I # In an intertial frame of reference (i.e. a frame of reference undergoing zero acceleration), an object at rest stays at rest (or, similarly, keeps its initial velocity) unless acted upon by an outside force.
\begin{equation} \sum \mathbf{F} = 0 \Leftrightarrow \frac{d\mathbf{v}}{dt} = 0 \end{equation}
II # In the simplest of terms:
\begin{equation} \mathbf{F} = m\mathbf{a} \end{equation}
which states that
The vector sum of the forces on an object is equal to that objects mass times the acceleration
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Newtonian Spacetime differs from Relativistic spacetime in that time is considered an absolute (i.e. \(t^{\prime} = t\)). Transformations between reference frames can be achieved with Galilean transformations. A particle traveling along the x axis in \(S\) at a constant speed u, has a speed \(u^{\prime}\) found by:
\begin{equation} u^{\prime}_{x} = \frac{dx^{\prime}}{dt^{\prime}} = \frac{dx^{\prime}}{dt} = \frac{dx}{dt} - v = u_{x} - v\tag{1} \end{equation}
remembering that the first derivative of position is velocity, and the second derivative of motion is acceleration
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Isotropy means that direction is not preferred, and that there is uniformity in any orientation.
Specifically, in Astrophysics and Cosmology, it means that there is no preferred direction in space. It looks the same no matter the direction you study (look).
Isotropy in space is only true at large scales (>100Mpc), the converse is true at smaller scales (anisotropy)
Above are two frames in Cartesian coordinates, \(S\) and \(S^{\prime}\). We have coordinates \((x, y, z)\), which define our dimensions.
Inertial Frame # An inertial frame is a frame of reference for which acceleration is zero. In other words:
\begin{equation} \frac{d^{2}x}{dt^{2}} = \frac{d^{2}y}{dt^{2}} = \frac{d^{2}z}{dt^{2}} = 0 \end{equation}
In the absence of gravity if \(S\) and \(S^{\prime}\) are two inertial frames they can only differ from each other by (and/or):
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Homogeneous means that location is not preferred, that the physics are the same at any point.
Specifically, in Astrophysics and Cosmology, it means that there is no preferred direction in space, it behaves the same no matter where you (as an observer) stand.
This is subtly different from isotropy, which is only concerned with the physical characteristics of the object (universe) rather than the object’s behavior (wrt physical law).
Generalized coordinates are useful when calculating Lagrangians or Hamiltonians.
The term itself, refers to the parameters that describe the configuration of a system.
For instance, to describe the position of a point on a circle, one could use its rectangular (Cartesian) coordinates, spherical coordinates, or cylindrical coordinates. Buth with generalized coordinates a reference doesn’t need to be made to the specific coordinate system, rather it could be abstracted into something more general as coordinates \((q_{1}, q_{2},…q_{n})\).
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Concepts # Inertial Frames Newtons Laws of Motion
In Newton’s formulation of spacetime, time is an absolute; meaning that every observer experiences the same flow of time. Assuming an inertial frame with only translation across the x axis, an event \(P\), can be translated between \(S\) and \(S^{\prime}\) by the following linear transformations:
\begin{equation} t^{\prime} = t\tag{1} \end{equation}
\begin{equation} x^{\prime} = x - vt\tag{2} \end{equation}
\begin{equation} y^{\prime} = y\tag{3} \end{equation}
\begin{equation} z^{\prime} = z\tag{4} \end{equation}
Switching between coordinates (\(S\) v.
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Overview # In the Big Bang model there are multiple parameters which appear to be “fine tuned” in that, small changes to a given parameter have massive effects on the evolution of the universe as a whole.
Wrt the flatness problem itself, the questionable parameter is the Density Parameter (\(\Omega\)), which is defined as
\begin{equation} \Omega \equiv \frac{\rho}{\rho_{c}}\tag{1} \end{equation}
where the subscript \(c\) is in reference to the “critical value” of density (\(\rho\)).
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