Relativity

 

In 1905 space and time were united by Albert Einstein’s Special Theory of Relativity.  This theory solved a puzzling problem that had arisen during the nineteenth century.  James Clark Maxwell (1865) had shown that accelerating charges radiate electromagnetic waves and these waves travel at a constant velocity c that turned out to be the speed of light.  Now a velocity is always measured with respect to some frame of reference.  When we are talking about wave velocity the frame of reference is the medium through which it travels.  For example, the velocity of water waves is with respect to undisturbed water, while sound wave velocity is measured relative to the (still) air.  Thus the fact that the speed of light is a constant implies that there is some medium (given the name ether) through which it travels and that this medium serves an absolute frame of reference for the universe.

 

Herein lies the problem.  According to Newton’s first law of motion it is not possible to distinguish between a body in uniform (non-accelerating) motion and a body at rest.  For example, if you are in an airplane experiencing a vibration free, smooth flight and the windows are closed, there is no way of telling whether you are setting on the tarmac or 40,000 feet in the air traveling at 600 miles per hour.  Suppose you clear the aisles and sprint from the tail of the plane to the front.  A friend in the plane with a stopwatch could time your run and conclude that you were running at a speed of 15 miles per hour.  This speed is, of course, with respect to the plane.  If an observer on the ground with a powerful x-ray telescope could see and time your sprint, he would conclude that your speed was 600 mph + 15 mph = 615 mph.  This is your speed with respect to the ground – your speed relative to the plane plus the speed of the plane relative to the ground; the velocities add.  Note there is no way a person in the plane could come up with the 615 mph figure because the plane is traveling at a uniform velocity and, according to Newton’s first law of motion, for an observer in the plane this is the same as not moving.

 

Now suppose that the observer on the ground radios the message that your speed relative to the ground is 615 mph.  Then you could deduce that the plane is flying through the air at 600 mph (your speed relative to the ground minus your speed relative to the plane).  So what Maxwell’s equations seemed to imply this: you always know the “ground” speed for light, i.e., the speed of light referenced to the absolute frame of the universe.  Thus, you can measure the motion of any object relative to this absolute frame by first measuring the speed of light relative to that object, then subtracting the known constant speed.  So, contrary to Newton’s law, you can tell whether you are moving or standing still.  You are standing still if you measure the speed of light and it comes out c.  Otherwise, you are moving with respect to the absolute frame of the universe.

 

In 1887 two American physicists Albert Michelson and Edward Morley attempted to measure how the velocity of light changed as the Earth moved around its orbit.  They reasoned that when the Earth was traveling with the ether, the velocity of light would be greater than the simultaneously measured velocity in a direction perpendicular to this path.  Conversely, the velocity of light would be less when the Earth was traveling against the ether.  The assumptions here is that the measured velocity of light would be equal to the constant c plus or minus the velocity of the Earth.   Michelson and Morley came to the puzzling conclusion that it seemed to make no difference in which direction the Earth was traveling.  The speed of light always came out to be c (very nearly 300,000 km/s or 186,000 miles/s).  So, maybe you couldn’t measure your absolute speed.  But, (assuming Maxwell’s equations are right) why not?

 

The Special Theory of Relativity

 

Einstein answered with the Special Theory of Relativity.  According to Einstein, the speed of light is an absolute constant.  No matter who measures it, it always comes out the same c = 300,000 km/hr.  This is assuming that the measurement is made in a system moving at a uniform velocity and is made in a vacuum.  The theory is special because it deals with the special case of uniform velocity.  The bottom line of special relativity is simply this: for light, velocities do not add.  This seems astonishing.  If sprinters down airplane aisles (the example above) followed the same rule, the observer on the ground would measure 15 miles per hour, just like the observer in the plane.  The speed of the plane would not show up, because the velocities would not add.  This is very counter-intuitive.  It had never been noticed for light before, because light is so fast compared to any moving object in our everyday world – even the 600 mph of the plane is insignificant compared to c.  The consequences are equally startling:

 

 

 

What does this equation mean?  Consider again the runner in the plane.  The guy with the stopwatch times the sprinter’s run as taking 10 seconds.  On the ground, however, the observer records a longer time by the amount given in the equation.  In a plane traveling at 600 mph, this increase is insignificant.  To show some real difference, let’s speed the plane up to 87% the speed of light.   Then if you do the math you will find that tg=2 x tm.  In other words, the observer on the ground says that the run took 20 seconds.  Since in the plane only 10 seconds have passed, seconds on the plane must be longer than seconds on the ground.  Of course if you are in the plane you can consider the plane at rest and the Earth moving.  Then if you timed an event on the ground, you would conclude that time on the ground was advancing more slowly.   Who is right?  Both are – it simply depends on which reference frame is taken as the one “at rest”. 

 

 

Note as the velocity approaches the speed of light, the length approaches zero

 

·        The mass of a moving object increases.  If m0 is the rest mass, then

 

 

Here, as the velocity approaches the speed of light, the mass goes to infinity!  This is mechanism that caps the universal speed limit at c – no way is anything going to have an infinite mass.

·        Energy and mass are equivalent:

 

 

Here E is energy, m is mass and, as always, c is the constant speed of light in a vacuum.

 

Since neither the dimensions nor the passage of time in a moving body can be specified without reference to how its position changes with time (i.e., its velocity), time and space cannot  be separate entities.  Events must be always be described as occurring in space-time, the four dimensional coordinate system of the universe.  Specifically, a distance in space-time is given by

 

 

The General Theory of Relativity

 

So Einstein had shown that it is not possible to establish an absolute frame of reference or to establish an absolute standard of time in the special case of objects moving uniformly.  What about accelerating objects?  You can sure tell that you are accelerating, because you are thrown back against the seat of the plane or car or what ever is providing the acceleration.  Also, note that gravity provides acceleration in accordance with Newton’s laws:

 

  

 

Here, m is the object being accelerated by the mutual gravitational attraction between masses m and M.  Is it possible to find an absolute frame of reference in a gravitational field?  Einstein answered “No!”

First, he postulated the equivalence of gravitational fields and accelerations.  Suppose you are out in interstellar space.  With no engines on, you are weightless, since there is no gravitational pull anywhere near.  Now, suppose someone fires the rocket engines.  You will be pressed to the floor because it is accelerating (and now, so are you).  It is possible to adjust the rocket blast so that the ship accelerates with exactly the same acceleration it would experience if it fell to the surface of the Earth.  Einstein maintained that under these circumstances it would be impossible to tell whether you were out in space or on the ground.  This is known as the principle of equivalence – accelerations are indistinguishable from gravitational fields.  Because of this principle, you can describe the behavior of accelerating objects and extrapolate that behavior to gravitational fields. 

 

Einstein’s great insight was that a body falling through a gravitational field, even though accelerating, experiences no forces.  Thus, falling through a gravitational field is the same as being weightless, the same as being far out in space with no large masses around, the same as moving with a uniform velocity.  If the windows were closed, there would be no way for an astronaut in a freely falling space ship to determine that the ship was moving.  Again, there is no absolute frame of reference, not even in a gravitational field.  (This strictly true only if the dimensions of the body are small compared to the mass that produces the gravitational field.  If this is not true the situation is more complicated, but the principle still holds)

 

Consider an astronaut in a windowless spaceship drifting at a constant velocity.  There is no way from the inside of the ship to determine the vehicle’s velocity.  Now suppose the ship drifts into a planet’s gravitational field.  Still, no force is felt as long as the ship is falling freely, still no way to detect motion.  However, sooner or later the ship will crash into the planet.  What happened?  According to Newton, the ship was pulled on to the planet by the force of gravity, but according to Einstein, the ship was following the shortest path through space-time and space-time around a massive body is curved.  So Einstein rejects the concept of gravity as Newton’s action-at-a-distance attractive force and replaces it with the idea that a mass warps (curves) the fabric of space-time.

 

Some consequences of General Relativity:

 

 

So when we say that the universe is expanding, what exactly is it that expands?  It is the fabric of space-time.  Galaxies are localizations of energy (in the form of mass) that warp space-time.  The expansion of the universe is the increase of space-time between these localizations.