Tuesday, December 30, 2014

Solutions to General Relativity -Tensor Problems

1) Write out in long form the full sum (all terms) for the interval ds2   =  g mn dx m dx n

Solution:

We write the interval:   ds2   =  g mn dx m dx n
   = g 11 dx12    + g 22 dx22    + g 33 dx32    + g 44 dx42                           
+   2 g 12 dx1 dx+   2 g 13 dx1 dx3 +   2 g 14 dx1 dx4
+   2 g 24 dx2 dx+   2 g 34 dx3 dx4 +   2 g 41 dx4 dx1
+   2 g 42 dx4 dx2 +   2 g 43 dx4 dx3


2 (a)Using the appropriate relations, estimate  the density of the universe  at a time 0.01 second after the Big Bang

Solution:

a) The density is expressed:

r(t) = 3/  8 p G t 2

Where G =  6.6726 × 1011 m3s2kg1

t = 0.01 s

r(t) =   3/  8 p [6.6726 × 1011 m3s2kg1](0.01s) 2

r(t) =  1.78  x  10 13  kg/ m3


b) Repeat your computation if the Hubble constant is found to be H =   100 km/ sec/Mpc.

The result is unaffected because r(t) does not depend on H, the Hubble constant.

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