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 dx₂  +   2 g ₁₃ dx₁ dx₃ +   2 g ₁₄ dx₁ dx₄

+   2 g ₂₄ dx₂ dx₄  +   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 × 10⁻¹¹ m³s⁻²kg⁻¹

t = 0.01 s

r(t) =   3/  8 p [6.6726 × 10⁻¹¹ m³s⁻²kg⁻¹](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.