Planck Time

The Planck time has a value of 5.39121 × 10−44 second (current uncertainty 40 × 10−44). It is the smallest time that can be operationally defined, that is, measured even in principle. The Planck length has a value of 1.61624 × 10−35 (current uncertainty 12 × 10−35) meter and represents the smallest length that can be operationally defined.

By international agreement, the distance or length, L, between two points in space is defined as the time, t, it takes for light to travel between the points in a vacuum, multiplied by a constant, c

where c = 299,792,458 meters per second is the speed of light in a vacuum. (This number is exact by definition.) In order to measure t, and thus L, we need a clock with an uncertainty At no larger than t. The time-energy uncertainty principle says that the product of At and the uncertainty in a measurement of energy in that time interval, AE, can be no less than h/2, where h = h/2% and h = 6.6260693 × 10−34 joule-second (current uncertainty 11 × 10−34) is Planck's constant. That is,

Thus,

This energy is equivalent to the rest energy of a body of mass m,

Equation 5 implies that within a spherical region of space of radius L, we cannot determine, by any measurement, that it contains a mass less than

Now, a spherical body of mass M will be a black hole if its radius R is less than or equal to

where G = 6.6742 × 10−11 cubic meters per kilogram per square second (current uncertainty 10 × 10−11) is Newton's gravitational constant. This is called the Schwarzschild radius

Consider a body of mass m given in Equation 5. Its Schwarzschild radius will be

which is called the Planck length. Notice it is simply the length formed from the three basic constants in physics, h,c, and G. It represents the smallest length that can be operationally defined, [Page 997]that is, defined in terms of measurements that can be made by any instrument. If we tried to measure a smaller distance, the time interval would be smaller, the uncertainty in rest energy larger, the uncertainty in mass larger, and the region of space would be experimentally indistinguishable from a black hole. Because nothing inside a black hole can climb outside its gravitational field, we cannot see inside and thus cannot make a smaller measurement of distance.

Similarly, we can make no smaller measurement of time than the Planck time,

The Planck time and Planck length are the most basic units of time and space. Although distance and time are assumed continuous variables, they are fundamentally discrete. However, because physics experiments have not yet even come close to probing space and time on the Planck scale, treating them as continuous remains a good approximation.

General relativity, the theory of gravity introduced by Albert Einstein in 1915, has so far passed every empirical test to high precision. However, not being a quantum theory, it can be expected to break down at the Planck scale, where it will have to be replaced by a quantum theory of gravity, still not developed.

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