Can anyone explain to me the mathematics of General relativity?

Question

I need a mathematical insight on how the Rikki tensor and all other tensor equations are derived and used in General Relativity.

Answer 1

I am going to assume that you are familiar with a range of tensor notations and non-euclidean geometry. So I'll be brief: -

The Riemann tensor R can have a range of algebraic symmetries. For example, the Riemann processes differential symmetries called 'Bianchi identities': -

R^(α)(λ;μ;ν) = 0

Where the ^ indicates super-scripted and the other indices are sub-scripted!

Also, from the Riemann one can form other curvature tensors by contraction. The easiest to form is the 'Ricci curvature tensor': -

R(μν) Ξ R^(α)(μαν) Ξ Γ^(α)(μν,α) -  ÃŽÂ“^(α)(μα,ν) + Γ^(α)(βα)Γ^(β)(μν) - Γ^(α)(βν)Γ^(β)(μα)

and the scalar tensor

R Ξ R^(μ)(μ)

But the much greater geometric significance is the 'Einstein curvature tensor': -

G^(μ)(ν) Ξ ½ε^(μαβγ)R(βγ)^(ρσ)½ε(ναρσ) = R^(μ)(ν) - ½δ^(μ)(ν)R.

Of all second-rank curvature tensors one can form by contracting the Riemann, only the Einstein = G tensor retains part of the 'Bianchi identities'; such that it satisfies: -

G^(μν)(ν)

I hope this is of some use (if I have made any indices mistakes I apologise in advance!)

Answer 2

http://www.amazon.com/Gravitation-Physic...

Answer 3

I think yahoo answers isn't the best suitable place to teach an entire very complicated university course on. There are many free online resources though. See the links: