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Additional info for A critique of the determination of the energy-momentum of a system from the equations of motion of matter in the general theory of relativity

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Multiple valued Lipschitz functions A nearly flat piece of an integral varifold may be essentially multi-layered. To approximate varifolds with this behavior, we shall consider Lipschitz functions f : Rk → Mν where ν ∈ N and Mν is the quotient space of (Rn−k )ν under the equivalence relation (w1 , . . , wν ) ≡ (z1 , . . , zν ) if and only if (w1 , . . , wν ) is a permutation of (z1 , . . , zν ). For y ∈ Rk we let ( f (y)1 , . . , f (y)ν ) be any representative of f (y), and if f is differentiable at y, then we must understand D f (y) j in the sense of [D f (y)] j rather than D[ f (y) j ].

D) If t0 > 0 and ||Vt ||(φ) is discontinuous at t0 , then ||Vt ||(φ) has a jump decrease at t0 . (e) For any t > 0, Ω limt→t − ||Vt || exists and 0 Ωlim ||Vt || ≥ Ωlim ||Vt ||. t→t0+ t→t0− (f) ||Vt || is a continuous function of t at almost all t ≥ 0. (g) If ||Vt || is continuous at t0 and s1 , s2 , . . and m1 , m2 , . . are sequences with si ∈ Q mi and limi→∞ si = t0 , then ||Vt0 || = Ω lim ||Vmi ,si ||. i→∞ Remark: The full, definition of Vt for all t will have to wait until we can show rectifiability.

Since V0 ψ is rectifiable, at ||V0 || ψ almost all x there is a unique tangent plane, and so 2 (x) S⊥ (Dφ(x)) dV0 S − g(x) ψ(x)−1 d||V0 ||x = |S⊥ (Dφ(x)) − g(x)|2 ψ(x)−1 dV0 (x, S). 8) < η2 /16B. Since spt g ⊂ {x ∈ Rn : ψ(x) > 0} we have g(x) · h(V0 , x) d||V0 ||x = lim δVi (g) i→∞ = lim i→∞ g(x) · h(Vi , x) d||Vi ||x. 10) (g(x) − S⊥ (Dφ(x))) · h(Vi , x) dV (x, S) + (η /16B)1/2 B1/2 ≤ lim sup i→∞ S⊥ (Dφ(x)) · h(Vi , x) dVi (x, S) + lim sup i→∞ |h(Vi , x)|2 ψ(x) dVi (x, S) · ≤ lim sup i→∞ |S⊥ (Dφ(x)) − g(x)|2 ψ(x)−1 dVi (x, S) 1/2 1/2 + η/4 S⊥ (Dφ(x)) · h(Vi , x) dVi (x, S) + η/4 + η/4.

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A critique of the determination of the energy-momentum of a system from the equations of motion of matter in the general theory of relativity


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