Reference for coordinate view of equations of motion

[Jonathan Scott]

Some time in the 1980s when I first started studying relativistic gravity, for ease of comparison with Newtonian and Special Relativity gravity I worked through pages of geodesic equations for a general isotropic coordinate system with spherical symmetry, converting everything to terms relating to their Newtonian equivalents and using coordinate time instead of proper time. I discovered that if I moved some terms to the other side of the equation, so that the equation of motion was expressed in terms of coordinate momentum rather than coordinate velocity, it became dramatically simpler, making the difference from Newtonian theory very clear. (Unfortunately, I do not know what happened to those notes). I later discovered a much quicker derivation in about 3 lines starting from the Lagrangian for the coordinate momentum. As I was new to relativistic gravity at the time, I assumed that this simple mathematical transformation must be well-known, especially as I'm aware that isotropic coordinates are routinely used for space navigation.

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.

 

[Dale]

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?

Seems legitimate. Here is the direct link

https://arxiv.org/abs/1503.01970

The journal is pretty low impact, but it is published by Springer so it is not considered a predatory publisher.

Here's what my version looks like

I am a little uncomfortable with you posting a reference to a paper and then immediately posting your own unpublished material. Particularly considering your previous history.

I am temporarily closing the thread to give the mentors some time to review

EDIT: after reviewing, the mentors have decided to open this thread up for discussion of the article and other related scientific literature publications. This thread will not be allowed to degenerate into personal speculation, so all participants are expected to have sources for all of their claims.

 

[Jonathan Scott]

A couple of corrections which I couldn't edit in time because the thread was locked:

1. Where I said "approximately equal to the reciprocal of the corresponding space scale factors B and C" that's slightly garbled, because the author has actually used 1/B and 1/C as the radial and tangential space scale factors, to make B and C the same as A for the exponential case. It could for example say "scale factors 1/B and 1/C".

2. My definition for $\mathbf{g}$ is missing a minus sign. It should be $-(c^2/\Phi) (\nabla \Phi)$. For example, if $\Phi$ is the exponential $\exp(-GM/rc^2)$ as used in various approximations (where for this purpose $c$ denotes the standard value), then we have the following in the radial direction:
$$\mathbf{g} = -(c^2/\exp(-GM/rc^2)) (GM/r^2c^2) \exp(-GM/rc^2) = -GM/r^2$$

Also, to avoid confusion please note that by "my version" I simply mean what happens when I rewrite the referenced equation (58) using my preferred notation to make the Newtonian analogy even more obvious. That is, plain $c$ instead of $c(r)$ and using my stated expression for $\mathbf{g}$, which maps to his form as follows:
$$\frac{E}{c^2} \mathbf{g} = \frac{E}{c^2} \left (- \frac{c^2}{A} {\nabla A} \right ) = -E \frac{\nabla A}{A} = -E \frac{A'}{A} \mathbf{e}_\parallel$$

On the left hand side, equation (58) in the reference uses $\mathbf{f}_\parallel$ to denote the component of $d\mathbf{p}/dt$ which is parallel to the field, as defined earlier in equation (53). As previously mentioned, the other component is zero by equation (56), so the left is actually the whole rate of change of coordinate momentum.

For solar system purposes, the specific exponential metric mentioned above is a close enough approximation, as the resulting PPN parameters are the same as for GR, and for that particular metric the reference equation (58) and my Newtonian notation equivalent are exact.

I think that this is a neat and memorable result which I've found extremely useful in understanding motion in a gravitational field, and I've been quite shocked to find that it isn't better known.

I understand that my use of $c$ to denote the coordinate speed of light in this specific context is decidedly unconventional. The article author tries to avoid this by using $c(r)$ to make it clear this is a function of the radial coordinate, but that's true of other quantities too which are not similarly notated. For coordinate values such as $x$, $v$ and $t$ we do not normally use any special indication, so notating the coordinate speed of light as $c$ seems logically consistent, but perhaps a step too far. It does not seem appropriate to use a prime or similar for the coordinate value because that's more likely to refer to local values rather than coordinates (and the article author also uses a prime for derivatives). As the clarity of this equation relies on its Newtonian resemblance I can't see any obvious way to avoid this unusual notation, although of course it has to be made clear wherever it is used.

Of course, in Special Relativity $c$ is just the conversion factor between space and time units and can be set to 1, but for general relativity (particularly in the isotropic chart) there is a concept of the coordinate speed of light which can be very useful and which varies from 1, although it can obviously be scaled so that the standard value is still 1. If we don't retain a symbol for this, we have to start writing expressions which involve bits of the metric to express the coordinate speed of light, which is also the maximum magnitude of a coordinate velocity. Perhaps in cleaning up GR notation to eliminate $c$ (and often $G$) something useful has been lost in translation.

 

[Dale]

my preferred notation to make the Newtonian analogy even more obvious. That is, plain c instead of c(r) and

If you write c then people will assume a constant which depends only on your choice of units and not a function of r (I.e. people assume $\partial_r c=0$). You will cause confusion in your communication, so let's stick with the published notation.

I think that Winkler's metric is overly flexible. Carroll shows (7.12 and 7.13 at https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html ) that a general spherically symmetric metric needs only two free functions instead of the three used by Winkler. He doesn't derive or cite a reference for his metric.

 

[Jonathan Scott]

Your preferred notation is a bad idea. If you write c then people will assume a constant which depends only on your choice of units and not a function of r (I.e. people assume $\partial_r c=0$). You will cause confusion in your communication.

I'm perfectly aware of that, if I fail to call attention to the unusual convention. Do you think you have a better alternative?

I feel that <personal speculation deleted>

 

[Dale]

Do you think you have a better alternative?

While discussing Winkler's paper use Winkler's notation.

I feel that...

I remind you that this thread is not for discussing your personal opinions, speculations, or feelings. Stick to published material

 

[Jonathan Scott]

I think that Winkler's metric is overly flexible. Carroll shows (7.12 and 7.13 at https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html ) that a general spherically symmetric metric needs only two free functions instead of the three used by Winkler. He doesn't derive or cite a reference for his metric.

I presume his three functions are intended to give him the flexibility to express the same metric (e.g. Schwarzschild) in different charts. He later picks out special simple cases where $B = C$ (general isotropic) and $A = B = C$ (for example the exponential metric I previously mentioned). Equation (58) is only exact for this latter case.

 

[vanhees71]

Seems legitimate. Here is the direct link

https://arxiv.org/abs/1503.01970

Hint: The published version is at least readable, the arXiv version a typographical desaster (Word?).