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Special and General Relativity

Historical Background

    Ever since Isaac Newton, science had a success not previously seen.  Newton's gravitational theory gave an accurate account for practically all the macroscopic and massive objects on Earth and on the universe.  But that was not the whole of science.  Many fields were still developing within science.  Since 1808, with John Dalton, atomic theory started being considered a serious proposal to explain phenomena in chemistry.   Other studies had more to do with electricity and magnetism.

    One of the greatest discoveries in the XIXth century was the of the relation between electricity and magnetism.  This relation was discovered by Hans Christian Østerd (1777-1851) and developed by Michael Faraday (1791-1867) and Joseph Henry (1797-1878) with their famous invention:  the electric generator.   This relation came to be known as electromagnetic energy.  James C. Maxwell, in 1860, formulated his own theory on electromagnetism, in which he explained all the laws of electricity and magnetism in four equations known today as Maxwell equations.  One of the most amazing predictions of this theory is the existence of electromagnetic waves which could travel at the speed of 3.00 x 108 m/s (in other words 300 000 000 m/s, or 300,000 km/s).

    Before this prediction, there was a physical phenomenon that traveled almost at that speed:  light.  In 1676 Olaf Roemer (1644-1710) wanted to measure the speed of light using one of the moons of Jupiter, but the most exact value was obtained by Albert A. Michelson in 1879, whose result was 2.99729 ± 0.00004 × 108 m/s.  This value was very close to the results given by Maxwell equations.  Light was, in the eyes of the scientists, an electromagnetic energy.  Later, in 1888, Heinrich Hertz produced electromagnetic waves that seemed to imply beyond doubt that light was an electromagnetic wave.

    Maxwell's equations also implied that these electromagnetic waves didn't need any means for travel.  For example, we know that sound waves need air to travel, if there is no air, there is no sound.  Why wouldn't light have some means for travel?  Some scientists formulated the theory that light propagated through what they called luminopherrous ether, or simply ether.  This ether strange properties like to be transparent to light, of being of undefined density, but rigid like steel.   This "substance" or "electromagnetic means" turned out to be cosmologist's absolute reference frame.  That meant that any kind of velocity in the universe would be measured having ether as its criteria.  For scientists, the speed of electromagnetic waves or of light in this absolute reference frame was always the same.  According to Galilean transformations (equations measuring the velocity of objects with respect to others) and Newton's theory, the laws of motion are the same for any inertial reference frame (any object that moves).

    But, in spite of these views, one question remains, do electromagnetic waves obey Galilean transformations?

 

Michelson-Morley Experiment and the Lorentz Transformations

    Michelson, the scientist that measured light speed accurately, and E. M. Morley carried out an experiment in 1887 which refuted the ether theory.  This was a disappointment for many scientists because that meant that there was no absolute reference frame.

    However, it was also a surprise, because the Michelson-Morley results indicated that the speed of light is always constant independently of the inertial reference frame where it is measured.  What does this mean?  OK...   Let's suppose you are in Car A, and that there is a Car B traveling besides you.  You are going at 10 m/s while Car B goes at 15 m/s.  If we use Galilean transformations we can ask how far ahead is Car B traveling with respect to your car.  We can find out simply this way:

Velocity Car B - Velocity of Car A = Velocity which Car B travels with respect to Car A

15 m/s  - 10 m/s = 5 m/s

That means simply that with respect to your car, car B is going ahead of you at 5 m/s.  Very simple enough, right?

    Now, let's suppose that you are in Car A traveling at 250,000 km/s, and that there is a beam of light traveling beside you at 300,000 km/s.  Obviously our first impression to know how far ahead is the beam of light traveling ahead of us we apply Galilean transformations:

Velocity of Light - Velocity of Car A = Velocity which Light travels with respect to Car A

300,000 km/s - 250,000 km/s = 50,000 km/s

So you think that light travels 50,000 km/s with respect to your reference frame... right?   WRONG!   Because what Michelson-Morley experiment showed is that doesn't matter at what velocity you are traveling, or even at which direction you are traveling, when you measure light's speed it will ALWAYS be 300,000 km/s.   Astonishing  ... right?  Light's speed is always constant independently of the inertial reference frame (velocity which we are moving) where it is measured.  This came to be known as the principle of the constancy of the speed of light.  Galilean transformations could not be applied to electromagnetism.

    After these unexpected results, in 1890, H. A. Lorentz developed transformations different taking into account the Michelson-Morley results.  These transformations were compatible with Galilean ones, as long as it is applied to objects that don't move close to light's speed.  These transformations were the basis for Einstein's revolutionary findings and his formulation of his theories of relativity.

 

Einstein's Special Theory of Relativity

    In 1905, Albert Einstein (1879-1955), who was then a patent clerk, wrote an article called "On Electrodynamics of Moving Bodies" in his paper where he made several statements concerning the relation of the movement of objects at high speeds.  This theory was based in two postulates:

First Postulate:  The principle of equivalence =  The laws of physics are always the same for any inertial reference frame (for any object in motion).

Second Postulate:  The principle of the constancy of the speed of light:  The speed of light in an empty space is always constant (c = 300,000 km/s) and is independent of relative movement of inertial systems (objects in motion), its source and the observer.

From these postulates, Einstein elaborated Lorentz's equations.  What will  happen to the length of the object that travels in a straight line.  Let us define the ends of two rods.  Rod L has its ends "x1" y "x2" and the ends of rod L' are "x1'" y "x2'".  The length of L and L' would be the same.  We can establish then these equations as true:

L= x2 - x1        and        L' =  x2' - x1'

Let's suppose that L' travels at a speed close to light's speed, while L is resting.  Using Lorentz equations, Einstein could reach this conclusion:

Equation on Length

or

Equation of Length   where gamma

What does this equation mean?  It means that as long as L' travels at a speed closer to the speed of light, its length would appear to contract with respect to L and vice versa.  From here, Einstein arrived to the following conclusion.  The measure of an object is always at it maximum when it is measured by an observer which is at rest with respect with that object, and that this object will appear to contract by a "g" factor to an observer that moves in relation to the object, and viceversa.

    Einstein discovers that also time is affected by movement.  If by T we represent the change of time for the subject resting, and T' to represent the time of an object moving, we will arrive to the following equation:

Equation on Time

This equation means that time will pass more rapidly when it is at rest with respect to an observer, and will appear slower by a "g" factor when it is moving with respect to the observer, and viceversa.  This gave way to what is called today the twin paradox:  Let's suppose that twins are born and then one stays on Earth, while the other goes to space in a rocket at a speed close to the speed of light and comes back again to Earth; when the second twin returns he will be a lot younger than the one that stayed here on Earth.

    Velocity is an important factor for momentum (the product of velocity and mass), and we could then study the effects of velocity on mass (using the equation p = m v, where "p" is momentum, "m" is mass and "v" is velocity).  We could establish two equations of the momentum of the observer resting p=mv, and the momentum of the object moving  p' = m' v.  We could use the principle of the conservation of mass with Lorentz's equations, and Einstein arrived to this one:

Equation on Mass

From this, Einstein deduced that mass would always remain at its minimum when the observer is resting, and that it will augment when it is closer to the speed of light.

    Now, momentum is an essential element to understand force, according to the equation already established in physics F=p/t.  According to this equation, Einstein could define force in motion this way:

Equation on Force  where D is the change of the product of g and v

According to the consequences of this equation, energy (E), should be defined this way:

E = mc2

This is perhaps the best known equation of Einstein which established the energy-mass equivalence principle:  Mass can transform into energy and energy to mass, but neither can be created or destroyed.

    Einstein knew that Galilean transformations were not refuted, they were just simply corrected.  For him, the Lorentz transformations and all his deductions apply only to objects that are moving relatively fast to light speed, or are moving at a velocity close to light speed.  Therefore, his theory was called special theory of relativity.

 

General Theory of Relativity

    The special theory of relativity changed the view of classic Newtonian mechanics.  From this moment on scientists had to take into account that Newtonian laws could not be applied to all events, but only those objects that are not traveling relatively fast with respect to light's speed.  There is no absolute reference frame, everything had to be measured to moving objects.  Space and time there are not two separate objects, but one sole entity.  Space-time didn't have three dimensions, but four (length, height, width, time).  Depending on the velocity of the object, time is also altered with respect to the observer.

    Now, according to Einstein's and Lorentz's results, nothing could travel faster than the speed of light.  This conflicted with Newton's gravitational theory.  According to him, the gravitational mass of one object affects another instantaneously, at an infinite speed.  How can there be an infinite speed, if the speed of light is the limit?

    Albert Einstein, who believed in the harmony of the universe, developed how gravitational force "really" affects objects.  He formulated his general theory of relativity in 1915.

    He studied the equivalence of gravitational force with other types of forces, like for example, inertial and centrifugal forces.  Let's suppose for instance that you are at an elevator that is still.  You'll notice that everything around you is having normal gravitation normal.  You are obviously pulled by gravitation to the floor, and if you drop the apple, it will accelerate down at 9.81 m/s2. Now let's suppose that this elevator is in empty space (with no gravitation around you) and then it starts accelerating at 9.81 m/s2 upward, you will experience the same thing as if the elevator was on Earth being still, as if you had gravitation.  You will obviously be pulled to the floor, and if you drop an apple it will accelerate downward at 9.81 m/s2.  Einstein called these seemingly equivalent forces to gravitation as "fictitious forces."  For Einstein, inertial forces and gravitational forces are equivalent.

    Einstein also wished to extend this principle to observers, doesn't matter the velocity which they are traveling.  For this, he needed to adopt non-Euclidean geometry to apply it to space-time.  He also used Mach's principle:   all inertial forces are due to the distribution of matter in the universe.  This notion that a distant mass would determine the inertial forces of an object, pointed to the possibility of coincidence between non-Euclidean geometry and matter.

    If there is an equivalence between inertia and gravitation, then we can talk too about the equivalence between gravitation and geometry, specially when we refer to curved surfaces.  Let's imagine a smooth rubber surface and we put on it a heavy sphere.  The surface would then deform or distort creating a curved surface surrounding it.  If we throw a marble to the curved surface created by the sphere, it will change its direction due to the distortion.  It would move hyperbolically, parabolically or elliptically.  In our little experiment it is obvious that the marble wouldn't be trapped in the elliptical orbit.  Because of the friction of the rubber, the marble would eventually fall into the curved surface.  However, in space-time there is no friction.

    Let us substitute then the heavy sphere with the sun, and the marble with the Earth.  We could then understand why the Earth is on an elliptical orbit around the sun.  The sun creates a curvature in space-time and the Earth is trapped in an elliptical orbit around the sun because it is on the curved space-time created by the sun. Under this notion of space-time, the gravitational effects from one object to another never exceeds the speed of light, because the distortion of space-time wouldn't travel at a higher speed than light's..  This seems odd enough, but a theory has to be put to the test.

    The first experiment was carried out to find out if eventually space-time is curved around the sun.  Einstein proposed in 1911 that his general theory of relativity implied that due to the curvature of space-time, light would change direction around massive objects.  This experiment was verified on May 29, 1919 in South America and Asia.  Two British expeditions confirmed during a solar eclipse that a certain star appeared to be out of its position when they saw its light close to the sun.  The only explanation for this apparent position was that the light of the star was deviated due to the distortion of space-time caused by the sun mass.

    Also general relativity could predict something that Newtonian mechanics could not explain.  It was a known fact by astronomers that Mercury's orbit was strange, because there is a shift in Mercury's perihelion.  General relativity predicted precisely that a body like Mercury, close to a massive object like the sun, could experience a shift in the perihelion, something which Newton never could predict with his theory of gravity.

    Another prediction made by general relativity was also the gravitational red shift.  According to relativity, any radiation that escapes a body, like a star or planet, loses energy because the pull of gravitational is less and the wavelength increases.  A beam of light that falls into a body gains energy because there is more gravitational force and due to this its wavelength increases.  This is a direct consequence of the principle of equivalence of inertia and gravitation.  Let's suppose that a scientist is in his lab, which has two holes, one on the floor and one in the ceiling.  Let's suppose to that a beam of light that enters the lab from the floor to the ceiling.  When the gravitational effect is abolished, he observes that the ray of light never changes its wavelength.  But if we take into account the acceleration of the lab, he will see that the velocity of light increases as the ray of light shines from the floor to the ceiling.  Therefore, the wavelength decreases because of the acceleration of the lab and has a constant value for the scientist that is inside of it.  From this it follows that any observer standing outside on the gravitational body, the wavelength increases when the light travels upward.  Therefore, if we register in space a beam of light that comes out of the surface, the specter light would be shifting to red; if they are registered from Earth, its frequency would increase and the light specter would be closer to blue.  This was confirmed by the discovery of Joseph Taylor and Russel Hule of Massachusetts University, in which they could study the gravitational effect on radiation in a pulsar.

    Another corroboration of this is the second twin paradox.  If frequency decreases as the beam of light goes away from Earth, and the formula for frequency is f = n/t, where "n" is the number of waves and "t" is time, if "n" doesn't vary, then time must vary.  Therefore, time would be pass faster when we are further away from the center of gravitation, than when we are closer to it.  In 1962, an experiment was carried out to verify this.  A pair of clocks were used to install them on a superior and inferior part of a deposit of water.  It was found that in the inferior part (the one closest to the center of the Earth) time passed slower, than the one at the top, just as general relativity predicted.

    Because of these remarkable discoveries, contemporary cosmology is founded in special and general relativity.

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