bridge derivation of Newtonian Gravity from the Geodesic Equation in a Weak, Static Gravitational Field

[bhpayne]

As suggested in #22

[bhpayne]

prompt

In mathematical Physics a derivation is comprised of steps and equations.

Is there a derivation that shows how the geodesic equation of General Relativity, which describes motion in curved spacetime, reduces to Newton's law of gravity in the limit of weak gravitational fields and low velocities? If yes, provide the steps of the derivation. Format the answer using Latex as a single file.

Latex is a software system for typesetting documents. LaTeX is widely used for the communication and publication of scientific documents and technical note-taking in many fields, owing partially to its support for complex mathematical notation.

Do not have equations inline with the text. Equations should be separate and inside a math environment for Latex:

\begin{equation}
\end{equation}

Each equation should have a left-hand side and right-hand side. Do specify equations, but avoid using equations that contain multiple equal signs.

Explicitly state the equations being used in each step. Also define each variable used in the derivation.

Provide explanatory text about each step of the derivation. Document the mathematical transformations that relate each equation in the file to other equations within the Latex file.

thoughts

The user is asking for a derivation in LaTeX that shows how the geodesic equation of General Relativity reduces to Newton's law of gravity in the Newtonian limit (weak gravitational fields and low velocities).

I need to:

1. Start with the geodesic equation.
2. Define the Newtonian limit conditions: weak field and low velocities.
3. Simplify the geodesic equation using these conditions.
4. Show that the simplified equation reduces to Newton's law of gravity.
5. Format the derivation in LaTeX as a single file.
6. Use separate math environments for equations.
7. Ensure each equation has LHS and RHS and avoids multiple equal signs.
8. Explicitly state the equations used in each step.
9. Define all variables used.
10. Provide explanatory text for each step and document mathematical transformations.

Step 1: Geodesic Equation
The geodesic equation describes the motion of particles in curved spacetime. In General Relativity, it 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

Step 2: Newtonian Limit Conditions

• Weak gravitational field: This means the spacetime metric $g_{\mu\nu}$ is close to the Minkowski metric $\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$. We can write the metric as a perturbation of the Minkowski metric:
  $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, where $|h_{\mu\nu}| \ll 1$.
• Low velocities: This means the velocities of particles are much smaller than the speed of light, i.e., $\frac{dx^i}{dt} \ll c$. In natural units where $c=1$, this means $\frac{dx^i}{dt} \ll 1$. Also, we can approximate the proper time $\tau$ with coordinate time $t$, i.e., $d\tau \approx dt$.

Step 3: Simplify the Geodesic Equation
We need to simplify the Christoffel symbols $\Gamma^\mu_{\alpha\beta}$ in the weak field limit and simplify the velocities in the low velocity limit.

• Christoffel Symbols in Weak Field Limit:
  The Christoffel symbols are given by:
  $\Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\nu} (\partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta})$
  In the weak field limit, $g^{\mu\nu} \approx \eta^{\mu\nu} - h^{\mu\nu} \approx \eta^{\mu\nu}$ (to first order in $h$) and $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$.
  So, $\Gamma^\mu_{\alpha\beta} \approx \frac{1}{2} \eta^{\mu\nu} (\partial_\alpha h_{\nu\beta} + \partial_\beta h_{\nu\alpha} - \partial_\nu h_{\alpha\beta})$

• Low Velocity Limit for Geodesic Equation:
  For low velocities, we consider motion in the non-relativistic limit. We are interested in the spatial components of the geodesic equation ($\mu = i = 1, 2, 3$).
  For slow motion, $\frac{dx^i}{d\tau} \ll \frac{dx^0}{d\tau} = \frac{dt}{d\tau}$. Also, we can approximate $\frac{d}{d\tau} \approx \frac{dt}{d\tau} \frac{d}{dt} \approx \frac{d}{dt}$ if $d\tau \approx dt$. For non-relativistic motion in a weak field, $d\tau \approx dt$.
  The geodesic equation becomes (for $\mu = i$):
  $\frac{d^2x^i}{d\tau^2} + \Gamma^i_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$
  $\frac{d^2x^i}{dt^2} + \Gamma^i_{\alpha\beta} \frac{dx^\alpha}{dt} \frac{dx^\beta}{dt} = 0$ (approximating $d\tau \approx dt$)

  Consider the terms $\frac{dx^\alpha}{dt} \frac{dx^\beta}{dt}$:

  • If $\alpha = 0, \beta = 0$: $\frac{dx^0}{dt} \frac{dx^0}{dt} = (\frac{dt}{dt})^2 = 1$
  • If $\alpha = 0, \beta = j$ or $\alpha = j, \beta = 0$: $\frac{dx^0}{dt} \frac{dx^j}{dt} = \frac{dx^j}{dt} \ll 1$ (low velocity)
  • If $\alpha = j, \beta = k$: $\frac{dx^j}{dt} \frac{dx^k}{dt} \ll 1$ (low velocity squared, even smaller)
    So, the dominant term is when $\alpha = 0, \beta = 0$. The geodesic equation approximately becomes:
    $\frac{d^2x^i}{dt^2} + \Gamma^i_{00} (\frac{dx^0}{dt})^2 = 0$
    $\frac{d^2x^i}{dt^2} + \Gamma^i_{00} = 0$ (since $\frac{dx^0}{dt} = 1$)

• Simplify $\Gamma^i_{00}$ in weak field limit:
  $\Gamma^i_{00} \approx \frac{1}{2} \eta^{i\nu} (\partial_0 h_{\nu 0} + \partial_0 h_{\nu 0} - \partial_\nu h_{00}) = \frac{1}{2} \eta^{i\nu} (2 \partial_0 h_{\nu 0} - \partial_\nu h_{00}) = \eta^{i\nu} (\partial_0 h_{\nu 0} - \frac{1}{2} \partial_\nu h_{00})$
  For a static weak field, the metric components are time-independent, i.e., $\partial_0 h_{\mu\nu} = 0$.
  So, $\Gamma^i_{00} \approx \eta^{i\nu} (-\frac{1}{2} \partial_\nu h_{00}) = -\frac{1}{2} \eta^{ij} \partial_j h_{00}$ (since $\nu$ must be a spatial index for $\eta^{i\nu} \neq 0$ if $i$ is spatial).
  $\eta^{ij}$ is the spatial part of Minkowski metric, $\eta^{ij} = \delta^{ij}$.
  $\Gamma^i_{00} \approx -\frac{1}{2} \delta^{ij} \partial_j h_{00} = -\frac{1}{2} \partial_i h_{00}$

  Thus, the geodesic equation becomes:
  $\frac{d^2x^i}{dt^2} - \frac{1}{2} \partial_i h_{00} = 0$
  $\frac{d^2x^i}{dt^2} = \frac{1}{2} \partial_i h_{00}$

Step 4: Relate to Newtonian Potential
In Newtonian gravity, the gravitational acceleration is given by $\vec{a} = -\nabla \Phi$, or in component form, $\frac{d^2x^i}{dt^2} = -\partial_i \Phi$, where $\Phi$ is the Newtonian gravitational potential.
We need to relate $h_{00}$ to the Newtonian potential $\Phi$.

From the weak field approximation of Einstein's field equations, for a static, weak field sourced by non-relativistic matter, we have $h_{00} = -2\Phi$.
(More precisely, $g_{00} = -(1 + 2\Phi) \approx \eta_{00} + h_{00} = -1 + h_{00}$, so $h_{00} = -2\Phi$.)

Substitute $h_{00} = -2\Phi$ into the geodesic equation:
$\frac{d^2x^i}{dt^2} = \frac{1}{2} \partial_i (-2\Phi) = -\partial_i \Phi$

This is exactly Newton's law of gravity!

Step 5: Write in LaTeX
Need to organize the steps, define variables, and write in LaTeX.

Variables to define:

• $x^\mu$: spacetime coordinates
• $\tau$: proper time
• $\Gamma^\mu_{\alpha\beta}$: Christoffel symbols
• $g_{\mu\nu}$: metric tensor
• $\eta_{\mu\nu}$: Minkowski metric
• $h_{\mu\nu}$: perturbation to Minkowski metric in weak field
• $t$: coordinate time
• $x^i$: spatial coordinates
• $\Phi$: Newtonian gravitational potential

Equations to explicitly mention:

• Geodesic equation
• Definition of Christoffel symbols
• Weak field metric approximation
• Newtonian law of gravity

Review and Refine:
Check all steps for correctness and clarity. Make sure the LaTeX formatting is correct. Ensure all variables are defined and all equations are explained. The approximation $d\tau \approx dt$ is valid for weak fields and low velocities. The relation $h_{00} = -2\Phi$ comes from comparing the Newtonian limit of Einstein's field equations with Poisson's equation.

response

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}

\title{Derivation of Newton's Law of Gravity from the Geodesic Equation}
\date{\today}

\begin{document}
\maketitle

This document provides a derivation showing how the geodesic equation of General Relativity reduces to Newton's law of gravity in the limit of weak gravitational fields and low velocities.

\section{Geodesic Equation}
In General Relativity, the motion of a test particle in a gravitational field is described by the geodesic equation. This equation dictates the path of shortest spacetime interval in a curved spacetime. The geodesic equation is given by:
\begin{equation}
\label{eq:1}
\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0
\end{equation}
where:
\begin{itemize}
    \item $x^\mu$ are the spacetime coordinates, with $\mu = 0, 1, 2, 3$. Conventionally, $x^0 = ct$, but we will use units where the speed of light $c=1$, so $x^0 = t$.
    \item $\tau$ is the proper time, the time measured by an observer moving along the geodesic.
    \item $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols, which represent the gravitational force field in General Relativity.
\end{itemize}

\section{Christoffel Symbols in a Weak Gravitational Field}
The Christoffel symbols $\Gamma^\mu_{\alpha\beta}$ are derived from the spacetime metric $g_{\mu\nu}$, which describes the geometry of spacetime. The relationship is given by:
\begin{equation}
\label{eq:2}
\Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\nu} \left( \partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta} \right)
\end{equation}
where $g^{\mu\nu}$ is the inverse metric tensor, satisfying $g^{\mu\nu}g_{\nu\lambda} = \delta^\mu_\lambda$, and $\partial_\alpha = \frac{\partial}{\partial x^\alpha}$.

In the case of a weak gravitational field, the spacetime metric $g_{\mu\nu}$ is approximately the Minkowski metric $\eta_{\mu\nu}$, plus a small perturbation $h_{\mu\nu}$. The Minkowski metric in Cartesian coordinates is $\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$. Thus, we can write:
\begin{equation}
\label{eq:3}
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}
\end{equation}
where $|h_{\mu\nu}| \ll 1$.  For weak fields, we can approximate $g^{\mu\nu} \approx \eta^{\mu\nu} - h^{\mu\nu} \approx \eta^{\mu\nu}$ to first order in $h$.

Substituting Equation \eqref{eq:3} into Equation \eqref{eq:2}, and using the approximation $g^{\mu\nu} \approx \eta^{\mu\nu}$, we get the Christoffel symbols in a weak gravitational field:
\begin{equation}
\label{eq:4}
\Gamma^\mu_{\alpha\beta} \approx \frac{1}{2} \eta^{\mu\nu} \left( \partial_\alpha h_{\nu\beta} + \partial_\beta h_{\nu\alpha} - \partial_\nu h_{\alpha\beta} \right)
\end{equation}

\section{Low Velocity Limit}
We consider the motion of a particle at low velocities in this weak gravitational field.  Low velocity means that the spatial components of the particle's four-velocity are much smaller than the time component, i.e., $|\frac{dx^i}{d\tau}| \ll |\frac{dx^0}{d\tau}| = |\frac{dt}{d\tau}|$ for $i = 1, 2, 3$.  For non-relativistic motion, we can also approximate the proper time derivative with the coordinate time derivative, $\frac{d}{d\tau} \approx \frac{dt}{d\tau} \frac{d}{dt}$. Furthermore, for weak, static gravitational fields and slow motion, we can approximate $\frac{dt}{d\tau} \approx 1$, so $\frac{d}{d\tau} \approx \frac{d}{dt}$ and $\frac{d^2}{d\tau^2} \approx \frac{d^2}{dt^2}$.

We are interested in the spatial components of the geodesic equation, i.e., $\mu = i$, where $i = 1, 2, 3$.  For $\mu = i$, Equation \eqref{eq:1} becomes:
\begin{equation}
\label{eq:5}
\frac{d^2x^i}{d\tau^2} + \Gamma^i_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0
\end{equation}
Using the low velocity approximation $\frac{d^2}{d\tau^2} \approx \frac{d^2}{dt^2}$, we have:
\begin{equation}
\label{eq:6}
\frac{d^2x^i}{dt^2} + \Gamma^i_{\alpha\beta} \frac{dx^\alpha}{dt} \frac{dx^\beta}{dt} = 0
\end{equation}
In the low velocity limit, the dominant term in the sum $\Gamma^i_{\alpha\beta} \frac{dx^\alpha}{dt} \frac{dx^\beta}{dt}$ is when $\alpha = 0$ and $\beta = 0$, because $\frac{dx^0}{dt} = \frac{dt}{dt} = 1$, while $\frac{dx^i}{dt} \ll 1$ for spatial components. Thus, we can approximate Equation \eqref{eq:6} by considering only the $\alpha = 0, \beta = 0$ term:
\begin{equation}
\label{eq:7}
\frac{d^2x^i}{dt^2} + \Gamma^i_{00} \left(\frac{dx^0}{dt}\right)^2 = 0
\end{equation}
Since $\frac{dx^0}{dt} = 1$, Equation \eqref{eq:7} simplifies to:
\begin{equation}
\label{eq:8}
\frac{d^2x^i}{dt^2} + \Gamma^i_{00} = 0
\end{equation}

\section{Simplifying $\Gamma^i_{00}$ for a Static Weak Field}
For a static gravitational field, the metric components are independent of time, i.e., $\partial_0 h_{\mu\nu} = 0$.  Using Equation \eqref{eq:4} for $\Gamma^i_{00}$:
\begin{equation}
\label{eq:9}
\Gamma^i_{00} \approx \frac{1}{2} \eta^{i\nu} \left( \partial_0 h_{\nu 0} + \partial_0 h_{\nu 0} - \partial_\nu h_{00} \right)
\end{equation}
Since the field is static, $\partial_0 h_{\nu 0} = 0$. Thus, Equation \eqref{eq:9} reduces to:
\begin{equation}
\label{eq:10}
\Gamma^i_{00} \approx \frac{1}{2} \eta^{i\nu} \left( 0 + 0 - \partial_\nu h_{00} \right) = -\frac{1}{2} \eta^{i\nu} \partial_\nu h_{00}
\end{equation}
Since $\eta^{i\nu}$ is non-zero only when $\nu$ is a spatial index (i.e., $\nu = j = 1, 2, 3$), and $\eta^{ij} = \delta^{ij}$, we have:
\begin{equation}
\label{eq:11}
\Gamma^i_{00} \approx -\frac{1}{2} \delta^{ij} \partial_j h_{00} = -\frac{1}{2} \partial_i h_{00}
\end{equation}
Substituting Equation \eqref{eq:11} into Equation \eqref{eq:8}, we get:
\begin{equation}
\label{eq:12}
\frac{d^2x^i}{dt^2} - \frac{1}{2} \partial_i h_{00} = 0
\end{equation}
Rearranging Equation \eqref{eq:12}:
\begin{equation}
\label{eq:13}
\frac{d^2x^i}{dt^2} = \frac{1}{2} \partial_i h_{00}
\end{equation}

\section{Relating $h_{00}$ to Newtonian Potential}
In the weak field limit, the $g_{00}$ component of the metric is related to the Newtonian gravitational potential $\Phi$ by:
\begin{equation}
\label{eq:14}
g_{00} \approx -(1 + 2\Phi)
\end{equation}
Comparing Equation \eqref{eq:14} with Equation \eqref{eq:3} for $\mu = \nu = 0$, we have $g_{00} = \eta_{00} + h_{00} = -1 + h_{00}$. Therefore,
\begin{equation}
\label{eq:15}
-1 + h_{00} \approx -(1 + 2\Phi)
\end{equation}
From Equation \eqref{eq:15}, we find the relation between $h_{00}$ and $\Phi$:
\begin{equation}
\label{eq:16}
h_{00} \approx -2\Phi
\end{equation}

\section{Reduction to Newton's Law of Gravity}
Substitute $h_{00} = -2\Phi$ from Equation \eqref{eq:16} into Equation \eqref{eq:13}:
\begin{equation}
\label{eq:17}
\frac{d^2x^i}{dt^2} = \frac{1}{2} \partial_i (-2\Phi)
\end{equation}
Simplifying Equation \eqref{eq:17}, we obtain:
\begin{equation}
\label{eq:18}
\frac{d^2x^i}{dt^2} = -\partial_i \Phi
\end{equation}
Equation \eqref{eq:18} is the component form of Newton's law of gravity, which states that the acceleration of a particle in a gravitational field is given by the negative gradient of the Newtonian gravitational potential $\Phi$: $\vec{a} = -\nabla \Phi$.

Thus, we have shown that in the limit of weak gravitational fields and low velocities, the geodesic equation of General Relativity reduces to Newton's law of gravity.

\end{document}