Linearized gravity

In theoretical physics, linearized gravity is the application of perturbation theory to the metric tensor in Einstein's field equations. Linearized gravity is useful for modeling the effects of gravity when the gravitational field is weak — in particular for studying gravitational waves and weak-field gravitational lensing.

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Standard cosmological models rely on an approximate treatment of gravity, utilizing solutions of the linearized Einstein equations as well as physical approximations. In an era of precision cosmology, we should ask: are these approximate predictions sufficiently accurate for comparison to observations, and can we draw meaningful conclusions about properties of our Universe from them? In this work we examine the accuracy of linearized gravity in the presence of collisionless matter and a cosmological constant utilizing fully general relativistic simulations. We observe the gauge dependence of corrections to linear theory, and note the amplitude of these corrections. For perturbations whose amplitudes are in line with expectations from the standard Λ cold dark matter model, we find that the full, general relativistic metric is well described by linear theory in Newtonian and harmonic gauges, while the metric in comoving-synchronous gauge is not. For the most extreme observed structures in our Universe, such as supervoids, our results suggest that corrections to linear gravitational theory can reach or surpass the percent level in all gauges.
- John T. Goblin Jr., James B. Mertens, Glenn D. Starkman, and Chi Tian, (2019). "Limited accuracy of linearized gravity". Physical Review D 99 (2). DOI:10.1103/PhysRevD.99.023527. -arxiv.org preprint

Both classical and quantum corrections can be analyzed by a perturbative approach based on the so-called weak field approximation. The heart of Einstein’s theory is represented by its ten coupled partial differential equations. The solutions of these equations, i.e., the gravitational potentials, are the metric tensor components. In the weak field approximation, the metric tensor can be decomposed in two terms: the flat Minkowski metric and the small perturbation multiplied by the gravitational constant. The solution of the non-linear equations can be considered the sum of infinite terms, and Newton’s theory emerges in the linear order. Léon Rosenfeld ... used this approximation to investigate the infinities that emerge by applying the early QFT techniques to quantize the gravitational interaction (Rosenfeld, 1930). Different kinds of divergent quantities had already appeared in the context of QED at the end of the 1920s: their correct treatment will be clarified only after the Second World War.
- Alessio Rocci and Thomas Van Riet, (2024). "The quantum theory of gravitation, effective field theories, and strings: yesterday and today". The European Physical Journal H 49 (1; article number 7). DOI:10.1140/epjh/s13129-024-00069-4. arxiv.org preprint (quote from p. 7 of preprint)

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