Refer to the following links, if necessary:

- Hyperspace and Warped Space Help
- Hyper Objects Help
- Beyond Four Dimensions Help
- Parallel Principle Help
- Curved Space Help

**Hyperspace and Warped Space Practice Test**

**Directions**: A good score is eight correct.

1. Suppose there are two points *P* and *Q* in deep space, separated by exactly 1 second-equivalent (or 1 light-second). At point *P* there is a space traveler with a powerful pulsed laser. At point *Q* there is a mirror oriented so the space traveler can shine the laser at it and see the reflected pulse exactly 2 seconds later. Suppose an extremely dense, dark neutron star passes almost (but not quite) between points *P* and *Q* , producing a pronounced negative curvature of space in the region. If the space traveler shines the laser toward point *Q* while the neutron star is near the line of sight, the return pulse will

(a) be seen after exactly 2 seconds

(b) be seen after a little more than 2 seconds

(c) be seen after a little less than 2 seconds

(d) never be seen

2. If there are no pairs of parallel geodesics on a surface, that surface is

(a) Lobachevskian

(b) Euclidean

(c) four-dimensional

(d) Riemannian

3. Einstein’s “four-sphere” cosmic model describes a universe with

(a) finite volume and finite radius

(b) finite volume and infinite radius

(c) infinite volume and finite radius

(d) infinite volume and infinite radius

4. Considered with respect to the speed of light, one second-equivalent represents a distance of about

(a) 300,000 meters

(b) 300,000,000 meters

(c) 300,000,000,000 meters

(d) 300,000,000,000,000 meters

5. If the universe is portrayed as Cartesian four-space with three spatial dimensions and one time dimension, then the 4D path of a stationary point that “lasts forever” is

(a) a straight line

(b) a circle

(c) a helix

(d) a spiral

6. How many line-segment edges does a rectangular four-prism have?

(a) 16

(b) 24

(c) 32

(d) 48

7. The distance between (0,0,0,0,0,0) and (1,1,1,1,1,1) in Cartesian six-space is

(a) equal to 1

(b) equal to the square root of 2

(c) equal to the sixth root of 2

(d) equal to the square root of 6

8. Suppose a light bulb is switched on, and the photons travel outward in ever-expanding spherical paths, as depicted in the dimensionally reduced drawing of Fig. 11-6. In this graph, the flare angle of the cone (the angle between the positive *t* axis and any ray extending from the origin outward along the cone’s surface) is

(a) 30°

(b) 45°

(c) 90°

(d) impossible to determine without more information

9. The 4D hypervolume of a rectangular four-prism consisting of a cube measuring 200,000 kilometers on each edge, and whose life is represented by a single point in time, is equal to

(a) zero

(b) 2/3 of a second-equivalent

(c) 4/9 of a second-equivalent

(d) 8/27 of a second-equivalent

10. Suppose *P* and *Q* are points on a surface that is positively curved. The length of the line segment *PQ* in 3D space

(a) is the same as the length of the geodesic *PQ*

(b) is greater than the length of the geodesic *PQ*

(c) is less than the length of the geodesic *PQ*

(d) might be equal to, less than, or greater than the length of the geodesic *PQ* , depending on the locations of *P* and *Q*

**Answers**

1. b

2. d

3. a

4. b

5. a

6. c

7. d

8. b

9. a

10. c

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