Einstein's Equivalence Postulate and Instantaneous Gravitational Action

EINSTEIN'S EQUIVALENCE POSTULATE AND SPACELIKE WAVES

James Constant

Constantrcs@cs.com

Einstein's Equivalency Postulate Must be Abandoned for Study of Spacelike Wave Phenomena Such as Gravitation

Einstein's Equivalence Postulate Revisited

During the end of the 19th century, theoretical physics was based on three areas of knowledge: Newton's equations, Maxwell's equations, and the Galilean transformation. While the Galilean transformation (GT) predicted that reference frames in relative motion preserve the form of Newton's equations, in sharp contrast Maxwell's equations change their form. We say that while Newton's equations are invariant in form only under a GT, Maxwell's equations are not. The GT predicts that the velocity of light should be different in the two systems. Einstein's theory of special relativity did two things. First, it adopted the Lorentz transformation (LT) which preserves the velocity of light and Maxwell's equations in all systems and, second, it required that all the laws of physics should be examined as to their transformation properties under the LT. Those laws which do not keep their form invariant are to be generalized so as to obey an "equivalence postulate" which requires that all physical laws must be phrased in an identical manner for all uniformly moving systems.¹ I question the all-inclusiveness of Einstein's equivalence postulate (EEP).

There is no doubt that there exist physical phenomena that obey EEP. Abundant experimental verification has by now been obtained confirming EEP, and this supports justification for Einstein's postulate for many phenomena. However, the question is whether there are some other phenomena which do not obey EEP and can be described by another postulate. Here, I begin by noting that Newton's equations are invariant under a GT and while they can be generalized to obey EEP, and perhaps provide new insights for phenomena, I see no reason why they should be exclusively made Lorentz invariant. Indeed, making them Lorentz dependent restricts them to behavior subject to the invariant speed of light. I further note that, for some phenomena, some other wave, other than light, may exist which preserves its speed in all systems.

Mathematically, if or and the LT becomes the GT. Thus, while the LT restricts the speed of sublight and light waves the GT admits superlight wave speeds. That such speeds have not been detected does not mean that they do not exist. Gravity waves come to mind. I now postulate that there exists a wave with speed which preserves its speed in all systems. Of course, we must await experimental verification to justify this postulate. In the meantime, I resort to wave theory to obtain some leads. In the following discussion of waves, I treat the wave speed c as having values c=light (for LT phenomena) and c=c_g =gravity (for GT phenomena).

Wave Energy and Momentum

The relativistic Energy-Momentum law ² is

, ................. ................................................. (1)

For quantum, mechanical, and electromagnetic wave phenomena (energy is frequency) and (momentum is wavenumber), m_p=proper mass, and c=speed of light. By setting , equation (1) corresponds ³ to the Lorentz invariant d'Alembertian operator ⁴

, .......... ...................... (2)

in which k_p is a constant, since m_p is finite positive for a real mass and is zero for a photon. Since a sublight or light wave is Lorentz invariant, the sublight wave equation and light wave equation each is invariant under a LT.

While sublight and light time-like wave equations () preserve the LT a superlight space-like wave equation v>c (E<Pc) does not (spatial and time coordinates reverse). Moreover, since Newton's equations are invariant under a GT they cannot be invariant under a LT. These difficulties mean that we must abandon the LT for the study of superlight space-like wave phenomena. For example, if , equation (2) becomes the Galilean invariant operator

, .............. ........................ (3)

in which k_p is assumed constant. This assumption is made for two reasons. First, because m_p is unknown. For example, if m_p=0 then k_p is indeterminate. And, second, because when (3) operates on a scalar wave function it is the product that determines the wave reality.

Equations and are called wave equations because they have periodic solutions . While sublight and light time-like wave equations describe physical waves, superlight space-like wave equations describe mathematical waves and have no corresponding physical equation (1). For example, superlight space-like wave equation has a mathematical correspondence in which the space-time frequency (proportional to energy E)=0 and the space-time wave-number k (proportional to momentum P) is imaginary. Since a superlight wave is Galilean invariant, the superlight wave equation is invariant under a GT. In summary, while sublight and light waves involve Einstein's Energy-Momentum law (1), superlight waves have no energy and have imaginary momentum. Looking for superlight waves with energy detectors is, therefore, a waste of time. Such waves can be detected only by spatial effects such as tides.

Wave Theory

Using Witham's mathematical theory of waves, in particular beginning with the linear Klein-Gordon wave equation ⁵

, .......... , .......................... (4)

in which operates on the scalar wavefunction with frequency (proportional to E) and wavenumber k (proportional to P), k_p=proper mass (rest P=0, rest E=0), and V=potential energy, it has been shown that the Energy-Momentum law (1) leads to Schroedinger's time-like wave equation (when v<c=light) and to the wave equation of light (when v=c=light), both amply confirmed by observation and experiment. It has also been shown, that equation (1) leads to Poisson and Laplace space-like wave equations (3) involving the gravitational potential (when )⁶ , as suggested by observation and experiment ⁷ The connection between potentials and Poisson and Laplace equations is also obtained independently from the classical theory of potential functions.⁸ Accordingly =2(m_pc/h)²V (when v<<c); =0 (when v=0); and =G (when for radial fields).

The linear Klein-Gordon wave equation (4) is the parent equation which unifies sublight, light and gravitational effects.

¹ Herbert Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., 1951 page 187 (Program of Special Relativity). The task of examining the laws of physics for invariance in form is greatly facilitated by writing them in terms of 4-vectors. page 194 (Covariant 4-dimensional formulations); page196 (invariant time interval)

² Id page 204 Equation (6-44).

³ G.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, 1974 page 367 (Correspondence Between Equation and Dispersion Relation).

⁴ Goldstein above pages 198 -199 Equations (6-26)-(6-28).

⁵ Whitham above Part II Dispersive Waves.

⁶ James Constant, Gravitational Action, Pioneer Publishing Co., 1978 Chapters 5 (Planetary Motion), 6 (Photon Motion), 7 (Graviton Motion).

⁷ Tom Van Flandern, The Speed of Gravity-What the Experiments Say. http://www.metaresearch.org/cosmology/speed_of_gravity.asp

⁸ A.P.Wills, Vector Analysis, Prentice Hall, Inc. Pages 114-115.

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