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Why is Planck time scaled by c^5?

  1. Mar 1, 2014 #1

    apeiron

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    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. Mar 1, 2014 #2

    phyzguy

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    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. Mar 1, 2014 #3
    Yes, that's the only combination of physical constants that gives you a constant with time units
     
  5. Mar 1, 2014 #4

    apeiron

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    Thanks. Beautifully simple.
     
  6. Mar 2, 2014 #5

    mfb

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    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. Mar 2, 2014 #6

    Meir Achuz

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    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. Mar 2, 2014 #7

    ChrisVer

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    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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