SPECIAL RELATIVITY
``Common sense is the collection of prejudices acquired by age eighteen.''
- Albert Einstein
Key Concept: Special Relativity in One Sentence.
• All speeds are relative, except for the speed of light, which is absolute.
Both the theory of Special Relativity and the theory of General Relativity were formulated by Albert Einstein (born 1879, died 1955). The theories of relativity have received the reputation of being incomprehensible. Actually, this is something of a bum rap. The theory of Special Relativity, in particular, is very simple from a mathematical viewpoint. Why, then, has Special Relativity attained a reputation for being incomprehensible? Mainly because it violates ``common sense''. In everyday life, we deal with objects moving much slower than the speed of light. Special relativity deals with objects moving at speeds close to the speed of light. It is not too surprising, then, that Special Relativity doesn't conform to the collection of prejudices that we have accumulated from our observations of slow-moving objects.
The theory of Special Relativity, published by Einstein in 1905 (when he was 26 years old), describes how objects behave when they have a constant velocity. Ten years later, in 1915, Einstein published his theory of General Relativity, which describes how objects move when they are accelerated by gravity. (General Relativity is the subject of tomorrow's lecture; today we'll stick to the simpler case of Special Relativity.)
Special Relativity can be summed up in one brief sentence:
All speeds are relative, except for the speed of light, which is absolute.
A concise sentence -- but what does it mean? Let's start by examining what is mean by ``relative speed''. A professor paces across a lecture platform.
Her speed relative to platform=
1 meter/second
Her speed relative to center of Earth=
360 meters/second
Her speed relative to center of Sun=
30,000 meters/second
Her speed relative to center of Galaxy=
220,000 meters/second
Which of the above speeds is the ``correct'' speed? They are all correct. When you state the speed of a material object, like a professor or a star, you are stating the speed relative to some other object. For massive objects, all speeds are relative.
As another example of the relative nature of speeds, consider a criminal barreling down High Street, driving his getaway car. As seen by an innocent bystander, the car has a velocity v = 30 meters/second (about 67 mph). The criminal draws his gun, and shoots a bullet in the direction he is traveling. Relative to the car, the bullet has a velocity u = 250 meters/second (the muzzle speed of a bullet from a .45 automatic).
To summarize the situation:
Speed of car relative to bystander = v = 30 meters/second
Speed of bullet relative to car = u = 250 meters/second
To find the speed of the bullet relative to the bystander, just add the speeds together:
Speed of bullet relative to bystander = v + u = 280 meters/second.
The question ``What is the speed of the bullet?'' doesn't have a single answer. The speed of the bullet relative to the car is 250 meters/second. The speed of the bullet relative to the bystander is 280 meters/second. For that matter, the speed of the bullet relative to the center of the galaxy is 220,000 meters/second.
There is a critical caveat attached to the theory of Special Relativity: all speeds are relative, except for the speed of light, which is absolute.
As an example of the absolute nature of the speed of light, consider the same criminal roaring down High Street in his getaway car. Relative to a bystander, the car has the same speed v = 30 meters/second. Now, however, the criminal draws a laser gun. A laser produces electromagnetic radiation, so relative to the car, the laser beam will travel at the speed of light: c = 300,000,000 meters/second.
To summarize the new situation:
Speed of car relative to bystander = v = 30 meters/second
Speed of light beam relative to car = c = 300,000,000 meters/second
What is the speed of the light beam relative to the bystander? A classical physicist, like Galileo or Newton, would say the speed is v+c = 300,000,030 meters/second. This, however, is WRONG. The correct answer, given by Einstein, is that the speed of the light beam relative to the bystander is c = 300,000,000.
The speed of light is absolute; that means it is the same seen by any observer, no matter how fast the observer is moving relative to the light source. THE OBSERVED SPEED OF LIGHT IN A VACUUM IS ALWAYS 299,792.459 KILOMETERS PER SECOND. (Parenthetical comments: it is necessary to add the qualification ``in a vacuum'' since interactions with matter can slow down a light beam. The exact value of the speed of light is usually rounded off to 300,000 km/sec for practical purposes. The speed of light is the speed of all electromagnetic radiation, from radio to gamma-rays.)
The fact that the speed of light is constant has been experimentally verified, first by a pair of physicists in Cleveland in 1887. The fact that the speed of light is absolute, while all other speeds are relative, has some bizarre consequences. Suppose I hand you a light bulb, and send you away from Earth with a speed equal to 99% the speed of light. You say:
``The light bulb is stationary. The light from the bulb is moving at a speed c.''
On the other hand, I say:
``The light bulb is moving at a speed 0.99c. The light from the bulb is moving at a speed c.''
Two observers are moving at a speed 0.99c relative to each other. Each observer, using his own yardstick and clock, measures the speed of a particular beam of light to be the same. The only way the two observers to observe the same speed for the beam of light, Einstein concluded, is for odd things to be happening to the yardsticks and clocks with which they measure the speed of light.
Relativistic Time Dilation
If two observers are in motion relative to each other, each sees the other's clock run more slowly.
Whose clock is correct? Both are correct. THERE IS NO SUCH THING AS ABSOLUTE TIME. The rate at which time flows is different for different observers.
Relativistic Length Contraction
If two observers are in motion relative to each other, each sees the other shortened in the direction of motion.
Whose yardstick is correct? Both are correct. THERE IS NO SUCH THING AS ABSOLUTE SPACE. The distance between two points is different for different observers.
Please note that the relativistic effects mentioned above (time dilation and length contraction) are all very small unless the relative speed is close to the speed of light.
Suppose you accelerate an object by applying a constant force to it. As the speed of the object approaches the speed of light,
• its length (as measured by you) approaches zero
• the time between ticks of its clock (as measured by you) approaches infinity
• its acceleration (as measured by you) approaches zero
• and thus its mass (defined as force/acceleration) approaches infinity.
No matter how much energy you use to accelerate an object, it will never reach the speed of light relative to you.
An insight of Einstein: Mass and energy are interconvertible. `Common sense' tells us that mass and energy are two distinct entities. However, Einstein revealed that mass can be converted to energy and vice versa. The ``exchange rate'' between energy and mass is given by the famous formula:
E = m c2
Where E is energy, m is mass, and c is the speed of light. Since the speed of light is large, a small amount of mass can be converted to a large amount of energy. One kilogram of matter is equivalent to 1017 joules, or 25 billion kilowatt-hours.
Further insight of Einstein: Space and time are interconvertible. `Common sense' tells us that space and time are two distinct entities (space is what we measure with yardsticks; time is what we measure with clocks). However, Einstein revealed that it is more useful to think of the three dimensions of space and the one dimension of time as being component parts of a four-dimensional spacetime. Two observers will not agree on the spatial distance between two points (like the two ends of a yardstick). They will not agree on the time interval between two events (like two ticks of a clock). However, squashing (or contracting) of space is accompanied by stretching (or dilating) of time. Thus, two observers will agree on the four-dimensional spacetime distance between two events.
GENERAL RELATIVITY
``Nothing puzzles me more than time and space, and yet nothing puzzles me less, for I never think about them.'' - Charles Lamb
Key Concept: General Relativity in One Sentence
• Mass-energy tells spacetime how to curve; the curvature of spacetime tells mass-energy how to move.
Yesterday's lecture dealt with Special Relativity, which describes how objects behave when they have a constant velocity relative to each other. Today's lecture deals with General Relativity, which describes how objects behave when they are accelerated by gravity.
Einstein's view of how gravity works is very different from Newton's view of how gravity works.
Newton's view:
Mass tells gravity how to exert a force ( F = G M1 M2 / d2 );
Force tells mass how to accelerate (F = M2 a).
Einstein's view:
Mass-energy tells spacetime how to curve;
Curvature tells mass-energy how to move.
What do we mean when we talk about the curvature of spacetime? Spacetime has four dimensions (3 space dimensions, 1 time dimension); the curvature of a four-dimensional volume is very difficult to visualize, to put it mildly!
For the purposes of visualization, consider a two-dimensional surface.
On a flat surface, like a tabletop:
The shortest distance between two points is a straight line.
Parallel lines never meet.
On a curved surface, like a globe:
The shortest distance between two points is a curved line (on a spherical globe, the shortest distance is a great circle).
Parallel lines converge or diverge (on a spherical globe, parallel lines converge).
Just as you can have flat or curved two-dimensional surfaces, you can have flat or curved four-dimensional volumes.
Curvature is amusing, but what does it have to do with gravity? Newton would say, ``Curvature is irrelevant. A massive object exerts a force on other massive objects, and makes them follow curved orbits.'' But if you asked Newton, ``What carries the force from one object to another across empty space?'', he would have to answer, ``I don't know''.
Einstein would say, ``A massive object curves the spacetime around it. Freely falling objects will follow the shortest path between two points, and hence will follow curved lines in the curved spacetime.'' Einstein would also point out that photons follow curved paths in spacetime as well, even though they have no mass.
As an analogy to Einstein's view of gravity, consider a rubber sheet which is held taut on a frame. It is flat, so if you roll a marble across it, the marble will follow a straight line. Now drop a heavy ball bearing onto the rubber sheet. The ball bearing is massive enough to cause a depression in the rubber sheet around it. Now roll a marble across the rubber sheet, close to the ball bearing. The curvature of the rubber sheet will cause the marble to follow a curved path. A distant observer, unable to see the curvature in the rubber sheet, would say ``The ball bearing must be exerting a force on the marble, deflecting it from its straight path.''
Thus, an object which is following the shortest path on a curved surface looks as if a force is being exerted on it. Similarly, an object following the shortest path in a curved four-dimensional spacetime will look as if a force (the force of gravity) is being exerted on it.
How can we tell whether Newton's view (gravity as a force) or Einstein's view (gravity as curvature) is correct? If there were no observations we could make to distinguish between the two theories, then the debate between them would be futile. However the predictions of Einstein's General Relativity theory differ from the predictions of Newton's theory in the region very close to a very massive object (where the escape speed is very close to the speed of light). Thus, General Relativity is crucial for understanding black holes, and very important for understanding neutron stars (where the escape speed from the surface is half the speed of light).
There have been three major experimental tests of the theory of General Relativity, ALL of which support Einstein over Newton.
Test One: Bending of starlight.
Photons follow the shortest path between two points. Near a massive object, such as the Sun, the shortest path will be curved. Thus, as shown in the illustration below, the apparent position of a star close to the Sun in the sky will differ from its true position.
The actual prediction of Einstein is that starlight passing close to the Sun will be deflected by only 1.75 arcseconds (a small but measurable amount). During a total solar eclipse in 1919, astronomers tested Einstein's prediction by photographing the apparent position of stars close to the Sun and comparing them to the true position, measured 6 months earlier when the Sun was on the opposite side of the celestial sphere. The results are shown below. The small blue circles represent the true positions of stars; the arrows point to their apparent positions when the Sun was in their midst.
The observations agree (within experimental error) with Einstein's prediction.
Score: Einstein 1, Newton 0.
As an aside: Much more spectacular examples of the bending of light are seen when light from a distant galaxy passes through a nearby galaxy or cluster of galaxies. The nearby galaxy or cluster acts as a `gravitational lens' when can cause magnified images, multiple images, and distorted images of the distant galaxy. Click on the small picture below, for instance, to see a larger image of a cluster of galaxies called Abell 2218. The extremely elongated arcs of light which you see encircling the cluster are actually badly distorted images of galaxies which lie far beyond the cluster Abell 2218.
[Image credit: W. Couch (U. New South Wales) & NASA]
Test Two: Precession of the perihelion of Mercury's orbit.
The major axis of Mercury's orbit precesses - that is, it changes its position slowly with time. The effect is shown (in a greatly exaggerated form) in the illustration below:
Part of the precession is explained by the perturbing force of the gravity of the other planets. However, the amount of the precession is 43 arcseconds/century greater than is predicted by Newton's theory. This doesn't sound like a large effect; it will take over 3 million years to twist the orbit of Mercury through 360 degrees. However, it was large enough to throw astronomers into a tizzy at the end of the 19th century. (They actually hypothesized the presence of an additional planet, called Vulcan, with an orbit smaller than that of Mercury. )
The additional precession of 43 arcsec/century, however, is precisely equal to the precession predicted by Einstein's theory (when Mercury is at perihelion, the greater spacetime curvature gives the orbit of Mercury a little extra twist). The hunt for the planet Vulcan could be called off.
Score: Einstein 2, Newton 0.
Test Three: Gravitational Redshift.
It takes energy to move away from a massive object. Since photons are affected by gravity, as they move away from a massive object, they must
• Decrease in energy
• Decrease in frequency
• Increase in wavelength.
Thus, light is redshifted as it moves away from a massive object. This prediction of General Relativity has been experimentally confirmed. In 1960, a pair of physicists shot a beam of gamma-rays from the bottom of a tower (23 meters tall) to a detector at the top. The redshift observed was small (only 1 part in 400 trillion!) since the Earth's gravitational field is weak, but they could detect it with the extremely sensitive detector they were using.
Score: Einstein 3, Newton 0.
It looks like Einstein wins in a shutout! Does this mean we should toss Newton in the garbage?
Not so fast...Newton's theory gives highly accurate results when speeds are slow and gravity is weak (this is true almost everywhere). Moreover, computing Newtonian forces is simpler than computing the full four-dimensional relativistic curvature, so astronomers, who like a simple life, use the theory of Newton whenever possible, resorting to the accuracy of Einstein's Theory of General Relativity only when absolutely necessary.
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