Graduate Einstein Field Equations: Covariant vs Contravariant

[Prez Cannady]

Depending on the source, I'll often see EFE written as either covariantly:

$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi GT_{\mu\nu}$$

or contravariantly

$$R^{\alpha\beta} - \frac{1}{2}Rg^{\alpha\beta} = 8 \pi GT^{\alpha\beta}$$

Physically, historically, and/or pragmatically, is there a reason for this? Or is it just the result of habit and preference with the understanding that you can raise and lower indices as required once you've solved for mass-energy or curvature?

I mean, mathematically I might use the bottom formulation when playing with covectors, but when would a relativist do that in practice in his every day work? Or do physicists not really concern themselves about the convention just so long as the indices line up at the end of the day?

 

[PeterDonis]

is it just the result of habit and preference

Yes. As long as you have a metric, you can raise and lower indexes whenever you like, so which way to put them in equations is just preference.

 

[robphy]

It might be good to make a scorecard of which author uses which form.
My guess is that a more-mathematically minded author would write the first form
since that is purer... using the inverse-metric fewer times.

 

[Orodruin]

It might be good to make a scorecard of which author uses which form.
My guess is that a more-mathematically minded author would write the first form
since that is purer... using the inverse-metric fewer times.

I don’t know about purer. To me it might seem ”purer” to vary the action with respect to the metric rather than its inverse, which would give you the second form ... In the end, to each his/her own.

 

[Matterwave]

I prefer $\mathbf{Ric}-\frac{1}{2}\mathbf{g}\text{Tr}_\mathbf{g}(\mathbf{Ric})=8\pi G\mathbf{T}$ because it looks fancier.