Intro to Differential Geometry and General Relativity - S Warner Episode 8 ppsx

... the coordinates of this base smoothly, the smoothness follows. ❄ Example In E 3 , the Levi-Civita tensor coincides with the totally antisymmetric third-order tensor œ ijk in Exercise Set 5. In the Exercises, ... refer to such a geodesic as timelike. Looking at the discussion before Definition 7.1, we see that this corresponds, in Minkowski space, to a particle traveling at sub-light spee...

Ngày tải lên: 12/08/2014, 16:20

... we chose? Answer Yes. Question But how can we interpret this strange object? Answer Just as a covariant vector field converts contravariant fields into scalars (see Section 3) we shall see that ... is a skew-symmetric tensor of type (0, 2), show that the quantities B rst defined by B rst = ∂A st ∂x r + ∂A tr ∂x s + ∂A rs ∂x t (a) are the components of a tensor; and (b) are skew-sy...

Ngày tải lên: 12/08/2014, 16:20

... light c is a universal constant, it seems logical to use c as this conversion factor. Now, if we happen to be living in a Riemannian 4-manifold whose metric diagonalizes to something with signature ... other axes. Now, to convert units of time to units of distance, we need to multiply by something with units of distance/time; that is, by a non-zero speed. Since relativity holds...

Ngày tải lên: 12/08/2014, 16:20

... are also orthogonal. 59 Neither of these gizmos are tensors, but instead transform as follows (Which you will prove in the exercises!) Transformation Law for Christoffel Symbols of the First Kind [hk, ... correctly. This allows us to say whether a field is parallel and of constant length by seeing whether this quantity vanishes. This claim is motivated by the following. Proposition 8. 3...

Ngày tải lên: 12/08/2014, 16:20

... Frames” In “flat space” E s all the Christoffel symbols vanish, so the following question arises: Question Can we find a chart (local coordinate system) such that the Christoffel symbols vanish—at ... coordinate system such that all geodesics are in fact straight lines? Answer Not in general; if you make some geodesics straight, then others wind up curved. It is the curvature tensor that i...

Ngày tải lên: 12/08/2014, 16:20

... is a symmetric tensor. Definition 11.4 Classically, a fluid has no viscosity if its stress tensor is diagonal in an MCFR (viscosity is a force parallel to the interfaces). Thus, for a viscosity-free ... å 2 FF FF is the resultant force on the scaled version of the prism, whereas its mass is proportional to å 3 . Thus its acceleration is proportional to 1/å (using Newton&apos ;s law). T...

Ngày tải lên: 12/08/2014, 16:20

... small indeed. Also, we could take that term over to the right-hand side of the equation and incorporate it into the stress-energy tensor, thus regarding - g ab /8 as the stress-energy tensor of empty space. Following ... ahead of ourselves Finally, we generalize the (second order differential) operator Ô to some yet -to- be- determined second order differential operator ∆. This...

Ngày tải lên: 12/08/2014, 16:20

... (g rk ,s + g ks,r - g sr,k ) P r P k But the sum of the second and third terms in parentheses is skew-symmetric in r and k, whereas the term outside is symmetric in them. This results in them canceling ... d˙ 2 . We see immediately that something strange happens when 2M = r, and we look at two cases. Case 1 (Not-So-Dense Stars) Radius of the star, r s > 2M. If we recall that...

Ngày tải lên: 12/08/2014, 16:20

... 1 28 and protons combine to form neutrons (and neutrinos which are nearly massless and noninteracting). A sufficiently dense star is unstable against such an interaction and all electrons and ... pressure becomes less and less effective against gravitational collapse. In most stars, this becomes a serious problem when the core has reached the carbon rich phase but the temperatur...

Ngày tải lên: 12/08/2014, 16:20

... and Proper Time 94 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 100 13. Two Basic Premises of General Relativity 109 14. The Einstein Field Equations and Derivation of Newton&apos ;s ... = r sin x 1 sin x 2 cos x 3 … y n-1 = r sin x 1 sin x 2 sin x 3 sin x 4 … cos x n-1 y n = r sin x 1 sin x 2 sin x 3 sin x 4 … sin x n-1 cos x n y n+1 = r sin x 1 s...

Ngày tải lên: 17/03/2014, 14:28