Q2.33:

A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics ($c$, $e$, mass of electron, mass of proton) and the gravitational constant $G$, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe ($\sim$15 billion years). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of ). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?

Step 1: Dimensional Analysis and Parameter Identification

To construct a quantity with the dimension of time $[T]$ from the fundamental constants $G$ (gravitational constant), $c$ (speed of light), $m_p$ (mass of proton), $m_e$ (mass of electron), and $e$ (elementary charge), we must analyze their dimensions:

• $G = [M^{-1} L^3 T^{-2}]$
• $c = [L T^{-1}]$
• $m_p, m_e = [M]$
• $e^2 / (4\pi\epsilon_0) = [M L^3 T^{-2}]$ (using the electrostatic force dimension)

Step 2: Deriving the Time Constant

Dirac observed that the ratio of the electrostatic force to the gravitational force between a proton and an electron is a dimensionless number of enormous magnitude:

$$N = \frac{e^2}{4\pi\epsilon_0 G m_p m_e} \approx \frac{9 \times 10^9 \times (1.6 \times 10^{-19})^2}{6.67 \times 10^{-11} \times 1.67 \times 10^{-27} \times 9.1 \times 10^{-31}} \approx 2.3 \times 10^{39}$$

To obtain a quantity with the dimension of time, we combine the gravitational constant $G$, the mass of the proton $m_p$, and the speed of light $c$. The time constant related to the scale of the universe (the light-travel time across the range defined by these constants) is given by:

$$T \approx \frac{G m_p}{c^3} \times N \approx \frac{G m_p}{c^3} \times \left( \frac{e^2}{4\pi\epsilon_0 G m_p m_e} \right)$$

Step 3: Numerical Calculation

Substituting the values ($G \approx 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$, $m_p \approx 1.67 \times 10^{-27} \, \text{kg}$, $c \approx 3 \times 10^8 \, \text{m/s}$):

$$t \approx \frac{6.67 \times 10^{-11} \times 1.67 \times 10^{-27}}{(3 \times 10^8)^3} \times 2.3 \times 10^{39}$$

$$t \approx 1.37 \times 10^{-52} \times 2.3 \times 10^{39} \approx 3.15 \times 10^{-13} \, \text{s} \times 2.3 \times 10^{39} \approx 7.2 \times 10^{26} \, \text{s}$$

Converting to years ($1 \, \text{yr} \approx 3.15 \times 10^7 \, \text{s}$):

$$T \approx \frac{7.2 \times 10^{26}}{3.15 \times 10^7} \approx 2.2 \times 10^{19} \, \text{years}$$

Step 4: Logical Implications

The magnitude calculated depends heavily on the chosen parameters (the "large number hypothesis"). If $N$ grows with the age of the universe, and $G$ is held constant, $e$ or $m$ would need to evolve over cosmological timescales. Modern measurements have constrained such variations to be extremely small, placing doubt on the direct dynamical coupling between these microphysical constants and the expansion of the universe.

Final Solution: The constructed time constant using fundamental constants yields a magnitude on the order of $10^{19}$ years. The existence of such large dimensionless numbers suggests that if their coincidence with the universe's age is meaningful, it would necessitate that the fundamental constants of nature are functions of cosmic time, rather than fixed values.