How Can I Understand Slowly Rotating Perturbations of the Whittaker Metric?

[MstoneHall]

I am interested in considering slowly rotating perturbations of the Whittaker metric. Unfortunately I have very little knowledge of this metric. Could someone help me with the exact form of this metric and maybe where to get more material? I understand this may be out of place here but...

 

[robphy]

http://www.journals.royalsoc.ac.uk/openurl.asp?genre=article&id=doi:10.1098/rspa.1968.0133

J. M. Whittaker,
An Interior Solution in General Relativity
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934-1990)
ISSN: 0080-4630
Issue: Volume 306, Number 1484 / July 30, 1968

has links to other papers that reference it.

http://www.jstor.org/view/00804630/di002761/00p00437/0 [via JSTOR]


http://adsabs.harvard.edu/abs/1968RSPSA.306...1W
lists additional citations.

 

[Chris Hillman]

Wont' say!

Well, a moment with Google yields the paper http://arxiv.org/abs/gr-qc/9910001, from which it seems that the so-called Whittaker metrics are obtained from the coframe (first covector field timelike, rest spacelike, all pairwise orthogonal)
$$\sigma^1 = -\sqrt{1+2 \, h(r,\theta)} \, A(r) \, dt$$
$$\sigma^2 = \sqrt{1+2 \, m(r,\theta)} \, B(r) \, dr$$
$$\sigma^3 = \sqrt{1 + 2 \, k(r,\theta)} \, C(r) \, d\theta$$
$$\sigma^4 = \sin(\theta) \, \left( d\theta + (\Omega - \omega(r)) \, dt) \right)$$
via
$$g = -\sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3 + \sigma^4 \otimes \sigma^4$$
I believe the motivation is Hartle's approximation scheme for slowly rotating axisymmetric fluid balls in gtr, you can look up the citations they give and see if any of the earlier papers clarify this.

(While I was composing this reply, robphy posted some relevant information. Rob, I don't have on-line access to that journal here--- do you know if this is an exact fluid solution or, as I guess, an approximation valid to say second order in a rotation parameter?)

Picking up the counting theme, the metric functions here comprise three functions of two variables and four functions of one variable, and all are independent of time. One can ask: how many metric functions of so many variables are required to represent an arbitrary stationary axisymmetric Lorentzian spacetime? An arbitrary stationary axisymmetric fluid solution?

 

[MstoneHall]

Much thanks Chris and robphy. Your explanations and links have been of tremendous help.