In Minkowski spacetime, the set of all events separated from the origin by a time like interval a² is a 3-surface, the hyperboloid t² − x² − y² − z² = a², where {t , x, y, z} are Lorentz coordinates of some inertial reference frame. On this hyperboloid, introduce coordinates {χ, θ , φ} such that

Note that χ is a radial coordinate and (θ , ∅) are spherical polar coordinates. Denote by V₃ the portion of the hyperboloid with radius χ ≤ b.(a) Verify that for all values of (χ , θ , ∅), the points defined by Eqs. (2.63) do lie on the hyperboloid.

(b) On a spacetime diagram, draw a picture of V₃, the {χ, θ , ∅} coordinates, and the elementary volume element (vector field) d ∑(vector)[Eq. (2.55)].

(c) Set A(vector) ≡ e(vector)₀ (the temporal basis vector), and express ∫V₃ A(vector) · d∑(vector) as an integral over {χ, θ , ∅}. Evaluate the integral.

(d) Consider a closed 3-surface consisting of the segment V₃ of the hyperboloid as its top, the hypercylinder {x² + y² + z² = a² sinh² b, 0 2 + y² + z² ≤ a² sinh² b, t = 0} as its bottom. Draw a picture of this closed 3-surface on a spacetime diagram. Use Gauss’s theorem, applied to this 3-surface, to show that ∫V₃ A(vector) · d ∑(vector) is equal to the 3-volume of its spherical base.

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t = a cosh x, x = a sinh x sin y = a sinh x sin 0 sin , cos , z = a sinh x cos 0 . (2.63)