3.4.3 Space-Time Diffeomorphism and Mass-Energy Equivalence

When Hilbert applied the principle of least action he obtained these equations, thus the action is called the Einstein-Hilbert action. This allowed for easy unification of General Relativity with other classical field theories, such as Maxwell’s theory of Electromagnetism, which are also formulated in terms of an action. The action also allows for the easy identification of conserved quantities, through Noether’s theorem, by studying the symmetries of the action, which is usually assumed to be a functional of the metric and matter fields.

In General Relativity, space-time symmetries are used in studying exact solutions of Einstein’s field equations, and they are different from internal symmetries. For example, the role of spherical symmetry is important in deriving the Schwarzschild solution and studying its physical consequences, such as the nonexistence of gravitational radiation in a spherically pulsating star. In the cosmological principle, symmetry restricts the type of universes that are consistent with large-scale observations, such as the Friedmann-Lemaître-Robertson-Walker metric. There various corresponding preserved properties, including: geodesics of the space-time, the metric tensor and the curvature tensor.

A rigorous definition of symmetries in General Relativity has been given by Hall in 2004, based on the idea of using smooth vector fields whose local flow diffeomorphisms preserve some property of the space-time. Because of this restricting diffeomorphism, the behavior of objects may not be as manifestly symmetric on large scales. A symmetry on the space-time is a smooth vector field whose local flow diffeomorphisms preserve some geometrical feature of the space-time, such as the metric, the energy-momentum tensor, or other aspects of the space-time, including its geodesic structure. The vector fields are called collineations or symmetries, and the set of all symmetry vector fields forms a Lie algebra under the Lie bracket operation, if the smoothness condition is fulfilled.

It is commonly believed that the equivalence between mass and energy arose from the Theory of Relativity as described by Poincaré in 1900, and later by Einstein in 1905, in his famous paper “Does the inertia of a body depend upon its energy-content?” , in which he proposed the equivalence of mass and energy as a general principle and a consequence of the symmetries of space and time. However, in the previous and many other related papers that he published in the next fifty years, Einstein gave various heuristic arguments for this relation without ever being able to prove it in any theoretical way (Hecht, 2011), as no one else also ever did. Based on Doppler effect and Maxwell’s theory of radiation, the reasoning that he gave in his 1905 original derivation was questioned by Planck and shown to be faulty. In 1907, Einstein acknowledged the controversy over his derivation. He later produced more than half dozen proofs that all suffer from unjustified assumptions or approximations. He never succeeded in producing a valid general proof (Ohanian, 2009). In 1955 he wrote in a letter to Carl Seelig (1894-1962): “I had already previously found that Maxwell’s theory did not account for the micro-structure of radiation and could therefore have no general validity.” (Capria, 2005).

As we already explained in Chapter V of Volume II, an exact derivation of this experimentally verified relation is not possible without postulating the inner levels of time, since it incorporates motion at the speed of light which leads to infinities on the physical level. Therefore, the Duality of Time is the only way that leads to exact mathematical derivation of this relation directly from the principles of classical Newtonian Mechanics, as we shall describe with further details in the following simple methods that could not have been possible without the metaphysical behavior in the inner levels of time, where the velocities of each individual geometrical points are perpetually and sequentially changing abruptly from rest to the speed of light, and vice versa, in literally “zero time” on the outer level. This is obviously not allowed on the normal level of time when dealing with physical objects that have mass, because it will lead to infinite acceleration and infinite energy.

The relation between mass and energy dates back to the 17th century, when the term vis viva, from the Latin for “living force” , is used for describing the kinetic energy in an early formulation of the principle of conservation of energy, proposed by Leibniz during the period 1676-1689. He noticed that in many mechanical systems of several masses, the quantity was conserved, and he called this quantity the vis viva, or living force of the system. It was later realized that this observation is only accurate for the conservation of kinetic energy in elastic collisions, and it is independent of the conservation of momentum. In 1807, Thomas Young (1773-1829) was the first to use this term as energy, but it was later calibrated to include the coefficient of a half:

Einstein was not the first to have related energy with mass. In 1717, Newton speculated that light particles and matter particles were inter-convertible. In 1734, in his own Principia, Emanuel Swedenborg (1688-1772) speculated that matter is ultimately composed of dimensionless points of “pure and total motion” .

Additionally, in the end of the 19th century, there were many attempts to understand how the mass of a charged object depends on the electrostatic field. The concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well, thus the object may have different longitudinal and transverse electromagnetic masses. One year before Einstein, in 1904, Lorentz expressed transverse electromagnetic mass as:

Shortly after that, Planck defined the relativistic momentum and gave the correct values for the longitudinal and transverse masses, but this concept of mass was redefined in 1909 as the ratio of momentum to velocity, instead of the ratio of force to acceleration. However, the concept of relativistic mass is not used anymore in Relativity, but it is considered as an invariant quantity, whereas the energy is relativistic.

After World War II, when huge energies released from nuclear fission, as demonstrated by the atomic bombs of Hiroshima and Nagasaki in 1945, the mass-energy equation E=mc² became linked by the public with the power of nuclear weapons, but the equation was not strictly necessary to develop the weapons. As Robert Serber (1909-1997) put it:

“Somehow the popular notion took hold long ago that Einstein’s Theory of Relativity, in particular his famous equation E=mc², plays some essential role in the theory of fission. Albert Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of Relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly.” (Serber and Rhodes, 1992, 7)

It can be readily seen from Figure III.5 that the transmutability between mass and energy can only occur in the inner levels of time, because it must involve motion at the speed of light that appears on the normal level of time as instantaneous, hence the same Relativity laws become inapplicable since they prohibit massive particles from moving at the speed of light, so the mass would be infinite and also the energy. In the inner levels of time, however, this would be the normal behavior because the geometrical points are still massless, and their continuous coupling and decoupling is what generates mass and energy on the inner and outer levels of time, respectively.

Figure III.5: Gradual versus Abrupt Change of Speed - Arrow 1: gradual change of speed in the outer level of time, leading to the classical equation of kinetic energy. Arrow 2: abrupt change from zero to c in the inward levels of time, leading to the mass-energy equivalence relation, which cannot happen on the normal level of time because physical objects may not move at the absolute speed of light.

The normal limited velocities of massive physical particles and objects are a result of the spatial and temporal superposition of the various dual-state velocities of their individual points. This superposition occurs in the inner levels of time, where individually each point is massless and it is either at rest or moving at the speed of creation, but collectively they have some non-zero inertial mass m, energy E, and limited total apparent velocity v, which can be calculated mathematically as demonstrated in Chapter V of Volume II. When we consider this imaginary velocity as being real, the Duality of Time Theory reduces to General Relativity, but when we consider its imaginary character we will uncover the hidden discrete space-time symmetry and we will automatically obtain Lorentz transformation, without introducing the principle of invariance of physics laws. For the same reason, we can see here that the mass-energy equivalence E=mc² can only be derived based on this profound discreteness that is manifested in dual-state velocity, which then allows the square integration in Figure III.5, because the change in speed is occurring abruptly from zero to c. Otherwise, when we consider v to be real continuous in time, we will get the gradual change which produces the triangular integration with the factor of half that gives the normal kinetic energy.