Albert Einstein was awarded the Nobel Prize in Physics 1921 mostly for his explanation of the photoelectric effect and of Brownian motion. In his Nobel lecture 1923, it became clear that he had a dream of linking gravity and electromagnetism. He could not "rest content with the assumption that there exist two distinct fields totally independent of each other by their nature."

The research by him and others has not resulted in a generally accepted unified theory solving, for example, the mystery of some so-called large number coincidences which were first recognized in 1919 and which are today interpreted by anthropic arguments. As a result of a more scientific interpretation that can make Einstein's dream come true, a long-sought missing link for unification was found. It has been overlooked by the scientific community, but the finding is included in a text written by me and published in 2009 by the Swedish Mathematical Society.

The text below complements with easy-to-follow mathematics at the undergraduate level both the above-mentioned paper* and additional papers** written by me and published in 2012 by the Swedish Mathematical Society.

The square root of the number of protons in the universe is a large number estimated to be in the order of 10⁴⁰ which happens to be equal to the size ratio between the universe and the proton as well as to the strength ratio of the electromagnetic and gravitational forces between the proton and the electron in the omnipresent hydrogen atom. According to the inverse square law in classical field theories, the size of the universe can be represented by the field defined radius r_g = GM/c² determined by its estimated total mass M, the gravitational constant G and the velocity c of light in vacuum. In correspondence thereto, the size of the proton can be represented by the field defined radius r_e = e²/(m_ec²) determined by the elementary charge e with the characteristic mass m_e. It is then possible to express the above-mentioned large number coincidences as r_g/r_e = e²/(Gm_pm_e) and (r_g/r_e)² = M/m_p where m_p is the proton mass.

The latter relation can be expressed as M/r_g² = m_p/r_e² and given the interpretation that the number of protons in the universe is in the expanding space-time model simply the result of a cristallization process on a cooling horizon surface r_g² with the cell area r_e² per proton. Anyway, it is quickly proved that the two so-called coincidences above are connected since (r_g/r_e)² = (r_g/r_e)e²/(Gm_pm_e) = (r_g/r_e)r_em_ec²/(Gm_pm_e) = (r_g/r_e)Mr_em_ec²/(r_gc²m_pm_e) = M/m_p. And it is also easily proved that both represent a mass ratio since r_g/r_e= e²/(Gm_pm_e) = r_e m_ec²/(Gm_pm_e) = m_e/(Gm_p m_e/(r_ec²)). The definition of a mass m_g as equal to Gm_pm_e/(r_ec²) results in the equation r_gm_g = r_em_e where it can be interpreted to represent the characteristic mass of an elementary particle connecting m_g and r_g to the force of gravitation in the same way as the electromagnetic force is connected to m_e and r_e.

The strength ratio of the two forces becomes here related to the ratio between the characteristic masses of their respective elementary particles with the masses m_g and m_e. If these are interpreted as wave groups, a motion velocity v can be expressed as c² divided by the phase velocity u defined as the product of the wavelength l and the frequency f. A chosen numerical value for Planck's constant h makes Louis de Broglie's equation h/v = l m usable for defining the mass m. The above-mentioned relation v l f = c², where c is today a chosen numerical value for defining distance from time, results then in the Einstein-Planck equation mc² = hf which can relate m_g to a wave frequency f_g in the order of 10^-20 Hz and m_e to a wave frequency f_e in the order of 10²⁰ Hz. While the existence is still not proved, de Broglie conjectured almost a hundred years ago that the elementary charge e has an internal clock.

Since (r_g/r_e)³ = (r_g/r_e)M/m_p = (m_e/m_g)(M/m_p), it is obvious that (M/m_g)/r_g³ = (m_p/m_e)/r_e³ so the universe has, as counted per volume measure r_e³, an average density of m_p/m_e particles with the mass m_g. Besides their role in the proton creation as described by the relation m_g = Gm_pm_e/(r_ec²), they offer an explanation of dark matter so my preferred nickname has been "darkinos" ***

Recognizing the equation r_gm_g = r_em_e as a long-sought missing link for unification, a most simple wave model of time was proposed by me for connecting both gravity and electromagnetism to a varying scale ratio of time dt/dt_r = sin(2π t/T) where T is the repetition cycle and dt is a measure that is related to a flow of time t representing either one of the two forces and that has a maximum dt_r as a measure related to a reference flow of time t_r representing then the other force. In this model, it is of interest to note that the direction of the flow of time t changes at the zero-crossing points inside the cycle T due to alternating positive and negative values for dt/dt_r, and that electric charges of opposite polarities possess opposite phases.

With distance defined from time, the size ratio r_g/r_e can be linked to the scale ratio of time dt_g/dt_e = sin(2π t_g/T) where it is easy to derive that dt_g/dt_e = sin(2 arctan e^{2π t_e/T} = 2/(e^{2π t_e/T} + e^{-2π t_e/T}). In summary, it is possible to use the proposed wave model of time for describing an apparent accelerated expansion of space-time with a constant density of "darkinos". 

* The paper published 2009 was written by me in English with a Swedish abstract:

http://www.swe-math-soc.se/pdf/utskick0910.pdf pp 69-73   

** Additional papers:

http://www.swe-math-soc.se/pdf/SMSbull1205.pdf p 31and http://www.swe-math-soc.se/pdf/SMSbull1210.pdf p 27.

*** https://www.livescience.com/fluffy-ball-darkinos-center-milky-way.htm