- #1

Dick Taid

- 6

- 0

## Homework Statement

Hi,

From the Kerr metric, in geometrized units,

[tex]\left(1 - \frac{2M}{r}\right) \left(\frac{dt}{d\lambda}\right)^2

+ \frac{4Ma}{r} \frac{dt}{d\lambda}\frac{d\phi}{d\lambda}

- \frac{r^2}{\Delta} \left(\frac{dr}{d\lambda}\right)^2

- R_a^2 \left(\frac{d\phi}{d\lambda}\right)^2 = 0[/tex]

where [itex]R_a^2 = r^2 + a^2 + \frac{2Ma^2}{r}[/itex] is the reduced circumference, [itex]a \equiv \frac{J}{M}[/itex] is the spin parameter and [itex]\lambda[/itex] is some affine parameter. I need to calculate the equations of motion.

## Homework Equations

I want to solve the Lagrange equations

[tex]-\frac{d}{d\sigma}\left(\frac{\partial L}{\partial\left(dx^\alpha/d\sigma\right)}\right)

+ \frac{\partial L}{\partial x^\alpha} = 0[/tex]

for the Lagrangian

[tex]\mathcal{L}\left(\frac{dx^{ \alpha}}{d\sigma},x^{\alpha}\right)

= \left(-g_{\alpha\beta}\frac{dx^{\alpha}}{d\sigma}\frac{dx^{\beta}}{d\sigma}\right)^{1/2}[/tex]

## The Attempt at a Solution

The problem is, that the metric is null so the Lagrangian is as well (?). Is it possible to calculate the equations of motion using this approach, or am I "forced" to do it the hard way, using

[tex]\frac{d^2x^{\alpha}}{d\lambda^2}

= -\Gamma_{\beta\gamma}^{\alpha}\frac{dx^{\beta}}{d \lambda}\frac{dx^{\gamma}}{d\lambda}[/tex]

finding the Christoffel symbols, and so forth?