The effective potential for particle orbits in the equatorial plane of a Schwarzschild black hole in units $G=M=c=1$ is given by $$V_{\textrm{eff}}=\sqrt{\left(1-\frac{2}{r}\right)\left(1+\frac{l^2}{r^2}\right)} \qquad\qquad\qquad(1)$$

When we are interested in studying accretion flows in the equatorial plane of a Schwarzschild black hole, we need to analyze the relativistic hydrodynamic equations. If we consider radial accretion of matter (with no angular momentum) in the equatorial plane, the relativistic Euler equation can be written as [page-569 of Shapiro and Teukolsky with $M=1$ due to chosen convention] $$uu'+\frac{1}{r^2}+\left(1-\frac{2}{r}+u^2\right)a^2\frac{n'}{n}=0$$ where $n$ is the baryon density and $u'=\dfrac{du}{dr}$.

From the above equation, we can easily identify the second term as the gravitational force experienced by the matter when the matter falls radially and there is no angular momentum.

I tried to derive similar results considering non-zero angular momentum and I found the equation $$uu'+\frac{1}{r^2}\frac{r^3-l^2(r-2)^2}{r^3-l^2(r-2)}+\left(1-\frac{2}{r}+u^2\right)a^2\frac{n'}{n}=0$$

From this equation, we observe that the radial force is of the form $$f_r=\frac{1}{r^2}\frac{r^3-l^2(r-2)^2}{r^3-l^2(r-2)} \qquad\qquad\qquad(2)$$ In the limit $l=0$, we obtain the form given in Shapiro and Teukolsky.

I integrated Eqn.$(2)$ to obtain the effective potential suitable for studying fluid motion in the equatorial plane of a Schwarzschild black hole. I found that the potential nearly matches with $V_{\textrm{eff}}$ given by Eqn.$(1)$, but with a deviation (though negligible, but increases with $l$) close to the black hole.

However, I am curious about the following:

The deviation is very negligible and we can easily use the effective potential obtained by integrating $f_r$. However, since both the expressions are derived from the exact equations (one from relativistic particle dynamics and the other from relativistic hydrodynamics), I think that this deviation is due to the transition from particle dynamics to hydrodynamics, incorporating which both the potentials would match exactly. So my questions are:

  1. Is the deviation really due to the use of two different approaches i.e., particle dynamics and hydrodynamics?
  2. If true, then is it possible to find a mathematical expression of this deviation from General Relativity?

N.B.: If related stuffs are already available in the literature, it would be very helpful if they are mentioned here.

  • $\begingroup$ (Other high rep users correct me if I'm wrong) I've removed the link to the book, as I believe on SE we don't allow links to pirated material. $\endgroup$ – JamalS Mar 5 at 12:35
  • $\begingroup$ I've added a question mark as otherwise this looks like a mere assertion. $\endgroup$ – Mozibur Ullah Mar 5 at 12:43
  • $\begingroup$ @Richard How did you derive equation (2)? $\endgroup$ – Alex Trounev Mar 5 at 14:01
  • $\begingroup$ @AlexTrounev Equation (2) can be derived from the radial Euler equation $(p+\rho)u^\mu\nabla_\mu u^r + (g^{\mu r}+u^\mu u^r)\partial_\mu p=0$ and using the Schwarzschild metric with the angular momentum defined by $l=\dfrac{\tilde{l}}{\mathscr{E}}=-\dfrac{hu_\phi}{hu_t}$, where $h$ is the enthalpy density. The angular momentum $\tilde{l}$ is conserved for particles and fluids, whereas $l$ is conserved only for fluids. $\endgroup$ – Richard Mar 6 at 8:03
  • $\begingroup$ @Richard If there is $u_{\phi}\ne 0$, then how did you separate the equations? $\endgroup$ – Alex Trounev Mar 6 at 12:02

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