# Pre-Lecture Homework

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Published 2022-10-09

# 7.1 Pre-Lecture Homework

# 7.1.1 Hubble's data

In the 1920-1930s, Edwin Hubble and Milton Humason collated and measured the distances and velocities of galaxies. Below is a table comprising some of their early data. Analyse the data and deduce a relationship between velocities and distance (if you have not already done so in previous IS).

# 7.1.2 The Cosmological Principle

The cosmological principle states that the Universe is homogeneous and isotropic.

What do the words in bold mean?

Reason that the cosmological principle implies that any observer looking at the motion of galaxies will find the radial velocities proportional to the distance between her and the galaxy.

# 7.1.3 Pythagoras theorem in spherical coordinates

Some intense mathy stuffs below. Just read, no need to work out.

Spherical coordinates are used to specify points on the sphere and are related to the Cartesian coordinates by

\begin{align}
x & =R\sin\theta\cos\phi\nonumber \\
y & =R\sin\theta\sin\phi\\
z & =R\cos\theta\nonumber 
\end{align}

The corresponding differentials are

\begin{align}
dx & =\sin\theta\cos\phi dR+R\cos\theta\cos\phi d\theta-R\sin\theta\sin\phi d\phi\nonumber \\
dy & =\sin\theta\sin\phi dR+R\cos\theta\sin\phi d\theta+R\sin\theta\cos\phi d\phi\\
dz & =\cos\theta dR-R\sin\theta d\theta\nonumber 
\end{align}

The infinitesimal length between two spatial points is given by

(dl)^{2}=(dx)^{2}+(dy)^{2}+(dz)^{2}

To write this in spherical coordinates, we'll use Eq. 2, and some elbow grease.

\begin{align*}
(dl)^{2} & =(\sin\theta\cos\phi dR+R\cos\theta\cos\phi d\theta-R\sin\theta\sin\phi d\phi)^{2}\\
 & \phantom{=}+(\sin\theta\sin\phi dR+R\cos\theta\sin\phi d\theta+R\sin\theta\cos\phi d\phi)^{2}\\
 & \phantom{=}+(\cos\theta dR-R\sin\theta d\theta)^{2}\\
 & =(dR)^{2}(\sin^{2}\theta\cos^{2}\phi+\sin^{2}\theta\sin^{2}\phi+\cos^{2}\theta)\\
 & \phantom{=}+(d\theta)^{2}(R^{2}\cos^{2}\theta\cos^{2}\phi+R^{2}\cos^{2}\theta\sin^{2}\phi+R^{2}\sin^{2}\theta)\\
 & \phantom{=}+(d\phi)^{2}(R^{2}\sin^{2}\theta\sin^{2}\phi+R^{2}\sin^{2}\theta\cos^{2}\phi)\\
 & \phantom{=}+2dRd\theta(R\sin\theta\cos^{2}\phi\cos\theta+R\sin\theta\sin^{2}\phi\cos\theta-R\sin\theta\cos\theta)\\
 & \phantom{=}+2dRd\phi(-R\sin^{2}\theta\cos\phi\sin\phi+R\sin^{2}\theta\sin\phi\cos\phi)\\
 & \phantom{=}+2d\theta d\phi(-R^{2}\cos\theta\cos\phi\sin\theta\sin\phi+R^{2}\cos\theta\sin\phi\sin\theta\cos\phi)\\
 & =(dR)^{2}+R^{2}(d\theta)^{2}+R^{2}\sin^{2}\theta(d\phi)^{2}
\end{align*}

Yeah finally done! This equation will be used later.

Oh one note, it is really troublesome to write the parenthesis for the squares of the differentials. From this point onward, we will not write them. Hence we have

\begin{align}
dl^{2} & =dx^{2}+dy^{2}+dz^{2}\nonumber \\
 & =dR^{2}+R^{2}d\theta^{2}+R^{2}\sin^{2}\theta d\phi^{2}
\end{align}