Theory of Gravity as Theory of Local Inertial Frames

Abstract

It is proved that the affine theory of the full Riemann tensor constitutes the extended theory of gravity. The fundamental object here is a spacetime symmetric connection. Its physical interpretation is that of a field of local inertial frames. In this approach, gravity arises, in a natural way, as a local version of Newton's First Law. A variational formulation of the theory is presented, based on the results introduced in the article co-authored by the present author and one of the supervisors. The general framework is supported by several examples. A central result of the dissertation is the transition from the affine picture to the metric picture. A simplified version of this procedure -- valid for a specific class of Lagrangians. It was proved there that the non-metricity in the affine picture can be interpreted as a matter field in the metric picture. This is precisely Hermann Weyl's interpretation of electromagnetism. Here, the transformation from the full Riemann tensor theory to the conventional metric theory is examined in full generality for the first time and illustrated using the previously introduced examples.

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BibTeX RIS

BibTeX

@misc{bak2026theory,
  title = {{Theory of Gravity as Theory of Local Inertial Frames}},
  author = {Bartłomiej Bąk},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22354},
  eprint = {2609.22354},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22354},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Theory of Gravity as Theory of Local Inertial Frames
AU  - Bartłomiej Bąk
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.22354
UR  - https://arxiv.org/abs/2609.22354
AB  - It is proved that the affine theory of the full Riemann tensor constitutes the extended theory of gravity. The fundamental object here is a spacetime symmetric connection. Its physical interpretation is that of a field of local inertial frames. In this approach, gravity arises, in a natural way, as a local version of Newton's First Law. A variational formulation of the theory is presented, based on the results introduced in the article co-authored by the present author and one of the supervisors. The general framework is supported by several examples. A central result of the dissertation is the transition from the affine picture to the metric picture. A simplified version of this procedure -- valid for a specific class of Lagrangians. It was proved there that the non-metricity in the affine picture can be interpreted as a matter field in the metric picture. This is precisely Hermann Weyl's interpretation of electromagnetism. Here, the transformation from the full Riemann tensor theory to the conventional metric theory is examined in full generality for the first time and illustrated using the previously introduced examples.
ER  -

Abstract

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