- #1
Thales Castro
- 11
- 0
- Homework Statement
- A rocked describes a circular orbit around a black hole with angular velocity $\Omega$ (measured by a static observer at infinity) and Schwarzschild radius $r=R$. Calculate the 4-acceleration felt by the rocket.
- Relevant Equations
- Schwarzschild metric:
$$
ds^{2} = -\left( 1 - \frac{2M}{r} \right) + \frac{1}{1-\frac{2M}{r}} dr^{2} + r^{2}d\Omega^{2}
$$
Kepler law for GR:
$$
\Omega^{2} = \frac{M}{r^{3}}
$$
Christoffel symbols:
$$
\Gamma ^{\alpha}_{\mu \nu} = \frac{1}{2}g^{\alpha \beta}\left( \partial_{\mu}g_{\nu \beta} + \partial_{\nu}g_{\mu \beta} - \partial_{\beta}g_{\mu \nu}\right )
$$
4-acceleration:
$$
a^{\mu} = u^{\alpha} \nabla_{\alpha} u^{\mu} = \frac{d u^{\mu}}{d\tau} + \Gamma^{\mu}_{\alpha \beta}u^{\alpha}u^{\beta}
$$
In a circular orbit, the 4-velocity is given by (I have already normalized it)
$$
u^{\mu} = \left(1-\frac{3M}{r}\right)^{-\frac{1}{2}} (1,0,0,\Omega)
$$Now, taking the covariant derivative, the only non vanishing term will be
$$
a^{1} = \Gamma^{1}_{00}u^{0}u^{0} + \Gamma^{1}_{33}u^{3}u^{3}
$$
Evaluating the Christoffel symbols, we have:
$$
\Gamma^{1}_{00} = \frac{M}{r^2}\left( 1 -\frac{2M}{r} \right )
$$
$$
\Gamma^{1}_{33} = -r\left( 1 -\frac{2M}{r} \right )
$$
By putting these values in the equation for a^1, I get
$$
a^{1} = \left( 1 - \frac{2M}{R} \right )\left(1 - \frac{3M}{R} \right )^{-1} \left[\frac{M}{R^{2}} - R\Omega^{2} \right ] = 0
$$
Now, I don't see why the acceleration should be zero in this problem, but still I can't find what I have done wrong in my calculations. Can anyone help me? Thanks in advance.
$$
u^{\mu} = \left(1-\frac{3M}{r}\right)^{-\frac{1}{2}} (1,0,0,\Omega)
$$Now, taking the covariant derivative, the only non vanishing term will be
$$
a^{1} = \Gamma^{1}_{00}u^{0}u^{0} + \Gamma^{1}_{33}u^{3}u^{3}
$$
Evaluating the Christoffel symbols, we have:
$$
\Gamma^{1}_{00} = \frac{M}{r^2}\left( 1 -\frac{2M}{r} \right )
$$
$$
\Gamma^{1}_{33} = -r\left( 1 -\frac{2M}{r} \right )
$$
By putting these values in the equation for a^1, I get
$$
a^{1} = \left( 1 - \frac{2M}{R} \right )\left(1 - \frac{3M}{R} \right )^{-1} \left[\frac{M}{R^{2}} - R\Omega^{2} \right ] = 0
$$
Now, I don't see why the acceleration should be zero in this problem, but still I can't find what I have done wrong in my calculations. Can anyone help me? Thanks in advance.