course index
Recall from last lecture:
· Light is an electromagnetic wave created by oscillating electric and magnetic fields
· The speed of light is c = 3.0×108 m/s.
· These characteristics of electromagnetic waves (and light) result from the equations of electromagnetics (Maxwell's equations).
Ch. 26: Relativity
The topics of the remaining chapters deal with modern physics, the term used to describe developments that have occured since about 1896, when Roentgen discovered x-rays. The developments of modern physics include, special and general relativity, an understanding of the structure of the atom, quantum mechanics, nuclear physics, particle physics, astrophysics, transistors, and lasers.
Many popular books exist on these topics. Some recent ones accessible to the interested student are:
· A Brief History of Time by Stephen Hawking.
· The Elegant Universe by Brian Green.
· Hyperspace by Machio Iku.
· The Jaguar and the Quark by .
· The God Particle by Leon Lederman.
26.1 Introduction
Albert Einstein is irrevocably linked with relativity. In 1906 he published papers on the special theory of relativity, the focus of most of this chapter, and ten years later on the general theory of relativity, which we will discuss briefly at the end of this chapter. The special theory of relativity, or just relativity for short, knits space and time together into the fabric of our existence. Basic beliefs about time and space must be reexamined in the context of relativity, as well as our concepts of past and future. The general theory builds on the special theory, adding gravity to the mix, resulting in a description of our world where gravity is just a manifestation of the geometry of space. Mass produces a curvature of space, like the way a bowling ball will deform and curve a taught sheet.
Why does relativity force us to rethink our understanding of space and time, and does this mean that all the time spent learning Newton's laws and their consequences was wasted?
I'll answer the second question first. No.
Well maybe a little more explanation is required. The effects of relativity manifest themselves when things move at speeds near to the speed of light, 3×108m/s. Everyday objects on Earth move at speeds much less than the speed of light, so that the difference between the results determined including the effects of relativity and without differ by minuscule amounts. Said another way, the best way to solve problems of falling balls, masses on inclined planes, and pendulums is exactly the way you learned.
Now to answer the first question. Relativity is based on two postulates:
- The laws of physics are the same in all inertial reference systems.
- The speed of light (in vacuum) is always measured to be 3×108m/s, independent of the motion of the observer or of the source of light.
You used the first postulate throughout your study of mechanics. That is, Newton's laws apply equally well in a smoothly flying airplane as on the surface of the earth. Note that an inertial reference system is one which is not accelerating.
The second postulate is the new element. It's not too hard to see that if the speed of light is the same in all reference systems, then the elapse of time will not be the same. Though it doesn't seem connected, this postulate will eventually lead to that most famous of equations, E = mc².
26.2 The Principle of Relativity
In order to describe a physical event, it is necessary to choose a frame of reference. For example, for experiments performed on the surface of the Earth can use the local surface as the reference frame. Someone passing by in a fast moving vehicle might choose the vehicle as her reference frame. Would these two see a dramatic difference in an event? Although they may not agree precisely on what occurred, they would both agree that whatever happened followed Newton's laws.
To be more specific, consider a scenario with two observers, one on a fast moving jet aircraft, and the other on the ground. Th observer on the airplane tosses a ball straight upward, and catches it when it falls back down. According to him, the ball moves according to Newton's laws, rising and falling in gravity.
The observer on the ground sees the ball go up and down as well, but according to him, the ball is also moving forward at the same speed as the aircraft, so it follows a parabolic path. Although the path reported is different, both observers agree that the ball moves according to Newton's laws. This is the first postulate of relativity.
Example: C26.2
What two speed measurements will two observers in relative motion always agree on?
Both will always agree on the speed of light (in vacuum). It is always measured to be c. The second is their relative speed. They must agree on their relative speed, otherwise their would be a difference between their inertial frames.
© Robert Harr 2000
course index
Recall from last lecture:
The two postulates of special relativity:
- The laws of physics are the same in all inertial (non-accelerating) reference systems.
- The speed of light (in vacuum) is always measured to be c = 3.0×108 m/s irrespective of the motion of the observer or the source.
26.3 The Speed of Light
The second postulate of relativity is easy to state, but rather subtle in its implications. You see, never before had something like the constancy of the speed of light occurred. In fact, after physicists realized that light is an electromagnetic wave, they believed that there must be a medium in which the wave propagates. All other waves -- sound, water, string vibrations -- propagated in a medium. So, went the thinking, there must be a medium, which they called the ether, for the propagation of electromagnetic waves. If such an ether existed, then the speed of light would equal c only when measured from a reference frame at rest relative to the ether. In reference frames that are moving with respect to the ether, the speed of light could be greater than or less than c.
This situation is akin to an observer moving towards or away from a stationary source of sound. As seen in the discussion of the Doppler effect in section 14.6, the speed of the observer adds or subtracts from the speed of sound in air, resulting in a different relative speed. This results in a change in the number of cycles of the wave that reach the observer in a given time, causing the frequency of the sound to change according to the Doppler formula.
26.4 The Michelson-Morley Experiment
The most famous experiment to detect the ether is the Michelson-Morley experiment. The experiment was performed in Cleveland, at what is now Case Western Reserve University, and to get to the punch line, the result was negative, no ether was detected.
The experiment is based on an interferometer, a device capable of detecting small changes along one light path relative to a second light path. The change is detected by observing a shift of the interference pattern produced by recombining the light from the two paths. In this interferometer, the two light paths are aligned perpendicularly. A measurement is made by rotating the entire interferometer through 90° and watching for a small change due to the change of orientation with respect to the velocity of the earth relative to the ether. No change of the interference pattern is observed.
Michelson interferometers are once again at the forefront of Physics research. Two large interferometers are being built for the LIGO experiment. LIGO stands for Laser Interferometer Gravitational-Wave Observatory. The facility housing one of the interferometers is pictured. This facility is in Hanford, Washington, and the second is in Livingston, Louisiana. In the photo you can see two long tubes emerging from the central building. Each tube is a 4 kilometer (2-½ mile) long vacuum for the light to travel in. Laser light is used for its superior properties in interferometry.
As the name implies, the LIGO experiment will search for gravitational waves, a prediction of Einstein's general theory of relativity which will be briefly discussed at the end of this chapter.
26.5 Einstein's Principle of Relativity
The null result of the Michelson-Morley experiment pretty much killed the ether hypothesis. (There were proposed modifications to make the results consistent, but as Einstein pointed out, the simple conclusion is 'you don't need the ether; it doesn't add anything to the Physics, so drop it'.
Maxwell's equations predict one speed for electromagnetic waves, and without an ether to provide a universal reference frame, the speed of light has to be the same in all valid (that is, inertial) reference frames. In order to make sense of the fact that everyone measures the same speed of light, we will have to alter our basic notions of space and time.
Is it worth it?
I mean, isn't this stuff speculative, not well proven, and subject to change at almost any moment?
No. Special relativity is very well established and relied upon daily by millions of people around the world. The Global Positioning System (GPS) satellites in orbit about the Earth carry very precise clocks; it is the precision of the clocks that enable the GPS system to accurately determine your location. Effects from special relativity cause the clocks to run slow by 7.11ms per day. That's not much, but it must be corrected for the system to function properly.
One of the results of special relativity is that no object or particle can travel faster than the speed of light. In my research work, we accelerate protons and electrons to extremely high speeds, nearly the speed of light, every day. The protons are accelerated to a speed of 0.9999995c inside of a large ring of magnets. If special relativity were not correct, the speed of the protons would reach 44 times the speed of light, and we wouldn't be able to follow them properly around the ring -- distance traveled = speed × time would be 44 times greater!
Special relativity is now deeply ingrained in Physics and technology.
26.6 Consequences of Special Relativity
There are a number of consequences of special relativity, some of which you may be familiar with:
· Objects cannot travel faster than the speed of light.
· Relativistic energy: E = mc².
· Time and length are not absolute, invariant quantities.
· Events that are simultaneous in one frame of reference may not be in another frame.
o Corollary: If event A occurs before event B in one frame of reference, it is possible that in another frame of reference, event B occurs before event A.
We will begin by exploring the last two items in this list.
© Robert Harr 2000
course index
Recall from last lecture:
- The laws of Physics are the same in all inertial reference frames.
- The speed of light is always measured to be c = 3.0×108 m/s.
The Consequences of Special Relativity
Simultaneity and the Relativity of Time
Consider the rail car pictured in Figure 26.7, with two observers, one in the middle of the moving rail car at O', and the other on the ground at location O. Lightning strikes both ends of the rail car, leaving marks at A, A', B, and B', timed such that the observer on the ground at O sees the light from both strikes simultaneously.
Does the observer in the rail car see the strikes simultaneously? No. The rail car is moving forward at speed v. The light from the lightning strikes propagates from A' and B' towards O'. Since O' is moving towards B', the light from that end reaches the observer first; the light from the other end arrives a bit later.
The observer on the ground says that the lightning strikes were simultaneous, and the observer in the rail car says that lightning hit at B' before A', and both are correct. In relativity, simultaneity is not absolute.
Note from this example that an observation is made when light (or some other signal) reaches an observer. This is an important subtlety in relativity.
Time Dilation
To understand time dilation, let's do what Einstein called a "thought experiment". Imagine two observers, observer 1 inside a vehicle which is moving with respect to observer 2, as shown in Figure 26.8. Observer 1 makes a simple clock by sending a pulse of light from a bulb, reflecting it from a mirror on the top of the vehicle, and measuring the time when it returns to where it started. The time it takes for the light to travel from bulb to mirror to bulb is Dt1 = 2d / c. Think of this as one tick of this clock.
How long does it take for the light to travel from bulb to mirror to bulb for observer 2 who is outside the moving vehicle? According to observer 2, the light leaves the bulb at an angle, travels up to the mirror, reflects at an angle, and returns to the bulb. Let's say that the time it takes is Dt2. We can find Dt2 in terms of d and v, by considering the path of the light as seen by observer 2. The path is a triangle, and if we divide it in half, we have two identical right triangles (Figure 26.8c). The length of one hypotenuse is cDt2/2, since it takes the light equal times to travel to the mirror and back. The lengths of the sides are d, and vDt2/2, the distance the vehicle moves in time Dt2/2. Using Pythagorean's theorem:
(cDt2/2)² = (vDt2/2)² + d²
This can be solved for Dt2. First, move all the terms with Dt2 to the left side, and factor out Dt2²
(Dt2)²((c/2)² - (v/2)²) = d²