in their hubris might fool her - and other humans - part of the time, they cannot do it all of the time. Eventually reality bursts out from the PR or propaganda shadows, then ordinary citizens suddenly see things for what they are and false claims are often exposed. The more such reality hiding games are played by a nation, the more it runs the risk of insanity - simply because insane people or nations are not governed by reality, which then comes back to exact revenge.

The Iraq invasion which was hailed as necessary - was a case in point - as no WMD were ever found, despite the Foxites (the epitome of insane freaks) claiming they're still there. In the end, when reality emerged, we learned it was an Iraqi rodent nicknamed ''curveball" (GOOGLE!) who made the whole thing up to try to gain some future advantage with the U.S. occupiers. And the cost exacted? More than $1.3 trillion in added debt, 4400 U.S. lives lost as well as 600,000 Iraqis.

Reality also burst onto the scene with the release of the Senate's CIA Torture report which anyone with the interest and a grain of sense can access online. You can now read what these state-sponsored sadists did in our name, and you can go ahead and decide if it is torture or not (again, a good test for sanity and sound judgment).

Finally, reality most recently infiltrated the U.S. entertainment industry as actual theater chains balked at exhibiting a foolish, inflammatory flick - claimed as comedy- but which in fact could be called neither art nor comedy by any stretch of a rational mentality or perspective. It was the basest filth, masquerading as film art, produced by two morons who really showed the total sterility of the nation's film product - that this is the best they could come up with to show at Christmas, 2014. (And we won't even go into the 'Dumb and Dumber' movies which encapsulate the low grade, common denominator appeal for too many in this country.)

But the reality adjustment I am writing about here in this post concerns the U.S. effort to fool everyone into believing we are really on the way to helping the planet via greater fuel efficiency and positive carbon reduction.  Thus, being led to believe that somehow carbon reduction, CO2 gas emission reduction is hardly a zero sum game - such that if we do it here we do it for the whole planet.

According to an AP report appearing in The Denver Post ('U.S. Exports Pollution, Offsets Advances' Dec. 9, p. 15A) it's all a royal lie. Citing the Associated Press investigation, the Post observes:

"Under President Barack Obama, the U.S. has reduced more carbon pollution from any other nation, about 475 million tons between 2008 and 2013, according to U.S. Depth. Of Energy data.

Yet the U.S. is sending more (high carbon) fuel than ever to other parts of the world, where efforts to reduce resulting pollution are just getting underway- if advancing at all. "

The report then goes on to note these U.S. exports used abroad have actually "released roughly one billion tons of carbon pollution into the atmosphere".

In other words, doing the math, there is a negative offset of:

1 billion - 475 million = 525 million tons

IN other words, the U.S. climate -energy gamesmanship has led to an additional 525 million tons of carbon released in the atmosphere during the same period of the U.S. claimed reduction. Zero sum game anyone? Anyone?

Look, it is clear that based on reality - the reality we all inhabit one planet which is affected by what any given part does - that this sort of chicanery cannot work, is not sustainable. It definitely bodes ill for our world, if we somehow think we can fob our filthiest products off on other nations then beat our breasts claiming energy efficiency and moral superiority. No, this way madness lies, because again, it rejects reality. Look at any psychotic, what is his defining attribute? It is the detachment from reality! So also Chris Hedges has pointed out for whole nations (like ours)  in his book Empire of Illusion.

In addition, PR and lies about the effects are also part of the ongoing national madness, the inability to reason or the belief that (as the Bushies tried to make us believe) "we create reality". No you don't and again, when that task mistress enters we find out- as you do - the hard way.  In a similar way, according to the Post report, it is unacceptable for the White House to claim "exports do not add more carbon to the atmosphere because they replace fuel that would come from someplace else"

A neat turn of phrase, but too cute by half. First, it is coming from us - our worst mess, and degraded fuels, so one cannot assume the same cheap degraded fuels (in the same amounts) would necessarily be purchased from other sources. Second, as other experts have noted, when energy is plentiful and cheap then demand increases across the board so you cannot claim that demand for your product is somehow - on balance- to be taken off any market reckoning.

Third, one cannot claim a carbon rectitude in one's own nation if the most degraded fuel is offloaded elsewhere so it can't be reckoned into your own mix. That would be like me claiming my sump pump is finally working - after I offloaded 4 extra tons of crap into my neighbor's yard to ease the burden.

Again, a genuine zero sum game cannot be made to look like a non-zero sum game if we are serious about moving to lower carbon deposition. And let's bear in mind there is a sense of urgency here to be realistic and to level with the people.

A new study published in the journal Environment Research Letters posits that climate change could have a devastating effect on the world’s food supply well within (most of) our lifespans. According to the report, our food supply could be cut by 18 percent by the year 2050–that’s only 35 years from now.
In a statement, the study’s co-author Michael Obersteiner said that water “may become dramatically scarcer much earlier than previously thought.”
A large part of the problem will be managing water supply, write the study’s authors. Our current irrigation infrastructure is struggling to deal with changing weather patterns, so significant investments in expanding and strengthening that infrastructure will have to be made.

The prediction is a significant increase in loss from previous studies, one of which anticipated a 2.3 percent drop in the food supply by the same year.

If climate change is managed correctly, food production could even rise 3 percent by 2050, the study said, as a higher concentration of carbon dioxide in the atmosphere has a fertilizing effect on plants. But this demands a grounding in reality, as opposed to confecting ever more lies, PR and disinformation.. As long as we are allied with make believe, we will not achieve what is needed to avoid a food as well as greenhouse warming disaster.

Friday, December 19, 2014

Hello, Obama! SONY Had NO CHOICE!

President Barack Obama is an intelligent guy, but his harsh reprimand of Sony today shows he can also be guilty of making some dumbo claims, rebukes and statements. We saw this earlier in the spring when he actually went on record as saying Russia was a "second class Power". Uh, no, dude. Not when they have nearly half all of the planet's nuclear weapons!

Earlier today Obama again showed he can sometimes be as clueless as Dumbya Bush when he rebuked Sony for withdrawing its year-end assassination flick, 'The Interview'.  The Prez said "I think they made a mistake" then followed it up with:

 "I'd wish they'd spoken to me first. I would have told them 'do not get into the pattern in which you are intimidated by these kind of criminal attacks".

And what would your speaking to them have accomplished, sir?  Let's think about this carefully, not parroting the bollocks and insanity of most of the Neoliberal, paper patriot, pseudo free speech crowd in the U.S. who are also insane and no longer able to reason properly.

First, even if Sony HAD NOT withdrawn this reprehensible flick - which depicts the head of a living leader of a sovereign state (never mind you may hate it)  being blown up in its final scene WHERE the hell could they have shown it? Alas, it was the theater chains like Cinemark, Carmike and Regal Cinema which pulled it after the physical threats were announced. This represented something like 19,200 screens across the country which would have meant the most Sony could have hoped for was maybe getting a few dozen indie flick screens in the malls. Now, seriously, do you really really believe those small indie theaters would have cooperated with Sony after all the majors said 'No'? If so you are more ensconced in the proverbial la la land than your CIA head Brennan - not willing to call the CIA torture, torture.

Second, as Lawrence O'Donnell pointed out last night on "All In", for the benefit of a clueless substitute for Chris Hayes, it was ultimately about possible litigation - given the FBI itself could not 100 percent promise the theater owners that no threat would materialize. (Even a nutso creep like James Eagen Holmes, who committed the Aurora CO theater massacre,  might have used it as a pretext for an incident.)  Given this lack of 100 % assurance, had the theater owners shown the film and ANY kind of violent incident occurred THEY would have been out their with their asses in the lawsuit wind, having to pay up to injured patrons. The repercussions would be similar to what the Aurora, CO Cinemark is facing from the families of the dozen slain and nearly five dozen wounded victims of that theater massacre 2 years ago. (A Denver Judge denied Cinemark's plea to avoid the 20 lawsuits because it "had no foreknowledge". Imagine the consequences for any violent episode in a theater showing 'The Interview')

So, let's get a bearing here on reality, which appears to be a commodity in short supply in the US of A: If you really would like to rebuke anyone or any entity, it is the actual cinema companies, theater owners and litigious lawyers you would have to go after before Sony. The fact you did not do this discloses your analog to "hostage taking" is simply empty rhetoric and posturing.

Obama continued his fulsome, self-righteous bloviation by saying:

"We cannot have a society in which some dictator some place can start imposing censorship here in the United States."

Sounds real testy with a tinge of humbug, and if I didn't know who had said the words I'd have attributed them to Bill O'Reilly or Sean Hannity - because those two don't think before disengaging mouth or brains (And let's get it clear it isn't a case of  "libtards" being "over sensitive" - as we see many flocking to the film's defense, yelping they want to see this crappola too - or are four square for Sony putting it up online.  See George Clooney. This is a case of rational people who can still exhibit some mode of discretion and sense, as opposed to jingoistic twits so hyped up on paper patriotism that it's gutted their febrile brains..)

Because when you make a sweeping remark like that as if in some existential vacuum you are revealing yourself oblivious to: 1) the fact that Sony's contract with theater owners allowed their withdrawal from exhibition, and 2) was ultimately because of our litigious society - something which you as a constitutional lawyers ought to know  - because you also must have taken some tort courses at Hah-vahd to get there.

If Obama wanted to be honest in attributing mistakes to Sony, it would have been in green lighting that irresponsible dreck in the first place. Even their own execs, for god's sake, were nervous about the violent ending. Even Obama's own Dept. of Homeland Security(according to an LA Times report) warned Sony that the film "could provoke retaliation."

Ultimately, this crap film masquerading as a comedy deserved to be "assassinated" as one commenter put it in a response at, because it was neither art or free speech. Showing a living leader's head being exploded and burned up in gory full screen color is not anything any normal, sane human grants as "speech" or "art" - any more than money is speech, and a color photo of a heaping pile of hog shit is art.

But it does show how far along our people - including our leaders- have lost their bearings, their good sense and judgment. More evidence as a whole that our nation - thanks to being pumped by PR and propaganda - is gradually losing its collective mind.  The critical thought test currently is only passed by those who agree that this film never ought to have seen the light of day - and to date- I have seen only two lone voices in the media, which is otherwise overwhelmed with this 'free speech' and "artistic license" BS.

One more point: It would not be wise to instigate a "cyber war" over what one Harvard law prof correctly described as a criminal attack on a lone corporation. Especially as the perceived victim nation does not launch such a war unless it is 100 percent certain all its ducks are cyber-secure and "in a row", including for infrastructure control,  power plants and water systems!

See also:




"emails suggest Sony execs remained nervous about the film’s content, especially the ending. On July 9, Lynton wrote to Pascal, “we cannot be cute here,” saying they could work with “no melting face and actually not seeing [Kim Jong-un] die. A look of horror as the fire approaches is probably what we need.” Rogen, while willing to play ball, appears reluctant to completely scuttle the visuals in this Sept. 25 email to Pascal: “There are currently four burn marks on his face. We will take out three of them, leaving only one. We reduce the flaming hair by 50% … The head explosion can’t be more obscured than it is because we honestly feel that if it’s any more obscured you won’t be able to tell its exploding and the joke won’t work.”

Another reason not to back this crappo film:

Colorado - The 'Promised Land' For Marijuana Cures?

The Denver Post's four part series on Ana Watson and her son Preston was heartbreaking, as it documented the pair's move from North Carolina (referring to themselves as "refugees") in an effort to find some kind of relief  or treatment for Preston's seizures. (Preston is among a rare group of kids across the U.S. suffering from Dravet syndrome - a rare form of epilepsy which completely disables a child's language and social skills - as each seizure destroys more brain tissue).

The search for a cure, according to the Post (Dec. 9, p. 7A) has led to "hundreds of families moving to Colorado to treat their children with medical marijuana"

This, despite the fact there is no guarantee of a cure or even successful long term treatment that can allow symptoms to abate.  A large part of the problem too, is the dearth of research into MJ's medical benefits (assuming they exist) when the feds themselves remain against it.  This is despite the fact that one form of MJ oil, called "CBD"  (cannabidiol) has been researched independently on animals for years - and found effective. ( Cannabidiol is that compound in cannabis that confers medical benefits but has no psycho-active effects, indeed it counteracts the effects of THC.)

"CBD" is the MJ-derived substance referred to  as 'Charlotte's Web'. The Post in a side story ('Medical Marijuana Credited With Progress for Charlotte Figi', ibid.)  described how five year old Charlotte Figi's Dravet syndrome seizures were ultimately controlled by its administration. Since infancy, little Charlotte suffered from debilitating seizures and practically lived at the hospital. Then in February, 2012, when she was five - her parents gave her CBD and everything "changed literally overnight".

To be specific, according to the Post account, "she slept soundly for the first time in years. She went seven days without a seizure. Over time the seizures dropped fro thousands  a month to just a few. And after not speaking for six months, she started talking again."

Miracle cure? NO!  CBD.

While CBD has been described as a "strain of marijuana" the fact is, it is not. According to MJ researcher Joel Stanley ('What Exactly is Charlotte's Web?', op. cit., p. 9A):

"Charlotte's Web is the brand name of the oil that Stanley makes from marijuana plants."

He's identified two strains, CW1a and CW2a,  which go into making 'Charlotte's Web'. Apart from the two named strains, there are 14 others - all with identifiers beginning with 'CW'.  According to the Post account:

"Stanley and his brothers now engineer the Charlotte's Web oil to try to ensure a consistent 30:1 ratio of CBD to THC (the active component in marijuana) in every dose."

This fixed ratio info has been of immense benefit to Ana Watson who has been able to vary the CBD to THC proportions for son Preston to elicit the optimal results. While Preston still struggles, he is finally able to say some words, and also ride a bike (at least his own way). Watson, like the other Dravet parents, knows that successes may only come slowly if at all, but they're not prepared to give up.

Two organizations currently serve the parents, Realm of Caring based in Colorado Springs and Flowering Hope Foundation in Boulder County. Currently the Realm serves 350 in Colorado, and 10,000 names are on the waiting list. Flowering Hope has more than 120 patients.

Currently, the parents from the two groups, when  not at their child's side or attending meetings with providers and medical experts, are out campaigning via a political movement (led by both organizations) to push for pro- CBD laws in state legislatures across the nation. They reason, correctly, that all families struggling with Dravet syndrome should not have to move here to Colorado to obtain hope or treatment.

Already, in states such as Alabama and Utah, some of the parents' stories have broken through paving the way for discussions to liberalize cannabis laws.

Until that is done, and prohibitions taken off the books, families will continue to flock to Colorado as the only "Promised land". In the meantime, Colorado is prepared to fight off lawsuits filed by Nebraska and Oklahoma which claim that Colorado's MJ is "spilling over into their states" and corrupting their upstanding people.(Who drive to CO to get the retail, psychoactive products.)

Maybe these yahoos need to do more research on CBD so families don't have to flee like refugees to get to Colorado for relief of  their kids' Dravet syndrome.

Thursday, December 18, 2014

A Look at General Relativity and Tensors (Part 1)

1. The Principle of Equivalence.

This could be called the heart of Einstein's General Theory of Relativity.  In its most elementary form, the principle states that gravitational and inertial masses are indistinguishable. Inertial mass refers to that which   initially moves uniformly in a straight line and is then subjected to some acceleration (a) producing a force F (and deviation from uniform motion) such that F = ma (or the mass m = F/a).  In principle, it should be possible to design an experiment to distinguish between inertial masses, say m1 and m2, on the basis of the accelerations imparted to them by the same force F. In general, one can say:

a2/ a1 = m1/ m2

     Thus, if one measures the ratio of accelerations: a2/a1 = 2, then the masses are in the exact opposite relation: m2/m1 = 1/2.  In other words, if m2 accelerates at twice the rate of m1, it must have only half the inertial mass of m1.
 Gravitational mass, on the other hand, refers to that measured with respect to the force of  attraction (weight or W) of earth's gravity field, thus m = W/g where g is the acceleration of free fall. Thus, the principle of equivalence maintains that the following are interchangeable for all reference frames:

W/g  =  F/a

In other words, the laws of physics are applicable to all reference frames. More importantly, there is no experiment that can be devised to discriminate between F and W in any of these reference frames. In a broader context, the principle means dispensing with the action of mysterious, long range forces (e.g. force of gravitation) and replacing them with the natural local action due to a deformation in spacetime.[1]  This is a major insight, which hasn’t been fully appreciated.

     For example, it implies that optical, electric and magnetic phenomena are subject to the laws of physics. As a case in point, light waves passing near a massive star accelerate by virtue of the fact their energy has mass, thereby disclosing a curved trajectory or deflection. Such ‘bending’ of starlight has been confirmed by measurements from photographic plates made during total solar eclipses, and compared to images of the same star without the Sun in the line of sight.
     At a deeper level, the Equivalence Principle motivated the search for a refined mathematical infrastructure, resulting in tensor calculus. This tool enabled easier transformations between differing coordinate systems and reference frames. In the course of tensor analysis of Einstein’s field equations, the Big Bang emerged naturally as a solution with matter present. We will see tensors in the next section and the field equations in the last.

      It helps to explore further misconceptions to see a more general applicability of the equivalence principle, and this is done using the comparative of two rockets, A and B, shown below.  In rocket (A) one stipulate an “inertial” or non-accelerating reference frame in which there is a gravitational field of intensity g acting. For rocket (B) one has a case of the rocket accelerating uniformly at the rate g i.e. equal to the rate of free fall in Earth’s g-field, or 9.81 ms-2

     Einstein’s Principle of Equivalence requires the complete physical equivalence of the two rocket systems.

Fig. 1. Does the Principle of Equivalence apply to two rockets A and B? (The depiction of one critic - Stephen Gift - that it does not- see his explanation below)

According to critic Stephen Gift, referring to the Principle of Equivalence:[2]

In rocket A the observer experiences a force W as a result of the gravitational field. There is also an equal and opposite floor upthrust F that acts at the soles of his feet. These two forces W and F acting on the observer keep him inertial. In rocket B only the upthrust F acts on the observer.  There is no gravitational force as in rocket A. The observer in B has only the force F acting on him thereby rendering the two systems dynamically different. This is completely contrary to Einstein’s Equivalence Principle and it therefore must be wrong.”

But is it really wrong?  Perhaps the most basic error made is in creating an additional force (in rocket A) where only one is needed. Not processed is the fact that it is precisely the upthrust or reaction force which creates the weight W!

For a person standing on a support in such a rocket, we have W = mg. But what if he’s in an elevator instead and the elevator is in free fall?  In this case there is no support so that the acceleration of the elevator a = g  and the relevant force equation is:

F = m(g – a) = m(g – g) = 0

So the observer is in free fall.  If there is an upward acceleration (similar to rocket B) then of course we have, assuming a < g:

F = m(g – a)  = mg – ma

If a is g/ 4,  for example, then:

F = m(g – a)  = mg – m(g/4) = 3mg/4

Again, the existence of an upthrust or reaction force is simply incorporated into the problem with the recognition that the weight is the reaction force from U. This is a basic principle such that forces always occur in pairs! You cannot single out or separate an upthrust from the weight.  In this case, it doesn’t matter if the critic only stipulates a single force F acting on the observer in B, because we know it must be paired with its companion force, W. Hence, the representation is in error.

    Thus, the observer in rocket A will be unaware of anything different from the observer in rocket B.  Even though no field intensity downward has been assigned, a uniform acceleration opposite to the direction F (which equates to the weight) has by the very application of Newton’s 3rd law.  The 3rd law again states that “for every action (force) there is an equal but opposite reaction (counter force). It can be illustrated as shown below for a body resting on a flat surface.

Fig. 2. Illustrating Newton’s Third Law of Motion.

Here: F(AB) = m(A)g (the weight of block A) and F(BA) is the normal force, N acting against it.
Although the critic attempted to portray the conditions of the two rockets differently by resort to Fig. 1, from the point of view of the observers and the forces acting they are equivalent.

2. An Introduction to Tensors.

Before we can proceed much farther it is necessary to introduce some basic features of tensors – which mathematical entities proved to be the keys to Einstein’s creation of general relativity. This introduction is not meant to be in any way comprehensive, only to show the basic properties and then how they are used in his field equations.

We consider, to fix ideas, the motion along some defined curve s as shown below:

Fig. 3. Curve in space associated with particle motion and an element ds.

Along the curve we also find an element ds, we can write for s:

S = s (x, y, z) = s (x1, x2, x3) = s(x i)

Further: ds2 = d x i  d x i = dx2 + dy 2 + dz2

Which can also be expressed:

ds2 x i / q j  x i / q k dqj dqk = g ik  dqi dqk

where the superscripts are used to denote particular contravariant operations. Then  g ik is a matrix which we call the “metric tensor”.  Or:

g ik   =



(0.....0........r2 sin

Further,   g ik  dqi dqk

(1.....0...............0)  (dr2 )

(0.....r2...............0) (d
q 2)

(0.....0........r2 sin
f) (df2)

So the operations applied to matrices can be applied to tensors.

In using tensors we take care with the subscripts and superscripts and use the first for  covariant tensors and the second for contra-variant tensors.

Basic terms:

A tensor of rank 2 is a dyad.

A tensor of rank 1 is a vector.

A tensor of rank 0 is a scalar.

The most basic tensor of all is the unit tensor, defined:

=  i^ i^ + j^ j^ + k^ k^  =





1 × C  =  i^ Cx + j^ Cy + k^ Cz  

In all the above, of course, we have yet to introduce time, but will in the next section to do with tying geodesics to the Principle of Equivalence.

Further properties:

A tensor is symmetric if:  T i j   = T j  i

A tensor is anti- symmetric if:  T i j   = -  T j  i

The latter will look something like this:

(0 ……a12………a 13)


(- a13……-a23……0)

Doubled dummy indices, e.g. ii, jj, kk refer to the trace of a matrix, or the sum of the diagonal elements.  For example, if: i^ i^ + j^ j^ + k^ k^  =





1. If  =  i^ i^ + j^ j^ + k^ k^

Write out the expression for 1 × D  

2. a) Provide a matrix which satisfies:  i^ i^ + j^ j^ + k^ k^  = 7/2

b) Write out the trace for the metric tensor.  

c) Give one example of  3 x 3  tensor, then show how it might contain an anti-symmetric and symmetric  tensor (also how to go from one form to the other). 

3. Find the trace (Tr) of each of the following matrices:


(0......8 p.......0)



(2 r......0........0)


(0........0.......    r /2 )

(p ..........0  ........0 .........0)

(0 ........ p/ 3....... 0........0)

(0.........0 ......p/ 6........0)


1. For example, the Earth revolving around the Sun because of a long range gravitational force emanating from the Sun. In the Einstein view, one  visualizes a deep ‘pit’ or ‘well’ surrounding the Sun - as a result of its mass. An analogy would be a lead ball placed in the middle of a suspended rubber sheet. In either case local space-time is deformed by the mass.
[2] Journal of the Barbados Astronomical Society, Vol. XIII, p. 50.