# Why is Planck time scaled by c^5?

1. ### apeiron

2,432
Just curious.

The Planck length is lpl = (hG/c3)1/2 = 10-33cm

And it seems intuitive that it's c cube because space has three dimensions for the action.

But the Planck time is tpl = (hG/c5)1/2 = 10-43s

So is there some obvious physical reason why c is to the power of five here?

2. ### phyzguy

The Planck time is just the Planck length divided by c. This adds two powers of c because they are inside the square root.

3. ### dauto

Yes, that's the only combination of physical constants that gives you a constant with time units

4. ### apeiron

2,432
Thanks. Beautifully simple.

### Staff: Mentor

I don't see how a power of 3/2 looks natural.
The planck volume has c9/2.

Those odd factors just show how "unnatural" the SI-units (where c, h, G, k are not nice numbers) are in terms of fundamental physics.

6. ### Meir Achuz

2,058
Newton said Gmm'/r=energy. In natural units, that means G is a (length)^2.
The hbar and c are just put in to get G in cm^2. This is always unique.

7. ### ChrisVer

2,289
what does "space has 3 dimensions for the action"? I mean that it's kind of weird, we don't know whether at Planck scale you need more than 3 spatial dimensions, so it's not so intuitive...
On the other hand, everything seems normal under what is called dimensional analysis... So you have some constants ($G, c, \hbar$) and you want to build characteristic quantities out of them .... So for everything, you just write:
$[X]= [c]^{a} [\hbar]^{b} [G]^{d}$
and you solve for $a,b,d$