Study Questions for Philosophy 2234 Test 1. I. Special Relativity 1. Explain the role of the axioms in Euclid=s geometry. Why did so many mathematicians attempt to prove the fifth axiom from the other four? 2. Describe Gauss= experiment re. the correct geometry of physical space. What problem did we raise regarding the conclusions to be drawn from such an experiment? 3. Why did Poincarré hold that our choice of geometry for the physical world was a matter of convention? 4. What is a universal force, and how does the idea of a universal force affect our understanding of the relation between observations/measurements and geometry? 5. What does the procedure Einstein mentions for determining if 3 points all lie on the same line assume? 6. What does Einstein’s procedure for measuring distance assume? 7. How is a system of coordinates related to a body of reference? 8. Why does Einstein insist that our two clocks be ‘of identical construction’? 9. How can we tell that our usual body of reference (the surface of the earth) does not provide us with a Galilean system of co-ordinates? 10. What is the ‘principle of relativity’ (in the restricted sense)? 11. Why does Einstein insist on a real procedure for measuring simultaneity, instead of relying simply on the (apparently obvious) intuitive understanding of what we mean by ‘simultaneity’? 12. What is an invariant? Give an example of an invariant in special relativity. 13. Explain how Einstein’s assumption that the velocity of light is c for all observers (in Galilean frames) leads to the result that simultaneity is relative to which frame you’re in. 14. Explain how Einstein’s assumption that the velocity of light is c for all observers leads to the result that moving clocks run slow, as observed from the “rest” frame. 15. Explain the role of the relativity of simultaneity in resolving the “twins paradox”: How is it that, though both twins see the other’s watch running slow relative to their own, the twin who leaves and later returns finds his watch behind the watch of the “stay-at-home” twin? 16. What did Michelson and Morley find when they tried to measure the earth’s motion relative to the “ether”? 17. Explain how relativity of simultaneity makes it coherent for observers in two different inertial frames to both claim that the other’s clock is running slow. 18. Why did we use three clocks (instead of two) in our special-relativity based account of the “twins paradox”? Explain how the effect of the round-trip is captured this way (and why we use this round- about approach). 19. Explain why the existence of a limiting velocity for signals affects attempts to establish observational criteria or the simultaneity of distant events. 20. Describe how simultaneity comes into one method for measuring the length of moving bodies (in the direction of their motion). 21. Explain how time dilation (the slowness of moving clocks) affects another method of measuring the length of moving bodies. II. General Relativity: 1. How does the principle of equivalence guarantee the equality of gravitational and inertial mass? 2. What is the principle of equivalence? 3. What problems afflict attempts to set up time and space measurements in a rotating reference frame? 4. Explain how Einstein argues, based on the special-relativistic treatment of a rotating reference frame, that gravitational fields will act to slow clocks? 5. Explain how Einstein argues, based on the special-relativistic treatment of a rotating reference frame, that gravitational fields will act on measuring rods, causing them to shrink in the direction perpendicular to the field’s direction. 6. What is Einstein’s initial, rough, formulation of the principle of general relativity? 7. What is a universal force? 8. What tells us that our efforts to impose a Euclidean coordinate system on the marble slab have gone awry? 9. What is a “reference mollusc,” and what does it have to do with Einstein’s appeal to Gaussian coordinates in GTR? 10. The Gaussian coordinate approach assumes something fundamental about the space-time we are dealing with. What is it? (Hint: see pp. 98-99.) 11. What is the form of the distance relation in a two-dimensional Gaussian coordinate system? 12. What is the exact formulation of the general principle of relativity? 13. What restriction does Einstein impose on the operation of the “clocks” that produce a reference mollusc’s time coordinates? 14. What are the four constraints that act together to produce Einstein’s general law of gravity? 15. Describe how we can use the principle of equivalence to solve the “balloon in a plane on takeoff” puzzle. 16. What features of events are ‘objective’, according to Einstein (and preserved in all Gaussian coordinate systems)? 17. What are the three observational results that Einstein mentions in support of GTR? 18. Describe how the decaying orbit of a pair of closely-orbiting pulsars produced a test of GTR. 19. How does a ‘material point’ get ‘characterized’ within a Gaussian coordinate system? 20. What limit constrains a Newtonian cosmology based on infinite Euclidean space? 21. Why doesn’t Einstein like Seeliger’s proposed resolution for this problem? 22. Describe a surface that is finite but unbounded (has no limits). 23. How does the possibility of a 3-dimensional space like this allow for a more satisfactory cosmology than that of a ‘quasi-Euclidean’ 3-dimensional space? III. Essay questions: Take care to give details, drawing on what you’ve learned, and develop your argument to address all of the question! 1. What was the “ether” supposed to be? Explain how Einstein’s proposal, to simply dispense with the ether once and for all, radically altered Newtonian ideas about the velocity of light, and hence about space, time, and the rest of classical mechanics. 2. Discuss objectivity, relativity, and relativism. Does the lack of agreement between observers on the length of objects, or the time between two events, show that there really is no one objective physical world? Explain why, or why not. If there is an objective world, what features really describe it? 3. Discuss the progress of science. Why is it difficult (at least) to describe it as a matter of giving better and better theories of a shared body of data (observations)? Do you think science does progress? Explain. Does the progress of science (if it occurs) suggest (or prove?) that scientific theories somehow get at the truth about the world? Explain, again! 4. Without a standard of simultaneity for distant events, the one-way velocity of light can never be directly measured (though in special relativity, the two-way average velocity can be measured, and always comes out to be c). Einstein regards the hypothesis that the one-way velocity is c as a kind of definition. Explain how this definition is linked to his proposed definition of simultaneity. Is there a real matter of fact here at all? Discuss (think here about how observers in other Galilean frames will describe the velocity of light with respect to observers in our frame).
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