Length Contraction & Time Dilation Beyond Planck Scale: Correct?

[Flatland]

If an object was traveling fast enough relative to an observer such that it's length is contracted down to the Planck scale (as with time), I would imagine that any further increase in speed would result in no more observable relativistic effects? Please correct me if I'm wrong.

 

[Ibix]

No. There's no limit to speed in relativity except that it must be below c for a massive object and exactly c for a massless one. There is no problem with something being shorter than the Planck length (except that you might want to calculate the speed needed to shrink even an atom down to that size).

The Planck length is not "the shortest possible length". It's merely a "natural" unit of length composed of various fundamental constants.

 

[Flatland]

But if it did contract beyond the Planck length it would be physically impossible to measure?

 

[julian]

You want to evoke quantum gravity theories that predict no physical relevance to scales below the Planck scale?

There are the doubly special relativity theories (DSR). These are modifications of special relativity in which some particular value of energy/momentum, in addition to the speed of light, is an invariant.

However, Carlo Rovelli has argued (in the context of loop quantum gravity) that a minimal length (or area) doesn't contradict Lorentz invariance. Length and area operators are not classical quantities. They are quantum observables. If an observer measures a system as having the Plank length, it means that the system is in an eigenstate of the length operator $L$. A boosted observer who measures the length of the same system is measuring a different observable $L'$, which generally does not commute with $L$. If the system is in an eigenstate of $L$, then generally it will not be in an eigenstate of $L'$. The eigenvalues of $L'$ will however be the same as the eigenvalues of $L$ (including the minimal value).

 

[Ibix]

I have no idea how you'd directly measure the length of something even many orders of magnitude larger than that, even if it weren't moving at 0.999999999c or whatever. One could presumably build a chain of rulers moving at progressively higher speeds, each measuring the apparent length of the next, until you have one traveling at a comparable speed to your Planck-length-contracted atom that can measure it.

Relativity has no problem with things being as length contracted as you want. As far as beyond-relativity models go, you should ask in the Beyond the Standard Model forum.