User:Tohline/SR/PoissonOrigin

Origin of the Poisson Equation

In deriving the,

Poisson Equation

<math>\nabla^2 \Phi = 4\pi G \rho</math>

we will follow closely the presentation found in §2.1 of [BT87].

According to Isaac Newton's inverse-square law of gravitation, the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho(\vec{x}^{~'}) d^3x'</math>, as follows:

<math>~\vec{a}(\vec{x})</math>	<math>~=</math>	<math>~ \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' \, , </math>
[BT87], p. 31, Eq. (2-2)

where, <math>~G</math> is the Newtonian gravitational constant.

Now, in the astrophysics literature, it is customary to adopt the following definition of the,

Scalar Gravitational Potential
<math>~ \Phi</math>	<math>~\equiv</math>	<math>~ -G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
[BT87], p. 31, Eq. (2-3)

[Note that, as we have detailed in a separate discussion, throughout [EFE] Chandrasekhar adopts a different sign convention as well as a different variable name to represent the gravitational potential.]

Drawn from Other Wiki Pages

It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of <math>~\vec{v}</math> to represent the inertial velocity field. More importantly, he adopts a different variable name and a different sign convention to represent the gravitational potential, specifically,

<math>~ - \Phi = \mathfrak{B} </math>

<math>~=</math>

<math>~ G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>

Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [EFE] into this H_Book.

… which expresses simply the conservation of the angular momentum of the system. The symmetric part of the tensor expression gives what is generally referred to as (see [EFE] for details) the,

Tensor Virial Equation

<math>~\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>	<math>~=</math>	<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>
[EFE], p. 23, Eq. (51) [BT87], p. 213, Eq. (4-78)

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation