levi

PS2 Octob er 18, 2011 1. Op erations on tensors a. LetS andT b e(p;q) tensors. Sho w that S+T is also a (p;q) tensor. b. LetS b e a(p;q) tensor andT b e an (r;s) tensor. Sho w that S T is a(p+r;q+s) tensor. c. LetT b e a (p;q) tensor. Sho w that T1:::p11:::q1 is a(p1;q1) tensor. c. LetT b e a (p;0) tensor. Sho w that T(1:::p) is a symmetric tensor and T[1:::p] is an an tisymmetric tensor, i.e. sho w that the symmetry/an tisymmetry is pre- serv ed b y co ordinate transformations. d. LetT b e a (p;q) tensor. Sho w that @T1:::p::: q is a(p;q+ 1) tensor when re- stricted to inertial frames, i.e. when the co ordinate transformations are P oincare transformations. 2. Levi-Civita tensor The Levi-Civita sym b ol ~1:::n is totally an tisymmetric and dened suc h that ~01:::(n1)= 1 in all co ordinate systems. Although the Levi-Civita sym b ol is in v arian t under Loren tz transformations, it do esn't trasform as a tensor under general co ordinate transformations. On the other hand, the follo wing ob ject is a tensor in all co ordinates systems: 1:::n=pg~1:::n 1 whereg is the determinan t of the metric g . V erify that 0 1:::0n=@x1 @~x0 1:::@xn @~x0n1:::n where1:::n is the Levi-Civita tensor in the x cordinate system and 0 1:::0nis the Levi-Civita tensor in the x0cordinate system. Sho w that the v olume elemen t 1:::ndx1 ::: dxnpgdx0^dx2:::^dxn1 is in v arian t under co ordinate transformations. 3. Maxw ell's equations in inertial co ordinates Sho w that in an inertial co ordinate system, Maxw ell's equations can b e written as follo ws: @F=J; @F +@F +@ F= 0 (in appropriate units) where J is the curren t 4-v ector. The Maxw ell stress- energy tensor is dened as T=FF 1 4FF whereF =F,F=F. Assuming that J v anishes, sho w that @T= 0: 4. Energy-momen tum tensor of a p erfect uid The energy momen tum tensor of a p erfect uid in Mink o wski space is giv en b y T= (+p)UUp where is the rest-frame energy densit y , Uis the 4-v elo cit y of the uid, and p is the rest-frame pressure. The motion of the uid is describ ed b y the conserv ation equation @T= 0 By pro jecting this equation on to U (U@T= 0) , sho w that @(U) +p@U= 0: (1) By pro jecting on to the the 3d space p erp endicular to U using the tensor P=UUg; sho w that (+p)dU d=P@p (2) where is the prop er time of the uid. Appro ximating the v elo cit y of the uid b yU(1;~ w) sho w that in the non-relativistic limit, eqs. 1 and 2 reduce to @t+~r
(~ w) = 0; @t~ w+ ~ w
~r ~ w=~rp: 2