Feynman Diagrams in General Relativity

© 2005, Uday Patil

Prerequisite: familiarity with the basic tensor formulation of general relativity.
To contact the author, email to udaypatil@qubitArt.com

Contents
Introduction
Diagrammatic Representation of Tensors in Reiman Space
The Basic Tensors of Curved Space
Schwarzschild Spacetime Geometry

Introduction

Feynman Diagrams are essentially a pictorial representation of formulae involving contractions of tensors and tensor-like entities. Feynman invented them for QED where the equations contain infinitely many contractions, each contraction happening in an infinite dimensional vector space. Such a contractions (in an infinite dimensional space) result in Path Integrals (the other thing Feynman became famous for). It would be interesting to use diagrams, just like Feynman did for QED, to represent the tensor equations of General Relativity. The obvious, qualitative difference being the finiteness of the number of contractions involved as well as that of the dimensionality of the underlying vector space.

This article is just such an attempt.

Diagrammatic Representation of Tensors in Reiman Space

A tensor is represented by a closed geometric shape (like a polygon or a circle) with one or more lines or leads coming out. Each lead represents an index of the tensor. The geometric shape itself has an interior made up of hollow and solid (filled) areas. A lead coming out of a hollow area is a covariant index while one coming out of a solid area is a contravariant index. The first diagram in Figure 1 shows a tensor with four leads (corresponding to two contravariant and two covariant indices).

A contraction between a covariant index and a contravariant index is denoted by joining the open ends of the corresponding leads. The second and third diagrams in Figure 1 show a tensor named S and a product of two Ss with two contractions respectively. Diagrammatically, a contraction is a line joining a hollow area to a solid area.

Figure 1 Tensors and Contractions

A derivative with respect to the coordinates is represented by a circle enclosing the entities being differentiated and a lead attached to the circle denoting the covariant index (this is not a tensor-index though). Multiple derivatives of the same quantity are represented by multiple leads coming out of the same circle. The first diagram in Figure 2 represents a tensor named T being differentiated twice. The remaining part of Figure 2 is a diagrammatic equation which brings together all the above concepts in capturing a simple rule of differentiation.

Figure 2 Coordinate Derivatives of Tensors

The Basic Tensors of Curved Space

Figure 3, Figure 4, Figure 5 and Figure 6 list the basic tensors used in describing curved space along with some of their properties. Clearly these diagrams make it easy for us to identify symmetries as well as to spot identical terms (and club them together or cancel them out).

Figure 3 The Metric Tensor

Note that the diagram for g (or its inverse) is symmetric in terms of its leads. The diagram for Γ is chosen to be symmetric with respect to its covariant leads as its expansion in terms of g exhibits that symmetry.

Figure 4 The Christoffel Symbol

Similarly, the expression for the Ricci Curvature Tensor is symmetric in its two leads, which leads us to choose a diagram reflecting this symmetry.

Figure 5 The Ricci Curvature Tensor

Each Γ has three terms in its expansion in terms of the metric. Naively speaking, this leads us to 24 terms (3+3+9+9) for a similar expansion of R. By using the differentiation rule (mentioned above) followed by clubbing together or canceling identical terms, the expression is simplified to 13 distinct terms. Similarly, the expansion of the Total Curvature (scalar) in terms of the metric has only 7 distinct diagrams.

Figure 6 The Total Curvature (scalar)

Schwarzschild Spacetime Geometry

Schwarzschild geometry is a free space solution stationary in time and radially symmetrical in space. The coordinates chosen (t, r, θ, φ) look like the familiar time and radial coordinates, but only θ and φ retain their usual interpretation, while t and r are coordinates along the time and radial axes respectively. Further, r is chosen to satisfy ds = rdθ for an arc with fixed r.

Under this choice of the coordinate system, the equation for the proper length element looks like

ds² = -h(r) dt² + f(r) dr² + r² (dθ² + sin²θ dφ²)

with the functions h and f yet to be determined through the equation Rij = 0. Figure 7 shows the structure of the metric tensor in the form of matrices (in the t, r, θ, φ coordinate system).

Figure 7 Structure of the Schwarzschild Metric in the (t, r, θ, φ) coordinate system

It is easy to identify the vanishing diagrams for different components of the Ricci tensor by looking at its diagrammatic representation (Figure 5) and using the following facts

g is diagonal
g is independent of t and φ
g_φφ is the only component of g that varies with θ
all diagonal terms of g vary with r

(The vanishing diagrams are those that are identically equal to zero regardless of the form of the functions h and f.)

Figure 8 The non-vanishing components of the Swarzschild (Ricci) Curvature

The non-vanishing diagrams of the Ricci tensor are given in Figure 8. Equating these components to zero gives us the following equations:

R_tt = 0 ⇒ (h’/h) + (f’/f) = 2(h’’/h’) + (4/r) . . . ( ⇒ hf = const h’²r⁴ )

R_θθ = 0 ⇒ (h’/h) - (f’/f) = 2(f-1)/r . . . ( R_φφ = 0 naturally yields the same equation)

R_rr = 0 ⇒ (f’/f) + (h’/h) = 0 . . . ( ⇒ fh = constant )

Combining the above three conditions, we get

f(r) = f_∞/(1 - r₀/r), h = h_∞(1 - r₀/r), . . . choose f_∞ = h_∞ = 1 (flat space at infinity)

The explicit form for the proper-length element (in the Schwarzschild coordinates) becomes:

ds² = -(1 - r₀/r) dt² + dr²/(1 - r₀/r) + r²(dθ² + sin²θ dφ²)

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