Lecture PowerPoint Chapter 26 Physics: Principles with Applications, 6th edition Chapter 26 The Special Theory of Relativity Giancoli © 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Units of Chapter 26 Units of Chapter 26 • Galilean-Newtonian Relativity • Relativistic Momentum and Mass • Postulates of the Special Theory of Relativity • The Ultimate Speed • Simultaneity • E = mc2; Mass and Energy • Time Dilation and the Twin Paradox • Relativistic Addition of Velocities • Length Contraction • The Impact of Special Relativity • Four-Dimensional Space-Time 26.1 Galilean-Newtonian Relativity Definition of an inertial reference frame: One in which Newton’s first law is valid: 26.1 Galilean-Newtonian Relativity Relativity principle: The basic laws of physics are the same in all inertial reference frames. No force => no acceleration! Earth is rotating and therefore not an inertial reference frame, but can treat it as one for many purposes A frame moving with a constant velocity with respect to an inertial reference frame is itself inertial 1 26.1 Galilean-Newtonian Relativity Classically: Time intervals are absolute 26.1 Galilean-Newtonian Relativity This principle works well for mechanical phenomena. However, Maxwell’s equations yield the velocity of light; it is 3.0 x 108 m/s. Space is absolute Two inertial frames x s = x + wt Motion in frame 1 x = vt Motion in frame 2 x s = (v + w)t So, which is the reference frame in which light travels at that speed? Scientists searched for variations in the speed of light depending on the direction of the ray, but found none. Velocity changes, acceleration is the same! 26.2 Postulates of the Special Theory of Relativity 1. The laws of physics have the same form in all inertial reference frames. 2. Light propagates through empty space with speed c independent of the speed of source or observer. 26.3 Simultaneity One of the implications of relativity theory is that time is not absolute. Distant observers do not necessarily agree on time intervals between events, or on whether they are simultaneous or not. In relativity, an “event” is defined as occurring at a specific place and time. This solves the problem – the speed of light is in fact the same in all inertial reference frames. 26.3 Simultaneity 26.3 Simultaneity Thought experiment: Lightning strikes at two separate places. One observer believes the events are simultaneous – the light has taken the same time to reach her – but another, moving with respect to the first, does not. Here, it is clear that if one observer sees the events as simultaneous, the other cannot, given that the speed of light is the same for each. 2 26.4 Time Dilation and the Twin Paradox A different thought experiment, using a clock consisting of a light beam and mirrors, shows that moving observers must disagree on the passage of time. 26.4 Time Dilation and the Twin Paradox In the spaceship frame: ∆t 0 = 2D c In the earth’ frame: ∆t = v2 2 D 2 + L2 4 D 2 4 L2 = + 2 = ∆t 02 + 2 ∆t 2 2 c c c c 26.4 Time Dilation and the Twin Paradox 26.4 Time Dilation and the Twin Paradox Calculating the difference between clock “ticks,” we find that the interval in the moving frame is related to the interval in the clock’s rest frame: The factor multiplying t0 occurs so often in relativity that it is given its own symbol, . (26-1a) (26-2) It is called the Lorentz factor. We can then write: (26-1b) 26.4 Time Dilation and the Twin Paradox 26.4 Time Dilation and the Twin Paradox To clarify: It has been proposed that space travel could take advantage of time dilation – if an astronaut’s speed is close enough to the speed of light, a trip of 100 light-years could appear to the astronaut as having been much shorter. • The time interval in the frame where two events occur in the same place is t0. • The time interval in a frame moving with respect to the first one is t. The astronaut would return to Earth after being away for a few years, and would find that hundreds of years had passed on Earth. 3 26.4 Time Dilation and the Twin Paradox 26.4 Time Dilation and the Twin Paradox This brings up the twin paradox – if any inertial frame is just as good as any other, why doesn’t the astronaut age faster than the Earth traveling away from him? The solution to the paradox is that the astronaut’s reference frame has not been continuously inertial – he turns around at some point and comes back. 26.5 Length Contraction 26.6 Four-Dimensional Space-Time If time intervals are different in different reference frames, lengths must be different as well. Length contraction is given by: (26-3a) or (26-3b) Space and time are even more intricately connected. Space has three dimensions, and time is a fourth. When viewed from different reference frames, the space and time coordinates can mix. Length contraction occurs only along the direction of motion. 26.7 Relativistic Momentum and Mass Expressions for momentum and mass also change at relativistic speeds. Momentum: (26-4) 26.8 The Ultimate Speed A basic result of special relativity is that nothing can equal or exceed the speed of light. This would require infinite momentum – not possible for anything with mass. Gamma and the rest mass are sometimes combined to form the relativistic mass: (26-5) 4 26.9 E = mc2; Mass and Energy At relativistic speeds, not only is the formula for momentum modified; that for energy is as well. The total energy can be written: 26.9 E = mc2; Mass and Energy Combining the relations for energy and momentum gives the relativistic relation between them: (26-7b) Where the particle is at rest, (26-8) 26.9 E = mc2; Mass and Energy E 2 = p 2 c 2 + m02 c 4 = m02 c 4 γ 2 Small velocity: 2 v + 1 = m02 c 4 c2 1 v2 1− 2 c All the formulae become the usual Newtonian formulae when the speeds are much smaller than the speed of light. 26.10 Relativistic Addition of Velocities Relativistic velocities cannot simply add; the speed of light is an absolute limit. The relativistic formula is: (26-11) 1 E ≅ m0 c 2 + m0 v 2 2 There is no rule for when the speed is high enough that relativistic formulas must be used – it depends on the desired accuracy of the calculation. 26.11 The Impact of Special Relativity The predictions of special relativity have been tested thoroughly, and verified to great accuracy. The correspondence principle says that a more general theory must agree with a more restricted theory where their realms of validity overlap. This is why the effects of special relativity are not obvious in everyday life. Summary of Chapter 26 • Inertial reference frame: one in which Newton’s first law holds • Principles of relativity: the laws of physics are the same in all inertial reference frames; the speed of light in vacuum is constant regardless of speed of source or observer • Time dilation: 5 Summary of Chapter 26 Summary of Chapter 26 • Length contraction: • Relativistic momentum: • Gamma: • Relativistic mass: Summary of Chapter 26 Summary of Chapter 26 • Mass-energy relation: • Kinetic energy: • Relationship between energy and momentum: • Total energy: 6