*“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.