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 $m_p^{(3d)}$?

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    $\begingroup$ Maybe this can help. $\endgroup$ Apr 9 at 8:41
  • $\begingroup$ 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. $\endgroup$ Apr 9 at 13:12
  • $\begingroup$ $L$ 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. $\endgroup$ Apr 9 at 13:32
  • $\begingroup$ 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. $\endgroup$ Apr 9 at 13:40
  • $\begingroup$ 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. $\endgroup$ Apr 9 at 13:56

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