Linear Motion # (under constant \(\hat{\mathbf{a}}\))
\begin{equation} \mathbf{v} = \mathbf{a}t + \mathbf{v_{0}}\tag{1} \end{equation}
\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \mathbf{v_{0}}t + \frac{1}{2}\mathbf{a}t^{2}\tag{2} \end{equation}
\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \frac{1}{2}(\mathbf{v} + \mathbf{v_0})t\tag{3} \end{equation}
\begin{equation} \mathbf{v^{2}} = \mathbf{v^{2}_0} + 2\mathbf{a}(\mathbf{r} - \mathbf{r_{0}})\tag{4} \end{equation}
\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \mathbf{v}t - \frac{1}{2}\mathbf{a}t^{2}\tag{5} \end{equation}
where:
\(\mathbf{r_{0}}\) is the initial position \(\mathbf{r}\) is the final position \(\mathbf{v_{0}}\) is the initial velocity \(\mathbf{v}\) is the final velocity \(\mathbf{a}\) is the acceleration \(t\) is the time interval
Any coordinate system which cannot describe an entirety of a space
As an example, polar coordinates \(r,\psi)\) are degenerate at \((0, \psi)\)
TODO #
Inflation # Inflation (Cosmic) is a theory put forth (originally, by Alan Guth in his 1980 paper) to explain why, if the universe is full of mass, is it not curved in some way? Instead, it’s flat (see: Curvature).
This isotropy in its temperature/geometry seems to contradict Thermodynamics in that we should see varying pockets of temperature around us since it would take time for the heat to dissipate throughout the universe.
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The vector space defined by generalized coordinates.
Example:
The position of a particle in Euclidean 3-space is defined by the generalized coordinate \(q = (x, y, z)\), therefore its configuration space is \(Q = \mathbb{R}^{3}\)
source: Physicists Argue That Black Holes From the Big Bang Could Be the Dark Matter
Primordial Black Holes # A “Primordial Black Hole” (PBH), is a black hole which has been around since nearly the time of the Big Bang.
PBH as Dark Matter # The hypothesis that PBHs could be the dark matter cosomolgists tangentially observe was put forth in many papers (like this one from Johns Hopkins U.
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High-level # The result of the marrying of QM with GR. (Initially, Hawking’s) calculations suggested that information would permanently disappear in a black hole. This runs afoul with the postulates of the Copenhagen Interpretation, in that information about a system is encoded in its wave function \(\Psi\) and, further, that that information is conserved.
TODOmore technical details #