SOLVED Physics 396 Homework Set 6

1. Consider the 4D spacetime, spanned by coordinates (v, r, ✓, ), with a line element ds2 = ✓ 1 Rs r ◆ dv2 + 2dvdr + r2 (d✓2 + sin2 ✓d2 ), (1) where Rs ⌘ 2GM/c2 is called the event horizon (a.k.a. Schwarzschild radius). It is noted that the above line element describes the 4D spacetime outside a static, spherically symmetric object of mass M in Eddington-Finkelstein coordinates. Consider a radial null curve (✓ = const., = const., ds2 = 0). a) Calculate the slopes of the light cone at a point (v, r). b) Evaluate the slopes of the light cones at points (v, r) = (0, Rs/2), (0, Rs), (0, 2Rs), (0, 3Rs), and (0, 4Rs). Draw the corresponding (v, r) spacetime diagram with the apex of the light cones positioned at these locations. This diagram should fill a standard piece of paper. Use a ruler to make sure that your drawn curves are consistent with your calculated values. c) How do the slopes of the light cones change with v for a given value of r? 2. In lecture, we studied the 4D wormhole spacetime with a line element of the form ds2 = c2 dt2 + dr2 + (b2 + r2 )(d✓2 + sin2 ✓d2 ). (2) We found the embedding diagram for a 2D equatorial slice of the 4D wormhole and that this surface has two asymptotically flat regions connected by a throat of circumference 2⇡b. We also found that this spacetime is spherically symmetric since a surface with constant r and t has the geometry of a sphere. Consider a t = const. slice of the wormhole geometry bounded by two spheres of coordinate radius R centered on r = 0, which reside on each side of the throat. a) Calculate the circumference of the equator of these spheres. b) Calculate the distance S from one sphere to the other, through the throat, along a ✓ = const., = const. line. 1 c) Calculate the area of the two-sphere of coordinate radius r = R. d) Calculate the 3D volume bounded by the two spheres of coordinate radius R. 3. In lecture, we arrived at the 2D non-Euclidean line element of a two-sphere of radius R from 3D Euclidean space by performing the coordinate transformation x = R sin ✓ cos y = R sin ✓ sin z = R cos , (3) where 0  ✓ < ⇡ and 0  < 2⇡. It is noted that the above transformation obeys the equation of constraint x2 + y2 + z2 = R2 , (4) which e↵ectively constrains the radial coordinate to take on a constant value, r = R. a) Here we wish to arrive at a 3D non-Euclidean line element of a three-sphere of radius R from a fictitious 4D Euclidean space. Consider the coordinate transformation x = R sin sin ✓ cos y = R sin sin ✓ sin z = R sin cos ✓ w = R cos , (5) where 0  < ⇡, 0  ✓ < ⇡, and 0  < 2⇡. Show that the above transformation obeys the equation of constraint x2 + y2 + z2 + w2 = R2 , (6) which e↵ectively constrains the radial coordinate to take on a constant value, r = R. b) Calculate dx, dy, dz, dw. c) Calculate dx2 + dy2. d) Calculate dx2 + dy2 + dz2. 2 e) Show that the 4D Euclidean line element dS2 = dx2 + dy2 + dz2 + dw2 (7) becomes that of a 3D non-Euclidean line element of a three-sphere of radius R of the form dS2 = R2 ⇥ d2 + sin2 (d✓2 + sin2 ✓d2 ) ⇤ (8) under the above coordinate transformation. f) By defining r ⌘ sin , show that the above line element takes the form dS2 = R2  dr2 1 r2 + r2 (d✓2 + sin2 ✓d2 ) . (9) Notice that the line element of Eqs. (8) and (9) equate to a t = const. slice of a homogeneous closed universe, which was analyzed in Example 7.6. g) What is the range of this r coordinate? 4. In lecture, we found an embedding diagram for the 4D wormhole. This equated to finding a curved 2D surface in 3D Euclidean space with the same intrinsic geometry as a 2D equatorial slice (t = const., ✓ = ⇡/2) of the 4D wormhole line element. a) Consider the t = const. slice of a homogeneous closed universe given by Eq. (9). Construct the corresponding 2D equatorial slice (✓ = ⇡/2) for this homogenous closed universe, analogous to Eq. (7.41) of your text. b) Following a procedure similar to that of Section 7.7 of your text, show that the curved 2D surface obeys the di↵erential equation dz dr = ±R r p1 r2 . (10) c) By integrating Eq. (10), show that the solution obeys the expression z2 R2 + r2 = 1. (11) d) Plot Eq. (11) on a z(r)/R vs. r set of axes. 3