Are principal curvatures equal?

  • Thread starter Thread starter Thinkor
  • Start date Start date
Thinkor
Messages
49
Reaction score
1
This is a question about Newtonian gravity. I post it here, because there seems little interest in gravity under the classical physics section.

Principal curvatures on surfaces of equal potential around an isolated spherically symmetric orb are equal at every point by symmetry, but are they equal generally in Newtonian gravity? If not, do you have an example where they would not be equal? If so, do you have a reference that proves it?
 
Physics news on Phys.org
Actually, immediately after posting this I thought of a counterexample. Imagine two flat walls in space close to one another but with a gap between them. The field is flat between them if sufficiently far from the edges of the walls. Now, crease both walls equally along the same line. The principal curvature along the midline between the creased walls will be zero (if far enough from the edges), but the other principal curvature for points on the midline will be nonzero.
 
I started reading a National Geographic article related to the Big Bang. It starts these statements: Gazing up at the stars at night, it’s easy to imagine that space goes on forever. But cosmologists know that the universe actually has limits. First, their best models indicate that space and time had a beginning, a subatomic point called a singularity. This point of intense heat and density rapidly ballooned outward. My first reaction was that this is a layman's approximation to...
Thread 'Dirac's integral for the energy-momentum of the gravitational field'
See Dirac's brief treatment of the energy-momentum pseudo-tensor in the attached picture. Dirac is presumably integrating eq. (31.2) over the 4D "hypercylinder" defined by ##T_1 \le x^0 \le T_2## and ##\mathbf{|x|} \le R##, where ##R## is sufficiently large to include all the matter-energy fields in the system. Then \begin{align} 0 &= \int_V \left[ ({t_\mu}^\nu + T_\mu^\nu)\sqrt{-g}\, \right]_{,\nu} d^4 x = \int_{\partial V} ({t_\mu}^\nu + T_\mu^\nu)\sqrt{-g} \, dS_\nu \nonumber\\ &= \left(...
In Philippe G. Ciarlet's book 'An introduction to differential geometry', He gives the integrability conditions of the differential equations like this: $$ \partial_{i} F_{lj}=L^p_{ij} F_{lp},\,\,\,F_{ij}(x_0)=F^0_{ij}. $$ The integrability conditions for the existence of a global solution ##F_{lj}## is: $$ R^i_{jkl}\equiv\partial_k L^i_{jl}-\partial_l L^i_{jk}+L^h_{jl} L^i_{hk}-L^h_{jk} L^i_{hl}=0 $$ Then from the equation: $$\nabla_b e_a= \Gamma^c_{ab} e_c$$ Using cartesian basis ## e_I...

Similar threads

Replies
40
Views
6K
Replies
13
Views
2K
Replies
9
Views
2K
Replies
13
Views
2K
Replies
37
Views
4K
Replies
5
Views
2K
Back
Top