I GR: Can Units Tell You if Quantity is Covariant or Contravariant?

[Twigg]

Note: My GR is very cursory and rusty. The probability that I say something bogus in stating my question is very high. Thanks for your help in advance!

I have a dumb question. I'm only casually familiar with GR, and I have a hard time telling if quantities should be covariant or contravariant. Can I tell just by looking at the units? For example, I remember 4-velocity and 4-momentum are contravariant and both have units that include meters in the numerator (in the convention where ($x^0 = ct$). I remember that the 4-gradient $\partial_{\mu}$ is a covariant quantity and its units have meters in the denominator. Can I get away with using this trend as a mnemonic or will I get into trouble? If no, can you give counter-examples?

I know covariant vectors can be transformed into contravariant vectors via the metric, but when I'm doing a problem I need to know whether I should write $X_{\mu} = (A,B,C,D)$ or $X_{\mu} = (-A,B,C,D)$ given that I know the values of the components A,B,C,D. For instance, I know that I can write 4-momentum as a covariant vector $P_{\mu} = \eta_{\mu \nu} P^{\nu}$, however when I'm doing a problem I have to know that P is naturally contravariant (in other words, $P^{\mu} = (E/c, p_x, p_y, p_z)$ and $P_{\mu} = (-E/c, p_x, p_y, p_z)$, and not the other way around). Just to make sure, I'm not misunderstanding this, right?

Thanks for bearing with me, all!

 

[Ibix]

If no, can you give counter-examples?

Four velocity has dimensions of speed, and the metric is usually dimensionless. $g_{ab}U^aU^b=U_bU^b$ has dimensions of speed squared, so $U_b$ must have the same dimensions as $U^a$.

I'm not sure of a good mnemonic for what's co/contra variant, but perhaps others can suggest something.

 

[Twigg]

Oh yep, that's valid. I done goofed

 

[Twigg]

Just saw another one: the 4-wavevector is contravariant though it has units of radians per meter.