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 (~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 of Fundamental Constants

To construct a quantity with the dimension of time $[T]$ using the fundamental constants of nature: speed of light ($c$), electronic charge ($e$), mass of electron ($m_e$), mass of proton ($m_p$), and the gravitational constant ($G$), we must equate dimensions. The SI units and dimensions are as follows:

Step 2: Constructing the Dimensionless Ratio

Dirac observed the ratio of the electrostatic force to the gravitational force between an electron and a proton. [Per Coulomb's Law and Newton's Law of Gravitation]:

$F_e = \frac{e^2}{4\pi\epsilon_0 r^2}$ and $F_g = \frac{G m_e m_p}{r^2}$

The dimensionless ratio $N$ is defined as:

$N = \frac{F_e}{F_g} = \frac{e^2}{4\pi\epsilon_0 G m_e m_p}$

Using standard values: $e \approx 1.6 \times 10^{-19} C$, $G \approx 6.67 \times 10^{-11} m^3kg^{-1}s^{-2}$, $m_e \approx 9.1 \times 10^{-31} kg$, $m_p \approx 1.67 \times 10^{-27} kg$, and $\frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9 Nm^2/C^2$:

$N \approx \frac{(9 \times 10^9)(1.6 \times 10^{-19})^2}{(6.67 \times 10^{-11})(9.1 \times 10^{-31})(1.67 \times 10^{-27})} \approx 2.3 \times 10^{39}$

Step 3: Deriving the Time Scale

To obtain a quantity with dimensions of time $[T]$, we multiply this dimensionless ratio by the time it takes light to traverse the classical radius of the electron ($r_e \approx \frac{e^2}{4\pi\epsilon_0 m_e c^2} \approx 2.8 \times 10^{-15} m$):

$T_{Dirac} = N \cdot \frac{r_e}{c} \approx 10^{39} \cdot \frac{2.8 \times 10^{-15}}{3 \times 10^8} \approx 10^{16} s$

Given $1 \text{ year} \approx 3.15 \times 10^7 s$, then $15 \text{ billion years} \approx 4.7 \times 10^{17} s$. The calculated $T_{Dirac}$ is of the same order of magnitude ($10^{16}$ to $10^{17} s$) as the age of the universe.

Step 4: Implications for Fundamental Constants

If the coincidence $T_{Dirac} \approx T_{Universe}$ is not accidental, it suggests that the fundamental "constants" of nature may not be strictly constant over cosmic time. [By the Large Number Hypothesis]:

• If $T_{Universe}$ increases linearly with time, then $G$ must decrease as $G \propto \frac{1}{t}$.
• Alternatively, the strength of interactions (determined by $e$ or $m_e$) might vary with the expansion of the universe.
• Modern astrophysical measurements (e.g., spectral analysis of distant quasars) generally constrain the variation of these constants to be extremely small, suggesting that Dirac's Large Number Hypothesis remains a fascinating theoretical speculation rather than a confirmed physical law.

Final Solution: The constructed time $T \approx N \frac{e^2}{4\pi\epsilon_0 m_e c^3} \approx 10^{16} - 10^{17} s$. This implies that if the coincidence is significant, the Gravitational Constant $G$ or other fundamental constants may vary inversely with the age of the universe.