Research

Modified-theory of Gravity : An lternative approach

Content

Introduction
Why we need an alternative theories?
Notes
Reference

Introduction

In the current standard ΛCDM cosmological model, the gravitational interaction is described by Einstein’s theory of General Relativity (GR). The main reason for this is perhaps related to its remarkable agreement with a wealth of precision tests of gravity done in the Solar System. These include the classical tests of gravitational redshift, the lensing of the light from background stars by the Sun and the anomalous perihelion of Mercury, as well as other tests such as the Shapiro time-delay effect measured by the Cassini spacecraft and Lunar laser ranging experiments which meausure the rate of change of the gravitational strength in the Solar system. Outside of the Solar system, GR is also in good agreement with the tests that involve changes in the orbital period of binary pulsars due to the emission of gravitational waves. More recently, GR was proven via the direct detection of gravitational waves from the mergers of the binary black holes and binary neutron stars by the advanced LIGO and advanced virgo detectors.

Why we need an alternative theories?

The detection of gravitational waves from the mergers of binary black holes and binary neutron stars by the advanced LIGO and Virgo detectors confirmed key predictions of General Relativity (GR) and provided the first direct evidence of stellar-mass black holes. However, the presence of singularities at the centers of black holes suggests limitations in GR, as the equivalence principle breaks down at these points. This implies that GR may not be a universal theory of space-time. In the low-energy regime, both theoretical and observational challenges with the ΛCDM model further suggest the need to explore alternatives to GR as the fundamental theory of gravity. Unlike GR, which involves second-order derivatives in its field equations, modified gravity theories often incorporate higher-order derivatives of the Ricci or Riemann tensors, leading to potentially significant deviations from GR. With various approaches to modifying GR, particularly in the contexts of strong gravity and cosmological scales, each alternative model offers unique characteristics.

This leads to the following crucial question: "Are there a set of unique signatures that distinguish GR from modified gravity (MG) theories?"

There are several reasons why alternative theories to General Relativity (GR) are considered, especially in the context of high-energy physics, cosmology, and astrophysics:

Dark Matter and Dark Energy: GR explains the large-scale structure of the universe, but it requires the existence of dark matter and dark energy to account for certain observations, such as the rotation curves of galaxies and the accelerated expansion of the universe. These unknown components suggest that either new forms of matter/energy exist or that GR may not fully describe gravity at cosmic scales.
Singularities: The formation of singularities, such as those inside black holes or the initial Big Bang singularity, indicates a breakdown of GR, where physical quantities like density and curvature become infinite. This suggests that GR is incomplete and cannot describe the true nature of space-time under extreme conditions.
Quantum Gravity: GR is incompatible with quantum mechanics. A consistent theory of quantum gravity is needed to describe phenomena at the Planck scale, where the effects of both quantum mechanics and gravity are significant. Modifying or extending GR is necessary to reconcile gravity with quantum theory.
Cosmic Inflation: The early universe’s rapid expansion (inflation) is not directly explained by GR. While the inflationary model fits observations, it introduces hypothetical fields (inflaton) and mechanisms, motivating alternative theories that might naturally explain this phase without additional assumptions.
Galactic Rotation Curves: The observed rotation curves of galaxies do not match the predictions of GR when only visible matter is considered. Modified gravity theories, like Modified Newtonian Dynamics (MOND), attempt to explain these discrepancies without invoking dark matter.
Unification with Other Forces: GR only describes gravity, while the Standard Model explains the other three fundamental forces (electromagnetic, weak, and strong). Efforts to unify gravity with these forces, such as string theory or loop quantum gravity, often involve modifying or extending GR.

These challenges motivate physicists to explore alternative theories of gravity that could address these unresolved issues and provide a more complete understanding of the universe.

Basic Concept

The extensions of General Relativity (GR) are designed to address some of the issues mentioned earlier, such as dark matter, dark energy, and quantum gravity. Certainly! Here's an overview of some basic concepts in modified gravity theories, expressed mathematically:

Extended Gravity Theories:In standard General Relativity (GR), the gravitational field is described by the Einstein field equations: \[ G_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu} \] where $ G_{\mu\nu} $ is the Einstein tensor, $ T_{\mu\nu} $ is the stress-energy tensor, $ G $ is the gravitational constant, and $ c $ is the speed of light. Modified gravity theories extend or alter this framework.
Here $G_{\mu\nu}$ is defined as:
$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu}R +\Lambda g_{\mu\nu}$$ where:
- $R_{\mu\nu}$ is the Ricci curvature tensor,
- $R$ is the Ricci scalar (trace of the Ricci tensor),
- $g_{\mu\nu}$ is the metric tensor of space-time,
- $\Lambda$ is the cosmological constant.
- $T_{\mu\nu}$ is the energy-momentum tesnor,
- $G$ is the gravitational constant, and $c$ is the speed of light.
F(R) Theories: One of the simplest modifications is the $ f(R) $ theory, where the Einstein-Hilbert action is replaced with a function of the Ricci scalar $ R $: \[ S = \frac{1}{16 \pi G} \int \left[ f(R) \sqrt{-g} \, d^4x \right] + S_m(g_{\mu\nu}, \psi) \] where $ f(R) $ is a function of the Ricci scalar $ R $, $ g $ is the determinant of the metric tensor $ g_{\mu\nu} $, and $ S_m $ is the action for matter field $\psi$. The field equations for $ f(R) $ theories become: \[ f'(R) R_{\mu\nu} - \frac{1}{2} f(R) g_{\mu\nu} + \left( \nabla_\mu \nabla_\nu - g_{\mu\nu} \Box \right) f'(R) = \frac{8 \pi G}{c^4} T_{\mu\nu} \] where $ f'(R) $ is the derivative of $ f(R) $ with respect to $ R $, and $ \Box $ is the d'Alembertian operator.
f(R, T) Theories: In $ f(R, T) $ theories, the action is modified to include the stress-energy tensor $ T $ as well: \[ S = \frac{1}{16 \pi G} \int \left[ f(R, T) \sqrt{-g} \, d^4x \right] + S_m \] The field equations are: \[ f_R R_{\mu\nu} - \frac{1}{2} f(R, T) g_{\mu\nu} + \left( \nabla_\mu \nabla_\nu - g_{\mu\nu} \Box \right) f_R = \frac{8 \pi G}{c^4} \left( T_{\mu\nu} + T_{\mu\nu}^{\text{(extra)}} \right) \] where $ f_R $ is the partial derivative of $ f $ with respect to $ R $, and $ T_{\mu\nu}^{\text{(extra)}} $ represents additional terms arising from the modification.
Scalar-Tensor Theories: In scalar-tensor theories, the gravitational interaction is mediated by both a tensor field (the metric) and a scalar field $ \phi $. The action is: \[ S = \frac{1}{16 \pi G} \int \left[ \phi R - \frac{\omega(\phi)}{\phi} (\nabla \phi)^2 \sqrt{-g} \, d^4x \right] + S_m \] The field equations for scalar-tensor theories are: \[ \phi R_{\mu\nu} - \frac{1}{2} \left( \phi R - \frac{\omega(\phi)}{\phi} (\nabla \phi)^2 \right) g_{\mu\nu} + \nabla_\mu \nabla_\nu \phi - g_{\mu\nu} \Box \phi = \frac{8 \pi G}{c^4} T_{\mu\nu} \] where $ \omega(\phi) $ is a function describing the coupling between the scalar field and the Ricci scalar.
TeVeS Theory: The Tensor-Vector-Scalar (TeVeS) theory is a relativistic generalization of MOND (Modified Newtonian Dynamics). It introduces a vector field $ A_\mu $ and modifies the Einstein-Hilbert action as follows: \[ S = \frac{1}{16 \pi G} \int \left[ R \sqrt{-g} \, d^4x - \frac{1}{4} \lambda (\nabla^\mu A_\mu)^2 \sqrt{-g} \, d^4x \right] + S_m \] The field equations include contributions from both the tensor and vector fields: \[ G_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu} - \frac{1}{2} \left( \nabla_\mu A_\nu \nabla^\mu A^\nu - \frac{1}{2} (\nabla^\mu A_\mu)^2 \right) g_{\mu\nu} \] These equations describe how the scalar and vector fields interact with gravity and matter. Each of these modifications to GR introduces new dynamics and predictions that can be tested against observations. They aim to address the limitations of GR or provide explanations for phenomena that GR cannot fully account for.
Higher-Derivative Theories: Modified gravity theories may involve higher-order derivatives of the metric. For example, in $ f(R) $ gravity, the action is modified to: \[ S = \frac{1}{16 \pi G} \int d^4x \sqrt{-g} \left[ f(R) + \mathcal{L}_\text{matter} \right] \] where $ \mathcal{L}_\text{matter} $ is the matter Lagrangian and $ g $ is the determinant of the metric tensor $ g_{\mu\nu} $. The function $ f(R) $ introduces additional terms into the field equations that include higher derivatives of the metric.
Brans-Dicke Theory: In Brans-Dicke theory, the field equations are modified to: \[ G_{\mu\nu} = \frac{8 \pi G}{\phi} \left( T_{\mu\nu} - \frac{1}{2} T g_{\mu\nu} \right) + \frac{1}{\phi} \left( \nabla_\mu \nabla_\nu \phi - g_{\mu\nu} \Box \phi \right) \] where $ \phi $ is a scalar field, and $ \nabla_\mu \nabla_\nu \phi $ represents the covariant derivative of $ \phi $.
K-essence Theory: In K-essence theories, the action is: \[ S = \int d^4x \sqrt{-g} \left[ \frac{R}{16 \pi G} - \frac{1}{2} \left( \nabla_\mu \phi \nabla^\mu \phi \right) - V(\phi) + \mathcal{L}_\text{matter} \right] \] where $ \phi $ is a scalar field, and $ V(\phi) $ is a potential term. The scalar field $ \phi $ can drive cosmic acceleration, similar to dark energy. These mathematical formulations illustrate how modified gravity theories extend or alter the standard framework of GR to address various physical phenomena and unresolved issues in cosmology and astrophysics.

These mathematical formulations illustrate how modified gravity theories extend or alter the standard framework of GR to address various physical phenomena and unresolved issues in cosmology and astrophysics.

Notes

For more detail on radiative transfer equations, see my notes:

References

Arun