Intro to Differential Geometry and General Relativity - S Warner Episode 4 ppsx

... this strange object? Answer Just as a covariant vector field converts contravariant fields into scalars (see Section 3) we shall see that a type (1,1) tensor converts contravariant fields to ... 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-sym...

Ngày tải lên: 12/08/2014, 16:20

... 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

... D 1 4 ) 2 + (-cD 1 2 + D 2 4 ) 2 + (-cD 1 3 + D 3 4 ) 2 - c 2 (-cD 1 4 + D 4 4 ) 2 = 0 …(**) Noting that this only effects cross-terms, subtracting and dividing by 4c gives D 1 1 D 1 4 ... D 1 3 - c 2 D 1 4 D 1 4 ] + [D 4 1 D 4 1 + D 4 2 D 4 2 + D 4 3 D 4 3 - c 2 D 4 4 D 4 4 ] = 0, showing that c 2 “column 1, column 1‘ = - column...

Ngày tải lên: 12/08/2014, 16:20

... D 2 3 - c 2 D 1 4 D 2 4 = 0, showing that columns 1 and 2 are also orthogonal. 59 Neither of these gizmos are tensors, but instead transform as follows (Which you will prove in the exercises!) Transformation ... 56 6. If the x– i -system is moving with a velocity v in a certain direction with resepct to the x i - system, we call this a boost in the given direction. Show that...

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 ... would like to generalize the stress tensor to 4- dimensional space. First we set the scenario for our discussion: We now work in a 4- manifold M whose metric has signature (1, 1, 1,...

Ngày tải lên: 12/08/2014, 16:20

... (exercise set) T 13 |3 = 0. Finally, a = 1, b = 4: T 14 |4 = ∂T 14 ∂x 4 + ¶ 4 1 4 T 44 + ¶ 4 4 1 T 11 = 1 2 g 11 (-g 44 ,1 ) T 44 + 1 2 g 44 (g 44 ,1 )T 11 = 1 2 e -2 ¡ (2∞'(r)e 2∞ )®e -2 ∞ ...           2 r ∞'e -4 ¡ - 1 r 2 e 2¡ (1-e -2 ¡ )0 00 0e -2 ¡ [∞''+(∞') 2 + ∞' r - '¡&apos ;- ¡&ap...

Ngày tải lên: 12/08/2014, 16:20

... thickness is dr. Thus, dM(R) dr = 4 2 ®(r). Here, ® is energy density, and by our choice of units, energy is equal to rest mass, so we interpret m(r) as the total mass of the star enclosed by a sphere ... g ik (g rk ,s + g ks,r - g sr,k ) P r P i m 0 dP s d† = 1 2 (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-s...

Ngày tải lên: 12/08/2014, 16:20

... 128 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 protons ... 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 temp...

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...

Ngày tải lên: 17/03/2014, 14:28