# No Such Thing as the Plank Length?

I've been tossing this one in my head for a while now, and every time I think of it I convince myself that there is no such thing as the Plank Length (i.e. smallest unit of measurement) because of this following situation:

We have two observers, we will consider one of these frames of reference to be stationary and the other travelling at half the speed of light. From the stationary frame of reference, the observer measures the plank length to be 1.616×10^-35 m. Lets assume the stationary observer is very clever and can accuratly measure a plankian distance on the second frame of reference travelling at a sizeable portion of the speed of light. He would measure the plankian length to be 1.399×10^-35 m, a small but noticably smaller length than his. He continues to measure, and each time the speed of the second frame of reference increases until at 0.999999c, the measured plank length is 2.285×10^-38 m. Now all that has to happen for a smaller distance to be measured is for the second frame of reference to bump up its speed a little (disregarding the huge amount of energy needed to do so). Therefore, the plank length can be as small as we like, so whats the point? I know that the plank length is not measured but calculated, but supposing it could be measured, would this not happen? Do we make the plank length immune to length contraction?

It would be great to discuss this

## Answers and Replies

Bill_K
Science Advisor
Nobody said the Planck length (note spelling) was the smallest possible length. And nobody said it was an exact value. It's just an approximate scale, an order of magnitude. It's the length scale at which quantum mechanics and general relativity merge.

The Planck length is where the Compton wavelength of a particle is equal to its gravitational radius: r = ħ/Mc = GM/c2. If you juggle this a bit you get r = √ħG/c3

CompuChip
Science Advisor
Homework Helper
Also note the crucial point that Bill made: at the Planck scale a new type of physics kicks in. It is (sort of by definition) the length scale at which general relativity (and by extension, special relativity) are no longer necessarily applicable.

Nobody said the Planck length (note spelling) was the smallest possible length.

Ok yes, if the Planck length is where general relativity and quantum mechanics are supposed to be unified, wouldn't this mean at relativistic speeds it would appear to be unified at a much smaller scale - and following from that, if the speed is great enough, wouldn't there be no need to have a unification? (since the Planck length is infentesibly small)

Also, if you are interested in Quantum General Relativity (and not strings) Canonical Quantum Gravity (also called Loop Quantum Gravity) derives a fundamental length scale, that is, quantized area, volume, etc... You might find it interesting.