What Are P^{μν} and \bar{P}^{μν} in Relativity?

[TriTertButoxy]

Hi, I'm a bit stuck on the interpretation of the following tensors:

$$P^{\mu\nu}=g^{\mu\nu}-\frac{q^\mu q^\nu}{q^2}\hspace{5mm}\text{and}\hspace{5mm}\bar{P}^{\mu\nu}=\frac{q^\mu q^\nu}{q^2}$$​

Here, $q^\mu$ is a 4-vector, and $g^{\mu\nu}$ is the metric that goes (1, -1, -1, -1).

I've calculated the following relations:

$$q_\mu P^{\mu\nu}=0$$

$$P^{\mu\nu} g_{\mu\nu}=-3$$

$$P^{\mu\alpha}P_{\alpha}^{\phantom{\alpha}\nu}=P^{\mu\nu}$$

$$q_\mu \bar{P}^{\mu\nu}=q^\nu$$

$$\bar{P}^{\mu\nu} g_{\mu\nu}=1$$

$$\bar{P}^{\mu\alpha}\bar{P}_{\alpha}^{\phantom{\alpha}\nu}=\bar{P}^{\mu\nu}$$, and

$$P^{\mu\alpha}\bar{P}_{\alpha}^{\phantom{\alpha}\nu}=0\,.$$

I see some symmetry going on here, like -3 corresponding to the three spatial directions, and +1 corresponding to the time direction. And a 'product' of one another yields 0. But, I can't quite pinpoint what $P^{\mu\nu}$ and $\bar{P}^{\mu\nu}$ are in words.

 

[robphy]

If q is a timelike vector, then P and P-bar are projection operators, the first in the spatial subspace orthogonal to q, the other parallel to q. P and P-bar can also be interpreted as a decomposition of g into degenerate spatial and temporal metrics, as decomposed by an observer with tangent vector q.

 

[TriTertButoxy]

Thanks a lot! Is there a place on the internet which has practice problems with the relativistic notation? I seem to have a really hard time doing math with it, and would like to be more fluent.

 

[robphy]

You could try
http://vishnu.mth.uct.ac.za/omei/gr/
http://pancake.uchicago.edu/~carroll/notes/
http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/tc.html
http://www.ima.umn.edu/nr/abstracts/arnold/einstein-intro.pdf
http://www.glue.umd.edu/~tajac/spacetimeprimer.ps
http://www.pma.caltech.edu/Courses/ph136/yr2004/
http://www.lps.uci.edu/home/fac-staff/faculty/malament/FndsofGR.html
http://www.lps.uci.edu/home/fac-staff/faculty/malament/geometryspacetime.html
... these last few show some explicit calculations and provide good physical and operational interpretation.

I'd strongly suggest that you learn to use the "abstract index notation" (see the last few urls ). While "coordinates" and "components" are helpful in doing numerical and functional calculations, abstract index notation is superior for keeping track of geometrical objects (and thus their physical interpretation)... not to mention that most of the modern relativity textbooks [e.g. MTW, Wald] use it.

One of the best ways I found to practice with relativistic notation is to obtain the usual Maxwell Equations in vector calculus form starting from its tensorial formulation.