The Newtonian Limit of General Relativity

General Relativity (GR) is a revolutionary theory of gravity that describes gravity not as a force, but as a curvature of spacetime caused by mass and energy. While GR is incredibly successful in describing phenomena like black holes and gravitational waves, it must also reduce to Newtonian gravity in the limit of weak gravitational fields and low velocities. This transition is known as the Newtonian limit, and understanding it is crucial for bridging the gap between classical and relativistic physics.

Why is the Newtonian Limit Important?

The Newtonian limit serves as a vital consistency check for General Relativity. If GR did not reproduce the well-established predictions of Newtonian gravity in everyday scenarios (like planetary orbits or falling objects), it would be considered a flawed theory. It also provides a conceptual bridge, showing how the more complex relativistic framework encompasses the simpler, yet highly successful, classical description of gravity.

Key Concepts for the Transition

Weak Field Approximation: GR simplifies significantly when spacetime curvature is small.

In the weak field approximation, we assume that the metric tensor is only slightly different from the flat Minkowski spacetime metric. This allows us to linearize Einstein's field equations.

The metric tensor, $g_{\mu\nu}$ , describes the geometry of spacetime. In flat Minkowski spacetime, $g_{\mu\nu} = \eta_{\mu\nu}$ , where $\eta_{\mu\nu}$ is the Minkowski metric. In the weak field approximation, we write $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ , where $h_{\mu\nu}$ is a small perturbation. This linearization is a key step in deriving the Newtonian limit.

Low Velocity Approximation: GR reduces to Newtonian mechanics when speeds are much less than the speed of light.

When objects move at speeds significantly lower than the speed of light ( $v \ll c$ ), relativistic effects like time dilation and length contraction become negligible. The geodesic equation, which describes motion in GR, simplifies considerably.

The geodesic equation in GR is given by $\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$ . For low velocities, the proper time $\tau$ is approximately proportional to the coordinate time $t$ . The Christoffel symbols ( $\Gamma^\mu_{\alpha\beta}$ ), which represent the gravitational field, also simplify, leading to an equation that resembles Newton's second law of motion.

Deriving Newtonian Gravity from GR

The process of deriving Newtonian gravity involves applying the weak field and low velocity approximations to Einstein's field equations and the geodesic equation. A common approach involves considering a static, spherically symmetric mass distribution, which leads to the Schwarzschild metric. By applying the approximations to this metric, one can recover Newton's law of universal gravitation.

The geodesic equation in General Relativity describes the path of a freely falling particle in curved spacetime. In the weak field and low velocity limit, this equation simplifies to Newton's second law of motion, F=ma. The Christoffel symbols, which encode the gravitational field, become proportional to the gradient of the Newtonian potential. Specifically, the $\Gamma^0_{00}$ component, when linearized and considering a static spacetime, relates to the time-time component of the metric perturbation and leads to the acceleration term in Newton's equation.

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The key insight is that the Newtonian gravitational potential, $\Phi$ , is related to the time-time component of the metric perturbation, $h_{00}$ , by $\Phi \approx \frac{1}{2}c^2 h_{00}$ in the weak field limit.

Observational Evidence

The success of Newtonian gravity in describing most everyday phenomena is itself a testament to the validity of the Newtonian limit of GR. Phenomena like the perihelion precession of Mercury, the bending of starlight by the Sun, and gravitational redshift are all predictions of GR that deviate slightly from Newtonian predictions, and these deviations are precisely what GR predicts when moving beyond the Newtonian limit.

What are the two primary approximations used to derive the Newtonian limit from General Relativity?

The weak field approximation and the low velocity approximation.

How is the Newtonian gravitational potential related to the metric perturbation in the weak field limit?

The Newtonian gravitational potential $\Phi$ is approximately $\frac{1}{2}c^2 h_{00}$ .

Learning Resources

General Relativity - Wikipedia(wikipedia)

Provides a comprehensive overview of General Relativity, including sections on its foundations and experimental tests, which implicitly cover the Newtonian limit.

The Newtonian Limit of General Relativity - Lecture Notes(documentation)

Detailed lecture notes from a graduate-level physics course explaining the derivation of the Newtonian limit from GR.

Introduction to General Relativity - Chapter 4: The Newtonian Limit(documentation)

A chapter from an online textbook dedicated to General Relativity, focusing specifically on how GR reduces to Newtonian gravity.

Gravitation and Spacetime - Newtonian Limit(documentation)

Part of a comprehensive set of lecture notes on gravitation, this section explicitly discusses the transition from GR to Newtonian gravity.

Einstein's Field Equations - Wikipedia(wikipedia)

Explains Einstein's field equations, the core of GR, and touches upon approximations and limits, including the Newtonian limit.

The Schwarzschild Metric - Wikipedia(wikipedia)

Details the Schwarzschild metric, a key solution to Einstein's field equations, which is often used to demonstrate the Newtonian limit.

General Relativity: A Tutorial(tutorial)

A tutorial that covers the basics of GR, including how it relates to Newtonian gravity in certain regimes.

Understanding General Relativity: A Visual Approach(video)

A video that uses visualizations to explain concepts in General Relativity, potentially offering intuitive insights into the Newtonian limit.

Relativistic Effects in the Solar System(video)

This video discusses observational evidence for GR, such as the precession of Mercury's orbit, which highlights the differences from Newtonian predictions.

The Principle of Equivalence - Stanford Encyclopedia of Philosophy(wikipedia)

Explores the Principle of Equivalence, a foundational concept in GR that helps bridge the gap to understanding gravity as spacetime curvature, relevant to the Newtonian limit.

Newtonian Limit of GR