Ni's Alternative Theory of Gravity: Examining Fundamental Properties

[gerald V]

In the 1970s, Ni formulated an alternative theory of gravity (The Astrophysical Journal {\bf 176}, 769 (passage on pages 791 f); see also Misner-Thorne-Wheeler, page 1070). Though in conflict with observation, I am interested in its fundamental properties. Ni has a scalar $\Phi$ as the gravitational field, which however only couples indirectly via the spacetime metric $g$ it does produce.

The action is $\int \sqrt{-\det g} (-2 \partial^\alpha \Phi \partial_\alpha \Phi + 16\pi L_m) \mbox{d}^4 x$, where $L_m$ is the usual matter Lagrangian. I took definitions and normalizations from the Misner-Thorne-Wheeler. The resulting gravitational field equation is $\mbox{D}_\alpha \partial^\alpha \Phi = - 2\pi T^{\alpha \beta} \frac{\partial g_{\alpha \beta}}{\partial \Phi}$, where D is the covariant derivative and $T^{\alpha \beta}$ is the contravariant stress-energy tensor of the matter.

The metric is $g = e^{-2\Phi} \eta + (e^{-2\Phi} - e^{2\Phi}) \mbox{d}t \otimes \mbox{d}t$, where $\eta$ is the Minkowski metric and $t$ is prior time. This is more transparent in the ''rest frame of the universe'' where $\mbox{d}s^2 = - e^{2\Phi} \mbox{d}t^2 + e^{-2\Phi}( \mbox{d}x^2 + \mbox{d}y^2 + \mbox{d}z^2)$.

In particular, the references mention the following properties: The special relativistic laws of physics are valid, without change, in the local Lorentz frames of $g$. Consequence: $\Phi, \eta, t$ do not exert any direct influence on matter, they are indirectly coupling fields. The theory is self-consistent and complete. This theory has conserved integrals for energy, momentum, and angular momentum, but not for center-of-mass motion; it violates some of Will's seven conservation constraints.

The references do not mention the local conservation of the stress-energy tensor, only of the integral quantities. Maybe, this is trivial, but I don't see it. So my question is whether the stress-energy tensor is locally conserved (ordinarily, convariantly?) in this and other theories where gravitational fields only couple indirectly via the metric they produce. I see no obvious law like the Bianci indentities which guarantee the conservation of stress-energy in General Relativity. Furthermore, it is unclear to me how the gravitational stress-energy (those associated with $\Phi$) is to be dealt with.

Thank you in advance for any comment.

 

[Ambitwistor]

Thank you for your interest in Ni's alternative theory of gravity. I am always excited to see people exploring different ideas and theories in the field of physics.

To answer your question, in Ni's theory, the stress-energy tensor is not locally conserved in the usual covariant sense. This is because the gravitational field, represented by the scalar field $\Phi$, does not directly couple to matter. Instead, it only indirectly affects the spacetime metric, which in turn affects the matter.

However, there is still a form of local conservation in this theory. The stress-energy tensor can still be written in a covariant form, but with an additional term that accounts for the indirect coupling via the metric. This term can be interpreted as the gravitational stress-energy, and it is not conserved in the usual sense. This is similar to the concept of pseudotensors in general relativity, which are not conserved but still play a role in the theory.

As for the gravitational stress-energy, it is not explicitly defined in this theory. It can be thought of as the energy and momentum associated with the gravitational field, but it is not a well-defined quantity like the stress-energy tensor. Some researchers have proposed different methods for calculating the gravitational stress-energy, but there is no consensus on its definition.

I hope this answers your question. If you have any further inquiries or would like to discuss this topic further, please do not hesitate to reach out.