- #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.