Einstein's Deflected Starlight Test Of General Relativity - As Momentous Now As It Was 100 Years Ago.

Einstein soon after publication of his paper on special relativity

One hundred years ago one of the most astounding tests of a physical theory (the general theory of relativity) was carried out, and its audacity still echoes through the decades.   The experiment entailed making a prime test: measuring the deflection of distant starlight during a total eclipse, with the light having passed by the Sun (and being influenced by its gravitational field).

But how can light - radiant energy - be affected by a gravity field? To find out one must adopt the view that light consists of particles which we call photons.  And although photons have zero rest mass there is still a way to tweak the equations of motion peculiar to general relativity to be able to fashion meaningful results. Remember also that Einstein received the Nobel Prize specifically for his work on the photoelectric effect which showed light can behave as particles.

The effect was first observed by Heinrich Hertz in 1887, but it was left for Einstein to explain (and for which he won the Nobel Prize) in 1905. The effect at the time, was most directly observed when a + charged zinc plate (in a Braun -type electroscope) was exposed to x-rays or ultraviolet radiation which caused an increased deflection of the electroscope leaf.  Conversely, a negatively charged plate exposed to the same high frequency radiation caused a decreased deflection showing a loss of potential. Hertz demonstrated the effect using an apparatus such as shown below:



Here, an evacuated tube contained two electrodes connected to an external circuit with the anode being the metal plate on which the radiation was incident. The photo-electrons emerging from the surface thus had sufficient kinetic energy to reach the cathode despite its negative potential. These electrons formed the current (photo-current) measured by the ammeter. 

To measure the maximum kinetic energy of the photo-electrons one applies a retarding voltage V, gradually increasing it until the most energetic photo-electrons are stopped so the photo-current becomes zero.

At this point::  eV_s = K _max=  ½ mv _max²

According to Einstein's explanation a beam of radiation consists of bundles of energy of size hf called “photons”. When such photons collide with electrons at or on a metal surface, they transfer an energy hf. The electrons on the metal surface either get all of this energy or none at all. In leaving the surface, electrons lose an amount of energy f which is the work function of the surface. The maximum energy with which an electron can emerge is:

(Energy gained from work function) – (work function)

In Einstein's general relativity paper, ‘On the Influence of Gravitation on the Propagation of Light’   he computed a deflection of only 1.7 seconds of arc.  This is a tiny amount indeed, and some idea can be grasped that the angular diameter of the full Moon or 1/2 degree, and one arcminute is 1/30 the width of the full Moon, and one arcsecond is 1/60th of an arcminute.

The basic concept illustrating deflection of starlight is given below:

Here the light from a star at an actual position S2 is seen to deflect by some angle a    thereby altering the image position to that seen at S1. This is a direct result of the effect of the gravitational field of the Sun on the light rays. The true direction is thus alone the ray ES2 while the deflected position is along the ray ES1.

Theoretically and quantitatively, one get obtain an estimate of the magnitude of deflection by incorporating another parameter – call it b   ("impact parameter") as shown in the diagram below:

Then we obtain for the deflection angle, a:

 a  = - 4 GM/ b   or (in cgs units): 

 a       = - 4 GM/ b c²  

The backstory to the experimental test validating at least one prediction of the general theory (though some nitpickers remain unconvinced because of measurement errors) is itself interesting.  We learn, for example, that in the midst of World War I in 1917, one of the astronomers involved (Frank Dyson - Astronomer Royal),  persuaded the British government to budget   £ 1,000 for a team of 4 astronomers - led by Sir Arthur Eddington of Cambridge - to make the critical observations during a total solar eclipse on May 29, 1919. 

As a sidenote, Eddington's book, 'Space, Time and Gravitation', remains one of the most readable accounts including the tie in to general relativity. It is also available as a pdf book online for those interested, e.g.

[PDF]

Space, Time and Gravitation

Anyway, one team of two astronomers was stationed at Principle, an island off the coast of West Africa, and the other team of two went to Sobral, a city in northeast Brazil.   As Eddington recorded in his diary:

"The first ten photographs show practically no stars.  The last six show a few images which I hope will give us what we need, but it is very disappointing."

Disappointed though he was, once back in England Eddington developed four more Principe plates. He detected in them Einstein's value for the deflection of starlight though within a rather large margin of error. Fortunately, the Sobral plates provided conclusive support for Einstein's theory. 

 Ciufolini and Wheeler, in their (1995) Princeton monograph 'Gravitation and Inertia (p. 120) note that the Eddington- Dyson measurements "confirmed the general relativistic predictions with about 30 percent accuracy".  However, since then (p. 121): "Long baseline interferometry and very long baseline interferometry "have dramatically improved the accuracy of photon deflection measurements" .   The amount of enhancement in accuracy is on the order of 50 times using VLBI instruments. 

In November, 1919 Sir Arthur presented his conclusions to a joint meeting of the Royal Society and the Royal Astronomical Society in London.  The greatest personage then involved in British physics - J.J. Thomson, declared:

"This is the most important result obtained in connection with the theory of gravitation since Newton's day. If it is sustained....it is the result of one of the highest achievements of human thought."

Most recently (last June) Dr. Thomas Collett of the Institute of Cosmology and Gravitation at the  University of Portsmouth, along with an international team of astronomers,  conducted what they say was the first test of general relativity on a large astronomical scale. The paper entitled:  A Precise Extragalactic Test of General Relativity,  was published in the journal Science and found that gravity’s behavior in distant galaxies reflects that way gravity behaves in our solar system, just as Einstein’s theory predicted.  

So far, Einstein's general theory is indeed one of the highest achievements of the human mind.

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A Welcome Addition To Basic Astrophysical Literature: "Welcome To The Universe"

As Black Friday dawned yesterday it was almost a certainty that 99 percent of people would be flooding big box stores to snatch up the latest electronic wonders including the new 55"  "Ultra HD" TV sets to make their debuts as Walmart and Best Buy. But a much more sensible option,  as opposed to more electronic titillation,  would have been to invest in Neil De Grasse Tyson's new book, Welcome to the Universe: An Astrophysical Tour.'

The book, a breathtaking overview of astrophysics and especially how we obtain the knowledge we do, is indispensable to getting to know the universe as science knows it   It is distinct from most other basic texts because most of those - as De Grasse Tyson noted on a recent CBS Early Show appearance -  are "a mile wide and an inch deep".  By contrast, this 472 -page book is a "mile wide and a mile deep."  Well, not quite but pretty close and hey, it even includes algebraic equations where necessary - so ignores the old rule that each equation given "decreases sales by one percent" - or some such jabberwocky.

Some of the topics covered include: stellar evolution (the lives and deaths of stars), the search for life in the galaxy, the interstellar medium, the Milky Way, the expansion of the universe, the early universe, quasars and Einstein's special and general theories of relativity. Also: black holes, cosmic strings, inflation and the shape of the universe.

The content actually represents a division of labor between Neil De Grasse Tyson, Michael A. Strauss and J. Richard Gott. For example, the three chapters on special and general relativity were very ably done by Gott.  If you truly wish to get a handle on these abstruse theories there is perhaps no place better then Gott's trio of chapters, so long as you're ok with some intermediate algebra to assist in the task.

Tyson himself is responsible for chapters:  1, 2, 4, 5, 6, 7, 9 and 10. The only one I really had issues with was Chapter 9, 'Why Pluto Is Not A Planet'.   For one thing, Pluto is not specifically an object for astrophysical inquiry but rather planetary astronomy. Hence, the chapter's inclusion can only be attributed to Tyson having already spent a lot of ink on the subject, so why not include it (since it's firmly in his academic 'wheelhouse') in a new book?

But as I noted in previous posts the arguments given don't make any physical sense. For example, Alan Stern, executive director of the Space Science & Engineering Division of the Southwest Research Institute – and Principal Investigator for NASA’s New Horizons mission to Pluto- observed that the new planet definition was “sadly flawed, particularly due to the vagueness of the third condition- clearing the neighborhood around its orbit - which might also disqualify Earth”.

He added: “A lot of people are going to ignore the (new) definition because it doesn’t make sense.” (Source: Eos Transactions of the American Geophysical Union, Vol. 87, 29 August, p. 350). The other aspect is that Pluto’s dethroning was done by a subjective vote at an IAU meeting, not by any objective measures or consistency. The problem is the IAU definition overlooks the neighborhood of the object.  Pluto just happens to be in the Edgeworth –Kuiper belt with thousands of other objects. This is a condition of happenstance, not of actual intrinsic property.

If Earth were to interchange its position with Pluto it would also be in the same belt, and because other substantial objects (e.g. Neptune) crossed its path, it would have to be demoted – by De Grasse Tyson and the IAU's definition. So what to call it, a “demi-Earth”? A jumbo dwarf planet or jumbo shrimp planet? Give me a break!

Tyson is on much firmer ground - as well within the proper astrophysical sphere -  when he explores the lives and deaths of stars, as well as the scale of the cosmos, how stars radiate energy and stellar spectra. The last two are especially important cornerstones of astrophysics, and Tyson does a stand up job. One also hopes that when curious readers finish his chapters they will move on to Lawrence Aller's  (and now also Leo Goldberg's) 'Atoms, Stars and Nebulae' - which remains the standard for elementary astrophysics in terms of deciphering the nature of stars.

But Tyson and company excel in their own approach presenting the (usual) less advanced reader with numerous examples borne out by superb diagrams such as Fig. 2.4 showing Kepler's laws, and Fig. 6.2 showing the energy levels and associated spectral line series for hydrogen. I also liked how Tyson delved into the historical background, e.g. for the Balmer series, discovered in 1885.

The H-R diagram (Fig. 7.1) is also one of the best, most beautiful I've seen in any astrophysics text, with the relative sizes of specific stars - like Betelgeuse and Aldebaran - in color and projected against their evolutionary tracks. It's also interesting just how the famous diagram was composed: "From a catalog of stellar luminosities and temperatures Hertzsprung and Russell started filling in the diagram and discovered stars did not occupy just any place". Well, no, because it plots the stellar spectra (or surface temperature) vs luminosity so because each stars has distinct pairs of these, it will have differing positions on the H-R grid.

One warning here: Tyson uses an unorthodox form for describing nuclear fusion reactions in stars. For example (p. 101):

ppn + ppn  =  ppnn + p + p

As opposed to say,

³ He + ³ He ®  ⁴ He + ¹H + ¹H  + 12.85 MeV

with the energy given off.    But truth be told, this notation of his would probably only irritate purists or pedants

What really sets the text apart is all the calculations shown, all very basic, such as the total energy a star will generate over its lifetime by multiplying the star's luminosity L by the lifetime. T. In fact, all the math used which leads to so many fascinating insights, is fairly basic. None of it ought to be beyond a person who has taken at least first year algebra though the derivation of  E = m c²       in Appendix I will likely require at least an intermediate algebra background to follow. .

What makes this book so special is that it offers the reader so many more useful insights and "tidbits" - for lack of a better word- than most astrophysics texts would at this level. One learns, for example (p. 51) "Superman can indeed throw an object 5 miles per second . The object will also fall under the influence of gravity but its curved trajectory now matches the curvature of the Earth such that it never hits the surface and ends up in a circular orbit."

De Grasse Tyson's conversational style, which served him so well in his  2014 reprise of 'COSMOS'  e.g.

http://brane-space.blogspot.com/2014/03/new-cosmos-series-is-well-worth-look.html


is apparent throughout his chapters in the current work. For example, on p. 88, we read:

"You can think of the outermost layer of a star as a tree. Do you know what is coming toward the tree (from inside the star)?  Mixed nuts. We have a mixed nut cannon (the interior of the star) firing mixed nuts (photons at different frequencies) into the tree, and in the tree we have squirrels. My squirrels like acorns (the photons) - these are acorn squirrels. They see all these mixed nuts coming through but they are grabbing only the nuts they like, the acorns, on the other side. (outside the star) come mixed nuts minus the acorns (the thermal radiation minus the H-alpha  photons)"

The reader has to understand that this sort of description is uniquely Tyson's, You won't find it in the chapters by Gott or Strauss, though theirs do remain in a colloquial mode.

The takeaway here is that if you enjoyed the COSMOS series you should find a lot  to enhance your reading in this book. There is all one could ask for to gain a foothold in grasping the basic principles of astrophysics - as well as other topics not specifically related to it, such as how the zodiacal constellations came to be (including why astrology is off) and, of course, why Tyson doesn't regard Pluto as a planet.

Black Holes - Where There Is No Space

It seems incomprehensible, and more than a few people are unable to get their heads around it, but there are astronomical objects where space does not exist. Let me rephrase that by saying the space is not that which is conformal space-time that can support the laws of physics. Since the space  literally occupies a one-dimensional point at the hole's singularity, and an infinitely small point is occupied by a collapsed, stellar scale mass, then the laws of physics (which incorporate causality) do not apply.

The threshold for making this cut is given by the well known Schwarzschild radius or:

R(s) = 2GM/c²

where G is the Newtonian gravitational constant, c is the speed of light in vacuo, and M is the gravitating mass. Once a stellar remnant collapses within this radius, light cannot escape and the object is no longer visible, hence effectively "air tight" to use the words of the article. It is a characteristic radius associated with every mass of macroscopic scale

Such is the case with the stellar- collapsed black hole., but we need to explore the specific conditions for the zero space singularity in more detail. It depends really on a spherically symmetric solution    (Kurt Schwarzschild's  solution)  of Einstein's general relativity field equations, viz.

ds² = -(1- 2M/r)dt²    + dr² / (1 - 2M/r)  + r²  (dq² + sin²  q dφ² )   

Where r, q,  φ  are spherical coordinates, for this conventional Schwarzschild metric..

As noted by Ciufolini and Wheeler ('Gravitation and Inertia', Princeton University Press, p. 69), for the above defined Schwarzschild metric the r = 2M region is a "mere coordinate singularity".  However, they note that the r = 0 region (where g₀₀   = - g₁₁  referring to the Einstein field potentials). is "a true geometrical singularity". In other words, expressing a region of no causality-based space time that we recognize. Of course, there are many different solutions, among which we find those well -behaved. The authors point out a specific example on p. 67 where "it is possible to extend analytically the Schwarzschild  solution to cover the whole Schwarzschild  geometry."

To fix ideas, reference is then made to a diagram on p. 66 showing an "Einstein -Rosen Bridge" or "wormhole connecting two regions in one Euclidean space."   It is emphasized that the sequence of diagrams (Fig. 2,9a, 2.9b, 2.9 c) denote "alternative interpretations of the three-dimensional maximally extended metric of Kruskal at time t' = 0"  For those interested, this referenced metric is given as (p. 67):

ds² =   (32 M³  / r) [exp (-r/ 2M)] (- dt'²  + dx'²  )  + r²  (t', x') (dq² +  sin²  q dφ² )

This is after transformation of the original Schwarzschild metric using the Kruskal -Szekeres coordinates given at the top of page 67.

The point is that a suitable mathematical approach can be used where one avoids a "spaceless" (e.g. true singularity) outcome, and instead engenders one with a space-time "escape hatch" - the Einstein-Rosen Bridge.

Obviously, in this post, many details are being left out including for the wide array of divergent black hole conditions (e.g. for angular momentum, rotation - rotating or not) and geometry, e.g. for Kerr black holes, see e.g.

http://www.daviddarling.info/encyclopedia/K/Kerr_black_hole.html

I only touch the bare surface here to encourage readers to investigate many more aspects on their own.

Granted, no one has ever seen a black hole but we know how to recognize the physical evidence for its existence: very powerful and periodic bursts of x-rays, registered on sensitive satellite detectors. Mathematically, the very brief periods of less than a millisecond betray an extremely compact volume. The x-rays indicate accretion to a large mass. Together, these can be matched to predictions given in the Einstein general relativity equations and Voila! the black hole emerges as an object consistent with the observations.

In general black hole identification is predicated upon observing its effects as a member of a binary (double) star system. Thereby, the black hole presence is inferred from x-rays given off when the companion star’s gaseous layers are sucked into it. As those accreting gaseous layers are pulled through the hole's event horizon, they are condensed and the impacting plasma leads to intense heating and x-rays.

The most convincing recent find which puts the question of black holes existence to rest once and for all has been via the Laser Interferometer Gravitational Wave Observatory (LIGO). e.g.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102


The recent LIGO detection provides the first direct evidence for gravitational waves but also opens the door to using them to study the powerful cosmic events that create them, in this case two colliding black holes. Based on the paper cited in the above link, the two black holes are  each roughly 30 times the mass of the Sun. They evidently merged some 1.3 billion light years from Earth. The  gravitational waves themselves were generated in the final moments before the black holes merged. The signal was brief but definitive and we on Earth have now received it.

It is in fact the most direct observation for black holes ever made.

Meanwhile, the total current assay of black holes may be dramatically underestimated. This is a result of difficulty of detection, as is the case with all forms of dark matter. Despite that, the research thrust goes on, along with black hole identifications in a variety of formats and scenarios.

Those who want to investigate more, can consult any or all of this sampling of research papers, from The Astrophysical Journal, on the American Astronomical Society website:

1) On the Correlations of Massive Black Holes with Their Host Galaxies

http://iopscience.iop.org/0004-637X/637/1/96/fulltext

2) The Jet Power, Radio Loudness, and Black Hole Mass in Radio-loud Active Galactic Nucleihttp://iopscience.iop.org/0004-637X/637/2/669/fulltext

3) Binary Mergers and Growth of Black Holes in Dense Star Clusters
http://iopscience.iop.org/0004-637X/637/2/937/fulltext

4) Black Hole Advective Accretion Disks with Optical Depth Transition

http://iopscience.iop.org/0004-637X/637/2/968/fulltext

5) Black Hole Masses and Eddington Ratios at 0.3 less than z less than 4http://iopscience.iop.org/0004-637X/648/1/128/fulltext

Dozens and dozens of other papers are also available, which show in concert that the black hole is no mere macguffin created by some math genius fantasizing in his parents' basement, but a valid object of inquiry worthy of serious scientific investigation.

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  inertial 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 Part 3.

      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 3^rd law.  The 3^rd 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 dq^j dq^k = g _ik  dqⁱ dq^k

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   =

(1.....0...............0)

(0.....r²...............0)

(0.....0........r² sin f)

Further,   g _ik  dqⁱ dq^k

  =

(1.....0...............0)  (dr² )

(0.....r²...............0) (dq 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:

1  =  i^ i^ + j^ j^ + k^ k^  =

(1......0........0)

(0........1.......0)

(0........0.......1)

Also:

1 × C  =  i^ C_x + j^ C_y + k^ C_z  

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)

(-a₁₂…..0……….a₂₃)

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

(0........1.......0)

(0........0.......1)

Then Tr =  (1 + 1 + 1) = 3

Problems:

1. If  1  =  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:

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

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

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

----------------------------------------

(2 r......0........0)

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

(0........0......r /2 )

-------------------------------------------

(p ..........0  ........0 .........0)

(0 ........ p/ 3....... 0........0)

(0.........0 ......p/ 6........0)

(0.........0........0......p/7)

---

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.