# John F. N. Salik

## Quantization of Space-Time: Planck Time

In Propulsion Physics, Space-Time Physics on December 2, 2012 at 8:25 pm

The structure of space at quantum or sub-quantum scales can be conceived as a collection of disconnected elements in space which cannot be distinguished by any observations.  Using principles from quantum theory, special relativity, and general relativity, one can demonstrate the existence of a quantized space-time (resulting in the Planck scales).  This is to say, there is the possibility that space-time is not continuous, but rather a collection of irreducible chunks.

Heisenberg’s uncertainty principle highlights the indeterminacies that relate time and energy via Planck’s constant:$\Delta t \Delta E > \hbar$. For an arbitrarily large energy uncertainty of $\Delta E$, we can have an arbitrarily small $\Delta t$.  If we wish to demonstrate the discrete nature of space-time, first need we need to carefully examine the time measurement process.

Let us consider the use of a clock with mass $m$, and length $l$ for time observation. Furthermore, let us assume that the smallest unit of time measure is $\delta t$.

1. From Special Relativity, we know that the total energy of a material object is given by $E=mc^2$.  We know therefore that a clock of mass $m$ cannot have $\Delta E > mc^2$.

The time uncertainty has a lower bound because of the upper bound on the total energy of the measurement clock.

2. Heisenberg’s uncertainty relationship is expressed as $\Delta t \Delta E > \hbar$, which means that $\Delta t$ can get arbitrarily small for an arbitrarily large $\Delta E$.
3. Because $\Delta E$ has an upper bound established by the clock’s mass property, we see that $\Delta t$ has a lower bound: $\Delta t> \frac{\hbar}{mc^2}$.
4. This lower bound suggests that $\delta t > \Delta t$.
5. In order for the clock to make a measurement, we expect that $c \delta t >l$ or the clock will not be able to receive its measurement signal.
6. The Schwarzchild radius of the clock gives us the characteristic length of the clock, which is where its total energy is equal to its gravitational energy: $r_s = \frac{Gm}{c^2}$.  Because our clock does not represent a black hole where signals cannot escape, we expect that the clock mass has an upper bound: $m < \frac{r_s c^2}{G}$.  This in turn suggests $r_s< \frac{Gm}{c^2}$.
7. Taking $l$ to be the Schwarzchild radius, we obtain $\delta t > \frac{Gm}{c^3}$the lowest bound on time measurement.
8. Finally, we observe the inequality while removing dependence on the clock mass to obtain: $\delta t^2 > \frac{G\hbar}{c^5}$.  We define the Planck Time as $t_P = \delta t$.

This reasoning provides us with a fundamental lower limit on time measurement and is termed Planck time.  Using similar reasoning, the smallest units of length and mass can be found as well.  Using the fundamental constants of Nature ( $\hbar$, $c$, and $G$), we have the Planck units:

Planck Mass
$\displaystyle m_P = \sqrt{\frac{\hbar c}{G}} = 2.176 \times 10^{-8} kg$

Planck Length
$\displaystyle l_P = \sqrt{\frac{G \hbar}{c^3}} = 1.615 \times 10^{-35} m$

Planck Time
$\displaystyle t_P = \sqrt{\frac{G \hbar}{c^5}} = 5.389 \times 10^{-44} s$

Using principles from quantum theory, special relativity, and general relativity, a quantized space-time can be demonstrated. This implies the existence of a fundamental lower limit on arbitrarily small time intervals.

According to this formulation, the classical notion of points in a continuous space-time does not make physical sense, and the physics below the Planck units requires the differentiation between vacuum and matter which this is yet not possible through known physics as measurements are not possible at these scales.