//07 Physics for Majors Today s Class The Arrow of Time Entropy/Memory/Expanding Universe Wormholes and Time Travel Unification Class Ideas from Hawking Section Why is space symmetric and time not? Last Class Solving hard problems with differential equations and Mathematica. The Lagrangian Orbital Equation and Potential Well Coordinate System Let s take a coordinate system with going upward.
//07 Symmetry of space We can move an object freely in the and directions. There is no real difference between and. We can move an object in the and directions, but there is an asymmetry here caused by gravity. An object at rest in space will only move in the direction. X-Y Trajectory In space, an object moves in the direction of its momentum vector. Symmetry of time If we think in terms of space-time and not just space, then what happens? Every object moves forward in time. The rate at which an object moves forward in time depends on the observer. Is there something like a force that moves objects forward in time? X-Y Trajectory Note how the y- component of momentum increases while the x- component remains the same size. Section World Lines T-X Trajectory Without invoking relativity, we can treat time the same way we would a spatial dimension. The time component of momentum is just total energy ( ).
//07 T-X Trajectory In spacetime, an object moves in the direction of its energymomentum four-vector. The World Line Recall that the world line is the trajectory of an object in space-time. Every point object has a world line in space-time. As an object moves in time, it goes one direction or the other along this world line. The arrow of time determines which direction an object moves on its world line. T-X Trajectory Note how the x- component of the energymomentum four-vector increases while the t- component remains the same size. An electron The slope of the electron line must be less than if time ( ) and space are both in units of m. T-X Trajectory If we treat the kinematics relativistically (not just treat time like a fourth variable), the t-component of the energymomentum four-vector increases in size as the kinetic energy increases. An electron What would be different about an electron if it were going backwards in time?
//07 An electron One interpretation of the mathematics of QED is that an electron going backward in time is a positron. Three Arrows of Time. Thermodynamic (Entropy). Psychological (Memory). Cosmological (Expanding Universe) Pair Production If an electron going backward in time undergoes an interaction that reverses its arrow of time, we produce an electron positron pair. Section Three Arrows of Time Entropy We usually hear of as disorder and that the nd Law of Thermodynamics says the universe never moves to greater order. More accurately, is related to probability. The universe is more likely to inhabit a state of greater probability.
//07 Entropy Think of two films, one of a pendulum swinging and the other of someone shooting a basketball. How would they appear if you ran them backwards? Which has an increase in? Which has the largest increase in? Balls in a Box What configuration is most likely? You put,000,000 atoms a box, let them move around for a while, and then put a partition in the middle of the box. What can you conclude about the pressure in either side? Why? Balls in a Box You put balls in a box and at a later time you slide a partition down the center. What is the probability of finding all the balls on the left? ball on the left? Entropy and the Arrow of Time The future is the time where has increased or the universe is found in a state of greater probability. Balls in a Box 5
//07 Memory When something happens it is recorded by a device or by our minds. Whatever is recorded is the past. What is not recorded is the future. Expanding Universe and Entropy Assume the beginning of the universe had no sharp boundaries or singularities. There are still quantum mechanical fluctuations in the density of matter and the velocity of particles, but these fluctuations would have to be small. Small lumpiness would lead to larger lumpiness so would increase as the universe expands. Memory and Entropy To order (on a computer, on an abacus, or in your brain), of the decreases, but of the universe increases. Therefore the direction of s arrow of time is the same as s arrow of time. Expanding Universe The future is the time where the universe becomes larger. What does this mean in terms of the open model? What does that mean in terms of the big crunch model? What does that mean in terms of the flat model? But Expanding Universe Expansion and contraction of the universe are not symmetric. It now appears that will continue to increase even in a collapsing universe. 6
//07 The Anthropic Principle Now the universe is increasing rapidly. Eventually the increase will be very slow. Life as we know it relies on to transfer energy to local order, but largescale disorder. To maintain life, we must have strong. If we can t increase order, life can t evolve. The only universe in which life can exist and the question of can be considered is an. Antimatter Quantum Mechanics In QM, antimatter can be thought of as having negative energy. Negative energy particles can be thought of as going backward in time. Section Wormholes and Time Travel Antimatter General Relativity In GR, there combined with the uncertainty principle, there can be regions with negative energy density. Negative energy causes a negative curvature of space-time. That can lead to unusual structures like tubes. Normal Matter Normal matter moves forward in time. Cause happens before effect. Entropy increases (or at least does not decrease) in time. Wormholes A wormhole is a tube that joins two points in otherwise flattish space-time. Wikipedia 7
//07 Wormholes Wormholes require negative energy matter, which only can be allowed by uncertainty principle fluctuations. That implies wormholes have a finite lifetime. They pinch off. Fundamental Interactions Gravity Weak Electromagnetic Strong Visitors from the Future Wormholes don t stay open long enough. Single particles may travel backward in time with small probability. Large objects require aggregate time travel. There used to be visitors from the future, but it s illegal now in the future world. Fundamental Interactions Gravity Weak Electroweak Electromagnetic Strong Color Section 5 Fundamental Interactions Gravity Grand Unification Weak Electromagnetic Strong Electroweak Color Standard Model 8
//07 Standard Model Lagrangian Density Section 6 Merry Christmas! Problems Quantum gravity requires spin gravitons and doesn t seem to work Schedule Final exam: Next Wednesday at :00 pm EM treated like general relativity requires 5-dimensional space and strong and weak forces don t work String Theory Particles are vibrations on strings Different vibrations correspond to different particles Requires 0+ or 6+ dimensions Seems to allow for gravitation Different universes could have completely different laws but there s probably no one there to notice. 9