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abby11

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In summary: The covariant derivative of a scalar function ##\phi## is given by ##\nabla_\nu \phi = \partial_\nu \phi##, where ##\partial_\nu## is the ordinary partial derivative. But for vectors and tensors, the covariant derivative includes additional terms from the connection, which will contribute to the final answer.

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abby11

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PeterDonis

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abby11 said:I am trying to derive the radial momentum equation in the equatorial Kerr geometry obtained from the equation

$$

(P+\rho)u^\nu u^r_{;\nu}+(g^{r\nu}+u^ru^\nu)P_{,r}=0 \qquad

$$

Where are you getting this equation from? It doesn't look anything like a "radial momentum equation" in Kerr geometry, since Kerr geometry is a vacuum geometry so ##\rho = P = 0## everywhere.

abby11 said:since the equation only involves radial derivatives

The equation has a covariant derivative, which includes terms in the connection coefficients that involve other values of ##\nu## besides ##r##.

The Kerr geometry is a mathematical model used to describe the spacetime around a rotating black hole. It was first developed by the physicist Roy Kerr in 1963.

Radial momentum is the component of momentum that is directed along the radial direction, or towards or away from the center of a system. In the context of Kerr geometry, it refers to the momentum of a particle moving in the radial direction around a rotating black hole.

The radial momentum equation in Kerr geometry is derived using the equations of motion for a particle in a curved spacetime, known as the geodesic equation. By solving this equation for a particle moving in the radial direction around a rotating black hole, we can obtain the expression for radial momentum.

The key factors that affect radial momentum in Kerr geometry are the mass and spin of the black hole, as well as the angular momentum and energy of the particle. These parameters determine the shape of the spacetime around the black hole and thus influence the motion of particles in its vicinity.

The radial momentum equation in Kerr geometry is used in various astrophysical and cosmological applications, such as studying the behavior of matter and radiation around black holes, understanding the dynamics of accretion disks, and predicting the properties of gravitational waves emitted by rotating black holes.

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