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Jonathan Scott

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However, I've had problems whenever I mention the results on the forums because in practice it doesn't seem to be well known and it seems that my own mathematical derivation is not considered an acceptable reference.

Today I discovered a paper on the ArXiv where someone has gone along exactly the same path that I did, with very similar notation, but apparently considered it sufficiently novel to be worth publishing: The article is arXiv:1503.01970 "Diagonal Metrics of Static, Spherically Symmetric Fields The Geodesic Equations and the Mass-Energy Relation from the Coordinate Perspective" by F-G Winkler, published in Int. J. Theor. Phys. 52, 3045-3056 (2013). Is this considered an acceptable reference?

The fact that someone would consider this sufficiently novel to be worth publishing was a surprise to me, as I had assumed that people would naturally have already looked for the closest possible analogy with Newtonian gravity. Winkler's notation is in some ways even more controversial than mine, mainly because he also covers the case of non-isotropic coordinate systems, where the coordinate speed of light is different in directions, which means that coordinate mass values also vary according to whether they are perpendicular or parallel to the field. However, this additional complication is not needed in the case of isotropic coordinates.

Here's what my version looks like for the motion of a test mass in a central field in the approximation where the metric factors for space and time are exact reciprocals of one another. Here ##c## is the coordinate speed of light (which is a function of ##r## and the same in all directions at any point in isotropic coordinates) and ##\mathbf{g}## represents the equivalent of the Newtonian gravitational field, ##(c^2/\Phi) (\nabla \Phi)## where ##\Phi## is the time dilation factor from the metric (the square root of the ##dt^2## coefficient, called ##A## in equation (4) of the referenced paper, approximately equal to the reciprocal of the corresponding space scale factors ##B## and ##C## in the paper).

$$\frac{d\mathbf{p}}{dt} = \frac{d}{dt} \left ( \frac{E\mathbf{v}}{c^2} \right ) = \frac{E}{c^2} \mathbf{g} \left ( 1 + \frac{v^2}{c^2} \right )$$

This version is equivalent to equation (58) in the referenced paper, as this approximation is equivalent to assuming the exponential metric, and the perpendicular component of force is zero because ##B = C## as mentioned in equation (56). For a general isotropic metric this can be made exact even for strong fields by using two separate ##\mathbf{g}## values which depend on the time and space factors in the metric separately:

$$\frac{d\mathbf{p}}{dt} = \frac{E}{c^2} \left ( \mathbf{g_t} + \mathbf{g_{xyz}} \, \frac{v^2}{c^2} \right )$$

For the specific case of General Relativity, one can substitute the appropriate values (for the isotropic coordinate form of the Schwarzschild solution) in the metric and then use the orbit equations combined with this equation of motion to confirm the perihelion precession of Mercury and compare it with any other theory that can be expressed in isotropic coordinates for the spherical static case.

As the total energy ##E## is constant in these coordinates, one can also divide both sides by it to give an equation of motion in terms of ##v/c^2##, making it clear that this does not depend on the energy of the test mass.

I personally find this form of the equation of motion very simple and very enlightening. For example, it is clear that the coordinate force is always aligned with the field (so it cannot change the coordinate angular momentum) and does not depend on the direction of motion, only on the speed. This means for example that the rate of change of momentum of a pulse of light is the same regardless of direction, so even if it is going upwards or downwards relative to the field, the rate of change of momentum is still downwards. In contrast, the rate of change of velocity does not behave so neatly; for motion in a general direction it is not necessarily aligned with the field, and the coordinate speed of a falling pulse of light decreases rather than increases.