User:Guy vandegrift/Blog/GR

Wikipedia resources
Partial derivative, Ricci_calculus Connection_(mathematics)

simple
$$m\vec a = -\vec\nabla \Phi$$

These are consistent with observation, are missing a feature we require for a fundamental law of physics:

$$\Phi=-gy$$ or $$\Phi=-\frac{M}{r^n}=-\frac{M}{(x^2+y^2)^{n/2}}$$,

where $$n$$ is some integer. In our universe with three spatial dimensions, $$n=2$$ (inverse square law).

$$\Phi = {\sum_j}^\prime\frac{m_j}{(\vec r-\vec r_j)^n}$$

pde
Mass density and mass

$$M=\int \rho (\vec r) d\tau$$

$$d\tau = dxdydz$$ or $$d\tau = dxdy$$

$$\partial \Phi / \partial x \equiv \partial_x\Phi$$

$$(t,x,y,z)\rightarrow (x_0,x_1,x_2,x_2)\rightarrow x_j$$

The need for symmetry
$$\partial_x \Phi = \rho$$ leads to $$\Phi = \int_K^x \rho(x',y) dx'$$

$$x=x_0+v_{x0}t$$ and $$y=y_0+v_{y0}t-\frac 1 2 g t^2$$

$$\Phi_{xy}=\rho$$

While this seems like the simplest possible second order partial differential equation relating to functions, $$\Phi=\Phi(x,y)$$ and $$\rho=\rho(x,y$$, it suffers from a major flaw: The equation is not invariant under a rotation of the coordinate system. In plain language, that means that if we observe nature in a rotated coordinate system, we get a different equation.

Rotating a two-dimensional coordinate system
Example: $$\theta = \frac \pi 4 = 40^\circ$$

The following is shorthand for writing two equations in two unknowns. If it doesn't make sense to you, don't worry because we will demonstrate an example for rotating coordinates by 45 degrees using conventional notation:

$$\begin{bmatrix} x' \\ y' \end{bmatrix}= \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \cdot\begin{bmatrix} x \\ y \end{bmatrix}$$

If $$\theta= \pi / 4 $$ (45 degrees) we have these two equations that express the coordiantes $$(x^\prime,y^\prime)$$ in terms of the unprimed coordinates:

$$\begin{alignat}{3} x' & = \;\;\frac 1 \sqrt 2 x && + \frac 1 \sqrt 2 y          \\ y' & = -\frac 1 \sqrt 2 x && + \frac 1 \sqrt 2 y \\ \end{alignat} $$

PROJECT: Create homework and quiz questions proving and interpretating facts such the following:
 * The line $$y=0$$ corresponds to the line $$y'=?$$.
 * $$x^2+y^2=1$$ if and only if $$(x')^2+(y')^2=1$$

$$xx$$ $$xx$$ $$xx$$ $$xx$$ $$xx$$ $$xx$$