You can calculate this quantity yourself using the formulas provided in the answer to the question "Why isn't an electron a black hole?", found here:
The significance of this mass is that it is the energy scale at which the quantum properties of the object (remember, this is a point particle!) are as important as the general relativity properties of the object. Therefore it is likely to be the mass scale at which quantum gravity effects start to matter. Turning this into a mass is as simple as using the formula for the Compton wavelength given in the above link and plugging in the Planck mass. Thus, the Planck length is the typical quantum size of a particle with a mass equal to the Planck mass. As you point out, the Planck time is then just the Planck length divided by the speed of light.
Since the Planck length has this special property of being the length scale where we can't ignore quantum gravity effects, it is typically taken to be the size of a fundamental string, in string theory. Alternatively, if we consider more general theories of quantum gravity, one might speculate that it is the typical size of the "fuzziness" of spacetime. It's a length scale (or energy scale) we are unlikely to probe in any future experiments so we tend to interpret it as the length scale at which classical general relativity (GR) "breaks down" -- i.e. at which classical GR fails to provide an accurate description of nature. This is very similar to the way that the speed of light is considered the velocity scale at which Newtonian mechanics "breaks down" and special relativity is called for.
Answered by: Brent Nelson, Ph.D., Research Fellow, University of Michigan
There are two pillars of study in physics: Quantum Mechanics and Relativity. All matter is affected by these two pillars to some degree, but they typically can be considered as mutually exclusive. If a particular mass is of a size to be greatly affected by relativity, there usually is little quantum affects. Or, if a particular mass has large quantum affects, there usually is little relativistic affects.
For a given mass, the distance over which relativistic affects are dominant is called the Schwarzschild radius. If a mass is squeezed smaller than its Schwarzschild radius it becomes a Black Hole. The Schwarzschild radius is given by Gm/c2.
The distance over which quantum affects are dominant is called the Compton length. For a given mass, anything smaller than its Compton length is strongly quantum. The Compton length is given by h/mc.
If we compare the Schwarzschild radius and Compton length formulas, as the mass varies, the two lengths are inversely proportional to each other. This is why the two affects seem mutually exclusive.
The question becomes, ï¿½Is there a particular mass with the same Schwarzschild radius and Compton length?ï¿½ The answer is yes. If we equate the two formulas and solve for m, we find that at a mass of (hc/G)1/2 (about 10-5 g) the Schwarzschild radius and the Compton length are equal. This mass is called the Plank mass. The length at which the Schwarzschild radius and the Compton length are equal is called the Plank length. At the Plank length, both relativistic and quantum affects are equally dominant.
Answered by: Randy MacDougall, President, Expert Systems Resources Inc.