>[!warning] >This content has not been peer reviewed. # From AQUAL to TeVeS — Bekenstein's relativistic MOND **Bekenstein and collaborators** built a full relativistic framework for MOND (Modified Newtonian Dynamics): first **AQUAL** (A Quadratically Modified Lagrangian), then **TeVeS** (Tensor–Vector–Scalar), which recovers MOND in the weak-field limit and is consistent with gravitational lensing and cosmology. So “MOND-like” dynamics is not just a curve-fit; it can be derived from a covariant theory with a transition scale $a_0$. --- ## What it is - **AQUAL (1984):** A modified Poisson equation that yields the MOND force law $F \propto \sqrt{a_0 M}$ in the low-acceleration regime (Bekenstein & Milgrom 1984). The transition is smooth (e.g. interpolating function). - **TeVeS (2004):** A relativistic completion: metric = tensor + vector + scalar (Bekenstein 2004). It reproduces MOND phenomenology, explains lensing without dark matter, and has a cosmological limit. The constant $a_0 \approx c H_0/(2\pi)$ appears as a fundamental scale. - **Implication:** There exists a consistent, relativistic theory whose weak-field limit is MOND. RST does not *replace* it but can **interpret** the transition scale and the exponent in terms of the substrate (e.g. $n_0$, resolution, relational distance). --- ## How RRT/RST uses it - **SPARC and RST:** The RST fits to SPARC ([[SPARC Evaluation Verification]]) use the same low-acceleration regime (flat rotation curves, BTFR). Bekenstein’s framework shows that this regime can come from a covariant theory with one scale $a_0$. RST adds: $a_0$ and the **[[Transition Sharpness]]** $n_0$ can be tied to the **[[Relational distance]]** and backbone dimension of the format — i.e. the transition is where the substrate’s resolution/backbone scaling kicks in. - **No extra parameters:** In RST, $n_0 \approx d_B$ (backbone dimension) and $a_0$ can be linked to the substrate’s refresh/scale; the “MOND sector” does not introduce new free constants beyond what the format already implies. --- ## Links | Concept | Note | |:---|:---| | RST fit to SPARC, BTFR | **[[SPARC Evaluation Verification]]** | | Exponent $n_0$ and backbone | **[[Transition Sharpness]]**, **[[Backbone Dimension]]** | | Distance and cost in the substrate | **[[Relational distance]]** | --- ## References - Bekenstein, J. D. & Milgrom, M. (1984). *Does the missing mass problem signal the breakdown of Newtonian gravity?* Astrophys. J. **286**, 7–14. [DOI](https://doi.org/10.1086/162570) (AQUAL.) - Bekenstein, J. D. (2004). *Relativistic gravitation theory for the modified Newtonian dynamics paradigm.* Phys. Rev. D **70**, 083509. [DOI](https://doi.org/10.1103/PhysRevD.70.083509) [arXiv:astro-ph/0403694](https://arxiv.org/abs/astro-ph/0403694) (TeVeS.) - Famaey, B. & McGaugh, S. S. (2012). *Modified Newtonian Dynamics (MOND): Observational phenomenology and relativistic extensions.* Living Rev. Relativ. **15**, 10. [DOI](https://doi.org/10.12942/lrr-2012-10) [arXiv:1112.3960](https://arxiv.org/abs/1112.3960) (Review of MOND and relativistic completions.)