View Single Post
 Quote by clamtrox Don't do that! The term you added already contains the second term. There are many ways of writing the same thing, and I think everyone benefits if we don't try to confuse things further by using angular momentum density in one term and velocity in the next. So I will stick to my earlier assumptions that particle velocity is relativistic, but the mass of the gravitating object is not: this means we don't get a circular orbit, and you can't use the "usual" formalism that one uses with for example Mercury precession. The equation for acceleration to order vē/cē is $\mathbf{a} = -\frac{GM\hat{\mathbf{r}}}{r^2} + \frac{4GM(\hat{\mathbf{r}} \cdot \dot{\mathbf{r}}) \dot{\mathbf{r}}}{c^2 r^2} - \frac{GMv^2 \hat{\mathbf{r}}}{c^2 r^2}$ If you have difficulties with vectors, it's very easy to write it without them: Let me define $a_r = \mathbf{a} \cdot \hat{\mathbf{r}}$ and $a_{\perp} = |\mathbf{a} - a_r \hat{\mathbf{r}}|$ and likewise for velocity v. Then $a_r = -\frac{GM}{r^2} (1 + \frac{v^2-4 v_r^2}{c^2}) = -\frac{GM}{r^2} (1 + \frac{v_{\perp}^2-3 v_r^2}{c^2})$ and $a_{\perp} = \frac{4 GM v_r v_{\perp}}{c^2 r^2}$ Did that make it clearer?