- Derivation of the Galilean Velocity Transformations >. - Derivation of the Lorentz Transformations >. - Proof of the Invariance of the Spacetime Interval >. - Argument to Show the Orientation of the x',t' Axes. - surface of the moon. - The other approach makes use of the idea of parallel worlds. - But the table and the Time Traveller’s house are sitting on the surface of the earth. - What are the coordinates of the same event in the train frame?. - All that remains is to determine the relation be- tween the x coordinates of the events in the two frames. - The coordinate axes of the S(track) frame are des- ignated X and Z. - We see from the diagram that the corresponding x coordinate of the explosion is simply x = x. - horizontal distance which the origin of the S' (train) frame has moved during the time t.. - the speed of the train with respect to the track (v), namely, u = 1 mph + 60 mph = 61 mph.. - ment there was a crest of the wave at point P. - The exclamation point at the end of the preceding paragraph is well deserved.. - This is one of the most remarkable and beautiful results in the history of theo-. - Then the v in the Galilean velocity transformation equations will be the speed of the earth rela-. - direction of the ether’s motion relative to the earth. - One beam moves at right angles to the direction of the earth’s motion through the ether. - The Michelson-Morley experiment is one of the truly seminal experiments in the history of physics. - We remind the reader of the situation under consideration. - Almost everything we do in the rest of the book follows from this equation. - That is, the interval has just the same form when ex- pressed in terms of the coordinates in the ship frame. - Observers in the two reference frames do not agree on the time of the second event. - As we discussed, observers in the two frames disagree on the relative speed of the earth and the light pulse. - Well, the future for me is already a thing of the past.. - all of the points we consider have y = z = 0. - worldline of the earth c t E. - The bottom and top portions of the light cone lie in the regions where t is, respectively, negative and positive. - Because of the light barrier, we must have u / c = x 1 / ct 1 ≤ 1. - That is, the slope of the ct 1. - The second station is right on the edge of the light cone. - worldline of the earth t = 0. - waves, warning of the attack. - 1 because of the light barrier restriction on the speed of iner- tial frames. - compared to clocks at rest, where v is the speed of the clock. - Reggie has remained at rest in the reference frame of the earth. - This is not true, however, of the reference frame of Jackie’s spaceship in the twin paradox. - The invariance of the spacetime interval can be expressed as s 2 = –(ct. - So Jackie is the younger of the twins when they reunite. - That was about the limit of the accuracy of the experiment. - Remember that in these equations, v is the speed of the ship and is subluminal, that is, u <. - Hence, the speed of the return tachyon must satisfy u r >. - sir arthur stanley eddington, The Nature of the Physical World. - west to east into the hitherto empty eastern half of the container. - This is called the microstate of the system.. - But this vi- olation of the second law does not occur spontaneously. - The shattering of the window is thus consistent with the second law. - It is important to emphasize the inclusion of the word “isolated” in the statement of the second law. - The total entropy of the earth increases as it absorbs solar radiation. - 15 of the American Institute of Physics, edited by William B. - The same is true in the case of the expanding universe. - “fl oat” relative to the walls of the ship. - surface of the earth. - surface of the earth 1g. - surface of the Earth. - Two ball bearings start out at each end of the car. - toward the center of the earth. - Therefore, a part of the balls’. - Second, only a part of the curva- ture of spacetime is contained in the left-hand side. - Each of the circles on the sheet represents a sphere in three-dimensional space. - It turns out that the other part of the eff ect comes from the contribution due to the warp- age of time in the sun’s gravitational fi eld. - M 3 , where M is the mass of the black hole. - At the horizon, the outgoing leg of the light cone is “parallel” to the horizon. - sa muel taylor coler idge, The Rime of the Ancient Mariner. - Each sphere is called a “mouth” of the wormhole. - radial distance from the center of the bubble inner. - The speed of a light beam in the interior of the bubble, inside the bubble wall, and outside the bubble. - “digging” of the Krasnikov tube—on the outbound journey. - parts of the light cones remain unmodifi ed inside the tube. - They are parallel to the forward-pointing parts of the light cones outside the tube. - (Incidentally, the crossing of the two dashed lines is simply the result x’. - Place one mouth, A, of the worm- hole on earth. - that occur prior to the formation of the fi rst closed null curve. - The billiard ball version of the grandfather paradox is illustrated in Figures 9.7. - The wormhole version of the grandfather paradox.. - Let us now consider the situation from the viewpoint of the billiard ball.. - M + m b , which was the original mass of the wormhole plus billiard ball. - They suggested that the complexity of the “jinnee” that emerges from the time machine may. - Novikov, “The Jinn of the Time Machine: Non-Trivial Self-Consistent Solu- tions,” Classical and Quantum Gravity . - A billiard ball enters mouth B of the wormhole at 5:00 p.m. - thus never reaches mouth B of the wormhole. - Remember, this doesn’t have any connection to the internal length of the wormhole. - All of this is just analogous to our discussion of the grandfather paradox. - It is the measurement that causes the sudden branching of the two worlds. - This is the copy of the bug that fi nds himself in the 0-world. - The 1-world thus contains two copies of the bug. - This is the copy of the bug that we have been following up until now. - Thus, we have the 1-world containing only half of the ball. - the minus sign indicates that the direction of the force is downward.. - the plus sign indicates that the sign of the gravitational force is upward. - So the direction of the gravitational force on the positive mass is upward. - 2 mv 2 , where v is the speed of the mass m.. - This diff erence in the position of the lines can be shown to be due to vacuum fl uctuations.. - to give a fl avor of the physical implications of such a bound. - A depiction of the physical interpretation of a quantum inequality.. - Let’s examine the right-hand side of the quantum inequality. - In the following chapter, we will discuss the implications of the quantum inequalities for wormholes and warp drives.. - It turns out that the thickness of the bubble wall is restricted by the relation. - Chris Van Den Broeck (then at the Katholieke Universitat Leuven in Belgium) proposed an ingenious idea to dramatically reduce the amount of negative en- ergy required for a warp bubble, to (only!) a few times the mass of the sun.. - Draw a spaceship on the bottom of the (two-dimensional) surface of the infl ated balloon