Newton's Constant in Higher Dim. Spacetimes, Velocity of Light=1

[gerald V]

TL;DR

In our spacetime, Newtons constant has dimension length/energy (the velocity of light set unity). How is this in a different number of dimensions?

In the following, I set the velocity of light unity.
I refer to theories of gravities in higher-dimensional spacetimes.

Newton` s constant converts the curvature scalar with dimension $lenght^{-2}$ into the matter Lagrangian with dimension $energy/length^3$. So its dimension is $length/energy$. But in $d$-dimensional spacetime the dimension of the curvature scalar would remain unchanged, whereas the matter Lagrangian would have dimension $energy/length^{d-1}$. So, for $d \ne 4$, Newtons constant would have a different dimension. Is this correct?

 

[robphy]

In d-dimensional space, Newton's Law of Gravity would be F_g=\displaystyle G\frac{m_1m_2}{r^{d-1}}
(which follows from Laplace-Poisson's equation in d-dimensions).
By equating that with Newton's Second Law, which is taken to be dimensionally-independent,
we get
in MKS units, for d-dimensional space, [G]=\displaystyle \frac{1}{kg} m^d s^{-2}.

https://en.wikipedia.org/wiki/Gravitational_constant

See: J.D. Barrow's
(1983) Dimensionality
Philosophical Transactions of the Royal Society of London.
Series A, Mathematical and Physical Sciences 310: 337–346 http://doi.org/10.1098/rsta.1983.0095
(See also https://www.jstor.org/stable/37418 )

It seems this following blog summarizes some of the details:
https://thespectrumofriemannium.wordpress.com/2012/11/18/log054-barrow-units/