Chapter 37 Relativity PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun
37. Relativity 1. Maxwell s equations (and especially the wave equation) are valid in all inertial reference frames (observers in constant t relative motion u) ) suggesting that t the speed of light in vacuum c is the same in all inertial reference frames(irf s). 2. This special constancy in the speed of light leads to time dilation/length contraction (a clock appears to run slower, and a rod appears to be shorter, according to an observer in constant motion u relative to the clock, and with the rod s length l oriented parallel to u). l 0 1 Δ t = γδt 0 3. Two postulates: (1) The laws of physics are the same in all IRF s. (2) The speed of light in vacuum c is the same in all IRF s. 4. Lorentz transformation (position, velocity, ): A linear transformation, relating quantities in one reference frame to those of any IRF, consistent with (or derived from) the postulates of relativity. For simplicity, assume that u points along the +ve x-axis.. 2 vx u vy vz x ' = γ( x ut), y' = y, z' = z, t' = γ( t xu/ c ) vx' =, v ', '. 2 y = v 2 z = 2 l = γ γ = 1 u / c 5. The Doppler effect: This is an observed phenomenon that may be described by the Lorentz transformation for period T, as time interval, and the wavelength λ, as displacement, of a sinusoidal wave. The emitted frequency f_0 and received ed frequency f are related ed by c+ u c u f = f 0 (moving closer) c u f = f 0 (moving apart) c+ u 2 2 1 uv / c γ(1 uv / c ) γ(1 uv / c ) x x x 6. Special relativistic mechanics (Newton s 2 nd law and the work-energy relation): The first postulate requires that Newton s 2 nd law of motion be revised. dp net or d ( ) net, de = γf γmv = F = v Fnet, dτ = dt (proper time) dτ dt dt γ. 2 2 2 2 2 2. E = γ mc = mc + Ke, p= γ mv E = ( mc ) + ( pc) 7. General Relativity: The laws of physics need to be valid not just for inertial observers but for all observers (ie. In all local coordinate systems). Now, just as one could not experimentally distinguish between IRF s, locally it is impossible to experimentally distinguish between a force field ( eg. gravitational field) in an IRF and a non-irf (curved space-time coordinate system). Therefore, gravity may be seen as a result of the curving of space-time by matter.
Goals for Chapter 37 To consider the invariance of physical laws, simultaneity To study time dilation To study length contraction To see how the Lorentz transformation can show how different frames of reference explain vexing observations To consider how the Doppler effect applies not only to sound but also to EM waves To explain how relativistic motion changes momentum To see where Newtonian Mechanics fit into the Special and General Theories of Relativity
Introduction Imagine a gedanken experiment where two identical twins in their early 30s meet at the launch pad of a new spacecraft. John-boy puts on his helmet and shakes hands with Billy-bob. John-boy flies away in a ship that travels at 0.999c for a one-year journey to a distant planet. When he comes back, a year older, he meets Billybob for lunch and sees a man across the table who s approaching retirement. WHAT? Experiments at Brookhaven can accelerate particles to near the speed of light. Unfortunately, none of them are wearing timepieces.
Einstein s First Postulate The laws of physics apply in the same fashion everywhere. Newton s Laws work no matter what your point of view. A magnet moving in a coil of wire will induce a current, but who s to say if the coil is moving over the magnet or if the magnet is moving through the coil.
Einstein s Second Postulate The speed of light is always the same. It matters not how fast you are going or in which direction you travel, the speed of light is always the same.
Simultaneity watching lightning Perception of one reference frame from another makes it hard to correctly interpret perceptions of physical events.
Relativity of time intervals A light clock can be used to gain a graphic understanding of the gamma term that modifies the size of any quantity modified by relativistic effects. Refer to Figure 37.6 below.
The gamma factor s exponential impact As velocity approaches c, the value of gamma grows quickly. Refer to Figure 37.8 below. Read Problem-Solving Strategy 37.1. Refer to Example 37.1 effect of travel at 0.99c. Refer to Example 37.2 effect of travel at aircraft velocities. Refer to Example 37.3 travel at 0.6c.
Lengths parallel or perpendicular to motion Motions perpendicular to the relativistic velocities are not contracted. Motions parallel are directly involved. Figure 37.12 (below) illustrates the effect. Read Problem-Solving Strategy 37.2.
Contraction examples Follow Example 37.4, illustrated by Figure 37.13 below. Follow Example 37.5 length contraction. Follow Conceptual Example 37.6.
Appearance of objects in relativistic motion How does an object moving at velocities appear to an observer? Refer to Figure 37.14.
The Lorentz transformation The transformation allows us to determine adjustments for each dimension (x, y, or z) that s involved in the problem. Refer to Figure 37.15 below at left. Read Problem-Solving Strategy 37.3. Follow Example 37.7. 7 Follow Example 37.8, illustrated by Figure 37.16, below at right.
The Doppler effect applied to EM waves The Doppler effect can cause enough change in the velocity of light to change the wavelength perceived by an observer at a distance. See Figure 37.17 along the bottom of the slide.
Relativistic momentum As velocities near the speed of light, the gamma factor would cause momentum/mass/kinetic energy needed to cause the motion to approach infinity. It s simply impossible. This is explained graphically in Figure 37.20 in the lower left corner for momentum and Figure 37.21 in the lower right corner for kinetic energy. Follow Example 37.10.
Relativistic examples Consider Example 37.11 motion of an electron. Consider Example 37.12 relativistic collision, illustrated by Figure 37.23 below.
Newtonian mechanics and relativity How would an astronaut perceive travel at (or near) the speed of light? See Figure 37.24.
Newtonian mechanics and relativity II Figure 37.25 speculates on the effect of motion near c on a 2-D map of outer space.