- 3

- 0

I'd like to understand how to calculate the components

of Newtonian tidal accelaration tensor in polar coordinates.

Is any available Internet source which clearly explains the

technique with details?

Reading James B. Hartle "Gravity" textbook I stumbled on the following

Example from Chapter 21.

=======================================================

Example 21.1 Tidal Acceleration Oitside a Spherical Mass.

The Newtonian gravitational potential outside a spherically symmetric

distribution of mass is (G = 1 units)

[tex]

\Phi = \frac {- M }{r} (21.6)

[/tex]

where

[tex]

r = \sqrt {x^2+y^2+z^2 }

[/tex]

is the distance from the center od symmetry.

Evaluating the tidal gravitational acceleration tensor using

the rectangular coordinates gives:

[tex]

a_{ij} \equiv - \frac{\partial^2\Phi}{\partial x^i\partial x^j} =

-(\delta_{ij}-3n_{i}n_{j})\frac{M}{r^3} (21.7)

[/tex]

where

[tex]

n_{i} \equiv \frac {x^i}{r}

[/tex]

are the components of a unit vector in a radial direction.

In an orthonormal basis

[tex]

\vec{e_{\hat{r}}} , \vec{e_{\hat{\theta}}} , \vec{e_{\hat{\phi}}}

[/tex]

oriented along coordinate directions of a polar coordinates (r,\theta,\phi)

the nonvanishing components of the tidal acceleration tensor are

[tex]

a_{\hat{r}\hat{r}} = \frac{2M}{r^3},

a_{\hat{\theta}\hat{\theta}}=a_{\hat{\phi}\hat{\phi}}=-\frac{M}{r^3} (21.8)

[/tex]

=======================================================

Q1. From (21.6) I see that gravitational potential does depend only

on 'r' but not on theta and phi. Why in that case its theta, phi

partial derivatives described in (21.8) are non-zero?

Looks like (21.8) are calculated in some other polar coordinates,

but how those coordinates related to another ones desribed in (21.6)?

Q2. Let's assume that I'm wrong in Q1, and (21.6) does represent

gravitational potential only in Cartesian coordinates, but not

in a polar ones. Then I tried another way. I applied Chain Rule of

differentiation to (21.7) to calculate (21.8) based

on standard Cartesian --> Polar transformations:

[tex]

x=r\sin{\theta}\cos{\phi}, y=r \sin{\theta} \sin{\phi}, z=r\cos{\theta}

[/tex]

I calculated

[tex]

a_{{\hat{\theta}}{\hat{\theta}}}

[/tex]

from (21.8) applying Chain Rule to (21.7) but got

[tex]

- \frac {2 M} {r}

[/tex]

which differs from (21.8).

Did I make a mistake in my calculations, or Chain Rule

simply cannot be applied here?

of Newtonian tidal accelaration tensor in polar coordinates.

Is any available Internet source which clearly explains the

technique with details?

Reading James B. Hartle "Gravity" textbook I stumbled on the following

Example from Chapter 21.

=======================================================

Example 21.1 Tidal Acceleration Oitside a Spherical Mass.

The Newtonian gravitational potential outside a spherically symmetric

distribution of mass is (G = 1 units)

[tex]

\Phi = \frac {- M }{r} (21.6)

[/tex]

where

[tex]

r = \sqrt {x^2+y^2+z^2 }

[/tex]

is the distance from the center od symmetry.

Evaluating the tidal gravitational acceleration tensor using

the rectangular coordinates gives:

[tex]

a_{ij} \equiv - \frac{\partial^2\Phi}{\partial x^i\partial x^j} =

-(\delta_{ij}-3n_{i}n_{j})\frac{M}{r^3} (21.7)

[/tex]

where

[tex]

n_{i} \equiv \frac {x^i}{r}

[/tex]

are the components of a unit vector in a radial direction.

In an orthonormal basis

[tex]

\vec{e_{\hat{r}}} , \vec{e_{\hat{\theta}}} , \vec{e_{\hat{\phi}}}

[/tex]

oriented along coordinate directions of a polar coordinates (r,\theta,\phi)

the nonvanishing components of the tidal acceleration tensor are

[tex]

a_{\hat{r}\hat{r}} = \frac{2M}{r^3},

a_{\hat{\theta}\hat{\theta}}=a_{\hat{\phi}\hat{\phi}}=-\frac{M}{r^3} (21.8)

[/tex]

=======================================================

Q1. From (21.6) I see that gravitational potential does depend only

on 'r' but not on theta and phi. Why in that case its theta, phi

partial derivatives described in (21.8) are non-zero?

Looks like (21.8) are calculated in some other polar coordinates,

but how those coordinates related to another ones desribed in (21.6)?

Q2. Let's assume that I'm wrong in Q1, and (21.6) does represent

gravitational potential only in Cartesian coordinates, but not

in a polar ones. Then I tried another way. I applied Chain Rule of

differentiation to (21.7) to calculate (21.8) based

on standard Cartesian --> Polar transformations:

[tex]

x=r\sin{\theta}\cos{\phi}, y=r \sin{\theta} \sin{\phi}, z=r\cos{\theta}

[/tex]

I calculated

[tex]

a_{{\hat{\theta}}{\hat{\theta}}}

[/tex]

from (21.8) applying Chain Rule to (21.7) but got

[tex]

- \frac {2 M} {r}

[/tex]

which differs from (21.8).

Did I make a mistake in my calculations, or Chain Rule

simply cannot be applied here?

Last edited: