Milne’s Expansion Clock: Why the Big Bang Is “Infinitely Long Ago”

This explains how, in Milne’s kinematic relativity, using the rate of expansion (or its integral) as the universal clock sends the big bang to infinite negative time. All equations below are LaTeX-compatible.

1) Milne Universe in One Line

Milne’s model is the empty (\rho=0) FRW universe with negative curvature (k=-1). The metric is

    \[ ds^2 \;=\; -\,dt^2 \;+\; a(t)^2\Big[d\chi^2+\sinh^2\!\chi\; d\Omega^2\Big], \qquad a(t)=t,\quad t>0. \]

Here t is the usual cosmic proper time of comoving observers. The Hubble rate is

    \[ H(t)\;\equiv\;\frac{\dot a}{a}\;=\;\frac{1}{t}. \]

As t\to 0^+, FRW language calls this a “big-bang surface.” In Milne, this boundary reflects a coordinate edge of flat Minkowski spacetime, but kinematically it looks like expansion with a=t.

2) Using the Expansion as a Universal Clock

Count the e-folds of expansion:

    \[ N(t)\;:=\;\int^t H(t')\,dt'\;=\;\int^t \frac{dt'}{t'}\;=\;\ln a(t)\;=\;\ln t \;+\; \text{const}. \]

  • N increases by 1 whenever the scale factor grows by a factor e.
  • As a\to 0,

        \[ N\;=\;\ln a\;\longrightarrow\;-\infty. \]

    Thus, on the N-clock, the big bang lies at infinite negative time.

3) Conformal Time Shows the Same Logarithm

Define conformal time \eta by

    \[ d\eta\;=\;\frac{dt}{a(t)}. \]

For Milne, a(t)=t, so

    \[ \eta \;=\;\int^t \frac{dt'}{t'}\;=\;\ln t \;+\; \text{const}, \qquad t\to 0^+ \;\Rightarrow\; \eta\to -\infty. \]

Hence the bang is also at infinite negative conformal time in Milne.

4) Why This Isn’t a Paradox

  • In proper time t, the interval from the bang to any later event is finite (it is just t).
  • In the logarithmic expansion clock N=\ln a (and, for Milne, in conformal time), the same boundary is at -\infty.

Different, physically motivated time variables slice the same spacetime differently. A clock tied to multiplicative growth (rates) naturally uses logarithms, pushing the origin to the infinite past.

5) Quick Generalization Near a Big Bang

Suppose near the bang

    \[ a(t)\sim t^{p},\qquad p>0. \]

E-fold time: N=\ln a \to -\infty as a\to 0 for any p>0. Thus the bang is infinitely far in the past in the N-clock for any big-bang FRW model.

Conformal time:

    \[ \eta \;=\;\int \frac{dt}{a(t)} \;\sim\; \int \frac{dt}{t^{p}} \;\propto\; \begin{cases} t^{\,1-p} & (p\neq 1),\\[4pt] \ln t & (p=1). \end{cases} \]

  • \eta\to -\infty for p\ge 1 (Milne has p=1).
  • For radiation p=\tfrac12 and matter p=\tfrac23, \eta remains finite at the bang.
  • But N=\ln a still sends the bang to -\infty in all cases.

TL;DR (Formulas)

    \[ \text{Milne: } a(t)=t,\quad H=1/t;\qquad N=\int H\,dt=\ln a \xrightarrow[a\to 0]{} -\infty;\qquad \eta=\int \frac{dt}{a}=\ln t \xrightarrow[t\to 0^+]{} -\infty. \]

Therefore, when the expansion rate (or its integral) is adopted as the universal time variable, the big bang is at infinite negative time.