f. How many dimensions do theoretical physicists predict? (two possible answers)
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Because we don't yet have a quantum theory of gravity, we cannot analyze the properties of the universe before the Planck time, about 10^-43 s. If we assume that the properties of the universe during that era were determined by quantum theory, relativity, and gravity, the Planck time should be characterized by the fundamental constants of those three theories: h, c, and G. We can therefore write t ∝ h^i c^j G^k, where i, j, and k are exponents to be determined. (a) Do a dimensional analysis to determine i, j, and k. (b) Assuming the proportionality parameter is of order unity, evaluate t. (c) What was the size of the observable universe at the Planck time?
Adi S.
Three fundamental constants of nature-the gravitational constant $G,$ Planck's constant $h,$ and the speed of light $c-$ have the dimensions of $\left[L^{3} / M T^{2}\right],\left[M L^{2} / T\right],$ and $[L / T]$ respectively. (a) Find the mathematical combination of these fundamental constants that has the dimension of time. This combination is called the "Planck time" $t_{\mathrm{P}}$ and is thought to be the earliest time, after the creation of the universe, at which the currently known laws of physics can be applied. (b) Determine the numerical value of $t_{\mathrm{P}}$ . (c) Find the mathematical combination of these fundamental constants that has the dimension of length.This combination is called the "Planck length" $\lambda_{\mathrm{P}}$ and is thought to be the smallest length over which the currently known laws of physics can be applied. (d) Determine the numerical value of $\lambda_{\mathrm{P}}$ .
Because we don't yet have a quantum theory of gravity, we cannot analyze the properties of the universe before the Planck time, about $10^{-43}$ s. If we assume that the properties of the universe during that era were determined by quantum theory, relativity, and gravity, the Planck time should be characterized by the fundamental constants of those three theories: $h, c,$ and $G .$ We can therefore write $t \propto h c^{\prime} G^{k},$ where $i, j,$ and $k$ are exponents to be determined. (a) Do a dimensional analysis to determine $i, j,$ and $k$. (b) Assuming the proportionality parameter is of order unity, evaluate $t$. (c) What was the size of the observable universe at the Planck time?
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