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I am trying to derive the radial momentum equation in the equatorial plane of Kerr geometry obtained by Lasota (1994) which reads (eqn. 6 in page-343; I am using units in which $M=1$) as follows: $$uu'+\frac{1}{r\Delta}\left(a^2-r-\frac{A\gamma^2K}{r^3}\right)u^2-\frac{A\gamma^2K}{r^6}+\frac{1}{P+\rho}\left(\frac{\Delta}{r^2}+u^2\right)P'=0 \qquad (1)$$ where $K=\dfrac{(\Omega-\Omega_K^+)(\Omega-\Omega_K^-)}{\Omega_K^+\Omega_K^-},\qquad \Omega_K^\pm=\pm\dfrac{1}{r^{3/2}\pm a},\qquad u\equiv u^r$

The primes in the above equation refer to derivative w.r.t. the coordinate r. This is obtained from the equation $$(P+\rho)u^\nu u^r_{;\nu}+(g^{r\nu}+u^ru^\nu)P_{,r}=0 \qquad (2)$$ which represents the projection of the covariant derivative of perfect fluid energy momentum tensor on the hypersurface orthogonal to the four-velocity.

I tried to derive it as follows:

First term in eqn.(2): $$(P+\rho)u^\nu u^r_{;\nu}=(P+\rho)u^r u^r_{;r}=(P+\rho)u^r\frac{1}{\sqrt-g}(\sqrt-g u^r)_{,r}=(P+\rho)uu'+(P+\rho)\frac{u^2}{r}$$ where I had used $A^i_{;i}=\dfrac{1}{\sqrt-g}(\sqrt-g A^i),\quad \sqrt-g=r,\quad u\equiv u^r$

Second term in eqn.(2): $$(g^{r\nu}+u^ru^\nu)P_{,\nu}=(g^{rr}+u^ru^r)P_{,r}=\frac{\Delta}{r^2}P'+u^2P'$$ where I had used $g^{rr}=\dfrac{\Delta}{r^2}$

Adding these two terms, we obtain $$uu'+\dfrac{u^2}{r}+\dfrac{1}{P+\rho}\left(u^2+\dfrac{\Delta}{r^2}\right)P'=0$$

Comparing this with eqn.(1), it can be observed that only the first and the last terms match. However, in the non-matched terms there is no factor of $P'$. This means that I am missing something in the calculation of the first term.

Can anyone please point out what I am missing?

EDIT:

I missed the $u^t$ and $u^\phi$ terms in the expansion of the first term in eqn.(2). I had now expanded the term as $$(P+\rho)u^\nu u^r_{;\nu}=(P+\rho)(u^r u^r_{;r}+\Gamma^r_{\phi\phi}u^\phi u^\phi+\Gamma^r_{t\phi}u^tu^\phi+\Gamma^r_{\phi t}u^\phi u^t+\Gamma^r_{tt}u^tu^t)$$ and using the expressions for the Christoffel connections, I get the following equation: $$uu'+\frac{1}{r\Delta}\left(-\frac{A\gamma^2K}{r^3}\right)u^2-\frac{A\gamma^2K}{r^6}+\frac{1}{P+\rho}\left(\frac{\Delta}{r^2}+u^2\right)P'=0$$ Comparing this with eqn.(1), I am still missing two terms.

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  • $\begingroup$ Your equation (2) has index conservation issues... $\endgroup$ – mmeent Sep 5 at 15:36
  • $\begingroup$ Anyway, you error seems to stem from the fact that you forgot about the t and phi components of u. $\endgroup$ – mmeent Sep 5 at 15:42
  • $\begingroup$ @mmeent I had now calculated using the t and phi components of u. But still two terms are missing. Can you please check the new information I added? $\endgroup$ – Richard Sep 6 at 12:05

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