Introduction:

        The Special Theory of Relativity was published in 1905. It was one of the three great concepts put forth by Einstein in that year. The others included the Theory of the Photoelectric Effect and Brownian Movement. Note that Einstein was never awarded the Nobel Prize for any Relativity Theory. They have been very controversial. (He received the Nobel Prize for his work on the Photoelectric Effect.) It is interesting that in his divorce decree he agreed to give his first (ex) wife any monies he might win should he be awarded the Nobel Prize.

        The Special Theory of Relativity describes the motion of one reference frame relative to another at a constant speed. The main contribution of this theory was not the discovery of a new set of equations that described relativistic motion. Indeed, the Lorentz Transformations were already in place. Rather, the Special Theory set forth the notion of Space-Time and showed that time, length and mass were not absolute quantities. It more or less helped to interpret existing known facts. Motion at a uniform speed  may not always be the case, so a more general theory was developed and published in 1916. The basic ideas of the General Theory are even farther reaching than the limited Special Theory. They give us an entirely new perspective of the universe, completely unlike anything we may have experienced before.

Twin Paradox:

            Look back at the Special Theory. Suppose twins grow up on the Earth. They are identical, so that even their hearts beat at exactly the same rate. Now, allow one to take off in a rocket ship flying near the speed of light. The twin in the space ship ages according to his or her heartbeat, let us say 72 times per minute. Now, the twin on the Earth sees that the moving twin's heart beats "slow" (moving clocks run slow.)  So the moving twin ages less than the twin on the Earth. The twin on the space ship comes back after ten years and finds his or her twin may have aged 25 years! But, wait a minute. According to the twin in the space ship, the coordinate in the space ship is stationary (there is no absolute reference frame) and the twin on the Earth is moving relative to the space ship. So the Earth bound twin's heart beats slower! Who ages more rapidly? The Special Theory of Relativity gave us this paradox. The Special Theory does not consider accelerated reference frames, only ones that move at constant speed relative to each other. This is a classical problem, called the twin paradox, and Einstein gave us the answer. It has to do with acceleration. The twin in the space ship will age less than the twin on earth because of the effects of acceleration. Let us consider the effects of acceleration and what it means.
 
 

        Suppose we were to perform an experiment in an enclosed laboratory with no windows. In fact, we could not tell anything about what happens outside of our laboratory. Now, we can drop a ball or toss it and predict the trajectory due to the influence of gravity. The acceleration due to gravity is commonly recognized as 9.8 m/s²  on the surface of the Earth. What if instead of leaving the lab on the surface of the Earth and instead were to accelerate in some enclosed laboratory at that same rate? It would experience the acceleration of 9.8 m/s and so would the ball that we predicted the trajectory thereof. Since we could not look outside to see if we were being accelerated or if we were motionless on the surface of the Earth, we would not know the difference. In fact, as Einstein pointed out to us, the effects on a mass are the same. This is called the Principle of Equivalence. There is no difference between the inertial and gravitational masses. The trajectories of the ball would be the same in both cases.

Equivalence of Matter and Energy:

        This idea can be traced back to Newton where conceptually the first problems with Newton's Theory of Gravitation surfaced. We observe that all objects fall at the same rate, regardless of their masses. Galileo dropped big and small objects to verify this. Newton's Theory (Equation 5-1) tells us that the Earth attracts more massive objects with a greater force than the smaller ones. At the same time his Law of Inertia suggests that it requires the extra force to overcome the greater inertia of the more massive object. We see here that we really need to consider a gravitational mass and- an inertial mass which are indistinguishable.

        If you are in free fall, the lab and all the objects in it fall at the same rate. Suppose we were in an elevator and were measuring our weight on a bathroom scale. Before we moved, the scale would read our normal weight. When we started upwards, accelerating, we would measure an increased weight. Once we hit a specific. speed and were not accelerating any more, the scale would read our normal weight. As the elevator slowed up (acceleration in the downwards direction) the scale would read something less than normal. At rest we read our normal weight; there is no acceleration. Now, we start down; we again experience a downwards acceleration, and the weight is less than normal. In fact, we may feel light in our stomaches. Suppose the cables snap and we undergo free fall. What does the scale read? Nothing! We are subjected to the same effects as the scale and the elevator. We "float" inside the elevator (until that very sudden impact) just like the astronauts do in the space shuttle. We really can't tell if we are in a gravitational field, or we are being accelerated.
 

        According to the Principle of Equivalence, we can eliminate the need for the concept of weight and gravity all together. The Greeks had said that natural motion is circular. Newton said that natural motion is in a straight line unless an external force casues the motion to deviate from it. Additionally, a force causes an acceleration directly proportional to the the force. The constant of proportionality, or the inertia effect, is the mass. Einstein said that "free fall" is natural. He depended heavily on geometry. He pointed out that we need to examine the path of natural motion. He said :"Physics after all must make use of geometry in the establishment of its concepts; the empirical content of geometry can be stated and tested only in the framework of the whole of physics."

        Einstein's  Thoery provided a perspective that showed us that not only are gravity and acceleration equivalent, but that the effect of gravity is a natural consequence of the geometry of the universe rather than a force as Newton treated it. From this perspective, gravity essentially has a cause (not evident in Newton's treatment) rather than just an effect. i.e., philosophically, in comes closer to telling us why we see the effect of gravity rather than just how it acts.

        What was more important to Einstein than the forces were in fact the courses that objects followed. His was an entirely new way of looking at the universe. We should keep in mind from the very beginning that when we describe our world at the classical level, the one we experience every day, we need only classical physics to do it. Newton's Laws and the theories of Electrodynamics work very well. They can get us around the solar system and provide us, in a very practical sense with the standard of living we enjoy today. But on the quantum level, the same theories fail miserably. We need the quantum theory at the microscopic level. And the quantum theory does not help us at the classical level. The Principles of Correspondence and Complementarity help us understand this. At the Universal Scale, the very, very large scale of the universe itself, where galaxies are our particles rather than Galileo's spheres, we need the Theory of Relativity. It does not help to talk classically about forces, but instead we need courses to describe the natural motion of objects.

        Recall if you will, that Einstein pointed out to us the equivalance of matter and energy (in Einstein's own words) in the Special Theory of Relativity. He adds now, that the distribution of matter-energy in the universe actually warps the courses that objects follow in space-time. This is something like putting a grid on your bedsheet and constraining objects to follow those lines. Now, if we drop a bowling ball on the bedsheet we observe that the sheet is somewhat distorted. Yet, if we still constrain the objects to travel on the courselines, (you might have to imagine that you have a plaid bedsheet with "straight lines on it, distorted or warped by the weight of the bowling ball)  the paths are distorted too. This is in fact equivalent to Newton's idea that object move in straight lines and the masses cause forces to disturb the original paths of the objects. Both descriptions are equivalent.

Warped Space-Time
 
 

         Newton was limited because he only worked with Euclidean geometry. This is essentially what we would call a flat space. In 2 dimensions it would be a flat paper. In three, it would be described by standard Cartesian Coordinate systems. Admittedly, it is difficult for the reader to extend a four dimensional space onto a 2-dimensional paper. What has more value is the comparison of features of the different geometric systems. But the real world has other possible geometries. It is difficult to represent three dimensional space in two dimensions, but we still attempt to demonstrate some effects. Consider the Euclidean Flat Space.  On it, the three angles of a triangle must add up to 180^o as shown below. Additionally, two parallel lines drawn outward must remain equidistant distance apart and never meet or diverge. This is the same kind of geometry we studied in 9th or 10th grade. It describes reasonably well, the everyday, normal experiences we observe.

        The world may in fact, be one of several geometries. We know now (although there is still a Flat Earth Society) that the earth is somewhat round. It is not exactly spherical, being somewhat oblate, but that is a pretty good model. However, when we are close to it we can see only a flat horizon. Our normal experiences do not permit us to discern the difference.  It wasn't until after Newton's time that other geometries had been developed. in 1829 Nikolai Lobachevski (a Russian Mathematician) showed in fact that another kind of geometry can exist. It is called a hyperbolic system and looks something like a horse saddle. on this kind of surface (again, remember that we are using a two dimensional representation for the space that is really three dimensions in space and one in time, or four dimensions. So the Geometry is not so simple.) we can see that if we were to draw a triangle that the sum of the angles is in fact not 180 , but something less than that. If two lines were drawn parallel to each other in some local area, as they moved out in space (space-time for the real universe) they actually would not remain the same distance apart, but would diverge!

In 1854 George Riemann showed the world how to do spherical geometry. Draw a triangle on this surface and we see that here the sum of the angles actually exceeds the 180° that was a necessary condition on a flat surface. If two lines are drawn parallel, as they move away, they do not remain the same distance apart, but converge! In an extreme example, consider two places on the equator of the earth, or some other spherical surface. Let these places be separated by one-fourth of the circumference. Allow individuals at these locations walk directly northward, being sure to keep at right angles with the equator. They intersect at the north pole, and the sum of the angles of this surface triangle is actually 270 . In general, we can say for spherical geometries, the sum of the angles of a triangle exceeds 180^o .
 
 

        If we examine these three geometries, we should observe that the spherical geometry is "closed" because two parallel lines would converge. The other two kinds of geometries are called "open" because the parallel lines would not converge. We need to decide which of these geometries is appropriate to describe the universe. This is one of the major tasks in general relativity and cosmology. One needs to understand from the very beginning that a description of the world is typically in terms of a model. If a model is any good, one should be able to adequately describe events and be able to make realistic predictions about related phenomena. One should be cautioned that the model is not really the world. It only describes the world. The world is what it is. Remember that now we are trying to describe space-time, not just the kind of space we have always grown up with. The description must be appropriate too.

        Consider the quantity, space-time. We saw from the Special Theory of Relativity that one of the consequences of simultaneous events was the fact that an event must be described in terms of a four dimensional world. This is a profound statement, that we do not independently have space or time, but that it is a continuum. Think of what it means to be an observer is space-time. If another reference frame moves at nearly the speed of light the length is contracted to a small amount and the time dilated to a very large value. To help describe the world let us look at something called a Space-Time Diagram. It is a two dimensional map of the four dimensional space-time.

         Look at the moon. What do you see? You see how it was about 1 second ago. Look at the sun. You see how it looked eight minutes ago. You are always looking back, not just in time or space, but in space-time. As we look at a remote galaxy we do not see how it is, but how it was thousands, or millions, or even billions of years ago.

        A line on the Space-Time Diagram is called a World Line and an Event is a Point  on the World Line, in spacetime. Light travels in a straight line on the diagram at 45 . The future is above, constrained by the lines of light and the past is below, again constrained to be within the light cone. This idea was first presented by Minkowski at the turn of the century (about 1900). Events on, this world line are said to be causally connected.

        In the Theory of General Relativity, the distribution of matter-energy determines the geometry of the world. A massive object produces curvature in the geometry, at least in a localized area. This is like accelerated motion in Mewton's world. Try not to get stuck on the classical ideas of forces and the like. Remember that they are only concepts that help us explain how things happen, not why. They are actually no better in describing the world than other models that give the same answers.

Simplicity and Power

        The basis of this theory is embodied in viewing the universe as a 4 dimensional Space-Time (3 dimensions of space and one time dimension which goes from past, through the present, and into the future) so that events are described with all four coordinates. The effect of combining Space-Time with the Principle of Equivalence is to arrive at a set of ten equations that can be simplified in one subtle, elegant statement which comes from a statment of Conservation of Energy:

        Note that the second term, the Gravitatiobnal Potential Energy of a system,  is negative. This means it is "bounded" or that energy would have to be added (it could be a form of Kinetic Energy or energy of motion) for an object to escape a gravitational field. But by Hubble's Law

M = r times Volume or we can say that:

 

rcrit    = 3 H2 / 8p G

Supporting Evidence:

    From the equation of General Relativity a wide range of physical phenomena can be explained and predicted.

Light Deflected by a Massive object

        The first prediction put to test was the apparent bending of light as it passes near a massive body. This effect was conclusively observed during the solar eclipse of 1919, when the Sun was silhouetted against the Hyades star cluster, for which the positions were well known. 

        Sir Arthur Eddington stationed himself on an island off the western coast of Africa and sent another group of British scientists to Brazil. Their measurements of several  of the stars in the cluster showed that the light from these stars was indeed bent as it grazed the Sun, by the exact amount of Einstein's predictions.  (Deflection of Light can be as much as 1.75 arc seconds for light which just grazes the sun.) Einstein became a celebrity overnight when the results were announced.

        The apparent displacement of light results from the warping of space in the vicinity of the massive object through which light travels. The light never changes course, but merely follows the curvature of space. Astronomers now refer to this displacement of light as gravitational lensing. But the Sun's gravity is relatively weak compared with what's out there in the depths of space. In the dramatic example of gravitational lensing below, the light from a quasar (a young, distant galaxy that emits prodigious amounts of radio energy) 8 billion light years away is bent round by the gravity of a closer galaxy that's "only" 400 million light years distant from Earth.
 

Light is "Slowed" in Space-Time

Not only is light deflected as it passes near a massive object, but because the actual path of radiation is in fact longer than the "straight-line" distance in the absence of Space-Time curvature, it takes longer to reach us. In 1966, Irwin Shapiro measured a relativistic time delay from radar signals bounced off of mercury and Venus. Signals from the Viking spacecraft were delayed about 200 microseconds (.2 seconds) when passing the sun to reach Earth.
 
 

Gravitational Redshift

        According to General Relativity, the wavelength of light (or any other form of electromagnetic radiation) passing through a gravitational field will be shifted towards redder regions of the spectrum. To understand this gravitational redshift, think of a baseball hit high into the air, slowing as it climbs. Einstein's theory says that as a photon fights its way out of a gravitational field, it
loses energy and its color reddens. Gravitational redshifts have been observed in diverse settings.

Earthbound Redshift

        In 1960, Robert V. Pound and Glen A. Rebka demonstrated that a beam of very high energy gamma rays was ever so slightly  redshifted as it climbed out of Earth's gravity and up an elevator shaft in the Jefferson Tower physics building at Harvard  University. The redshift predicted by Einstein's Field Equations for the 74 ft. tall tower was but two parts in a thousand trillion. The gravitational redshift detected came within ten percent of the computed value. Quite a feat!

Solar Redshift

        In the 1960s, a team at Princeton University measured the redshift of sunlight. Though small, given the Sun's mass and density, the redshift matched Einstein's prediction very closely.

White Dwarf Redshift

        Take a star like the white dwarf star, Sirius B that is 61,000 times denser than the Sun. Its gravitational field is correspondingly much stronger and so is the redshift for the light it emits: 30 times greater, according to earlier observations from the Mount Wilson Observatory taken by W.S. Adams way back in 1924. Still larger redshifts have more recently been detected in studies of so-called neutron stars -- collapsed stars that are even denser. What about the redshift caused by a black hole? It can be thought of as infinite. In other words, photons inside the hole are so redshifted they can never get out!
 

Precession of the Perihelion of Mercury's Orbit

        In another test, the theory explained slight alterations in the  perihelion of Mercury's orbit around the Sun. The Perihelion is the point in the elliptical orbit that most closely approaches the sun. Since almost two centuries earlier astronomers had been aware of a small flaw in Mercury's orbit around the Sun, as predicted by Newton's laws. The Perihelion is the point in the elliptical orbit that most closely approaches the sun. As the closest planet to the Sun, Mercury orbits a region in the solar system where spacetime is disturbed by the Sun's mass.  There is a small amount, an excess of 43 seconds of arc per century, observed by astronomers.(Gravitational influences of planets, etc. predict 531 arc seconds) Using Newtonian mechanics, scientists predicted the existence of the planet Vulcan (home of Dr. Spock for all you Trekkies) somewhere between the orbit of Mercury and the Sun. It doesn't really exist, but can only be explained, within a few percent, by the General Theory of Relativity, not by Newtonian Mechanics.Einstein's predictions exactly matched the observation.

Figure 10-11. Precession of the Perihelion of Mercury.

Gravitational Redshift

I        In1960, Rebka and Pound performed an experiment at Yale, showing that a photon acted relativisitically as if it had mass. Basically, if a photon was "dropped" in a gravitational field, it should, like a mass, gain energy. Photons travel at the speed of light, so the velocity is constant, but a shift in the frequency was actually detectedl The energy was increased.

        As light leaves a massive object, it loses energy. Since light travels at a constant speed, this loss of energy is manifested in a decreased frequency or longer wavelength. Thus we can say that it is shifted to a longer wavelength, towrds the reddish end of the spectrum, i.e., red-shifted. This effect is analagous to a doppler shift of an object receding from us. (this also gives a reddening effect.) According to the Principle of Equivalence, these effects are the same. We have looked to astronomers to verify such effects.
 
 

 Studies of high density stars such as white dwarfs (one solar mass concentrated in a volume about the size of the
Earth, abouth one-millionth that of the sun) have verified this  effect. The use of atomic clocks on the Earth
demonstrate the same delays. Black Holes are some of the most bizarre consequences of the previously
mentioned effects is the Black Hole. A black hole is an object so massive thatblight uses all of its energy trying
to escape the surface. Its frequency is shifted to zerol And the remaining energy is ( E - hf) also zero. Speculations
about the nature of these objects, although not confirmed, suggests that-our own milky way galaxy may itself, harbour a black hole of a million solar masses in its center. Several candidates for black holes exist, but since they can't be
seen, they are difficult to confirm. Only their effects can be seen.

Gravity Waves

        If the Theory of Relativity is correct, it is possible to observe gravity waves just like light waves. They would be ripples in the curvature of our space-time system that propogates outward at the speed of light (like all other waves.) Since the distribution of matter-energy do in fact determine the curvature of space, any disturbance in this system should theoretically result in a "gravity" wave. Such effects are very difficult to observe because they are so weak. The best candidate for this would be a Super Nova explosion. These are not common events, but February 23, 1987, SN1987a was observed in the LMC (Large Magellenic Cloud), a companion galaxy to our Milky Way, about 163,000 light years distant. This was the first super nova seen with the naked eye since Kepler's of 1604. See Figure 10-13.

 Summary:

The major difference between the classical picture of the world and Einstein's is that Newton stresses FORCES while Einstein stresses COURSES. Both models are appropriate to the scale of the world they are trying to describe. Neithert works outside their own realm. We will some day find out that Relativity is probably not complete in itself, but it is a useful and reasonably accurate description of the universe on the large scale. We turn our attention now to the application of the General Theory of Relativity in dealing with the origin of, evolution and ultimate destiny of the universe itself. We call this interesting branch of science, COSMOLOGY.

Tiwn Paradox

Acceleration of reference frames

Principle of Equivalence

Space-Time

"Proofs" of The General Theory

QUESTIONS

1. Explain what is meant by the Principle of Equivalence.

2. What will happen to the Universe if it is Open?

3. What will happen to the Universe if it is Closed?

4. What will happen to the Universe if it is Flat?

5. What is the "critical mass?"

Dear Sir:

        I am writing in response to your request for additional information. In block #3 of the accident reporting form, I put "poor planning" as the cause of my accident. You said in your letter that I 'should explain more fully and I trust that the following details will be sufficient. I am a bricklayer by trade. On the day of the accident, I was working alone on the roof of a new six-story building. When I completed my work, I discovered that I had about 500 pounds of brick left over. Rather than carry the bricks down by hand, I decided to lower them in a barrel by using a pulley which, fortunately, was attached to the side of the building at the sixth floor.

        Securing the rope at ground level, I went up to the roof, swung the barrel out and loaded the brick into it. Then I went back to the ground and untied the rope, holding it tightly to insure a slow descent of the 500 pounds of bricks. You will note in block #11 of the accident reporting form that I weigh 135 pounds. Due to my surprise at being jerked off the ground so suddenly, - I lost my presence of mind and forgot to let go of the rope. Needless to say, I proceeded at a rather rapid rate up the side of the building. In the vicinity of the third floor, I met the barrel coming down. This explains the fractured skull and broken collarbone.

        Slowed only slightly, I continued my rapid ascent, not stopping until the fingers of my right hand were two knuckles deep into the pulley. Fortunately, by this time I had regained my presence of mind and was able to hold tightly to the rope in spite of my pain. At approximately the same time, however, the barrel of bricks hit the ground - and the bottom fell out of the barrel. Devoid of the weight of the bricks, the barrel now weighed approximately 50 pounds.
I refer you again to my weight in block #11. As you might imagine, I began a rapid descent down the side of the building.

In the vicinity of the third foor, I met the barrel coming up. This accounts for the two fractured ankles and the lacerations of my legs and lower body. The encounter with the barrel slowed me enough to lessen my injuries when I fell onto the pile of bricks and, fortunately, only three vertebrae were cracked.

        I am sorry to report, however, that as I lay there on the bricks, in pain, unable to stand, and watching the empty barrel six stories above me, I again lost my presence of mind - I LET GO OF THE ROPE!