Scalar theory of gravitation

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## A scalar theory of gravitation

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The basic field equation of a scalar theory of gravitation from Poth published in 2012/2013 [1] is

( 1 )       1 c 2 ∂ 2 Φ ∂ t 2 − ∇ 2 Φ = κ ⋅ ρ , {\displaystyle (1)~~~{\frac {1}{c^{2}}}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}-\nabla ^{2}\Phi =\kappa \cdot \rho ,}

wherein  Φ {\displaystyle \Phi } is the scalar gravitational potential and  ρ {\displaystyle \rho } the proper time density of the mass being the source of the gravitational potential and  κ {\displaystyle \kappa } is a constant incorporating the gravitational constant G.  κ {\displaystyle \kappa } is given by

( 2 )       κ = 4 π h c 2 G , {\displaystyle (2)~~~\kappa =4\pi {\frac {h}{c^{2}}}G,}

wherein h is Planck's constant and c is the veleocity of light.

The proper time density  ρ {\displaystyle \rho } is defined by

( 3 )       ρ = ν M Δ V , {\displaystyle (3)~~~\rho ={\frac {\nu _{M}}{\Delta V}},}

wherein  ν M {\displaystyle \nu _{M}} is the frequency of mass M being regarded as an oscillator according to de Broglie and  Δ V {\displaystyle \Delta V} is the volume in which that mass is situated. Thus  ρ {\displaystyle \rho } is a scalar. Hence, (3) is Lorentz invariant.

Thus, the gravitational interaction becomes essentially an interaction between oscillators or between clocks representing masses. That yields directly the dependency of clock rates on the gravitational potential. And as the frequencies for the masses are velocity dependent, the gravitational interaction becomes dependent on the velocities of the interacting masses.

Thus, the heavy mass  m h {\displaystyle m_{h}} of an inertial rest mass  m 0 {\displaystyle m_{0}} is given by

( 4 )       m h = m 0 1 − v 2 c 2 {\displaystyle (4)~~~m_{h}=m_{0}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}

Usually, that can not be noticed, in particular not under common laboratory conditions for determining the gravitational constant G.

The relativistic action integral for a mass  m 0 {\displaystyle m_{0}} moving at velocity v in the gravitational potential of an other mass M at rest becomes

( 5 )       A = − ∫ ( m 0 c 2 − G M r m 0 1 − v 2 c 2 ) 1 − v 2 c 2 ⋅ d t , {\displaystyle (5)~~~A=-\int \left(m_{0}\,c^{2}-{\frac {GM}{r}}m_{0}\,{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)\,{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\cdot dt,}

That results eventually in the energy H given by

( 6 )       H = m 0 c 2 1 − v 2 c 2 − G M m 0 r ( 1 + v 2 c 2 ) , {\displaystyle (6)~~~H={\frac {m_{0}c^{2}}{\sqrt {1-\displaystyle {\frac {v^{2}}{c^{2}}}}}}-{\frac {GMm_{0}}{r}}\left(1+{\frac {v^{2}}{c^{2}}}\right),}

Hence, the gravitational potential has become „velocity dependent“. From that can be derived for low velocities in polar coordinates

( 7 )       H = m c 2 + m 2 r ˙ 2 + l 2 2 m r 2 − G M m r − G M m r r ˙ 2 c 2 − G M c 2 l 2 m r 3 {\displaystyle (7)~~~H=mc^{2}+{\frac {m}{2}}{\dot {r}}^{2}+{\frac {l^{2}}{2mr^{2}}}-{\frac {GMm}{r}}-{\frac {GMm}{r}}{\frac {{\dot {r}}^{2}}{c^{2}}}-{\frac {GM}{c^{2}}}{\frac {l^{2}}{m\,r^{3}}}}

The known last term in that equation causes the perihelion precession [2]. However, the second last term causes a correspondingly small reduction of the orbital period, for Mercury of about 0,19 sec per orbit which can be translated in a correspondingly small reduction of the orbital distance from the sun. The geodesic precision and the frame dragging effect as observed by the Gravity B Probe follow also and more.

If two stars of equal mass M orbit each other at the radius r from their center of mass as in a binary star under essentially nonrelativistic conditions, a gravitational wave is created from their retardation difference as seen from a distant observer. That is not a dipole radiation but corresponds to a quadrupole radiation. The radiant power P of that wave is given up to a factor by

( 5 )       P ∝ G c 5 ω 6 M 2 r 4 , {\displaystyle (5)~~~P\propto {\frac {G}{c^{5}}}\,\omega ^{6}\,M^{2}\,r^{4},}

like in the general theory of relativity. The factor appears to be 3/2 times of that obtained from the general theory of gravitation cf. e.g.[3] [4].

In view of the basic field equation (1) a corresponding Lagrangian can be drafted and, hence, also the energy density of the gravitational waves which could be directly quantized. As they are scalar waves, their spin is zero. There is however no direct determination of the Lagrangian possible.

## References

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1. ↑ 'A new Theory of Gravitation and its Quantization - Version 1.1', from Hartwig Poth on 20.01.2013 under ISBN 978-3-8442-4654-4
2. ↑ Albert Einstein, Erklärung der Periheldrehung des Merkur aus der Allgemeinen Relativitätstheorie (Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1915), p.838
3. ↑ Torsten Fließbach, 'Allgemeine Relativitätstheorie' (Spektrum Akademischer Verlag, Heidelberg Berlin, 2003, ISBN 3-8274-1356-7), p.199, 202-205
4. ↑ L. D. Landau E. M. Lifschitz, 'Lehrbuch der theoretischen Physik II Klassische Feldtheorie' (Akademie Verlag, Berlin, 1967), p.357 'Aufgabe'

## Comments

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### Feynman's arguments

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In the scalar theory the field equation simply coincides with the common wave equation. In fact, when being applied it is not so simple. It appears that Feynman had such a scalar model in mind [1] but eventually dismissing it; at that time he had apparently no physical entity in mind providing for the gravitation interaction and being porportional to  1 − v 2 / c 2 {\displaystyle {\sqrt {1-v^{2}/c^{2}}}} . In particular he didn't consider the proper time of a mass particle  m {\displaystyle m} as a possible entity, albeit it had been commonly know that the Lagrangian  L m {\displaystyle L_{m}} of a mass particle is given by [2]

L m = − m c 2 1 − v 2 / c 2 {\displaystyle L_{m}=-mc^{2}{\sqrt {1-v^{2}/c^{2}}}}

All the less some kind of spatial density of such an entity had he considered. Nevertheless, the scalar theory can be drafted, at least as such, abeit not every mathematical model has to be implemented by nature.

1. ↑ Richard P. Feynman et al., 'Feynman Lectures on Gravitation' (Addison-Wesley Publishing Company, Reading Massachusetts, 1995, ISBN 0-201-62734-5), p.30 bottom
2. ↑ Herbert Goldstein, 'Klassische Mechanik' (Akademische Verlagsgesellschaft, Frankfurt am Main, 1963) eq. (6-49) on p.228

### The classical particles concept

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It should be noted that rest mass particles are considered in that theory as classical single particles. That means, they have a positive rest energy and there are no conrresponding antiparticles of them. The relationship between the relativistic energy of the rest mass particle and its energy in the gravitational potential in principle does not appear to imply negative energy solutions. Therefore, that theory would not comply with the Dirac equation comprising anti-electrons having negative energies. Nevertheless, such a theory should be some approximation for very weak gravitational potentials and very low velocities.