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If the metric $g_{\mu\nu}$ is dimensionless (i.e. does not have associated dimensional units) 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 $x^\mu$. Surely the coefficients in Pythagoras' theorem don't have 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 potential 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 $[M]$.

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    $\begingroup$ GR is a classic theory and QM is not a classic theory. Hard to represent this classic field to be a quantum field. $\endgroup$ – user256968 Mar 26 at 11:36
  • $\begingroup$ The dimensionality of the metric is a convention. I prefer the convention where the coordinates are dimensionless and the metric has dimensions of length^2 $\endgroup$ – Dale Mar 26 at 11:55
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    $\begingroup$ The title question is as nonsensical as say “Could a human being be dimensionless?” Gravitons are not physical quantities. They do not have a property “physical dimension” defined for them. $\endgroup$ – A.V.S. Mar 26 at 15:53
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If the metric gμν is dimensionless and gravitons are quantum excitations of the metric does that mean that gravitons themselves are dimensionless?

It has dimensions , at least look here to a particular metric , the matrix elements have the dimensions of meter square.

Is graviton energy included in the stress-energy tensor Tμν?

In phsyics there are several frames where different theories describe mathematically the data and observations, and also predict new systems, but in mainstream physics they are consistent with one another in the region of overlap.

Metrics of General Relativity belong to large dimensions large masses . GR is a classical theory, which means it is not quantized. The graviton is the hypothetical gauge boson of a quantized gravitational theory, expected in the future. At the moment only effective quantization of gravitation is used in mainstream cosmological models.

So it is mixing up apples and oranges to require GR to have gravitons included.

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

It is not clear what you mean by dimensionless. All the particles in the standard model are point particles , thus have no dimensions. They are described when interacting with an energy and momentum four vector.

In string theories, which can quantize gravity, the graviton is part of the particle spectrum. If a definite theory is found then the graviton will be like the photon a zero dimension elementary particle.

Note that it is expected that mathematically, the quantized gravity theory will reduce to GR for large masses and dimensions. Here one can see the equivalent for electromagnetism, how classical electromagnetic fields emerge from the quantized fields of QED.

This emergent behavior happens in regions of phase space overlap between theories, for example: thermodynamics emerges from classical statistical mechanics. Otherwise physics would not be a consistent theoretical system.

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  • $\begingroup$ I mean in the sense that the Newtonian gravitational potential field $\Phi$ does not have associated dimensional units. If gravitons are excitations of the $\Phi$ field then they are dimensionless themselves unlike other fields/particles with dimensions of mass/energy. $\endgroup$ – John Eastmond Mar 27 at 9:17
  • $\begingroup$ Newton and quantum mechanics match only through emergence. Natural units have no meaning in Newtonian mechanics, as they are connected with lorenz transformations and quantum mechanics. . I do not know what you mean the potential is dimensionless, as it has the units of energy afaik . $\endgroup$ – anna v Mar 27 at 11:33
  • $\begingroup$ In SI units the Newtonian gravitational potential $\Phi$ has dimensions of $c^2$. I think you can set $c=1$ without using quantum mechanics. $\endgroup$ – John Eastmond Mar 27 at 12:14
  • $\begingroup$ You can set anything equal one. Does it make sense in the rest of the theoretical framework? Not in classical physics. It is just a change in units and it still carries its units if you do any calculations. en.wikipedia.org/wiki/Natural_units natural units means instead of having meters /seconds/ etc to have as units the choice. It just means that if you want to know how far the bus stop is, you have to express it in the chosen units. $\endgroup$ – anna v Mar 27 at 14:01

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