# Planck's Length, Length Contraction?

Would Planck's length be subject to length contraction if an object hypothetically of the length had a velocity near c?

I suspect that it won't becuase Planck's length is defined by the invariant, physical constants of G, c and the reduced Planck's constant. Thus, Planck's length would not be subject to relativistic effects like length contraction.

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phinds
Gold Member
2021 Award
Would Planck's length be subject to length contraction if an object hypothetically of the length had a velocity near c?

I suspect that it won't becuase Planck's length is defined by the invariant, physical constants of G, c and the reduced Planck's constant. Thus, Planck's length would not be subject to relativistic effects like length contraction.
Planks Length would be affected exactly the same (percentage wise) as would a meter or a mile or any other measurement amount.

Ibix
Planck's length would not be subject to relativistic effects like length contraction.
What do you mean by this? The Planck length is just a unit. It neither expands nor does it not expand - it's not a physical thing.

My metre ruler has a length measurable as a rather large number of Planck lengths. Are you saying that an observer who sees it length contracted to one centimetre will see the same number of Planck lengths as I do despite seeing the ruler being 1% as long as I see it? Or are you saying that an object that is one Planck length long will not undergo length contraction?

The former is absurd. The latter has no theoretical support as far as I'm aware, although the Beyond the Standard Model guys may have something to say about it. Certainly we haven't probed that scale experimentally, and certainly SR would predict nothing unusual about that scale.

Or do you mean something else?

What do you mean by this? The Planck length is just a unit. It neither expands nor does it not expand - it's not a physical thing.

My metre ruler has a length measurable as a rather large number of Planck lengths. Are you saying that an observer who sees it length contracted to one centimetre will see the same number of Planck lengths as I do despite seeing the ruler being 1% as long as I see it? Or are you saying that an object that is one Planck length long will not undergo length contraction?

The former is absurd. The latter has no theoretical support as far as I'm aware, although the Beyond the Standard Model guys may have something to say about it. Certainly we haven't probed that scale experimentally, and certainly SR would predict nothing unusual about that scale.

Or do you mean something else?
Planck's length is the smallest meaningful scale of length; it seems to me that if it could be contracted then it wouldn't be the smallest meaningful scale of length; thus by definition it shouldn't be subject to contraction. And I'm referring to your latter point.

Also, surely SR wouldn't predict anything unusual at the scale because SR was not derived with the consideration of QM.

Planks Length would be affected exactly the same (percentage wise) as would a meter or a mile or any other measurement amount.
May I ask what evidence there is for this?

Ibix
Planck's length is the smallest meaningful scale of length; it seems to me that if it could be contracted then it wouldn't be the smallest meaningful scale of length; thus by definition it shouldn't be subject to contraction. And I'm referring to your latter point.
You're going to have to find an object of that scale to measure, then, or a theory that does predict something odd at that scale. See my previous post.

Also, surely SR wouldn't predict anything unusual at the scale because SR was not derived with the consideration of QM.
If you don't think relativity can answer your questions, why are you asking in the relativity forum...? Again, see my previous post - Beyond the Standard Model might be a better bet.

• Einstein's Cat
Ibix
Planck's length is the smallest meaningful scale of length
Incidentally, I think this is rather an overstatement. Some quantum theories ascribe significance to the length. Others do so in an order-of-magnitude kind of way. See, for example, the "Theoretical significance" section of https://en.m.wikipedia.org/wiki/Planck_length

PeterDonis
Mentor
Would Planck's length be subject to length contraction if an object hypothetically of the length had a velocity near c?

According to standard classical SR/GR, no; length contraction applies the same way on all scales.

There are speculative hypotheses (for example, Google "doubly special relativity") along the lines of: when you get down to length scales of around the Planck length, standard Lorentz invariance breaks down, so that, for example, an object that is one Planck length long in its rest frame would still be one Planck length long in any frame (instead of contracting). This means that the transformation between frames can no longer be a standard Lorentz transformation on these length scales.

AFAIK no speculation along these lines has led to anything fruitful in terms of either guiding possible experiments to test it (not surprising since the current smallest length scale we can probe is about 20 orders of magnitude larger than the Planck length) or being able to account for something theoretically that is hard to account for in standard SR/GR.

• Einstein's Cat
Nugatory
Mentor
May I ask what evidence there is for this?
That's like asking what evidence there is that if there are a thousand millimeters in a meter then there will be ten millimeters in a stick one one-hundredth of a meter long.

The planck length is about ##1.6\times{10}^{-35}## meters, the same way that a millimeter is ##10^-{3}## meters. Just as there are a thousand millimeters in a meter, or twelve inches in a foot, there are ##6.25\times{10}^{34}## Planck lengths in a meter. Length-contract that meter down to a centimeter and it's only one one-hundredth as long, so now it's ten millimeters or .01 meters or ##6.25\times{10}^{32}## Planck lengths long.

That's like asking what evidence there is that if there are a thousand millimeters in a meter then there will be ten millimeters in a stick one one-hundredth of a meter long.

The planck length is about ##1.6\times{10}^{-35}## meters, the same way that a millimeter is ##10^-{3}## meters. Just as there are a thousand millimeters in a meter, or twelve inches in a foot, there are ##6.25\times{10}^{34}## Planck lengths in a meter. Length-contract that meter down to a centimeter and it's only one one-hundredth as long, so now it's ten millimeters or .01 meters or ##6.25\times{10}^{32}## Planck lengths long.
Hypothetically if an object's length was to be contracted so that its length was Planck's length and if the objects velocity increased, would the object's length contract further?

Nugatory
Mentor
Hypothetically if an object's length was to be contracted so that its length was Planck's length and if the objects velocity increased, would the object's length contract further?
According to special relativity, the answer is "Yes, of course".

However now would be a good time for a sort of fun mathematical exercise. There is a meter stick floating in space. You are floating next to it in a spaceship, and if the spaceship were to be placed in motion then the stick would be moving relative to you and it would be length contracted.
1) How fast would the ship have to be moving before the meter stick was contracted down to one planck length?
2) Assume the ship, with you in it, has a total mass of 1000 kg. What is the kinetic energy of the spaceship moving at the speed you calculated in #1? Compare this value to some other energy sources in the universe.

It would be not amazing to find some new and interesting physics under these conditions..... But none of that changes the fact that the Planck length is just a unit of distance like meters and inches and fathoms and furlongs.

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Mister T