GR Relation Reduces to Newtonian: Limits Explained

[center o bass]

In GR for orbits about a central mass in the Schwarzschild metric one can show that

$$\dot r^2 = \frac{E^2}{m^2 c^2} - (1-\frac{r_s}{r})(c^2 + \frac{p_\phi^2}{r^2}).$$

where $E=-p_t$, $r_s$ is the Schwarzschild radius and 'dot' represent differentiation with respect to proper time. Similarly for the Newtonian case one gets

$$\frac{1}{2}m \dot r^2 = E - V_{eff}(r)$$

with

$$V_{eff}(r) = \frac{p_\phi^2}{2mr^2} - \frac{GmM}{r}.$$

This time with dot representing time differentiation. I would think that the relation for GR should reduce to the newtionan relation in some limit, but I'm having problems deriving it. Especially I'm concerned about the square (E^2) appearing in the above relation.

Does the GR relation reduce to the Newtonian (I would think it had to?) and what would be the proper limits to take to obtain it?

 

[Bill_K]

Not the same E. The part of your first equation that's independent of r is (E² - m²c⁴)/m²c², and if you let E = mc² + ε where ε is the Newtonian energy, and expand, you'll get the right Newtonian limit.