- #76

marcus

Science Advisor

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lets put c and hbar in explicitly and see what the planck units

actually are in (1+2)D

thing to notice is that in 4D we have

[tex]GM^2 = \hbar c[/tex]

because GM^2 has to equal the unit force x area (inverse sq. law)

and that equation defines the pl. mass in 4D

but in 3D GM^2 will equal the unit force x distance!

and that is the unit energy in the system: Mc^2, so we have instead

[tex]GM^2 = M c^2[/tex] which solves to

[tex]M = \frac{c^2}{G}[/tex]

After that, easy, unit energy is

[tex]E = \frac{c^4}{G}[/tex]

and unit freq is

[tex]\omega = \frac{c^4}{G\hbar}[/tex]

That makes unit time

[tex]T = \frac{G\hbar}{c^4}[/tex]

and unit distance

[tex]L = \frac{G\hbar}{c^3}[/tex]

actually are in (1+2)D

thing to notice is that in 4D we have

[tex]GM^2 = \hbar c[/tex]

because GM^2 has to equal the unit force x area (inverse sq. law)

and that equation defines the pl. mass in 4D

but in 3D GM^2 will equal the unit force x distance!

and that is the unit energy in the system: Mc^2, so we have instead

[tex]GM^2 = M c^2[/tex] which solves to

[tex]M = \frac{c^2}{G}[/tex]

After that, easy, unit energy is

[tex]E = \frac{c^4}{G}[/tex]

and unit freq is

[tex]\omega = \frac{c^4}{G\hbar}[/tex]

That makes unit time

[tex]T = \frac{G\hbar}{c^4}[/tex]

and unit distance

[tex]L = \frac{G\hbar}{c^3}[/tex]

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