Astronomy 162: Professor Barbara Ryden

Monday, February 10

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.

Prof. Barbara Ryden (ryden@astronomy.ohio-state.edu)

Updated: 2003 Feb 10