In 1929, Edwin Hubble studied the relationship between the radial velocity and distance of extragalactic nebulae. At that time, radial velocities were available for only 46 extragalactic nebulae, and only 24 of those had estimated distances. Hubble derived a roughly linear proportional relationship between radial velocity and distance.
Many cosmology books mention the title of this post. So, why is Hubble’s Law a manifestation of cosmic isotropy? Or rather, why does cosmic isotropy necessarily lead to Hubble’s Law?
First, we need to understand the Cosmological Principle, which tells us that the universe is homogeneous and isotropic on a large scale. Based on this principle, we arrive at some peculiar conclusions, such as the idea that every point in the universe is the center of the universe. Additionally, we can deduce that the (overall) motion of the universe should take the same form in every direction.
Let us use this Cosmological Principle to study the problem of cosmic expansion. Suppose an observer at point C sees the position vector of object A as \vec{r} and its velocity as \dot{\vec{r}}=f(\vec{r}). They see the position vector of object B as \vec{R}. According to the principle of isotropy, the velocity of B, \dot{\vec{R}}, must also be the same function f() with respect to \vec{R}; therefore, we also have \dot{\vec{R}}=f(\vec{R}).
Now, let’s switch to the observer at point B looking at object C. Obviously, the position vector is \vec{R}-\vec{r}, and the velocity is \frac{d(\vec{R}-\vec{r})}{dt}=\dot{\vec{R}}-\dot{\vec{r}}. Due to isotropy, the observation of the motion of C from point B should be the same as the observation from point A. Therefore, the velocity \dot{\vec{R}}-\dot{\vec{r}} must also be the same function f() with respect to \vec{R}-\vec{r}, namely: \dot{\vec{R}}-\dot{\vec{r}}=f(\vec{R}-\vec{r})=f(\vec{R})-f(\vec{r})
Next, we will see that only a direct proportion function satisfies the above equation. In fact, this is equivalent to solving the mathematical functional equation f(x-y)=f(x)-f(y). By setting x=y=0, we immediately get f(0)=0. By setting x=a+1 and y=1, we get f(a+1)-f(a)=f(1). This is similar to the arithmetic progressions we have studied, so we can write f(x)=(x-1)\cdot f(1)+f(1)=f(1)\cdot x=kx, where k is a constant. Although this result is derived specifically for x as a natural number, this "partial" evidence from natural numbers already yields a unique result. That is to say, we have proven that only direct proportion functions satisfy f(x-y)=f(x)-f(y). Similarly, the only function satisfying f(\vec{R}-\vec{r})=f(\vec{R})-f(\vec{r}) is the direct proportion function \vec{v}(\vec{r})=H_0 \vec{r}. Thus, Hubble’s Law is inevitable.
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