Are length contractions limited by Planck length?

Question

While we are getting closer to speed of light our length in the direction of the movement is according to Lorentz transformation getting shorter. But we can not (even theoretically) consider length shorter than Planck length. So is speed in the universe limited more than we thought before quantum physics?

Answer 1

But we can not (even theoretically) consider length shorter than Planck length.

This is a popular misconception. Treating Planck units as special is really more numerology than anything else. For example, the Planck mass is about the mass of a single biological cell. Does that mean physics doesn't apply to anything smaller (or is it larger?) than a cell?

Furthermore, the whole idea of relativity is that you can perform any subluminal frame transformation. If an object is 1 meter in size in one frame, then traveling at $(1-0.5\times10^{-70})\ c$ relative to that frame will contract it to $10^{-35}\ \mathrm{m}$. That's just how the moving observer would see things. It's not as though the object itself feels squished to such a small size.

For a proof by contradiction, suppose there were a speed limit based on not being able to length-contract anything to less than a Planck length. Would you accept that there could be objects of size 10 Planck lengths? If so, then we only need to boost by $0.995c$ to contract such an object to a Planck length. But we are easily able to achieve such relative speeds in the laboratory, and they happen throughout the universe all the time (e.g. with cosmic rays or quasar jets).

Answer 2

While we are getting closer to speed of light our length in the direction of the movement is according to Lorentz transformation getting shorter.

This are two misconceptions here. One is that the way this is written implies that velocity is absolute. This is not the case. The "relativity" in relativity theory means exactly the opposite. Velocity is relative.

The other misconception is with regard to the concept of length contraction. Suppose you are in a spaceship moving toward the Earth, traveling at a relativistic speed with respect to the Earth. Someone on Earth looking at you will see you as length contracted, but you won't see anything different with your own body. What you will see is that see distance between the spacecraft and Earth appears to be length contracted.

Answer 3

The Planck length is a quantum effect (among others). SO the answer is: yes, speeds in the universe are limited by c, and also by the Planck length, indirectly. However, this latter limit has never been measured nor achieved, so that in practice, the limit by c is sufficient.

All Planck values are limits. There is no way to get a measurement result lower than the Planck length. This has been discussed at length in the literature. As an additional example, the Planck mass limits the energy that any elementary particle can have. The answer claiming the opposite is plain wrong. Planck values are not numerology, they are actual limits. Most famously, the Planck speed c, the speed of light, is an actual limit for speed of energy.

By the way: nobody has ever boosted a particle to a speed that makes it appear shorter than the Planck length, despite what the answers above claim. (Just put the numbers in and see by yourself.)