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 (seeSection 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 thequantities Brst defined byBrst = ∂Ast∂xr + ∂Atr∂x s + ∂Ars∂xt (a) are the components of a tensor; and (b) are skew-symmetric ... skew-symmetric in all pairs in indices.(c) How many independent components does Brst have?7. Cross Product(a) If X and Y are contravariant vectors, then their cross-product is defined as the...

... thecoordinates of this base smoothly, the smoothness follows. ❄Example In E3, the Levi-Civita tensor coincides with the totally antisymmetric third-ordertensor œijk in Exercise Set 5. In the Exercises, ... refer to such a geodesic as timelike. Looking at the discussion beforeDefinition 7.1, we see that this corresponds, in Minkowski space, to a particle traveling atsub-light speed. It follows that ... there, because in the eyes of the observer, spacetime should be flat.)Question Does parallel transport preserve the relationship of these vectors to the curve.That is, does the vector V (4) remain...

... D1 4 )2 + (-cD12 + D2 4 )2 + (-cD13 + D3 4 )2 - c2(-cD1 4 + D 4 4)2 = 0 …(**)Noting that this only effects cross-terms, subtracting and dividing by 4c givesD11D1 4 ... D13 - c2D1 4 D1 4 ]+ [D 4 1D 4 1 + D 4 2 D 4 2 + D 4 3 D 4 3 - c2D 4 4 D 4 4] = 0,showing thatc2“column 1, column 1‘ = - column 4, column 4 .So, if we write 47 Speed2 ... somethingwith units of distance/time; that is, by a non-zero speed. Since relativity holds that thespeed of light c is a universal constant, it seems logical to use c as this conversion factor.Now,...

... D23 - c2D1 4 D2 4 = 0,showing that columns 1 and 2 are also orthogonal.59Neither of these gizmos are tensors, but instead transform as follows (Which you willprove in the exercises!)Transformation ... 566. If the x–i-system is moving with a velocity v in a certain direction with resepct to the xi - system, we call this a boost in the given direction. Show that successive boosts in twoperpendicular ... is the speed of light, and ∫ is a certain constant. (The meaning of ∫ willemerge in due course). Its norm-squared is (1 - ∫2), and we want this to be 1, so wereplace the vector by“1 1- 2...

... 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 symbolsvanish—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 is responsible ... this. This involves the derivatives of theChristoffel symbols, and we can't make it vanish.Question If I throw a ball in the air, then the path is curved and also a geodesic. Does thismean...

... is a symmetric tensor.Definition 11 .4 Classically, a fluid has no viscosity if its stress tensor is diagonal in anMCFR (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 thescenario for our discussion:We now work in a 4- manifold M whose metric has signature (1, 1, 1, -1 ).We have ... å2FFFFis the resultant force on the scaled version of the prism, whereas its mass is proportional to å3. Thus its acceleration is proportional to 1/å (using Newton&apos ;s law). This means that, aså...