# Why is Planck time scaled by c^5?

by apeiron
Tags: planck, scaled, time
 PF Gold P: 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?
 P: 2,179 The Planck time is just the Planck length divided by c. This adds two powers of c because they are inside the square root.
Thanks
P: 1,948
 Quote by apeiron 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?
Yes, that's the only combination of physical constants that gives you a constant with time units

PF Gold
P: 2,432
Why is Planck time scaled by c^5?

 Quote by 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.
Thanks. Beautifully simple.
Mentor
P: 11,589
 Quote by apeiron 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.
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.
 Sci Advisor HW Helper PF Gold P: 1,991 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.
 P: 754 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 $a,b,d$

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