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# Einstein’s Universe Final Exam

What is “special” about special relativity? What is the most general principle of relativity?
-It’s special in the sense that it is limited to the special case of uniform motion.

-The most general principle of relativity would be that the laws of physics are the same in all frames of reference.

Newton’s Theory of Gravity
Newton’s law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

What are two reasons that Newton’s Theory of Gravity is not consistent with special relativity?
-It is ambiguous as to which distance between two objects to use in the mathematical expression of the law of gravity. Since there is relative motion between the two objects, the distance between the objects will be different as measured in the two reference frames of these objects. That will give two different values of the gravitational force. Which is correct?

-Another reason it is not consistent with special relativity is that no information can be transferred at a speed greater than the speed of light (in a vacuum). But that mathematics of Newton’s Theory of Gravity predicts that the gravitational force is propagated instantaneously: the Moon instantly knows if there is a change in the distance between it and the Earth. How can anything or any information be transmitted at infinite speed?

What astronomical observation existed in Einstein’s day that Newton’s Theory of Gravity was not correctly explaining?
-There was very little such experimental evidence. The only observation that Newton’s Theory of Gravity could not explain involved the motion of the planet Mercury.

Did the astronomical observation represent a large or small discrepancy between observation and Newton’s Theory of Gravity?
This was a very small discrepancy. Only very careful astronomical observations detected it.

Was the discrepancy enough to convince physicists that Newton’s Theory of Gravity was not the final correct theory of gravity?
No. Some thought that some other body might be detected that exerted a gravitational influence on Mercury producing the unpredicted deviation from Newton’s gravitational theory. There were too many things that Newton’s Theory of Gravity explained: to abandon it meant abandoning the solutions to a large number of solutions.

Describe Galileo’s experiment involving falling objects near the surface of the Earth.
Supposedly, Galileo dropped objects of different weights from the Leaning Tower of Pisa All objects fall with the same acceleration near the surface of the Earth: 32 feet per second squared or 9.8 meters per second squared.

Define inertial mass and gravitational mass in Newton’s laws of physics.
Inertial mass is measured by applying a force to an object and producing an acceleration of that object. Take the applied force and divide it by the resulting acceleration and you get the inertial mass. This is an operational definition of inertial mass, using Newton’s Second Law of Motion, F = m a. So inertial mass is a measure of an object’s resistance to having its motion changed.

Gravitational mass is a measure of how strongly the Earth (or, by extension any planet or star) pulls on an object through the gravitational force. This is the mass that appears in the formula for Newton’s Law of Universal Gravitation: F = G m_1 m_2/r^2, where r is the distance between the two objects of mass m_1 and m_2.

What is meant by the term “weightlessness”? Astronauts in orbit around Earth in the International Space Station are said to be weightless because gravity is not present. Is this true that gravity is not present on the International Space Station or in the Space Shuttle when it orbits the Earth?
The term weightlessness applies when one is in a state of falling towards a large object like a planet or a star. That state of falling includes a satellite or spacecraft in orbit around the Earth. Though one feels as if they are weightless, as shown by a scale’s reading of 0 if one stood on it while falling, gravity is still present. If it wasn’t then the spacecraft would not stay in orbit around the Earth: it would fly off into outer space.

Is there a frame of reference one can go into that seems to eliminate gravity as Newton described it?
Yes such a frame exists: a free-fall (free-float frame) frame. This frame of reference is subject only to gravity and no forces such as electromagnetic forces or nuclear forces. For example, it is like being in the elevator that had the cable cut. You and everything in the elevator fall toward the earth with the same acceleration so everything in the elevator appears to float – as if everything was weightless.

What is the principle of equivalence?
-One statement of the principle of equivalence is that in any small freely falling reference frame anywhere in our real, gravity-endowed universe, the laws of physics must be the same as they are in an inertial reference frame in an idealized, gravity-free universe. This asserts that small, freely falling frames in the presence of gravity are equivalent to inertial frames in the absence of gravity.

-Newton achieved this in his theory (explaining Galileo’s result that all objects fall near the surface of the Earth with the same acceleration) by equating gravitational mass and inertial mass for every object.

If the inertial and gravitational mass of an object was not numerically equal, then present an argument to show that the effects of gravity and acceleration would be distinguishable.
-Consider the situation shown in the figure below, modified so that each person has two objects that they can release. If gravity did not produce identical accelerations in all objects then in the situation in the figure on the left, one ball would reach the ground before the other ball, assuming that each was dropped from the same height and released at the same time. On the right side of the figure, in the accelerating frame in the absence of gravity, if a person released two objects from the same height above the floor, both objects would collide with the floor at the same time. This can be seen by viewing the situation from a frame that observes the acceleration of the rocket. The rocket moves upwards while the two objects are at rest and eventually the rising floor collides with the two objects

-So the result of this would be that the same experiment done in the two frames shown in the figure would produce different results: in other words, the two frames of reference would be distinguishable by virtue of the different results of the same experiment done in the two frames of reference.

What is tidal gravity?
Tidal gravity is caused by differences in the strength of gravity and the direction of the pull of gravity from one place to another. Gravity gets weaker the farther you are from the center of the Earth (or Moon or Sun or any object) and it is directed towards the center of the Earth (or the center of the Moon or Sun or any roughly spherical object). So the side of the Earth closest to the Moon feels a stronger gravitational pull towards the Moon than the opposite side of the Earth; this is what causes the ocean tides and why this is called tidal gravity. The moon, instead of the sun, causes waves because the sun’s gravitational pull is so strong on the Earth that it effects Earth in its entirety, while the moon’s gravitational exertion on Earth is much smaller and thus, it only affects the ocean (as far as we can tell).

Can tidal gravity be transformed away by going into a different frame of reference?
No. This is why Einstein thought that tidal gravity was indicative of the true nature of gravity. For a tall building situated on Earth, gravity is stronger at the bottom of the building than at the top of the building, though the building is at rest on Earth. So, on the penthouse of the building, time moves faster, meaning the person would age faster.

Why are effects of acceleration are indistinguishable from effects of gravity? In other words, why does Einstein’s Theory of General Relativity become a theory of gravity?
-One of stating this is because gravity affects every object in the same way, regardless of the objects mass.

-So this comes back to the principle of equivalence, which asserts that inertial and gravitational mass are numerically equal for all bodies, results in a reference frame in free fall (a frame in which the objects within it are subject only to gravitation) is indistinguishable from a small reference frame subject to a uniform gravitational field.

If we discover some observation that takes place as a result of acceleration, what does Einstein say must happen if the same experiment is done in a uniform gravitational field?
The experiment must give the identical result if it is done in a uniform gravitational field, by virtue of the principle of equivalence.

If we discover an affect of acceleration on light, why did Einstein feel justified in extending the cause of the effect to include gravitation as well?
Because of the principle of equivalence and its statement that the laws of physics are the same in an accelerating frame of reference as they are in a frame with gravity and without acceleration. This is a specific example of what is being asked in question 1.

What is Einstein’s interpretation of gravity? Does Einstein consider gravity to be a force as Newton did?
-“The word curvature is an analogy, a visual way of extending ideas about three-dimensional space to the four dimensions of spacetime. Travelers detect curvature – in both three and four dimensions – by the gradual increase or decrease of the “distance” between “straight lines” that are initially parallel. In three space dimensions, the actual paths in space converge or diverge. Think of two travelers who start near one another at the equator of Earth and march “straight north”. Neither traveler deviates to the right or to the left, yet as they continue northward they disoer that the distance between them decreases, finally reaching zero as they arrive at the north pole. They can use this deviation to describe the curved spherical surface on which they travel. Similarly, in four-dimensional spacetime, travelers detect the deviation from parallelism of nearby worldliness of free particles, each of which follows an ideally stragith spacetime path, often called a geodesic. This curvature can be measured by travelers and varies from place to place in spacetime.” – from “Exploring Black Holes: An Introduction to General Relativity” by Edwin F. Taylor and John Archibald Wheeler.

In General Relativity what is the natural state of motion?
-The natural state of motion is called free-fall/free-float and is produced when there is no force (not electromagnetic, not nuclear) on an object. I would say only gravity acts on the object, but that is not a force in Einstein’s view, it is spacetime curvature. So the natural state of motion is produced when someone moves through spacetime, subject to no forces – this is free-fall/free-float.

-In this frame of reference the laws of special relativity hold true; the law of inertia holds true. While it may be impossible or difficult to determine if one is really in a uniform state of motion, it is one can determine if one is in a free-fall/float frame. Just determine if there are any non-gravitational forces present. Such forces do not affect all bodies equally: gravity does and that is the key to the equivalence principle.

What effect does the presence of matter have on spacetime?
Matter causes spacetime to curve. The analogy is a heavy ball resting on a trampoline located on the surface of the Earth. The ball distorts the shape of the trampoline: the trampoline is no longer flat but becomes a curved surface.

What does a “straight” line in a curved space mean? What is a geodesic?
A straight line in a curved space would be the shortest distance between two points. This is a geodesic. Note that in the case of a closed surface, like a sphere, the route from one point to another could be along the geodesic, but you can take one of two paths (since the geodesic is a closed line) one of which is shorter than the other.

How does spacetime tell matter how to move?
Spacetime tells matter to move along the shortest possible path between any two points in space. It is just that in a spacetime that is curved (by mass and energy). Since spacetime is curved the shortest distance between two points is a curved line, so matter moves along a curved line, like the Earth moving through the curved spacetime around it.

“The word curvature is an analogy, a visual way of extending ideas about three-dimensional space to the four dimensions of spacetime. Travelers detect curvature – in both three and four dimensions – by the gradual increase or decrease of the “distance” between “straight lines” that are initially parallel. In three space dimensions, the actual paths in space converge or diverge. Think of two travelers who start near one another at the equator of Earth and march “straight north”. Neither traveler deviates to the right or to the left, yet as they continue northward they disoer that the distance between them decreases, finally reaching zero as they arrive at the north pole. They can use this deviation to describe the curved spherical surface on which they travel. Similarly, in four-dimensional spacetime, travelers detect the deviation from parallelism of nearby worldliness of free particles, each of which follows an ideally stragith spacetime path, often called a geodesic. This curvature can be measured by travelers and varies from place to place in spacetime.” – from “Exploring Black Holes: An Introduction to General Relativity” by Edwin F. Taylor and John Archibald Wheeler.

When an object moves through space with no force (electromagnetic, nuclear, or friction force) on it, what path does the object follow?
It follows a geodesic. It is in a free-fall/free-float reference frame anything in such a frame follows a geodesic. Any two events that occur at the same place in several reference frames will take place with the longest separation in time in the free-fall/free-float reference frame.

How do we conclude the effect that gravity has on light? What is the role of the equivalence principle in making this conclusion?
The idea is to analyze the trajectory of light in a reference frame that is accelerating. The conclusion made in the accelerating frame must also be true in the reference frame containing the gravitational field by virtue of the equivalence principle.

Describe the 1919 astronomical observation that convinced the scientific community that Einstein’s General Theory of Relativity was accurate and that made Einstein famous among people outside of the scientific community.
Light from a distant star comes directly to Earth. When the sun and the star are in the same region of the sky, the Sun’s gravity deflects the starlight. Observers on Earth then see the star at a different apparent position in the sky. Eddington went carried out eclipse expedition at two locations.

Does Einstein’s Theory of General Relativity correctly predict the orbit of Mercury?
It does. This gave Einstein a reason to believe that his new theory of gravity was correct.

Interpret the orbit of the Earth around the Sun using General Relativity.
The Earth moves along the straightest possible path through a spacetime curved by the mass of the Sun. The Earth is in a state of free-fall/free-float since there are no forces acting on it. If the spacetime around the Earth were flat then the Earth would follow what we think of as a straight line.

What is the gravitational red shift?
When light travels away from a large mass object, such as a planet or a star, the wavelength of light undergoes a shift in wavelength: the wavelength increases, and since red light has a greater wavelength than blue light this is called a gravitational red shift.

What is gravitational time dilation?
-You can use the wavelength of light emitted by a subatomic process as a clock. This is facilitated by using a source of light that gives off a single frequency. The time is kept by counting the number of peaks of the light wave that passes by. The longer the time interval, the more peaks of the wave pass you by.

-When the light travels upwards, away from a star, the wavelength increases (the gravitational red shift). The person far from the star can compare the frequency of this gravitationally red shifted light to light emitted at her location by the same precise process. Since the light from the surface is gravitationally red shifted, the person far above the surface will say that the clock at the surface is running slow: gravity slows down the rate at which clocks keep time. The stronger the gravity the slower the clock will run.

State ways in which gravitational time dilation differs from time dilation in special relativity?
Imagine two observers, one on the surface of a planet and the other some distance above the surface of the planet. Each person will claim that their clock is running normally. But the person on the planet will say that the clock far from the planet is running fast, while the person above the surface of the planet will say that the clock on the surface of the planet is running slow. This effect is not reciprocal like time dilation in special relativity where observers in relative motion agree that clocks moving with respect to each of them run slow.

Also note the difference between gravitational time dilation and the time dilation in special relativity. The latter is that the shortest time between two events is in the frame in which the two events happen at the same place. There is only one observer (one reference frame) who sees the two events as happening at the same place. Other observers in relative motion to this observer do not record the two events as happening in the same place.

What Wolfson calls “The principle of cosmic laziness” on page 197 refers to a pair of events with many observers who are in relative motion, but who are also present at both events, meaning that they all see the two events as happening in the same place, even though they are in relative motion (in different frames of reference.). The only way that one can have observers in different reference frames observing pairs of events that all agree happened at the same place is if one allows accelerating reference frames (acceleration due to a force, and remember gravity is not a force according to Einstein).

Using the equivalence between a uniform gravitational field and uniform acceleration reinterpret the Twin paradox.
For any two events with more than one observer present at the two events, the longest time interval will be measured by the observer in the free-float reference frame. All other observers in different reference frames who are present at the two events will measure shorter times. The twin who stays on Earth is in the free-float frame (neglecting the weak gravity that produces the weak normal force upwards on the person) and so measures the longest time between the events, the twin leaves Earth on the spaceship and the twin returns to Earth on the spaceship. The twin on the spaceship was not in a free-float frame so time flowed at a slower rate for them).

What observation of 1925 verified the gravitational red shift?
This was observed in the spectrum of the sun. The dark absorption lines were shifted to slightly greater wavelengths in the just the amount predicted by general relativity for light leaving a body with the mass of the Sun.

Describe the experiment done in the early 1960’s that verified gravitational red shift/gravitational time dilation.
Physicists at Harvard University studied radiation emitted by a particular species of nuclei. They had two identical experiments set up: one at the bottom of a tower and one at the top of the same tower. Each measured the same wavelength for the radiation emitted by their respective nuclei. But when the radiation emitted by the nucleus located at the bottom of the tower was sent up the tower, the person at the top of the tower measured a slightly longer wavelength: that is the gravitational red shift effect. Since the radiation constitutes a clock, the person at the top of the tower observes that time at the bottom of the tower runs slightly slower than time kept by his own clock at the top of the tower.

What is the escape velocity of an object?
The escape velocity is the smallest velocity an object can have such that it escapes from any body that is exerting a gravitational influence on it. In other words, an object leaving at the escape velocity will never “fall back down to the object”.

What distinguishes between strong gravity and weak gravity?
The escape velocity and how it compares to the speed of light. The closer the escape velocity gets to the speed of light the stronger the gravity, and globally speaking, the more curved the space is.

Is there any region of our solar system in which we find strong gravity? Where in our solar system would we find the strongest gravity?
The strongest gravity is found near the surface of the sun, but even there the escape velocity, though larger than on Earth or on any other planet of the solar system, is tiny compared to the speed of light. Thus there is no place in our solar system in which we would find strong gravity.

In what limit does Einstein’s and Newton’s theories of gravity give virtually identical results?
Einstein’s and Newton’s Theories of Gravity give near-identical results in the limit of weak gravity, or in other words, in the limit of escape velocities far below the speed of light.

Where do we find strong gravity?
Strong gravity is found outside of our solar system, near neutron stars and black holes.

What is the significance of astronomical observations for general relativity?
In space there exist denser objects that produce stronger spacetime curvature due to a greater escape velocity. It is in this regime that one finds phenomena that greatly disagree with predictions of Newtonian Gravitation. Observations can be compared to the Newton and Einstein’s theories of gravity in an effort to see which is consistent with observation.

What is gravitational lensing?
-A galaxy can distort the space around it, deflecting light coming from a further galaxy (or quasar) producing multiple images, or in highly symmetric arrangements of mass, a ring (called an Einstein Ring).

-From Wolfson: “Any massive body whose gravitation, or spacetime curvature in general relativity, bends light from more distant objects. The result may be brightened, distorted, or multiple images.”

How does gravitational lensing verify predictions made by general relativity?
Einstein used his General Theory of Relativity to predict the phenomenon of gravitational lensing in the late 1930’s. Gravitational lensing was discovered decades later and has the properties predicted by Einstein.

What is the binary pulsar?
-This is a system of two stars that are both neutron stars. Each has 50% more mass than the sun and they move in elliptical orbits. They orbit each other in 8 days as they are close to each other: at closest approach they are about half the Sun’s diameter apart. These neutron stars rotate with high speeds: one of them rotates 16.94 times every second (thought this rate is changing).

-The orbital period is changing because the neutron stars’ orbit is shrinking because the stars are losing energy as predicted by general relativity. They are losing energy through the creation of gravitational waves, ripples in spacetime, that they create by moving so quickly through spacetime.

What predications of general relativity do observations of it verify?
-The orbits of the pulsars are seen to precess at a much greater rate than the orbit of Mercury. The higher precession rate of the two pulsars is just what general relativity predicts given the masses of the pulsars and their separations.

-General Relativity also predicts the neutron stars should be losing energy as they orbit each other and this has been measured in careful observations of the binary pulsar. Where does this energy go? Supposedly into gravitational waves, but those waves have never been detected. There are currently experiments under development to detect gravity waves.

Why do we not notice the bending of light in our everyday environment?
We occupy too small a reference frame (any frame on a small enough scale looks flat) and gravity is too weak.

Would someone on top of a building age more or less than someone at the street level?
Someone on the top of a building would age at a greater rate than someone at the street level of a building. Of course on Earth, the difference in the rate of aging is very very tiny.

Why don’t we experience gravitational time dilation in any noticeable way?
Because gravity is weak on Earth, and there is little change in the strength of gravity between the surface of the Earth and the upper floors of the tallest buildings and the heights above the surface of the Earth that airplanes fly at.

What is a star?
A star is an astronomical object that has such large self-gravity that the central core is at such a high temperature that nuclear reactions take place that convert lighter elements to heavier elements, such as converting hydrogen nuclei into helium nuclei. It is the energy generated/released in the nuclear reactions that support the star against collapse due to its self gravity.

What is a galaxy?
A large collection of stars held together by their mutual gravitational attraction. Galaxies also include large clouds of gas and dust.

What is the cosmological constant? Why did Einstein introduce it into his general theory of relativity when applying general relativity to cosmology?
“A number Einstein introduced into the general theory of relativity so the theory would predict a static Universe. Einstein abandoned the cosmological constant when observations showed that the Un9iverse was in fact expanding. Discoveries at the end of the twentieth century suggest the constant might be necessary after all.” From Wolfson, page 246.

What is the Doppler effect?
The Doppler effect is the increase or decrease in the wavelength and frequency of waves when the source of the waves is moving toward or away from the observer.

What is a red shift? What is a blue shift?
When the wavelength of the waves decreases that is called a blue shift and when the wavelength of the waves increases that is called a red shift. This nomenclature is due to red light waves having a greater wavelength than blue light waves.

What is the spectrum of a star? What are the dark lines in a star’s spectrum?
The spectrum is the light emitted by the star broken down by wavelength, so that the intensity of light of each wavelength is measured. A graph of intensity vs wavelength is often drawn. The spectrum of stars, including the sun, shows that there are missing or dimmer (lower intensity) wavelengths present due to absorption of light of these wavelengths by atoms in the star’s atmosphere.

What is Hubble’s Law?
Hubble found that most of the galaxies are moving away from us, and the speed of recession is proportional to the distance from us. The farther a galaxy is from us the faster it is moving away from us. In mathematical form it is v = Ho d, where Ho is called the Hubble constant (or Hubble parameter).

What is the meant by the phrase “the expanding universe”?
-Hubble studied the spectra of galaxies and noticed that the bulk of them had spectra that were redshifted. He used the Doppler effect to interpret the results and concluded that galaxies with redshifted spectra were moving away from us.

-But the question remains are the galaxies moving away from us through a fixed space, or is the space itself between galaxies expanding, or stretching, which also produces this motion of all galaxies away from us?

-Note that some nearby galaxies were found to be blueshifted. Since the closer galaxies feel a stronger gravitational pull to our galaxy, these galaxies are moving towards us because of the local gravitational attraction with our galaxy. The question of whether the expansion will continue forever or reverse itself depends on the rate of the expansion of the Universe and the density of matter in the universe.

What connection is there between the expanding universe and general relativity?
The expanding universe refers to the expansion of space between galaxies (really that is between galaxies that belong to different clusters of galaxies). But it is general relativity that describes space as being affected by matter and energy, and as being dynamic. Thus it is general relativity that provides the basis for interpreting Hubble’s Law.

Contrast the Doppler effect as observed on Earth with sound waves and light waves, with the red shift discussed in the context of the expanding universe.
The red shift is not a Doppler effect as the galaxies are not moving through space. The light is red shifted when it travels long distances through expanding space. This is not the source of the Doppler effect.

What is the Big Bang?
-From Wolfson, page 245 – “The cosmic explosion that began the Universe.”

-Since the majority of the galaxies were moving away from us, as if we were at the center of a great explosion of matter in our Universe. This means that the farther back in time you go, the closer the galaxies, and all matter, are together. Keep going back farther in time and eventually all the matter seems to have been at the same place. At this place there was some sort of expansion or explosion that caused space to expand and carry matter away from other clumps of matter. We call this instant the Big Bang.

Does Hubble’s Law necessarily mean that as a galaxy moves further from us, the faster it will be moving?
No. The Hubble’s Law describes the cosmological expansion as it is now. The speed of recession of a galaxy might be changing as time goes on (it might be speeding up or slowing down) but that is not reflected in the current value of the Hubble constant.

Is the Hubble constant a constant in space or in time? Explain.
The Hubble constant has the same value in space but not in time. If the cosmological principle is true (the universe is homogeneous and isotropic) then we would expect the Hubble constant to be the same everywhere at any instant of time. But due to the universal attraction of gravity we would expect the expansion to be slowing down as time goes on; this would change the value of the Hubble constant as time goes on.

What is the interpretation of the Hubble constant? How can you use the Hubble constant to estimate the age of the Universe (the time since the Big Bang)?
From Hubble’s Law, we can deduce that the unit of the Hubble constant is 1/time. This means that the reciprocal of the Hubble constant has units of time. The reciprocal of the Hubble constant is called the Hubble time. If the rate of expansion of the universe were constant since the big bang then the Hubble time would be the age of the Universe, i.e., the amount of time that has passed since the Big Bang. If the rate of expansion is decreasing, due to gravitation, then the Hubble time would be an upper limit to the time since the Big Bang (an upper limit to the age of the Universe). This is because if the expansion rate was greater in the past, then a shorter amount of time has passed since the Big Bang, since less time would be needed to expand to its present size.

What is the Hubble time? Under what condition would the Hubble time be the age of the Universe?
This is the reciprocal of the Hubble constant. In a universe in which the rate of expansion is constant (so the speeds of each galaxy does not change) the Hubble time would be the amount of time that has passed since the Big Bang.

Suppose the rate of expansion of the universe was greater in the past. What would that mean for the age of the Universe?
The Universe would be younger than the Hubble Time.

How does light emitted by distant galaxies acquire a redshift?
The light gets redshifted as it moves across large distances in space. The longer distance the light travels the greater the redshift. This is different than the Doppler shift observed on Earth with, for example, radio waves emitted by speed detectors. These Doppler shifts are acquired as the wave reflects off of a car moving with respect to the source of the radio waves. There is no expansion involved in this case since the distance between the car and the source of radio waves is so small.

Most galaxies are moving away from us. So are we the center of the Big Bang?
The idea that space is expanding resolves the idea that we alone are at the center of the expanding universe, and hence occupy some special location. As the example of the expanding balloon shows, regardless of your location, you would observe that the farther something is from you, the faster it moves away from you in the manner described by Hubble’s Law (meaning a linear relation between recession speed and distance).

Is the Milky Way galaxy expanding (is the distance between stars within the galaxy increasing)? If not, what prevents it from expanding?
The Milky Way galaxy is not expanding. No individual galaxy is expanding as gravity holds the galaxy together and the dimensions of a typical galaxy (100,000 light-year diameter) are not large enough to detect the expansion. The expansion is not even detected between galaxies bound together through gravity in a cluster. The expansion is evident over the large distances between clusters of galaxies.

What were conditions like in the first few seconds after the Big Bang?
The density of matter was much higher than it is now. There was a time when there were no large structures like galaxies and stars. At earlier times there may have only been neutral atoms and at even earlier times there were only protons, neutrons and electrons, not bound together.

Independent of the expansion of the Universe, how could you form a lower limit to the age of the Universe?
The Universe must be as old as the objects contained within the Universe. So the idea is to find the oldest objects in the Universe. For example, there are Moon rocks 4 billion years old, so the age of the Universe must be at least 4 billion years. There are clusters of stars, called globular clusters, that seem to be about 11 to 14 billion years old, so the Universe must be at least this old.

What is the cosmological principle?
The cosmological principle consists of two parts: (i) the universe looks the same, on very large distance scales, regardless of the location of the observer in the Universe (this property is called homogeneity) and (ii) the universe looks the same in all directions (this property is called isotropy.)

What determines which of the three models might describe our Universe?
(1). When the average density of matter in the Universe is greater than the critical density then the Universe expands and contracts.

(2). When the average density of matter in the Universe is less than the critical density then the Universe expands forever with a non-zero limiting speed.

(3). When the average density of matter in the Universe is equal to the critical density then the Universe expands forever with a zero limiting speed.

What is the fate of the universe in the three models derived from General Relativity using the cosmological principle?
(1). When the average density of matter in the Universe is greater than the critical density then the all the matter in the Universe will return to the same location. The density and temperature of the Universe will increase and all matter will be broken down to tiny fundamental particles.

(2). When the average density of matter in the Universe is less than the critical density then all of the matter in the Universe will cool down and get farther apart. The average density of matter will continue to decrease.

(3). When the average density of matter in the Universe is equal to the critical density then then all of the matter in the Universe will cool down and get farther apart. The average density of matter will continue to decrease.

What does the observational evidence have to say about the rate of expansion of the Universe?
The latest evidence suggests that the rate of expansion of the Universe is increasing, and not decreasing as had always been assumed. The evidence comes from observations of supernovas in very distant galaxies. The power of the supernovae is known and they are dimmer than they should be in the very distant galaxies, indicating they are farther away than they would be if the Universe’s rate of expansion were decreasing.

As the size of an object of fixed mass is decreased, what happens to the escape velocity of the object?
The escape velocity increases as the size of a fixed amount of mass decreases. If you had two spheres of equal mass, the smaller of the two would have a larger escape velocity from its surface.

What is the Schwarzschild radius? How big is a black hole?
The formula is Rs = 2 G M/c2. When an object of mass M has a radius smaller then the Schwarzschild radius the escape velocity from the object’s surface surpasses the speed of light. Usually the size of a black hole (a non-rotating black hole) is identified with a sphere of radius equal to the Scwarzschild radius.

What would the radius of an object whose mass equals that of the Sun need to be for it to have an escape velocity equal to that of light? What about something with the mass of the Earth?
Use the formula for the Schwarzschild radius Rs = 2 G MSun/c2 which comes to about 3 km, while for the Earth it would be Rs = 2 G MEarth/c2 which comes out to about 0.89 cm.

How would the Earth’s orbit be altered if the Sun were to suddenly shrink to an object with a radius less than the Schwarzschild radius? Does a black hole swallow everything around it?
The Earth’s orbit would not be altered if the Sun was suddenly replaced by a black hole (at the same location as the Sun) with a mass equal to that of the Sun. Black holes do not swallow everything around them; something has to be pretty close to the black hole for it to be in a situation where it would fall into the black hole. If the Sun were replaced by a black hole with the Sun’s mass, everything outside the present solar radius would follow the same trajectories they currently follow.

What is the event horizon of the black hole?
This is the location where the escape velocity equals the speed of light. Closer to the black hole and one has escape velocities exceeding the speed of light. Hence all events that occur within the event horizon can’t be observed by anyone outside the event horizon. This is where the event horizon gets its name.

A light-bulb within the event horizon is suddenly switched on. What happens to the light emitted by the bulb?
The light moves towards the center of the black hole. It does not move away from the center, and it never crosses the event horizon, for then someone outside the event horizon could detect it.

Does the term radius have the usual meaning when it comes to a black hole? Is the distance from the center of the black hole to the event horizon equal to the radius?
Since space is curved within the event horizon (it is also curved outside of it, provided one is not too far from it) the distance from the center of the black hole to the event horizon is not equal to the circumference of the event horizon divided by 2 pi. This relation is valid for flat space (Euclidean space as it is sometimes called) but not for curved space, and near and inside the event horizon of a black hole, space is strongly curved.

Can one distinguish between black holes made of different substances?
Since no information can cross from within the event horizon to outside the event horizon, you cannot get any information about what the black hole is made of. The only information one can use to distinguish one black hole from another are the mass, charge and rotation of the black hole.

What is the photon sphere of a black hole?
This is the sphere of “radius” 1.5 Rs at which light can orbit the collapsing star or black hole.

A “photon sphere” is defined by the set of points surrounding a black whole where light which is emitted tangentially (at right angles) to the surface of such a sphere, would travel in a circle around the black hole, forever. If the light is emitted just slightly toward the hole, it will spiral in. If just slightly outward, it will escape. The radius of the photon sphere is three-halves the size of the event horizon (“Schwarzchild radius”), where no light can escape.

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