3.2 KiB
title, date
| title | date |
|---|---|
| Introduction to General Relativity | 2025-01-04 |
The Introduction to the Introduction
I pulled all of this from a certain immensely helpful textbook1 . The need for special relativity arises from the limitations of Newton's theory of gravitation. We've all seen the following equation:
\textbf{F}=-\frac{MmG}{r^3}\textbf{r}
\textbf{r} is the direction vector, or unit direction vector from masses M and m. The problem is that most of these quantities vary over time. For example, if M was a function of t:
\textbf{F}(t)=-\frac{M(t)mG}{r^3}\textbf{r}
Note
Gis the gravitational constant.
The problem here is that a variation in mass creates a delayed reaction in gravitation, but our formula with Newtonian mechanics does not provide for that. Above, a variation in any quantity yields instant variation in force. In reality, gravity moves at the speed of light.
Why you can't just use t-r/c as the parameter will be covered sometime later. For now, just know Einstien Enstiened here. Such remodeling of Newton's law was attempted and then met with failure. The intensity of the gravitational field at any distance r from the center of mass and direction \hat{r} is provided by:
\textbf{g}=-\frac{GM}{r^2}\hat{r}
This is not entirely accurate, since what is actually being described here is the gradient field of attractive forces, written so:
\textbf{g}=-\nabla\phi, \phi(r)=-\frac{GM}{r}
The function \phi(r) is the gravitational potential, a scalar field. Each scalar value in the field represents the work needed to be done to move mass m there from r=0. Specifically, this is the change in potential energy, meaning an upward motion out of the gravitational well caused by a mass M results in a positive change in potential energy.
The potential \phi satisfies field equations.
\text{(Laplace, in a vacuum) } \nabla^2\phi=0
\text{(Poisson, with matter) } \nabla^2\phi=4\pi G \rho
The field \textbf{g} depends on r but not t, meaning the partial derivative with respect to r equals 0.
Equivalence Principle
[!NOTE] definition The Equivalence Principle states: No experiment in mechanics can distinguish between a gravitational field and an accelerating frame of reference.
After reading this, I realized that the textbook has it worded horribly. It means that the two — gravity and acceleration are locally indistinguishable. There are three relevant equations. The Force acting on an object:
\vec{F}=m_i \vec{a}
The above equation describes force opposite to inertia. The below equation describes force appeared to have been exerted by the gravitational field:
\vec{F}=m_g \vec{g}
Here, m_g is the gravitational mass of the particle. Gravitational Mass is a measure of the response of a particle to the gravitational field. These are conceptually different things. The gravitational mass depends on the scalar gravitational potential field. Of course, there are several experiments to prove this, but the equation of note is as follows:
\vec{a}=\frac{\vec{F}}{m_i}=\frac{m_g}{m_i}\vec{g}
Einstien's Train
#physics
-
Ryder, L. (2009). Introduction to General Relativity. Singapore: Cambridge University Press. ↩︎