Why is Planck time scaled by c^5?

  1. apeiron

    apeiron 2,366
    Gold Member

    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. jcsd
  3. phyzguy

    phyzguy 2,489
    Science Advisor

    The Planck time is just the Planck length divided by c. This adds two powers of c because they are inside the square root.
  4. Yes, that's the only combination of physical constants that gives you a constant with time units
  5. apeiron

    apeiron 2,366
    Gold Member

    Thanks. Beautifully simple.
  6. mfb

    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.
  7. Meir Achuz

    Meir Achuz 2,076
    Science Advisor
    Homework Helper
    Gold Member

    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.
  8. ChrisVer

    ChrisVer 2,403
    Gold Member

    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 ([itex]G, c, \hbar [/itex]) and you want to build characteristic quantities out of them .... So for everything, you just write:
    [itex] [X]= [c]^{a} [\hbar]^{b} [G]^{d} [/itex]
    and you solve for [itex]a,b,d[/itex]
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