Special Relativity

Introduction

I do not see any reason to assume that the heuristic significance of the principle of general relativity is restricted to gravitation and that the rest of physics can be dealt with separately on the basis of special relativity, with the hope that later on the whole may be fitted consistently into a general relativistic scheme. I do not think that such an attitude, although historically understandable, can be objectively justified. The comparative smallness of what we know today as gravitational effects is not a conclusive reason for ignoring the principle of general relativity in theoretical investigations of a fundamental character. In other words, I do not believe that it is justifiable to ask: What would physics look like without gravitation?

— Albert Einstein On the Generalized Theory of Gravitation

Your whole world is wobbling. Every time something accelerates, it creates a ripple in the very fabric of the universe. That supernova on the far side of the galaxy, that pair of neutron stars orbiting one another in deep space, the tennis ball in the men's final at Wimbledon, all of them create gravitational waves. Those waves spread out through the cosmos, stretching and compressing space and time like the ripples on a pond. Eventually, they will pass through you, and your space and time will wobble. Only a bit, of course. You will shrink and stretch in height, but imperceptibly. Your watch will run fast and slow, but by such a minute amount as to be unnoticeable. And then the wave will pass, and your space and your time will be still again.

At least, that was the conclusion of Albert Einstein. His general theory of relativity tells us that in most cases, the wobbling of space and time caused by gravitational waves will be so small as to be undetectable. To see the effects, you need to accelerate something really big. And the universe is pretty big. Just after the big bang, it had a rapid expansion, faster than the speed of light. And when you accelerate something as big as the universe as much as that, you are going to get some pretty big gravitational waves. It's the after-effects of these waves that Bicep2 has detected.

— Ben Miller, author of It's Not Rocket science, on the discovery of gravity waves by the BICEP2 Telescope at the Dark Sector Lab, South Pole, March 2014

Distance, Time, and the Speed of Light

Galilean-Newtonian Relativity

Galilean-Newtonian Relativity is the first approximation to the behavior of matter with respect to space and time.

Recall that a reference frame is a framework, often a coordinate system, against which we map the position of an object with respect to time. The frame can be fixed with respect to some external point, while the object moves through space, or it can be on the object, in which case the environment will appear to move. Your Interactive Physics workbook scenarios allow you to chose which frame of reference you use to "see what happens" when objects undergo accelerating forces from different initial conditions. While the actual paths described by the objects may be plotted differently in different inertial frames of reference, two principles hold true for all inertial frames of reference:

The laws of physics are the same in all inertial reference frames. (The relativity principle)

All inertial frames are equivalent.

The nature of space and time

In Newtonian mechanics, we assume that space and time are absolute: we measure them the same way in every frame of reference. Thus, ultimately, all inertial frames of reference are equally valid--we get the same result based on the same assumptions. This is very useful, because it means that you can chose the most convenient frame of reference for a particular situation, increasing clarity and reducing calculations, if possible, in finding the solution to a given set of problems.

Contemplate, for example, a ball thrown by mycroft from a car careening down the road at 65 mph. Assume three situations:

mycroft throws the ball forward, in which case it has the initial velocity of 65 mph plus the speed mycroft puts on it (90 mph). The car driver sees the ball speed out ahead of the car at 90mph. The observer at the side of the road sees the ball speed out ahead of the car at 90+65=155 mph, and ducks.

mycroft throws the ball out the side window. The ball now has a forward direction of 65 mph and a sideways direction of 90mph. The driver sees the ball move directly away from the car to the side at 90 mph. The sidewalk observer sees the ball move diagonally away from the car--forward and to the side-- at a combined velocity of sqrt (90^2 + 65^2), per the laws of vector addition.

mycroft throws the ball out the back window. The ball has a forward motion of 65 mph, and a backward motion of 90mph. The driver sees the ball moving backward at 90mph. The sidewalk observer sees the ball moving backward at 90-65 = 25 mph.

In all three cases, the ball's motion relative to the car and the ground observer ARE DIFFERENT, based on the speed of the point of origin (the car). If the car's velocity were zero, it would share the same frame of reference as the ground observer, and the path observed would be the same. Since it is moving, the path observed by the car driver and the ground observer are different, but the ball winds up in the same physical location.

Light

The difficulty arose when Maxwell showed that the speed of electromagnetic waves was itself absolute: a wave travels at approximately 3 * 10^5 km/sec regardless of the speed of the observer or the speed of the wave source. If we replace mycroft's ball with mycroft's flashlight, and somehow mark the first wave crest from the flash light so that we can identify it, then the car driver will see the crest move away from the car at 3 * 10^5 km/per sec, regardless of the direction of the light wave with respect to the moving car. This makes sense--he saw mycroft's ball move away at 90mph regardless of the direction of the car. The difference comes when we look at what the ground observer sees. Instead of lightspeed + car speed or light speed - car speed = observed speed of light, the ground observer ALSO sees the light move at 3*10^5 km/sec. Light speed is unaffected by the velocity of the source or the observer.

When this was first discovered, it was as though light had its own personal reference frame. When Michelson and Morley tried to determine the speed of the earth relative to this "fundamental" reference frame, they discovered that the relative speed of the earth was zero.

The Michelson-Morley Experiment

They basically did a mycroft-ball-car experiment, but with light. They projected light along the motion of the earth in space, and measured the speed. They then projected light perpendicular to the motion of the earth in space, and measured its speed. If light traveled in some kind of medium (the ether) which was stationary (to explain the fundamental unchanging speed of light as observed), then there would be an "ether wind" as observed from earth: a position in the ether would appear to move backwards as the earth actually moved forwards. Light would be affected by this ether wind and would also appear to move backwards as the earth moved forwards. In order to get the light to reflect back to the point of origin, they would have to aim it at a reflection device "ahead" of the point of origin along the earth's path through space. Light traveling at this angle would have to move through a greater distance than light aimed directly forward (parallel to the wind); it would be out of phase then, and the phase shift could be measured.

But when they did the experiment, there was no phase shift. They got a null result, possibly the most famous experimental failure in the history of science.

Einstein's Solution

In 1905, Albert Einstein proposed a solution, accepting both the relativity principle and the absolute speed of light as postulates. As a result, the speed of light as predicted by Maxwell is the speed of light in any reference frame. The "collateral damage" is to our sense of space and time. In adopting the speed of light as an absolute, rather than as relative to a particular frame of reference, we must give up the notion that space and time are absolutes. Space and time can contract; simultaneity is a fiction. The world we live in is not the world we like to think we live in.

Simultaneity

The first victim is simultaneity. Consider two "simultaneous" events: Mt. Hood and Mt. Adams both erupt. A hiker in the wilderness of the Gifford Pinchot National Forest sits down to eat his lunch precisely halfway between the two eruptions, and part way through the second half of his ham and cheese sandwich, sees the flash from each eruption in the same instant. Since he knows that it is precisely the same distance to each mountain (a careful hiker, he has taken along his GPS meter), and that each beam of light travels at the same speed, he assumes that the events are simultaneous--they happened at the same time.

Now let's look at the astronauts in Mir orbiting above the volcanoes, following a path which takes them along a line above and connecting Mt. Adams with Mt. Hood. They have a velocity v relative to the hiker on the ground, and they are directly overhead. The light from Mt. Adams starts out at the same time as the light from Mt. Hood (according to the hiker). To the astronauts on Mir, however, the light from Mt. Adams must travel farther to reach them (since they are moving away from Mt. Adams) and the light from Mt. Hood must travel less far (since they are moving toward it. Mt. Hood appears to erupt before Mt. Adams: the two events are not simultaneous.

Who is right? They both are--but we have to take into account their physical frames of reference to account for the difference. Time is not absolute: the frame of reference matters. The other ramification of this is that space is not absolute either....

The Twin Paradox

One of the most famous thought experiments involves the story of the relativistic twins. Robert Heinlein uses this idea in Time for the Stars. Unfortunately, my copy is in a box in the garage, so I can't check the names of the characters, but I think they were Tom and Pat. Pat stays on Earth, while Tom leaves on a starship capable of traveling at near-light speeds.

Tom's ship contains a light clock, which uses a reflected beam of light bouncing back and forth to keep track of time. The mirrors on the clock are a distance D apart. One cycle is a bounce out and back, which means it travels 2D. Since the speed of light is constant, the time t for one cycle is $t = \frac{2 D}{c}$

From earth, Pat watches Tom's ship cross the sky at .5c. He has a super-good telescope (based on the principles of chapter 25) and he aims it at the ship. He sees the light move across the distance D and back BUT he also sees the light move with the component of the ship's velocity at .5c. In the time t₀ it takes the light to make one trip between mirrors, the ship moves .5c/t₀ along its trajectory, which we'll call the distance L. Thus for one cycle, Pat thus sees the light travel a combination of the distance 2D (across the clock and back) and 2L (the distance the ship and clock travel through space). This combined distance is $total distance = 2 \sqrt{D^{2} + L^{2}}$

Now, the velocity of light c has to be distance/time = constant.

Tom sees the speed of light as 2D/t = c.

Pat sees the speed of light as 2 * √(D² + L²)/t₀ = c.

In other words, $\frac{2 D}{t} = c = \frac{2 \sqrt{D^{2} + L^{2}}}{t_{0}}$

The times observed by the two boys, t and t₀, cannot be the same, because the speed of light must be the same. When we solve for t in terms of t₀ (a similar derivation is in your text), we get

t_{0} = \frac{t}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

We use this form because it points out the relative importance of the size of v with respect to c. If v << c, v²/c² << 1, and we can ignore it. At small v, the equation reduces to

t₀ = t/√(1) = t

which is what we normally observe; the times are the same. But if v ~ c, say, .5 c, then the problem becomes more obvious:

t₀ = t/(√(1 - (.5c)²/c²)

t₀ = t/(√(1 - .25) = t/√(.75) = t/.866

OR t = .866t₀

In other words, if Tom is traveling at half the speed of light, the time he experiences (t) is LESS than the time Pat experiences (t₀) during the same event. The closer to c that the ship travels, the less time t passes for a given t₀.

Now, as long as the earth travels at constant speed and the ship travels at constant speed, both frames are inertial and equivalent. Tom can claim that Pat has aged less just as Pat can claim that Tom has aged less. The twin paradox arises when Tom's ship turns around and comes back--or even just when it slows down and stops at Alpha Centuri. A frame of reference is an inertial frame only if it is traveling at constant speed with respect to another inertial frame (earth's speed is constant with respect to the universe). But as soon as the frame of reverence accelerates or decelerates, it is no longer inertial, and no longer equivalent and equally valid with respect to any other inertial frame. Tom will return to earth approximate .86 the age of his twin (give or take a little for the periods of acceleration/deceleration and the time spent studying in the library of Alpha Centauri).

Practice with the Concepts

Discussion Points

You are in a windowless car in an exceptionally smooth train moving at constant velocity. Is there any physical experiment you can do in the train car to determine whether you are moving? If so, how would you do it?
If you were on a spaceship traveling at 0.63c away from a star, at what speed would the starlight pass you?
Does time dilation mean that time actually passes more slowly in moving reference frames or that it only seems to pass more slowly?

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Physics

Chapter 26: 1-5