Mathematical Physics   Einstein's Field Theory is Wrong and Nordstr\"om's Correct

Authors: Jorma Jormakka

Gunnar Nordstr\"om published his second gravitation theory in 1913. This theory is today considered to be inconsistent with observations. At this time Einstein was working on his field theory, the General Relativity Theory. Einstein's theory has been accepted as the only theory of gravitation consistent with measurements. The article reconsiders Nordstr\"om's theory and proves the following claims. 1) If gravitation is caused by a scalar field, then the theory is Nordstr\"om's second gravitation, which in a vacuum outside a point mass reduces to his first gravitation theory. Nordstr\"om's scalar field theory gives proper time values that fully agree with gravitational redshift in the Pound-Rebka experiment and with the Shapiro time delay in Shapiro's radar bouncing experiment. Gravitation in Schwarzschild's solution is not a field but a deformed geometry. If proper time is calculated via the General Relativity formula, Schwarzschild's solution fails both the Pound-Rebka redshift and Shapiro time delay tests because the ball in Schwarzschild's solution is deformed and light as measured by an external clock can exceed \$c\$. 2) The third classical tests of Einstein's theory is the movement of the perihelion of Mercury. Calculations from Schwarzschild's exact solution to Einstein's equations gave a correction that very well fitted the unexplained part of Mercury's movement. However, Schwarzschild's solution as a stationary solution it fails to explain why the orbit of Mercury, or any planet, is an ellipse. It is shown that the customary proof of Kepler's law stating that the orbit is an ellipse is incorrect: under a central stationary Newtonian force the orbit of a two mass system can only be a circle or (almost) a hyperbole because of conservation of energy. This observation invalidates the movement of Mercury as a test of General Relativity: Schwarzschild's solution cannot produce an elliptic orbit, therefore it is not the solution and that it gives a correct size modification to the movement of the perihelion is just a coincidence. Nordstr\"om's theory remains inconclusive in the Mercury test because calculating the orbit is difficult and cannot be done in this article. Nordstr\"om's theory, however, offers a possibility for explaining elliptic orbits: some energy is needed for waves in time-dependent solutions to Nordstr\"om's field equation and this loss of potential energy from the radial potential can lead to elliptic orbits. 3) The fourth classical test is the light bending test. Light bends in Nordstr\"om's theory as light behaves as a test mass in a gravitational field. Calculation of the amount of light bending in Norstrs\"om's theory is similar to calculation of the orbit of planets and beyond the scope of this article. Theoretical consideration of bending of light leads to the conclusion that the stress-energy tensor in the General Relativity is incorrect: the diagonal entries should contain the energy of a stationary gravitational field in the vacuum outside a point mass and therefore diagonal Ricci tensor entries cannot be zeroes. Nordstr\"om's theory passes this theoretical consideration while Einstein's theory fails it.