Title: Temperature-Time equation!

Wanchung Hu

PostDoc Fellow

Correspondence to: lukluk73_2006@yahoo.com.tw

Abstract

Dark energy is a mystery in current cosmological research. Professor Albert Einstein originally thought the universe is static. Thus, he added a cosmological constant in his universe equation to prevent the collapse due to gravity produced by mass in universe. However, due to Professor Hubble’s observation, our universe is actually expending. Based on his observation, he obtained a physic law called Hubble’s law: V=H*R (H=Hubble constant, V=recession velocity, R=comoving proper distance). Currently, we still don’t know what the reason to cause universe expansion is. A term: dark energy is given to explain the universe expansion. Dark energy is still a puzzle in Astronomy. No one knows what dark energy is. In this article, we propose that radiation pressure is actually dark energy. It has several important characteristics to be the best candidate of dark energy. In addition, we will use radiation pressure to derive Hubble’s law. However, further experiments and observations are helpful for confirming that radiation pressure is dark energy.

Introduction

Dark energy is a mystery in current cosmological research. Professor Albert Einstein originally thought the universe is static. Thus, he added a cosmological constant in his universe equation to prevent the collapse due to gravity produced by mass in universe. However, due to Professor Hubble’s observation, our universe is actually expending. Based on his observation, he obtained a physic law called Hubble’s law:

v=H*R (H=Hubble constant, v=recession velocity, R=comoving proper distance).

Currently, we still don’t know what the reason to cause universe expansion is. A term: dark energy is given to explain the universe expansion. Dark energy is still a puzzle in Astronomy.

Radiation pressure is originally derived from Professor James Clerk Maxwell. It was derived from the momentum change of light. The formula of radiation pressure is:

P=(uT^4)/c (u=Stefan-Boltzmann constant=5.67*10^-8 JS^-1m^-2K^-4; T=absolute temperature; c=lightspeed)

The basic concept for deriving the formula is from the momentum of light: p=E/c. When light is arriving on the surface of substance, it will cause momentum change. Thus, radiation pressure is resulted. The radiation pressure from Sun to Earth is 4.6uPa. It is a very tiny amount compared to the gravity from Sun to Earth. It can almost be neglected compared to gravity. However, we propose here that radiation also causes pressure on space-time dimension. And, due to the high temperature of all stars of universe and the fourth magnitude of radiation pressure(~T^4), radiation pressure exceeds gravity and plays a dominant role in universe expansion. Due to the T^4 magnitude, our universe is expanding accelerately. Radiation pressure has greater impact on space-time dimension than gravity. Gravity is a force to cause space-time distortion. It is not a pure inward force to cause space-time contraction. Thus, we don’t need to balance gravity with radiation pressure. We believe that radiation pressure is actually the dark energy.

Mach’s principle pointed out there is no absolute time and space. Substance in space-time will affect the surrounding space-time. Einstein was inspired by Mach’s principle and developed general relativity. There must be an outward force to cause universe expansion. Radiation pressure is the best candidate. The other fundamental forces cannot account for the large dimensional universe expansion. Strong force and weak force are mediated from particles and only work in a very short range inside the atom. Electromagnetism can be transmitted from a long distance. However, almost all the large substances are neutralized in electricity in our universe. Electrostatic force is from rest charges and magnetic force is from moving and spinning charges. However, when there is no net charge in stars or planets, there is no electromagnetism. Thus, electromagnetism cannot be the cause of universe expansion. Radiation (Heat and Light) becomes the best candidate of the force of dark energy.

Due to the second law of thermodynamics, heat has a very close relationship with time. Entropy is always increasing from high temperature to low temperature to achieve the maximal magnitude of randomness when time is moving ahead. In addition, lightspeed is very important in space and time due to relativity. Radiative wave can cause entrophy increase, and convergent wave can cause entrophy decrease. Thus, we think heat and light play dominant roles in our four dimensional universe. The heat’s moving arrow is very correlated with the space-time moving arrow(universe expansion arrow or comsmological arrow). It means that heat and light decides the moving of space-time. Radiation pressure is actually from heat and light. Thus, heat and light can cause space-time movement due to the mediation of radiation pressure. Radiation is from the central substance and radiating to all the outward direction. Thus, radiation from stars or from galaxy can expend the space-time outwardly and evenly in every direction. It is very important to synchronize all the time arrows in universe. All the time arrows must be the same and have only one meaning. If radiation pressure is dark energy, then entrophy arrow=radiation wave arrow=cosmological universe expansion arrow=time arrow=causal arrow. Thus, there is no contradicts in these time arrows. They can be well synchronized. Only when dark energy is radiation pressure, all the time arrows can be synchronized. Thus, radiation pressure is the best candidate of dark energy. It is also important to know that universe will cease to expand if the whole universe reaches heat death(maximal entrophy). In addition, universe won’t contract when it reaches heat death because radiation wave won’t become convergent wave to let universe contraction and to let time arrow fly back. This concept is very important. It means that the time arrow won’t be reversed to disobey the causal-effect relationship.

In order to test if radiation pressure is the dark energy, we tried to derive Hubble’s law from radiation pressure. We found that Hubble’s law can be derived from assuming radiation pressure is dark energy. Below is our deduction:

In this deduction, we need to assume that observable universe is the actual universe. We can imagine this from the following example. When a light is emitted from our solar system to the actual universe boundary to cause universe expansion, the time it spends is the same as a light emitted from the observable universe boundary to travel to our solar system. Thus, observable universe is the actual universe. The volume of observable universe is given by Hubble volume:

Hubble Volume V=(c^3/v^3)R^3 (c=lightspeed, R=distance, v=recession velocity)

Hubble Length R’=cR/v

Thus, we can derive the relation between heat and distance first:

Universe entropy per comoving volume(Hubble Volume) is:

s=(2pi^2/45)GsT^3

There is conservation of total entropy:

Thus, S=(2pi^2/45)GsT^3*c^3R^3/v^3=constant

Thus, T -> (v/c)R

temperature(T) is inverse proportional to cR/v

Second, we assume the whole universe is a close system. Based on the conservation of energy, the total energy of the universe is constant.

Radiation outward pressure=uT^4/c,

Thus, Total E=P*V=(uT^4/c)*V=constant

Applying the concept of Hubble Volume and Hubble Length

And, T is inverse proportional to cR/v

Total E={kv^4/(c^4R^4)}* [(c^3/v^3)R^3]=constant

Total E=(k)v/cR=constant (k, c are both constants)

Thus, v is direct proportional to R

That is Hubble’s law : v=HR (H: Hubble’s constant)

In the second part of this article, we will derive why our universe is accelerated expanding. There is a relation between temperature and acceleration which is called Unruh effect. The formula is:

T=h’a/2pi*k*c (T=absolute temperature. h'=reduced planck constant, a=acceleration, k=Boltzman constant, c=lightspeed)

This formula can be derived by Lorentz transformation, Planck’s law, and Doppler shift. This formula is a natural consequence of time dependent Doppler shift seen by the accelerated observer. Originally, the formula is explained, “A observer undergoing uniform acceleration a in vacuum responds as though it were immersed in thermal radiation T (T=h’a/2pi*k*c)”.[1] However, we can also explain that temperature can cause uniform acceleration in vacuum.

In one spatial dimension, T is in direct proportion to a. Thus, it means that absolute temperature T can cause vacuum to expand at a acceleration in X-axis. In addition, the radiation pressure is:

P=(uT^4)/c, (T=h’a/2pi*k*c)

Thus, P=(u/c)*(h’^4*a^4/16pi^4*k^4*c^4)

Radiation pressure P is in direct proportion to acceleration a^4. It means that radiation pressure can cause accelerated universe expansion in 4 dimensional space-time. Based on current observation, our universe is accelerated expanding. If radiation pressure is dark energy, it can perfectly fulfill our observation.

Then, we introduce u=(pi^2/60)k^4/h'^3c^2 into the above equation, then we get:

P=(pi^2/60)h'a^4/16pi^4*c^7

We know c=at, then a=c/t (t=time,x=space distance),

P=(1/pi^2*960)(h'/t^4c^3), let x=ct

Thus, P=(1/pi^2*960)(h'/t*x^3)=(1/pi^2*960)(h'c/x^4)

Compared to the Casimir effect F=P/A=(pi^2/240)(h'c/x^4)

There is only a difference of factor 4pi^4, thus we can infer that radiation pressure is the cause of Casamir force

In Carnot cycle, we find out that it disobeys energy conservation. Small amount work can produce higher amount of heat. That is heat pump. By using the above equation, we can know that heat is a function of spacetime. In the initial universe, space-time is so small with greatest amount of heat(temperature). In addition, we can expain why friction can induce heat. It is because heat production is from spacetime compression movement. Thus, we can solve the paradox of energy conservation problem in Carnot cycle.

Then, we let P=(1/pi^2*960)(h'/t^4*c^3)==(uT^4)/c==(pi^2/60)k^4T^4/h'^3c^3

Then, h'^4/16pi^4*t^4=k^4T^4

Thus, h'/2pi*t=KT, since h'=h/2pi and t=1/f and w=2pi/t,

Then, we get hf/4pi^2=h'w/4pi^2=kT that is temeprature=time equation

Time is a function of temperature

In summary, we believe radiation pressure is the best candidate of dark energy. It can be well correlated with space-time movement. We suggest further experiments and observations to confirm this hypothesis.

Reference

1. Alsing P. M. and Milonni P. W. Simplified derivation of the Hawking-Unruh temperature for an accelerated observer in vacuum arXiv:quant-ph/0401170v2 Aug27,2004