GR stands alone in its ability to pass both weak and strong field tests of gravity fields.
From 1905 to 1915, there was renewed interest in a somehow modified scalar field theory. Here is the action of a scalar gravity model, with $z^{\mu}$ a worldline parameterized by $\lambda$:
$$
\begin{align*}I_{sg} =& - m\int d \lambda \sqrt{-\left(e^\Phi \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\left(e^\Phi \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\eta} \\ =& - m\int d \lambda \sqrt{e^{2\Phi} \left( \frac{d z_0}{d \lambda}\right )^2 - e^{2\Phi} \left( \frac{d z_1}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_2}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_3}{d \lambda}\right )^2} \end{align*}
$$
The terms under the square root look just like a metric (completely flat in this case). Physicists knew the $g_{00}$ term from Newton's law. Physicists completely guessed at the $g_{uu}$ terms [see footnote], and that guess was wrong. When a path is varied in the presence of a gravity field, the changes in $g_{uu}$ should be approximately inverse of $g_{00}$.
I am exploring an action (scalar gravity coupling) that at the very least is consistent with experimental tests unknown to physicists prior to 1919:
$$
\begin{align*}I_{sgc} =& - m\int d \lambda \sqrt{-\left(\frac{1}{e^\Phi} \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\left(\frac{1}{e^\Phi} \frac{d z_0}{d \lambda}, e^\Phi \frac{d z_1}{d \lambda}, e^\Phi \frac{d z_2}{d \lambda}, e^\Phi \frac{d z_3}{d \lambda}\right)\eta} \\ =& - m\int d \lambda \sqrt{e^{-2\Phi} \left( \frac{d z_0}{d \lambda}\right )^2 - e^{2\Phi} \left( \frac{d z_1}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_2}{d \lambda}\right )^2 - e^{2\Phi} \left(\frac{d z_3}{d \lambda}\right )^2} \end{align*}
$$
The terms under the square root look like the Rosen metric which will pass all weak field tests (take the velocity equal to square root of $g_{00}$ to $g_{uu}$ to calculate light diffraction, or get the equations of motion, and use the constants of motion to get the same value). Rosen's bi-metric tensor theory fails strong tests of GR, specifically it allows for dipole modes of gravity wave emission which are not consistent with observations of energy loss in binary pulsars.
In the scalar gravity coupling model, the $g_{00}$ must be exactly the inverse of $g_{uu}$. It predicts slightly more bending that GR to second order PPN accuracy (11.7 versus 11.0 $\mu$arcsecond).
New proposals for gravity are usually pretty easy to shoot down. Do you see a flaw in this one? It looks like the Rosen metric and thus curves spacetime for weak fields (good), and is simpler than a bi-metric theory for strong field tests (good). It sure is simpler than GR. Does GR finally have a real competitor?
[footnote]: Metrics can be written in an unending variety of coordinates. I have written this action presuming a point source and Cartesian coordinates. Had I chosen spherical coordinates, then there would be no gravity term in front of $z_2$ or $z_3$. Note added in response to an issue raised by @Trimok.
This question has been put on hold.
For the record, I certainly do believe I "have specific questions evaluating new theories in the context of established science are usually allowed". There is both an action, and a metric associated with the scalar gravity coupling model. That is the canonical method for doing gravity research. The outcome is specific down to 0.7 $\mu$arcseconds (details below), something established physics like work with strings never achieves. It is a rare, but should be a valuable bird of a model that could respect the strong equivalence principle, meaning that gravity is only about making the interval dynamic.
Given a metric for any model, the post post Newtonian deflection can be calculated using a formula from Epstein and Shapiro (Phys. Rev. D, 22:12, p. 2947, 1980):
$$
\begin{align*}
d\tau^2 =& \left(1 - 2 \gamma \frac{G M}{c^2 R} + 2 \beta \left(\frac{G M}{c^2 R}\right)^2 \right) dt^2 - \left( 1 + 2 \gamma \frac{G M}{c^2 R} + \frac{3}{2} \epsilon \left(\frac{G M}{c^2 R}\right)^2 \right) dR^2/c^2 \\
\theta_{_{ppN}} =& ~\pi (2 + 2 \gamma - \beta + \frac{3}{4} \epsilon) \left(\frac{G M}{c^2 R} \right)^2 \\
\theta_{_{GR}} =& ~\frac{15}{16} \pi \left(\frac{G M}{c^2 R} \right)^2 \\
\theta_{_{sgc}} =& ~4 \pi \left(\frac{G M}{c^2 R} \right)^2
\end{align*}
$$
The difference between the two will be about 6%.
A longer blog on this subject is online. Feel free to contact me if you any technical issues with the proposal, large or small.
This post imported from StackExchange Physics at 2014-03-05 14:56 (UCT), posted by SE-user sweetser
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