Prove Yourself Euler’s Formula what Feynman Described As Most Remarkable  

Today morning, while pondering over first and second derivative of a curve, I wandered to the subject of Euler’s formula. Such is the beauty of this formula that Richard P. Feynman considered it as the most remarkable formula in mathematics. Even readers of the Mathematical Intelligencer voted it “the Most Beautiful Mathematical Formula Ever”.  

Euler’s Formula

In simple words, it is e^ix = cos x + i sin x. If x is pi then e^i.pi = -1 or e^i.pi +1 = 0 where e is natural number and i square root of -1.
This formula owed to great man Leonhard Euler who was a Swiss mathematician, physicist, astronomer, logician and engineer. His contribution to mathematics is so broad based, if we were to minus his contributions, mathematics would come to its nascent stage.   

Euler’s  formula relates to complex numbers which have real and imaginary parts. Real part is realisable, measurable and practical whereas imaginary part as the name says, is unreal, unmeasurable and impractical. Imaginary parts pertains to square root of minus one. This imaginary quantity, appearing in complex analysis, is denoted by alphabet, ‘i’. Further if a quantity has only imaginary part, it is called pure imaginary, if it has only real part it is called real and if both, it is called complex quantity.Two multiplied by two or 2^2 is solvable as 4 but 2^i that is real quantity raised to imaginary power, is not solvable but Euler made it possible.

Expression Of A Quantity In Hyperbolic Functions And Trigonometric Functions

A thought comes to ones mind, if “e” raised to power imaginary quantity say i. x, is solvable to real and imaginary parts as cos x, i sin x, respectively then “e” raised to power some real quantity may also be solvable to some real parts like cos x, sin x or some other functions. Also sin x and cos x are quantities which can have maximum value 1, therefore, solving e^x may result in too large a quantity to be coped up by  sin and cosine of an angle. Hyperbolic functions sinh x, cosh x constituted of e^x and e^-x, it seems, can solve the purpose in such cases. We will examine this aspect in due course.

Independent Approach

I submit, I am attempting to solve Euler’s formula by independent approach which is altogether different from settled methods, I have not perused the method, the great mathematician adopted and I am also not using Taylor’s or Maclaurin’s series for proving the formula. Expansions of functions in power series are based on my own method and I also subscribe to the notion that all functions are expandable in power series without using settled methods except a few. Those few I am not discussing here, being off the topic.

Defining “e” And “e^x”

With this background and information, I proceed to define ‘e’. ‘e’ is a natural number which equals the quantity (1 + 1/n)^n when n tends to infinity. By applying Binomial theorem,

the quantity (1 + 1/n)^n can be expanded as

limit n—–infinity, [ 1 +n.1/.n +1/n^2 (n)(n-1)/2! + 1/n^3 (n)(n-1)(n-2)/3! +1/n^4(n)(n-1)(n-2)(n-3)/4! +…upto infinity].

Limit n—- infinity or 1/n – 0, [ 1+ 1+ (1-1/n)/2! + (1-1/n)(1-2/n)/3! …..up to infinity,

= [1+ 1+ ½! 1/3! +……….upto infinity]. 

Therefore, e = 1+ 1+ ½! 1/3! +……….upto infinity.

Similarly, e^x can be expressed as

 limit n–> infinity, (1 + 1/n)^n.x.

 By applying Binomial theorem,

limit n—–>infinity, [ 1 +n.x.1/.n +1/n^2 (nx)(nx -1)/2! + 1/n^3 (n.x)(nx- 1)(nx -2)/3! +1/n^4(nx )(nx -1)(nx -2)(nx-3)/4! +…upto infinity].

Limit n—-> infinity or 1/n –> 0, [ 1+ x+ x(x-1/n)/2! +x. (x -1/n)(x-2/n)/3! …..up to infinity

= [1+ x +x^2. ½! + x^3.1/3! +……….upto infinity]. Therefore,

e^x = 1+ x + ½!.x^2 +1/3! .x^3.+……….upto infinity…,…………………(1).

 Hyperbolic Functions

This e^x along with 1/e^x are the constituents of hyperbolic functions where

Sinh x = (e^x – e^-x)/2,…………………………………….(2)

Cosh x = (e^x + e^-x)/2,…………………………………….(3)

Tanh x = (e^x / e^-x)/(e^x + e^-x).

Other functions I am not mentioning as these will not be used in our discussion and analysis. But for the knowledge of learned readers, these are more or less analogous to trigonometric functions.

If we add equation (3) to (2) then

Cosh x + Sinh x = (e^x + e^-x)/2 + (e^x – e^-x)/2 = e^x………..(4)

Therefore, e^x can be written as Cosh x + Sinh x. Coming to the discussion where I was probing the feasibility of solving natural number raised to power real quantity in terms of hyperbolic functions, it is clear now that it is solvable in hyperbolic functions of cosh x and sinh x as hyperbolic functions can have values more than 1. 

 
Expressing Any Quantity In Hyperbolic Function 

I am giving here examples to illustrate how e raised to power any quantity real or imaginary is expressible in cosh and Sinh functions. Let n be the quantity and say y = n.

On taking log from both sides, we get

Log y = log n,

or y = e^log n = cosh (log n) + sinh (log n) using equation (4). This is also equal to 

 cos (i log n) – i sin (i log n) using equations (12) and (13) as we will see later on. 

If quantity is y = n^i then log y = i log n. Taking antilog, 

y = e^(i. log n) Repeating the same procedure of earlier case,

y = cosh (i. log n) + sinh (i.log n)

 = cos (log n) +i sin (log n) using equations (10) and (11). This will prove in due course. 

Expressing e^ix In Hyperbolic Functions

If we put x as i.x, then from equation 4, 

 e^i.x = cosh i.x + sinh i.x………………………………….(4/1)

On differentiating both sides, we get

i.e^i.x = i.sinh x + i cosh ix

At x = 0,

i = 0 + i. cosh 0.

That means cosh 0 must be real as coefficient of i on left side is 1 which is real. That further means cosh ix must be real.

If we put x as i.x in equation (1), we get

e^ix = 1+ ix – ½!.x^2 – ix^3.1/3! .x^3.+ ix^4/4!…………..upto infinity and on separating real and imaginary parts,

e^i.x = (1- ½!.x^2 + ¼!.x^4 – 1/6!.x^6 +…………upto infinity)

+ i ( x -1/3!.x^3 + 1/5!.x^5- 1/7!.x^7 +…….up to infinity)…………(5)

Expansion Of Cos x

I submit that any function can be expressed as power series. Let

Cos x = a0. + a1.x + a2.x^2 + a3.x^3 + a4.x^4+………. up to infinity…………………………………(6). This is general form of power series for all functions but here it is being considered for cos x. 

When x =0, 1 = a0, on putting x = 0 in  Equation (6).

On differentiating equation (6) and putting x = 0, we get

0 = a1.

On differentiating second time and putting x = 0, we get

-1 = 2.a2 or a2 = – ½!.

On differentiating third time and putting x = 0, we get

 a3 = 0.

On differentiating fourth time and putting x = 0, we get

 a4 = 1/4! …………….so on

Putting these values of a0, a1, a2, a3, a4……… in equation (6), we get

Cos x = 1- x^2/2! + x^4/4! + ………………..upto infinity …..(7),

On differentiating equation (7), we get

-Sin x = -x + x^3/3!- x^5/5!+ ………………up to infinity.

Or sin x = = x – x^3/3! + x^5/5!+ ……………up to infinity…(8).

On putting these values of cos x and sin x in equation (5), we get

e^i x = cos x + i sin x………………………………………….(9).

That proves Euler’s formula by independent method.

Also from equation 9,

(cos x + i sin x)^2 = (e^ix)^2 = cos 2x + i.sin 2x.

Or (cos x + i sin x)^2 = cos 2x + i sin 2x. By mathematical induction, 

(cos x + i sin x)^n = cos n.x+ i sin n.x.

This proves De Moivre’s theorem.

Hyperbolic Identities

We have already proved in equation (4/1) that

e^i.x = cosh i.x + sinh i.x and we have also proved cosh ix is a real part.

On comparing real and imaginary parts of equation (9) with equation (4/1), we get

cosh ix = cos x …………………………………………….(10)

and sinh ix = i sin x…………………………………….(11)

If we put ix = z, in equations  (10) and (11), we get 

cosh z = cos z/i = cos (-iz) = cos iz……………….(12)

sinh z = i sin z/i =i. sin (-iz) = – i.sin iz……………..(13)

To avoid the article from being lengthy, I am not proving other identities which otherwise can easily be proved using the above Identities.

Conclusions

1.Euler’s formula that e^ix = cos x + i sin x, has been proved in equation 9 independently without resorting to the method adopted by great mathematician Euler.

2. e^x, cos x and sin x has been expanded without resorting to Taylor’s or Maclaurin’s series expansion.

3. Hyperbolic identities

cosh ix = cos x,

sinh ix = i.sin x,

cosh x = cos ix and

sinh x = -i sin ix, 

have been proved based on independent derivation of Euler’s formula.

4. It is stated that Euler’s formula that e^ix = cos x + i sin x, is applicable only when power of natural number e is imaginary but if the power is real, e^x can not be expanded in cosines and sine of angle. Whereas formula e^x = cosh x + sinh x as derived in equation (4), is applicable whether x is real or imaginary. If x is real, e^x = cosh x + sinh x = cos ix – i sin ix. And if x is imaginary, then e^ix = cosh ix + sinh ix = cos x + i sin x.

5. Any quantity real or imaginary can be expressed in hyperbolic functions of cosh and sinh and also in trigonometric ratios of cosine and sine of angle.

Before parting with these derivations, I submit for information of my learned readers that all functions can be expressed in power series the way I did for e^x, cos x and sin x. Once expanded, desired operations like additions, subtractions, multiplications, divisions, differentiation, integration can be performed. For example, from the power series expansion of e^x, cos x, sin x as solved  by me, one can determine, summation of (e^x + sin x), subtraction of (e^x – sin x), multiplication of (e^x.sin x), division of (e^x/sin x), differentiation ie d/dx(e^x.cos x), Integration of (e^x.sin x) with respect to x. It is easy, one can do it provided one starts doing it on ones note book.

Reference: 

Three-dimensional figure of Euler’s formula image curtesy https://commons.m.wikimedia.org/wiki/File:Euler%27s_Formula_c.png#mw-jump-to-license

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Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Physics, Mathematics, Yoga, Humanity etc.
 

 

Goodbye To Log Table And Calculator! Determining Logarithm Is Easy Now

Solving exponential quantity has always been cumbersome, it is that field where intuitive calculations generally fail. It reminds me of the time when mathematics wizard Smt Shankutla Devi* visited my college and asked us to put questions for their instantaneous solutions by her. Knowing well that exponential terms are difficult to simplify, I wrote on a piece of paper what the value of x is, when x^x equals 16. This was the most intriguing and difficult question to me at that time and still it is, except that I can solve it by method of approximation. Although she replied most of questions, unluckily my question remained unanswered. 

Logarithm Of A Quantity 
I was taught in school how to use log tables to solve exponential terms, complex multiplications and divisions. What the mathematics behind it is, was not taught. Two terms, exponenl and mantissa and use of tables were sufficient for solving numerical for me at that time.But today, with God’s blessing, I want to discuss all about logarithms with special emphasis on how to calculate its value when calculators and log tables are beyond our reach.


                                     A nautilus displaying a logarithmic spiral

With this background, I come to the basic point what the logarithm is. A hundred can be written as 10 multiplied by 10, there are two ten, first ten is multiplied with second ten. Number of times, ten is being multiplied, is called its power. In other words, I can say, 100 has power 2 and 100 can also be stated as 10 raised to the power two. 10000 is 10 raised to power 4. Power indicates number of times 10 must be multiplied to get that given number. This power of ten is the logarithm (to the base 10) of the given number. Therefore log 1000 (to base 10) is equal to 3. If I ask you to find what is the logarithm of 25, in short log 25, you will have to find the power of 10 which makes it 25.
But it is not easy to find as it was in earlier cases. How to find the power of ten or logarithm of 25 or any other number, is the subject of my discussion.

Changing Base Of Logarithm

I further submit that base needs not necessarily be ten, it can be any other quantity also. It can be 5, 2 or natural number “e”. If base is five then we will have to consider power of 5, ie log 25 (to the base 5) is 2. Generally, in our arithmetic calculations, base is taken as 10 universally. If base is not mentioned, it is assumed that base is ten. For example log 100 means, log 100 to the base 10 and is equal to 2. 

But in scientific calculations, base is taken as natural number ‘e’ and logarithm of 100 to the base e is written as ln 100 where ln means logarithm to the base e or natural logarithm. For your kind knowledge ln 100 is not equal to 2 since base here is not 10. But we can always change base from one quantity to the other by simple base changing formula. 

(Log a to the base b) x (log b to the base c) = log a to the base c……(1)

Also (log a to the base 10) x (log 10 to the base e)= log a to the base e

or log a to the base 10
= (log a to the base e)/(log 10 to the base e)…….(2). 

Above formulas are called base changing formula.

It can be memorised by considering log a to the base b as a/b. 

Applying this to equation (1), 

we get a/b x b/c = a/c 

and applying to equation (2), 

we get a/10 = (a/e)/ (10/e) = a/10.

What is natural number “e?”

 This is the sum of following infinite convergent series

e = 1 + 1/1!+ ½! + 1/3! + …………….1/n! +………… up to infinity.

On addition of a few terms, its value cam be calculated as 2.7182818284590452353602874713527 or up to three decimals as 2.718.

Method Of Determining Logarithm Of A Quantity

How can we determine the value of log of a quantity “n” to the base e or ln n. As stated earlier, I am finding the power of e which makes it equal to n. Let us consider this ln n as equal to y where is the quantity to be determined. Mathematically,

 y = ln n and it can also be written as

e^y = n where sign ^ denotes raised to power.

e^y can be expanded in power series as

e^y = 1+ y/1! + y^2/2! + y^3/3! + y^4/4! +…………..up to infinity.

Since e^y = n, we can write as

n = 1 + y/1! + y^2/2! + y^3/3! + y^4/4! +…………..up to infinity………(3)

This is an equation which has infinite terms but if y is made so small that terms y^3/3!, y^4/4!,………so on can be neglected then equation (3) can be approximated as

n = 1 + y/1! + y^2/2!.

Or y^2 + 2y + 2(1-n) = 0.

This is a quadratic equation in y and has two roots as

 -1 + (2n – 1)^1/2 or -1 – (2n – 1)^1/2……….. (4)

Out if these two roots, one will be applicable and other not and which one is applicable will be discussed in due course.

Value of the root so obtained will equal to ln n and it will be approximate as we have considered only three terms of equation (3) and rest ignored. However we can reduce the error by transforming the quantity to value nearby 1. Such transformation will make value of ln n so small that its cube and higher power powers can be ignored.

Alternate Method For Determining Logarithm Of A Quantity

In addition to expansion of e^y for calculating ln n, we can also find its value from expansion of ln (1+x)

Ln (1 + x) = x -x^2/2 + 1/3.x^3 -1/4.x^4 +……….up to infinity……….(5)

where n is equal to 1+x.

Transforming Quantity By Factorisation Or Otherwise To Bring Its Magnitude Near One

Our endeavour is to make the value of ln n as as close to 0 as possible. We also know that  ln 1 = 0, therefore if we can make first integer of quantity n as near to 1 as possible, ln n would contain less error.  

But n can have first integer of any number or quantity not under our control.
How can then we reduce first integer if n to a number nearing 1?   I give hereunder some ways to transform the quantity by factorisation or its taking its inverse.  

1) Let us say n is any number that starts with integer 5 or more. Say is 800 then 8 can be brought near 1 by simple fraction 10^2 x (1/8)^(-1)

 Or 10^3 x (1.25)^(-1)

2) Let us say n is .003 which can be written as approximately 10^(-3) x (1.3 x 1.3x 1.3 X1.3 x1.05)

Example 1

Determining value of log 1.3 to the base 10.

On putting the value of 1.3 in equation (4), we get

Ln 1.3 = -1 + 1.265 = .265.

Also ln 1.3 = -1 – 1.265 = -2.265 but this can not be true 1.3 is more than 1 and ln 1.3 must be positive. Hence this value -2.265 is ignored.

Converting it to logarithm of base 10 by using equation (2),

Log 1.3 = .265/2.303 = .1150, since log e to the base 10 = 1/2.303

Actual value from log table is .1139.

 There is an error of + 1 pc which is quite likely as this method gives approximate value.

Value of ln 1.3 using equation (5)

Ln (1+.3) = .3 – .09/2 + .027/3 = .264 taking only three terms.

Converting it to log to the base 10,

log 1.3 =.264/2.303 = .1146.

This method has .6 pc error and is preferable over the first one.

Example 2

Calculation of log .003.

.003 can be written as 10^(-3) x (1.3 x 1.3x 1.3 X1.3 x1.05) as has already been explained in previous paragraph.

Ln 3 = Ln (1.3 x 1.3 x 1.3 x 1.3x 1.05) = 4 ln 1.3 + ln 1.05 as ln a.b = ln a + ln b.

Ln 1.3 equals to .265. It has already been calculated in example 1.

From equation (4), Ln 1.05 = -1 + 1.04881= .04881

Therefore, Ln 3 = 4 x .265 + .04881 = 1.10881.

Converting it to base 10, we get

Log 3 = 1.10881/2.303 = .4814,

Log .003 = -3 + .4814 = -2.5186.

Actual log 3 from log table is .4771. Error = 1 pc.

We can also calculate from eqn 5,

Ln 1.05 = ln (1+.05) = .05 – .0025/2 + .00125/3 = .04922
, taking only three terms.

Ln 3 = .264 x 4 + .04922 = 1.0522

Log 3 = 1.10522/2.303 = .4799. Error is .5 pc.

Log .003 = -3 + .4799 = -2.5211.

Conclusions

Examples given above prove that logarithm of any quantity can be determined by suitably transforming it by factorisation or otherwise so that its first integer must be near unity. Had this transformation not done, results can be highly erratic. Also after transformation, result using expansion of ln (1 + x) are found more approximate, meaning thereby that this series converges sooner. Also the logarithmic value as determined by these methods will always be approximate as higher power terms of the series are ignored. 

Now you can pick up your pen and a piece of paper to determine the logarithm of any quantity whatsoever that comes to your mind. If you are careful enough in calculations, your answer will match (within small error) with that of log table. Can you not then prepare your own log table without using calculator or any other device? If you can, it would be a bliss unparalleled with worldly pleasures.  Good luck!

Note: 1) Title image courtesy https://en.m.wikipedia.org/wiki/Logarithm.

2) Image of a nautilus displaying a logarithmic courtesy spiralhttps://en.m.wikipedia.org/wiki/Logarithm#/media/File%3ANautilusCutawayLogarithmicSpiral.jpg

2) Smt Shakuntala Devi: Details are available at https://en.m.wikipedia.org/wiki/Shakuntala_Devi. 

                                                   End

Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.

A Circle Always Has Corners

A circle is a round figure like wheel of a bicycle or a disc. It is continuous, curves smoothly and joins the point it started from. If one moves palm around the circular rim, there is no discontinuity or corner. Does that mean, a circle has no corner? Probably from above, it seems so. 

For probing the mater further, requires definition of a circle. Mathematics defines a circle as the locus of a point that moves in such a way that its distance from a fixed point is always constant. The fixed point is called centre, constant distance its radius. In simple words, it is the path traced by a point which moves in such a way that its distance from a fixed point is always constant. If the moving point is (x’, y’) and constant distance, say from fixed point (0, 0) is r,

then x’^2 + y’^2 = r^2 and its locus is x^2 + y^2 = r^2
and it is an equation of the circle

When one fixes one end of a divider say at point O, and rotate its other end, the moving end traverses a circle with centre at O. Clearly, such circular figure does not seem to have a sharp bend or corner. Hence this definition also leads us to believe, a circle is corner-less.  

Also if a cone is cut by a plane parallel to horizontal plane, cross sectional area of the truncated cone is a circle.

                            Figure of Regular Hexagonal ABCDEF.

Now I take the case of a regular polygon say Regular Hexagonal of six equal sides. One may take regular polygon of as many sides as one likes. The figure, ABCDEF represents this regular hexagonal with external angle CBB’ = 2.pi/n and each internal angle = (1- 2/n).pi whee n is number of sides of regular polygon. 

Area of triangle OAB = ½. perpendicular x base

 = ½. ( r sin x . 2 r cos x ) = ½. r^2.Sin 2x

= 1/2.r^2.Sin (1-2/n)pi = ½.r^2.sin 2.pi/n. …………….(1)

where O is point of intersection of diagonals AD, BE, CF, and OA = OB= OC = OD = OE = OF = r. 2x is the internal angle equal to (1- 2/n).pi.

Area of regular hexagon = number of triangles multiplied by area of each triangle

= n. ½.r^2.sin 2.pi/n………………(2)

When the number of sides n increases to infinity, area of polygon 

 = limit n tends to Infinity, n. ½.r^2.sin 2.pi/n

= pi.r^2.(sin 2pi/n)/(2.pi/n) on rearranging.

When n tends to infinity, 2.pi/n tends to zero,

(sin 2pi/n)/(2.pi/n) will then be equal to 1.

In that condition, area of regular polygon

 = pi.r^2.1 = pi.r^2…………(3).

But pi.r^2 is the area of a circle of radius r. That means when number of sides of regular polygon is increased Infinitely large, it no longer remains regular polygon but takes the shape of a circle.

Explanatory argument for the above is that when the number of sides of regular polygon increases to infinity, length of each side will tend to zero and the regular polygon will exactly take the shape of a circle.

This leads us to the conclusion that a circle, in fact, is a regular polygon with infinite number of sides. But how can one count infinite sides? Infinity is not a finite quantity or a finite number, it is ever increasing continuously. Counting Infinity* has been explained in my earlier article and learned reader may refer to it for understanding its concept.

In the opening paragraphs, it was mentioned that a circle has no side, hence no corner but latter paragraphs makes mention that it has infinite sides and hence, infinite corners. Which one is true, requires further analysis. It is mentioned in foregoing paragraphs, when number of sides of regular polygon tends to infinity, with that 1/n or 2.pi/n tends to zero. Kindly read between the lines that 2.pi/n tends to zero but it is not zero. There is a huge difference between tending to zero and zero. Tending to zero is infinitesimal small, so small that it is smaller than the smallest and it is proceeding to value zero. Or in other words, the quantity is going to lose its magnitude to become zero.

On the other hand, zero is nothing or null and is absolutely devoid of magnitude. The quantity itself is zero, it needs not reach zero. 

Now let us say, circle has side of zero length (as that means no side), then 2.pi/n will be zero, please mind it, in this case 2pi/n is not tending to zero as was considered earlier, here it is perfect zero. In that eventually, (sin 2pi/n)/(2pi/n) will be 0/0**.  But dividing zero by zero is indeterminate. In that case, area of polygon would be indeterminate but in practice, area is calculable and finite. Therefore, our assumption that a circle has sides of zero length or no side is not correct.  

But it has already been proved, if 2.pi/n tends to zero then regular polygon transforms to a circle and has finite area pi.r^2. Therefore, number of side of regular polygon must be infinite and length of side tending to zero. That proves, a circle always has infinite sides, hence infinite corners and it is never without corners. 

Notes:

1. Counting Infinity may be read at https://narinderkw.wordpress.com/2017/03/12/counting-infinity/

2. 0/0** Solving division of zero by zero may be read at https://narinderkw.wordpress.com/2016/11/27/solving-division-of-zero-by-zero/ 

3. Title Image courtesy https://commons.m.wikimedia.org/wiki/File:Sphere_wireframe.svg#mw-jump-to-license 

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Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc. 

Facebook Can Secretly Watch You  

 

On reading the news that Facebook can use the camera of your phones and computers for spying on you, reminded me an incident of the era of Cold War between Soviet Union and United States of America. Both nations were trying to surpass the other in technology and warships to establish their dominance. In the race of oneupmanship, a renowned physicist of Soviet Unions was assigned the task of spying the embassy of USA in Moscow using electronic devices that too without battery.  The physicist at the very onset, showed his helplessness as all the electronic devices need energy from some active source like battery to operate. However, use of battery was unwarranted for the fear of the detection of the device. Soviet Union was displeased with the negative response of the physicist. There was one option with the physicist either he should research on feasibility of a device using no battery or face the wrath of death from the rulers.

Necessity is the mother of invention. This necessity which weighed against the life of the physicist, invented a passive bugging device that worked without use of battery. You may please hold your breath and think over whether there can be an electronic device which can work without the use of battery or power. Connect some integrated circuits, transistors, diodes, resistances, capacitances whatever electronic components are, but you can never get output till you energise the circuit by the battery. Some learned readers may think of utilising vacuum energy as demonstrated by the physicist Casimir but probably, that didn’t come into being at that time when Soviet physicist was commanded to do it. Otherwise also, energy from Casimir effect is so low that it has not been practically utilised.

Ultimately, that Soviet Physicist did a bright thing by inventing the bugging device nicknamed “The Thing,” that operated battery less but it utilised the energy of the remote battery kept far away. How can one use remote direct current battery for operating an electronic device kept far away? Battery does not radiate power, it gives electrical dc power at its anode and cathode by converting chemical energy to electrical energy.

The Thing” or the bugging device as was popularly known at that time, has a cavity membrane with an adjustable post for regulating its frequency. This arrangement worked as a cavity oscillator with the radio frequency energy transmitted from the control station that resonated the cavity oscillator. In other words, at control station, radio frequency was generated and transmitted in air and the radiated r f energy resonated the membrane of The Thing. The audio signal generated on account of the conversation of the persons being spied, modulated the frequency of the cavity resonator and reradiated it through the monopole antenna. For the information of my learned readers, I submit that every receiving antenna simultaneously acts as a transmitting antenna. That antenna of the bugging device first received r f signals at its resonating frequency from control station and after modulation with spied audio signal retransmitted these to the control station. At the control station, these signals were received and demodulated with diode rectifiers and heard with speakers reproducing the conversations of the persons being spied.

The seal opened exposing the Soviet bugging device, on display at the NSA’s National Cryptologic Museum.

Though this tiny bugging device weighed app 30 gms yet it was difficult to instal these inside the American embassy. Hence these were installed on outer windows or even on nearby trees in the periphery of the embassy. The Thing was designed by Soviet Russian inventor Léon Theremin, whose best-known invention is the theremin, an electronic musical instrument. The incident is also connected with the “Great Seal” a gift given by the Soviets to the US Ambassador to Moscow on August 4, 1945 and it had the Thing bug concealed in it.

Replica of the Great Seal which contained a Soviet bugging device, on display at the NSA’s National Cryptologic Museum.

However, this bug was detected only in the year 1951 ie after six years of its secret operations. The passive device is so small in size that it can be concealed anywhere and it continues working for years as it does not need a battery.  

I submit, spying is not limited to recording ones conversations or videographing of ones activities, it can control ones activities including activities of ones brain using infrasound waves. Infrasound is mechanical vibration of medium at frequencies below 20 Hz. When infrasound propagates, it causes mechanical vibrations and if the frequency of the objects is equal to that of the infrasound, the source imparts energy to the object and both are joined and vibrate at same frequency. This is called resonance.

That means, if one increases the energy of the source, energy of the vibrating object will also increase when in resonance. In a way, one can control the energy of the object while sitting at a place away from the object. It is submitted that our body organs also are in the state of vibrations. In fact all biological organs vibrate at frequencies within infrasound range. It is, therefore, natural that infrasound can affect and control vibration of our limbs.

But if the vibrating limbs are parts of the brain, it would seriously affect its functioning. That may culminate into change of mood, unexpected depression, unexplained mood swings likewise. In a controlled manner, this property of infrasound can cure the neurological problems and is a blessing in disguise. I submit, the infrasound can not be heard, can not be felt, can not be seen and the cause of the affliction will not be explainable. Thus these undetected extremely low frequencies go on adversely affecting the person. By manipulating the energy and frequency of infrasound, one can control the emotions of a person or in other words, the mind can be controlled with the infrasound without the knowledge of the person affected from distance.

When a person receives infrasound, it causes the neurological disturbance unknown to the person and his brain searches for the stimulus. On not finding the cause of disturbance, the brain tries to find more options but in vain. A stage is then reached where he goes in extreme fright and is not able to decipher what is happening to him. For details one can go through the article written by me, “Infrasound That Kills,” at link https://narinderkw.wordpress.com/2016/09/22/infrasound-inaudible-sound-that-kills/

Coming to the spying by Facebook as mentioned in the news item, I submit, you would have observed that when you play a video clip say a crocodile tearing apart Impala, you will find a number of clips similar to the one watched by you appearing on the Facebook page. It is the algorithm of the programme that sequences similar clips and presents these to you. What I want to emphasise is that there is someone either the algorithm or some other eyes watching your activities.

I further submit that invariably in your mobile, there is a camera connected to the electronic circuit. It is energised as and when we switch to camera mode, otherwise for all intents and purposes, it remains passive although connected with the circuits videographing whatever is in its focus. But generally this videography is not saved in storing device and gets ineffective. But if there is a programme, that monitors the videography and generates a pulse when your face looks pleased or displeased with the contents you are watching on FB, that pulse will actuate the algorithm, resultantly, that pushes the similar contents on your mobile or laptop or personal computers on FB page. If your reaction is averse to some contents, reverse will happen and it will block such contents. Next is memory, once you have shown your liking or disliking by facial expressions, that is memorised by the software which will either preclude showing such contents or facilitate these on your Facebook page in future.

But if the FB programme is modified so as to reveal your preferences recoded in it on the basis of facial expressions likings, in that case, if these were divulged to someone whom these should not be, irreparable damage could be done to the user. Apart from this, if you are smart enough to camouflage your facial expressions, the programme will fail to detect your liking or disliking and will be misguided. But concealing ones expression is like concealing ones nature and is a Herculean task.

Will recording your reactions by FB be a boom for advertisers? I submit, we all have our liking and disliking for particular things. This is applicable to contents of advertisements also, some like advertisements charged with actions, others which are lessons conveying. Utilising stored information of our liking and disliking, programme will present a series of advertisements of our likings to the benefits of advertisers.

The worry what I can envisage now is that such software programmes may breach ones privacy if ones preferences are made public. Commission of such act would be serious in nature and hopefully can be avoided also. 

As the time passes, use of smartphones, laptops and personal computers is increasing manifolds, these devices are our obsessions and it is difficult to abstain from these. In such a situation, a simple way to block transmission of our facial expression is by disconnecting cameras and webcams of our devices from the circuitry or simply blocking them with opaque tapes.

However, I am also of the opinion that storing your facial expressions of smile or anger seems less damaging as compared to reading ones brain and auto typing the message on the basis of neurosignals as has already been proposed by FB Tech. Once your mind is revealed, there remains nothing to conceal particularly when we all have mens rea at one time or the other. This subject was dealt exhaustively by me “Facebook Tech May Let You Type With Your Brain,” at link https://narinderkw.wordpress.com/2017/04/21/facebook-tech-may-let-you-type-with-your-brain/

What is the remedy available then? From the description given above, it appears, private is no private now. Situation is so alarming that there are allegations of interference in the poll process of the world largest democracy which has the world’s best technology. It reminds me a principle which I learnt during my revenue training. It lays down, “Mushtri Hoshiar Bash,” that is “Purchaser, be aware.” If I change the word purchaser to user it squarely fits for FB. “Users be aware,” you are being watched and spied.

                                         End

Notes:

1) Webcam image courtesy https://commons.m.wikimedia.org/wiki/File:Webcam000c1.jpg#mw-jump-to-license

2)
Soviet Seal Bugging Device image courtesy https://commons.m.wikimedia.org/wiki/File:Bugged-great-seal-open.jpg#mw-jump-to-license 

3) Replica of the Great Seal which contained a Soviet bugging device, on display at the NSA’s National Cryptologic Museum Image courtesy https://en.m.wikipedia.org/wiki/The_Thing_(listening_device)#/media/File%3ABugged-great-seal-closed.jpg

Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.

 

 Enjoy Stress Free Life With Five Minutes Of Dhyan Mudra Daily


Sitting in the grassy ground of Meditation Centre, an asset to this city situated in the foothills of Shiwalik, Himalaya, I religiously go there and do “Dhyan and Yoga,” mediation and Yoga under the able guidance and supervision of my Guru or teacher. My Yoga Guru is a simple man with no greed for material gains. He often jolts me why I don’t do Dhyan early in the morning at 3 AM Amrit Vela, pious time. He advises me, spending twenty minutes upon Dhyan here in Yoga session was by no means adequate and sufficient for connecting with Him.But for me, early morning with cool ambience, in otherwise roasting heat of 48 degree Celsius, tempts me and I toss in bed waiting for 4.15 AM alarm when I get up. But my inner self always says, “Narinder, it is high time, lead a mediative life before it is too late. When Kaal, or death will hover over me, it is Dhyan that mitigates my sufferings.” Once I adopted Meditation but alas! I could not maitain it for long.

Today in Yoga session, I asked Guru Ji, “Sir, is there a way to combat depression and free oneself from ones irritating nature?” As at my senior age, depression was inevitable particularly when my mind was not occupied except devotion of few hours to pressing the alphabets of this shining glass screen of the device for writing what my mind chose worth writing or going for long walks in the morning or answering questions on quora and at times, pedalling the bicycle safely on forlorn roads in suburbs.

Based upon his spiritual solution to this malady, I am going to share one solution out of many to unleash oneself from falling and failing mood. It is Gyan Mudra Or Dhyan Mudra. It is so simple that it can be performed by one and all irrespective of ones body shape or age.

What is Mudra?

Mudra, in general means expression or way of expression or body gestures to communicate to describe or tell something. In Indian classical dances, different Mudras of the artists may be so exhaustive as to depict the entire ancient epic withiut the artists opening his or her lips. But in Yog Mudra, this expression is limited to specific positions of hands or fingers of hands. I submit that process of conducting signal to the brain starts with slight touch of one finger with the other and there starts another process of offsetting the afflictions our body suffers from. Before going into the method of healing by Yog Mudra, I will explain what our fingers represent in balancing the body system.

Five Elements


                                     Fingers representing five elements 

In Ayurveda, a healthy body has balanced Jal (Water), Prithvi (Earth), Vayu (Air), Akash (Space or Sky) and Surya Or Agni (Sun or Fire). If there is imbalance amongst these, one is likely to fall ill. In other words, illness originates with imbalance of these elements.

How can then we correct and maintain this balance?

Our fingers are the probes that make the balance. Little finger is a probe for controlling Jal (Water), next, ring finger controls Prithvi (Earth), central finger Vayu (Air), fore finger Akash (Space or Sky) and finally thumb controls Surya Or Agni (Sun or Fire). Method or way of touching one finger with the other is a Yog Mudra and has healing effect that will be explained in proceeding paragraphs.

General Sitting Posture for All Mudras


                                          My Guru Ji in Gyan Mudra or Dhyan Mudra

For all Mudras and Pranayama unless otherwise stated, one sits in crisscross position with palms on knees and back straight. It will be better to spread a mat or blanket to sit upon it in crisscross position. A yoga suit or comfortable dress will serve the purpose of Yog Mudra. I submit that before one starts Yog Mudras or Yoga, it is advisable to chant Aum and offer prayer to Sun in the form of Gayatri Mantra.

Prerequisites of Gyan or Dhyan Mudra

A. Chanting of Aum

  Before doing Yog Mudra or Yoga, it would be better to mingle with omnipresent that encompasses everything and is present everywhere . That is Aum . It is primordial Mantra or Beej Mantra or Basic Mantra which precedes most of the Vedic Mantras . It has been found by NASA that Aum is in fact the sound of Sun and the vibrations found in deep cosmos on compression or rarefaction, give humming sound of Aum . Even the sounds generated from different sources cocktail to give humming sound of Aum . Those who are interested to know details are referred to link at https://priyankawadhawan.wordpress.com/2016/05/23/international-yoga-day-2016/ 

Therefore, sit on folded blanket placed over spread mat with legs crisscrossed in Gyan Mudra ( it will be explained in proceeding paragraphs) and concentrate upon third eye. Take a deep breath and fill your lungs then chant Aum which is formed by A+U+M as instructed in the linked article for once or three times .

B. Gayatri Mantra


                                                           AUM

  We owe our existence to Sun , it gives us energy , light , oxygen , water , food from vegetation and everything . It is the generator of our life . Therefore , it is our foremost duty to pray to him to continue giving us life force and also enlighten us as its rays dispel darkness and light our life. Gayatri Mantra is dedicated to Lord Sun . With eyes closed , sitting in Gyan Mudra ( it will be explained in proceeding paragraphs) concentrating upon third eye where Sun should be sketched with inner eyes or it should be visualised , we should chant Gayatri Mantra . It is reproduced in Hindi with Devanagri Script followed by English below.

“ॐ भूर्भुवस्व: |

तत्सवितुर्वरेण्यम् |

भर्गो देवस्य धीमहि |

धियो यो न: प्रचोदयात् ||”

“ Om bhūr bhuvaḥ svaḥ

tát savitúr váreṇ(i)yaṃ

bhárgo devásya dhīmahi

dhíyo yó naḥ prachodayat.”

Gayatri Mantra likewise may be chanted once or thrice depending upon your wish. When above prerequisites are over, we come to Gyan or Dhyan Mudra.

Gyan Mudra or Dhyan Mudra


                              Gyan Mudra or Dhyan Mudra

Sit in crisscross position with back straight, touch tip of forefinger with tip thumb. These should not press each other but should be just touching each other. Kindly see the image above. Place your palms in such a way that these just lie upon the knees and it will automatically straighten up your back and this is the requirement for this Mudra. Also three fingers adjacent to fore finger of both hands must be straight and pointing outwards. The palms must be facing upward, these should not be tilted sideways. First and foremost point is that there should be a feeling of happiness, wellness and bliss on your face. Yog Mudra or Yoga should never be done in tensed up and exerting conditions. But before sitting in Gyan or Dhyan Mudra, one must chant Aum as already explained. One must concentrate upon the space between two eyes called Agya Chakra. For distracting oneself from worries, unnecessary thoughts occupying our minds, it would be better to scribe Aum as shown in the photograph. One may scribe symbol of Jesus Christ or any other religious symbol, one feels like.

I further submit that those who can not sit in this position for long, they can sit in any other position taking care that back should be straight.

Duration of Gyan Mudra or Dhyan Mudra

Minimum duration for this Mudra should be two minutes and maximum depends upon ones practice and ease with which one can sit in crisscross position and concentrate upon Agya Chakra. Care should be taken that back should be straight. The effect of a Mudra may be perceived immediately or only after a certain amount of time. You start to feel warm, the sense of unwellness and pain fade away, your mood improves, and your mind is refreshed. But exactly the opposite may occur at the start. You become tired, or start to feel cold and shiver. This is also a positive sign of the effect.

How does it affect?

I briefly stated earlier that mismatch in level of five elements represented by our five fingers are the basis of our afflictions, according to ancient Indian healing system. Mere touching of forefinger with thumb (or Vayu or air) with Agni (or Surya or Fire or Sun), an electric circuit is completed and that current triggers the generation of neurotransmitters and more the neurotransmitters, better the mood. By adopting Yog Mudra as daily routine, the concentration reaches the required value of neurotransmitters which eliminates totally or reduce the depressive feeling.

Effects of Gyan Dhyan Or Dhyan Mudra

Though this is the basic Mudra yet it has charismatic effect on wellness particularly functioning of the brain which controls every body part. This Mudra produces a sense of general wellness also that one is not suffering from any affliction and if there exists some, there comes a feeling that it is being purged out by this Mudra while chanting Aum. Depression, down mood, feeling of incapability, incapacitation, swaying of mood likewise neurological disorders, become the things of the past. It is highly usefulness for the students and others who are required to concentrate. Forgetfulness, insomnia, laziness, fear psychosis, indecisiveness are also controlled if we sit in Gyan Mudra. Bad temper and irritation are also connected with low neurotransmitters and Yog Mudra builds these to the required level to set right the mood.

Yoga, Dhyan and Allopathy

Allopathy and Yoga are not on cross roads but are in fact convergent leading to wellness. I have seen allopathic doctors prescribing Yoga and Dhyan for betterment of neurotic conditions of the patients. Those doctors are not the ones sitting in a shop in an unfrequented alley but they are well salaried appointed in multinational health care. You will agree with me, they rightly prefer Yoga over addictive antidepressants. It is also my personal experience that the situation which earlier seemed unsolvable and grave to one, after sitting in Dhyan Mudra regularly, seemed trivial and immaterial. That is the reason I get up early at dawn and while enjoying the psychedelic patterns of multi hued eastern sky, proceed to Meditation Centre and start my day with chanting of Universal Sound of Aum. Would you then not like to sit in Dhyan Mudra to lead stress free life? Do it, it is easier than explained.

Namastey!

May peace prevail in the universe!

                                                 End

Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.

Notes:

1. Yog or Yoga deote to same meaning. In Sanskrit, Yoga is pronounced as Yog.

2. Images credit to https://commons.m.wikimedia.org/wiki/File:Aum_Om_navy_blue_circle_coral.svg#mw-jump-to- 

https://sites.google.com/site/ncs976089/mudra-yoga 

 

Will Trump’s Exit From Paris Accord “Make America Great Again?”

It is understood that President Donald Trump may pull out from Paris Accord on climate change basically on the issues that reducing consumption of fossil fuel including coal is detrimental to the U S economy and will affect job creation. Fear of climate change is overstated and commitment to address the non existent threat is destroying the economy and coal sector. On reading between the lines the object is to usher the economy particularly in view of the fact that he does not appear to apprehend much danger from climate change. But moot question that emerges out is whether climate change is farce, baseless and uneventful. If that be so, it will lift all curbs on carbon oxide emissions. On the other hand, if concentration of carbon dioxide is increasing and f it is driving the temperature upwards, pulling out from agreement would not only be unfortunate but prove disastrous for the planet also.

What Is Global Warming Or Climate Change? 

Global warming is considered as rise in the planet earth’s temperature due to excessive emissions of gases namely methane, carbon dioxide along with water vapours, generally called “Green House Gases.” 


 These gases are considered to absorb rays reflected back from earth and thus act as opaque layer to the rays. Blockage of radiations raises the temperature of the atmospheric layer which in turn is considered to raise the temperature of the earth. Green house gases thus are considered to block the heat escaping out to space and warms the earth, is the concept behind global warming. It is argued that more the concentration of green house gases, here I am considering only carbon dioxide, more will be blockage of heat resulting in further warming. This theory and concept that increased concentration of carbon dioxide in the atmosphere causes global warming, is supported by most of atmospheric scientists, hereinafter, after called warmists. They believe that natural sources of carbon dioxide although, contribute major share as compared to manmade, the carbon dioxide liberated from these sources is eventually absorbed back and does not cause any disturbance whereas human sources ie emissions from fossil fuels, coal, oil and natural gas although do not contribute much, are considered major culprits of increasing carbon dioxide. Warmists argue that carbon stored during millions of years in these fuels, is being converted to carbon dioxide in few years and is causing step change in its concentration.

Does increased concentration of carbon dioxide elevate atmospheric temperature?

In preceding paragraph, it was mentioned that green house gases act as opaque layer that does not allow the radiations from earth to escape. If these were opaque to outgoing radiations then these should be opaque to incoming radiations also. That will then resist temperature rise, will then these not act against global warming? Yes, these will but such arguments are ignored by the warmists. In the year 1940 when second World War was also fought, rivers of oil were burnt. But data of the National Aeronautics and Space Administration, reveal that carbon dioxide levels remained same during the decade of 1940 to 1950 even after burning of whopping oil in Second World War. That proves, burning of fossil fuel did not increase carbon dioxide level. 

Solar Activities And Atmospheric Temperature

.

 

The above graph shows, temperature anomaly on y axis and time in years on x axis. Different colours indicate temperature measured by different agencies at different locations. Mr Al Gore the then Vice President of America who had interest in atmospheric science pointed out from this curve that fossil fuel was burnt maximum when industrial activities increased and increased concentration of carbon dioxide led to exponential increase in temperature after 1850. However on examining this curve called hockey stick for 2000 years in stead of narrow period of 1850 to the year 2004, it is observed that from the year 1000 to 1400, temperature remained almost flat and normal, crops flourished and sun spots which are indicators of strong solar activities and intense heat radiations, were also observed during this period. But from 1400 to 1900 years, graph shows a drop in temperature called little ice age. Also sun spots were not seen during this period which is called Maunder Minimum. This gives a strong indication that temperature of earth depends upon solar activities. When there are sun spots, temperature increases and vice versa.

 

On examining the hockey Blade curve between 1850 to 2004, it is observe that from 1850 to 1880, temperature rose then from 1880 to 1910, it fell and from 1910 to 1950, it again rose. It is, therefore, clear that temperature had not been increasing exponential like that of hockey blade but it undulated.The graph shows temperature fell from the year 1850 to 1910 and also from the year 1940 to 1950 when the use of oil was at peak being used in Second World War. This is also not in conformity with theory and concept of warmists. Either the theory of Global warming is incorrect or the graph is wrong. But the graph is based on the temperature actually recorded, therefore, correctness of the global warming theory is at stake.

Temperature Causes Change In Carbon Dioxide Level Not Carbon Dioxide Increases Temperature

 Al Gore also corelated concentration of carbon dioxide in the atmosphere with temperature and deduced that increased carbon dioxide level drives the temperature higher.

 

Figure 3 is a graph showing variations of carbon dioxide, temperature and methane (other green house gas) with ice core records in distant past. It is apparent that statistically, shapes of curves of carbon dioxide and temperature match each other. That is variations of temperature and carbon dioxide has the same trend. To ascertain whether temperature follows concentration of carbon dioxide or concentration of carbon dioxide follows temperature, the graph is examined at expanded scale.

 

Above graph shows variations of methane, carbon dioxide and temperature measured at arctic with ice core record of distant past. At 17000 age (Yr BP), carbon dioxide level is at minimum, if carbon dioxide is driving the temperature, temperature at 17000 age (Ye BP) should be minimum but it is not so as is clear from the graph. However, temperature at Antarctic is minimum at 18000 age (Yr BP), therefore, there is a lag of 800 years. If carbon dioxide were causing change in temperature, it would have been immediate but it is not so. Conversely, it proves that it was the temperature that was causing change in concentration of carbon dioxide.

Temperature Regulates Carbon Dioxide Level

 Apart from this graph, it is submitted that at higher temperatures, solubility of carbon dioxide in water decreases and carbon dioxide bubbles out from water and this increases the concentration of carbon dioxide in atmosphere. Similarly, lower temperature increases the solubility of water and this lowers the concentration of carbon dioxide in atmosphere. It has already been proved that it is not the concentration of carbon dioxide that drives change in temperature but it is the temperature that causes changes in concentration of carbon dioxide. Thus increase in level of carbon dioxide if any, will not cause rise in temperature but changes in temperature due to solar activities or otherwise will vary the level of carbon dioxide.

That falsifies the theory of global warming which states that increase in concentration of carbon dioxide in atmosphere causes global warming.

Where Is The Problem?

A hotchpotch has been created by a number of factors, that increased concentration of carbon dioxide raises the temperature was examined by atmospheric scientists on different assumptions and erratic mathematical models on the basis of different data from different agencies. The global warming hypothesis was originated from the United Nationsʼ Intergovernmental Panel on Climate Change (IPCC) whereas the best scientific data available is with the National Aeronautics and Space Administration and that shows, carbon dioxide levels “flat-lined” during the decade of 1940 to 1950 but such data is ignored and findings made are narrow based to arrive at predecided result of global warming.

Evidence Against Global Warming

John Coleman, founder of weather channel, strongly oppose theory of global warming and categorically stated, “The ocean is not rising significantly. The polar ice is increasing, not melting away. Polar Bears are increasing in number. Heat waves have actually diminished, not increased. There is not an uptick in the number or strength of storms (in fact storms are diminishing). Global warming has become a political and environment agenda item, but the science is not valid.”

Wasteful Expenditure

Steps to cut down carbon emissions are not only sheer wastage of funds but are also causing threat to aquatic life. One such step is sequestering of carbon from oil, sequestered carbon does not react synergically with alkaline reservoirs forming harmful byproducts to aquatic life. When ingested by aquatic creatures, it creates a vicious circle of misery for human beings. An increased presence of carbon reduces the metabolism rate and the ability to synthesise protein. Over curiosity among scientific community about carbon warming does more harm than good. Carbon Sequestration is about losing an eye for an eye.

Donald Trump Words: Commitment To Address The Non Existent Threat Is Destroying Economy

Growth of plants for biofuel is decreasing land acreage under edible crops resulting in inflation of prices of agricultural produce. Heavy cost of biofuel as compared to fossil fuel also burdens the common man. Resorting to wood as fuel for domestic purposes is causing eyes ailments in addition to adding general pollution due to smoke. The Regulators have imposed a ceiling on allowable emissions. Emission Units are defined by Least Common Multiple (LCM) among the competing organisations. These units are auctioned among the organisations or served to the organisations through a First Come, First Serve basis or through a convenient administrative procedure. Ultimately the common man has to pay the price. To reduce emissions, carbon tax is imposed and that also burden general public. Had such expenditures to cut down carbon emissions been avoided, the savings would have ensured the food for the unprivileged.

Is the carbon dioxide a villain for environment?

Entire plant kingdom synthesises its food from carbon dioxide, water and sunlight. If carbon dioxide were reduced from atmosphere, food synthesis by plants would slow down. There would be less plants growth, less agricultural produce and famine. That situation would cause hunger, starvation and deaths. Notwithstanding, carbon dioxide being a life supporting gas, is considered synonymous with pollutant. Honourable Supreme Court while deciding a case in America, has termed carbon dioxide as pollutant irrespective of the fact it is essential for photosynthesis, a process which forms hydrocarbons in plants. It would have been better if it were termed a pollutant above certain level but condemning it as pollutant without consideration to its concentration, is a big set back to rational environmentalists. But there is a silver lining in this murky affair. Documentary “An inconvenient Truth,” of Mr Al Gore has been found inconvenient for its acceptance as scientific truth. London, England in October 2007, the High Court identified nine errors in Al Gore’s movie An Inconvenient Truth that were very much outside accepted scientific truth. So much so that the Government of the UK sent notices to all secondary schools in England pointing out the errors in his film.

Even the argument that global warming will melt the polar ice raising sea level and extinguishing low level islands does not stand the test of scientific scrutiny. Ice exists submerged at poles called polar caps. It displaces equal volume of water. Had the ice melted, it would have displaced the same volume. Further same mass of ice has more volume than the water. Water at 4 degrees Celsius occupies minimum volume being the heaviest. All these scientific facts prove, there would not be change in ocean level, had the ice melted. It is argued that thickness of ice at arctic is affected by oscillations of pressure in Iceland and Azores.This effect is similar to that of a build-up of a monsoon or a cyclone. Further, it is observed that global temperature is decreasing and we are proceeding towards ice age. Global warming is unscientific and a hoax.

What President Donald Trump considers of global warming is logical and scientific. Is it then not the right time that expenditures on the projects meant for reducing carbon emission may be diverted to projects for feeding the hungry and starved of unprivileged nations? I hope President is in the process of doing it.
Such a step will not only  feed unprivileged nations but make America great again. 

                         End

N B: 1) This article is purely based on science and may not be construed as political.

2) Image Courtesy https://upload.wikimedia.org/wikipedia/commons/7/7e/Donald_Trump%27s_hair_fashion_2011.jpg
                                                         

                                         End

Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.

How Fast Is Gravity? 


How much time does a mass take to experience gravitational attraction due to another mass? Is it instantaneous or time delayed? Will it take some time for a person to feel the weight of a jug full of water? Sir Isaac Newton although did not mention the time in experiencing gravity in his popular Laws Of Gravitation yet it is universal that gravity is felt instantaneously without loss of any time. The gravitational laws simply state the force of attraction between two masses is proportional to their product divided by square of the distance between them.

Precession Of The Orbit Of Mercury

It was observed that planet mercury precesses by a small angle in its orbit and Newton’s laws of gravitation were not able to explain this behaviour. Different hypothesis were put forth to explain precession of the orbit but in vain.This anomalous precession of mercury was successfully explained by French astronomer Urbain Le Verrier. He theorised that force of attraction between two masses not only depends upon inverse square of distance but also upon the cube of distance between them.

 It was a big set back to Newton’s inverse square law. These were reduced to the status of approximate laws. It, not only hit laws of gravitation but also fortified the theory of General Relativity as dependence of gravitational force upon the cube of distance was a mere deduction from the laws of general relativity. Dependence of gravitational force upon GR was the grand success of theory of General Relativity GR and it elated Einstein to the extent that he felt heart palpitations with the happiness.

Laplace Derivation Of Velocity Of Gravity

In modern physics, force is attributed to interaction of particles of the the field. Gravitational force is interactions of hypothetical massless particles called gravitons. It appears obvious that for exerting instantaneous attraction, the gravitons must propagate at infinite velocity.

Is it possible to have superluminal (moving with velocity higher than that of light) particles? Before Einstein’s Theory of relativity, there was no bar upon the value of maximum achievable velocity of a particle. Gravitational laws as postulated by Newton were impregnable. There was no bar upon highest achievable velocity.

That led physicists to give their hypothesis based on velocities that far exceeded that of light. First attempt to determine the speed of gravity was made by Laplace, he considered the gravitational field as radiation field whereby changes in the velocity of the attracting masses transmit some sort of waves. While examining orbital motion of a planet, he proposed that speed of gravity must be appreciably high to cancel the effect of delay in time to cause gravitational attraction as this delay was likely for interacting particles to reach the masses which were situated apart. He determined the velocity of gravity to be at least 7×10^6 times the velocity of light.

This hypothesis was also rejected as Einstein theory of Special relative was proved experimentally correct on the fateful day of solar eclipse. With that, the limitation of maximum achievable velocity by any object was fixed as that of light abbreviated as ‘c.’ Now hypothesis of infinite speed of particles gravitons constituting gravity was unacceptable and unscientific.

Theoretical Value Of Velocity Of Gravity According To GR

If we consider solutions of field equations of GR, these provide theoretically the value of velocity of gravity as that of speed of light. That means gravitons interactions propagate at velocity c. If such were the case, would it not take some time to experience gravity due to heavy object situated at an astronomical distance? To consider the effect of delay in time in experiencing gravity, we take the case of solar system where light rays reach earth in 500 seconds from sun. Let us take the hypothetical case where the sun disappears suddenly at time T. Light rays which started at time T from sun will continue falling on earth for 500 seconds more after disappearance of the sun. Let us examine the case of gravity, considering velocity of gravity as that of light, will the earth also continue experiencing gravitational attraction for 500 seconds more after the disappearance of sun?  

As it is stated earlier, solutions to General Relativity provide velocity of gravity equal to that of light c. In that situation, Gravity would not cease immediately and  earth orbital motion must continue for 500 seconds more. If it were assumed that the earth would continue orbiting 500 seconds more then that would mean, changes in gravitating mass would not cause immediate change in gravitational field at a distance. But this explanation is too simple to prove speed of gravity and its interaction and it requires further corroboration.  

Instability Of Orbit

Take the orbital motion of Jupiter around the sun, at any time, force due to gravity felt by the sun would pertain to gravitational attraction that took place earlier by time ‘t’at Jupiter where t is time taken by gravitational interactions to reach from Jupiter to the sun. That is, the sun is experiencing gravity from the Jupiter at its present position earlier by time t. Similarly, the Jupiter will also experience gravity from the sun at its earlier position. In such a situation, the sun is attracting the Jupiter at its earlier position say ghost position. And the Jupiter is attracting the sun from its ghost position.
 

 We know, for a stable orbit, line of action of the force between the sun and the Jupiter must be directed towards them. But in the case being discussed, there are two separate parallel forces and we know that such parallel forces yield a couple which will destabilise the system. Therefore, effect of the gravitational force must be experienced towards their present positions instantaneously irrespective of the fact that the sun and the Jupiter are situated at a distance. That warrants infinite velocity of gravity to cause its effect instantaneously but GR warrants velocity of gravity to be equal to that of light.

Concept Of Spacetime

This is a paradoxical situation which needs resolution. General Relativity lays down that mass and energy warps or curves the spacetime. Spacetime in simple words can be considered as expression of space by four variables including time. We know, a point came be expressed with three coordinates ie length, width and height or x, y and z. But the position of point may change with time t, we will have to add one more variable of time t. Spacetime of a point then can be expressed as (x,y,z,t). A ship in spacetime coordinates can be expressed at “Mumbai at time 8.00 PM on May 24, 2017.”

 Coming to warping or curving of space, I submit, massive the object or energy, more warped or curved spacetime will be. In simple words, if the spacetime is highly curved, it denotes higher the value of gravity. Consider bike race in circus in a conical chamber where the bikers ply bikes all around the chamber. A biker who runs bike slow will be pulled downward and the fastest can run bike at the top rim of the chamber. If a biker stops accelerating the vehicle, he will spiral down to the floor of conical chamber. We can safe assume that the conical chamber is warped spacetime, bikers celestial bodies which orbit in warped spacetime according to their kinetic energies that depend upon their velocities. Bodies wth higher velocities will be in outer orbits and bodies with smaller velocity in inner orbits.

Instantaneous Effect Of Gravity

Having explained the basic concepts of spacetime of GR, we revert to the example where we considered the hypothetical case of disappearance of the sun and its effect on orbital motion of the earth. When the sun was there, it had the warped spacetime, when it disappeared, it is to be examined whether the warped spacetime also disappeared instantaneously with infinite speed or with finite speed of light. I submit, it is the subtle nature of the interactions that tries to annul the effect of propagation delay. That means, the characteristics nature of interactions, annul time delay in changing the warped spacetime to flat provided there is no external mass to interfere.If there were some external mass or energy, the spacetime, strictly speaking would take the form depending upon that external energy and mass.

 It is worth mentioning here that these are not only the planets that experience gravitational force and orbit around the sun but the sun also experiences gravitational force and orbits the planets depending upon its mass as compared to that of planets. If total mass of the planets is very small, the star (here sun) may neither orbit nor spin.

Lorentz Invariance And Velocity Of Gravity

Before proceeding further with the subject, I state that for instantaneous disappearance of spacetime (or gravity) in the example of earth and the sun, it is necessary that there should be constant velocity between the sun and the earth. In this case the velocity is almost constant.

Gravitational field of masses which have constant speed are associated with static and continuous field that does not propagate. Lorentz invariance demands that what a moving body in a static field sees and what a moving body that emits that field sees be symmetrical. Therefore, two celestial bodies moving at constant velocity with respect to each other, each feels a force instantaneously towards the present position of the other body and therefore, there is no speed of gravity delay. In the example of the Jupiter and the sun, gravity between the two will be according to their current positions and speed of gravity delay will never be observed.

Experimental Determination Of Velocity Of Gravity

We know the gravitational force between two masses is too weak to measure, it is not possible in laboratory to measure speed of gravity. But in case of binary pulsar PSR 1913 + 16, the orbit of the binary system is slowly decreasing. The decrease in radius of the orbit is attributed to the slowing of the speed of binary stars on account of loss of energy. This loss is accrued to gravitational radiation and its lowest order is quadrupole radiation where the radiated power depends on the third time derivative of the mass quadrupole moment.

But in any field theory, radiation is related to the finite velocity of field propagation and the orbital changes due to gravitational radiation can equivalently be viewed as damping caused by the finite propagation speed. On computing the rate of damping, one finds that it depends upon the speed of gravity. If the calculations framework of GR is accepted, speed of gravity can be calculated in the basis of damping. Based on the data of this binary, the speed of gravity calculated is found equal to that of light with in 1 pc error.

I submit, speed of gravity is, in fact, speed of graviton interaction that propagates as gravitational waves and speed of gravity is, therefore, equal to speed of gravitational waves. Whenever there is disturbance in spacetime, it creates gravitational waves, obviously massive the objects, more distinctly gravitational waves can be detected.

In September 2015, pair of blackholes merged and that caused disturbance in spacetime generating gravitational waves. We have already stated that gravity propagates due to interactions of gravitons. If gravitons interaction propagates at speed of light, special relativity warrants, its rest mass must be zero. If the gravitons are found to possess mass, then they can not move with velocity of light and speed of gravity in that case would be less than that of light. And if the velocity of gravity were less than that of light, corresponding lower frequencies of gravitons would cause dispersion of light at the point of merger of the blackholes. But no such dispersion of light was observed and that also proved, the velocity of gravitational waves or gravity equals that of light with in experimental result.

In this way, experimental results and theoretical calculations concurred the velocity of gravity to be finite and equal to that of light. It is interesting to note that velocity of gravity though finite, its effect is experienced instantaneously. When a mass disappears, its spacetime also transforms to flat line instantaneously provided there is no other interfering energy or mass thanks to Lorentz invariance and subtle characteristics of spacetime. Gravitational force being always attractive, leads us to physically prove that force of attraction between two charges is always more than force of repulsion between them. It is heartening to note the characteristics of gravity that although its velocity is that of light, its effect of attraction is felt instantaneously, satisfy the works of both great physicists Sir Isaac Newton and Albert Einstein. But the fact still remains that Newton’s laws of gravitation are approximation of Einstein GR in weak gravitational field. But the gravity will continue propagating at velocity of light, 3 x 10^8 metres per second.

                                                               End

Notes:

1.References a) The Speed of Gravity – What the Experiments Say by Tom Van Flandern

b) Does Gravity Travel at the Speed of Light? by Steve Clip

2. a) Image Courtesy
Two-dimensional illustration of how mass in the Universe distorts space-time (Credit: NASA) 

     b) Image (Figure 1) courtesy  The Speed of Gravity – What the Experiments Say by Tom Van Flandern

Writer Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.