Why does gravity couple with a negative dimension?

I was reading this post where it states that

"gravity does have a coupling with negative dimension and it's the Newton constant"

However, I don't understand at all why Newton's constant leads to gravity coupling with negative dimension. Could it be possible for someone to show why gravity couples with negative dimensions?

• Have you learned about "natural units"? The "negative dimension" of $G$ is $[M]^{-2}$ in the sense of dimensional analysis using natural units where $\hbar=c=1$. The quote isn't talking about spacetime dimensions. Commented Aug 24, 2022 at 4:22

To expand on @Ghoster's comment, $$c=\hbar=1$$ implies $$1/m=\hbar/(mc)$$ is a length, i.e. length and time each have mass dimension $$-1$$, and $$G$$ has dimension $$\mathsf{L}^3\mathsf{M}^{-1}\mathsf{T}^{-2}$$ so has mass dimension $$-3-1+2=-2$$. By contrast, comparing the GPE $$-Gm_1m_2/r$$ to its electrostatic counterpart $$kq_1q_2/r$$ shows $$k$$, like charge, has mass dimension $$0$$.

• Thanks! But I just want to ask, the gravitational constant has dimensions $L^3 M^{-1} T^{-2}$ but why would that then lead to $-3-1+2$? Commented Aug 24, 2022 at 9:14
• @JungwoonSong As I said, length & time each have mass dimension $-1$. Alternatively, note $[G/(c\hbar)]=\mathsf{M}^{-2}$.
– J.G.
Commented Aug 24, 2022 at 9:20
• Thanks so much! Commented Aug 24, 2022 at 9:42
1. Let's for simplicity work in units where $$\hbar=1=c$$. We will then classify physical quantities $$Q$$ according to their mass dimension $$[Q]$$, cf. Ref. 1.

2. Recall that Newton's law of gravitation in $$d$$ spacetime dimensions is $$F~=~G\frac{m_1m_2}{r^{d-2}},\tag{1}$$ up to possible dimensionless constants. The fact that $$[F]~=~2, \qquad [m_1]~=~1~=~[m_2], \qquad [r]~=~-1,\tag{2}$$ implies that the gravitational constant $$G$$ has mass dimension $$[G]~=~2-d,\tag{3}$$ which is negative for $$d>2$$, cf. OP's title question. See also e.g. this Phys.SE post.

3. Now, the underlying reason OP asks their question is apparently because this fact leads to the non-renormalizability of perturbative quantum gravity, cf. their linked post. This is e.g. shown in my Phys.SE answer here.

References:

1. M. Srednicki, QFT, 2007; Chapter 12. A prepublication draft PDF file is available here.