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4.2 Distributions in curved spacetime

The distributions introduced in Section 4.1.5 can also be defined in a four-dimensional spacetime with metric gα β. Here we produce the relevant generalizations of the results derived in that section.

4.2.1 Invariant Dirac distribution

We first introduce ′ δ4(x, x), an invariant Dirac functional in a four-dimensional curved spacetime. This is defined by the relations

∫ √ --- ∫ ∘ ---- f (x )δ4(x, x′) − g d4x = f(x′), f(x′)δ4(x,x′) − g′d4x′ = f (x), (258 ) V V′
where f(x) is a smooth test function, V any four-dimensional region that contains x′, and V ′ any four-dimensional region that contains x. These relations imply that ′ δ4(x,x ) is symmetric in its arguments, and it is easy to see that
δ (x − x′) δ (x − x′) δ4(x,x ′) = -4-√------ = -4√------- = (gg′)−1∕4δ4(x − x′), (259 ) − g − g′
where δ (x − x′) = δ(x0 − x′0)δ(x1 − x′1)δ(x2 − x′2)δ(x3 − x′3) 4 is the ordinary (coordinate) four-dimensional Dirac functional. The relations of Equation (259View Equation) are all equivalent because f (x )δ4(x,x′) = f(x′)δ4(x,x′) is a distributional identity; the last form is manifestly symmetric in x and x ′.

The invariant Dirac distribution satisfies the identities

Ω...(x,x′)δ4(x, x′) = [Ω...]δ4(x,x′), ( ′ ) (260 ) (gαα′(x,x ′)δ4(x,x′));α = − ∂α′δ4(x, x′), gαα(x′,x)δ4(x, x′);α′= − ∂αδ4(x,x′),
where ′ Ω...(x,x ) is any bitensor and α ′ g α′(x,x ), α′ ′ g α(x,x ) are parallel propagators. The first identity follows immediately from the definition of the δ-function. The second and third identities are established by showing that integration against a test function f(x) gives the same result from both sides. For example, the first of the Equations (258View Equation) implies
∫ f(x)∂ ′δ (x,x′)√ − g-d4x = ∂ ′f(x ′), V α 4 α

and on the other hand,

∫ ∮ α ′ √ --- 4 α ′ α ′ − f(x )(g α′δ4(x,x ));α − g d x = − f (x)g α′δ4(x,x )dΣ α + [f,αg α′] = ∂α′f(x ), V ∂V

which establishes the second identity of Equation (260View Equation). Notice that in these manipulations, the integrations involve scalar functions of the coordinates x; the fact that these functions are also vectors with respect to ′ x does not invalidate the procedure. The third identity of Equation (260View Equation) is proved in a similar way.

4.2.2 Light-cone distributions

For the remainder of Section 4.2 we assume that ′ x ∈ 𝒩 (x ), so that a unique geodesic β links these two points. We then let σ(x,x ′) be the curved spacetime world function, and we define light-cone step functions by

′ θ±(− σ) = θ±(x,Σ )θ(− σ), x ∈ Σ, (261 )
where θ+(x,Σ ) is one if x is in the future of the spacelike hypersurface Σ and zero otherwise, and θ− (x,Σ) = 1 − θ+ (x, Σ). These are immediate generalizations to curved spacetime of the objects defined in flat spacetime by Equation (246View Equation). We have that θ (− σ) + is one if x is an element of + ′ I (x ), the chronological future of ′ x, and zero otherwise, and θ− (− σ) is one if x is an element of I− (x′), the chronological past of x ′, and zero otherwise. We also have θ+ (− σ ) + θ− (− σ ) = θ(− σ ).

We define the curved-spacetime version of the light-cone Dirac functionals by

′ δ± (σ) = θ±(x,Σ )δ(σ), x ∈ Σ, (262 )
an immediate generalization of Equation (247View Equation). We have that δ+ (σ ), when viewed as a function of x, is supported on the future light cone of x′, while δ− (σ ) is supported on its past light cone. We also have δ+ (σ ) + δ− (σ) = δ(σ), and we recall that σ is negative if x and ′ x are timelike related, and positive if they are spacelike related.

For the same reasons as those mentioned in Section 4.1.5, it is sometimes convenient to shift the argument of the step and δ-functions from σ to σ + ε, where ε is a small positive quantity. With this shift, the light-cone distributions can be straightforwardly differentiated with respect to σ. For example, δ±(σ + ε) = − θ′±(− σ − ε), with a prime indicating differentiation with respect to σ.

We now prove that the identities of Equation (248View Equation, 249View Equation, 250View Equation) generalize to

lε→i0m+ εδ± (σ + ε) = 0, (263 ) lim εδ′ (σ + ε) = 0, (264 ) ε→0+ ± lim εδ′′(σ + ε) = 2π δ (x,x′) (265 ) ε→0+ ± 4
in a four-dimensional curved spacetime; the only differences lie with the definition of the world function and the fact that it is the invariant Dirac functional that appears in Equation (265View Equation). To establish these identities in curved spacetime we use the fact that they hold in flat spacetime – as was shown in Section 4.1.5 – and that they are scalar relations that must be valid in any coordinate system if they are found to hold in one. Let us then examine Equations (263View Equation, 264View Equation) in the Riemann normal coordinates of Section 3.1; these are denoted ˆxα and are based at x′. We have that σ (x,x′) = 1η ˆxαxˆβ 2 αβ and ′ ′ ′ ′ δ4(x,x ) = Δ (x,x )δ4(x − x ) = δ4(x − x ), where ′ Δ (x,x ) is the van Vleck determinant, whose coincidence limit is unity. In Riemann normal coordinates, therefore, Equations (263View Equation, 264View Equation, 265View Equation) take exactly the same form as Equations (248View Equation, 264View Equation, 250View Equation). Because the identities are true in flat spacetime, they must be true also in curved spacetime (in Riemann normal coordinates based at ′ x); and because these are scalar relations, they must be valid in any coordinate system.
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