Let t_pl be the Planck time. It is well-known that cosmology is related to the following coupling constants: quantum mechanics (h), gravity (G), and lightspeed (c).
We assume that t_pl = h^αG^βc^γ. Since [px ]=[ h], the dimension of h is [m_vx] = [mx^2t], where m, v, x, and t represent mass, velocity, distance, and time, respectively. Using Newton's law of universal gravitation, we obtain that G = F_r^2 / m^2. Therefore, the dimension of G is [max^2m^2] = [mx^3m^2t^2] = [x^3mt^2], given that the dimension of acceleration is [a] = [xt^2]. The dimension of c is [x t].
By the above, we get:
[t] = [(mx^2t)^α(x^3mt^2)^β(x t)^γ] = [m^α-βx^(2α+3β+γ)t^(-α-2β-γ)].
We need to solve the following system of equations to determine the values of α, β, and γ:
{α - β = 0 2α + 3β + γ = 0 -α - 2β - γ = 1}
Using Gaussian elimination, we find that α = 1/2, β = 1/2, and γ = -5/2. Therefore,
t_pl = h^(1/2) G^(1/2) c^(-5/2) = (hG/c^5)^(1/2).
The first step in this dimensional analysis problem is to write down the dimensions for the constants h, G, and c. I will use slightly different notation for the dimensional analysis compared to what is used in the lectures. For the dimensions, I will set [M] to be the dimension of mass, [L] to be the dimension of length, and [T] to be the dimension of time.
Firstly, we know that c is a velocity, so it has dimensions [M][L][T]^-1. The force = G = F_r^2 / r^2 m^2 = [M^-1][L^3][T^-2]. We also know that h = E / ν = [M][L^2][T^-2]^-1 = [M][L^2][T^-1].
So tpl = h^x G^y c^z = [T]. Thus, we have the following equation for the dimensions:
[M^0][L^0][T^1] = [M^-x+y][L^3x+2y][T^-2x-y-z].
We can now obtain the system of three equations:
-x + y = 0 3x + 2y + z = 0 -2x - y - z = 1
The first equation immediately gives that x = y, and by solving the second and third equations, we obtain z = -5/2 and x = y = 1/2. So we have derived the desired relation: tpl = h^(1/2) G^(1/2) c^(-5/2).
