Time May Disappear From The Universe

    “As gravitational gamma reaches infinity with ageing of the universe, rate of time will go down to zero. Time will freeze and disappear. Universe will be devoid of time.”

 According to Big Bang model of the universe, huge quantity of energy is necessitated to maintain its density in the wake of newly created space by cosmological expansion. What is the generator of such huge energy, baffles the cosmologists particularly when there is no such source to replenish the energy deficit. Or another situation may arise where the energy density of the universe may decrease at an alarming high rate reaching zero. These are some unexplainable situations associated with Big Bang theory which otherwise is mostly accepted in scientific circles. But quantum model negates the aspect of decreasing oenergy density. According to this model, the energy density is ridiculously high of the range of 10^113 Joules per metre cube which remains constant. 

“Is it the room that is expanding or are we diminishing in size?” as said Alice in “Alice In Wonderland” appears befitting statement here. Quite probable, Spacetime itself may be undergoing transformation and changing its properties in such a manner that these unexplainable questions get answered themselves satisfactorily. Also units and dimensions which are otherwise constant, may be undergoing continuous changes giving the universe relative cosmological expansion.
According to Theory of General Relativity, mass and energy warps the spacetime curvature, giving birth to gravity. Field equations of General Relativity as solved by Schwarzschild, have the term Г known as gravitational gamma which is defined as the rate of time in coordinate clock to the local rate of time in gravity. A stationery clock infinitely far away from the mass in Schwarzschild’s universe is designated as the “coordinate clock” with a rate of time designated dt or one can say, it is rate of time in hypothetical universe with zero gravity. In the same stationary frame of reference, the local rate of time in gravity near the mass is designated dτ.   
Mathematically, gravitational gamma is given by equation,                                                            Г = dt/dτ = 1/[1-2Gm/(c^2.R)]^1/2———-(1)                                                  where G is universal gravitational constant, m mass of particle, R circumferential radius and sign ^ denotes raised to power. In weak gravity, Г approximately equals 1 + β, where β called gravitational magnitude is equal to 2Gm/(c^2.R). Gravitational gamma Г can also be expressed in terms of velocities as Г = 1/[1 – (Ve/c)^2]^1/2————————(2).                                                    where Ve is escape velocity equal to (2Gm/R)^1/2 and c velocity of light. This is analogous to the term gamma γ that appears in Special Relativity where this γ is equal to 1/[1 –(V/c)^2]^1/2. Here only difference in GR is that velocity V in SR is replaced by escape velocity Ve. At the time of Big Bang, since there was no gravity, the term 2Gm/(c^2.R) was 0 and gravitational gamma Г was 1. The maximum value Г at other extreme would be at the event horizon of blackhole as ∞ and the particle would be pulled in the blackhole with escape velocity when 2Gm/(c^2.R) equals 1.                                                                      
Our location in the universe is similar to a cavity surrounded by mass. The universe can be assumed as if surrounded by a shell that is increasing its mass every second due to expansion. If we consider a cavity at the centre of the earth, escape velocity from the cavity would be 22 pc more than that from surface of the earth. This increase in escape velocity in turn will increase the value of gravitational gamma as is evident from equation (2). That means a point surrounded by mass, will experience increased gravitational gamma as the universe ages and mass surrounding it increases. It is submitted that this background gravitational gamma Гu attributed to the matter outside, affects the rate of time, the gravitational potential, the escape velocity etc. However the matter inside the point of reference, affects acceleration due to gravity.                                                                
Gravitational gamma is also equal to dL/dR where R is coordinate length in zero gravity and L is proper length in gravity. Obliviously, increase in the value of gravitational gamma due to increase in surrounding mass, will increase proper length and volume of the universe or in other words cause cosmological expansion. With expansion, surrounded mass increases and this further increases the gravitational gamma. Cosmological expansion can, therefore, be considered as regenerative action which accelerates as the universe ages.                                    
From equation (1), gravitational gamma or Г = dt/dτ. Since Г is always more than 1, dt/dτ will also always be more than 1 or dt is always more than dτ or dτ = dt/Г. That is rate of proper time is equal to 1/ Г time rate of coordinate time. In other words, rate of passing of time is less than the rate of passing of time in hypothetical universe with zero gravity or at the time of Big Bang. Value of gravitational gamma at present is 2.6×10^31* and this indicates that rate of passing of time has slowed down by 2.6×10^31 times from the rate of time as it were at the time of Big Bang. Time, therefore, is passing slower now or in other words, it is dilating. Time to complete an event now will consume 2.6×10^31 times more than the time, it had taken to complete the same at the instant of Big Bang.                                              
But can slowing of time be checked experimentally to remove all doubts? For that, conditions of hypothetical (empty) universe with zero gravity which were prevalent at the instant of Big Bang are required to be simulated for comparison. Simulation of such conditions are not feasible because the universe is homogeneously surrounded by matter and background gravitational gamma is all pervasive. At the most, there may be some places in the universe where gravity is relatively less but effect of background gravitational gamma can not be avoided. No place with zero gravity can thus be located. It is, therefore, not practically possible to check time dilation or slowing of time due to cosmological expansion.                          
Admittedly, light rays from galaxies take more time now to reach us than what these took earlier. These are tell tale signs that the galaxies are now moving farther away from us than their situation earlier. Spacetime model explains this, “As on account of dilation of time, more time is now being taken for the rays to reach us from galaxies and more time justifies the longer distance, hence cosmological expansion.”                                                                                                                         
Up to what maximum limit, can the gravitational gamma enlarge? Gravitational gamma is increasing with cosmological expansion, there is no stoppage and it may finally end up at infinity. At that instant, rate of passing of time will go down to zero.Time will freeze and disappear. Universe would be devoid of time.” After reaching its maximum, gravitational gamma would go back to its initial value 1, the value it had at Big Bang. Again, the evolution of universe might start again.
“Time is running out,” now means time is slowing down and ultimately disappearing.                                           End
Notes:                                                      1. * Detail derivation of figure    is given in the article at https://niume.com/post/271300          2. Image Antenna Galaxy NGC 4038 courtesy https://commons.m.wikimedia.org/wiki/File:Antennae_Galaxies_reloaded.jpg#mw-jump-to-license

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.   

Unseen Side Of Equilateral Triangle 

Ever wondered what the altitude of an equilateral triangle equals other than sine sixty degrees time its side? Perhaps, no as the aforesaid formula is the easiest and the commonest. Quest to know more of it, revealed me the fascinating aspect of trigonometry the other day while I was pedalling bicycle on a forlorn road in the outskirts of Chandigarh, a few days back. I was occupied with calculations, if I drop a perpendicular to the base of an equilateral triangle from its vertex and then drop another from its meeting point with base on its adjacent side, then drop another to the base, again to the side and continue it infinitely, and sum of all such perpendiculars may equal altitude of the main equilateral triangle.
 Another stumbling block in this exercise was whether it was possible to drop infinite perpendiculars as infinite number is not achievable, counting Infinity* is not possible and as such, drawing infinite perpendiculars would also be beyond doable. It was a situation similar to what Zeno faced in his famous footrace between Achilles and Tortoise. At the same time, I was also assured where physical explanation fails, mathematics wins and undoubtedly mathematics is an unsubstitutable tool to solve mysteries of physics. 


On reaching home, with these thoughts in my mind, I took few blank papers and a fountain pen and drew a triangle ABC and dropped a perpendicular from its vertex A to the base BC to meet it at a point D. In this way, another triangle ADC ( here in after such triangles were called subsequent triangles) was formed and from point D, another perpendicular was drawn on side AC meeting it at a point E. Another subsequent triangle DEC was formed and another perpendicular was dropped from point E on DC where it met at point F forming another subsequent triangle FCE, from point F, another perpendicular was drawn from point F upon line EC where it met it at point G. In this way perpendiculars were continuously drawn on the lines BC and AC alternatively. 

Evidently, the perpendiculars continued diminishing in magnitude and nearing point C. If infinite perpendiculars were drawn the way as explained, these would merge with point C. Apparently, these perpendiculars took the form of a Saw tooth waves with diminishing amplitude which ultimately reached zero at the vertex C. But as the number of perpendiculars were infinity, mathematical treatment was supplemented to compute their summation.
From triangle ADC, its altitude AD can be found by use of trigonometry as AD/AC or perpendicular divided by hypotenuse = Sin C, where angle C denotes angle ACB, angle B denotes angle ABC, angle A denotes angle BAC, ‘a’ denotes length of side BC, ‘b’ denotes length of side AC, ‘c’ denotes length of side AB. Therefore altitude of equilateral triangle ABC,

AD = AC. Sin C = b Sin C ……….(1)
Altitudes of subsequent triangles, DE = AD. Cos C = b Sin C. Cos C ………………………………………….(2)
EF = DE.Cos C. Cos C = b Sin C. Cos C. Cos C = b Sin C. Cos^2 C ………(3) where sign ^ denotes raised to power.
FG = EF.Cos C = b Sin C. Cos^2 C. Cos C = b Sin C. Cos^3 C ……(4)
Similarly, GH = b Sin C Cos^4 C …………………..(5)
—————————————————————————————————————————–
On adding these altitudes of subsequent triangles ie DE + EF + FG + GH + HI +……………up to infinity, sum equals
 = b Sin C. Cos C + b Sin C. Cos^2 C + b Sin C. Cos^3 C + b Sin C. Cos^4 C + ……………………….. up to infinity
= b Sin C. Cos C ( 1 + Cos C + Cos^2 C + Cos^3 C +…………… up to infinity )………………..(6). 
It is obvious from equation (6) that it is a geometrical progression with common ratio of Cos C. Magnitude of Cos C from basic trigonometry, is always less than or equal to 1. If Cos C is 1 then angle C becomes 0 degree. In such a situation, triangle ABC will reduce to a straight line and will never be a triangle. In other cases, Cos C is always less than 1, that makes square, cube and higher power of Cos C even further less than 1. Therefore, term of the series corresponding to infinity would be a quantity less than 1, raised to power infinity. And such a quantity reduces to zero, is a well known fact in mathematics. Therefore series as given in (6) is infinite with decreasing magnitude or convergent series and such a series has a finite sum given by formula first term of series divided by (1 – common ratio) or T1/(1-r).
Therefore applying this formula, sum of all altitudes of subsequent triangles                                    = b Sin C.Cos C/(1 – Cos C). = b Sin C.Cos C/(2.Sin^2 C/2). = b. 2 Sin C/2. Cos C/2. Cos C/(2.Sin^2 C/2). = b Cos C.Cot C/2 as Cos C = 1 – 2. Sin^2 C/2 and Sin C = 2. Sin C/2. Cos C/2.

Similarly, drawing perpendiculars of subsequent triangles upon side AB, sum of altitudes of subsequent triangles = c. Cos B. Cot B/2. Total sum of altitudes of subsequent triangles = b Cos C.Cot C/2 + c. Cos B. Cot B/2
Similarly, if perpendiculars are drawn from vertex B, total sum of altitudes of subsequent triangles = c Cos A Cot A/2 + a.Cos C.Cot C/2 Similarly, if perpendiculars are drawn from vertex C, total sum of altitudes of subsequent triangles = a Cos B. Cot B/2 + b Cos A .Cot A/2 
Sum of total altitudes of all subsequent triangles from vertices A, B, C. = b Cos C.Cot C/2 + c. Cos B. Cot B/2 + c Cos A Cot A/2 + a.Cos C Cot C/2 + a Cos B Cot B/2 + b Cos A Cot A/2 = Cos C. Cot C/2. ( b + a ) + Cos B Cot B/2. ( c + a ) + Cos A. Cot A/2. ( b + c ) ———————————–(7)
Sum of main altitudes of triangles drawn from vertices A, B, C = b Sin C + c Sin A + a Sin B. In equation (7), from each vertex altitudes on both sides have been taken, therefore, twice the sum of altitudes of triangle drawn from vertices A, B, C = bSin C + c Sin B + c.Sin A + a.Sin C + a. Sin B + b. Sin A. = Sin C (b + a) + Sin B (c + a) + SinA ( a + b) —————-(8). 

 If the triangle ABC is equilateral then each side is equal and each angle at its vertices is also equal (to 60 degree). That is a = b = c = l say. On putting these values in equation (8), it becomes = 2 ( l Sin 60 + l Sin 60 + l Sin 60 ) = 6.l. square root 3/2 And each altitude becomes equal to 1/6 of 6.l.square root 3/2 or l.Sin 60 or l.square root of 3/2 ———–(9). 

 On putting values of a = b = c = l and angles A= B = C = 60 degree in equation (7), sum of total altitudes of all subsequent triangles from vertices A, B, C, becomes = 6.l.1/2. Square root 3. Drawn from one vertex, sum of altitudes of infinite subsequent triangles = (1/6).6.l.square root 3/2 = l. square root 3/2—————-(10). 

 This magnitude in equation (10) is equal to altitude of equilateral triangle as derived in equation (9). Hence altitude of an equilateral triangle is equal to sum of altitudes of infinite subsequent triangle. Conversely also, to make equation (7) equal to equation (8), coefficients of (b+a), (c+a), (a+b) of both equations must be equal. 

 That is Cos C. Cot C/2                                                                                                                                        = Sin C or Cot C/2                                                                                                                                                   = tan C or tan C = cot (90-C/2) or C = 90-C/2 or C = 60 degrees.                                                      Similarly, Cos A. Cot A/2 = Sin A or A = 60 degrees. And Cos B. Cot B/2 = Sin B or B = 60.                                                                                                 Therefore conversely, if each angle of triangle is equal to 60 degree or triangle is equilateral, then its altitude is equal to sum of altitudes of infinite subsequent triangles.

Also if altitude of a triangle ABC, drawn from vertex A is greater than sum of altitudes (drawn on sides AC and BC) of infinite subsequent triangles then Sin C > Cos C. Cot C/2 from equations (7) and (8). Or tan C > Cot C/2. Or tan C > tan (90 – C/2) or C > 90 – C/2. Or C > 60 degree.              That means if at vertex C of a triangle, angle is more than 60 degree than amplitude of triangle will be larger than sum of altitudes (drawn on sides AC and BC) of infinite subsequent triangles. If this angle is less than 60 degree then altitude of triangle will be less than sum of infinite altitudes (drawn on sides AC and BC) of infinite subsequent triangles. If angle at vertex say C, is 90 degree then no subsequent triangles will be formed and altitude of triangle will be considered highest. If angle at vertex say C is more than 90 degree then sum of altitudes of infinite subsequent triangle will be negative as these are drawn on side BC which is extended.
From this it is conclusively proved that sum of altitudes of subsequent triangles is a measure which on comparison with main altitude, indicates the magnitude of angle at that vertex. If from all vertices, sum of altitudes of subsequent triangles equals to altitude of main triangle, then it is equilateral. 

Next time, when someone asks you what angle a line makes with amother line in a given figure, draw five, six or perpendiculars as explained above and add their magnitudes. If the sum is more than main altitude, the angle is less than 60 degrees. If it is less, angle is more than 60 degrees and if equal, it is 60 degrees. Necessity of memorising the values of sine or cosine of different angles is eliminated as far as assessment of an angle is limited to finding its value whether more or less than 60 degrees is concerned.                                                 End   

Notes: 1.Image credit to https://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/1136px-Triangle.Equilateral.svg.png. 2. Counting Infinity* is available at https://niume.com/post/281021                                                      

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 short stories and also on subjects, Astronomy, Mathematics, Yoga, Humanity etc.                                                           

Counting Infinity


In Science and Mathematics, Infinity that denotes a quantity which is ever increasing, enlarging, getting bigger and bigger to the extent it can not be measured or mentioned by numerical figure, finds repeated use in Science and Mathematics. This quantity which is endless, numberless and not finite, was assigned the symbol ∞ by Great mathematician, John Wallis* to denote it for its use in problems of mathematics. How large infinity can continue to grow, is the extent of ones imagination of the the largest quantity and it surpasses that as it is bigger than what one imagined. It is larger than the largest.

Defining Infinity In Simple Terms

Consider an apple which is cut in pieces of one fourth size, each piece then can be distributed to four persons. If the apple is sliced to pieces of the size 1/100000, it can be distributed to 100000 persons. Likewise, if each piece is cut to very very small size, it can be given to very very large number. As the size of the piece reduces, number of persons served increases. If each piece is cut to very very small or to extent of zero size, it can be distributed to persons as large as ones imagination and beyond and that number would be infinite. Infinity is thus defined as a quantity divided by zero. Mathematically, ∞ = Q/0 where Q is any quantity other than 0.

                             Twin Jet Nebula Resembling Infinity

Multiplying Or Dividing infinity With Any Quantity.

Since infinity is beyond imagination and endless, nothing can be bigger than it, two times ∞ will again be endless or ∞.Three times ∞ will again be as endless and large as ∞. Or any quantity multiplied with infinity results in ∞. Conversely, infinity divided by any quantity is infinity.

Mathematically, n.x ∞ = ∞. And ∞/N = ∞ where N is any quantity other 0.

If anything is added to endlessly growing quantity, it will remain endlessly growing. Similarly, anything is subtracted from endlessly growing quantity, it is not affected and remains same. Mathematically, n + ∞ = ∞ and ∞ – n = ∞.

If endless quantity is multiplied with another endless quantity, once or twice or any number of times, it remains endless or infinity.

Mathematically, ∞ x ∞ = ∞, ∞ x ∞ x ∞ = ∞,∞ x ∞ x ∞ x ∞……….any number of times = ∞.

If infinity is divided by infinity, resultant quantity is in determinant, If infinity is subtracted from infinity, it does not result in zero, since infinity is endlessly growing and never attains a finite value and two endlessly growing quantities if subtracted from one another, results in any quantity to ones imagination. Therefore, subtraction of infinity from infinity is in determinate.

Mathematically, ∞/∞ = In determinate, ∞ – ∞ = In determinate.

 However, negative infinity exists and will be explained in appropriate paragraph in this article.

 Time Arrow Paradox

Zeno of Elea, a pre-Socratic Greek philosopher of southern Italy, considered when an arrow is shot from a point A1 to hit target A∞, at any instant, during its flight, arrow will be found stationary because the instant does not have any magnitude of time variation. No time variation means no displacement. Accordingly, if the arrow is observed at any other instant, it would again be found stationary or at rest. There will be such infinite instants during flight of arrow and at each instant, it would be observed as if it were at rest. But in fact, the arrow during its flight, always remains in motion and also has also specific time of flight. But there would be infinite instants during flight of arrow as each instant has zero time duration. If that be so, its time during fight should be zero as instants of zero time duration added infinite times would always be zero. But there is finite time of flight. This self contradicting situation is called Time Arrow Paradox.

In the example of slicing of apple in to infinite parts, each part had zero size. In that case also, if one adds all parts of zero size, these must restore to the original size of apple from which these were cut but zero added infinitely remains zero. Infinity defined in foregoing paragraph was any quantity divided by zero. Also infinite parts of size zero, do not add up to result in original quantity. There is then something amiss and it needs further explanation to define infinity and solve Time Arrow Paradox.

Achilles And Tortoise Paradox

Zeno considered another case of a footrace between Achilles** and tortoise. Achilles challenged tortoise of a footrace and gave him a head start of half distance and both started at the same instant Achilles from point A1 and tortoise from mid point A2. When Achilles reached point A2, the tortoise, in the mean time, had gone ahead to point A3. When Achilles reached point A3, the tortoise had gone ahead to point A4. In this way, as Achilles reached the point left by the tortoise, the tortoise had gone ahead in the intervening period. Thus, there were infinite such points left by the tortoise which had to be covered by Achilles to overtake the tortoise. Crossing infinite points left behind by the tortoise was not physically possible as infinite points being larger than the largest number, can never be covered by Achille . But Achilles practically overtook the tortoise. There was thus contradiction between theory and practice. This contradicting situation is called Zeno paradox.

The situation that Achilles would overtake the tortoise as correct, means Achilles had crossed infinite points left by the tortoise. Is crossing infinite points by Achilles correct when it is known that infinite means larger than the largest which is beyond our imagination?

Mathematical Examination Of Achilles Tortoise Paradox

This paradox is examined mathematically by considering Achilles runs the distance 1/2, in the mean time, the tortoise runs 1/4 ahead, Achilles crosses that ¼ left by the tortoise, the tortoise in the meantime runs 1/8 more so on and it continues up to infinity.

   A1——————->A2———–>A3——–A5—->A6– -A ∞

   (Achilles).                             (Tortoise)

       |——–1/2———-|—-1/4——–|—-1/8—-|-1/16–|– -|

The distance covered by Achilles can be written mathematically as

 1/2 + 1/4 + 1/8 + 1/16 + ——————- so on up to infinity.

Inspection of these terms reveals that it is a Geometrical Progression whose terms are decreasing in magnitude endlessly. As second term 1/4 is less than first term 1/2, third term 1/8 is less than fourth 1/16 so on and has a common ratio r, ½ which is less than 1, such an infinite terms series is called an infinite convergent series and has a finite sum.

Computer Fails

If a computer programme for adding these terms ab initio without using mathematical formula for summation of such series, is fed, the computer will take infinite time to solve it because it will calculate the magnitude of each terms which extend to infinity and then will add up. Calculation of magnitude of endless terms will take endless time and endless time means an unaccomplishing job. The computer will continue calculating endlessly. If the computer is commanded to state the intermittent status, it will display, “calculating.” That means, summation of infinite terms abinitio without use of mathematical formula, is impossibility for computer and is in conformity with prediction of Zeno.

Summation Of Convergent Infinite Series Using Mathematical Formula

However, such an infinite convergent series can be summed up by simple formula a/(1 -r) where a is the first term and r is mean ratio. In the above series, a = 1/2 and r = 1/2. Therefore its sum or distance covered is 1/2. [1/(1-1/2)] which equals1. In other words, impossibility of crossing infinite points by Achilles is solvable by a simple mathematical formula. Zeno paradox if examined from the point of view of performing infinite acts, is a physical impossibility but it is not so, from the point of view of mathematics. That leads to the concept, though infinity can not be determined yet under certain situations infinite acts can be performed resulting in finite output.

There are number of infinite convergent series which sums up to finite value. One example is given below.

Natural number, e = 1+1/2!+1/3!+1/4!+……………..up to infinity.

Here also infinite terms are summed up to give value e where ! is sign of factorial. All decreasing infinite series not necessarily adds up to finite sum. A series given below does not have a finite sum although magnitude of terms is decreasing.

1+1/2+1/3+1/4+1/5………………….so on up to infinity. It does not have finite sum as sum of its terms is infinity.

But series 1-1/2+1/3-1/4+1/5………so on up to infinity, is a convergent series and sums up to log 2. Here also infinite terms have been summed up to give finite result. Thus endless repetitive acts or infinite acts are possible under certain circumstances.

Infinitely Repetitive Number And Infinity.

Coming to the case of Achilles and the tortoise with regard to summation of the terms of the series by a computer, if the computer is commanded to give result at the stage where it was, the result displayed would be 0.999999999999999999999999—— so on and it will never be 1. But it is seen above that summation of this series results in one. Sum as 1 will only be achieved if the computer is let continue summation till infinite time. Achieving infinite time is impossible. In ideal case, sum or distance covered as displayed by computer would be

0.9999999 repeating 9 up to ∞.

D = 0.99999—————– up to ∞, where D is the total distance.

Or D = 0.9 + 0.0999999999999 —– up to ∞

Or D = 0.9 + D/10

Or 0.9.D = 0.9

Or Distance D = 1.

It is deduced from above that computer would display the distance as 0.999999999999 —– repeating 9 very large times. If infinity were achievable, the display would have been 1. Infinity thus corresponds to the number of times 9 repeats after decimal point to make the figure1.

Irrational Number And Infinity

It is not true with the figure 0.9999 —— up to infinity only to define infinity but it can also be extended to any irrational number. An example of irrational number 1/3 is given here. 1/3 is generally written as 0.33 or 0.33333 but its actual value is different. Strictly speaking, 1/3 is 0.33333——-with 3 repeating up to infinity. Again infinity can be defined here as the number of times 3 must repeat after decimal point to make the figure equal to 1/3.

Consider another case where say y = 3.535353535 ———up to infinity. On examination, it is clear that 35 is repeating up to infinity. If y is divided by 100, then y/100 = .0353535——–35 repeating up to infinity

Or y – y/100 = 3.5 or y = 350/99. Though 35 was appearing in 3.5353535 infinitely but it has a finite value equal to 35/99. Infinity can therefore be defined as number of times 35 should repeat after 3.5 to make the results ant figure equal to 35/99. In this way, infinity can be defined by each irrational number.

Defining Infinity From Mathematical Terms Extending Infinitely

Consider a fraction 2 + 1/(2 +1/(2+ 1 which continues infinitely. Here also, infinite operations are required to solve it and it is known performance of infinite operations is impossibility.Therefore, this fraction involving infinite terms can not be solved by the method of elementary additions.

However this can be written as y = 2 + 1/(2 +1/(2+ 1 continuing infinitely

Or y = 2 + 1/y since the terms of y are repeating infinitely and are equal to y itself.

Or y^2 – 2.y – 1= 0 where sign ^ indicates raised to power. This quadratic equation is solvable for y and gives values as 1+ square root of 2 or 1 – square root of 2. Value 1 – square root of 2 being negative, will have to be ignored as y can not have negative value. That leaves us with solution as y equal to positive value ie 1+ square root of 2. Therefore fractions involving infinite numbers are solvable.

Consider another case where x is less than 1 and is raised to power ∞.

Let y = x^∞ = x.x.x.x.x.—————up to infinity where x is positive and less than 1. As x is multiplied by x, it becomes x^2 which has value further less than 1. Again if x^2 is multiplied by x, resultant x^3 has value further less than 1. If it x is continuously multiplied by x up to infinity, value of product x^∞ will diminish to zero. Infinity can thus be defined as that power of a positive quantity less than 1 which reduces it to zero.

In mathematics (algebra, calculus, integration etc) operations of ∞ in certain cases are allowed and are possible and give finite result also. But operations of ∞ are not permissible as rule in all cases.

Negative And Positive Infinity

Negative and positive infinity is used in solving mathematical and physics problems. Negative infinity is different from positive infinity in the sense that it is assumed to extend in the direction opposite to the direction of positive infinity. Physically, positive infinity with regard to length of a line can be visualised as a line going vertically up endlessly and negative infinity as a line going downward endlessly. If magnetic field due to a current carrying line conductor of length l is to be calculated at a point situated at a perpendicular distance d from the centre of conductor, an infinite small length delta x is considered along the conductor to calculate its magnetic field, then the magnetic field due to length l is calculated by integrating the field with respect to distance varying from -1/2.l to +1/2.l. But if the length is infinitely long, the limits will change to from – ∞ to + ∞.

Limitation Of Length

While defining infinity earlier, cutting apple endlessly till size of each piece reduces to zero thus making the number of pieces infinite was considered. Mathematically, it is possible to cut the piece to the size as small as zero but laws of physics do not allow to reduce the size of a piece further below a fixed length. Below that limit, length is indivisible. This limiting length is Planck length which is equal to 1.6 x 10^-35 m. Therefore, physics does not allow division of length infinitely but stops further subdivision at Planck length. However, mathematics allows subdivision of length below Planck length even up to zero. Also in case of Achilles, Tortoise race, when the tortoise ultimately runs ahead by Planck length and Achilles thereafter covers this length, the tortoise is stopped by laws of Physics to run further. That point is the finishing point and the act of running ahead by the tortoise is not infinite. But what Zeno said was mathematical, not precluded by Physics. According to mathematics, acts of reaching the point left by the tortoise were infinite as is evident from infinite terms of the series. But Physics limits it to that number where the final distance to be covered by the tortoise is reduced to Planck length. Therefore length consists of small units of indivisible Planck length. Limiting the minimum achievable length to that of Planck, makes the number of acts finite. But mathematics does not have any such limitations and the acts can be infinite but such infinite acts are possible as the resultant convergent infinite series was summable.

Limitation Of Time Duration

Again coming to time arrow paradox, it is submitted, Zeno considered an instant (of time) is of zero second and infinite instants constitute the time of Arrow flight as of zero time. As instant has zero time variation, the arrow could not be seen moving in zero time variation and would appear stationary. Therefore at all such instants of time during flight, the arrow should be stationary. Physics does not allow division of time below the limit of Planck time, 5.39 x 10^-44 second. Duration less than Planck time will tantamount to zero duration of time. Therefore, in this case, time duration of instant can be minimised to Planck time but can never be zero. Thus time of flight is quantised or consists of smallest duration of time which is Planck time that is indivisible. Number of instants would be finite equal to time of flight divided by Planck time and will not be infinite. However, mathematics allows infinite instants where each instant corresponds to zero time. But Physics is based on laws of nature and allows motion of Arrow in an instant of time which can nit be less than Planck time.

Infinity And Metaphysics

By now it is abundantly clear, mathematics allows endless quantity bigger than the biggest one can imagine of. No matter infinity is endless, it can be visualised even in small number because, it is perpetual endless continuation of an act. For example, even between 0 and .1, infinite fractions exist like .01, .001, .0001, .00001, .000001,………….so on endlessly. Infinity can be visualised as extending endlessly beyond observable universe of 13.8 billion light years. Infinity can also be considered as endless points of zero volume confined to the volume of photon as infinite points of zero volume do not constitute any volume. Nevertheless, infinity pertaining to infinite points which can be accommodated in a photon is no smaller than infinity pertaining to infinite length extending beyond the observable universe. Thus infinity from the perspective of Infinite points those can be accommodated in a minuscule particle is as good an infinity as that from the point of of view of endless length beyond observable universe. Infinity does not have beginning and also does not have end, it is endless. It can not be eliminated, it remains same if it is multiplied with, divided by, added to, or subtracted from any quantity, it is always unaffected. No physical parameter can change it.

These are all the qualities which one and only one possesses and He is Almighty who is omnipresent, indestructible, unborn, immortal, unaffected by space or time. Infinity is synonymous with God. Nature represents it in the form of infinity nebula. Infinity will continue endlessly unstoppably, unfathomably, unmeasurably even when the time stops.                                            

                                                             End

Notes: 

1 * John Wallis was an English mathematician who gave symbol ∞ to denote infinity. He also defined 1/∞ as infinitesimal. He also gave formula to calculate the value of pi.

2** Achilles was a great Greek warrior and had his heel as vulnerable part of his body. His mother dipped him with feet in her hands in holy river Styx to immortalise him. The feet which remained out of holy water became his vulnerable body parts where he was hit by Alexander Paris. Archives died of wounds at his heels and that is why vulnerable part is called, “Achilles’ heel.”

3*** Tortoise is a personification of sluggishness and slowness.

4. Image Courtesy  https://en.m.wikipedia.org/wiki/List_of_planetary_nebulae#/media/File%3APlanetary_Nebula_M2-9.jpg

https://en.m.wikipedia.org/wiki/Zeno’s_paradoxes

5. I have not dealt bigger infinity and smaller infinity from the point of view of set theory as propounded by Cantor.  

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 short stories and also on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
 
 

 

 

Force Is Nothing But Asymmetry In Quantum Vacuum


Galaxies, stars, planets, you name an object randomly that exists in the universe, all are moving. Some are receding, some orbiting around, some approaching the other. Nothing is at absolute stationary state. Life will come to halt if the motion ceases. Have you ever thought, behind every such motion, there is a force which keeps the dynamicity of the universe intact? That universal force is nothing but it is the asymmetrisation of quantum vacuum. Quantum vacuum an ever pervading ocean of infinite virtual energy density is omnipresent, is symmetrical and has the maximum entropy. To harness observable energy from it, summery needs interference in the form of its alignment. This process is called symmetry breaking, making it aligned and orderly by reducing its entropy called negentropy. Now when I am using iPad, electrochemical energy in its battery is asymmetrising the quantum vacuum and this asymmetric Spacetime of quantum vacuum is creating the electromagnetic force. This electromagnetic force moves the electrons in its circuitry and functionalises the device. All forces in the universe are manifestation of asymmetric spacetime of quantum vacuum. In other words, it is the asymmetry of spacetime of quantum vacuum that generates force and moves the universe.

Then what is this quantum vacuum and what is its symmetry! Vacuum is presumed to be void, empty or nothing but is in fact, everything. According to quantum mechanical model, vacuum is filled with ‘virtual particles’ and ‘anti virtual particles’ which emerge into existence from nowhere, stay for a brief time and then disappear. In the words of John Archibald Wheeler, “Empty space is not empty… The density of field fluctuation energy in the vacuum argues that elementary particles represent percentage‐ wise almost completely negligible change in the locally violent conditions that characterise the vacuum.”

 The virtual particles are not real and have virtual energy that is not observable. Observable energy can be felt and utilised. But these virtual particles can be transformed to real or observable particles on interaction with observable objects. Quantum vacuum is spread infinitely where annihilations of virtual particles and virtual antiparticles take place randomly. In other words the space time on the whole, is spread uniformly all along randomly. This vacuum is symmetrical and non aligned. 

According to Professor Stephen Hawking, the meaning of quantum fluctuations is that “even empty space-time is filled with virtual particle pairs. These virtual particle pairs emerge from space- time, then the two charges move a little bit from each other and then again come together. After that they annihilate each other. One partner of the virtual particle pair has positive energy, while the other partner has negative energy.”

Quantum vacuum has the lowest observable energy but that does not mean, the vacuum does not have energy. On the contrary, the quantum vacuum has energy density of 10^113 Joules per metre cube and fluctuations in energy density can be observed at distances in the range of Planck length and these follow uncertainty principle. There is a simple reason why we do not observe this huge density energy as it is completely evenly or symmetrically distributed and has no quantised momentum whereas energy with quantised momentum gives birth to fermions and Bosons building of observable matter.This energy has the properties of superfluidity which does not interact. Thus, the quantum vacuum describes a lowest or zero (observable) state of energy and has maximum symmetry. For details on energy density of vacuum, you may read article , “Resolution of largest numerical discrepancy in all of physics.”* Symmetry  is a state where the virtual particles are orientated unevenly, randomly and has maximum entropy.  

                Figure 1,   Alignment of virtual particles by breaking symmetry of Spacetime 

When an electron is kept in vacuum, the virtual particles which have polarity of positive and negative charge, rotate and align themselves in such a way that positive charge faces the electron and there after second virtual particle’s positive charge side faces the negative charge of the first particle and so on. Kindly see figure 1 above. This alignment is accomplished in such a way that number of particles nearest to the electron is more or denser than number of particles away from it. This reduction in density is inversely proportional to the distance squared from the electron. The distribution of virtual particles is in dynamic equilibrium meaning thereby that distribution remains same although some virtual particles appear and disappear on annihilation. Formation of particle, antiparticle pairs, their subsequent annihilation and reduction in density with density away from the electron, tantamounts to radiation of energy from the electron. It is submitted that a certain length and volume of a virtual particle so aligned in space-time can be considered as a classical photon.

It is reiterated that the charge of the electron has aligns the particles in such a way that that the degree of alignment goes on reducing as the distance increases from the electron. Therefote the Spacetime near the electron is asymmetrised more than the spacetime away from the electron. Thus asymmetry goes on decreasing with the distance away from the electron. The electron has broken the symmetry but at a large distance the symmetry is still intact. Thus there is a gradient of asymmetry as the distance increases from the electron which is its static aspect but annihilation of the particles are still taking place which is its dynamic aspect of the electromagnetic field. The situation is similar to a waterfall which when flowing is in dynamic state but when frozen also exhibit the same situation as that of running although it is in static state. It is thus concluded, the alignment of the particles near the charge is dynamic in nature but also fulfils static conditions.            Figure 3, Symmetric and Asymmetric Spacetime of quantum vacuum.

How the asymmetry of Spacetime causes change in pressure is explained by the example of a diver who goes down deep in water. But he does not experience any pressure on his body underwater although he is quite deep down. Explanation for this phenomenon is simple, pressure inside equals that outside his body or in other words pressure inside his body equals outside and cancels each other being acting on opposite directions. But when he ties a pressure gauge around his wrist and dives underwater, the gauge indicates pressure. Explanation again is sample, in this case, only pressure due to water head is being measured. In quantum vacuum also, if the Spacetime is symmetrical or the particles are not aligned in Spacetime, there is no flow of potential but if it is asymmetric, particles are aligned and even, there is flow of potential. Broken symmetry leads to flow of potential which creates a real force and acceleration upon a real particle. Thus, a real force and energy only emerge when a broken symmetry interacts with a real particle. Just as the difference in water pressure inside and outside the pressure gauge causes the pointer to move, the broken symmetry in space-time creates a force depending upon difference in energy density. So a virtual particle’s virtual energy receives an observable energy characteristic only when it interacts with an observable particle. That is how every atom receives its energy from the volume of space-time occupied by that atom. More the asymmetry, more the potential flow and more the pressure difference associated with quantum vacuum. Conversely, more the symmetry, less the potential flow and less the difference of pressure associated. 

It is evident from above that for creation of force, the quantum vacuum will have to be asymmetrised. How can then quantum vacuum  be asymmetrrised?  It is submitted that such a situation can be created by any object as all objects consist of atoms which have electrons in their orbits. Their orbital electrons facilitate alignment of the particles of quantum vacuum thus breaking the symmetry. Such objects are called passive objects.
Figure 3, Force of attraction Asymmetry when plates are kept near each other in Casimir Effect.

 If two conducting plates are brought near each other in vacuum, the electrons of the plates align the particle polarity in its proximity thus asymmetrising the quantum vacuum near it. Both the plates in this experiment show force of attraction.This force attraction is created by a difference of energy densities, which exist between the inner and outer sides of the plates due to asymmetrisation.

       Figure 4, No Force of Attraction, no asymmetry when plates are kept at a large distance 

But if the plates are laced at a large distance apart, the asymmetrisation gets reduced at longer distance and no force of attraction is observed.This phenomenon of attraction between two conducting plates kept near each other is called Casimir Effect. This is a good example of transforming virtual energy to observable energy and creation of observable force.

Figure 5, Assymetrisation of spacetime of quantum vacuum by battery,

In case of a battery, electrochemical energy of the cells, aligns the particles in vacuum and this asymmetrises the quantum vacuum. Extent of asymmetry of quantum vacuum is the indicator of voltage of the battery. The batteries are, therefore, active objects for breaking symmetry of quantum vacuum. 

Summing up, broken symmetry causes difference in energy density in Spacetime and this results in creation of force. All observable forces whether mechanical, electrical, magnetic, electrochemical, electromechanical or of any type in the universe are creation of asymmetrisation of Spacetime of quantum vacuum. We asymmetrise the quantum vacuum and that creates force to perform our activities. We are not performing the activities but are breaking symmetry of the quantum and that asymmetrisation is translating into action but certainly we are the causers as without causer, asymmetrisation can not take place. If the Spacetime is symmetrical and not aligned, no observable force will be created. We have vacuum energy density of 10^113 joules per metre cube which is a huge quantity keeping in view the fact that the universe has 10^80 particles only. Surprisingly but truthfully, energy density of observable universe is infinitesimal small of the range of 10-9 joule per metre cube. Transforming virtual energy to observable as in case of Casimir Effect is the need of the hour. I learnt that Reid Cell which transforms virtual energy to observable by asymmetrisation of quantum vacuum, has already been tested. Hopefully, vacuum energy would be the best candidate for making up energy deficit.

                                 End.

Notes:

1) Reference: Virtual particles in electromagnetism by Marcus Reid. All figures courtesy Marcus Reid. 

2) * Article, “Resolution of largest numerical discrepancy in all of physics,” is available at https://niume.com/post/267726. 

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 short stories and also on subjects, Astronomy, Mathematics, Yoga, Humanity etc.

 

 
 

   

 

 

 
 
 

 

 

 
        

 

  

Metre Stick InThe Universe Is Contracting By .00000000000000000229 Metre Per Second


 General Relativity lays down that energy warps the spacetime curvature which gives rise to gravity. Spacetime is smooth on macroscopic scale and there are no fluctuations as are observed in quantum mechanical model at Planck length.Minimum distance between two particles is limited to the extent so as not to forma a blackhole. Field equations of General Relativity were solved by Schwarzschild and the solution has the term gravitational gamma Г defined as the rate of time in coordinate clock to the local rate of time in gravity and this is given by equation  
Г ≡ dt/dτ = 1/[1-2Gm/(c^2.R)]^1/2—————(1)

where G is universal gravitational constant, m mass of particle, R circumferential radius and sign ^ denotes raised to power. A stationery clock infinitely far away from the mass in Schwarzschild’s universe is designated as the “coordinate clock” with a rate of time designated dt or one can say that it is rate of time in hypothetical universe where there is zero gravity. In the same stationary frame of reference, the local rate of time in gravity near the mass is designated dτ.

In equation (1), the term 2Gm/(c^2.R) is always between 0 and 1 otherwise denominator of this equation will become imaginary quantity and that is impossibility. If 2Gm/(c^2.R) is small then gravitational field is weak and Г can be expanded by Binomial Theorem as           Г ≡ dt/dτ = [1-2Gm/(c^2.R)^-1/2]                                                                                                                 = 1 + ½.2Gm/(c^2.R)+ —                                                                                                      ≡1+Gm/(c^2.R) + neglecting other terms, being small in value.                                                        ≡ 1+ β, where β is Gravitational Magnitude equal to Gm/(c^2.R).                                                                   Gravitational gamma Г can also be expressed in terms of velocities as                                          Г = 1/[1 – (Ve/c)^2]^1/2————————(2)                                                                where Ve is escape velocity equal to (2Gm/R)^1/2 and c velocity of light. In special relativity also, the term gamma γ appears and is equal to 1/[1 – (V/c)^2]^1/2. Here only difference in GR is that velocity V in SR is replaced by escape velocity Ve.

 It is submitted that at the time of Big Bang, there was no gravity and the term 2Gm/(c^2.R) was 0. Therefore at that time, gravitational gamma Г was 1. The maximum value Г at other extreme would be at the event horizon of blackhole as ∞ and the particle would be pulled in the blackhole with escape velocity when 2Gm/(c^2.R) equals 1. 1 + 7x 10^-10 . 

Our location in the universe is similar to a cavity surrounded by mass. The universe is surrounded by shell that is increasing its mass every second due to expansion. If we consider a cavity at the centre of the universe, escape velocity from the cavity would be 22 pc more than that from surface of the earth. This increase in escape velocity in turn will increase the value of gravitational gamma as is evident from equation (2). That means a point surrounded by mass, will experience increased gravitational gamma as the universe ages. It is submitted that this background gravitational gamma Гu is attributed to the matter outside and affects the rate of time, the gravitational potential, the escape velocity etc. However the matter inside the point of reference, affects acceleration due to gravity. This increase in the value of gravitational gamma due to surrounding mass, is causing the cosmological expansion.

Present value of gravitational gamma can be calculated from various methods but its value here is calculated from cosmic microwave background CMB temperature. Assuming the model where the universe started from Planck sphere at Planck temperature Tps ( ½ Ep/k ) 7.08 x 10^31 degree K and cooling down to its present temperature Tcmb 2.725 degree K, background gravitational Гu can be calculated as Tps/Tcmb = 7.08 x 10^31/2.725 = 2.6 x 10^31. 

In GR, scaling factor au which equals Гu is defined as dL/dR where R is coordinate length in zero gravity and L is proper length in gravity. At Planck Spacetime au and Гu are 1, coordinate length R is equal to proper length L. When a unit of proper length L is smaller than a unit of coordinate length then progressing in the radial direction the rate of change in proper length dL will be greater than the rate of change in proper length. Since Spacetime is homogeneous and isotopic, progressing in any direction will lead to same result. From above, scaling factor or gravitational gamma can be written in length ratio as au = Гu = R/L. This equation clarifies that increase in gravitational gamma Гu decreases proper length whereas R is constant being length in hypothetical universe with no gravity. This contraction will be all pervasive meaning thereby that it will affect dimensions of all objects including standard unit of length.  

Rate of length contraction with time depends upon rate of change of gravitational gamma with respect to proper time dГu/dτ. If rate of change of gravitational gamma with respect to proper time dГu/dτ is divided by observed value of gravitational gamma Гuo, it becomes equal to rate of expansion divided by total expansion or (dl/dt)/l or v/l where v is velocity and l its length at that velocity. It is, in fact, the definition of Hubble’s constant H.                                    In other words, (dГu/dτ)/ Гuo = v/l = velocity of light/length of observable universe = H————————(3).                                                                                                                   Velocity of the farthest receding galaxy can approach velocity of light c and maximum distance at which this farthest galaxy is situated, is the observable length of the universe (about 13.7 billion light years). On substituting these values in equation (3),                             H = .00000000000000000229 metre per second per metre. Or rate of metre stick contraction (dГu/dτ)/ Гuo = .00000000000000000229 metre per second.                                                                                                     

Contraction of metre stick appears counter intuitive in view of accepted theory of cosmological expansion. The contraction in length can only be measured if the present day metre stick is compared with standard coordinate length of metre in zero gravity.There are locations in the universe where the gravity is very feeble approaching zero but such locations are still surrounded by shell of matter and that has the effect of background gravitational gamma. Thus there is no location in the universe where there is zero gravity and such universe can only be hypothetical beyond our reach. Given the situation, a metre stick can not be checked with metre stick of universe with zero gravity and contraction can never be confirmed.

Can then contraction in metre stick be checked from primordial metre stick if we happen to locate it? If we could locate such metre stick, that primordial metre stick would have also experienced the contraction and would have shrunk to the present metre length. This comparison is also not feasible. Spacetime transformation model convincingly proves red shift that is tell tale sign of cosmological expansion. Spacetime transformation also proves shrinking length as the universe ages. It will have to be examined whether it is the shrinking unit of length that proves cosmological explanations or there is some other phenomenon responsible.

(An isometric embedding of part of the visible universe over most of its history, showing how a light ray (red line) can travel an effective distance of 28 billion light years (orange line) in just 13 billion years of cosmological time)

According to Spacetime transformation and as discussed above, the unit of length or metre stick is contracting by .00000000000000000229 metre per second. This length contraction is inflating the multiplier of unit of length giving effect of relative expansion. It is submitted that Spacetime transformation with increasing gravitational gamma, also increases the time in conformity with equation To = Tu/ Гu. Since resent Гu is equal to 2.6 x 10^31, the time has dilated by this factor. Velocity of light is fixed but the time taken by the light rays to reach from the galaxies has increased by afore stated factor, this translates to increased distance or cosmic inflation. Simultaneously, unit of metre has contracted, this multiplies the expansion figure. These transformations do not change laws of physics which remain same. Shrinking unit of length, increase in time (or time dilation) get corroboration from Hindu Religious Scriptures also.

Bhagavata Purana, Vishnu Purana and Garga Samhita narrate the tale of Revati* the only daughter of King Kakudmi who took her to Brahmaloka for seeking advice from Lord Brahma for suitable groom. Lord Brahma was busy listening to Gandharv music so they waited patiently until the performance was finished. Then, Kakudmi presented his shortlist of candidates. Lord Brahma laughed loudly and explained that time runs differently on different planes of existence and that during the short time they had waited in Brahmaloka to see him, 27 chatur-yugas or 27 ages (a chatur-yuga is a cycle of four yugas, hence 27 chatur-yugas total 108 yugas, the Garga Samhita differs and says 27 yugas in total) had passed on Earth and all the candidates had died long ago. Lord Brahma then recommended Balram brother of Lord Krishna as worthy husband of Revati. On returning to planet Earth, Kaudmi and Revati were shocked to find many ages have passed during short period of their tour to Brahmaloka and the landscape and environment had changed on earth and the race of men had become short in stature, reduced in vigour, and enfeebled in intellect. These scriptures thus supports slowing of rate of time in Brahmloka due to high gravitational gamma and also reduction in height (of man) with ageing of the universe.

Properties of Spacetime transformation is contracting metre stick by .00000000000000000229 metre per second and has increased time by gravitational gamma 2.6 x 10^31 since the universe was one Planck time old. 

                                             End

                      Notes:

                      1) Reference: Universe Is Only Spacetime john@onlyspacetime.com

                     2) * https://en.m.wikipedia.org/wiki/Revati

                      3) Sign ^ denotes raised to power e.g. 10^3 = 10x10x10.

                      4) Image courtesy   https://en.m.wikipedia.org/wiki/Metric_expansion_of_space#/media/File%3AEmbedded_LambdaCDM_geometry.png

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.    

Resolution Of Largest Numerical Discrepancy In All Of Physics

In the observable universe, number of particles are estimated to be 10^80 and if there were some discrepancy in Physics with the explanations of the observable universe or with its each particle then it should confine to factor  10^80. I submit that 10^80 is a huge figure that forms if one puts eighty zeros after 1. But if the discrepancy is of the factor 10^120 then either it is beyond the total number of particles constituting the universe or the physicists might have gravely erred in their calculations. It might be a freak happening that resulted in such a huge quantity. After all, freaks are also the creations of nature or probably the nature itself has erred here.This discrepancy of 10^120 is the largest and worst cosmological confusion which can be abbreviated CC and rightly so for cosmological constant as it is the cosmological constant based on Quantum mechanical model. Quantum mechanical model says, energy density of the vacuum is in the the range of 10^113 Joules per metre cube whereas General Relativity calculates it in the range of 10^-9 Joule per metre cube. An attempt is made to resolve this discrepancy using Spacetime transformation and gravitational gamma  Г. Gravitational gamma  Г is a term that appears in Schwarzschild solution of general relativity equations. 
I submit that vacuum is not nothing but is everything and quantum mechanical model of the vacuum has very large energy density. In the words of John Archibald Wheeler, “Empty space is not empty… The density of field fluctuation energy in the vacuum argues that elementary particles represent percentage‐ wise almost completely negligible change in the locally violent conditions that characterise the vacuum.” That means there are violent conditions or fluctuations although vacuum on large scale appears smooth. Spacetime model has the capability of creating matter, forces, fields and particles. In fact, matter even the entire universe is assumed as spacetime as has been explained in my earlier article, “Matter Is No More Than Fluctuations In Vacuum*.”

I submit that the energy density of the vacuum, due to uncertainty principle, varies from point to point causing fluctuations. In other words, energy density at different points goes on varying even though there is no mass causing gravity. But even then at two points, due to fluctuations of energy density, rate of time may vary, causing difference in time by Planck time. These fluctuations can only be felt at a distance limited to Planck length or at microscopic scale but on macroscopic scale the space time, according to Quantum Mechanical model, appears smooth with no disturbances. ( Above figure shows simplified representation of chaotic dipole waves in Spacetime that form vacuum energy at age of the universe corresponding to Гu1.)                                                    These fluctuations at Planck length are assigned the term Quantum Foam and are also explained as creations of virtual particles or photons and their antiparticles which annihilate. Force of attraction between two conducting plates kept at a geometrically small distance, corroborates the factum of quantum fluctuations. (Above figure shows simplified representation of chaotic dipole waves in Spacetime that form vacuum energy at age of the universe corresponding to Гu2 where Гu2 = 3.Гu1).           This is force of attraction commonly called Casimir Effect has also been practically observed when such conditions get simulated between two sailing ships. Variation in energy level of hydrogen atom called Lamb shift also evidences presence of vacuum fluctuations. These fluctuations also cause variation in coulomb laws between two charged particles. All these phenomena conclusively prove existence of vacuum fluctuations and their effects.

These fluctuations can be visualised as being generated by harmonic oscillator of angular frequency w. Since minimum cavity length for this oscillator is half wavelength for reflection of the wave, energy associated with such oscillator is calculated by equation E = ½. h bar. w , commonly known as zero point energy where h bar is Planck constant divided by 2. pi or h/2π. This equation can also be written as E = ½. h bar.c/λ bar where λ bar is wavelength divided by 2π as c velocity of light is equal to product of λ bar and angular frequency. For this harmonic oscillator, volume where in, this energy is associated is a cube of length λ bar or its volume is λ bar^3 where sign ^ denotes raised to the power. Energy density or energy per unit volume is given by equation

  E = (½. h bar.c/λ bar)/ λ bar^3 = ½. h bar.c/ λ bar^4 = ½.h bar.w^4/c^3 ———–(1)

The above equation is obtained on substituting λ bar by c/w.

This angular frequency w may vary have all frequencies from 0 to infinity and in such case energy density would be infinite but w at the most can have maximum achievable angular frequency that corresponds to Planck time tp. Maximum achievable angular velocity is, therefore, Planck angular frequency wp corresponding to time period of Planck time tp. According to Milonni, in such a case of harmonic oscillator, energy density would correspond to Planck angular velocity divided by Planck volume. Planck angular frequency, wp = 1/tp = [c^5/(h bar.G)]^1/2 = 1.85 x 10^43 rev per second. Substituting this value in equation (1), we get Planck energy density of vacuum Up = 4 x 10^113 Joules/metre cube. This is shockingly highest energy density.

Here why Planck frequency and Planck length have been chosen for the purpose of calculation of energy density, is explained here under. It is noteworthy that minimum length and maximum angular velocity have been considered equal to Planck length and Planck angular frequency as it is the Planck distance which is minimum measurable and achievable. A length can not be measured if it is smaller than Planck length. A fraction of Planck length is not achievable, Planck distance can not be subdivided. It is the limiting minimum distance. If instead of Planck distance, zero distance is taken then energy density would scale to infinity, a quantity that can neither be achieved nor defined. In the same way, minimum time that can be measured or achieved is Planck time and this time can not be further subdivided. Angular frequency which is inverse of Planck time, is therefore, highest achievable. We can not take angular frequency as infinity as that will lead to indeterminate infinite energy of harmonic oscillator. Constant like Planck time, Planck length, Planck mass or extreme universal quantities which can be achieved.

So far, quantum vacuum at micro scale in relation to Spacetime was discussed. However, General Relativity lays down that energy warps the spacetime curvature which gives rise to gravity. Spacetime is smooth on macroscopic scale and there are no fluctuations as are observed in quantum mechanical model at Planck length with Planck frequency. The distance between the particles is limited to the extent of formation of blackhole meaning thereby that such distance is the limiting minimum distance so as to avoid formation of blackhole.This limiting distance is also in conformity with Quantum mechanical model. According to General Relativity, the universe would collapse to a blackhole if its energy density exceeds the critical value of energy density. The observed energy density of the universe 10^-9 Joules/metre cube is with in margin of error of 1% of equalling critical density.  

If on comparing the energy density of 4 x 10^113 Joules/metre cube as derived from Quantum Mechanical model with that 10^-9 Joules/metre cube calculated on the basis of General Relativity, the energy density given by Quantum mechanical model is observed ridiculously large. It is larger by the factor as large as 10^122,  in physics it is generally termed discrepancy factor of 10^120.

Assuming energy density on the basis of Quantum mechanical model as correct, then in this expanding universe, we need 4x 10^113 Joules of energy for one cubic metre expansion. Requirement of 4x 10^113 J of energy even for single cubic metre expansion is a huge quantity beyond imagination. To keep this energy density 4x 10^113 Joules/metre cube constant, we need a regular source to generate this much huge energy. If the existing mass were to be converted in to energy to make up the requirement caused by expanding universe, entire observable mass in the universe may fall short of as such huge energy would be required. It needs some other explanation to justify the constancy of this energy density.

To reconcile mammoth energy 4x 10^113 J/metre cube with minuscule figure of 10^-9 J/metre cube, support of Spacetime Transformation Model is needed. Einstein Field equations in General Relativity were solved by Schwarzschild wherein the term gravitational gamma Г appears. Gravitational gamma Г is defined as the rate of time in coordinate clock to the local rate of time in gravity and is given by equation  

 Г ≡ dt/dτ = 1/[1-2Gm/(c^2.R)]^1/2—————(2) 

where G is universal gravitational constant, m mass of particle, R circumferential radius. Stationary clock infinitely far away from the mass in Schwarzschild’s universe is designated the “coordinate clock” with a rate of time designated dt or one can say that it is rate of time in hypothetical universe where there is zero gravity. In the same stationary frame of reference, the local rate of time in gravity near the mass is designated dτ.

In equation (2), the term 2Gm/(c^2.R) is always between 0 and 1 otherwise denominator of this equation will become imaginary quantity and that is not a possibility. If 2Gm/(c^2.R) is small then gravitational field is weak and we can expand it by Binomial Theorem as

    Г ≡ dt/dτ = [1-2Gm/(c^2.R)^-1/2]= 1 + ½.2Gm/(c^2.R)+ —-

                         = 1+Gm/(c^2.R) + neglecting other terms being small

                      = 1 + β

A particle if thrown with a velocity equal to (2Gm/R)^1/2 from a gravitating mass m, it leaves the mass for ever with this escape denoted as Ve. Substituting this value in equation (2), we get Г equal to 1/[1 – (Ve/c)^2]^1/2————————(3).

In special relativity also, the term γ equals 1/[1 – (V/c)^2]^1/2. Only difference is that velocity in SR is replaced by escape velocity in General Relativity GR in equation of gravitational gamma.

It is submitted that at the time of Big Bang, there was no gravity and the term 2Gm/(c^2.R) was 0. Therefore at that time, gravitational gamma Г was 1. The maximum value Г at other extreme would be at the event horizon of blackhole as ∞ and the particle would be pulled in the blackhole with escape velocity when 2Gm/(c^2.R) equals 1. Value of Г at earth surface is 1 + 7x 10^-10 . Г, therefore, always has a value more than 1 where ever there is gravity.

Our location in the universe is similar to cavity surrounded by mass. In fact, the situation of the universe is like that which is surrounded by shell that is increasing its mass every second. If we consider a cavity at the centre of the universe, escape velocity from the cavity would be 22 pc more than that from surface of the earth. This increase in escape velocity in turn will increase the value of gravitational gamma as is evident from equation (3). That means at a point, there will be increased gravitational gamma if it is surrounded by mass. Our universe as submitted above is considered surrounding by shell with increasing mass which results in increasing gravitational gamma as the universe ages. It is submitted that this background gravitational gamma Гu is attributed to the matter outside and causes the rate of time, the gravitational potential, the escape velocity etc. However the internal matter affects acceleration due to gravity. Increase in value of gravitational gamma is causing the cosmological expansion. Present value of gravitational gamma can be calculated from various method but its value here is calculated from cosmic microwave background temperature. Assuming the model where the universe started from Planck sphere at Planck temperature ( ½ Ep/k ) 7.08 x 10^31 degree K and its present temperature 2.725 degree K, Г can be calculated as Tps/Tcmb = 7.08 x 10^31/2.725 = 2.6 x 10^31.

Reconciliation 

 We have already defined gravitational gamma as rate of time in in clock at hypothetical universe with zero gravity to rate of time in the universe with gravity or Г = dt/dτ or dt = Г.dτ. Now, a case is considered where an impulse of one second is emitted in hypothetical universe with zero gravity and simultaneously in the universe with gravity. Here dt is 1. Since the universe has gravity, Г will have some value more than 1. Therefore 1 = Г dτ and that indicates dτ should be less than 1 as Г is more than one. That means, rate of time is short or less in gravity or time is being dilated in gravity. From this, T0= Tg/Г where To is time in zero gravity and Tg is time in gravity. From conservation of momentum, we can calculate Mo = Mg/Г. The suffix 0 denotes parameters in hypothetical universe with zero gravity and suffix g denotes parameter in the universe with gravity. 

 Applying these transformation to the present universe which has gravitational gamma Гu (as 2.6×10^31 as calculated above), following equations hold good.                                                                                                          T1 = Tu/ Гu,                                                                                                                                                                                                      M1 = Mu/Гu,                                                                                                                                                                                            where suffix 1 denotes parameters in Planck Spacetime and u to parameters in the present universe. Similarly unit of length are also transformed by equation,                                            L1 = Гu.Lu.                                                                                                                                                              From these transformations of time, mass, length, transformations of energy, energy density can be calculated as given below.                                                                                                                

Energy transformation                                                                                                                                    Energy at Planck Spacetime E1 in mass, length and time dimensions is M1.L1^2/T1^2. On substituting the value of M1, L1, T1 and simplifying, energy at Planck Spacetime E1 = Гu^3. Mu.Lu^2/Tu^2 = Гu^3.Eu.                                                                                                                                                           That is E1 = Гu^3.Eu.—————————–(4)

Volume transformation                                                                                                                                                      Volume V1 at Planck space time = L1xL1xL1. On substituting value of L1 as Гu.Lu,                           V1 = L1^3 = Гu^3.Lu^3.————————–(5)

Energy density transformation                                                                                                                            Energy density at Planck Spacetime Up = E1/V1 = (Гu^3.Eu)/ Гu^3.Lu^3 = Eu/Vu = Uu                          or Up = Uu.———————————————-(6).                                          That is energy density at Planck Spacetime is same as energy density in the present universe and since there is no gravitational gamma in the equation, there is no effect of cosmological expansion on energy density of the universe and it remains constant. Invariance of energy density is explained as that with cosmological expansion, gravitational gamma increases and one joule of energy at Planck Spacetime gets transformed to Гu^3 times one joule in present universe. In the present universe Гu^3 is equal to 1.8 x 10^94. Гu and resultant Гu^3 are always more than 1 when there is cosmological expansion and that reduces the value of one joule in the present universe by 1/ Гu^3 although it is still called joule as it was called at Planck Spacetime. In fact, with cosmological expansion, unit of energy joule is shrunk but its numerical multiplier increases accordingly. In this way, one joule at Planck Spacetime is equal to Гu^3 or 1.8 x 10^94 joules in the present universe.

To illustrate it further, I give an example of a tourist who starts his journey from India with 10000 rupees in his pocket and goes to Pakistan where travel agents exchanges his money with Pakistani rupees and gives him equivalent 15000 Pakistani rupees. Now the tourist has 15000 rupees in stead of 10000 rupees. Here the unit of currency rupee has shrunk but numerical figure of rupees has increased. He then proceeds to Kathmandu, Nepal and the agent again exchanges the currency with Nepalese rupees and gives him equivalent 16000 rupees. Here again the unit of currency rupee has shrunk further but numerical figure of rupees has increased. In all these three countries, currency is rupee and tourist in a way can be said to have gained numerical figure of rupees from 10000 to 15000 to 16000 but rupee unit of currency has shrunk. In the same way as wheel of time rotates, cosmological expansion shows its effect by increasing numerical figure of unit of energy joule. Unit of rupees shrinks and numerical figure of rupees increases. Similarly with cosmological expansion, unit of length shrinks, numerical figure increases, unit of time expands, its numerical figure decreases, unit of mass expands, its figure decreases. Then how can we make distinction between units at different cosmological age of universe? Only distinction which we can make is by relating joule to that time or Гu when it is being measured. When we calculate energy density, enlarged multiplier of energy in numerator cancels the equally enlarged multiplier of length cube in denominator resulting in same energy density irrespective of time or Гu at which it is being measured.

Energy density calculated on the basis of Quantum mechanical model as 4x 10^113 Joules/metre cube remains same as the universe expands by Hubble’s law. Extra energy required for expanding space is countered by extra volume generated as elaborated in the preceding paragraph and as such creation of mammoth energy is not required for maintaining the energy density.

Next moot question is reconciliation of energy density of 4x 10^113 Joules/metre cube calculated on Quantum mechanical model with 10^-9 Joules/metre cube calculated on the basis of General Relativity or discrepancy of factor 10^120. It is submitted that at Planck time, when universe was a sphere of Planck length, its entire energy had the quantised angular momentum ie consisting of fermions of angular momentum of odd number divided by two. and Boson of angular momentum of whole number. But now almost all the vacuum energy does not have quantised angular momentum. The spacetime transformation model of the universe proposes that over the age of the universe there has been no change in the total energy density of the universe. Energy density at Planck Spacetime is equal to energy density of observable universe plus energy density of vacuum or Up = Uobs + Uvac**. Over time the transformation of spacetime has resulted in a dramatic decrease in the observable energy density of the universe and an equal increase in vacuum energy density of the universe. Now Uvac is approximately equal to 10^122.Uobs***. This vacuum energy, in fact, acts as a homogeneous superfluid which does not interact but prevents the universe from gravitational collapse. This vacuum energy can be envisaged as background shade and being devoid of angular momentum does not interact to form observable universe.                                                                                 Before paring with article, I sum up by submitting that although there is discrepancy factor of 10^122, this does not require generation of additional energy as according to Spacetime transformation model, the numerical value of energy increases by cube of gravitational gamma and that is offset by increase in numerical value of volume by same factor of cube of gravitational gamma and that keeps energy density constant. In Quantum mechanical model, the universe is considered created from Planck time and Planck sphere, entire energy at Planck time possessed quantised angular momentum but with passage of time, vacuum energy with no angular momentum formed. Energy with quantised angular momentum evolved observable universe, gravitational gamma increased with time and cosmological expansion occurred and that caused contraction of unit of length, expansion in unit of time and mass. At present energy density of vacuum is approximately equal to 10^122 times energy density of observable universe. Energy density on the basis of GR is energy density of observable universe with quantised momentum whereas energy density of vacuum on the basis of Quantum mechanical is vacuum energy density which has the properties of superfluid that is non quantised and does not interact but contributes to background gravitational gamma. Thus comparing energy density on the basis of Quantum mechanical model with energy density on the basis of GR is analogous to comparing apple with orange.                                                                                                                                                                  Reference:     The Universe Is Only Spacetime ©2012 john@onlyspacetime.com                                                                                                                                                                          Image Credit: The Universe Is Only Spacetime ©2012 john@onlyspacetime.com.                  End.                                                                                                                                                                                                            Notes:                                                                                                                                                                                            1. *Kindly read “Matter Is No More Than Fluctuations In Vacuum,” at https://niume.com/post/147680.                                                                                                                                     2. Sign ^ means raised to the power. For example λ bar^3 is equal to λ bar x λ bar x λ bar or λ bar cube.                                                                                                                                                                        3. ** Uvac means vacuum energy density.                                                                                                4. *** Uobs means energy density of observable universe.

 Writer:                                                                                                                                                                  He 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.   

     

 

 

 

 

 

 Red Shift: A Window Which Peeps To The Past

In optical astronomy, Redshift, Parallax, Luminosity, Spectrum analysis are tools which facilitate astronomers to explore the cosmos. In this ever expanding universe, redshift is that tool which helps determining the velocities of stars, galaxies and other light emitting celestial bodies which are in motion. Once velocities determined, we can compute how distant these are from us. Light is electromagnetic radiations which move in waves and simultaneously, are also comprised of particles photons which travel with velocity 30000000000 centimetres per second a hefty quantity which is maximum achievable as propounded by great Physicist of twentieth century Albert Einstein. An electromagnetic wave has cyclic varying electric field combined with another cyclic varying magnetic field perpendicular to the plane of electric field. The energy of the wave propagates in a direction perpendicular to the planes of electric and magnetic fields. A wave is characterised by frequency denoted by ‘f’ and is defined as number of vibrations or cycles that wave makes in one second. This is its fundamental feature that distinguishes it from radio waves of low frequency low energy to X rays or gamma rays of high frequency and high energy waves. That means higher the frequency, higher will be the energy of the wave. Related to frequency is the wavelength which is the distance travelled in one cycle by the wave. Since the wave completes ‘f’ cycles in one second, it travels a distance of ‘f’ multiplied by wavelength in one second and it is its velocity denoted by ‘c.’ Mathematically, it is written, c = f. λ. Natural corollary from above is, that higher the frequency, shorter the wavelength or lower the frequency, higher the wavelength. 

 
According to Physicist De Broglie, wavelength = h/p where ‘h’ is Planck’s constant and p is momentum a quantity which is product of mass and Velocity. And also energy of the wave = Planck’s constant multiplied by frequency or h.f. Waves can be of frequency as low as infrasound and as high as those of Planck frequency. Frequency in increasing order and also energy is given below.   

Infrasound frequencies ——> Sound or Audio frequencies ——–>Ultrasound frequencies—–>Radio frequencies (HF, VHF, VVHF, UHF)——> Infrared (frequencies)—–>Light rays (Optical range frequencies)—>UV frequencies——> Gamma rays including X rays and still higher frequencies.  

A ray of white sunlight consists of light of seven colours and when it passes through a prism, it splits up into seven colours (of frequency bands correspondingly), violet, indigo, blue, green, yellow, orange and red popularly known as VBGYOR. The colours violet to red have frequencies and energy in decreasing order ie violet light has maximum and red minimum energy in optical band.  

Christian Doppler in 1842, stated, if there is relative velocity between the sourced waves and the observer, then observed wavelength will be more or less from its source wavelength depending upon whether the source is receding or approaching the observer, respectively. Thumb rule is, if the distance between the source and the observer is increases, the observed wavelength will exceed the wavelength of the source but if the distance decreases, the observed wavelength will decrease.  

In case of receding source, increment in wavelength calculates as (v/c) multiplied by wavelength of source light. Thus if we know the increment in wavelength, we can calculate velocity of recession of the source. Distance of the source can then be calculated from the value of recession velocity as distance is proportion to velocity due to Hubble’s law. Increment in wavelength accordingly shifts the spectrum of the source towards red end and such shift towards red end is called redshift. However, if the source approaches the observer, wavelength shifts towards the blue end of the spectrum which is called blue shifted. 

How is redshift measured? Rays of light from the source star or Galaxy are passed through spectrometer which splits these into seven colour bands and this spectrum is analysed with regard to position of emission and absorption lines and shifts of the lines from standard spectrum measured. Magnitude of redshifted lines corresponds to increase in wavelength.

Image Redshift shows in right hand side, shifted lines towards red end of spectrum. (Image Courtesy https://upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Redshift.svg/339px-Redshift.svg.png)

In addition to redshift due to Doppler effect, rays emitted from sources also suffer loss in energy when these pass in the vicinity of massive objects in space. In such situations, the rays of light are pulled by the heavy gravitational field which bend the path of the rays towards the massive object. Massive objects act like an optical lens and forms dual images of the source. Such phenomenon is called gravitational lensing. According to Albert Einstein, the space around mass is warped and is not flat. That is also an explanation of loss in energy or red shift on the basis of Spacetime curvature. 

 Distances of sources like stars or galaxies have been determined by measuring the redshift of spectral lines. In order to explain how redshift corresponds to the distance of the Galaxy detected, I give below a table.

Redshift  Light Travel Time      Age Of Universe

    12          13.45 billion years      0.365 billion years

    15          13.55 billion years.      0.267 billion years

    18.         13.61 billion years.      0.206 billion years

    20.        13.64 billion years.     0.177 billion years

This is not the end, with better optical instruments, even farther located galaxies are expected to be located. Problem encountered in locating far flung galaxies is that distant galaxies are obscured by intergalactic matter and intervening objects in space and these obscure the image. With advanced instrumentation, it is hoped, we would be able to sight the farthest and faintest galaxies. It is quite probable, we may see the first light brightening the universe.

Recently, a Galaxy GN-z11 causing whopping redshift of 11.1 has been observed by Hubble space telescope. On calculation, it is found to be located at a distance of 13.4 billion light years away ie rays of light took 13.4 billion years to reach earth from that galaxy. In other words, this galaxy was born when the universe was just 400 million years old. To give better insight, I state that rays of light from this Galaxy started 13.4 billion years back when the universe was only 400 million years old and these rays have now reached us. We are looking at the Galaxy as it was 13.4 billion years back but can not say what we are seeing this Galaxy, would actually be present now. By this time, it would have gone far beyond our observable universe. We are looking at past the time which was 13.4 billion years back.  Redshift is thus making us look back and as far back as flash of first light in the universe. We can now even look back to the stage when the universe was baby in its infancy.  

If we imagine, there were a planet teeming with life in one of the stars of galaxy GN-z11 and inhabitants of that planet had transmitted information 13.4 billion years back, then the signals would reach us now. Although the signals were transmitted at the time the universe was just 400 million years old only, these would be ‘live’ to us as the delay occurred only in reception. These would be fresh and live irrespective of their transmission 13.4 billion years ago. Past and present are conjoined, thanks to redshift which has opened a window to look at the primordial 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.