Special Relativity
Special Relativity
Special relativity is the theory developed by Albert Einstein in 1905 to explain the observed fact that the speed of light is a constant regardless of the direction or velocity of one’s motion. Einstein laid down two simple postulates to explain this strange fact, and, in the process, derived a number of results that are even stranger. According to his theory, time slows down for objects moving at near light speeds, and the objects themselves become shorter and heavier. The wild feat of imagination that is special relativity has since been confirmed by experiment and now plays an important role in astronomical observation.
The Michelson-Morley Experiment
As we discussed in the chapter on waves, all waves travel through a medium: sound travels through air, ripples travel across water, etc. Near the end of the nineteenth century, physicists were still perplexed as to what sort of medium light travels through. The most popular answer at the time was that there is some sort of invisible ether through which light travels. In 1879, Albert Michelson and Edward Morley made a very precise measurement to determine at what speed the Earth is moving relative to the ether. If the Earth is moving through the ether, they reasoned, the speed of light should be slightly different when hitting the Earth head-on than when hitting the Earth perpendicularly. To their surprise, the speed of light was the same in both directions.
For people who believed that light must travel through an ether, the result of the Michelson-Morley experiment was like taking a ride in a boat and discovering that the boat crossed the wave crests at the same rate when it was driving against the waves as when it was driving in the same direction as the waves.
No one was sure what to make of the Michelson-Morley experiment until 1905, when Albert Einstein offered the two basic postulates of special relativity and changed forever the way we think about space and time. He asked all sorts of unconventional questions, such as, “What would I see if I were traveling at the speed of light?” and came up with all sorts of unconventional answers that experiment has since more or less confirmed.
The Basic Postulates of Special Relativity
Special relativity is founded upon two basic postulates, one a holdover from Newtonian mechanics and the other a seeming consequence of the Michelson-Morley experiment. As we shall see, these two postulates combined lead to some pretty counterintuitive results.
First Postulate
The laws of physics are the same in all inertial reference frames.
An inertial reference frame is one where Newton’s First Law, the law of inertia, holds. That means that if two reference frames are moving relative to one another at a constant velocity, the laws of physics in one are the same as in the other. You may have experienced this at a train station when the train is moving. Because the train is moving at a slow, steady velocity, it looks from a passenger’s point of view that the station is moving backward, whereas for someone standing on the platform, it looks as if the train is moving forward.
Einstein’s first postulate tells us that neither the passenger on the train nor the person on the platform is wrong. It’s just as correct to say that the train is still and the Earth is moving as it is to say that the Earth is still and the train is moving. Any inertial reference frame is as good as any other.
Second Postulate
The speed of light in a vacuum is a constant— m/s—in every reference frame, regardless of the motion of the observer or the source of the light.
This postulate goes against everything we’ve learned about vector addition. According to the principles of vector addition, if I am in a car moving at 20 m/s and collide with a wall, the wall will be moving at 20 m/s relative to me. If I am in a car moving at 20 m/s and collide with a car coming at me at 30 m/s, the other car will be moving at 50 m/s relative to me.
By contrast, the second postulate says that, if I’m standing still, I will measure light to be moving at m/s, or c, relative to me, and if I’m moving toward the source of light at one half of the speed of light, I will still observe the light to be moving at c relative to me.
By following out the consequences of this postulate—a postulate supported by the Michelson-Morley experiment—we can derive all the peculiar results of special relativity.
Time Dilation
One of the most famous consequences of relativity is time dilation: time slows down at high speeds. However, it’s important to understand exactly what this means. One of the consequences of the first postulate of special relativity is that there is no such thing as absolute speed: a person on a train is just as correct in saying that the platform is moving backward as a person on the platform is in saying that the train is moving forward. Further, both the person on the train and the person on the platform are in inertial reference frames, meaning that all the laws of physics are totally normal. Two people on a moving train can play table tennis without having to account for the motion of the train.
The point of time dilation is that, if you are moving relative to me in a very highspeed train at one-half the speed of light, it will appear to me that time is moving slower on board the train. On board the train, you will feel like time is moving at its normal speed. Further, because you will observe me moving at one-half the speed of light relative to you, you will think time is going more slowly for me.
What does this all mean? Time is relative. There is no absolute clock to say whether I am right or you are right. All the observations I make in my reference frame will be totally consistent, and so will yours.
We can express time dilation mathematically. If I were carrying a stopwatch and measured a time interval, , you would get a different measure, t, for the amount of time I had the stopwatch running.
The relation between these measures is:
So suppose I am moving at one-half the speed of light relative to you. If I measure 10 seconds on my stopwatch, you will measure the same time interval to be:
This equation has noticeable effects only at near light speeds. The difference between t and is only a factor of . This factor—which comes up so frequently in special relativity that it has been given its own symbol, —is very close to 1 unless v is a significant fraction of c. You don’t observe things on a train moving at a slower rate, since even on the fastest trains in the world, time slows down by only about 0.00005%.
Time Dilation and Simultaneity
Normally, we would think that if two events occur at the same time, they occur at the same time for all observers, regardless of where they are. However, because time can speed up or slow down depending on your reference frame, two events that may appear simultaneous to one observer may not appear simultaneous to another. In other words, special relativity challenges the idea of absolute simultaneity of events.
Example
A spaceship of alien sports enthusiasts passes by the Earth at a speed of 0.8c, watching the final minute of a basketball game as they zoom by. Though the clock on Earth measures a minute left of play, how long do the aliens think the game lasts?
Because the Earth is moving at such a high speed relative to the alien spaceship, time appears to move slower on Earth from the aliens’ vantage point. To be precise, a minute of Earth time seems to last:
Length Contraction
Not only would you observe time moving more slowly on a train moving relative to you at half the speed of light, you would also observe the train itself becoming shorter. The length of an object, , contracts in the direction of motion to a length when observed from a reference frame moving relative to that object at a speed v.
Example
You measure a train at rest to have a length of 100 m and width of 5 m. When you observe this train traveling at 0.6c (it’s a very fast train), what is its length? What is its width?
What is its length?
We can determine the length of the train using the equation above:
What is its width?
The width of the train remains at 5 m, since length contraction only works in the direction of motion.
Addition of Velocities
If you observe a person traveling in a car at 20 m/s, and throwing a baseball out the window in the direction of the car’s motion at a speed of 10 m/s, you will observe the baseball to be moving at 30 m/s. However, things don’t quite work this way at relativistic speeds. If a spaceship moving toward you at speed u ejects something in the direction of its motion at speed relative to the spaceship, you will observe that object to be moving at a speed v:
Example
A spaceship flying toward the Earth at a speed of 0.5c fires a rocket at the Earth that moves at a speed of 0.8c relative to the spaceship. What is the best approximation for the speed, v, of the rocket relative to the Earth?
(A) v > c
(B) v = c
(C) 0.8c < v < c
(D) 0.5c < v < 0.8c
(E) v < 0.5c
The most precise way to solve this problem is simply to do the math. If we let the speed of the spaceship be u = 0.5c and the speed of the rocket relative to the spaceship be = 0.8c, then the speed, v, of the rocket relative to the Earth is
As we can see, the answer is (C). However, we could also have solved the problem by reason alone, without the help of equations. Relative to the Earth, the rocket would be moving faster than 0.8c, since that is the rocket’s speed relative to a spaceship that is speeding toward the Earth. The rocket cannot move faster than the speed of light, so we can safely infer that the speed of the rocket relative to the Earth must be somewhere between 0.8c and c.
Mass and Energy
Mass and energy are also affected by relativistic speeds. As things get faster, they also get heavier. An object with mass at rest will have a mass m when observed to be traveling at speed v:
Kinetic Energy
Because the mass increases, the kinetic energy of objects at high velocities also increases. Kinetic energy is given by the equation:
You’ll notice that as v approaches c, kinetic energy approaches infinity. That means it would take an infinite amount of energy to accelerate a massive object to the speed of light. That’s why physicists doubt that anything will ever be able to travel faster than the speed of light.
Mass-Energy Equivalence
Einstein also derived his most famous equation from the principles of relativity. Mass and energy can be converted into one another. An object with a rest mass of can be converted into an amount of energy, given by:
We will put this equation to work when we look at nuclear physics.
Relativity and Graphs
One of the most common ways SAT II Physics tests your knowledge of special relativity is by using graphs. The key to remember is that, if there is a dotted line representing the speed of light, nothing can cross that line. For instance, here are two graphs of kinetic energy vs. velocity: the first deals with normal speeds and the second deals with relativistic speeds:
In the first graph, we get a perfect parabola. The second graph begins as a parabola, but as it approaches the dotted line representing c, it bends so that it constantly approaches c but never quite touches it, much like a  y = 1/graph will constantly approach the x-axis but never quite touch it.
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