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Relativity, Cosmology, and Time

Origin of Time (2019 Version)  Most models of early universe assume that time starts with the Big Bang. Study of a recent theory reveals that it may not be so. Time emerged gradually along with a sequence of events as illustrated in the followings. The novel features include a "rediscovery" of the Wheeler–DeWitt equation which suggests the disappearance of time in the formulation some 50 years ago. A simplified derivation for quantization of the Friedmann Equation (Matter-only) shows the same conclusion about timeless but also reveals a mechanism that can mimic the

Figure 02 Interactions, 4-types [view large image]

phenomenon of inflation. Here's the mathematical formalism for such interpretation.

The formulation starts from a very small universe at Planck scale (see Figure 01) with Planck mass/energy MPL/EPL and size r0 = LPL. Accordingly, its energy density PL ~ 10114 erg/cm3. Such entity is at the realm of quantum gravity. Unfortunately, there is no such universally accepted theory for now. One of the ad hoc solutions is to quantize the Friedmann Equation for cosmic expansion.

The classical Friedmann Equation is in the general form : (dR/dt)2 - 2GM / R(r0)3 = -kc2 ---------- (1a),

in terms of the Planck time tPL, it can be re-written as : (dR/dt)2 - 2 / R(tPL)2 = -kc2 ---------- (1b).

In first quantization to endow wave property to a particle the linear momentum px and position x has to satisfy the commutative relation :  Figure 03 H2 Energy Level & Cosmic Curvature See more cumbersome detail in the original derivation of "Quantization of the Friedmann Equation (Matter-only)" in 2011.

Some comments on the quantized Friedmann Equation :

1. Time is absent in Eq.(2), which can be interpreted as representing a static configuration. It is similar to an isolated hydrogen atom just sitting there alone until a photon comes along to induce transition to other state. Similar occurrence could happen to the Planckian universe via interaction with graviton. However, the transition is to smaller spatial curvature k (see Eq.(3)) manifesting as "inflation" of space. The "inflation" terminates as the transition approaching the continuum, i.e., as n and k 0, that's when time began and standard (classical) cosmology took over.

2. The Planck length LPL is the only parameter in the quantized Friedmann Equation. The form of mass/energy is not specified in
Eq.(2). Actually, "mass" is a misnomer; it did not exist until some time later. Anyway, invoking E = Mc2, the energy E can assume various form, it is not essential to pin-point which type(s) at this level. However, according to the conventional theory of inflation by Alan Guth, high energy particles were created near the end of the process.

3. The formulas in General Relativity is non-linear in the metric tensors ( ~ gravitational fields), this is the root of the problem for its quantization. The present scheme linearizes the Friedmann Equation and in the process enables the superposition of states - a key feature of quantum theory. For simplicity, it is assumed that the system is at its ground state with no mixture.

4. According to classical theory, this Planckian universe is a black hole since its radius LPL < rs = 2LPL. It is not known what's the effect of qunatization on this black hole. A string theorist suggests that it turns into a fuzzball with no sharp event horizon (see "The Fuzzball Fix for a Black Hole Paradox")

5.  6. When quantum fields are included into the vacuum of the Planckian universe (Figure 04), there would be virtual particles popping up and vanishing briefly and incessantly according to the Uncertainty Principle t E  (Figure 05). The t in this case is a time interval in the order of the Planck time tPL, it has no connection with the absolute time in the evolution of the universe.

Figure 05 Virtual Particles [view large image]

However, this quantum fluctuation has profound effect on the large structure such as super galactic cluster in the later epoch of the universe.

8. Since the quantum fields and virtual particles permeated throughout the space (Figures 04 and 05), there is no need to devise a mechanism for attaining a homogeneous and isotropic state.

9. The end of inflation marks the transition from quantum to classical cosmological model, which involves time (see Eq. (1)). This boundary is the "new" absolute zero time (Figure 01). It shifts the conventional cosmic time scale up by an negligible amount
~ 10-33 sec. This instance could be associated with the appearance of particles, the emergence of strong interaction, and chiral symmetry breaking (Figure 02, also see "Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking").

10. There is a discontinuity in time at the moment of transition. Time progresses forward only, there is no backward time at that point. Thus, time symmetry is violated at this point. Since charge and particle numbers also emerge at this moment (from zero to something), charge symmetry is also violated. Then the cardinal rule of CPT invariance in Quantum Field Theory is violated as each one is broken at that moment.

11. The entropy in the beginning was at its minimum (see "Cosmic Evolution of Entropy" and note : mass is the inverse of length in natural unit).
The emergent universe after the end of inflation turns out to be an era of radiation, i.e., with kinetic energy of particles >> rest mass. As a result, the Friedmann Equation (with time variable now) has a slightly different form in which k = 0 is implied as the result of inflation :   Figure 07 Cosmic History, Very Early Figure 06 shows the evaluation of the t0 and 0 parameters at the recombination point, while Figure 07 is a rough sketch for the evolution of the scale factor in very early universe.  The same sketch also shows a very brief period of the order 10-10 sec when all the particles are massless. The Standard Model (SM) of elementary particles dictates that before the so-called "Electro-weak Symmetry Breaking", the Higgs field existed in false vacuum and could not interact with other particles making them massless (Figure 08,a) and all of them move at the speed of light (Figure 09,a).

Figure 09 Mass, Origin of [view large image]

See "Largranian for WS Model of SM", WS stands for Weinberg-Salam - two of the 1979 Nobel prize recipients for formulating the Stand Model. The third one is SG, who produced the QCD.
BTW, As shown in Figure 08, this is also the point when the electro-weak interaction separated into the weak (with massive mediating bosons - making the force short range) and electromagnetic (with the massless photon - producing long range force) interactions.  Figure 10 Proper Time [view large image]

See more about "Special Relativity".

After "Symmetry Breaking", the Higgs field in the true vacuum acquires the ability to interact with some particles giving them the appearance of mass (Figures 08b, 09b). It was still very hot in the radiation era. Finally, the universe entered into the matter era emerging gradually with structures like nuclei, atoms, molecules, living organisms, planets, stars, galaxies, ... see Figure 01, and Main Menu.

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