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I have an expression where I have converted all units such that the answer is expressed in 4d Planck mass, $m_p^{(4d)}$.

In David Tong's lecture notes, Newtons gravitational constant in 4d is given by $ G \sim \frac{1}{m_p^{(4d)}}$.

How would I express this (or anything in terms of 4d Planck mass) in $m_p^{(3d)}$?

Nihar Karve
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Johan Hansen
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    Maybe this can help. – Davide Morgante Apr 09 '21 at 08:41
  • Thanks! There is an ambiguity with that answer. Because the "$L$" term can have dimensions. He refers to "$L$" having dimensions of "length". But it is not clear what dimension that $L$ have. I know how to relate $G_4$ and $G_3$ in different dimensions in terms of $m_p^{(4d)}. My d is not the dimension of the spacetime, it is the dimension of the Planck mass. – Johan Hansen Apr 09 '21 at 13:12
  • $L$ has dimensions of length, so meters in the SI. The statement " My d is not the dimension of the spacetime, it is the dimension of the Planck mass." does not make much sense to me. – Davide Morgante Apr 09 '21 at 13:32
  • Planck mass and Planck length is dimension dependent. The four-dimensional (4D) gravitational constant can be express in terms of e.g. $m_p^{(3d)}$ or $m_p^{(4d)}$. d should not be confused with D. I don't understand good enough to explain it better. – Johan Hansen Apr 09 '21 at 13:40
  • Clearly the two are dimension independent since one has dimensions of length and one dimensions of mass. In any given spacetime dimensions they have dimensions of length and dimension of mass respectively. Their functional form, constructed from various constants like $G$, changes in different spacetime dimensions.This functional constant depends upon the physical dimensions of the constants that build them, so you have to know how they change in different spacetime dimensions. – Davide Morgante Apr 09 '21 at 13:56
  • This makes sense. And it is what I am wondering as well. The reason the answer you linked is hard to apply to this problem is because L can take either $l_p^{(3d)}$ or $l_p^{(4d)}$, I think. – Johan Hansen Apr 09 '21 at 14:13
  • I suggest you so start from the definition of that quantities and work your way through various spacetime dimensions – Davide Morgante Apr 09 '21 at 15:10

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