Box 12.6 an orthonormal basis in schwarzschild coordinate consider the metric component o,.o= 께=-1. if the observer is at rest in schwarzschild coordinates, then the spatial schwarzschild components of 0-4. must all be zero. therefore, obs (12.24) where g , is the schwarzschild metric here. since the other components are zero, this leads to the first of equations 12.10. because this vector has no spatial components, the condition -8 implies that o, has zero schwarzschild time component. the same argument ap- plies to the other spatial basis vectors. (note that if the observer were moving in schwarzschild spacetime, then o, would have nonzero spatial components, and the spatial basis vectors would possibly have a nonzero time component.) exercise 12.6.1. argue in the space provided on the next page that the rest of equations 12.10 are correct if ono,, and o, point in the φ,-9, and r directions. respectively, in schwarzschild spacetime.

Let v1, v2, w be three linearly independent vectors in r 3 . that is, they do not all lie on the same plane. for each of the following (infinite) set of vectors, carefully sketch it in r 3 , and determine whether or not it is a vector space (i.e., a subspace of r 3 ). explain your reasoning

Follow these directions and answer the questions. 1. set up the ripple tank as in previous investigations. 2. bend the rubber tube to form a "concave mirror" and place in the ripple tank. the water level must be below the top of the hose. 3. generate a few straight pulses with the dowel and observe the reflected waves. do the waves focus (come together) upon reflection? can you locate the place where the waves meet? 4. touch the water surface where the waves converged. what happens to the reflected wave? 5. move your finger twice that distance from the hose (2f = c of c, center of the curvature) and touch the water again. does the image (the reflected wave) appear in the same location (c of c)? you may have to experiment before you find the exact location. sometimes it is hard to visualize with the ripple tank because the waves move so quickly. likewise, it is impossible to "see" light waves because they have such small wavelengths and move at the speed of light. however, both are examples of transverse waves and behave in the same way when a parallel wave fronts hit a curved surface.

Choose all the answers that apply. our solar system: is in a spiral galaxy no longer includes pluto is made mostly of empty space is in the andromeda galaxy is the only known system that supports life