# Weight of a tensor density

Is there any freedom in choosing the weight of a tensor density?

I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with the weight of $\frac{-1}{2}$ :

$$\overset{\sim}{g_{ab}} = \sqrt{({\det g})} ^{\frac{-1}{2}}~~g_{ab}$$

In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), on page 1599 there is an example.

You can also look at the relation $2.4$ of this article.

I don't understand why this special weight is chosen for the tensor density constructed from metric.

• Who is "they"?? – Kyle Kanos Sep 27 '13 at 19:09
• I mean some articles. – Dory Sep 27 '13 at 19:14
• Ah. Perhaps you could cite these papers? Maybe then someone could answer your question. – Kyle Kanos Sep 27 '13 at 19:19
• @KyleKanos:I added an example. – Dory Sep 27 '13 at 19:47
• Comment to the question (v4): Are you asking (i) if the same fixed physical tensor density could be assigned different tensor weights, or (ii) if it is possible to modify a physical tensor density, so that it has any weight, that one would like? – Qmechanic Sep 30 '13 at 22:08

In these particular cases, the authors are interested in the conformal structure, i.e. lightcone structure, of the manifold. A conformal structure can be defined by an equivalence class of metrics, all of which are related to each other by a conformal transformation, $$g_{ab}\sim e^{\omega(x)}\bar{g}_{ab}$$ A nice way to characterize a conformal structure is to say it is just the tensor density of weight $-1/2$ that you mention. Under a conformal transformation, the metric determinant changes by $$g\mapsto e^{4\omega}\bar{g},$$ so you can see that the density weight is chosen so that the tensor density $\tilde{g}_{ab}$ is invariant under the conformal transformation. If you were in dimension other than $4$, you would choose the weight to be $-2/d$, to maintain this invariance under conformal transformations.