# Could gravitons be dimensionless?

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## Main Question or Discussion Point

If the metric $g_{\mu\nu}$ is dimensionless and gravitons are quantum excitations of the metric does that mean that gravitons themselves are dimensionless?

I say this as locally the metric is just the flat metric $\eta_{\mu\nu}=\hbox{diag}(-1,1,1,1)$ with the dimensions in the co-ordinates $x^\mu$.

To put it another way:

Is graviton energy included in the stress-energy tensor $T_{\mu\nu}$?

Actually classical gravitational waves can be detected so does that imply that gravitons can't be dimensionless?

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phinds
Gold Member
2019 Award
Actually classical gravitational waves can be detected so does that imply that gravitons can't be dimensionless?
Photons can be detected - so does that imply that they can't be dimensionless?

ohwilleke
Gold Member
If the metric $g_{\mu\nu}$ is dimensionless and gravitons are quantum excitations of the metric does that mean that gravitons themselves are dimensionless?
Are you asking if they are point particles, or something deeper than that?

jtbell
Mentor
Are you asking if they are point particles, or something deeper than that?
Ah, you beat me to it!

I was about to ask whether there might possibly be some confusion between two meanings of "dimensionless":

1. Not having associated dimensional units (e.g. in SI a.k.a. MKS units). The fine structure constant is dimensionless in this sense.

2. Having zero size, in some sense (e.g. an electron versus a proton)

I mean (1) : not having dimensional units.

In terms of Newtonian gravitation we have the gravitational potential given by:

$$\Phi \sim -\frac{G M}{R}$$

In natural units, $\hbar=c=1$ (dimensionless), Newton's gravitational constant is $G=1/M_{pl}^2$ where $M_{pl}$ is the Planck mass. Therefore the dimensions of the gravitational field $\Phi$ is

$$[\Phi] = \frac{[M]^{-2}[M]}{[M]^{-1}}=1$$

If gravitons are excitations of $\Phi$ then they must themselves be dimensionless.

This is unlike other fields and their associated particles that have dimensions of mass/energy $[M]$.