General Relativity, Einstein and All That

GREAT

Coordinates, Metrics and Notation

We are interested in the study of General Relativity (as the field is called), because it provides the only generalization of Newton's laws that are consistent with everything we can observe in the macroscopic world. (That is, in processes the action of which is incommensurately larger than ℏ.)

The fundamental "philosophical postulate" of General Relativity is that spacetime is not an inert arena in which matter does its thing: the presence of matter affects spacetime in which it exists. To the mathematical expression of this, however, we need a little… well, math.

To begin with, the local geometry of spacetime is determined by how we measure distances. Pythagoras' theorem applies to flat, Euclidean space, where an infinitesima line element may be expressend in terms of Cartesian coordinates as:

d\(TraditionalForm\`s\^2 = \ dx\^2 + dy\^2 + dz\^2\)+⋯

More generally, the coordinates may not be orthogonal to each other, so we expect a more general quadratic combination, which we write:

\(TraditionalForm\`ds\^2 = \(g\_μν\) dx\^μ dx\^ν\)

with the strict (implicit!) Einstein summation convention in force. This convention simply means that there is an implicit summation to be performed upon every index that occurs precisely once as a subscript (down) and once as a superscript.

In point of fact, since df(
            x) = ∂(f/∂)x dx, we can easily change to non-cartesian coordinates:

\(TraditionalForm\`ds\^2 = \(\(g\_μν\) dx\^μ dx\^ν = \(\((\(g\_μν\) \(∂x\^μ\/∂ξ\^ρ\) ∂x\^ν\/∂ξ\^σ)\) dξ\^ρ dξ\^σ = \(g\_ρσ\) dξ\^ρ dξ\^σ\)\)\)

This tells two very important lessons in tensor calculus:

1. The quantity (Subscript[g, μ])ν is a rank-2, (twice covariant, i.e., type-(0,2)) tensor: when changing from the coordinate system ((x^μ)) to the system ((ξ^ρ)), it transforms by the double linear, homogeneous transformation by the matrices of partial derivatives ∂((x^μ)/∂)(ξ^ρ). (Old-by-new!)

2. The appearance of d(s^2) remains the same in any coordinate system; it is an invariant with respect to general coordinate transfomations. (Incidentally, it is this "general" that bestows its name upon "General Relativity"!)

Notice that coordinates are superscripted, so that a derivative with respect to a coordinate becomes subscripted. In vact, we routinely write (∂_μ) to mean (∂/∂)(x^μ). Indeed, this sub/super-scripting is not a frivolity! Contrary to what most "mathematical methods for physicists" pracise, (Yes, including Arfken and Webber's.), there is a world of information in index-positioning.

To wit, compare \(TraditionalForm\`dx\^μ = \(∂x\^μ\/∂ξ\^ρ\) dξ\^ρ\) with \(TraditionalForm\`∂\/∂x\^μ = \(∂ξ\^ρ\/∂x\^μ\) ∂\/∂ξ\^ρ\) . One transforms inversely from the other!!! The former is called contra-variant, the latter covariant.

Riemannian Calculus

Given a metric, (Subscript[g, μ])ν, that describes the spacetime, there are certain measures of the curvature of spacetime:

\(TraditionalForm\`Γ\_μν\%ρ := \ \(1\/2\) \(\(g\^ρσ\)(∂\_μ g\_σν + ∂\_ν g\_μσ - ∂\_σ g\_μν)\)\)

is the Christoffel connexion. It occurs in the definition of the covariant derivative of a contra-variant vector, (V^μ):

\(TraditionalForm\`\(D\_μ\) 
        V\^ν = \((∂\_μ V\^ν + \(Γ\_μσ\%ν\) V\^σ)\)\)

the first part of which is an ordinary (partial) derivative, the second a coordinate-dependent linear, homogeneous transformation.

With the aid of this, we define the Riemann curvature tensor:

\(TraditionalForm\`\(R\_μνρ\%σ;\) = ∂\_μ Γ\_νρ\%σ + \(Γ\_μρ\%τ\) Γ\_ντ\%σ - ∂\_ν Γ\_μρ\%σ + \(Γ\_νρ\%τ\) Γ\_μτ\%σ\)

From this, we define the Ricci tensor:

\(TraditionalForm\`R\_μρ := \(δ\_σ\%ν\) 
        R\_μνρ\%σ\)

With it, we can now write the Einstein equation as:

\(TraditionalForm\`R\_μν - \(1\/2\) \(\(g\_μν\)(\(g\^ρσ\) R\_ρσ)\)\  = \ T\_μν\) .

Invariants of the curvature are

\(TraditionalForm\`\((\(R\_μνρ\%σ\) \(g\^μλ\) \(g\^νκ\) 
        R\_λκσ\%ρ)\), \ \((\(R\_μν\) \(g\^μρ\) \(g\^νσ\) 
          R\_ρσ)\)\ \ and\ \ \(\((\(g\^μν\) 
            R\_μν)\)\(.\)\)\)

If we define

\(TraditionalForm\`R\_μνρσ := 
      R\_μνρ\%κ\ g\_κσ\)

you can (try it!) prove that

\(TraditionalForm\`R\_μνρσ = \(\(-R\_νμρσ\) = \(-R\_μνσρ\)\)\)

This brings the number of independent components from (4^4) to 6·6=36.

(Subscript[R, μ])νρσ=+(Subscript[R, ρ])σμν

This leaves 6+15=21 independent elements. In fact, there's one more identity:

(ε^μ)νρσ 
        (Subscript[R, μ])νρσ=0

This finally makes it 20.